Nonlinear_PoroViscoelasticity.h

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 *   
 * @endcode
 * 
 * We start by including all the necessary deal.II header files and some C++
 * related ones. They have been discussed in detail in previous tutorial
 * programs, so you need only refer to past tutorials for details.
 * 
 
 * 
 * 
 * @code
 *   #include <deal.II/base/function.h>
 *   #include <deal.II/base/parameter_handler.h>
 *   #include <deal.II/base/point.h>
 *   #include <deal.II/base/quadrature_lib.h>
 *   #include <deal.II/base/symmetric_tensor.h>
 *   #include <deal.II/base/tensor.h>
 *   #include <deal.II/base/timer.h>
 *   #include <deal.II/base/work_stream.h>
 *   #include <deal.II/base/mpi.h>
 *   #include <deal.II/base/quadrature_point_data.h>
 *   
 *   #include <deal.II/differentiation/ad.h>
 *   
 *   #include <deal.II/distributed/shared_tria.h>
 *   
 *   #include <deal.II/dofs/dof_renumbering.h>
 *   #include <deal.II/dofs/dof_tools.h>
 *   #include <deal.II/dofs/dof_accessor.h>
 *   
 *   #include <deal.II/grid/filtered_iterator.h>
 *   #include <deal.II/grid/grid_generator.h>
 *   #include <deal.II/grid/grid_tools.h>
 *   #include <deal.II/grid/grid_in.h>
 *   #include <deal.II/grid/grid_out.h>
 *   #include <deal.II/grid/manifold_lib.h>
 *   #include <deal.II/grid/tria_accessor.h>
 *   #include <deal.II/grid/tria_iterator.h>
 *   
 *   #include <deal.II/fe/fe_dgp_monomial.h>
 *   #include <deal.II/fe/fe_q.h>
 *   #include <deal.II/fe/fe_system.h>
 *   #include <deal.II/fe/fe_tools.h>
 *   #include <deal.II/fe/fe_values.h>
 *   
 *   #include <deal.II/lac/block_sparsity_pattern.h>
 *   #include <deal.II/lac/affine_constraints.h>
 *   #include <deal.II/lac/dynamic_sparsity_pattern.h>
 *   #include <deal.II/lac/full_matrix.h>
 *   #include <deal.II/lac/linear_operator.h>
 *   #include <deal.II/lac/packaged_operation.h>
 *   
 *   #include <deal.II/lac/trilinos_block_sparse_matrix.h>
 *   #include <deal.II/lac/trilinos_linear_operator.h>
 *   #include <deal.II/lac/trilinos_parallel_block_vector.h>
 *   #include <deal.II/lac/trilinos_precondition.h>
 *   #include <deal.II/lac/trilinos_sparse_matrix.h>
 *   #include <deal.II/lac/trilinos_sparsity_pattern.h>
 *   #include <deal.II/lac/trilinos_solver.h>
 *   #include <deal.II/lac/trilinos_vector.h>
 *   
 *   #include <deal.II/lac/block_vector.h>
 *   #include <deal.II/lac/vector.h>
 *   
 *   #include <deal.II/numerics/data_postprocessor.h>
 *   #include <deal.II/numerics/data_out.h>
 *   #include <deal.II/numerics/data_out_faces.h>
 *   #include <deal.II/numerics/fe_field_function.h>
 *   #include <deal.II/numerics/vector_tools.h>
 *   
 *   #include <deal.II/physics/transformations.h>
 *   #include <deal.II/physics/elasticity/kinematics.h>
 *   #include <deal.II/physics/elasticity/standard_tensors.h>
 *   
 *   #include <iostream>
 *   #include <fstream>
 *   #include <numeric>
 *   #include <iomanip>
 *   
 *   
 * @endcode
 * 
 * We create a namespace for everything that relates to
 * the nonlinear poro-viscoelastic formulation,
 * and import all the deal.II function and class names into it:
 * 
 * @code
 *   namespace NonLinearPoroViscoElasticity
 *   {
 *       using namespace dealii;
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Runtimeparameters"></a> 
 * <h3>Run-time parameters</h3>
 * 
 
 * 
 * Set up a ParameterHandler object to read in the parameter choices at run-time
 * introduced by the user through the file "parameters.prm"
 * 
 * @code
 *       namespace Parameters
 *       {
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-FiniteElementsystem"></a> 
 * <h4>Finite Element system</h4>
 * Here we specify the polynomial order used to approximate the solution,
 * both for the displacements and pressure unknowns.
 * The quadrature order should be adjusted accordingly.
 * 
 * @code
 *         struct FESystem
 *         {
 *           unsigned int poly_degree_displ;
 *           unsigned int poly_degree_pore;
 *           unsigned int quad_order;
 *   
 *           static void
 *           declare_parameters(ParameterHandler &prm);
 *   
 *           void
 *           parse_parameters(ParameterHandler &prm);
 *         };
 *   
 *         void FESystem::declare_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Finite element system");
 *           {
 *             prm.declare_entry("Polynomial degree displ", "2",
 *                               Patterns::Integer(0),
 *                               "Displacement system polynomial order");
 *   
 *             prm.declare_entry("Polynomial degree pore", "1",
 *                               Patterns::Integer(0),
 *                               "Pore pressure system polynomial order");
 *   
 *             prm.declare_entry("Quadrature order", "3",
 *                               Patterns::Integer(0),
 *                               "Gauss quadrature order");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 *         void FESystem::parse_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Finite element system");
 *           {
 *             poly_degree_displ = prm.get_integer("Polynomial degree displ");
 *             poly_degree_pore = prm.get_integer("Polynomial degree pore");
 *             quad_order = prm.get_integer("Quadrature order");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Geometry"></a> 
 * <h4>Geometry</h4>
 * These parameters are related to the geometry definition and mesh generation.
 * We select the type of problem to solve and introduce the desired load values.
 * 
 * @code
 *         struct Geometry
 *         {
 *           std::string  geom_type;
 *           unsigned int global_refinement;
 *           double       scale;
 *           std::string  load_type;
 *           double       load;
 *           unsigned int num_cycle_sets;
 *           double       fluid_flow;
 *           double       drained_pressure;
 *   
 *           static void
 *           declare_parameters(ParameterHandler &prm);
 *   
 *           void
 *           parse_parameters(ParameterHandler &prm);
 *         };
 *   
 *         void Geometry::declare_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Geometry");
 *           {
 *             prm.declare_entry("Geometry type", "Ehlers_tube_step_load",
 *                                Patterns::Selection("Ehlers_tube_step_load"
 *                                                    "|Ehlers_tube_increase_load"
 *                                                    "|Ehlers_cube_consolidation"
 *                                                    "|Franceschini_consolidation"
 *                                                    "|Budday_cube_tension_compression"
 *                                                    "|Budday_cube_tension_compression_fully_fixed"
 *                                                    "|Budday_cube_shear_fully_fixed"),
 *                                   "Type of geometry used. "
 *                                   "For Ehlers verification examples see Ehlers and Eipper (1999). "
 *                                   "For Franceschini brain consolidation see Franceschini et al. (2006)"
 *                                   "For Budday brain examples see Budday et al. (2017)");
 *   
 *             prm.declare_entry("Global refinement", "1",
 *                               Patterns::Integer(0),
 *                               "Global refinement level");
 *   
 *             prm.declare_entry("Grid scale", "1.0",
 *                               Patterns::Double(0.0),
 *                               "Global grid scaling factor");
 *   
 *             prm.declare_entry("Load type", "pressure",
 *                               Patterns::Selection("pressure|displacement|none"),
 *                               "Type of loading");
 *   
 *             prm.declare_entry("Load value", "-7.5e+6",
 *                               Patterns::Double(),
 *                               "Loading value");
 *   
 *             prm.declare_entry("Number of cycle sets", "1",
 *                               Patterns::Integer(1,2),
 *                               "Number of times each set of 3 cycles is repeated, only for "
 *                               "Budday_cube_tension_compression and Budday_cube_tension_compression_fully_fixed. "
 *                               "Load value is doubled in second set, load rate is kept constant."
 *                               "Final time indicates end of second cycle set.");
 *   
 *             prm.declare_entry("Fluid flow value", "0.0",
 *                               Patterns::Double(),
 *                               "Prescribed fluid flow. Not implemented in any example yet.");
 *   
 *             prm.declare_entry("Drained pressure", "0.0",
 *                               Patterns::Double(),
 *                               "Increase of pressure value at drained boundary w.r.t the atmospheric pressure.");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 *         void Geometry::parse_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Geometry");
 *           {
 *             geom_type = prm.get("Geometry type");
 *             global_refinement = prm.get_integer("Global refinement");
 *             scale = prm.get_double("Grid scale");
 *             load_type = prm.get("Load type");
 *             load = prm.get_double("Load value");
 *             num_cycle_sets = prm.get_integer("Number of cycle sets");
 *             fluid_flow = prm.get_double("Fluid flow value");
 *             drained_pressure = prm.get_double("Drained pressure");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Materials"></a> 
 * <h4>Materials</h4>
 * 
 
 * 
 * Here we select the type of material for the solid component
 * and define the corresponding material parameters.
 * Then we define he fluid data, including the type of
 * seepage velocity definition to use.
 * 
 * @code
 *         struct Materials
 *         {
 *           std::string  mat_type;
 *           double lambda;
 *           double mu;
 *           double mu1_infty;
 *           double mu2_infty;
 *           double mu3_infty;
 *           double alpha1_infty;
 *           double alpha2_infty;
 *           double alpha3_infty;
 *           double mu1_mode_1;
 *           double mu2_mode_1;
 *           double mu3_mode_1;
 *           double alpha1_mode_1;
 *           double alpha2_mode_1;
 *           double alpha3_mode_1;
 *           double viscosity_mode_1;
 *           std::string  fluid_type;
 *           double solid_vol_frac;
 *           double kappa_darcy;
 *           double init_intrinsic_perm;
 *           double viscosity_FR;
 *           double init_darcy_coef;
 *           double weight_FR;
 *           bool gravity_term;
 *           int gravity_direction;
 *           double gravity_value;
 *           double density_FR;
 *           double density_SR;
 *           enum SymmetricTensorEigenvectorMethod eigen_solver;
 *   
 *           static void
 *           declare_parameters(ParameterHandler &prm);
 *   
 *           void
 *           parse_parameters(ParameterHandler &prm);
 *         };
 *   
 *         void Materials::declare_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Material properties");
 *           {
 *             prm.declare_entry("material", "Neo-Hooke",
 *                               Patterns::Selection("Neo-Hooke|Ogden|visco-Ogden"),
 *                               "Type of material used in the problem");
 *   
 *             prm.declare_entry("lambda", "8.375e6",
 *                               Patterns::Double(0,1e100),
 *                               "First Lamé parameter for extension function related to compactation point in solid material [Pa].");
 *   
 *             prm.declare_entry("shear modulus", "5.583e6",
 *                               Patterns::Double(0,1e100),
 *                               "shear modulus for Neo-Hooke materials [Pa].");
 *   
 *             prm.declare_entry("eigen solver", "QL Implicit Shifts",
 *                               Patterns::Selection("QL Implicit Shifts|Jacobi"),
 *                               "The type of eigen solver to be used for Ogden and visco-Ogden models.");
 *   
 *             prm.declare_entry("mu1", "0.0",
 *                               Patterns::Double(),
 *                               "Shear material parameter 'mu1' for Ogden material [Pa].");
 *   
 *             prm.declare_entry("mu2", "0.0",
 *                               Patterns::Double(),
 *                               "Shear material parameter 'mu2' for Ogden material [Pa].");
 *   
 *             prm.declare_entry("mu3", "0.0",
 *                               Patterns::Double(),
 *                               "Shear material parameter 'mu1' for Ogden material [Pa].");
 *   
 *             prm.declare_entry("alpha1", "1.0",
 *                               Patterns::Double(),
 *                               "Stiffness material parameter 'alpha1' for Ogden material [-].");
 *   
 *             prm.declare_entry("alpha2", "1.0",
 *                               Patterns::Double(),
 *                               "Stiffness material parameter 'alpha2' for Ogden material [-].");
 *   
 *             prm.declare_entry("alpha3", "1.0",
 *                               Patterns::Double(),
 *                               "Stiffness material parameter 'alpha3' for Ogden material [-].");
 *   
 *             prm.declare_entry("mu1_1", "0.0",
 *                               Patterns::Double(),
 *                               "Shear material parameter 'mu1' for first viscous mode in Ogden material [Pa].");
 *   
 *             prm.declare_entry("mu2_1", "0.0",
 *                               Patterns::Double(),
 *                               "Shear material parameter 'mu2' for first viscous mode in Ogden material [Pa].");
 *   
 *             prm.declare_entry("mu3_1", "0.0",
 *                               Patterns::Double(),
 *                               "Shear material parameter 'mu1' for first viscous mode in Ogden material [Pa].");
 *   
 *             prm.declare_entry("alpha1_1", "1.0",
 *                               Patterns::Double(),
 *                               "Stiffness material parameter 'alpha1' for first viscous mode in Ogden material [-].");
 *   
 *             prm.declare_entry("alpha2_1", "1.0",
 *                               Patterns::Double(),
 *                               "Stiffness material parameter 'alpha2' for first viscous mode in Ogden material [-].");
 *   
 *             prm.declare_entry("alpha3_1", "1.0",
 *                               Patterns::Double(),
 *                               "Stiffness material parameter 'alpha3' for first viscous mode in Ogden material [-].");
 *   
 *             prm.declare_entry("viscosity_1", "1e-10",
 *                               Patterns::Double(1e-10,1e100),
 *                               "Deformation-independent viscosity parameter 'eta_1' for first viscous mode in Ogden material [-].");
 *   
 *             prm.declare_entry("seepage definition", "Ehlers",
 *                               Patterns::Selection("Markert|Ehlers"),
 *                               "Type of formulation used to define the seepage velocity in the problem. "
 *                               "Choose between Markert formulation of deformation-dependent intrinsic permeability "
 *                               "and Ehlers formulation of deformation-dependent Darcy flow coefficient.");
 *   
 *             prm.declare_entry("initial solid volume fraction", "0.67",
 *                               Patterns::Double(0.001,0.999),
 *                               "Initial porosity (solid volume fraction, 0 < n_0s < 1)");
 *   
 *             prm.declare_entry("kappa", "0.0",
 *                               Patterns::Double(0,100),
 *                               "Deformation-dependency control parameter for specific permeability (kappa >= 0)");
 *   
 *             prm.declare_entry("initial intrinsic permeability", "0.0",
 *                               Patterns::Double(0,1e100),
 *                               "Initial intrinsic permeability parameter [m^2] (isotropic permeability). To be used with Markert formulation.");
 *   
 *             prm.declare_entry("fluid viscosity", "0.0",
 *                               Patterns::Double(0, 1e100),
 *                               "Effective shear viscosity parameter of the fluid [Pa·s, (N·s)/m^2]. To be used with Markert formulation.");
 *   
 *             prm.declare_entry("initial Darcy coefficient", "1.0e-4",
 *                               Patterns::Double(0,1e100),
 *                               "Initial Darcy flow coefficient [m/s] (isotropic permeability). To be used with Ehlers formulation.");
 *   
 *             prm.declare_entry("fluid weight", "1.0e4",
 *                               Patterns::Double(0, 1e100),
 *                               "Effective weight of the fluid [N/m^3]. To be used with Ehlers formulation.");
 *   
 *             prm.declare_entry("gravity term", "false",
 *                               Patterns::Bool(),
 *                               "Gravity term considered (true) or neglected (false)");
 *   
 *             prm.declare_entry("fluid density", "1.0",
 *                               Patterns::Double(0,1e100),
 *                               "Real (or effective) density of the fluid");
 *   
 *             prm.declare_entry("solid density", "1.0",
 *                               Patterns::Double(0,1e100),
 *                               "Real (or effective) density of the solid");
 *   
 *             prm.declare_entry("gravity direction", "2",
 *                               Patterns::Integer(0,2),
 *                               "Direction of gravity (unit vector 0 for x, 1 for y, 2 for z)");
 *   
 *             prm.declare_entry("gravity value", "-9.81",
 *                               Patterns::Double(),
 *                               "Value of gravity (be careful to have consistent units!)");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 *         void Materials::parse_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Material properties");
 *           {
 * @endcode
 * 
 * Solid
 * 
 * @code
 *             mat_type = prm.get("material");
 *             lambda = prm.get_double("lambda");
 *             mu = prm.get_double("shear modulus");
 *             mu1_infty = prm.get_double("mu1");
 *             mu2_infty = prm.get_double("mu2");
 *             mu3_infty = prm.get_double("mu3");
 *             alpha1_infty = prm.get_double("alpha1");
 *             alpha2_infty = prm.get_double("alpha2");
 *             alpha3_infty = prm.get_double("alpha3");
 *             mu1_mode_1 = prm.get_double("mu1_1");
 *             mu2_mode_1 = prm.get_double("mu2_1");
 *             mu3_mode_1 = prm.get_double("mu3_1");
 *             alpha1_mode_1 = prm.get_double("alpha1_1");
 *             alpha2_mode_1 = prm.get_double("alpha2_1");
 *             alpha3_mode_1 = prm.get_double("alpha3_1");
 *             viscosity_mode_1 = prm.get_double("viscosity_1");
 * @endcode
 * 
 * Fluid
 * 
 * @code
 *             fluid_type = prm.get("seepage definition");
 *             solid_vol_frac = prm.get_double("initial solid volume fraction");
 *             kappa_darcy = prm.get_double("kappa");
 *             init_intrinsic_perm = prm.get_double("initial intrinsic permeability");
 *             viscosity_FR = prm.get_double("fluid viscosity");
 *             init_darcy_coef = prm.get_double("initial Darcy coefficient");
 *             weight_FR = prm.get_double("fluid weight");
 * @endcode
 * 
 * Gravity effects
 * 
 * @code
 *             gravity_term = prm.get_bool("gravity term");
 *             density_FR = prm.get_double("fluid density");
 *             density_SR = prm.get_double("solid density");
 *             gravity_direction = prm.get_integer("gravity direction");
 *             gravity_value = prm.get_double("gravity value");
 *   
 *             if ( (fluid_type == "Markert") && ((init_intrinsic_perm == 0.0) || (viscosity_FR == 0.0)) )
 *                 AssertThrow(false, ExcMessage("Markert seepage velocity formulation requires the definition of "
 *                                               "'initial intrinsic permeability' and 'fluid viscosity' greater than 0.0."));
 *   
 *             if ( (fluid_type == "Ehlers") && ((init_darcy_coef == 0.0) || (weight_FR == 0.0)) )
 *                 AssertThrow(false, ExcMessage("Ehler seepage velocity formulation requires the definition of "
 *                                               "'initial Darcy coefficient' and 'fluid weight' greater than 0.0."));
 *   
 *             const std::string eigen_solver_type = prm.get("eigen solver");
 *             if (eigen_solver_type == "QL Implicit Shifts")
 *               eigen_solver = SymmetricTensorEigenvectorMethod::ql_implicit_shifts;
 *             else if (eigen_solver_type == "Jacobi")
 *               eigen_solver = SymmetricTensorEigenvectorMethod::jacobi;
 *             else
 *             {
 *               AssertThrow(false, ExcMessage("Unknown eigen solver selected."));
 *             }
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Nonlinearsolver"></a> 
 * <h4>Nonlinear solver</h4>
 * 
 
 * 
 * We now define the tolerances and the maximum number of iterations for the
 * Newton-Raphson scheme used to solve the nonlinear system of governing equations.
 * 
 * @code
 *         struct NonlinearSolver
 *         {
 *           unsigned int max_iterations_NR;
 *           double       tol_f;
 *           double       tol_u;
 *           double       tol_p_fluid;
 *   
 *           static void
 *           declare_parameters(ParameterHandler &prm);
 *   
 *           void
 *           parse_parameters(ParameterHandler &prm);
 *         };
 *   
 *         void NonlinearSolver::declare_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Nonlinear solver");
 *           {
 *             prm.declare_entry("Max iterations Newton-Raphson", "15",
 *                               Patterns::Integer(0),
 *                               "Number of Newton-Raphson iterations allowed");
 *   
 *             prm.declare_entry("Tolerance force", "1.0e-8",
 *                               Patterns::Double(0.0),
 *                               "Force residual tolerance");
 *   
 *             prm.declare_entry("Tolerance displacement", "1.0e-6",
 *                               Patterns::Double(0.0),
 *                               "Displacement error tolerance");
 *   
 *             prm.declare_entry("Tolerance pore pressure", "1.0e-6",
 *                               Patterns::Double(0.0),
 *                               "Pore pressure error tolerance");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 *         void NonlinearSolver::parse_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Nonlinear solver");
 *           {
 *             max_iterations_NR = prm.get_integer("Max iterations Newton-Raphson");
 *             tol_f = prm.get_double("Tolerance force");
 *             tol_u = prm.get_double("Tolerance displacement");
 *             tol_p_fluid =  prm.get_double("Tolerance pore pressure");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Time"></a> 
 * <h4>Time</h4>
 * Here we set the timestep size @f$ \varDelta t @f$ and the simulation end-time.
 * 
 * @code
 *         struct Time
 *         {
 *           double end_time;
 *           double delta_t;
 *           static void
 *           declare_parameters(ParameterHandler &prm);
 *   
 *           void
 *           parse_parameters(ParameterHandler &prm);
 *         };
 *   
 *         void Time::declare_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Time");
 *           {
 *             prm.declare_entry("End time", "10.0",
 *                               Patterns::Double(),
 *                               "End time");
 *   
 *             prm.declare_entry("Time step size", "0.002",
 *                               Patterns::Double(1.0e-6),
 *                               "Time step size. The value must be larger than the displacement error tolerance defined.");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 *         void Time::parse_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Time");
 *           {
 *             end_time = prm.get_double("End time");
 *             delta_t = prm.get_double("Time step size");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Output"></a> 
 * <h4>Output</h4>
 * We can choose the frequency of the data for the output files.
 * 
 * @code
 *         struct OutputParam
 *         {
 *   
 *           std::string  outfiles_requested;
 *           unsigned int timestep_output;
 *           std::string  outtype;
 *   
 *           static void
 *           declare_parameters(ParameterHandler &prm);
 *   
 *           void
 *           parse_parameters(ParameterHandler &prm);
 *         };
 *   
 *         void OutputParam::declare_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Output parameters");
 *           {
 *             prm.declare_entry("Output files", "true",
 *                               Patterns::Selection("true|false"),
 *                               "Paraview output files to generate.");
 *             prm.declare_entry("Time step number output", "1",
 *                               Patterns::Integer(0),
 *                               "Output data for time steps multiple of the given "
 *                               "integer value.");
 *             prm.declare_entry("Averaged results", "nodes",
 *                                Patterns::Selection("elements|nodes"),
 *                                "Output data associated with integration point values"
 *                                " averaged on elements or on nodes.");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 *         void OutputParam::parse_parameters(ParameterHandler &prm)
 *         {
 *           prm.enter_subsection("Output parameters");
 *           {
 *             outfiles_requested = prm.get("Output files");
 *             timestep_output = prm.get_integer("Time step number output");
 *             outtype = prm.get("Averaged results");
 *           }
 *           prm.leave_subsection();
 *         }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Allparameters"></a> 
 * <h4>All parameters</h4>
 * We finally consolidate all of the above structures into a single container that holds all the run-time selections.
 * 
 * @code
 *         struct AllParameters : public FESystem,
 *                                public Geometry,
 *                                public Materials,
 *                                public NonlinearSolver,
 *                                public Time,
 *                                public OutputParam
 *         {
 *           AllParameters(const std::string &input_file);
 *   
 *           static void
 *           declare_parameters(ParameterHandler &prm);
 *   
 *           void
 *           parse_parameters(ParameterHandler &prm);
 *         };
 *   
 *         AllParameters::AllParameters(const std::string &input_file)
 *         {
 *           ParameterHandler prm;
 *           declare_parameters(prm);
 *           prm.parse_input(input_file);
 *           parse_parameters(prm);
 *         }
 *   
 *         void AllParameters::declare_parameters(ParameterHandler &prm)
 *         {
 *           FESystem::declare_parameters(prm);
 *           Geometry::declare_parameters(prm);
 *           Materials::declare_parameters(prm);
 *           NonlinearSolver::declare_parameters(prm);
 *           Time::declare_parameters(prm);
 *           OutputParam::declare_parameters(prm);
 *         }
 *   
 *         void AllParameters::parse_parameters(ParameterHandler &prm)
 *         {
 *           FESystem::parse_parameters(prm);
 *           Geometry::parse_parameters(prm);
 *           Materials::parse_parameters(prm);
 *           NonlinearSolver::parse_parameters(prm);
 *           Time::parse_parameters(prm);
 *           OutputParam::parse_parameters(prm);
 *         }
 *       }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Timeclass"></a> 
 * <h3>Time class</h3>
 * A simple class to store time data.
 * For simplicity we assume a constant time step size.
 * 
 * @code
 *       class Time
 *       {
 *           public:
 *             Time (const double time_end,
 *                   const double delta_t)
 *               :
 *               timestep(0),
 *               time_current(0.0),
 *               time_end(time_end),
 *               delta_t(delta_t)
 *             {}
 *   
 *             virtual ~Time()
 *             {}
 *   
 *             double get_current() const
 *             {
 *               return time_current;
 *             }
 *             double get_end() const
 *             {
 *               return time_end;
 *             }
 *             double get_delta_t() const
 *             {
 *               return delta_t;
 *             }
 *             unsigned int get_timestep() const
 *             {
 *               return timestep;
 *             }
 *             void increment_time ()
 *             {
 *               time_current += delta_t;
 *               ++timestep;
 *             }
 *   
 *           private:
 *             unsigned int timestep;
 *             double time_current;
 *             double time_end;
 *             const double delta_t;
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Constitutiveequationforthesolidcomponentofthebiphasicmaterial"></a> 
 * <h3>Constitutive equation for the solid component of the biphasic material</h3>
 * 
 
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Baseclassgenerichyperelasticmaterial"></a> 
 * <h4>Base class: generic hyperelastic material</h4>
 * The "extra" Kirchhoff stress in the solid component is the sum of isochoric
 * and a volumetric part:
 * @f$\mathbf{\tau} = \mathbf{\tau}_E^{(\bullet)} + \mathbf{\tau}^{\textrm{vol}}@f$.
 * The deviatoric part changes depending on the type of material model selected:
 * Neo-Hooken hyperelasticity, Ogden hyperelasticiy,
 * or a single-mode finite viscoelasticity based on the Ogden hyperelastic model.
 * In this base class we declare  it as a virtual function,
 * and it will be defined for each model type in the corresponding derived class.
 * We define here the volumetric component, which depends on the
 * extension function @f$U(J_S)@f$ selected, and in this case is the same for all models.
 * We use the function proposed by
 * Ehlers & Eipper 1999 doi:10.1023/A:1006565509095.
 * We also define some public functions to access and update the internal variables.
 * 
 * @code
 *       template <int dim, typename NumberType = Sacado::Fad::DFad<double> >
 *       class Material_Hyperelastic
 *       {
 *           public:
 *             Material_Hyperelastic(const Parameters::AllParameters &parameters,
 *                                   const Time                      &time)
 *               :
 *               n_OS (parameters.solid_vol_frac),
 *               lambda (parameters.lambda),
 *               time(time),
 *               det_F (1.0),
 *               det_F_converged (1.0),
 *               eigen_solver (parameters.eigen_solver)
 *              {}
 *             ~Material_Hyperelastic()
 *             {}
 *   
 *             SymmetricTensor<2, dim, NumberType>
 *             get_tau_E(const Tensor<2,dim, NumberType> &F) const
 *             {
 *               return ( get_tau_E_base(F) + get_tau_E_ext_func(F) );
 *             }
 *   
 *             SymmetricTensor<2, dim, NumberType>
 *             get_Cauchy_E(const Tensor<2, dim, NumberType> &F) const
 *             {
 *                 const NumberType det_F = determinant(F);
 *                 Assert(det_F > 0, ExcInternalError());
 *                 return get_tau_E(F)*NumberType(1/det_F);
 *             }
 *   
 *             double
 *             get_converged_det_F() const
 *             {
 *                 return  det_F_converged;
 *             }
 *   
 *             virtual void
 *             update_end_timestep()
 *             {
 *                 det_F_converged = det_F;
 *             }
 *   
 *             virtual void
 *             update_internal_equilibrium( const Tensor<2, dim, NumberType> &F )
 *             {
 *                 det_F = Tensor<0,dim,double>(determinant(F));
 *             }
 *   
 *             virtual double
 *             get_viscous_dissipation( ) const = 0;
 *   
 *             const double n_OS;
 *             const double lambda;
 *             const Time  &time;
 *             double det_F;
 *             double det_F_converged;
 *             const enum SymmetricTensorEigenvectorMethod eigen_solver;
 *   
 *           protected:
 *             SymmetricTensor<2, dim, NumberType>
 *             get_tau_E_ext_func(const Tensor<2,dim, NumberType> &F) const
 *             {
 *                 const NumberType det_F = determinant(F);
 *                 Assert(det_F > 0, ExcInternalError());
 *   
 *                 static const SymmetricTensor< 2, dim, double>
 *                       I (Physics::Elasticity::StandardTensors<dim>::I);
 *                 return  ( NumberType(lambda * (1.0-n_OS)*(1.0-n_OS)
 *                            * (det_F/(1.0-n_OS) - det_F/(det_F-n_OS))) * I );
 *             }
 *   
 *             virtual SymmetricTensor<2, dim, NumberType>
 *              get_tau_E_base(const Tensor<2,dim, NumberType> &F) const = 0;
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassNeoHookeanhyperelasticmaterial"></a> 
 * <h4>Derived class: Neo-Hookean hyperelastic material</h4>
 * 
 * @code
 *       template <int dim, typename NumberType = Sacado::Fad::DFad<double> >
 *       class NeoHooke : public Material_Hyperelastic < dim, NumberType >
 *       {
 *           public:
 *               NeoHooke(const Parameters::AllParameters &parameters,
 *                        const Time                      &time)
 *               :
 *               Material_Hyperelastic< dim, NumberType > (parameters,time),
 *               mu(parameters.mu)
 *              {}
 *             virtual ~NeoHooke()
 *             {}
 *   
 *              double
 *              get_viscous_dissipation() const override
 *              {
 *                  return 0.0;
 *              }
 *   
 *           protected:
 *             const double mu;
 *   
 *             SymmetricTensor<2, dim, NumberType>
 *             get_tau_E_base(const Tensor<2,dim, NumberType> &F) const override
 *             {
 *                static const SymmetricTensor< 2, dim, double>
 *                   I (Physics::Elasticity::StandardTensors<dim>::I);
 *   
 *                const bool use_standard_model = true;
 *   
 *                if (use_standard_model)
 *                {
 * @endcode
 * 
 * Standard Neo-Hooke
 * 
 * @code
 *                  return ( mu * ( symmetrize(F * transpose(F)) - I ) );
 *                }
 *                else
 *                {
 * @endcode
 * 
 * Neo-Hooke in terms of principal stretches
 * 
 * @code
 *                  const SymmetricTensor<2, dim, NumberType>
 *                   B = symmetrize(F * transpose(F));
 *                  const std::array< std::pair< NumberType, Tensor< 1, dim, NumberType > >, dim >
 *                   eigen_B = eigenvectors(B, this->eigen_solver);
 *   
 *                  SymmetricTensor<2, dim, NumberType> B_ev;
 *                  for (unsigned int d=0; d<dim; ++d)
 *                    B_ev += eigen_B[d].first*symmetrize(outer_product(eigen_B[d].second,eigen_B[d].second));
 *   
 *                   return ( mu*(B_ev-I) );
 *                }
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassOgdenhyperelasticmaterial"></a> 
 * <h4>Derived class: Ogden hyperelastic material</h4>
 * 
 * @code
 *       template <int dim, typename NumberType = Sacado::Fad::DFad<double> >
 *       class Ogden : public Material_Hyperelastic < dim, NumberType >
 *       {
 *           public:
 *             Ogden(const Parameters::AllParameters &parameters,
 *                   const Time                      &time)
 *             :
 *             Material_Hyperelastic< dim, NumberType > (parameters,time),
 *             mu({parameters.mu1_infty,
 *                 parameters.mu2_infty,
 *                 parameters.mu3_infty}),
 *             alpha({parameters.alpha1_infty,
 *                    parameters.alpha2_infty,
 *                    parameters.alpha3_infty})
 *              {}
 *             virtual ~Ogden()
 *             {}
 *   
 *              double
 *              get_viscous_dissipation() const override
 *              {
 *                  return 0.0;
 *              }
 *   
 *           protected:
 *             std::vector<double> mu;
 *             std::vector<double> alpha;
 *   
 *             SymmetricTensor<2, dim, NumberType>
 *             get_tau_E_base(const Tensor<2,dim, NumberType> &F) const override
 *             {
 *               const SymmetricTensor<2, dim, NumberType>
 *                B = symmetrize(F * transpose(F));
 *   
 *               const std::array< std::pair< NumberType, Tensor< 1, dim, NumberType > >, dim >
 *                eigen_B = eigenvectors(B, this->eigen_solver);
 *   
 *               SymmetricTensor<2, dim, NumberType>  tau;
 *               static const SymmetricTensor< 2, dim, double>
 *                 I (Physics::Elasticity::StandardTensors<dim>::I);
 *   
 *               for (unsigned int i = 0; i < 3; ++i)
 *               {
 *                   for (unsigned int A = 0; A < dim; ++A)
 *                   {
 *                       SymmetricTensor<2, dim, NumberType>  tau_aux1 = symmetrize(
 *                               outer_product(eigen_B[A].second,eigen_B[A].second));
 *                       tau_aux1 *= mu[i]*std::pow(eigen_B[A].first, (alpha[i]/2.) );
 *                       tau += tau_aux1;
 *                   }
 *                   SymmetricTensor<2, dim, NumberType>  tau_aux2 (I);
 *                   tau_aux2 *= mu[i];
 *                   tau -= tau_aux2;
 *               }
 *               return tau;
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassSinglemodeOgdenviscoelasticmaterial"></a> 
 * <h4>Derived class: Single-mode Ogden viscoelastic material</h4>
 * We use the finite viscoelastic model described in
 * Reese & Govindjee (1998) doi:10.1016/S0020-7683(97)00217-5
 * The algorithm for the implicit exponential time integration is given in
 * Budday et al. (2017) doi: 10.1016/j.actbio.2017.06.024
 * 
 * @code
 *       template <int dim, typename NumberType = Sacado::Fad::DFad<double> >
 *       class visco_Ogden : public Material_Hyperelastic < dim, NumberType >
 *       {
 *           public:
 *               visco_Ogden(const Parameters::AllParameters &parameters,
 *                           const Time                      &time)
 *               :
 *               Material_Hyperelastic< dim, NumberType > (parameters,time),
 *               mu_infty({parameters.mu1_infty,
 *                         parameters.mu2_infty,
 *                         parameters.mu3_infty}),
 *               alpha_infty({parameters.alpha1_infty,
 *                            parameters.alpha2_infty,
 *                            parameters.alpha3_infty}),
 *               mu_mode_1({parameters.mu1_mode_1,
 *                          parameters.mu2_mode_1,
 *                          parameters.mu3_mode_1}),
 *               alpha_mode_1({parameters.alpha1_mode_1,
 *                             parameters.alpha2_mode_1,
 *                             parameters.alpha3_mode_1}),
 *               viscosity_mode_1(parameters.viscosity_mode_1),
 *               Cinv_v_1(Physics::Elasticity::StandardTensors<dim>::I),
 *               Cinv_v_1_converged(Physics::Elasticity::StandardTensors<dim>::I)
 *               {}
 *               virtual ~visco_Ogden()
 *               {}
 *   
 *             void
 *             update_internal_equilibrium( const Tensor<2, dim, NumberType> &F ) override
 *             {
 *                 Material_Hyperelastic < dim, NumberType >::update_internal_equilibrium(F);
 *   
 *                 this->Cinv_v_1 = this->Cinv_v_1_converged;
 *                 SymmetricTensor<2, dim, NumberType> B_e_1_tr = symmetrize(F * this->Cinv_v_1 * transpose(F));
 *   
 *                 const std::array< std::pair< NumberType, Tensor< 1, dim, NumberType > >, dim >
 *                   eigen_B_e_1_tr = eigenvectors(B_e_1_tr, this->eigen_solver);
 *   
 *                 Tensor< 1, dim, NumberType > lambdas_e_1_tr;
 *                 Tensor< 1, dim, NumberType > epsilon_e_1_tr;
 *                 for (int a = 0; a < dim; ++a)
 *                 {
 *                     lambdas_e_1_tr[a] = std::sqrt(eigen_B_e_1_tr[a].first);
 *                     epsilon_e_1_tr[a] = std::log(lambdas_e_1_tr[a]);
 *                 }
 *   
 *                const double tolerance = 1e-8;
 *                double residual_check = tolerance*10.0;
 *                Tensor< 1, dim, NumberType > residual;
 *                Tensor< 2, dim, NumberType > tangent;
 *                static const SymmetricTensor< 2, dim, double> I(Physics::Elasticity::StandardTensors<dim>::I);
 *                NumberType J_e_1 = std::sqrt(determinant(B_e_1_tr));
 *   
 *                std::vector<NumberType> lambdas_e_1_iso(dim);
 *                SymmetricTensor<2, dim, NumberType> B_e_1;
 *                int iteration = 0;
 *   
 *                Tensor< 1, dim, NumberType > lambdas_e_1;
 *                Tensor< 1, dim, NumberType > epsilon_e_1;
 *                epsilon_e_1 = epsilon_e_1_tr;
 *   
 *                 while(residual_check > tolerance)
 *                 {
 *                     NumberType aux_J_e_1 = 1.0;
 *                     for (unsigned int a = 0; a < dim; ++a)
 *                     {
 *                         lambdas_e_1[a] = std::exp(epsilon_e_1[a]);
 *                         aux_J_e_1 *= lambdas_e_1[a];
 *                     }
 *   
 *                     J_e_1 = aux_J_e_1;
 *   
 *                     for (unsigned int a = 0; a < dim; ++a)
 *                         lambdas_e_1_iso[a] = lambdas_e_1[a]*std::pow(J_e_1,-1.0/dim);
 *   
 *                     for (unsigned int a = 0; a < dim; ++a)
 *                     {
 *                         residual[a] = get_beta_mode_1(lambdas_e_1_iso, a);
 *                         residual[a] *= this->time.get_delta_t()/(2.0*viscosity_mode_1);
 *                         residual[a] += epsilon_e_1[a];
 *                         residual[a] -= epsilon_e_1_tr[a];
 *   
 *                         for (unsigned int b = 0; b < dim; ++b)
 *                         {
 *                             tangent[a][b]  = get_gamma_mode_1(lambdas_e_1_iso, a, b);
 *                             tangent[a][b] *= this->time.get_delta_t()/(2.0*viscosity_mode_1);
 *                             tangent[a][b] += I[a][b];
 *                         }
 *   
 *                     }
 *                     epsilon_e_1 -= invert(tangent)*residual;
 *   
 *                     residual_check = 0.0;
 *                     for (unsigned int a = 0; a < dim; ++a)
 *                     {
 *                         if ( std::abs(residual[a]) > residual_check)
 *                             residual_check = std::abs(Tensor<0,dim,double>(residual[a]));
 *                     }
 *                     iteration += 1;
 *                     if (iteration > 15 )
 *                         AssertThrow(false, ExcMessage("No convergence in local Newton iteration for the "
 *                                                       "viscoelastic exponential time integration algorithm."));
 *                 }
 *   
 *                 NumberType aux_J_e_1 = 1.0;
 *                 for (unsigned int a = 0; a < dim; ++a)
 *                 {
 *                     lambdas_e_1[a] = std::exp(epsilon_e_1[a]);
 *                     aux_J_e_1 *= lambdas_e_1[a];
 *                 }
 *                 J_e_1 = aux_J_e_1;
 *   
 *                 for (unsigned int a = 0; a < dim; ++a)
 *                     lambdas_e_1_iso[a] = lambdas_e_1[a]*std::pow(J_e_1,-1.0/dim);
 *   
 *                 for (unsigned int a = 0; a < dim; ++a)
 *                 {
 *                     SymmetricTensor<2, dim, NumberType>
 *                     B_e_1_aux = symmetrize(outer_product(eigen_B_e_1_tr[a].second,eigen_B_e_1_tr[a].second));
 *                     B_e_1_aux *= lambdas_e_1[a] * lambdas_e_1[a];
 *                     B_e_1 += B_e_1_aux;
 *                 }
 *   
 *                 Tensor<2, dim, NumberType>Cinv_v_1_AD = symmetrize(invert(F) * B_e_1 * invert(transpose(F)));
 *   
 *                 this->tau_neq_1 = 0;
 *                 for (unsigned int a = 0; a < dim; ++a)
 *                 {
 *                     SymmetricTensor<2, dim, NumberType>
 *                     tau_neq_1_aux = symmetrize(outer_product(eigen_B_e_1_tr[a].second,eigen_B_e_1_tr[a].second));
 *                     tau_neq_1_aux *=  get_beta_mode_1(lambdas_e_1_iso, a);
 *                     this->tau_neq_1 += tau_neq_1_aux;
 *                 }
 *   
 * @endcode
 * 
 * Store history
 * 
 * @code
 *                 for (unsigned int a = 0; a < dim; ++a)
 *                     for (unsigned int b = 0; b < dim; ++b)
 *                         this->Cinv_v_1[a][b]= Tensor<0,dim,double>(Cinv_v_1_AD[a][b]);
 *             }
 *   
 *             void update_end_timestep() override
 *             {
 *                 Material_Hyperelastic < dim, NumberType >::update_end_timestep();
 *                 this->Cinv_v_1_converged = this->Cinv_v_1;
 *             }
 *   
 *              double get_viscous_dissipation() const override
 *              {
 *                  NumberType dissipation_term = get_tau_E_neq() * get_tau_E_neq(); //Double contract the two SymmetricTensor
 *                  dissipation_term /= (2*viscosity_mode_1);
 *   
 *                  return dissipation_term.val();
 *              }
 *   
 *           protected:
 *             std::vector<double> mu_infty;
 *             std::vector<double> alpha_infty;
 *             std::vector<double> mu_mode_1;
 *             std::vector<double> alpha_mode_1;
 *             double viscosity_mode_1;
 *             SymmetricTensor<2, dim, double> Cinv_v_1;
 *             SymmetricTensor<2, dim, double> Cinv_v_1_converged;
 *             SymmetricTensor<2, dim, NumberType> tau_neq_1;
 *   
 *             SymmetricTensor<2, dim, NumberType>
 *             get_tau_E_base(const Tensor<2,dim, NumberType> &F) const override
 *             {
 *                 return ( get_tau_E_neq() + get_tau_E_eq(F) );
 *             }
 *   
 *             SymmetricTensor<2, dim, NumberType>
 *             get_tau_E_eq(const Tensor<2,dim, NumberType> &F) const
 *             {
 *               const SymmetricTensor<2, dim, NumberType> B = symmetrize(F * transpose(F));
 *   
 *               std::array< std::pair< NumberType, Tensor< 1, dim, NumberType > >, dim > eigen_B;
 *               eigen_B = eigenvectors(B, this->eigen_solver);
 *   
 *               SymmetricTensor<2, dim, NumberType>  tau;
 *               static const SymmetricTensor< 2, dim, double>
 *                 I (Physics::Elasticity::StandardTensors<dim>::I);
 *   
 *               for (unsigned int i = 0; i < 3; ++i)
 *               {
 *                   for (unsigned int A = 0; A < dim; ++A)
 *                   {
 *                       SymmetricTensor<2, dim, NumberType>  tau_aux1 = symmetrize(
 *                             outer_product(eigen_B[A].second,eigen_B[A].second));
 *                       tau_aux1 *= mu_infty[i]*std::pow(eigen_B[A].first, (alpha_infty[i]/2.) );
 *                       tau += tau_aux1;
 *                   }
 *                   SymmetricTensor<2, dim, NumberType>  tau_aux2 (I);
 *                   tau_aux2 *= mu_infty[i];
 *                   tau -= tau_aux2;
 *               }
 *               return tau;
 *             }
 *   
 *             SymmetricTensor<2, dim, NumberType>
 *             get_tau_E_neq() const
 *             {
 *                 return tau_neq_1;
 *             }
 *   
 *             NumberType
 *             get_beta_mode_1(std::vector< NumberType > &lambda, const int &A) const
 *             {
 *                 NumberType beta = 0.0;
 *   
 *                 for (unsigned int i = 0; i < 3; ++i) //3rd-order Ogden model
 *                 {
 *   
 *                     NumberType aux = 0.0;
 *                     for (int p = 0; p < dim; ++p)
 *                         aux += std::pow(lambda[p],alpha_mode_1[i]);
 *   
 *                     aux *= -1.0/dim;
 *                     aux += std::pow(lambda[A], alpha_mode_1[i]);
 *                     aux *= mu_mode_1[i];
 *   
 *                     beta  += aux;
 *                 }
 *                 return beta;
 *             }
 *   
 *             NumberType
 *             get_gamma_mode_1(std::vector< NumberType > &lambda,
 *                              const int                 &A,
 *                              const int                 &B       ) const
 *             {
 *                 NumberType gamma = 0.0;
 *   
 *                 if (A==B)
 *                 {
 *                     for (unsigned int i = 0; i < 3; ++i)
 *                     {
 *                         NumberType aux = 0.0;
 *                         for (int p = 0; p < dim; ++p)
 *                             aux += std::pow(lambda[p],alpha_mode_1[i]);
 *   
 *                         aux *= 1.0/(dim*dim);
 *                         aux += 1.0/dim * std::pow(lambda[A], alpha_mode_1[i]);
 *                         aux *= mu_mode_1[i]*alpha_mode_1[i];
 *   
 *                         gamma += aux;
 *                     }
 *                 }
 *                 else
 *                 {
 *                     for (unsigned int i = 0; i < 3; ++i)
 *                     {
 *                         NumberType aux = 0.0;
 *                         for (int p = 0; p < dim; ++p)
 *                             aux += std::pow(lambda[p],alpha_mode_1[i]);
 *   
 *                         aux *= 1.0/(dim*dim);
 *                         aux -= 1.0/dim * std::pow(lambda[A], alpha_mode_1[i]);
 *                         aux -= 1.0/dim * std::pow(lambda[B], alpha_mode_1[i]);
 *                         aux *= mu_mode_1[i]*alpha_mode_1[i];
 *   
 *                         gamma += aux;
 *                     }
 *                 }
 *   
 *                 return gamma;
 *             }
 *       };
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Constitutiveequationforthefluidcomponentofthebiphasicmaterial"></a> 
 * <h3>Constitutive equation for the fluid component of the biphasic material</h3>
 * We consider two slightly different definitions to define the seepage velocity with a Darcy-like law.
 * Ehlers & Eipper 1999, doi:10.1023/A:1006565509095
 * Markert 2007, doi:10.1007/s11242-007-9107-6
 * The selection of one or another is made by the user via the parameters file.
 * 
 * @code
 *       template <int dim, typename NumberType = Sacado::Fad::DFad<double> >
 *       class Material_Darcy_Fluid
 *       {
 *          public:
 *            Material_Darcy_Fluid(const Parameters::AllParameters &parameters)
 *            :
 *            fluid_type(parameters.fluid_type),
 *            n_OS(parameters.solid_vol_frac),
 *            initial_intrinsic_permeability(parameters.init_intrinsic_perm),
 *            viscosity_FR(parameters.viscosity_FR),
 *            initial_darcy_coefficient(parameters.init_darcy_coef),
 *            weight_FR(parameters.weight_FR),
 *            kappa_darcy(parameters.kappa_darcy),
 *            gravity_term(parameters.gravity_term),
 *            density_FR(parameters.density_FR),
 *            gravity_direction(parameters.gravity_direction),
 *            gravity_value(parameters.gravity_value)
 *            {
 *              Assert(kappa_darcy >= 0, ExcInternalError());
 *            }
 *            ~Material_Darcy_Fluid()
 *            {}
 *   
 *            Tensor<1, dim, NumberType> get_seepage_velocity_current
 *                                (const Tensor<2,dim, NumberType> &F,
 *                                 const Tensor<1,dim, NumberType> &grad_p_fluid) const
 *            {
 *                const NumberType det_F = determinant(F);
 *                Assert(det_F > 0.0, ExcInternalError());
 *   
 *                Tensor<2, dim, NumberType> permeability_term;
 *   
 *                if (fluid_type == "Markert")
 *                    permeability_term = get_instrinsic_permeability_current(F) / viscosity_FR;
 *   
 *                else if (fluid_type == "Ehlers")
 *                    permeability_term = get_darcy_flow_current(F) / weight_FR;
 *   
 *                else
 *                    AssertThrow(false, ExcMessage(
 *                      "Material_Darcy_Fluid --> Only Markert "
 *                      "and Ehlers formulations have been implemented."));
 *   
 *                return ( -1.0 * permeability_term * det_F
 *                         * (grad_p_fluid - get_body_force_FR_current()) );
 *            }
 *   
 *            double get_porous_dissipation(const Tensor<2,dim, NumberType> &F,
 *                                          const Tensor<1,dim, NumberType> &grad_p_fluid) const
 *            {
 *                NumberType dissipation_term;
 *                Tensor<1, dim, NumberType> seepage_velocity;
 *                Tensor<2, dim, NumberType> permeability_term;
 *   
 *                const NumberType det_F = determinant(F);
 *                Assert(det_F > 0.0, ExcInternalError());
 *   
 *                if (fluid_type == "Markert")
 *                {
 *                    permeability_term = get_instrinsic_permeability_current(F) / viscosity_FR;
 *                    seepage_velocity = get_seepage_velocity_current(F,grad_p_fluid);
 *                }
 *                else if (fluid_type == "Ehlers")
 *                {
 *                    permeability_term = get_darcy_flow_current(F) / weight_FR;
 *                    seepage_velocity = get_seepage_velocity_current(F,grad_p_fluid);
 *                }
 *                else
 *                    AssertThrow(false, ExcMessage(
 *                      "Material_Darcy_Fluid --> Only Markert and Ehlers "
 *                      "formulations have been implemented."));
 *   
 *                dissipation_term = ( invert(permeability_term) * seepage_velocity ) * seepage_velocity;
 *                dissipation_term *= 1.0/(det_F*det_F);
 *                return Tensor<0,dim,double>(dissipation_term);
 *            }
 *   
 *          protected:
 *            const std::string  fluid_type;
 *            const double n_OS;
 *            const double initial_intrinsic_permeability;
 *            const double viscosity_FR;
 *            const double initial_darcy_coefficient;
 *            const double weight_FR;
 *            const double kappa_darcy;
 *            const bool   gravity_term;
 *            const double density_FR;
 *            const int    gravity_direction;
 *            const double    gravity_value;
 *   
 *            Tensor<2, dim, NumberType>
 *            get_instrinsic_permeability_current(const Tensor<2,dim, NumberType> &F) const
 *            {
 *              static const SymmetricTensor< 2, dim, double>
 *                  I (Physics::Elasticity::StandardTensors<dim>::I);
 *              const Tensor<2, dim, NumberType> initial_instrinsic_permeability_tensor
 *                  = Tensor<2, dim, double>(initial_intrinsic_permeability * I);
 *   
 *              const NumberType det_F = determinant(F);
 *              Assert(det_F > 0.0, ExcInternalError());
 *   
 *              const NumberType fraction = (det_F - n_OS)/(1 - n_OS);
 *              return ( NumberType (std::pow(fraction, kappa_darcy))
 *                        * initial_instrinsic_permeability_tensor );
 *            }
 *   
 *            Tensor<2, dim, NumberType>
 *            get_darcy_flow_current(const Tensor<2,dim, NumberType> &F) const
 *            {
 *              static const SymmetricTensor< 2, dim, double>
 *                 I (Physics::Elasticity::StandardTensors<dim>::I);
 *              const Tensor<2, dim, NumberType> initial_darcy_flow_tensor
 *                 = Tensor<2, dim, double>(initial_darcy_coefficient * I);
 *   
 *              const NumberType det_F = determinant(F);
 *              Assert(det_F > 0.0, ExcInternalError());
 *   
 *              const NumberType fraction = (1.0 - (n_OS / det_F) )/(1.0 - n_OS);
 *              return ( NumberType (std::pow(fraction, kappa_darcy))
 *                        * initial_darcy_flow_tensor);
 *            }
 *   
 *           Tensor<1, dim, NumberType>
 *           get_body_force_FR_current() const
 *           {
 *               Tensor<1, dim, NumberType> body_force_FR_current;
 *   
 *               if (gravity_term == true)
 *               {
 *                  Tensor<1, dim, NumberType> gravity_vector;
 *                  gravity_vector[gravity_direction] = gravity_value;
 *                  body_force_FR_current = density_FR * gravity_vector;
 *               }
 *               return body_force_FR_current;
 *           }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Quadraturepointhistory"></a> 
 * <h3>Quadrature point history</h3>
 * As seen in @ref step_18 "step-18", the <code> PointHistory </code> class offers a method
 * for storing data at the quadrature points.  Here each quadrature point
 * holds a pointer to a material description.  Thus, different material models
 * can be used in different regions of the domain.  Among other data, we
 * choose to store the "extra" Kirchhoff stress @f$\boldsymbol{\tau}_E@f$ and
 * the dissipation values @f$\mathcal{D}_p@f$ and @f$\mathcal{D}_v@f$.
 * 
 * @code
 *       template <int dim, typename NumberType = Sacado::Fad::DFad<double> > //double>
 *       class PointHistory
 *       {
 *           public:
 *               PointHistory()
 *               {}
 *   
 *               virtual ~PointHistory()
 *               {}
 *   
 *               void setup_lqp (const Parameters::AllParameters &parameters,
 *                               const Time                      &time)
 *               {
 *                   if (parameters.mat_type == "Neo-Hooke")
 *                       solid_material.reset(new NeoHooke<dim,NumberType>(parameters,time));
 *                   else if (parameters.mat_type == "Ogden")
 *                       solid_material.reset(new Ogden<dim,NumberType>(parameters,time));
 *                   else if (parameters.mat_type == "visco-Ogden")
 *                       solid_material.reset(new visco_Ogden<dim,NumberType>(parameters,time));
 *                   else
 *                       Assert (false, ExcMessage("Material type not implemented"));
 *   
 *                   fluid_material.reset(new Material_Darcy_Fluid<dim,NumberType>(parameters));
 *               }
 *   
 *               SymmetricTensor<2, dim, NumberType>
 *               get_tau_E(const Tensor<2, dim, NumberType> &F) const
 *               {
 *                   return solid_material->get_tau_E(F);
 *               }
 *   
 *               SymmetricTensor<2, dim, NumberType>
 *               get_Cauchy_E(const Tensor<2, dim, NumberType> &F) const
 *               {
 *                   return solid_material->get_Cauchy_E(F);
 *               }
 *   
 *               double
 *               get_converged_det_F() const
 *               {
 *                 return  solid_material->get_converged_det_F();
 *               }
 *   
 *               void
 *               update_end_timestep()
 *               {
 *                   solid_material->update_end_timestep();
 *               }
 *   
 *               void
 *               update_internal_equilibrium(const Tensor<2, dim, NumberType> &F )
 *               {
 *                   solid_material->update_internal_equilibrium(F);
 *               }
 *   
 *               double
 *               get_viscous_dissipation() const
 *               {
 *                   return solid_material->get_viscous_dissipation();
 *               }
 *   
 *               Tensor<1,dim, NumberType>
 *               get_seepage_velocity_current (const Tensor<2,dim, NumberType> &F,
 *                                             const Tensor<1,dim, NumberType> &grad_p_fluid) const
 *                {
 *                    return fluid_material->get_seepage_velocity_current(F, grad_p_fluid);
 *                }
 *   
 *               double
 *               get_porous_dissipation(const Tensor<2,dim, NumberType> &F,
 *                                      const Tensor<1,dim, NumberType> &grad_p_fluid) const
 *               {
 *                   return fluid_material->get_porous_dissipation(F, grad_p_fluid);
 *               }
 *   
 *               Tensor<1, dim, NumberType>
 *               get_overall_body_force (const Tensor<2,dim, NumberType> &F,
 *                                       const Parameters::AllParameters &parameters) const
 *               {
 *                   Tensor<1, dim, NumberType> body_force;
 *   
 *                   if (parameters.gravity_term == true)
 *                   {
 *                       const NumberType det_F_AD = determinant(F);
 *                       Assert(det_F_AD > 0.0, ExcInternalError());
 *   
 *                       const NumberType overall_density_ref
 *                           = parameters.density_SR * parameters.solid_vol_frac
 *                             + parameters.density_FR
 *                             * (det_F_AD - parameters.solid_vol_frac);
 *   
 *                      Tensor<1, dim, NumberType> gravity_vector;
 *                      gravity_vector[parameters.gravity_direction] = parameters.gravity_value;
 *                      body_force = overall_density_ref * gravity_vector;
 *                   }
 *   
 *                   return body_force;
 *               }
 *           private:
 *               std::shared_ptr< Material_Hyperelastic<dim, NumberType> > solid_material;
 *               std::shared_ptr< Material_Darcy_Fluid<dim, NumberType> > fluid_material;
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Nonlinearporoviscoelasticsolid"></a> 
 * <h3>Nonlinear poro-viscoelastic solid</h3>
 * The Solid class is the central class as it represents the problem at hand:
 * the nonlinear poro-viscoelastic solid
 * 
 * @code
 *       template <int dim>
 *       class Solid
 *       {
 *             public:
 *               Solid(const Parameters::AllParameters &parameters);
 *               virtual ~Solid();
 *               void run();
 *   
 *             protected:
 *               using ADNumberType = Sacado::Fad::DFad<double>;
 *   
 *               std::ofstream outfile;
 *               std::ofstream pointfile;
 *   
 *               struct PerTaskData_ASM;
 *               template<typename NumberType = double> struct ScratchData_ASM;
 *   
 * @endcode
 * 
 * Generate mesh
 * 
 * @code
 *               virtual void make_grid() = 0;
 *   
 * @endcode
 * 
 * Define points for post-processing
 * 
 * @code
 *               virtual void define_tracked_vertices(std::vector<Point<dim> > &tracked_vertices) = 0;
 *   
 * @endcode
 * 
 * Set up the finite element system to be solved:
 * 
 * @code
 *               void system_setup(TrilinosWrappers::MPI::BlockVector &solution_delta_OUT);
 *   
 * @endcode
 * 
 * Extract sub-blocks from the global matrix
 * 
 * @code
 *               void determine_component_extractors();
 *   
 * @endcode
 * 
 * Several functions to assemble the system and right hand side matrices using multithreading.
 * 
 * @code
 *               void assemble_system
 *                     (const TrilinosWrappers::MPI::BlockVector &solution_delta_OUT );
 *               void assemble_system_one_cell
 *                     (const typename DoFHandler<dim>::active_cell_iterator &cell,
 *                      ScratchData_ASM<ADNumberType> &scratch,
 *                      PerTaskData_ASM &data) const;
 *               void copy_local_to_global_system(const PerTaskData_ASM &data);
 *   
 * @endcode
 * 
 * Define boundary conditions
 * 
 * @code
 *               virtual void make_constraints(const int &it_nr);
 *               virtual void make_dirichlet_constraints(AffineConstraints<double> &constraints) = 0;
 *               virtual Tensor<1,dim> get_neumann_traction
 *                      (const types::boundary_id &boundary_id,
 *                       const Point<dim>         &pt,
 *                       const Tensor<1,dim>      &N) const = 0;
 *               virtual double get_prescribed_fluid_flow
 *                      (const types::boundary_id &boundary_id,
 *                       const Point<dim>         &pt) const = 0;
 *               virtual types::boundary_id
 *                        get_reaction_boundary_id_for_output () const = 0;
 *               virtual std::pair<types::boundary_id,types::boundary_id>
 *                        get_drained_boundary_id_for_output () const = 0;
 *               virtual std::vector<double> get_dirichlet_load
 *                       (const types::boundary_id   &boundary_id,
 *                        const int                  &direction) const = 0;
 *   
 * @endcode
 * 
 * Create and update the quadrature points.
 * 
 * @code
 *               void setup_qph();
 *   
 * @endcode
 * 
 * Solve non-linear system using a Newton-Raphson scheme
 * 
 * @code
 *               void solve_nonlinear_timestep(TrilinosWrappers::MPI::BlockVector &solution_delta_OUT);
 *   
 * @endcode
 * 
 * Solve the linearized equations using a direct solver
 * 
 * @code
 *               void solve_linear_system ( TrilinosWrappers::MPI::BlockVector &newton_update_OUT);
 *   
 * @endcode
 * 
 * Retrieve the  solution
 * 
 * @code
 *               TrilinosWrappers::MPI::BlockVector
 *               get_total_solution(const TrilinosWrappers::MPI::BlockVector &solution_delta_IN) const;
 *   
 * @endcode
 * 
 * Store the converged values of the internal variables at the end of each timestep
 * 
 * @code
 *               void update_end_timestep();
 *   
 * @endcode
 * 
 * Post-processing and writing data to files
 * 
 * @code
 *               void output_results_to_vtu(const unsigned int timestep,
 *                                          const double current_time,
 *                                          TrilinosWrappers::MPI::BlockVector solution) const;
 *               void output_results_to_plot(const unsigned int timestep,
 *                                           const double current_time,
 *                                           TrilinosWrappers::MPI::BlockVector solution,
 *                                           std::vector<Point<dim> > &tracked_vertices,
 *                                           std::ofstream &pointfile) const;
 *   
 * @endcode
 * 
 * Headers and footer for the output files
 * 
 * @code
 *               void print_console_file_header( std::ofstream &outfile) const;
 *               void print_plot_file_header(std::vector<Point<dim> > &tracked_vertices,
 *                                           std::ofstream &pointfile) const;
 *               void print_console_file_footer(std::ofstream &outfile) const;
 *               void print_plot_file_footer( std::ofstream &pointfile) const;
 *   
 * @endcode
 * 
 * For parallel communication
 * 
 * @code
 *               MPI_Comm                         mpi_communicator;
 *               const unsigned int               n_mpi_processes;
 *               const unsigned int               this_mpi_process;
 *               mutable ConditionalOStream       pcout;
 *   
 * @endcode
 * 
 * A collection of the parameters used to describe the problem setup
 * 
 * @code
 *               const Parameters::AllParameters &parameters;
 *   
 * @endcode
 * 
 * Declare an instance of dealii Triangulation class (mesh)
 * 
 * @code
 *               parallel::shared::Triangulation<dim>  triangulation;
 *   
 * @endcode
 * 
 * Keep track of the current time and the time spent evaluating certain functions
 * 
 * @code
 *               Time          time;
 *               TimerOutput   timerconsole;
 *               TimerOutput   timerfile;
 *   
 * @endcode
 * 
 * A storage object for quadrature point information.
 * 
 * @code
 *               CellDataStorage<typename Triangulation<dim>::cell_iterator, PointHistory<dim,ADNumberType> > quadrature_point_history;
 *   
 * @endcode
 * 
 * Integers to store polynomial degree (needed for output)
 * 
 * @code
 *               const unsigned int  degree_displ;
 *               const unsigned int  degree_pore;
 *   
 * @endcode
 * 
 * Declare an instance of dealii FESystem class (finite element definition)
 * 
 * @code
 *               const FESystem<dim> fe;
 *   
 * @endcode
 * 
 * Declare an instance of dealii DoFHandler class (assign DoFs to mesh)
 * 
 * @code
 *               DoFHandler<dim>     dof_handler_ref;
 *   
 * @endcode
 * 
 * Integer to store DoFs per element (this value will be used often)
 * 
 * @code
 *               const unsigned int  dofs_per_cell;
 *   
 * @endcode
 * 
 * Declare an instance of dealii Extractor objects used to retrieve information from the solution vectors
 * We will use "u_fe" and "p_fluid_fe"as subscript in operator [] expressions on FEValues and FEFaceValues
 * objects to extract the components of the displacement vector and fluid pressure, respectively.
 * 
 * @code
 *               const FEValuesExtractors::Vector u_fe;
 *               const FEValuesExtractors::Scalar p_fluid_fe;
 *   
 * @endcode
 * 
 * Description of how the block-system is arranged. There are 3 blocks:
 * 0 - vector DOF displacements u
 * 1 - scalar DOF fluid pressure p_fluid
 * 
 * @code
 *               static const unsigned int  n_blocks = 2;
 *               static const unsigned int  n_components = dim+1;
 *               static const unsigned int  first_u_component = 0;
 *               static const unsigned int  p_fluid_component = dim;
 *   
 *               enum
 *               {
 *                 u_block = 0,
 *                 p_fluid_block = 1
 *               };
 *   
 * @endcode
 * 
 * Extractors
 * 
 * @code
 *               const FEValuesExtractors::Scalar x_displacement;
 *               const FEValuesExtractors::Scalar y_displacement;
 *               const FEValuesExtractors::Scalar z_displacement;
 *               const FEValuesExtractors::Scalar pressure;
 *   
 * @endcode
 * 
 * Block data
 * 
 * @code
 *               std::vector<unsigned int> block_component;
 *   
 * @endcode
 * 
 * DoF index data
 * 
 * @code
 *               std::vector<IndexSet> all_locally_owned_dofs;
 *               IndexSet locally_owned_dofs;
 *               IndexSet locally_relevant_dofs;
 *               std::vector<IndexSet> locally_owned_partitioning;
 *               std::vector<IndexSet> locally_relevant_partitioning;
 *   
 *               std::vector<types::global_dof_index>   dofs_per_block;
 *               std::vector<types::global_dof_index>   element_indices_u;
 *               std::vector<types::global_dof_index>   element_indices_p_fluid;
 *   
 * @endcode
 * 
 * Declare an instance of dealii QGauss class (The Gauss-Legendre family of quadrature rules for numerical integration)
 * Gauss Points in element, with n quadrature points (in each space direction <dim> )
 * 
 * @code
 *               const QGauss<dim>                qf_cell;
 * @endcode
 * 
 * Gauss Points on element faces (used for definition of BCs)
 * 
 * @code
 *               const QGauss<dim - 1>            qf_face;
 * @endcode
 * 
 * Integer to store num GPs per element (this value will be used often)
 * 
 * @code
 *               const unsigned int               n_q_points;
 * @endcode
 * 
 * Integer to store num GPs per face (this value will be used often)
 * 
 * @code
 *               const unsigned int               n_q_points_f;
 *   
 * @endcode
 * 
 * Declare an instance of dealii AffineConstraints class (linear constraints on DoFs due to hanging nodes or BCs)
 * 
 * @code
 *               AffineConstraints<double>        constraints;
 *   
 * @endcode
 * 
 * Declare an instance of dealii classes necessary for FE system set-up and assembly
 * Store elements of tangent matrix (indicated by SparsityPattern class) as sparse matrix (more efficient)
 * 
 * @code
 *               TrilinosWrappers::BlockSparseMatrix tangent_matrix;
 *               TrilinosWrappers::BlockSparseMatrix tangent_matrix_preconditioner;
 * @endcode
 * 
 * Right hand side vector of forces
 * 
 * @code
 *               TrilinosWrappers::MPI::BlockVector  system_rhs;
 * @endcode
 * 
 * Total displacement values + pressure (accumulated solution to FE system)
 * 
 * @code
 *               TrilinosWrappers::MPI::BlockVector  solution_n;
 *   
 * @endcode
 * 
 * Non-block system for the direct solver. We will copy the block system into these to solve the linearized system of equations.
 * 
 * @code
 *               TrilinosWrappers::SparseMatrix tangent_matrix_nb;
 *               TrilinosWrappers::MPI::Vector  system_rhs_nb;
 *   
 * @endcode
 * 
 * We define variables to store norms and update norms and normalisation factors.
 * 
 * @code
 *               struct Errors
 *               {
 *                 Errors()
 *                   :
 *                   norm(1.0), u(1.0), p_fluid(1.0)
 *                 {}
 *   
 *                 void reset()
 *                 {
 *                   norm = 1.0;
 *                   u = 1.0;
 *                   p_fluid = 1.0;
 *                 }
 *                 void normalise(const Errors &rhs)
 *                 {
 *                   if (rhs.norm != 0.0)
 *                     norm /= rhs.norm;
 *                   if (rhs.u != 0.0)
 *                     u /= rhs.u;
 *                   if (rhs.p_fluid != 0.0)
 *                     p_fluid /= rhs.p_fluid;
 *                 }
 *   
 *                 double norm, u, p_fluid;
 *               };
 *   
 * @endcode
 * 
 * Declare several instances of the "Error" structure
 * 
 * @code
 *               Errors error_residual, error_residual_0, error_residual_norm, error_update,
 *                      error_update_0, error_update_norm;
 *   
 * @endcode
 * 
 * Methods to calculate error measures
 * 
 * @code
 *               void get_error_residual(Errors &error_residual_OUT);
 *               void get_error_update
 *                    (const TrilinosWrappers::MPI::BlockVector &newton_update_IN,
 *                     Errors                                   &error_update_OUT);
 *   
 * @endcode
 * 
 * Print information to screen
 * 
 * @code
 *               void print_conv_header();
 *               void print_conv_footer();
 *   
 * @endcode
 * 
 * NOTE: In all functions, we pass by reference (&), so these functions work on the original copy (not a clone copy),
 * modifying the input variables inside the functions will change them outside the function.
 * 
 * @code
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-ImplementationofthecodeSolidcodeclass"></a> 
 * <h3>Implementation of the <code>Solid</code> class</h3>
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Publicinterface"></a> 
 * <h4>Public interface</h4>
 * We initialise the Solid class using data extracted from the parameter file.
 * 
 * @code
 *       template <int dim>
 *       Solid<dim>::Solid(const Parameters::AllParameters &parameters)
 *           :
 *           mpi_communicator(MPI_COMM_WORLD),
 *           n_mpi_processes (Utilities::MPI::n_mpi_processes(mpi_communicator)),
 *           this_mpi_process (Utilities::MPI::this_mpi_process(mpi_communicator)),
 *           pcout(std::cout, this_mpi_process == 0),
 *           parameters(parameters),
 *           triangulation(mpi_communicator,Triangulation<dim>::maximum_smoothing),
 *           time(parameters.end_time, parameters.delta_t),
 *           timerconsole( mpi_communicator,
 *                         pcout,
 *                         TimerOutput::summary,
 *                         TimerOutput::wall_times),
 *           timerfile( mpi_communicator,
 *                      outfile,
 *                      TimerOutput::summary,
 *                      TimerOutput::wall_times),
 *           degree_displ(parameters.poly_degree_displ),
 *           degree_pore(parameters.poly_degree_pore),
 *           fe( FE_Q<dim>(parameters.poly_degree_displ), dim,
 *               FE_Q<dim>(parameters.poly_degree_pore), 1 ),
 *           dof_handler_ref(triangulation),
 *           dofs_per_cell (fe.dofs_per_cell),
 *           u_fe(first_u_component),
 *           p_fluid_fe(p_fluid_component),
 *           x_displacement(first_u_component),
 *           y_displacement(first_u_component+1),
 *           z_displacement(first_u_component+2),
 *           pressure(p_fluid_component),
 *           dofs_per_block(n_blocks),
 *           qf_cell(parameters.quad_order),
 *           qf_face(parameters.quad_order),
 *           n_q_points (qf_cell.size()),
 *           n_q_points_f (qf_face.size())
 *           {
 *            Assert(dim==3, ExcMessage("This problem only works in 3 space dimensions."));
 *            determine_component_extractors();
 *           }
 *   
 * @endcode
 * 
 * The class destructor simply clears the data held by the DOFHandler
 * 
 * @code
 *       template <int dim>
 *       Solid<dim>::~Solid()
 *       {
 *           dof_handler_ref.clear();
 *       }
 *   
 * @endcode
 * 
 * Runs the 3D solid problem
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::run()
 *       {
 * @endcode
 * 
 * The current solution increment is defined as a block vector to reflect the structure
 * of the PDE system, with multiple solution components
 * 
 * @code
 *             TrilinosWrappers::MPI::BlockVector solution_delta;
 *   
 * @endcode
 * 
 * Open file
 * 
 * @code
 *             if (this_mpi_process == 0)
 *             {
 *                 outfile.open("console-output.sol");
 *                 print_console_file_header(outfile);
 *             }
 *   
 * @endcode
 * 
 * Generate mesh
 * 
 * @code
 *             make_grid();
 *   
 * @endcode
 * 
 * Assign DOFs and create the stiffness and right-hand-side force vector
 * 
 * @code
 *             system_setup(solution_delta);
 *   
 * @endcode
 * 
 * Define points for post-processing
 * 
 * @code
 *             std::vector<Point<dim> > tracked_vertices (2);
 *             define_tracked_vertices(tracked_vertices);
 *             std::vector<Point<dim>> reaction_force;
 *   
 *             if (this_mpi_process == 0)
 *             {
 *                 pointfile.open("data-for-gnuplot.sol");
 *                 print_plot_file_header(tracked_vertices, pointfile);
 *             }
 *   
 * @endcode
 * 
 * Print results to output file
 * 
 * @code
 *             if (parameters.outfiles_requested == "true")
 *             {
 *                   output_results_to_vtu(time.get_timestep(),
 *                                         time.get_current(),
 *                                         solution_n           );
 *             }
 *   
 *             output_results_to_plot(time.get_timestep(),
 *                                    time.get_current(),
 *                                    solution_n,
 *                                    tracked_vertices,
 *                                    pointfile);
 *   
 * @endcode
 * 
 * Increment time step (=load step)
 * NOTE: In solving the quasi-static problem, the time becomes a loading parameter,
 * i.e. we increase the loading linearly with time, making the two concepts interchangeable.
 * 
 * @code
 *             time.increment_time();
 *   
 * @endcode
 * 
 * Print information on screen
 * 
 * @code
 *             pcout << "\nSolver:";
 *             pcout << "\n  CST     = make constraints";
 *             pcout << "\n  ASM_SYS = assemble system";
 *             pcout << "\n  SLV     = linear solver \n";
 *   
 * @endcode
 * 
 * Print information on file
 * 
 * @code
 *             outfile << "\nSolver:";
 *             outfile << "\n  CST     = make constraints";
 *             outfile << "\n  ASM_SYS = assemble system";
 *             outfile << "\n  SLV     = linear solver \n";
 *   
 *             while ( (time.get_end() - time.get_current()) > -1.0*parameters.tol_u )
 *               {
 * @endcode
 * 
 * Initialize the current solution increment to zero
 * 
 * @code
 *                 solution_delta = 0.0;
 *   
 * @endcode
 * 
 * Solve the non-linear system using a Newton-Rapshon scheme
 * 
 * @code
 *                 solve_nonlinear_timestep(solution_delta);
 *   
 * @endcode
 * 
 * Add the computed solution increment to total solution
 * 
 * @code
 *                 solution_n += solution_delta;
 *   
 * @endcode
 * 
 * Store the converged values of the internal variables
 * 
 * @code
 *                 update_end_timestep();
 *   
 * @endcode
 * 
 * Output results
 * 
 * @code
 *                 if (( (time.get_timestep()%parameters.timestep_output) == 0 )
 *                      && (parameters.outfiles_requested == "true") )
 *                 {
 *                         output_results_to_vtu(time.get_timestep(),
 *                                               time.get_current(),
 *                                               solution_n           );
 *                 }
 *   
 *                 output_results_to_plot(time.get_timestep(),
 *                                        time.get_current(),
 *                                        solution_n,
 *                                        tracked_vertices,
 *                                        pointfile);
 *   
 * @endcode
 * 
 * Increment the time step (=load step)
 * 
 * @code
 *                 time.increment_time();
 *               }
 *   
 * @endcode
 * 
 * Print the footers and close files
 * 
 * @code
 *             if (this_mpi_process == 0)
 *             {
 *                 print_plot_file_footer(pointfile);
 *                 pointfile.close ();
 *                 print_console_file_footer(outfile);
 *   
 * @endcode
 * 
 * NOTE: ideally, we should close the outfile here [ >> outfile.close (); ]
 * But if we do, then the timer output will not be printed. That is why we leave it open.
 * 
 * @code
 *             }
 *       }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Privateinterface"></a> 
 * <h4>Private interface</h4>
 * We define the structures needed for parallelization with Threading Building Blocks (TBB)
 * Tangent matrix and right-hand side force vector assembly structures.
 * PerTaskData_ASM stores local contributions
 * 
 * @code
 *       template <int dim>
 *       struct Solid<dim>::PerTaskData_ASM
 *       {
 *           FullMatrix<double>        cell_matrix;
 *           Vector<double>            cell_rhs;
 *           std::vector<types::global_dof_index> local_dof_indices;
 *   
 *           PerTaskData_ASM(const unsigned int dofs_per_cell)
 *             :
 *             cell_matrix(dofs_per_cell, dofs_per_cell),
 *             cell_rhs(dofs_per_cell),
 *             local_dof_indices(dofs_per_cell)
 *           {}
 *   
 *           void reset()
 *           {
 *             cell_matrix = 0.0;
 *             cell_rhs = 0.0;
 *           }
 *       };
 *   
 * @endcode
 * 
 * ScratchData_ASM stores larger objects used during the assembly
 * 
 * @code
 *       template <int dim>
 *       template <typename NumberType>
 *       struct Solid<dim>::ScratchData_ASM
 *       {
 *           const TrilinosWrappers::MPI::BlockVector &solution_total;
 *   
 * @endcode
 * 
 * Integration helper
 * 
 * @code
 *           FEValues<dim>     fe_values_ref;
 *           FEFaceValues<dim> fe_face_values_ref;
 *   
 * @endcode
 * 
 * Quadrature point solution
 * 
 * @code
 *           std::vector<NumberType>                  local_dof_values;
 *           std::vector<Tensor<2, dim, NumberType> > solution_grads_u_total;
 *           std::vector<NumberType>                  solution_values_p_fluid_total;
 *           std::vector<Tensor<1, dim, NumberType> > solution_grads_p_fluid_total;
 *           std::vector<Tensor<1, dim, NumberType> > solution_grads_face_p_fluid_total;
 *   
 * @endcode
 * 
 * shape function values
 * 
 * @code
 *           std::vector<std::vector<Tensor<1,dim>>>          Nx;
 *           std::vector<std::vector<double>>                 Nx_p_fluid;
 * @endcode
 * 
 * shape function gradients
 * 
 * @code
 *           std::vector<std::vector<Tensor<2,dim, NumberType>>>          grad_Nx;
 *           std::vector<std::vector<SymmetricTensor<2,dim, NumberType>>> symm_grad_Nx;
 *           std::vector<std::vector<Tensor<1,dim, NumberType>>>          grad_Nx_p_fluid;
 *   
 *           ScratchData_ASM(const FiniteElement<dim> &fe_cell,
 *                           const QGauss<dim> &qf_cell, const UpdateFlags uf_cell,
 *                           const QGauss<dim - 1> & qf_face, const UpdateFlags uf_face,
 *                           const TrilinosWrappers::MPI::BlockVector &solution_total    )
 *             :
 *             solution_total (solution_total),
 *             fe_values_ref(fe_cell, qf_cell, uf_cell),
 *             fe_face_values_ref(fe_cell, qf_face, uf_face),
 *             local_dof_values(fe_cell.dofs_per_cell),
 *             solution_grads_u_total(qf_cell.size()),
 *             solution_values_p_fluid_total(qf_cell.size()),
 *             solution_grads_p_fluid_total(qf_cell.size()),
 *             solution_grads_face_p_fluid_total(qf_face.size()),
 *             Nx(qf_cell.size(), std::vector<Tensor<1,dim>>(fe_cell.dofs_per_cell)),
 *             Nx_p_fluid(qf_cell.size(), std::vector<double>(fe_cell.dofs_per_cell)),
 *             grad_Nx(qf_cell.size(), std::vector<Tensor<2, dim, NumberType>>(fe_cell.dofs_per_cell)),
 *             symm_grad_Nx(qf_cell.size(), std::vector<SymmetricTensor<2, dim, NumberType>> (fe_cell.dofs_per_cell)),
 *             grad_Nx_p_fluid(qf_cell.size(), std::vector<Tensor<1, dim, NumberType>>(fe_cell.dofs_per_cell))
 *           {}
 *   
 *           ScratchData_ASM(const ScratchData_ASM &rhs)
 *             :
 *             solution_total (rhs.solution_total),
 *             fe_values_ref(rhs.fe_values_ref.get_fe(),
 *                           rhs.fe_values_ref.get_quadrature(),
 *                           rhs.fe_values_ref.get_update_flags()),
 *             fe_face_values_ref(rhs.fe_face_values_ref.get_fe(),
 *                                rhs.fe_face_values_ref.get_quadrature(),
 *                                rhs.fe_face_values_ref.get_update_flags()),
 *             local_dof_values(rhs.local_dof_values),
 *             solution_grads_u_total(rhs.solution_grads_u_total),
 *             solution_values_p_fluid_total(rhs.solution_values_p_fluid_total),
 *             solution_grads_p_fluid_total(rhs.solution_grads_p_fluid_total),
 *             solution_grads_face_p_fluid_total(rhs.solution_grads_face_p_fluid_total),
 *             Nx(rhs.Nx),
 *             Nx_p_fluid(rhs.Nx_p_fluid),
 *             grad_Nx(rhs.grad_Nx),
 *             symm_grad_Nx(rhs.symm_grad_Nx),
 *             grad_Nx_p_fluid(rhs.grad_Nx_p_fluid)
 *           {}
 *   
 *           void reset()
 *           {
 *             const unsigned int n_q_points = Nx_p_fluid.size();
 *             const unsigned int n_dofs_per_cell = Nx_p_fluid[0].size();
 *   
 *             Assert(local_dof_values.size() == n_dofs_per_cell, ExcInternalError());
 *   
 *             for (unsigned int k = 0; k < n_dofs_per_cell; ++k)
 *               {
 *                 local_dof_values[k] = 0.0;
 *               }
 *   
 *             Assert(solution_grads_u_total.size() == n_q_points, ExcInternalError());
 *             Assert(solution_values_p_fluid_total.size() == n_q_points, ExcInternalError());
 *             Assert(solution_grads_p_fluid_total.size() == n_q_points, ExcInternalError());
 *   
 *             Assert(Nx.size() == n_q_points, ExcInternalError());
 *             Assert(grad_Nx.size() == n_q_points, ExcInternalError());
 *             Assert(symm_grad_Nx.size() == n_q_points, ExcInternalError());
 *   
 *             for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
 *               {
 *                 Assert( Nx[q_point].size() == n_dofs_per_cell, ExcInternalError());
 *                 Assert( grad_Nx[q_point].size() == n_dofs_per_cell, ExcInternalError());
 *                 Assert( symm_grad_Nx[q_point].size() == n_dofs_per_cell, ExcInternalError());
 *   
 *                 solution_grads_u_total[q_point] = 0.0;
 *                 solution_values_p_fluid_total[q_point] = 0.0;
 *                 solution_grads_p_fluid_total[q_point] = 0.0;
 *   
 *                 for (unsigned int k = 0; k < n_dofs_per_cell; ++k)
 *                   {
 *                     Nx[q_point][k] = 0.0;
 *                     Nx_p_fluid[q_point][k] = 0.0;
 *                     grad_Nx[q_point][k] = 0.0;
 *                     symm_grad_Nx[q_point][k] = 0.0;
 *                     grad_Nx_p_fluid[q_point][k] = 0.0;
 *                   }
 *               }
 *   
 *             const unsigned int n_f_q_points = solution_grads_face_p_fluid_total.size();
 *             Assert(solution_grads_face_p_fluid_total.size() == n_f_q_points, ExcInternalError());
 *   
 *             for (unsigned int f_q_point = 0; f_q_point < n_f_q_points; ++f_q_point)
 *                 solution_grads_face_p_fluid_total[f_q_point] = 0.0;
 *           }
 *       };
 *   
 * @endcode
 * 
 * Define the boundary conditions on the mesh
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::make_constraints(const int &it_nr_IN)
 *       {
 *           pcout     << " CST " << std::flush;
 *           outfile   << " CST " << std::flush;
 *   
 *           if (it_nr_IN > 1) return;
 *   
 *           const bool apply_dirichlet_bc = (it_nr_IN == 0);
 *   
 *           if (apply_dirichlet_bc)
 *           {
 *             constraints.clear();
 *             make_dirichlet_constraints(constraints);
 *           }
 *           else
 *           {
 *             for (unsigned int i=0; i<dof_handler_ref.n_dofs(); ++i)
 *               if (constraints.is_inhomogeneously_constrained(i) == true)
 *                 constraints.set_inhomogeneity(i,0.0);
 *           }
 *           constraints.close();
 *       }
 *   
 * @endcode
 * 
 * Set-up the FE system
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::system_setup(TrilinosWrappers::MPI::BlockVector &solution_delta_OUT)
 *       {
 *           timerconsole.enter_subsection("Setup system");
 *           timerfile.enter_subsection("Setup system");
 *   
 * @endcode
 * 
 * Determine number of components per block
 * 
 * @code
 *           std::vector<unsigned int> block_component(n_components, u_block);
 *           block_component[p_fluid_component] = p_fluid_block;
 *   
 * @endcode
 * 
 * The DOF handler is initialised and we renumber the grid in an efficient manner.
 * 
 * @code
 *           dof_handler_ref.distribute_dofs(fe);
 *           DoFRenumbering::Cuthill_McKee(dof_handler_ref);
 *           DoFRenumbering::component_wise(dof_handler_ref, block_component);
 *   
 * @endcode
 * 
 * Count the number of DoFs in each block
 * 
 * @code
 *           dofs_per_block = DoFTools::count_dofs_per_fe_block(dof_handler_ref, block_component);
 *   
 * @endcode
 * 
 * Setup the sparsity pattern and tangent matrix
 * 
 * @code
 *           all_locally_owned_dofs = DoFTools::locally_owned_dofs_per_subdomain (dof_handler_ref);
 *           std::vector<IndexSet> all_locally_relevant_dofs
 *           = DoFTools::locally_relevant_dofs_per_subdomain (dof_handler_ref);
 *   
 *           locally_owned_dofs.clear();
 *           locally_owned_partitioning.clear();
 *           Assert(all_locally_owned_dofs.size() > this_mpi_process, ExcInternalError());
 *           locally_owned_dofs = all_locally_owned_dofs[this_mpi_process];
 *   
 *           locally_relevant_dofs.clear();
 *           locally_relevant_partitioning.clear();
 *           Assert(all_locally_relevant_dofs.size() > this_mpi_process, ExcInternalError());
 *           locally_relevant_dofs = all_locally_relevant_dofs[this_mpi_process];
 *   
 *           locally_owned_partitioning.reserve(n_blocks);
 *           locally_relevant_partitioning.reserve(n_blocks);
 *   
 *           for (unsigned int b=0; b<n_blocks; ++b)
 *             {
 *               const types::global_dof_index idx_begin
 *               = std::accumulate(dofs_per_block.begin(),
 *                                 std::next(dofs_per_block.begin(),b), 0);
 *               const types::global_dof_index idx_end
 *               = std::accumulate(dofs_per_block.begin(),
 *                                 std::next(dofs_per_block.begin(),b+1), 0);
 *               locally_owned_partitioning.push_back(locally_owned_dofs.get_view(idx_begin, idx_end));
 *               locally_relevant_partitioning.push_back(locally_relevant_dofs.get_view(idx_begin, idx_end));
 *             }
 *   
 * @endcode
 * 
 * Print information on screen
 * 
 * @code
 *           pcout  << "\nTriangulation:\n"
 *                  << "  Number of active cells: "
 *                  << triangulation.n_active_cells()
 *                  << " (by partition:";
 *           for (unsigned int p=0; p<n_mpi_processes; ++p)
 *             pcout  << (p==0 ? ' ' : '+')
 *                    << (GridTools::count_cells_with_subdomain_association (triangulation,p));
 *           pcout << ")"
 *                 << std::endl;
 *           pcout << "  Number of degrees of freedom: "
 *                 << dof_handler_ref.n_dofs()
 *                 << " (by partition:";
 *           for (unsigned int p=0; p<n_mpi_processes; ++p)
 *             pcout  << (p==0 ? ' ' : '+')
 *                    << (DoFTools::count_dofs_with_subdomain_association (dof_handler_ref,p));
 *           pcout << ")"
 *                 << std::endl;
 *           pcout   << "  Number of degrees of freedom per block: "
 *               << "[n_u, n_p_fluid] = ["
 *               << dofs_per_block[u_block]
 *               << ", "
 *               << dofs_per_block[p_fluid_block]
 *               << "]"
 *               << std::endl;
 *   
 * @endcode
 * 
 * Print information to file
 * 
 * @code
 *           outfile  << "\nTriangulation:\n"
 *                    <<  "  Number of active cells: "
 *                   << triangulation.n_active_cells()
 *                   << " (by partition:";
 *           for (unsigned int p=0; p<n_mpi_processes; ++p)
 *             outfile << (p==0 ? ' ' : '+')
 *                     << (GridTools::count_cells_with_subdomain_association (triangulation,p));
 *           outfile << ")"
 *                   << std::endl;
 *           outfile << "  Number of degrees of freedom: "
 *                   << dof_handler_ref.n_dofs()
 *                   << " (by partition:";
 *           for (unsigned int p=0; p<n_mpi_processes; ++p)
 *             outfile  << (p==0 ? ' ' : '+')
 *                      << (DoFTools::count_dofs_with_subdomain_association (dof_handler_ref,p));
 *           outfile << ")"
 *                   << std::endl;
 *           outfile << "  Number of degrees of freedom per block: "
 *               << "[n_u, n_p_fluid] = ["
 *               << dofs_per_block[u_block]
 *               << ", "
 *               << dofs_per_block[p_fluid_block]
 *               << "]"
 *               << std::endl;
 *   
 * @endcode
 * 
 * We optimise the sparsity pattern to reflect this structure and prevent
 * unnecessary data creation for the right-diagonal block components.
 * 
 * @code
 *           Table<2, DoFTools::Coupling> coupling(n_components, n_components);
 *           for (unsigned int ii = 0; ii < n_components; ++ii)
 *             for (unsigned int jj = 0; jj < n_components; ++jj)
 *   
 * @endcode
 * 
 * Identify "zero" matrix components of FE-system (The two components do not couple)
 * 
 * @code
 *               if (((ii == p_fluid_component) && (jj < p_fluid_component))
 *                   || ((ii < p_fluid_component) && (jj == p_fluid_component)) )
 *                 coupling[ii][jj] = DoFTools::none;
 *   
 * @endcode
 * 
 * The rest of components always couple
 * 
 * @code
 *               else
 *                 coupling[ii][jj] = DoFTools::always;
 *   
 *           TrilinosWrappers::BlockSparsityPattern bsp (locally_owned_partitioning,
 *                                                       mpi_communicator);
 *   
 *           DoFTools::make_sparsity_pattern (dof_handler_ref, bsp, constraints,
 *                                            false, this_mpi_process);
 *           bsp.compress();
 *   
 * @endcode
 * 
 * Reinitialize the (sparse) tangent matrix with the given sparsity pattern.
 * 
 * @code
 *           tangent_matrix.reinit (bsp);
 *   
 * @endcode
 * 
 * Initialize the right hand side and solution vectors with number of DoFs
 * 
 * @code
 *           system_rhs.reinit(locally_owned_partitioning, mpi_communicator);
 *           solution_n.reinit(locally_owned_partitioning, mpi_communicator);
 *           solution_delta_OUT.reinit(locally_owned_partitioning, mpi_communicator);
 *   
 * @endcode
 * 
 * Non-block system
 * 
 * @code
 *           TrilinosWrappers::SparsityPattern sp (locally_owned_dofs,
 *                                                 mpi_communicator);
 *           DoFTools::make_sparsity_pattern (dof_handler_ref, sp, constraints,
 *                                            false, this_mpi_process);
 *           sp.compress();
 *           tangent_matrix_nb.reinit (sp);
 *           system_rhs_nb.reinit(locally_owned_dofs, mpi_communicator);
 *   
 * @endcode
 * 
 * Set up the quadrature point history
 * 
 * @code
 *           setup_qph();
 *   
 *           timerconsole.leave_subsection();
 *           timerfile.leave_subsection();
 *       }
 *   
 * @endcode
 * 
 * Component extractors: used to extract sub-blocks from the global matrix
 * Description of which local element DOFs are attached to which block component
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::determine_component_extractors()
 *       {
 *           element_indices_u.clear();
 *           element_indices_p_fluid.clear();
 *   
 *           for (unsigned int k = 0; k < fe.dofs_per_cell; ++k)
 *             {
 *               const unsigned int k_group = fe.system_to_base_index(k).first.first;
 *               if (k_group == u_block)
 *                 element_indices_u.push_back(k);
 *               else if (k_group == p_fluid_block)
 *                 element_indices_p_fluid.push_back(k);
 *               else
 *                 {
 *                   Assert(k_group <= p_fluid_block, ExcInternalError());
 *                 }
 *             }
 *       }
 *   
 * @endcode
 * 
 * Set-up quadrature point history (QPH) data objects
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::setup_qph()
 *       {
 *           pcout     << "\nSetting up quadrature point data..." << std::endl;
 *           outfile   << "\nSetting up quadrature point data..." << std::endl;
 *   
 * @endcode
 * 
 * Create QPH data objects.
 * 
 * @code
 *           quadrature_point_history.initialize(triangulation.begin_active(),
 *                                               triangulation.end(), n_q_points);
 *   
 * @endcode
 * 
 * Setup the initial quadrature point data using the info stored in parameters
 * 
 * @code
 *           FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
 *           cell (IteratorFilters::LocallyOwnedCell(),
 *                 dof_handler_ref.begin_active()),
 *           endc (IteratorFilters::LocallyOwnedCell(),
 *                 dof_handler_ref.end());
 *           for (; cell!=endc; ++cell)
 *             {
 *               Assert(cell->is_locally_owned(), ExcInternalError());
 *               Assert(cell->subdomain_id() == this_mpi_process, ExcInternalError());
 *   
 *               const std::vector<std::shared_ptr<PointHistory<dim, ADNumberType> > >
 *                   lqph = quadrature_point_history.get_data(cell);
 *               Assert(lqph.size() == n_q_points, ExcInternalError());
 *   
 *               for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
 *                 lqph[q_point]->setup_lqp(parameters, time);
 *             }
 *       }
 *   
 * @endcode
 * 
 * Solve the non-linear system using a Newton-Raphson scheme
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::solve_nonlinear_timestep(TrilinosWrappers::MPI::BlockVector &solution_delta_OUT)
 *       {
 * @endcode
 * 
 * Print the load step
 * 
 * @code
 *           pcout  << std::endl
 *                  << "\nTimestep "
 *                  << time.get_timestep()
 *                  << " @ "
 *                  << time.get_current()
 *                  << "s"
 *                  << std::endl;
 *           outfile  << std::endl
 *                    << "\nTimestep "
 *                    << time.get_timestep()
 *                    << " @ "
 *                    << time.get_current()
 *                    << "s"
 *                    << std::endl;
 *   
 * @endcode
 * 
 * Declare newton_update vector (solution of a Newton iteration),
 * which must have as many positions as global DoFs.
 * 
 * @code
 *           TrilinosWrappers::MPI::BlockVector newton_update
 *               (locally_owned_partitioning, mpi_communicator);
 *   
 * @endcode
 * 
 * Reset the error storage objects
 * 
 * @code
 *           error_residual.reset();
 *           error_residual_0.reset();
 *           error_residual_norm.reset();
 *           error_update.reset();
 *           error_update_0.reset();
 *           error_update_norm.reset();
 *   
 *           print_conv_header();
 *   
 * @endcode
 * 
 * Declare and initialize iterator for the Newton-Raphson algorithm steps
 * 
 * @code
 *           unsigned int newton_iteration = 0;
 *   
 * @endcode
 * 
 * Iterate until error is below tolerance or max number iterations are reached
 * 
 * @code
 *           while(newton_iteration < parameters.max_iterations_NR)
 *             {
 *               pcout     << " " << std::setw(2) << newton_iteration << " " << std::flush;
 *               outfile   << " " << std::setw(2) << newton_iteration << " " << std::flush;
 *   
 * @endcode
 * 
 * Initialize global stiffness matrix and global force vector to zero
 * 
 * @code
 *               tangent_matrix = 0.0;
 *               system_rhs = 0.0;
 *   
 *               tangent_matrix_nb = 0.0;
 *               system_rhs_nb = 0.0;
 *   
 * @endcode
 * 
 * Apply boundary conditions
 * 
 * @code
 *               make_constraints(newton_iteration);
 *               assemble_system(solution_delta_OUT);
 *   
 * @endcode
 * 
 * Compute the rhs residual (error between external and internal forces in FE system)
 * 
 * @code
 *               get_error_residual(error_residual);
 *   
 * @endcode
 * 
 * error_residual in first iteration is stored to normalize posterior error measures
 * 
 * @code
 *               if (newton_iteration == 0)
 *                 error_residual_0 = error_residual;
 *   
 * @endcode
 * 
 * Determine the normalised residual error
 * 
 * @code
 *               error_residual_norm = error_residual;
 *               error_residual_norm.normalise(error_residual_0);
 *   
 * @endcode
 * 
 * If both errors are below the tolerances, exit the loop.
 * We need to check the residual vector directly for convergence
 * in the load steps where no external forces or displacements are imposed.
 * 
 * @code
 *               if (  ((newton_iteration > 0)
 *                   && (error_update_norm.u <= parameters.tol_u)
 *                   && (error_update_norm.p_fluid <= parameters.tol_p_fluid)
 *                   && (error_residual_norm.u <= parameters.tol_f)
 *                   && (error_residual_norm.p_fluid  <= parameters.tol_f))
 *                   || ( (newton_iteration > 0)
 *                       && system_rhs.l2_norm() <= parameters.tol_f) )
 *                 {
 *                   pcout   << "\n ***** CONVERGED! *****     "
 *                           << system_rhs.l2_norm() << "      "
 *                           << "  " << error_residual_norm.norm
 *                           << "  " << error_residual_norm.u
 *                           << "  " << error_residual_norm.p_fluid
 *                           << "        " << error_update_norm.norm
 *                           << "  " << error_update_norm.u
 *                           << "  " << error_update_norm.p_fluid
 *                           << "  " << std::endl;
 *                   outfile   << "\n ***** CONVERGED! *****     "
 *                           << system_rhs.l2_norm() << "      "
 *                           << "  " << error_residual_norm.norm
 *                           << "  " << error_residual_norm.u
 *                           << "  " << error_residual_norm.p_fluid
 *                           << "        " << error_update_norm.norm
 *                           << "  " << error_update_norm.u
 *                           << "  " << error_update_norm.p_fluid
 *                           << "  " << std::endl;
 *                   print_conv_footer();
 *   
 *                   break;
 *                 }
 *   
 * @endcode
 * 
 * Solve the linearized system
 * 
 * @code
 *               solve_linear_system(newton_update);
 *               constraints.distribute(newton_update);
 *   
 * @endcode
 * 
 * Compute the displacement error
 * 
 * @code
 *               get_error_update(newton_update, error_update);
 *   
 * @endcode
 * 
 * error_update in first iteration is stored to normalize posterior error measures
 * 
 * @code
 *               if (newton_iteration == 0)
 *                 error_update_0 = error_update;
 *   
 * @endcode
 * 
 * Determine the normalised Newton update error
 * 
 * @code
 *               error_update_norm = error_update;
 *               error_update_norm.normalise(error_update_0);
 *   
 * @endcode
 * 
 * Determine the normalised residual error
 * 
 * @code
 *               error_residual_norm = error_residual;
 *               error_residual_norm.normalise(error_residual_0);
 *   
 * @endcode
 * 
 * Print error values
 * 
 * @code
 *               pcout    << " |   " << std::fixed << std::setprecision(3)
 *               << std::setw(7) << std::scientific
 *               << system_rhs.l2_norm()
 *               << "        " << error_residual_norm.norm
 *               << "  " << error_residual_norm.u
 *               << "  " << error_residual_norm.p_fluid
 *               << "        " << error_update_norm.norm
 *               << "  " << error_update_norm.u
 *               << "  " << error_update_norm.p_fluid
 *               << "  " << std::endl;
 *   
 *               outfile  << " |   " << std::fixed << std::setprecision(3)
 *               << std::setw(7) << std::scientific
 *               << system_rhs.l2_norm()
 *               << "        " << error_residual_norm.norm
 *               << "  " << error_residual_norm.u
 *               << "  " << error_residual_norm.p_fluid
 *               << "        " << error_update_norm.norm
 *               << "  " << error_update_norm.u
 *               << "  " << error_update_norm.p_fluid
 *               << "  " << std::endl;
 *   
 * @endcode
 * 
 * Update
 * 
 * @code
 *               solution_delta_OUT += newton_update;
 *               newton_update = 0.0;
 *               newton_iteration++;
 *             }
 *   
 * @endcode
 * 
 * If maximum allowed number of iterations for Newton algorithm are reached, print non-convergence message and abort program
 * 
 * @code
 *           AssertThrow (newton_iteration < parameters.max_iterations_NR, ExcMessage("No convergence in nonlinear solver!"));
 *       }
 *   
 * @endcode
 * 
 * Prints the header for convergence info on console
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::print_conv_header()
 *       {
 *           static const unsigned int l_width = 120;
 *   
 *           for (unsigned int i = 0; i < l_width; ++i)
 *             {
 *               pcout     << "_";
 *               outfile   << "_";
 *             }
 *   
 *           pcout     << std::endl;
 *           outfile   << std::endl;
 *   
 *           pcout   << "\n       SOLVER STEP      |    SYS_RES         "
 *                   << "RES_NORM     RES_U      RES_P           "
 *                   << "NU_NORM     NU_U       NU_P " << std::endl;
 *           outfile << "\n       SOLVER STEP      |    SYS_RES         "
 *                   << "RES_NORM     RES_U      RES_P           "
 *                   << "NU_NORM     NU_U       NU_P " << std::endl;
 *   
 *           for (unsigned int i = 0; i < l_width; ++i)
 *             {
 *               pcout     << "_";
 *               outfile   << "_";
 *             }
 *           pcout     << std::endl << std::endl;
 *           outfile   << std::endl << std::endl;
 *       }
 *   
 * @endcode
 * 
 * Prints the footer for convergence info on console
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::print_conv_footer()
 *       {
 *           static const unsigned int l_width = 120;
 *   
 *           for (unsigned int i = 0; i < l_width; ++i)
 *             {
 *               pcout     << "_";
 *               outfile   << "_";
 *             }
 *           pcout     << std::endl << std::endl;
 *           outfile   << std::endl << std::endl;
 *   
 *           pcout << "Relative errors:" << std::endl
 *                 << "Displacement:  "
 *                 << error_update.u / error_update_0.u << std::endl
 *                 << "Force (displ): "
 *                 << error_residual.u / error_residual_0.u << std::endl
 *                 << "Pore pressure: "
 *                 << error_update.p_fluid / error_update_0.p_fluid << std::endl
 *                 << "Force (pore):  "
 *                 << error_residual.p_fluid / error_residual_0.p_fluid << std::endl;
 *           outfile << "Relative errors:" << std::endl
 *                   << "Displacement:  "
 *                   << error_update.u / error_update_0.u << std::endl
 *                   << "Force (displ): "
 *                   << error_residual.u / error_residual_0.u << std::endl
 *                   << "Pore pressure: "
 *                   << error_update.p_fluid / error_update_0.p_fluid << std::endl
 *                   << "Force (pore):  "
 *                   << error_residual.p_fluid / error_residual_0.p_fluid << std::endl;
 *       }
 *   
 * @endcode
 * 
 * Determine the true residual error for the problem
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::get_error_residual(Errors &error_residual_OUT)
 *       {
 *           TrilinosWrappers::MPI::BlockVector error_res(system_rhs);
 *           constraints.set_zero(error_res);
 *   
 *           error_residual_OUT.norm = error_res.l2_norm();
 *           error_residual_OUT.u = error_res.block(u_block).l2_norm();
 *           error_residual_OUT.p_fluid = error_res.block(p_fluid_block).l2_norm();
 *       }
 *   
 * @endcode
 * 
 * Determine the true Newton update error for the problem
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::get_error_update
 *                  (const TrilinosWrappers::MPI::BlockVector &newton_update_IN,
 *                   Errors                                   &error_update_OUT)
 *       {
 *           TrilinosWrappers::MPI::BlockVector error_ud(newton_update_IN);
 *           constraints.set_zero(error_ud);
 *   
 *           error_update_OUT.norm = error_ud.l2_norm();
 *           error_update_OUT.u = error_ud.block(u_block).l2_norm();
 *           error_update_OUT.p_fluid = error_ud.block(p_fluid_block).l2_norm();
 *       }
 *   
 * @endcode
 * 
 * Compute the total solution, which is valid at any Newton step. This is required as, to reduce
 * computational error, the total solution is only updated at the end of the timestep.
 * 
 * @code
 *       template <int dim>
 *       TrilinosWrappers::MPI::BlockVector
 *       Solid<dim>::get_total_solution(const TrilinosWrappers::MPI::BlockVector &solution_delta_IN) const
 *       {
 * @endcode
 * 
 * Cell interpolation -> Ghosted vector
 * 
 * @code
 *           TrilinosWrappers::MPI::BlockVector
 *                solution_total (locally_owned_partitioning,
 *                                locally_relevant_partitioning,
 *                                mpi_communicator,
 *                                /*vector_writable = */ false);
 *           TrilinosWrappers::MPI::BlockVector tmp (solution_total);
 *           solution_total = solution_n;
 *           tmp = solution_delta_IN;
 *           solution_total += tmp;
 *           return solution_total;
 *       }
 *   
 * @endcode
 * 
 * Compute elemental stiffness tensor and right-hand side force vector, and assemble into global ones
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::assemble_system( const TrilinosWrappers::MPI::BlockVector &solution_delta )
 *       {
 *           timerconsole.enter_subsection("Assemble system");
 *           timerfile.enter_subsection("Assemble system");
 *           pcout     << " ASM_SYS " << std::flush;
 *           outfile   << " ASM_SYS " << std::flush;
 *   
 *           const TrilinosWrappers::MPI::BlockVector solution_total(get_total_solution(solution_delta));
 *   
 * @endcode
 * 
 * Info given to FEValues and FEFaceValues constructors, to indicate which data will be needed at each element.
 * 
 * @code
 *           const UpdateFlags uf_cell(update_values |
 *                                     update_gradients |
 *                                     update_JxW_values);
 *           const UpdateFlags uf_face(update_values |
 *                                     update_gradients |
 *                                     update_normal_vectors |
 *                                     update_quadrature_points |
 *                                     update_JxW_values );
 *   
 * @endcode
 * 
 * Setup a copy of the data structures required for the process and pass them, along with the
 * memory addresses of the assembly functions to the WorkStream object for processing
 * 
 * @code
 *           PerTaskData_ASM per_task_data(dofs_per_cell);
 *           ScratchData_ASM<ADNumberType> scratch_data(fe, qf_cell, uf_cell,
 *                                                      qf_face, uf_face,
 *                                                      solution_total);
 *   
 *           FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
 *           cell (IteratorFilters::LocallyOwnedCell(),
 *                 dof_handler_ref.begin_active()),
 *           endc (IteratorFilters::LocallyOwnedCell(),
 *                 dof_handler_ref.end());
 *           for (; cell != endc; ++cell)
 *             {
 *               Assert(cell->is_locally_owned(), ExcInternalError());
 *               Assert(cell->subdomain_id() == this_mpi_process, ExcInternalError());
 *   
 *               assemble_system_one_cell(cell, scratch_data, per_task_data);
 *               copy_local_to_global_system(per_task_data);
 *             }
 *           tangent_matrix.compress(VectorOperation::add);
 *           system_rhs.compress(VectorOperation::add);
 *   
 *           tangent_matrix_nb.compress(VectorOperation::add);
 *           system_rhs_nb.compress(VectorOperation::add);
 *   
 *           timerconsole.leave_subsection();
 *           timerfile.leave_subsection();
 *       }
 *   
 * @endcode
 * 
 * Add the local elemental contribution to the global stiffness tensor
 * We do it twice, for the block and the non-block systems
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::copy_local_to_global_system (const PerTaskData_ASM &data)
 *       {
 *           constraints.distribute_local_to_global(data.cell_matrix,
 *               data.cell_rhs,
 *               data.local_dof_indices,
 *               tangent_matrix,
 *               system_rhs);
 *   
 *           constraints.distribute_local_to_global(data.cell_matrix,
 *               data.cell_rhs,
 *               data.local_dof_indices,
 *               tangent_matrix_nb,
 *               system_rhs_nb);
 *       }
 *   
 * @endcode
 * 
 * Compute stiffness matrix and corresponding rhs for one element
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::assemble_system_one_cell
 *                (const typename DoFHandler<dim>::active_cell_iterator &cell,
 *                 ScratchData_ASM<ADNumberType>                        &scratch,
 *                 PerTaskData_ASM                                      &data) const
 *       {
 *           Assert(cell->is_locally_owned(), ExcInternalError());
 *   
 *           data.reset();
 *           scratch.reset();
 *           scratch.fe_values_ref.reinit(cell);
 *           cell->get_dof_indices(data.local_dof_indices);
 *   
 * @endcode
 * 
 * Setup automatic differentiation
 * 
 * @code
 *           for (unsigned int k = 0; k < dofs_per_cell; ++k)
 *             {
 * @endcode
 * 
 * Initialise the dofs for the cell using the current solution.
 * 
 * @code
 *               scratch.local_dof_values[k] = scratch.solution_total[data.local_dof_indices[k]];
 * @endcode
 * 
 * Mark this cell DoF as an independent variable
 * 
 * @code
 *               scratch.local_dof_values[k].diff(k, dofs_per_cell);
 *             }
 *   
 * @endcode
 * 
 * Update the quadrature point solution
 * Compute the values and gradients of the solution in terms of the AD variables
 * 
 * @code
 *           for (unsigned int q = 0; q < n_q_points; ++q)
 *             {
 *               for (unsigned int k = 0; k < dofs_per_cell; ++k)
 *                 {
 *                   const unsigned int k_group = fe.system_to_base_index(k).first.first;
 *                   if (k_group == u_block)
 *                     {
 *                       const Tensor<2, dim> Grad_Nx_u =
 *                              scratch.fe_values_ref[u_fe].gradient(k, q);
 *                       for (unsigned int dd = 0; dd < dim; ++dd)
 *                         {
 *                           for (unsigned int ee = 0; ee < dim; ++ee)
 *                             {
 *                               scratch.solution_grads_u_total[q][dd][ee]
 *                                += scratch.local_dof_values[k] * Grad_Nx_u[dd][ee];
 *                             }
 *                         }
 *                     }
 *                   else if  (k_group == p_fluid_block)
 *                     {
 *                       const double Nx_p = scratch.fe_values_ref[p_fluid_fe].value(k, q);
 *                       const Tensor<1, dim> Grad_Nx_p =
 *                                 scratch.fe_values_ref[p_fluid_fe].gradient(k, q);
 *   
 *                       scratch.solution_values_p_fluid_total[q]
 *                                += scratch.local_dof_values[k] * Nx_p;
 *                       for (unsigned int dd = 0; dd < dim; ++dd)
 *                         {
 *                           scratch.solution_grads_p_fluid_total[q][dd]
 *                               += scratch.local_dof_values[k] * Grad_Nx_p[dd];
 *                         }
 *                     }
 *                   else
 *                     Assert(k_group <= p_fluid_block, ExcInternalError());
 *                 }
 *             }
 *   
 * @endcode
 * 
 * Set up pointer "lgph" to the PointHistory object of this element
 * 
 * @code
 *           const std::vector<std::shared_ptr<const PointHistory<dim, ADNumberType> > >
 *               lqph = quadrature_point_history.get_data(cell);
 *           Assert(lqph.size() == n_q_points, ExcInternalError());
 *   
 *   
 * @endcode
 * 
 * Precalculate the element shape function values and gradients
 * 
 * @code
 *           for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
 *             {
 *               Tensor<2, dim, ADNumberType> F_AD = scratch.solution_grads_u_total[q_point];
 *               F_AD += Tensor<2, dim, double>(Physics::Elasticity::StandardTensors<dim>::I);
 *               Assert(determinant(F_AD) > 0, ExcMessage("Invalid deformation map"));
 *               const Tensor<2, dim, ADNumberType> F_inv_AD = invert(F_AD);
 *   
 *               for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                 {
 *                   const unsigned int i_group = fe.system_to_base_index(i).first.first;
 *   
 *                   if (i_group == u_block)
 *                     {
 *                       scratch.Nx[q_point][i] =
 *                           scratch.fe_values_ref[u_fe].value(i, q_point);
 *                       scratch.grad_Nx[q_point][i] =
 *                           scratch.fe_values_ref[u_fe].gradient(i, q_point)*F_inv_AD;
 *                       scratch.symm_grad_Nx[q_point][i] =
 *                           symmetrize(scratch.grad_Nx[q_point][i]);
 *                     }
 *                   else if  (i_group == p_fluid_block)
 *                     {
 *                       scratch.Nx_p_fluid[q_point][i] =
 *                           scratch.fe_values_ref[p_fluid_fe].value(i, q_point);
 *                       scratch.grad_Nx_p_fluid[q_point][i] =
 *                           scratch.fe_values_ref[p_fluid_fe].gradient(i, q_point)*F_inv_AD;
 *                     }
 *                   else
 *                     Assert(i_group <= p_fluid_block, ExcInternalError());
 *                 }
 *             }
 *   
 * @endcode
 * 
 * Assemble the stiffness matrix and rhs vector
 * 
 * @code
 *           std::vector<ADNumberType> residual_ad (dofs_per_cell, ADNumberType(0.0));
 *           for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
 *             {
 *               Tensor<2, dim, ADNumberType> F_AD = scratch.solution_grads_u_total[q_point];
 *               F_AD += Tensor<2, dim,double>(Physics::Elasticity::StandardTensors<dim>::I);
 *               const ADNumberType det_F_AD = determinant(F_AD);
 *   
 *               Assert(det_F_AD > 0, ExcInternalError());
 *               const Tensor<2, dim, ADNumberType> F_inv_AD = invert(F_AD); //inverse of def. gradient tensor
 *   
 *               const ADNumberType p_fluid = scratch.solution_values_p_fluid_total[q_point];
 *   
 *               {
 *                 PointHistory<dim, ADNumberType> *lqph_q_point_nc =
 *                    const_cast<PointHistory<dim, ADNumberType>*>(lqph[q_point].get());
 *                 lqph_q_point_nc->update_internal_equilibrium(F_AD);
 *               }
 *   
 * @endcode
 * 
 * Get some info from constitutive model of solid
 * 
 * @code
 *               static const SymmetricTensor< 2, dim, double>
 *                   I (Physics::Elasticity::StandardTensors<dim>::I);
 *               const SymmetricTensor<2, dim, ADNumberType>
 *                   tau_E = lqph[q_point]->get_tau_E(F_AD);
 *               SymmetricTensor<2, dim, ADNumberType> tau_fluid_vol (I);
 *               tau_fluid_vol *= -1.0 * p_fluid * det_F_AD;
 *   
 * @endcode
 * 
 * Get some info from constitutive model of fluid
 * 
 * @code
 *               const ADNumberType det_F_aux =  lqph[q_point]->get_converged_det_F();
 *               const double det_F_converged = Tensor<0,dim,double>(det_F_aux); //Needs to be double, not AD number
 *               const Tensor<1, dim, ADNumberType> overall_body_force
 *                   = lqph[q_point]->get_overall_body_force(F_AD, parameters);
 *   
 * @endcode
 * 
 * Define some aliases to make the assembly process easier to follow
 * 
 * @code
 *               const std::vector<Tensor<1,dim>> &Nu = scratch.Nx[q_point];
 *               const std::vector<SymmetricTensor<2, dim, ADNumberType>>
 *                   &symm_grad_Nu = scratch.symm_grad_Nx[q_point];
 *               const std::vector<double> &Np = scratch.Nx_p_fluid[q_point];
 *               const std::vector<Tensor<1, dim, ADNumberType> > &grad_Np
 *                   = scratch.grad_Nx_p_fluid[q_point];
 *               const Tensor<1, dim, ADNumberType> grad_p
 *                   = scratch.solution_grads_p_fluid_total[q_point]*F_inv_AD;
 *               const double JxW = scratch.fe_values_ref.JxW(q_point);
 *   
 *               for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                 {
 *                   const unsigned int i_group = fe.system_to_base_index(i).first.first;
 *   
 *                   if (i_group == u_block)
 *                     {
 *                       residual_ad[i] += symm_grad_Nu[i] * ( tau_E + tau_fluid_vol ) * JxW;
 *                       residual_ad[i] -= Nu[i] * overall_body_force * JxW;
 *                     }
 *                   else if (i_group == p_fluid_block)
 *                     {
 *                       const Tensor<1, dim, ADNumberType> seepage_vel_current
 *                           = lqph[q_point]->get_seepage_velocity_current(F_AD, grad_p);
 *                       residual_ad[i] += Np[i] * (det_F_AD - det_F_converged) * JxW;
 *                       residual_ad[i] -= time.get_delta_t() * grad_Np[i]
 *                                         * seepage_vel_current * JxW;
 *                     }
 *                   else
 *                     Assert(i_group <= p_fluid_block, ExcInternalError());
 *                 }
 *             }
 *   
 * @endcode
 * 
 * Assemble the Neumann contribution (external force contribution).
 * 
 * @code
 *             for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face) //Loop over faces in element
 *               {
 *                 if (cell->face(face)->at_boundary() == true)
 *                   {
 *                     scratch.fe_face_values_ref.reinit(cell, face);
 *   
 *                     for (unsigned int f_q_point = 0; f_q_point < n_q_points_f; ++f_q_point)
 *                       {
 *                         const Tensor<1, dim> &N
 *                             = scratch.fe_face_values_ref.normal_vector(f_q_point);
 *                         const Point<dim>     &pt
 *                             = scratch.fe_face_values_ref.quadrature_point(f_q_point);
 *                         const Tensor<1, dim> traction
 *                             = get_neumann_traction(cell->face(face)->boundary_id(), pt, N);
 *                         const double flow
 *                             = get_prescribed_fluid_flow(cell->face(face)->boundary_id(), pt);
 *   
 *                         if ( (traction.norm() < 1e-12) && (std::abs(flow) < 1e-12) ) continue;
 *   
 *                         const double JxW_f = scratch.fe_face_values_ref.JxW(f_q_point);
 *   
 *                         for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *                           {
 *                             const unsigned int i_group = fe.system_to_base_index(i).first.first;
 *   
 *                             if ((i_group == u_block) && (traction.norm() > 1e-12))
 *                             {
 *                                 const unsigned int component_i
 *                                   = fe.system_to_component_index(i).first;
 *                                 const double Nu_f
 *                                   = scratch.fe_face_values_ref.shape_value(i, f_q_point);
 *                                 residual_ad[i] -= (Nu_f * traction[component_i]) * JxW_f;
 *                             }
 *                             if ((i_group == p_fluid_block) && (std::abs(flow) > 1e-12))
 *                             {
 *                                 const double Nu_p
 *                                   = scratch.fe_face_values_ref.shape_value(i, f_q_point);
 *                                 residual_ad[i] -= (Nu_p * flow) * JxW_f;
 *                             }
 *                           }
 *                       }
 *                   }
 *               }
 *   
 * @endcode
 * 
 * Linearise the residual
 * 
 * @code
 *           for (unsigned int i = 0; i < dofs_per_cell; ++i)
 *             {
 *               const ADNumberType &R_i = residual_ad[i];
 *   
 *               data.cell_rhs(i) -= R_i.val();
 *               for (unsigned int j=0; j<dofs_per_cell; ++j)
 *                 data.cell_matrix(i,j) += R_i.fastAccessDx(j);
 *             }
 *       }
 *   
 * @endcode
 * 
 * Store the converged values of the internal variables
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::update_end_timestep()
 *       {
 *             FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
 *             cell (IteratorFilters::LocallyOwnedCell(),
 *                   dof_handler_ref.begin_active()),
 *             endc (IteratorFilters::LocallyOwnedCell(),
 *                   dof_handler_ref.end());
 *             for (; cell!=endc; ++cell)
 *             {
 *               Assert(cell->is_locally_owned(), ExcInternalError());
 *               Assert(cell->subdomain_id() == this_mpi_process, ExcInternalError());
 *   
 *               const std::vector<std::shared_ptr<PointHistory<dim, ADNumberType> > >
 *                   lqph = quadrature_point_history.get_data(cell);
 *               Assert(lqph.size() == n_q_points, ExcInternalError());
 *               for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
 *                 lqph[q_point]->update_end_timestep();
 *             }
 *       }
 *   
 *   
 * @endcode
 * 
 * Solve the linearized equations
 * 
 * @code
 *        template <int dim>
 *        void Solid<dim>::solve_linear_system( TrilinosWrappers::MPI::BlockVector &newton_update_OUT)
 *        {
 *   
 *              timerconsole.enter_subsection("Linear solver");
 *              timerfile.enter_subsection("Linear solver");
 *              pcout     << " SLV " << std::flush;
 *              outfile   << " SLV " << std::flush;
 *   
 *              TrilinosWrappers::MPI::Vector newton_update_nb;
 *              newton_update_nb.reinit(locally_owned_dofs, mpi_communicator);
 *   
 *              SolverControl solver_control (tangent_matrix_nb.m(),
 *                                            1.0e-6 * system_rhs_nb.l2_norm());
 *              TrilinosWrappers::SolverDirect solver (solver_control);
 *              solver.solve(tangent_matrix_nb, newton_update_nb, system_rhs_nb);
 *   
 * @endcode
 * 
 * Copy the non-block solution back to block system
 * 
 * @code
 *              for (unsigned int i=0; i<locally_owned_dofs.n_elements(); ++i)
 *                {
 *                  const types::global_dof_index idx_i
 *                                 = locally_owned_dofs.nth_index_in_set(i);
 *                  newton_update_OUT(idx_i) = newton_update_nb(idx_i);
 *                }
 *              newton_update_OUT.compress(VectorOperation::insert);
 *   
 *              timerconsole.leave_subsection();
 *              timerfile.leave_subsection();
 *        }
 *   
 * @endcode
 * 
 * Class to compute gradient of the pressure
 * 
 * @code
 *       template <int dim>
 *       class GradientPostprocessor : public DataPostprocessorVector<dim>
 *       {
 *           public:
 *             GradientPostprocessor (const unsigned int p_fluid_component)
 *               :
 *               DataPostprocessorVector<dim> ("grad_p",
 *                                             update_gradients),
 *               p_fluid_component (p_fluid_component)
 *             {}
 *   
 *             virtual ~GradientPostprocessor(){}
 *   
 *             virtual void
 *             evaluate_vector_field
 *                  (const DataPostprocessorInputs::Vector<dim> &input_data,
 *                   std::vector<Vector<double> >               &computed_quantities) const override
 *             {
 *               AssertDimension (input_data.solution_gradients.size(),
 *                                computed_quantities.size());
 *               for (unsigned int p=0; p<input_data.solution_gradients.size(); ++p)
 *                 {
 *                   AssertDimension (computed_quantities[p].size(), dim);
 *                   for (unsigned int d=0; d<dim; ++d)
 *                     computed_quantities[p][d]
 *                       = input_data.solution_gradients[p][p_fluid_component][d];
 *                 }
 *             }
 *   
 *           private:
 *             const unsigned int  p_fluid_component;
 *       };
 *   
 *   
 * @endcode
 * 
 * Print results to vtu file
 * 
 * @code
 *         template <int dim> void Solid<dim>::output_results_to_vtu
 *                               (const unsigned int timestep,
 *                                const double current_time,
 *                                TrilinosWrappers::MPI::BlockVector solution_IN) const
 *         {
 *           TrilinosWrappers::MPI::BlockVector solution_total(locally_owned_partitioning,
 *                                                             locally_relevant_partitioning,
 *                                                             mpi_communicator,
 *                                                             false);
 *           solution_total = solution_IN;
 *           Vector<double> material_id;
 *           material_id.reinit(triangulation.n_active_cells());
 *           std::vector<types::subdomain_id> partition_int(triangulation.n_active_cells());
 *           GradientPostprocessor<dim> gradient_postprocessor(p_fluid_component);
 *   
 * @endcode
 * 
 * Declare local variables with number of stress components
 * & assign value according to "dim" value
 * 
 * @code
 *            unsigned int num_comp_symm_tensor = 6;
 *   
 * @endcode
 * 
 * Declare local vectors to store values
 * OUTPUT AVERAGED ON ELEMENTS -------------------------------------------
 * 
 * @code
 *           std::vector<Vector<double>>cauchy_stresses_total_elements
 *                                (num_comp_symm_tensor,
 *                                 Vector<double> (triangulation.n_active_cells()));
 *           std::vector<Vector<double>>cauchy_stresses_E_elements
 *                                (num_comp_symm_tensor,
 *                                 Vector<double> (triangulation.n_active_cells()));
 *           std::vector<Vector<double>>stretches_elements
 *                                (dim,
 *                                 Vector<double> (triangulation.n_active_cells()));
 *           std::vector<Vector<double>>seepage_velocity_elements
 *                                 (dim,
 *                                  Vector<double> (triangulation.n_active_cells()));
 *           Vector<double> porous_dissipation_elements
 *                                 (triangulation.n_active_cells());
 *           Vector<double> viscous_dissipation_elements
 *                                 (triangulation.n_active_cells());
 *           Vector<double> solid_vol_fraction_elements
 *                                 (triangulation.n_active_cells());
 *   
 * @endcode
 * 
 * OUTPUT AVERAGED ON NODES ----------------------------------------------
 * We need to create a new FE space with a single dof per node to avoid
 * duplication of the output on nodes for our problem with dim+1 dofs.
 * 
 * @code
 *           FE_Q<dim> fe_vertex(1);
 *           DoFHandler<dim> vertex_handler_ref(triangulation);
 *           vertex_handler_ref.distribute_dofs(fe_vertex);
 *           AssertThrow(vertex_handler_ref.n_dofs() == triangulation.n_vertices(),
 *             ExcDimensionMismatch(vertex_handler_ref.n_dofs(),
 *                                  triangulation.n_vertices()));
 *   
 *           Vector<double> counter_on_vertices_mpi
 *                           (vertex_handler_ref.n_dofs());
 *           Vector<double> sum_counter_on_vertices
 *                           (vertex_handler_ref.n_dofs());
 *   
 *           std::vector<Vector<double>>cauchy_stresses_total_vertex_mpi
 *                                     (num_comp_symm_tensor,
 *                                      Vector<double>(vertex_handler_ref.n_dofs()));
 *           std::vector<Vector<double>>sum_cauchy_stresses_total_vertex
 *                                     (num_comp_symm_tensor,
 *                                      Vector<double>(vertex_handler_ref.n_dofs()));
 *           std::vector<Vector<double>>cauchy_stresses_E_vertex_mpi
 *                                     (num_comp_symm_tensor,
 *                                      Vector<double>(vertex_handler_ref.n_dofs()));
 *           std::vector<Vector<double>>sum_cauchy_stresses_E_vertex
 *                                     (num_comp_symm_tensor,
 *                                      Vector<double>(vertex_handler_ref.n_dofs()));
 *           std::vector<Vector<double>>stretches_vertex_mpi
 *                                     (dim,
 *                                      Vector<double>(vertex_handler_ref.n_dofs()));
 *           std::vector<Vector<double>>sum_stretches_vertex
 *                                     (dim,
 *                                      Vector<double>(vertex_handler_ref.n_dofs()));
 *           Vector<double> porous_dissipation_vertex_mpi(vertex_handler_ref.n_dofs());
 *           Vector<double> sum_porous_dissipation_vertex(vertex_handler_ref.n_dofs());
 *           Vector<double> viscous_dissipation_vertex_mpi(vertex_handler_ref.n_dofs());
 *           Vector<double> sum_viscous_dissipation_vertex(vertex_handler_ref.n_dofs());
 *           Vector<double> solid_vol_fraction_vertex_mpi(vertex_handler_ref.n_dofs());
 *           Vector<double> sum_solid_vol_fraction_vertex(vertex_handler_ref.n_dofs());
 *   
 * @endcode
 * 
 * We need to create a new FE space with a dim dof per node to
 * be able to output data on nodes in vector form
 * 
 * @code
 *           FESystem<dim> fe_vertex_vec(FE_Q<dim>(1),dim);
 *           DoFHandler<dim> vertex_vec_handler_ref(triangulation);
 *           vertex_vec_handler_ref.distribute_dofs(fe_vertex_vec);
 *           AssertThrow(vertex_vec_handler_ref.n_dofs() == (dim*triangulation.n_vertices()),
 *             ExcDimensionMismatch(vertex_vec_handler_ref.n_dofs(),
 *                                  (dim*triangulation.n_vertices())));
 *   
 *           Vector<double> seepage_velocity_vertex_vec_mpi(vertex_vec_handler_ref.n_dofs());
 *           Vector<double> sum_seepage_velocity_vertex_vec(vertex_vec_handler_ref.n_dofs());
 *           Vector<double> counter_on_vertices_vec_mpi(vertex_vec_handler_ref.n_dofs());
 *           Vector<double> sum_counter_on_vertices_vec(vertex_vec_handler_ref.n_dofs());
 * @endcode
 * 
 * -----------------------------------------------------------------------
 * 
 
 * 
 * Declare and initialize local unit vectors (to construct tensor basis)
 * 
 * @code
 *           std::vector<Tensor<1,dim>> basis_vectors (dim, Tensor<1,dim>() );
 *           for (unsigned int i=0; i<dim; ++i)
 *               basis_vectors[i][i] = 1;
 *   
 * @endcode
 * 
 * Declare an instance of the material class object
 * 
 * @code
 *           if (parameters.mat_type == "Neo-Hooke")
 *               NeoHooke<dim,ADNumberType> material(parameters,time);
 *           else if (parameters.mat_type == "Ogden")
 *               Ogden<dim,ADNumberType> material(parameters,time);
 *           else if (parameters.mat_type == "visco-Ogden")
 *               visco_Ogden <dim,ADNumberType>material(parameters,time);
 *           else
 *               Assert (false, ExcMessage("Material type not implemented"));
 *   
 * @endcode
 * 
 * Define a local instance of FEValues to compute updated values required
 * to calculate stresses
 * 
 * @code
 *           const UpdateFlags uf_cell(update_values | update_gradients |
 *                                     update_JxW_values);
 *           FEValues<dim> fe_values_ref (fe, qf_cell, uf_cell);
 *   
 * @endcode
 * 
 * Iterate through elements (cells) and Gauss Points
 * 
 * @code
 *           FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
 *             cell(IteratorFilters::LocallyOwnedCell(),
 *                  dof_handler_ref.begin_active()),
 *             endc(IteratorFilters::LocallyOwnedCell(),
 *                  dof_handler_ref.end()),
 *             cell_v(IteratorFilters::LocallyOwnedCell(),
 *                    vertex_handler_ref.begin_active()),
 *             cell_v_vec(IteratorFilters::LocallyOwnedCell(),
 *                        vertex_vec_handler_ref.begin_active());
 * @endcode
 * 
 * start cell loop
 * 
 * @code
 *           for (; cell!=endc; ++cell, ++cell_v, ++cell_v_vec)
 *           {
 *               Assert(cell->is_locally_owned(), ExcInternalError());
 *               Assert(cell->subdomain_id() == this_mpi_process, ExcInternalError());
 *   
 *               material_id(cell->active_cell_index())=
 *                 static_cast<int>(cell->material_id());
 *   
 *               fe_values_ref.reinit(cell);
 *   
 *               std::vector<Tensor<2,dim>> solution_grads_u(n_q_points);
 *               fe_values_ref[u_fe].get_function_gradients(solution_total,
 *                                                          solution_grads_u);
 *   
 *               std::vector<double> solution_values_p_fluid_total(n_q_points);
 *               fe_values_ref[p_fluid_fe].get_function_values(solution_total,
 *                                                             solution_values_p_fluid_total);
 *   
 *               std::vector<Tensor<1,dim>> solution_grads_p_fluid_AD (n_q_points);
 *               fe_values_ref[p_fluid_fe].get_function_gradients(solution_total,
 *                                                                solution_grads_p_fluid_AD);
 *   
 * @endcode
 * 
 * start gauss point loop
 * 
 * @code
 *               for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
 *               {
 *                   const Tensor<2,dim,ADNumberType>
 *                     F_AD = Physics::Elasticity::Kinematics::F(solution_grads_u[q_point]);
 *                   ADNumberType det_F_AD = determinant(F_AD);
 *                   const double det_F = Tensor<0,dim,double>(det_F_AD);
 *   
 *                   const std::vector<std::shared_ptr<const PointHistory<dim,ADNumberType>>>
 *                       lqph = quadrature_point_history.get_data(cell);
 *                   Assert(lqph.size() == n_q_points, ExcInternalError());
 *   
 *                   const double p_fluid = solution_values_p_fluid_total[q_point];
 *   
 * @endcode
 * 
 * Cauchy stress
 * 
 * @code
 *                   static const SymmetricTensor<2,dim,double>
 *                     I (Physics::Elasticity::StandardTensors<dim>::I);
 *                   SymmetricTensor<2,dim> sigma_E;
 *                   const SymmetricTensor<2,dim,ADNumberType> sigma_E_AD =
 *                     lqph[q_point]->get_Cauchy_E(F_AD);
 *   
 *                   for (unsigned int i=0; i<dim; ++i)
 *                       for (unsigned int j=0; j<dim; ++j)
 *                          sigma_E[i][j] = Tensor<0,dim,double>(sigma_E_AD[i][j]);
 *   
 *                   SymmetricTensor<2,dim> sigma_fluid_vol (I);
 *                   sigma_fluid_vol *= -p_fluid;
 *                   const SymmetricTensor<2,dim> sigma = sigma_E + sigma_fluid_vol;
 *   
 * @endcode
 * 
 * Volumes
 * 
 * @code
 *                   const double solid_vol_fraction = (parameters.solid_vol_frac)/det_F;
 *   
 * @endcode
 * 
 * Green-Lagrange strain
 * 
 * @code
 *                   const Tensor<2,dim> E_strain = 0.5*(transpose(F_AD)*F_AD - I);
 *   
 * @endcode
 * 
 * Seepage velocity
 * 
 * @code
 *                   const Tensor<2,dim,ADNumberType> F_inv = invert(F_AD);
 *                   const Tensor<1,dim,ADNumberType> grad_p_fluid_AD =
 *                                             solution_grads_p_fluid_AD[q_point]*F_inv;
 *                   const Tensor<1,dim,ADNumberType> seepage_vel_AD =
 *                    lqph[q_point]->get_seepage_velocity_current(F_AD, grad_p_fluid_AD);
 *   
 * @endcode
 * 
 * Dissipations
 * 
 * @code
 *                   const double porous_dissipation =
 *                     lqph[q_point]->get_porous_dissipation(F_AD, grad_p_fluid_AD);
 *                   const double viscous_dissipation =
 *                     lqph[q_point]->get_viscous_dissipation();
 *   
 * @endcode
 * 
 * OUTPUT AVERAGED ON ELEMENTS -------------------------------------------
 * Both average on elements and on nodes is NOT weighted with the
 * integration point volume, i.e., we assume equal contribution of each
 * integration point to the average. Ideally, it should be weighted,
 * but I haven't invested time in getting it to work properly.
 * 
 * @code
 *                   if (parameters.outtype == "elements")
 *                   {
 *                       for (unsigned int j=0; j<dim; ++j)
 *                       {
 *                           cauchy_stresses_total_elements[j](cell->active_cell_index())
 *                             += ((sigma*basis_vectors[j])*basis_vectors[j])/n_q_points;
 *                           cauchy_stresses_E_elements[j](cell->active_cell_index())
 *                             += ((sigma_E*basis_vectors[j])*basis_vectors[j])/n_q_points;
 *                           stretches_elements[j](cell->active_cell_index())
 *                             += std::sqrt(1.0+2.0*Tensor<0,dim,double>(E_strain[j][j]))
 *                                /n_q_points;
 *                           seepage_velocity_elements[j](cell->active_cell_index())
 *                             +=  Tensor<0,dim,double>(seepage_vel_AD[j])/n_q_points;
 *                       }
 *   
 *                       porous_dissipation_elements(cell->active_cell_index())
 *                         +=  porous_dissipation/n_q_points;
 *                       viscous_dissipation_elements(cell->active_cell_index())
 *                         +=  viscous_dissipation/n_q_points;
 *                       solid_vol_fraction_elements(cell->active_cell_index())
 *                         +=  solid_vol_fraction/n_q_points;
 *   
 *                       cauchy_stresses_total_elements[3](cell->active_cell_index())
 *                         += ((sigma*basis_vectors[0])*basis_vectors[1])/n_q_points; //sig_xy
 *                       cauchy_stresses_total_elements[4](cell->active_cell_index())
 *                         += ((sigma*basis_vectors[0])*basis_vectors[2])/n_q_points;//sig_xz
 *                       cauchy_stresses_total_elements[5](cell->active_cell_index())
 *                         += ((sigma*basis_vectors[1])*basis_vectors[2])/n_q_points;//sig_yz
 *   
 *                       cauchy_stresses_E_elements[3](cell->active_cell_index())
 *                         += ((sigma_E*basis_vectors[0])* basis_vectors[1])/n_q_points; //sig_xy
 *                       cauchy_stresses_E_elements[4](cell->active_cell_index())
 *                         += ((sigma_E*basis_vectors[0])* basis_vectors[2])/n_q_points;//sig_xz
 *                       cauchy_stresses_E_elements[5](cell->active_cell_index())
 *                         += ((sigma_E*basis_vectors[1])* basis_vectors[2])/n_q_points;//sig_yz
 *   
 *                   }
 * @endcode
 * 
 * OUTPUT AVERAGED ON NODES -------------------------------------------
 * 
 * @code
 *                   else if (parameters.outtype == "nodes")
 *                   {
 *                     for (unsigned int v=0; v<(GeometryInfo<dim>::vertices_per_cell); ++v)
 *                     {
 *                         types::global_dof_index local_vertex_indices =
 *                                                       cell_v->vertex_dof_index(v, 0);
 *                         counter_on_vertices_mpi(local_vertex_indices) += 1;
 *                         for (unsigned int k=0; k<dim; ++k)
 *                         {
 *                             cauchy_stresses_total_vertex_mpi[k](local_vertex_indices)
 *                               += (sigma*basis_vectors[k])*basis_vectors[k];
 *                             cauchy_stresses_E_vertex_mpi[k](local_vertex_indices)
 *                               += (sigma_E*basis_vectors[k])*basis_vectors[k];
 *                             stretches_vertex_mpi[k](local_vertex_indices)
 *                               += std::sqrt(1.0+2.0*Tensor<0,dim,double>(E_strain[k][k]));
 *   
 *                             types::global_dof_index local_vertex_vec_indices =
 *                                                   cell_v_vec->vertex_dof_index(v, k);
 *                             counter_on_vertices_vec_mpi(local_vertex_vec_indices) += 1;
 *                             seepage_velocity_vertex_vec_mpi(local_vertex_vec_indices)
 *                               += Tensor<0,dim,double>(seepage_vel_AD[k]);
 *                         }
 *   
 *                         porous_dissipation_vertex_mpi(local_vertex_indices)
 *                           += porous_dissipation;
 *                         viscous_dissipation_vertex_mpi(local_vertex_indices)
 *                           += viscous_dissipation;
 *                         solid_vol_fraction_vertex_mpi(local_vertex_indices)
 *                           += solid_vol_fraction;
 *   
 *                         cauchy_stresses_total_vertex_mpi[3](local_vertex_indices)
 *                           += (sigma*basis_vectors[0])*basis_vectors[1]; //sig_xy
 *                         cauchy_stresses_total_vertex_mpi[4](local_vertex_indices)
 *                           += (sigma*basis_vectors[0])*basis_vectors[2];//sig_xz
 *                         cauchy_stresses_total_vertex_mpi[5](local_vertex_indices)
 *                           += (sigma*basis_vectors[1])*basis_vectors[2]; //sig_yz
 *   
 *                         cauchy_stresses_E_vertex_mpi[3](local_vertex_indices)
 *                           += (sigma_E*basis_vectors[0])*basis_vectors[1]; //sig_xy
 *                         cauchy_stresses_E_vertex_mpi[4](local_vertex_indices)
 *                           += (sigma_E*basis_vectors[0])*basis_vectors[2];//sig_xz
 *                         cauchy_stresses_E_vertex_mpi[5](local_vertex_indices)
 *                           += (sigma_E*basis_vectors[1])*basis_vectors[2]; //sig_yz
 *                       }
 *                 }
 * @endcode
 * 
 * ---------------------------------------------------------------
 * 
 * @code
 *               } //end gauss point loop
 *           }//end cell loop
 *   
 * @endcode
 * 
 * Different nodes might have different amount of contributions, e.g.,
 * corner nodes have less integration points contributing to the averaged.
 * This is why we need a counter and divide at the end, outside the cell loop.
 * 
 * @code
 *           if (parameters.outtype == "nodes")
 *           {
 *             for (unsigned int d=0; d<(vertex_handler_ref.n_dofs()); ++d)
 *             {
 *               sum_counter_on_vertices[d] =
 *                 Utilities::MPI::sum(counter_on_vertices_mpi[d],
 *                                     mpi_communicator);
 *               sum_porous_dissipation_vertex[d] =
 *                 Utilities::MPI::sum(porous_dissipation_vertex_mpi[d],
 *                                     mpi_communicator);
 *               sum_viscous_dissipation_vertex[d] =
 *                 Utilities::MPI::sum(viscous_dissipation_vertex_mpi[d],
 *                                     mpi_communicator);
 *               sum_solid_vol_fraction_vertex[d] =
 *                 Utilities::MPI::sum(solid_vol_fraction_vertex_mpi[d],
 *                                     mpi_communicator);
 *   
 *               for (unsigned int k=0; k<num_comp_symm_tensor; ++k)
 *               {
 *                 sum_cauchy_stresses_total_vertex[k][d] =
 *                     Utilities::MPI::sum(cauchy_stresses_total_vertex_mpi[k][d],
 *                                         mpi_communicator);
 *                 sum_cauchy_stresses_E_vertex[k][d] =
 *                     Utilities::MPI::sum(cauchy_stresses_E_vertex_mpi[k][d],
 *                                         mpi_communicator);
 *               }
 *               for (unsigned int k=0; k<dim; ++k)
 *               {
 *                 sum_stretches_vertex[k][d] =
 *                     Utilities::MPI::sum(stretches_vertex_mpi[k][d],
 *                                         mpi_communicator);
 *               }
 *             }
 *   
 *             for (unsigned int d=0; d<(vertex_vec_handler_ref.n_dofs()); ++d)
 *             {
 *                 sum_counter_on_vertices_vec[d] =
 *                     Utilities::MPI::sum(counter_on_vertices_vec_mpi[d],
 *                                         mpi_communicator);
 *                 sum_seepage_velocity_vertex_vec[d] =
 *                     Utilities::MPI::sum(seepage_velocity_vertex_vec_mpi[d],
 *                                         mpi_communicator);
 *             }
 *   
 *             for (unsigned int d=0; d<(vertex_handler_ref.n_dofs()); ++d)
 *             {
 *               if (sum_counter_on_vertices[d]>0)
 *               {
 *                 for (unsigned int i=0; i<num_comp_symm_tensor; ++i)
 *                 {
 *                     sum_cauchy_stresses_total_vertex[i][d] /= sum_counter_on_vertices[d];
 *                     sum_cauchy_stresses_E_vertex[i][d] /= sum_counter_on_vertices[d];
 *                 }
 *                 for (unsigned int i=0; i<dim; ++i)
 *                 {
 *                     sum_stretches_vertex[i][d] /= sum_counter_on_vertices[d];
 *                 }
 *                 sum_porous_dissipation_vertex[d] /= sum_counter_on_vertices[d];
 *                 sum_viscous_dissipation_vertex[d] /= sum_counter_on_vertices[d];
 *                 sum_solid_vol_fraction_vertex[d] /= sum_counter_on_vertices[d];
 *               }
 *             }
 *   
 *             for (unsigned int d=0; d<(vertex_vec_handler_ref.n_dofs()); ++d)
 *             {
 *               if (sum_counter_on_vertices_vec[d]>0)
 *               {
 *                 sum_seepage_velocity_vertex_vec[d] /= sum_counter_on_vertices_vec[d];
 *               }
 *             }
 *   
 *           }
 *   
 * @endcode
 * 
 * Add the results to the solution to create the output file for Paraview
 * 
 * @code
 *           DataOut<dim> data_out;
 *           std::vector<DataComponentInterpretation::DataComponentInterpretation>
 *             comp_type(dim,
 *                       DataComponentInterpretation::component_is_part_of_vector);
 *           comp_type.push_back(DataComponentInterpretation::component_is_scalar);
 *   
 *           GridTools::get_subdomain_association(triangulation, partition_int);
 *   
 *           std::vector<std::string> solution_name(dim, "displacement");
 *           solution_name.push_back("pore_pressure");
 *   
 *           data_out.attach_dof_handler(dof_handler_ref);
 *           data_out.add_data_vector(solution_total,
 *                                    solution_name,
 *                                    DataOut<dim>::type_dof_data,
 *                                    comp_type);
 *   
 *           data_out.add_data_vector(solution_total,
 *                                    gradient_postprocessor);
 *   
 *           const Vector<double> partitioning(partition_int.begin(),
 *                                             partition_int.end());
 *   
 *           data_out.add_data_vector(partitioning, "partitioning");
 *           data_out.add_data_vector(material_id, "material_id");
 *   
 * @endcode
 * 
 * Integration point results -----------------------------------------------------------
 * 
 * @code
 *           if (parameters.outtype == "elements")
 *           {
 *             data_out.add_data_vector(cauchy_stresses_total_elements[0], "cauchy_xx");
 *             data_out.add_data_vector(cauchy_stresses_total_elements[1], "cauchy_yy");
 *             data_out.add_data_vector(cauchy_stresses_total_elements[2], "cauchy_zz");
 *             data_out.add_data_vector(cauchy_stresses_total_elements[3], "cauchy_xy");
 *             data_out.add_data_vector(cauchy_stresses_total_elements[4], "cauchy_xz");
 *             data_out.add_data_vector(cauchy_stresses_total_elements[5], "cauchy_yz");
 *   
 *             data_out.add_data_vector(cauchy_stresses_E_elements[0], "cauchy_E_xx");
 *             data_out.add_data_vector(cauchy_stresses_E_elements[1], "cauchy_E_yy");
 *             data_out.add_data_vector(cauchy_stresses_E_elements[2], "cauchy_E_zz");
 *             data_out.add_data_vector(cauchy_stresses_E_elements[3], "cauchy_E_xy");
 *             data_out.add_data_vector(cauchy_stresses_E_elements[4], "cauchy_E_xz");
 *             data_out.add_data_vector(cauchy_stresses_E_elements[5], "cauchy_E_yz");
 *   
 *             data_out.add_data_vector(stretches_elements[0], "stretch_xx");
 *             data_out.add_data_vector(stretches_elements[1], "stretch_yy");
 *             data_out.add_data_vector(stretches_elements[2], "stretch_zz");
 *   
 *             data_out.add_data_vector(seepage_velocity_elements[0], "seepage_vel_x");
 *             data_out.add_data_vector(seepage_velocity_elements[1], "seepage_vel_y");
 *             data_out.add_data_vector(seepage_velocity_elements[2], "seepage_vel_z");
 *   
 *             data_out.add_data_vector(porous_dissipation_elements, "dissipation_porous");
 *             data_out.add_data_vector(viscous_dissipation_elements, "dissipation_viscous");
 *             data_out.add_data_vector(solid_vol_fraction_elements, "solid_vol_fraction");
 *           }
 *           else if  (parameters.outtype == "nodes")
 *           {
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_total_vertex[0],
 *                                      "cauchy_xx");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_total_vertex[1],
 *                                      "cauchy_yy");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_total_vertex[2],
 *                                      "cauchy_zz");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_total_vertex[3],
 *                                      "cauchy_xy");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_total_vertex[4],
 *                                      "cauchy_xz");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_total_vertex[5],
 *                                      "cauchy_yz");
 *   
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_E_vertex[0],
 *                                      "cauchy_E_xx");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_E_vertex[1],
 *                                      "cauchy_E_yy");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_E_vertex[2],
 *                                      "cauchy_E_zz");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_E_vertex[3],
 *                                      "cauchy_E_xy");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_E_vertex[4],
 *                                      "cauchy_E_xz");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_cauchy_stresses_E_vertex[5],
 *                                      "cauchy_E_yz");
 *   
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_stretches_vertex[0],
 *                                      "stretch_xx");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_stretches_vertex[1],
 *                                      "stretch_yy");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_stretches_vertex[2],
 *                                      "stretch_zz");
 *   
 *             std::vector<DataComponentInterpretation::DataComponentInterpretation>
 *              comp_type_vec(dim,
 *                            DataComponentInterpretation::component_is_part_of_vector);
 *             std::vector<std::string> solution_name_vec(dim,"seepage_velocity");
 *   
 *             data_out.add_data_vector(vertex_vec_handler_ref,
 *                                      sum_seepage_velocity_vertex_vec,
 *                                      solution_name_vec,
 *                                      comp_type_vec);
 *   
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_porous_dissipation_vertex,
 *                                      "dissipation_porous");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_viscous_dissipation_vertex,
 *                                      "dissipation_viscous");
 *             data_out.add_data_vector(vertex_handler_ref,
 *                                      sum_solid_vol_fraction_vertex,
 *                                      "solid_vol_fraction");
 *           }
 * @endcode
 * 
 * ---------------------------------------------------------------------
 * 
 
 * 
 * 
 * @code
 *           data_out.build_patches(degree_displ);
 *   
 *           struct Filename
 *           {
 *             static std::string get_filename_vtu(unsigned int process,
 *                                                 unsigned int timestep,
 *                                                 const unsigned int n_digits = 5)
 *             {
 *               std::ostringstream filename_vtu;
 *               filename_vtu
 *               << "solution."
 *               << Utilities::int_to_string(process, n_digits)
 *               << "."
 *               << Utilities::int_to_string(timestep, n_digits)
 *               << ".vtu";
 *               return filename_vtu.str();
 *             }
 *   
 *             static std::string get_filename_pvtu(unsigned int timestep,
 *                                                  const unsigned int n_digits = 5)
 *             {
 *               std::ostringstream filename_vtu;
 *               filename_vtu
 *               << "solution."
 *               << Utilities::int_to_string(timestep, n_digits)
 *               << ".pvtu";
 *               return filename_vtu.str();
 *             }
 *   
 *             static std::string get_filename_pvd (void)
 *             {
 *               std::ostringstream filename_vtu;
 *               filename_vtu
 *               << "solution.pvd";
 *               return filename_vtu.str();
 *             }
 *           };
 *   
 *           const std::string filename_vtu = Filename::get_filename_vtu(this_mpi_process,
 *                                                                       timestep);
 *           std::ofstream output(filename_vtu.c_str());
 *           data_out.write_vtu(output);
 *   
 * @endcode
 * 
 * We have a collection of files written in parallel
 * This next set of steps should only be performed by master process
 * 
 * @code
 *           if (this_mpi_process == 0)
 *           {
 * @endcode
 * 
 * List of all files written out at this timestep by all processors
 * 
 * @code
 *             std::vector<std::string> parallel_filenames_vtu;
 *             for (unsigned int p=0; p<n_mpi_processes; ++p)
 *             {
 *               parallel_filenames_vtu.push_back(Filename::get_filename_vtu(p, timestep));
 *             }
 *   
 *             const std::string filename_pvtu(Filename::get_filename_pvtu(timestep));
 *             std::ofstream pvtu_master(filename_pvtu.c_str());
 *             data_out.write_pvtu_record(pvtu_master,
 *                                        parallel_filenames_vtu);
 *   
 * @endcode
 * 
 * Time dependent data master file
 * 
 * @code
 *             static std::vector<std::pair<double,std::string>> time_and_name_history;
 *             time_and_name_history.push_back(std::make_pair(current_time,
 *                                                             filename_pvtu));
 *             const std::string filename_pvd(Filename::get_filename_pvd());
 *             std::ofstream pvd_output(filename_pvd.c_str());
 *             DataOutBase::write_pvd_record(pvd_output, time_and_name_history);
 *           }
 *         }
 *   
 *   
 * @endcode
 * 
 * Print results to plotting file
 * 
 * @code
 *         template <int dim>
 *         void Solid<dim>::output_results_to_plot(
 *                                   const unsigned int timestep,
 *                                   const double current_time,
 *                                   TrilinosWrappers::MPI::BlockVector solution_IN,
 *                                   std::vector<Point<dim> > &tracked_vertices_IN,
 *                                   std::ofstream &plotpointfile) const
 *         {
 *           TrilinosWrappers::MPI::BlockVector solution_total(locally_owned_partitioning,
 *                                                             locally_relevant_partitioning,
 *                                                             mpi_communicator,
 *                                                             false);
 *   
 *           (void) timestep;
 *           solution_total = solution_IN;
 *   
 * @endcode
 * 
 * Variables needed to print the solution file for plotting
 * 
 * @code
 *           Point<dim> reaction_force;
 *           Point<dim> reaction_force_pressure;
 *           Point<dim> reaction_force_extra;
 *           double total_fluid_flow = 0.0;
 *           double total_porous_dissipation = 0.0;
 *           double total_viscous_dissipation = 0.0;
 *           double total_solid_vol = 0.0;
 *           double total_vol_current = 0.0;
 *           double total_vol_reference = 0.0;
 *           std::vector<Point<dim+1>> solution_vertices(tracked_vertices_IN.size());
 *   
 * @endcode
 * 
 * Auxiliary variables needed for mpi processing
 * 
 * @code
 *           Tensor<1,dim> sum_reaction_mpi;
 *           Tensor<1,dim> sum_reaction_pressure_mpi;
 *           Tensor<1,dim> sum_reaction_extra_mpi;
 *           sum_reaction_mpi = 0.0;
 *           sum_reaction_pressure_mpi = 0.0;
 *           sum_reaction_extra_mpi = 0.0;
 *           double sum_total_flow_mpi = 0.0;
 *           double sum_porous_dissipation_mpi = 0.0;
 *           double sum_viscous_dissipation_mpi = 0.0;
 *           double sum_solid_vol_mpi = 0.0;
 *           double sum_vol_current_mpi = 0.0;
 *           double sum_vol_reference_mpi = 0.0;
 *   
 * @endcode
 * 
 * Declare an instance of the material class object
 * 
 * @code
 *           if (parameters.mat_type == "Neo-Hooke")
 *               NeoHooke<dim,ADNumberType> material(parameters,time);
 *           else if (parameters.mat_type == "Ogden")
 *               Ogden<dim,ADNumberType> material(parameters, time);
 *           else if (parameters.mat_type == "visco-Ogden")
 *               visco_Ogden <dim,ADNumberType>material(parameters,time);
 *           else
 *           Assert (false, ExcMessage("Material type not implemented"));
 *   
 * @endcode
 * 
 * Define a local instance of FEValues to compute updated values required
 * to calculate stresses
 * 
 * @code
 *           const UpdateFlags uf_cell(update_values | update_gradients |
 *                                     update_JxW_values);
 *           FEValues<dim> fe_values_ref (fe, qf_cell, uf_cell);
 *   
 * @endcode
 * 
 * Iterate through elements (cells) and Gauss Points
 * 
 * @code
 *           FilteredIterator<typename DoFHandler<dim>::active_cell_iterator>
 *             cell(IteratorFilters::LocallyOwnedCell(),
 *                  dof_handler_ref.begin_active()),
 *             endc(IteratorFilters::LocallyOwnedCell(),
 *                  dof_handler_ref.end());
 * @endcode
 * 
 * start cell loop
 * 
 * @code
 *           for (; cell!=endc; ++cell)
 *           {
 *               Assert(cell->is_locally_owned(), ExcInternalError());
 *               Assert(cell->subdomain_id() == this_mpi_process, ExcInternalError());
 *   
 *               fe_values_ref.reinit(cell);
 *   
 *               std::vector<Tensor<2,dim>> solution_grads_u(n_q_points);
 *               fe_values_ref[u_fe].get_function_gradients(solution_total,
 *                                                          solution_grads_u);
 *   
 *               std::vector<double> solution_values_p_fluid_total(n_q_points);
 *               fe_values_ref[p_fluid_fe].get_function_values(solution_total,
 *                                                             solution_values_p_fluid_total);
 *   
 *               std::vector<Tensor<1,dim >> solution_grads_p_fluid_AD(n_q_points);
 *               fe_values_ref[p_fluid_fe].get_function_gradients(solution_total,
 *                                                                solution_grads_p_fluid_AD);
 *   
 * @endcode
 * 
 * start gauss point loop
 * 
 * @code
 *               for (unsigned int q_point=0; q_point<n_q_points; ++q_point)
 *               {
 *                   const Tensor<2,dim,ADNumberType>
 *                     F_AD = Physics::Elasticity::Kinematics::F(solution_grads_u[q_point]);
 *                   ADNumberType det_F_AD = determinant(F_AD);
 *                   const double det_F = Tensor<0,dim,double>(det_F_AD);
 *   
 *                   const std::vector<std::shared_ptr<const PointHistory<dim,ADNumberType>>>
 *                       lqph = quadrature_point_history.get_data(cell);
 *                   Assert(lqph.size() == n_q_points, ExcInternalError());
 *   
 *                   double JxW = fe_values_ref.JxW(q_point);
 *   
 * @endcode
 * 
 * Volumes
 * 
 * @code
 *                   sum_vol_current_mpi  += det_F * JxW;
 *                   sum_vol_reference_mpi += JxW;
 *                   sum_solid_vol_mpi += parameters.solid_vol_frac * JxW * det_F;
 *   
 * @endcode
 * 
 * Seepage velocity
 * 
 * @code
 *                   const Tensor<2,dim,ADNumberType> F_inv = invert(F_AD);
 *                   const Tensor<1,dim,ADNumberType>
 *                     grad_p_fluid_AD =  solution_grads_p_fluid_AD[q_point]*F_inv;
 *                   const Tensor<1,dim,ADNumberType> seepage_vel_AD
 *                   = lqph[q_point]->get_seepage_velocity_current(F_AD, grad_p_fluid_AD);
 *   
 * @endcode
 * 
 * Dissipations
 * 
 * @code
 *                   const double porous_dissipation =
 *                     lqph[q_point]->get_porous_dissipation(F_AD, grad_p_fluid_AD);
 *                   sum_porous_dissipation_mpi += porous_dissipation * det_F * JxW;
 *   
 *                   const double viscous_dissipation = lqph[q_point]->get_viscous_dissipation();
 *                   sum_viscous_dissipation_mpi += viscous_dissipation * det_F * JxW;
 *   
 * @endcode
 * 
 * ---------------------------------------------------------------
 * 
 * @code
 *               } //end gauss point loop
 *   
 * @endcode
 * 
 * Compute reaction force on load boundary & total fluid flow across
 * drained boundary.
 * Define a local instance of FEFaceValues to compute values required
 * to calculate reaction force
 * 
 * @code
 *               const UpdateFlags uf_face( update_values | update_gradients |
 *                                          update_normal_vectors | update_JxW_values );
 *               FEFaceValues<dim> fe_face_values_ref(fe, qf_face, uf_face);
 *   
 * @endcode
 * 
 * start face loop
 * 
 * @code
 *               for (unsigned int face=0; face<GeometryInfo<dim>::faces_per_cell; ++face)
 *               {
 * @endcode
 * 
 * Reaction force
 * 
 * @code
 *                   if (cell->face(face)->at_boundary() == true &&
 *                       cell->face(face)->boundary_id() == get_reaction_boundary_id_for_output() )
 *                   {
 *                       fe_face_values_ref.reinit(cell, face);
 *   
 * @endcode
 * 
 * Get displacement gradients for current face
 * 
 * @code
 *                       std::vector<Tensor<2,dim> > solution_grads_u_f(n_q_points_f);
 *                       fe_face_values_ref[u_fe].get_function_gradients
 *                                                            (solution_total,
 *                                                             solution_grads_u_f);
 *   
 * @endcode
 * 
 * Get pressure for current element
 * 
 * @code
 *                       std::vector< double > solution_values_p_fluid_total_f(n_q_points_f);
 *                       fe_face_values_ref[p_fluid_fe].get_function_values
 *                                                  (solution_total,
 *                                                   solution_values_p_fluid_total_f);
 *   
 * @endcode
 * 
 * start gauss points on faces loop
 * 
 * @code
 *                       for (unsigned int f_q_point=0; f_q_point<n_q_points_f; ++f_q_point)
 *                       {
 *                           const Tensor<1,dim> &N = fe_face_values_ref.normal_vector(f_q_point);
 *                           const double JxW_f = fe_face_values_ref.JxW(f_q_point);
 *   
 * @endcode
 * 
 * Compute deformation gradient from displacements gradient
 * (present configuration)
 * 
 * @code
 *                           const Tensor<2,dim,ADNumberType> F_AD =
 *                             Physics::Elasticity::Kinematics::F(solution_grads_u_f[f_q_point]);
 *   
 *                           const std::vector<std::shared_ptr<const PointHistory<dim,ADNumberType>>>
 *                               lqph = quadrature_point_history.get_data(cell);
 *                           Assert(lqph.size() == n_q_points, ExcInternalError());
 *   
 *                           const double p_fluid = solution_values_p_fluid_total[f_q_point];
 *   
 * @endcode
 * 
 * Cauchy stress
 * 
 * @code
 *                           static const SymmetricTensor<2,dim,double>
 *                             I (Physics::Elasticity::StandardTensors<dim>::I);
 *                           SymmetricTensor<2,dim> sigma_E;
 *                           const SymmetricTensor<2,dim,ADNumberType> sigma_E_AD =
 *                             lqph[f_q_point]->get_Cauchy_E(F_AD);
 *   
 *                           for (unsigned int i=0; i<dim; ++i)
 *                               for (unsigned int j=0; j<dim; ++j)
 *                                  sigma_E[i][j] = Tensor<0,dim,double>(sigma_E_AD[i][j]);
 *   
 *                           SymmetricTensor<2,dim> sigma_fluid_vol(I);
 *                           sigma_fluid_vol *= -1.0*p_fluid;
 *                           const SymmetricTensor<2,dim> sigma = sigma_E+sigma_fluid_vol;
 *                           sum_reaction_mpi += sigma * N * JxW_f;
 *                           sum_reaction_pressure_mpi += sigma_fluid_vol * N * JxW_f;
 *                           sum_reaction_extra_mpi += sigma_E * N * JxW_f;
 *                       }//end gauss points on faces loop
 *                   }
 *   
 * @endcode
 * 
 * Fluid flow
 * 
 * @code
 *                   if (cell->face(face)->at_boundary() == true &&
 *                      (cell->face(face)->boundary_id() ==
 *                         get_drained_boundary_id_for_output().first ||
 *                       cell->face(face)->boundary_id() ==
 *                         get_drained_boundary_id_for_output().second ) )
 *                   {
 *                       fe_face_values_ref.reinit(cell, face);
 *   
 * @endcode
 * 
 * Get displacement gradients for current face
 * 
 * @code
 *                       std::vector<Tensor<2,dim>> solution_grads_u_f(n_q_points_f);
 *                       fe_face_values_ref[u_fe].get_function_gradients
 *                                                               (solution_total,
 *                                                                solution_grads_u_f);
 *   
 * @endcode
 * 
 * Get pressure gradients for current face
 * 
 * @code
 *                       std::vector<Tensor<1,dim>> solution_grads_p_f(n_q_points_f);
 *                       fe_face_values_ref[p_fluid_fe].get_function_gradients
 *                                                                (solution_total,
 *                                                                 solution_grads_p_f);
 *   
 * @endcode
 * 
 * start gauss points on faces loop
 * 
 * @code
 *                       for (unsigned int f_q_point=0; f_q_point<n_q_points_f; ++f_q_point)
 *                       {
 *                           const Tensor<1,dim> &N =
 *                                     fe_face_values_ref.normal_vector(f_q_point);
 *                           const double JxW_f = fe_face_values_ref.JxW(f_q_point);
 *   
 * @endcode
 * 
 * Deformation gradient and inverse from displacements gradient
 * (present configuration)
 * 
 * @code
 *                           const Tensor<2,dim,ADNumberType> F_AD
 *                               = Physics::Elasticity::Kinematics::F(solution_grads_u_f[f_q_point]);
 *   
 *                           const Tensor<2,dim,ADNumberType> F_inv_AD = invert(F_AD);
 *                           ADNumberType det_F_AD = determinant(F_AD);
 *   
 *                           const std::vector<std::shared_ptr<const PointHistory<dim,ADNumberType>>>
 *                               lqph = quadrature_point_history.get_data(cell);
 *                           Assert(lqph.size() == n_q_points, ExcInternalError());
 *   
 * @endcode
 * 
 * Seepage velocity
 * 
 * @code
 *                           Tensor<1,dim> seepage;
 *                           double det_F = Tensor<0,dim,double>(det_F_AD);
 *                           const Tensor<1,dim,ADNumberType> grad_p
 *                                             = solution_grads_p_f[f_q_point]*F_inv_AD;
 *                           const Tensor<1,dim,ADNumberType> seepage_AD
 *                             = lqph[f_q_point]->get_seepage_velocity_current(F_AD, grad_p);
 *   
 *                           for (unsigned int i=0; i<dim; ++i)
 *                               seepage[i] = Tensor<0,dim,double>(seepage_AD[i]);
 *   
 *                           sum_total_flow_mpi += (seepage/det_F) * N * JxW_f;
 *                       }//end gauss points on faces loop
 *                   }
 *               }//end face loop
 *           }//end cell loop
 *   
 * @endcode
 * 
 * Sum the results from different MPI process and then add to the reaction_force vector
 * In theory, the solution on each surface (each cell) only exists in one MPI process
 * so, we add all MPI process, one will have the solution and the others will be zero
 * 
 * @code
 *           for (unsigned int d=0; d<dim; ++d)
 *           {
 *               reaction_force[d] = Utilities::MPI::sum(sum_reaction_mpi[d],
 *                                                       mpi_communicator);
 *               reaction_force_pressure[d] = Utilities::MPI::sum(sum_reaction_pressure_mpi[d],
 *                                                                mpi_communicator);
 *               reaction_force_extra[d] = Utilities::MPI::sum(sum_reaction_extra_mpi[d],
 *                                                             mpi_communicator);
 *           }
 *   
 * @endcode
 * 
 * Same for total fluid flow, and for porous and viscous dissipations
 * 
 * @code
 *           total_fluid_flow = Utilities::MPI::sum(sum_total_flow_mpi,
 *                                                  mpi_communicator);
 *           total_porous_dissipation = Utilities::MPI::sum(sum_porous_dissipation_mpi,
 *                                                          mpi_communicator);
 *           total_viscous_dissipation = Utilities::MPI::sum(sum_viscous_dissipation_mpi,
 *                                                           mpi_communicator);
 *           total_solid_vol = Utilities::MPI::sum(sum_solid_vol_mpi,
 *                                                 mpi_communicator);
 *           total_vol_current = Utilities::MPI::sum(sum_vol_current_mpi,
 *                                                   mpi_communicator);
 *           total_vol_reference = Utilities::MPI::sum(sum_vol_reference_mpi,
 *                                                     mpi_communicator);
 *   
 * @endcode
 * 
 * Extract solution for tracked vectors
 * Copying an MPI::BlockVector into MPI::Vector is not possible,
 * so we copy each block of MPI::BlockVector into an MPI::Vector
 * And then we copy the MPI::Vector into "normal" Vectors
 * 
 * @code
 *           TrilinosWrappers::MPI::Vector solution_vector_u_MPI(solution_total.block(u_block));
 *           TrilinosWrappers::MPI::Vector solution_vector_p_MPI(solution_total.block(p_fluid_block));
 *           Vector<double> solution_u_vector(solution_vector_u_MPI);
 *           Vector<double> solution_p_vector(solution_vector_p_MPI);
 *   
 *           if (this_mpi_process == 0)
 *           {
 * @endcode
 * 
 * Append the pressure solution vector to the displacement solution vector,
 * creating a single solution vector equivalent to the original BlockVector
 * so FEFieldFunction will work with the dof_handler_ref.
 * 
 * @code
 *               Vector<double> solution_vector(solution_p_vector.size()
 *                                              +solution_u_vector.size());
 *   
 *               for (unsigned int d=0; d<(solution_u_vector.size()); ++d)
 *                   solution_vector[d] = solution_u_vector[d];
 *   
 *               for (unsigned int d=0; d<(solution_p_vector.size()); ++d)
 *                   solution_vector[solution_u_vector.size()+d] = solution_p_vector[d];
 *   
 *               Functions::FEFieldFunction<dim,Vector<double>>
 *               find_solution(dof_handler_ref, solution_vector);
 *   
 *               for (unsigned int p=0; p<tracked_vertices_IN.size(); ++p)
 *               {
 *                   Vector<double> update(dim+1);
 *                   Point<dim> pt_ref;
 *   
 *                   pt_ref[0]= tracked_vertices_IN[p][0];
 *                   pt_ref[1]= tracked_vertices_IN[p][1];
 *                   pt_ref[2]= tracked_vertices_IN[p][2];
 *   
 *                  find_solution.vector_value(pt_ref, update);
 *   
 *                  for (unsigned int d=0; d<(dim+1); ++d)
 *                  {
 * @endcode
 * 
 * For values close to zero, set to 0.0
 * 
 * @code
 *                      if (abs(update[d])<1.5*parameters.tol_u)
 *                          update[d] = 0.0;
 *                      solution_vertices[p][d] = update[d];
 *                  }
 *               }
 * @endcode
 * 
 * Write the results to the plotting file.
 * Add two blank lines between cycles in the cyclic loading examples so GNUPLOT can detect each cycle as a different block
 * 
 * @code
 *               if (( (parameters.geom_type == "Budday_cube_tension_compression_fully_fixed")||
 *                     (parameters.geom_type == "Budday_cube_tension_compression")||
 *                     (parameters.geom_type == "Budday_cube_shear_fully_fixed")                ) &&
 *                   ( (abs(current_time - parameters.end_time/3.)   <0.9*parameters.delta_t)||
 *                     (abs(current_time - 2.*parameters.end_time/3.)<0.9*parameters.delta_t)   ) &&
 *                     parameters.num_cycle_sets == 1 )
 *               {
 *                   plotpointfile << std::endl<< std::endl;
 *               }
 *               if (( (parameters.geom_type == "Budday_cube_tension_compression_fully_fixed")||
 *                     (parameters.geom_type == "Budday_cube_tension_compression")||
 *                     (parameters.geom_type == "Budday_cube_shear_fully_fixed")             ) &&
 *                   ( (abs(current_time - parameters.end_time/9.)   <0.9*parameters.delta_t)||
 *                     (abs(current_time - 2.*parameters.end_time/9.)<0.9*parameters.delta_t)||
 *                     (abs(current_time - 3.*parameters.end_time/9.)<0.9*parameters.delta_t)||
 *                     (abs(current_time - 5.*parameters.end_time/9.)<0.9*parameters.delta_t)||
 *                     (abs(current_time - 7.*parameters.end_time/9.)<0.9*parameters.delta_t) ) &&
 *                     parameters.num_cycle_sets == 2 )
 *               {
 *                   plotpointfile << std::endl<< std::endl;
 *               }
 *   
 *               plotpointfile <<  std::setprecision(6) << std::scientific;
 *               plotpointfile << std::setw(16) << current_time        << ","
 *                             << std::setw(15) << total_vol_reference << ","
 *                             << std::setw(15) << total_vol_current   << ","
 *                             << std::setw(15) << total_solid_vol     << ",";
 *   
 *               if (current_time == 0.0)
 *               {
 *                   for (unsigned int p=0; p<tracked_vertices_IN.size(); ++p)
 *                   {
 *                       for (unsigned int d=0; d<dim; ++d)
 *                           plotpointfile << std::setw(15) << 0.0 << ",";
 *   
 *                       plotpointfile << std::setw(15) << parameters.drained_pressure << ",";
 *                   }
 *                   for (unsigned int d=0; d<(3*dim+2); ++d)
 *                       plotpointfile << std::setw(15) << 0.0 << ",";
 *   
 *                   plotpointfile << std::setw(15) << 0.0;
 *               }
 *               else
 *               {
 *                   for (unsigned int p=0; p<tracked_vertices_IN.size(); ++p)
 *                       for (unsigned int d=0; d<(dim+1); ++d)
 *                           plotpointfile << std::setw(15) << solution_vertices[p][d]<< ",";
 *   
 *                   for (unsigned int d=0; d<dim; ++d)
 *                       plotpointfile << std::setw(15) << reaction_force[d] << ",";
 *   
 *                   for (unsigned int d=0; d<dim; ++d)
 *                       plotpointfile << std::setw(15) << reaction_force_pressure[d] << ",";
 *   
 *                   for (unsigned int d=0; d<dim; ++d)
 *                       plotpointfile << std::setw(15) << reaction_force_extra[d] << ",";
 *   
 *                   plotpointfile << std::setw(15) << total_fluid_flow << ","
 *                                 << std::setw(15) << total_porous_dissipation<< ","
 *                                 << std::setw(15) << total_viscous_dissipation;
 *               }
 *               plotpointfile << std::endl;
 *           }
 *         }
 *   
 * @endcode
 * 
 * Header for console output file
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::print_console_file_header(std::ofstream &outputfile) const
 *       {
 *               outputfile << "/*-----------------------------------------------------------------------------------------";
 *               outputfile << "\n\n  Poro-viscoelastic formulation to solve nonlinear solid mechanics problems using deal.ii";
 *               outputfile << "\n\n  Problem setup by E Comellas and J-P Pelteret, University of Erlangen-Nuremberg, 2018";
 *               outputfile << "\n\n/*-----------------------------------------------------------------------------------------";
 *               outputfile << "\n\nCONSOLE OUTPUT: \n\n";
 *       }
 *   
 * @endcode
 * 
 * Header for plotting output file
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::print_plot_file_header(std::vector<Point<dim> > &tracked_vertices,
 *                                               std::ofstream &plotpointfile) const
 *       {
 *               plotpointfile << "#\n# *** Solution history for tracked vertices -- DOF: 0 = Ux,  1 = Uy,  2 = Uz,  3 = P ***"
 *                             << std::endl;
 *   
 *               for  (unsigned int p=0; p<tracked_vertices.size(); ++p)
 *               {
 *                   plotpointfile << "#        Point " << p << " coordinates:  ";
 *                   for (unsigned int d=0; d<dim; ++d)
 *                     {
 *                       plotpointfile << tracked_vertices[p][d];
 *                       if (!( (p == tracked_vertices.size()-1) && (d == dim-1) ))
 *                           plotpointfile << ",        ";
 *                     }
 *                   plotpointfile << std::endl;
 *               }
 *               plotpointfile << "#    The reaction force is the integral over the loaded surfaces in the "
 *                             << "undeformed configuration of the Cauchy stress times the normal surface unit vector.\n"
 *                             << "#    reac(p) corresponds to the volumetric part of the Cauchy stress due to the pore fluid pressure"
 *                             << " and reac(E) corresponds to the extra part of the Cauchy stress due to the solid contribution."
 *                             << std::endl
 *                             << "#    The fluid flow is the integral over the drained surfaces in the "
 *                             << "undeformed configuration of the seepage velocity times the normal surface unit vector."
 *                             << std::endl
 *                             << "# Column number:"
 *                             << std::endl
 *                             << "#";
 *   
 *             unsigned int columns = 24;
 *             for (unsigned int d=1; d<columns; ++d)
 *                 plotpointfile << std::setw(15)<< d <<",";
 *   
 *               plotpointfile << std::setw(15)<< columns
 *                             << std::endl
 *                             << "#"
 *                             << std::right << std::setw(16) << "Time,"
 *                             << std::right << std::setw(16) << "ref vol,"
 *                             << std::right << std::setw(16) << "def vol,"
 *                             << std::right << std::setw(16) << "solid vol,";
 *               for (unsigned int p=0; p<tracked_vertices.size(); ++p)
 *                   for (unsigned int d=0; d<(dim+1); ++d)
 *                       plotpointfile << std::right<< std::setw(11)
 *                                     <<"P" << p << "[" << d << "],";
 *   
 *               for (unsigned int d=0; d<dim; ++d)
 *                   plotpointfile << std::right<< std::setw(13)
 *                                 << "reaction [" << d << "],";
 *   
 *               for (unsigned int d=0; d<dim; ++d)
 *                   plotpointfile << std::right<< std::setw(13)
 *                                 << "reac(p) [" << d << "],";
 *   
 *               for (unsigned int d=0; d<dim; ++d)
 *                   plotpointfile << std::right<< std::setw(13)
 *                                 << "reac(E) [" << d << "],";
 *   
 *               plotpointfile << std::right<< std::setw(16)<< "fluid flow,"
 *                             << std::right<< std::setw(16)<< "porous dissip,"
 *                             << std::right<< std::setw(15)<< "viscous dissip"
 *                             << std::endl;
 *       }
 *   
 * @endcode
 * 
 * Footer for console output file
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::print_console_file_footer(std::ofstream &outputfile) const
 *       {
 * @endcode
 * 
 * Copy "parameters" file at end of output file.
 * 
 * @code
 *              std::ifstream infile("parameters.prm");
 *              std::string content = "";
 *              int i;
 *   
 *              for(i=0 ; infile.eof()!=true ; i++)
 *              {
 *                  char aux = infile.get();
 *                  content += aux;
 *                  if(aux=='\n') content += '#';
 *              }
 *   
 *              i--;
 *              content.erase(content.end()-1);
 *              infile.close();
 *   
 *              outputfile << "\n\n\n\n PARAMETERS FILE USED IN THIS COMPUTATION: \n#"
 *                         << std::endl
 *                         << content;
 *       }
 *   
 * @endcode
 * 
 * Footer for plotting output file
 * 
 * @code
 *       template <int dim>
 *       void Solid<dim>::print_plot_file_footer(std::ofstream &plotpointfile) const
 *       {
 * @endcode
 * 
 * Copy "parameters" file at end of output file.
 * 
 * @code
 *              std::ifstream infile("parameters.prm");
 *              std::string content = "";
 *              int i;
 *   
 *              for(i=0 ; infile.eof()!=true ; i++)
 *              {
 *                  char aux = infile.get();
 *                  content += aux;
 *                  if(aux=='\n') content += '#';
 *              }
 *   
 *              i--;
 *              content.erase(content.end()-1);
 *              infile.close();
 *   
 *              plotpointfile << "#"<< std::endl
 *                            << "#"<< std::endl
 *                            << "# PARAMETERS FILE USED IN THIS COMPUTATION:" << std::endl
 *                            << "#"<< std::endl
 *                            << content;
 *       }
 *   
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-VerificationexamplesfromEhlersandEipper1999"></a> 
 * <h3>Verification examples from Ehlers and Eipper 1999</h3>
 * We group the definition of the geometry, boundary and loading conditions specific to
 * the verification examples from Ehlers and Eipper 1999 into specific classes.
 * 
 
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-BaseclassTubegeometryandboundaryconditions"></a> 
 * <h4>Base class: Tube geometry and boundary conditions</h4>
 * 
 * @code
 *       template <int dim>
 *       class VerificationEhlers1999TubeBase
 *             : public Solid<dim>
 *       {
 *           public:
 *             VerificationEhlers1999TubeBase (const Parameters::AllParameters &parameters)
 *               : Solid<dim> (parameters)
 *             {}
 *   
 *             virtual ~VerificationEhlers1999TubeBase () {}
 *   
 *           private:
 *             virtual void make_grid() override
 *             {
 *               GridGenerator::cylinder( this->triangulation,
 *                                        0.1,
 *                                        0.5);
 *   
 *               const double rot_angle = 3.0*numbers::PI/2.0;
 *               GridTools::rotate( Point<3>::unit_vector(1), rot_angle, this->triangulation);
 *   
 *               this->triangulation.reset_manifold(0);
 *               static const CylindricalManifold<dim> manifold_description_3d(2);
 *               this->triangulation.set_manifold (0, manifold_description_3d);
 *               GridTools::scale(this->parameters.scale, this->triangulation);
 *               this->triangulation.refine_global(std::max (1U, this->parameters.global_refinement));
 *               this->triangulation.reset_manifold(0);
 *             }
 *   
 *             virtual void define_tracked_vertices(std::vector<Point<dim> > &tracked_vertices) override
 *             {
 *               tracked_vertices[0][0] = 0.0*this->parameters.scale;
 *               tracked_vertices[0][1] = 0.0*this->parameters.scale;
 *               tracked_vertices[0][2] = 0.5*this->parameters.scale;
 *   
 *               tracked_vertices[1][0] = 0.0*this->parameters.scale;
 *               tracked_vertices[1][1] = 0.0*this->parameters.scale;
 *               tracked_vertices[1][2] = -0.5*this->parameters.scale;
 *             }
 *   
 *             virtual void make_dirichlet_constraints(AffineConstraints<double> &constraints) override
 *             {
 *               if (this->time.get_timestep() < 2)
 *               {
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          2,
 *                                                          Functions::ConstantFunction<dim>(this->parameters.drained_pressure,this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *               }
 *               else
 *               {
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          2,
 *                                                          Functions::ZeroFunction<dim>(this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *               }
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         0,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         (this->fe.component_mask(this->x_displacement)|
 *                                                          this->fe.component_mask(this->y_displacement)  ) );
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         1,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         (this->fe.component_mask(this->x_displacement) |
 *                                                          this->fe.component_mask(this->y_displacement) |
 *                                                          this->fe.component_mask(this->z_displacement) ));
 *             }
 *   
 *             virtual double
 *             get_prescribed_fluid_flow (const types::boundary_id &boundary_id,
 *                                        const Point<dim>         &pt) const override
 *             {
 *                 (void)pt;
 *                 (void)boundary_id;
 *                 return 0.0;
 *             }
 *   
 *             virtual types::boundary_id
 *             get_reaction_boundary_id_for_output() const override
 *             {
 *                 return 2;
 *             }
 *   
 *             virtual  std::pair<types::boundary_id,types::boundary_id>
 *             get_drained_boundary_id_for_output() const override
 *             {
 *                 return std::make_pair(2,2);
 *             }
 *   
 *             virtual std::vector<double>
 *             get_dirichlet_load(const types::boundary_id   &boundary_id,
 *                                const int                  &direction) const override
 *             {
 *                 std::vector<double> displ_incr(dim, 0.0);
 *                 (void)boundary_id;
 *                 (void)direction;
 *                 AssertThrow(false, ExcMessage("Displacement loading not implemented for Ehlers verification examples."));
 *   
 *                 return displ_incr;
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassSteploadexample"></a> 
 * <h4>Derived class: Step load example</h4>
 * 
 * @code
 *       template <int dim>
 *       class VerificationEhlers1999StepLoad
 *             : public VerificationEhlers1999TubeBase<dim>
 *       {
 *           public:
 *             VerificationEhlers1999StepLoad (const Parameters::AllParameters &parameters)
 *               : VerificationEhlers1999TubeBase<dim> (parameters)
 *             {}
 *   
 *             virtual ~VerificationEhlers1999StepLoad () {}
 *   
 *           private:
 *               virtual Tensor<1,dim>
 *               get_neumann_traction (const types::boundary_id &boundary_id,
 *                                     const Point<dim>         &pt,
 *                                     const Tensor<1,dim>      &N) const override
 *               {
 *                 if (this->parameters.load_type == "pressure")
 *                 {
 *                   if (boundary_id == 2)
 *                   {
 *                     return this->parameters.load * N;
 *                   }
 *                 }
 *   
 *                 (void)pt;
 *   
 *                 return Tensor<1,dim>();
 *               }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassLoadincreasingexample"></a> 
 * <h4>Derived class: Load increasing example</h4>
 * 
 * @code
 *       template <int dim>
 *       class VerificationEhlers1999IncreaseLoad
 *             : public VerificationEhlers1999TubeBase<dim>
 *       {
 *           public:
 *             VerificationEhlers1999IncreaseLoad (const Parameters::AllParameters &parameters)
 *               : VerificationEhlers1999TubeBase<dim> (parameters)
 *             {}
 *   
 *             virtual ~VerificationEhlers1999IncreaseLoad () {}
 *   
 *           private:
 *               virtual Tensor<1,dim>
 *               get_neumann_traction (const types::boundary_id &boundary_id,
 *                                     const Point<dim>         &pt,
 *                                     const Tensor<1,dim>      &N) const override
 *               {
 *                 if (this->parameters.load_type == "pressure")
 *                 {
 *                   if (boundary_id == 2)
 *                   {
 *                     const double initial_load = this->parameters.load;
 *                     const double final_load = 20.0*initial_load;
 *                     const double initial_time = this->time.get_delta_t();
 *                     const double final_time = this->time.get_end();
 *                     const double current_time = this->time.get_current();
 *                     const double load = initial_load + (final_load-initial_load)*(current_time-initial_time)/(final_time-initial_time);
 *                     return load * N;
 *                   }
 *                 }
 *   
 *                 (void)pt;
 *   
 *                 return Tensor<1,dim>();
 *               }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-ClassConsolidationcube"></a> 
 * <h4>Class: Consolidation cube</h4>
 * 
 * @code
 *       template <int dim>
 *       class VerificationEhlers1999CubeConsolidation
 *             : public Solid<dim>
 *       {
 *           public:
 *             VerificationEhlers1999CubeConsolidation (const Parameters::AllParameters &parameters)
 *               : Solid<dim> (parameters)
 *             {}
 *   
 *             virtual ~VerificationEhlers1999CubeConsolidation () {}
 *   
 *           private:
 *             virtual void
 *             make_grid() override
 *             {
 *                GridGenerator::hyper_rectangle(this->triangulation,
 *                                               Point<dim>(0.0, 0.0, 0.0),
 *                                               Point<dim>(1.0, 1.0, 1.0),
 *                                               true);
 *   
 *                GridTools::scale(this->parameters.scale, this->triangulation);
 *                this->triangulation.refine_global(std::max (1U, this->parameters.global_refinement));
 *   
 *                typename Triangulation<dim>::active_cell_iterator cell =
 *                        this->triangulation.begin_active(), endc = this->triangulation.end();
 *                for (; cell != endc; ++cell)
 *                {
 *                  for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
 *                    if (cell->face(face)->at_boundary() == true  &&
 *                        cell->face(face)->center()[2] == 1.0 * this->parameters.scale)
 *                    {
 *                      if (cell->face(face)->center()[0] < 0.5 * this->parameters.scale  &&
 *                          cell->face(face)->center()[1] < 0.5 * this->parameters.scale)
 *                          cell->face(face)->set_boundary_id(100);
 *                      else
 *                          cell->face(face)->set_boundary_id(101);
 *                    }
 *                }
 *             }
 *   
 *             virtual void
 *             define_tracked_vertices(std::vector<Point<dim> > &tracked_vertices) override
 *             {
 *               tracked_vertices[0][0] = 0.0*this->parameters.scale;
 *               tracked_vertices[0][1] = 0.0*this->parameters.scale;
 *               tracked_vertices[0][2] = 1.0*this->parameters.scale;
 *   
 *               tracked_vertices[1][0] = 0.0*this->parameters.scale;
 *               tracked_vertices[1][1] = 0.0*this->parameters.scale;
 *               tracked_vertices[1][2] = 0.0*this->parameters.scale;
 *             }
 *   
 *             virtual void
 *             make_dirichlet_constraints(AffineConstraints<double> &constraints) override
 *             {
 *               if (this->time.get_timestep() < 2)
 *               {
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          101,
 *                                                          Functions::ConstantFunction<dim>(this->parameters.drained_pressure,this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *               }
 *               else
 *               {
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          101,
 *                                                          Functions::ZeroFunction<dim>(this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *               }
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         0,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         this->fe.component_mask(this->x_displacement));
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         1,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         this->fe.component_mask(this->x_displacement));
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         2,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         this->fe.component_mask(this->y_displacement));
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         3,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         this->fe.component_mask(this->y_displacement));
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         4,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         ( this->fe.component_mask(this->x_displacement) |
 *                                                           this->fe.component_mask(this->y_displacement) |
 *                                                           this->fe.component_mask(this->z_displacement) ));
 *             }
 *   
 *             virtual Tensor<1,dim>
 *             get_neumann_traction (const types::boundary_id &boundary_id,
 *                                   const Point<dim>         &pt,
 *                                   const Tensor<1,dim>      &N) const override
 *             {
 *               if (this->parameters.load_type == "pressure")
 *               {
 *                 if (boundary_id == 100)
 *                 {
 *                   return this->parameters.load * N;
 *                 }
 *               }
 *   
 *               (void)pt;
 *   
 *               return Tensor<1,dim>();
 *             }
 *   
 *             virtual double
 *             get_prescribed_fluid_flow (const types::boundary_id &boundary_id,
 *                                        const Point<dim>         &pt) const override
 *             {
 *                 (void)pt;
 *                 (void)boundary_id;
 *                 return 0.0;
 *             }
 *   
 *             virtual types::boundary_id
 *             get_reaction_boundary_id_for_output() const override
 *             {
 *                 return 100;
 *             }
 *   
 *             virtual  std::pair<types::boundary_id,types::boundary_id>
 *             get_drained_boundary_id_for_output() const override
 *             {
 *                 return std::make_pair(101,101);
 *             }
 *   
 *             virtual std::vector<double>
 *             get_dirichlet_load(const types::boundary_id   &boundary_id,
 *                                const int                  &direction) const override
 *             {
 *                 std::vector<double> displ_incr(dim, 0.0);
 *                 (void)boundary_id;
 *                 (void)direction;
 *                 AssertThrow(false, ExcMessage("Displacement loading not implemented for Ehlers verification examples."));
 *   
 *                 return displ_incr;
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Franceschiniexperiments"></a> 
 * <h4>Franceschini experiments</h4>
 * 
 * @code
 *       template <int dim>
 *       class Franceschini2006Consolidation
 *             : public Solid<dim>
 *       {
 *           public:
 *           Franceschini2006Consolidation (const Parameters::AllParameters &parameters)
 *               : Solid<dim> (parameters)
 *             {}
 *   
 *             virtual ~Franceschini2006Consolidation () {}
 *   
 *           private:
 *             virtual void make_grid() override
 *             {
 *               const Point<dim-1> mesh_center(0.0, 0.0);
 *               const double radius = 0.5;
 * @endcode
 * 
 * const double height = 0.27;  //8.1 mm for 30 mm radius
 * 
 * @code
 *               const double height = 0.23;  //6.9 mm for 30 mm radius
 *               Triangulation<dim-1> triangulation_in;
 *               GridGenerator::hyper_ball( triangulation_in,
 *                                          mesh_center,
 *                                          radius);
 *   
 *               GridGenerator::extrude_triangulation(triangulation_in,
 *                                                     2,
 *                                                     height,
 *                                                     this->triangulation);
 *   
 *               const CylindricalManifold<dim> cylinder_3d(2);
 *               const types::manifold_id cylinder_id = 0;
 *   
 *   
 *               this->triangulation.set_manifold(cylinder_id, cylinder_3d);
 *   
 *               for (auto cell : this->triangulation.active_cell_iterators())
 *               {
 *                 for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
 *                 {
 *                   if (cell->face(face)->at_boundary() == true)
 *                   {
 *                     if (cell->face(face)->center()[2] == 0.0)
 *                         cell->face(face)->set_boundary_id(1);
 *   
 *                     else if (cell->face(face)->center()[2] == height)
 *                         cell->face(face)->set_boundary_id(2);
 *   
 *                     else
 *                     {
 *                         cell->face(face)->set_boundary_id(0);
 *                         cell->face(face)->set_all_manifold_ids(cylinder_id);
 *                     }
 *                   }
 *                 }
 *               }
 *   
 *               GridTools::scale(this->parameters.scale, this->triangulation);
 *               this->triangulation.refine_global(std::max (1U, this->parameters.global_refinement));
 *             }
 *   
 *             virtual void define_tracked_vertices(std::vector<Point<dim> > &tracked_vertices) override
 *             {
 *               tracked_vertices[0][0] = 0.0*this->parameters.scale;
 *               tracked_vertices[0][1] = 0.0*this->parameters.scale;
 * @endcode
 * 
 * tracked_vertices[0][2] = 0.27*this->parameters.scale;
 * 
 * @code
 *               tracked_vertices[0][2] = 0.23*this->parameters.scale;
 *   
 *               tracked_vertices[1][0] = 0.0*this->parameters.scale;
 *               tracked_vertices[1][1] = 0.0*this->parameters.scale;
 *               tracked_vertices[1][2] = 0.0*this->parameters.scale;
 *             }
 *   
 *             virtual void make_dirichlet_constraints(AffineConstraints<double> &constraints) override
 *             {
 *               if (this->time.get_timestep() < 2)
 *               {
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          1,
 *                                                          Functions::ConstantFunction<dim>(this->parameters.drained_pressure,this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *   
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          2,
 *                                                          Functions::ConstantFunction<dim>(this->parameters.drained_pressure,this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *               }
 *               else
 *               {
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          1,
 *                                                          Functions::ZeroFunction<dim>(this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *   
 *                 VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                          2,
 *                                                          Functions::ZeroFunction<dim>(this->n_components),
 *                                                          constraints,
 *                                                          (this->fe.component_mask(this->pressure)));
 *               }
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         0,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         (this->fe.component_mask(this->x_displacement)|
 *                                                          this->fe.component_mask(this->y_displacement)  ) );
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         1,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         (this->fe.component_mask(this->x_displacement) |
 *                                                          this->fe.component_mask(this->y_displacement) |
 *                                                          this->fe.component_mask(this->z_displacement) ));
 *   
 *               VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                         2,
 *                                                         Functions::ZeroFunction<dim>(this->n_components),
 *                                                         constraints,
 *                                                         (this->fe.component_mask(this->x_displacement) |
 *                                                          this->fe.component_mask(this->y_displacement) ));
 *             }
 *   
 *             virtual double
 *             get_prescribed_fluid_flow (const types::boundary_id &boundary_id,
 *                                        const Point<dim>         &pt) const override
 *             {
 *                 (void)pt;
 *                 (void)boundary_id;
 *                 return 0.0;
 *             }
 *   
 *             virtual types::boundary_id
 *             get_reaction_boundary_id_for_output() const override
 *             {
 *                 return 2;
 *             }
 *   
 *             virtual  std::pair<types::boundary_id,types::boundary_id>
 *             get_drained_boundary_id_for_output() const override
 *             {
 *                 return std::make_pair(1,2);
 *             }
 *   
 *             virtual std::vector<double>
 *             get_dirichlet_load(const types::boundary_id   &boundary_id,
 *                                const int                  &direction) const override
 *             {
 *                 std::vector<double> displ_incr(dim, 0.0);
 *                 (void)boundary_id;
 *                 (void)direction;
 *                 AssertThrow(false, ExcMessage("Displacement loading not implemented for Franceschini examples."));
 *   
 *                 return displ_incr;
 *             }
 *   
 *             virtual Tensor<1,dim>
 *             get_neumann_traction (const types::boundary_id &boundary_id,
 *                                   const Point<dim>         &pt,
 *                                   const Tensor<1,dim>      &N) const override
 *             {
 *               if (this->parameters.load_type == "pressure")
 *               {
 *                 if (boundary_id == 2)
 *                 {
 *                   return (this->parameters.load * N);
 *                   /*
 *                   const double final_load = this->parameters.load;
 *                   const double final_load_time = 10 * this->time.get_delta_t();
 *                   const double current_time = this->time.get_current();
 *   
 *   
 *                   const double c = final_load_time / 2.0;
 *                   const double r = 200.0 * 0.03 / c;
 *   
 *                   const double load = final_load * std::exp(r * current_time)
 *                                       / ( std::exp(c * current_time) +  std::exp(r * current_time));
 *                   return load * N;
 *                   */
 *                 }
 *               }
 *   
 *               (void)pt;
 *   
 *               return Tensor<1,dim>();
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-ExamplestoreproduceexperimentsbyBuddayetal2017"></a> 
 * <h3>Examples to reproduce experiments by Budday et al. 2017</h3>
 * We group the definition of the geometry, boundary and loading conditions specific to
 * the examples to reproduce experiments by Budday et al. 2017 into specific classes.
 * 
 
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-BaseclassCubegeometryandloadingpattern"></a> 
 * <h4>Base class: Cube geometry and loading pattern</h4>
 * 
 * @code
 *       template <int dim>
 *       class BrainBudday2017BaseCube
 *             : public Solid<dim>
 *       {
 *           public:
 *               BrainBudday2017BaseCube (const Parameters::AllParameters &parameters)
 *               : Solid<dim> (parameters)
 *             {}
 *   
 *             virtual ~BrainBudday2017BaseCube () {}
 *   
 *           private:
 *             virtual void
 *             make_grid() override
 *             {
 *               GridGenerator::hyper_cube(this->triangulation,
 *                                         0.0,
 *                                         1.0,
 *                                         true);
 *   
 *               typename Triangulation<dim>::active_cell_iterator cell =
 *                       this->triangulation.begin_active(), endc = this->triangulation.end();
 *               for (; cell != endc; ++cell)
 *               {
 *                 for (unsigned int face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
 *                   if (cell->face(face)->at_boundary() == true  &&
 *                       ( cell->face(face)->boundary_id() == 0 ||
 *                         cell->face(face)->boundary_id() == 1 ||
 *                         cell->face(face)->boundary_id() == 2 ||
 *                         cell->face(face)->boundary_id() == 3    ) )
 *   
 *                         cell->face(face)->set_boundary_id(100);
 *   
 *               }
 *   
 *               GridTools::scale(this->parameters.scale, this->triangulation);
 *               this->triangulation.refine_global(std::max (1U, this->parameters.global_refinement));
 *             }
 *   
 *             virtual double
 *             get_prescribed_fluid_flow (const types::boundary_id &boundary_id,
 *                                        const Point<dim>         &pt) const override
 *             {
 *                 (void)pt;
 *                 (void)boundary_id;
 *                 return 0.0;
 *             }
 *   
 *             virtual  std::pair<types::boundary_id,types::boundary_id>
 *             get_drained_boundary_id_for_output() const override
 *             {
 *                 return std::make_pair(100,100);
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassUniaxialboundaryconditions"></a> 
 * <h4>Derived class: Uniaxial boundary conditions</h4>
 * 
 * @code
 *       template <int dim>
 *       class BrainBudday2017CubeTensionCompression
 *             : public BrainBudday2017BaseCube<dim>
 *       {
 *           public:
 *             BrainBudday2017CubeTensionCompression (const Parameters::AllParameters &parameters)
 *               : BrainBudday2017BaseCube<dim> (parameters)
 *             {}
 *   
 *             virtual ~BrainBudday2017CubeTensionCompression () {}
 *   
 *           private:
 *             virtual void
 *             define_tracked_vertices(std::vector<Point<dim> > &tracked_vertices) override
 *             {
 *               tracked_vertices[0][0] = 0.5*this->parameters.scale;
 *               tracked_vertices[0][1] = 0.5*this->parameters.scale;
 *               tracked_vertices[0][2] = 1.0*this->parameters.scale;
 *   
 *               tracked_vertices[1][0] = 0.5*this->parameters.scale;
 *               tracked_vertices[1][1] = 0.5*this->parameters.scale;
 *               tracked_vertices[1][2] = 0.5*this->parameters.scale;
 *             }
 *   
 *             virtual void
 *             make_dirichlet_constraints(AffineConstraints<double> &constraints) override
 *             {
 *                 if (this->time.get_timestep() < 2)
 *                 {
 *                     VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                              100,
 *                                                              Functions::ConstantFunction<dim>(this->parameters.drained_pressure,this->n_components),
 *                                                              constraints,
 *                                                              (this->fe.component_mask(this->pressure)));
 *                 }
 *                 else
 *                 {
 *                     VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                               100,
 *                                                               Functions::ZeroFunction<dim>(this->n_components),
 *                                                               constraints,
 *                                                               (this->fe.component_mask(this->pressure)));
 *                 }
 *                 VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                           4,
 *                                                           Functions::ZeroFunction<dim>(this->n_components),
 *                                                           constraints,
 *                                                           this->fe.component_mask(this->z_displacement) );
 *   
 *               Point<dim> fix_node(0.5*this->parameters.scale, 0.5*this->parameters.scale, 0.0);
 *               typename DoFHandler<dim>::active_cell_iterator
 *               cell = this->dof_handler_ref.begin_active(), endc = this->dof_handler_ref.end();
 *               for (; cell != endc; ++cell)
 *                 for (unsigned int node = 0; node < GeometryInfo<dim>::vertices_per_cell; ++node)
 *                 {
 *                     if (  (abs(cell->vertex(node)[2]-fix_node[2]) < (1e-6 * this->parameters.scale))
 *                       &&  (abs(cell->vertex(node)[0]-fix_node[0]) < (1e-6 * this->parameters.scale)))
 *                         constraints.add_line(cell->vertex_dof_index(node, 0));
 *   
 *                     if (  (abs(cell->vertex(node)[2]-fix_node[2]) < (1e-6 * this->parameters.scale))
 *                       &&  (abs(cell->vertex(node)[1]-fix_node[1]) < (1e-6 * this->parameters.scale)))
 *                       constraints.add_line(cell->vertex_dof_index(node, 1));
 *                 }
 *   
 *               if (this->parameters.load_type == "displacement")
 *               {
 *                   const std::vector<double> value = get_dirichlet_load(5,2);
 *                   const FEValuesExtractors::Scalar direction(this->z_displacement);
 *   
 *                   VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                              5,
 *                                                              Functions::ConstantFunction<dim>(value[2],this->n_components),
 *                                                              constraints,
 *                                                              this->fe.component_mask(direction));
 *               }
 *             }
 *   
 *             virtual Tensor<1,dim>
 *             get_neumann_traction (const types::boundary_id &boundary_id,
 *                                   const Point<dim>         &pt,
 *                                   const Tensor<1,dim>      &N) const override
 *             {
 *                 if (this->parameters.load_type == "pressure")
 *                 {
 *                   if (boundary_id ==  5)
 *                   {
 *                       const double final_load   = this->parameters.load;
 *                       const double current_time = this->time.get_current();
 *                       const double final_time   = this->time.get_end();
 *                       const double num_cycles   = 3.0;
 *   
 *                       return  final_load/2.0 * (1.0 - std::sin(numbers::PI * (2.0*num_cycles*current_time/final_time + 0.5))) * N;
 *                   }
 *                 }
 *   
 *                 (void)pt;
 *   
 *                 return Tensor<1,dim>();
 *               }
 *   
 *             virtual types::boundary_id
 *             get_reaction_boundary_id_for_output() const override
 *             {
 *                 return 5;
 *             }
 *   
 *             virtual std::vector<double>
 *             get_dirichlet_load(const types::boundary_id   &boundary_id,
 *                                const int                  &direction) const override
 *             {
 *                   std::vector<double> displ_incr(dim,0.0);
 *   
 *                   if ( (boundary_id == 5) && (direction == 2) )
 *                   {
 *                       const double final_displ  = this->parameters.load;
 *                       const double current_time = this->time.get_current();
 *                       const double final_time   = this->time.get_end();
 *                       const double delta_time   = this->time.get_delta_t();
 *                       const double num_cycles   = 3.0;
 *                       double current_displ = 0.0;
 *                       double previous_displ = 0.0;
 *   
 *                       if (this->parameters.num_cycle_sets == 1)
 *                       {
 *                           current_displ  = final_displ/2.0 * (1.0
 *                             - std::sin(numbers::PI * (2.0*num_cycles*current_time/final_time + 0.5)));
 *                           previous_displ = final_displ/2.0 * (1.0
 *                             - std::sin(numbers::PI * (2.0*num_cycles*(current_time-delta_time)/final_time + 0.5)));
 *                       }
 *                       else
 *                       {
 *                           if ( current_time <= (final_time*1.0/3.0) )
 *                           {
 *                               current_displ  = final_displ/2.0 * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*current_time/(final_time*1.0/3.0) + 0.5)));
 *                               previous_displ = final_displ/2.0 * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*(current_time-delta_time)/(final_time*1.0/3.0) + 0.5)));
 *                           }
 *                           else
 *                           {
 *                               current_displ  = final_displ * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*current_time / (final_time*2.0/3.0)
 *                                                     - (num_cycles - 0.5) )));
 *                               previous_displ = final_displ * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*(current_time-delta_time) / (final_time*2.0/3.0)
 *                                                     - (num_cycles - 0.5))));
 *                           }
 *                       }
 *                       displ_incr[2] = current_displ - previous_displ;
 *                   }
 *                   return displ_incr;
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassNolateraldisplacementinloadingsurfaces"></a> 
 * <h4>Derived class: No lateral displacement in loading surfaces</h4>
 * 
 * @code
 *       template <int dim>
 *       class BrainBudday2017CubeTensionCompressionFullyFixed
 *             : public BrainBudday2017BaseCube<dim>
 *       {
 *           public:
 *             BrainBudday2017CubeTensionCompressionFullyFixed (const Parameters::AllParameters &parameters)
 *               : BrainBudday2017BaseCube<dim> (parameters)
 *             {}
 *   
 *             virtual ~BrainBudday2017CubeTensionCompressionFullyFixed () {}
 *   
 *           private:
 *             virtual void
 *             define_tracked_vertices(std::vector<Point<dim> > &tracked_vertices) override
 *             {
 *               tracked_vertices[0][0] = 0.5*this->parameters.scale;
 *               tracked_vertices[0][1] = 0.5*this->parameters.scale;
 *               tracked_vertices[0][2] = 1.0*this->parameters.scale;
 *   
 *               tracked_vertices[1][0] = 0.5*this->parameters.scale;
 *               tracked_vertices[1][1] = 0.5*this->parameters.scale;
 *               tracked_vertices[1][2] = 0.5*this->parameters.scale;
 *             }
 *   
 *             virtual void
 *             make_dirichlet_constraints(AffineConstraints<double> &constraints) override
 *             {
 *                 if (this->time.get_timestep() < 2)
 *                 {
 *                     VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                              100,
 *                                                              Functions::ConstantFunction<dim>(this->parameters.drained_pressure,this->n_components),
 *                                                              constraints,
 *                                                              (this->fe.component_mask(this->pressure)));
 *                 }
 *                 else
 *                 {
 *                     VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                               100,
 *                                                               Functions::ZeroFunction<dim>(this->n_components),
 *                                                               constraints,
 *                                                               (this->fe.component_mask(this->pressure)));
 *                 }
 *   
 *                 VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                           4,
 *                                                           Functions::ZeroFunction<dim>(this->n_components),
 *                                                           constraints,
 *                                                         (this->fe.component_mask(this->x_displacement) |
 *                                                          this->fe.component_mask(this->y_displacement) |
 *                                                          this->fe.component_mask(this->z_displacement) ));
 *   
 *   
 *               if (this->parameters.load_type == "displacement")
 *               {
 *                   const std::vector<double> value = get_dirichlet_load(5,2);
 *                   const FEValuesExtractors::Scalar direction(this->z_displacement);
 *   
 *                   VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                              5,
 *                                                              Functions::ConstantFunction<dim>(value[2],this->n_components),
 *                                                              constraints,
 *                                                              this->fe.component_mask(direction) );
 *   
 *                  VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                             5,
 *                                                             Functions::ZeroFunction<dim>(this->n_components),
 *                                                             constraints,
 *                                                             (this->fe.component_mask(this->x_displacement) |
 *                                                              this->fe.component_mask(this->y_displacement) ));
 *               }
 *             }
 *   
 *             virtual Tensor<1,dim>
 *             get_neumann_traction (const types::boundary_id &boundary_id,
 *                                   const Point<dim>         &pt,
 *                                   const Tensor<1,dim>      &N) const override
 *             {
 *                 if (this->parameters.load_type == "pressure")
 *                 {
 *                   if (boundary_id ==  5)
 *                   {
 *                       const double final_load   = this->parameters.load;
 *                       const double current_time = this->time.get_current();
 *                       const double final_time   = this->time.get_end();
 *                       const double num_cycles   = 3.0;
 *   
 *                       return  final_load/2.0 * (1.0 - std::sin(numbers::PI * (2.0*num_cycles*current_time/final_time + 0.5))) * N;
 *                   }
 *                 }
 *   
 *                 (void)pt;
 *   
 *                 return Tensor<1,dim>();
 *               }
 *   
 *             virtual types::boundary_id
 *             get_reaction_boundary_id_for_output() const override
 *             {
 *                 return 5;
 *             }
 *   
 *             virtual std::vector<double>
 *             get_dirichlet_load(const types::boundary_id   &boundary_id,
 *                                const int                  &direction) const override
 *             {
 *                   std::vector<double> displ_incr(dim,0.0);
 *   
 *                   if ( (boundary_id == 5) && (direction == 2) )
 *                   {
 *                       const double final_displ  = this->parameters.load;
 *                       const double current_time = this->time.get_current();
 *                       const double final_time   = this->time.get_end();
 *                       const double delta_time   = this->time.get_delta_t();
 *                       const double num_cycles   = 3.0;
 *                       double current_displ = 0.0;
 *                       double previous_displ = 0.0;
 *   
 *                       if (this->parameters.num_cycle_sets == 1)
 *                       {
 *                           current_displ  = final_displ/2.0 * (1.0 - std::sin(numbers::PI * (2.0*num_cycles*current_time/final_time + 0.5)));
 *                           previous_displ = final_displ/2.0 * (1.0 - std::sin(numbers::PI * (2.0*num_cycles*(current_time-delta_time)/final_time + 0.5)));
 *                       }
 *                       else
 *                       {
 *                           if ( current_time <= (final_time*1.0/3.0) )
 *                           {
 *                               current_displ  = final_displ/2.0 * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*current_time/(final_time*1.0/3.0) + 0.5)));
 *                               previous_displ = final_displ/2.0 * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*(current_time-delta_time)/(final_time*1.0/3.0) + 0.5)));
 *                           }
 *                           else
 *                           {
 *                               current_displ  = final_displ * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*current_time / (final_time*2.0/3.0)
 *                                                     - (num_cycles - 0.5) )));
 *                               previous_displ = final_displ * (1.0 - std::sin(numbers::PI *
 *                                                    (2.0*num_cycles*(current_time-delta_time) / (final_time*2.0/3.0)
 *                                                     - (num_cycles - 0.5))));
 *                           }
 *                       }
 *                       displ_incr[2] = current_displ - previous_displ;
 *                   }
 *                   return displ_incr;
 *             }
 *       };
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-DerivedclassNolateralorverticaldisplacementinloadingsurface"></a> 
 * <h4>Derived class: No lateral or vertical displacement in loading surface</h4>
 * 
 * @code
 *       template <int dim>
 *       class BrainBudday2017CubeShearFullyFixed
 *             : public BrainBudday2017BaseCube<dim>
 *       {
 *           public:
 *             BrainBudday2017CubeShearFullyFixed (const Parameters::AllParameters &parameters)
 *               : BrainBudday2017BaseCube<dim> (parameters)
 *             {}
 *   
 *             virtual ~BrainBudday2017CubeShearFullyFixed () {}
 *   
 *           private:
 *             virtual void
 *             define_tracked_vertices(std::vector<Point<dim> > &tracked_vertices) override
 *             {
 *               tracked_vertices[0][0] = 0.75*this->parameters.scale;
 *               tracked_vertices[0][1] = 0.5*this->parameters.scale;
 *               tracked_vertices[0][2] = 0.0*this->parameters.scale;
 *   
 *               tracked_vertices[1][0] = 0.25*this->parameters.scale;
 *               tracked_vertices[1][1] = 0.5*this->parameters.scale;
 *               tracked_vertices[1][2] = 0.0*this->parameters.scale;
 *             }
 *   
 *             virtual void
 *             make_dirichlet_constraints(AffineConstraints<double> &constraints) override
 *             {
 *                 if (this->time.get_timestep() < 2)
 *                 {
 *                     VectorTools::interpolate_boundary_values(this->dof_handler_ref,
 *                                                              100,
 *                                                              Functions::ConstantFunction<dim>(this->parameters.drained_pressure,this->n_components),
 *                                                              constraints,
 *                                                              (this->fe.component_mask(this->pressure)));
 *                 }
 *                 else
 *                 {
 *                     VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                               100,
 *                                                               Functions::ZeroFunction<dim>(this->n_components),
 *                                                               constraints,
 *                                                               (this->fe.component_mask(this->pressure)));
 *                 }
 *   
 *                 VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                           5,
 *                                                           Functions::ZeroFunction<dim>(this->n_components),
 *                                                           constraints,
 *                                                         (this->fe.component_mask(this->x_displacement) |
 *                                                          this->fe.component_mask(this->y_displacement) |
 *                                                          this->fe.component_mask(this->z_displacement) ));
 *   
 *   
 *               if (this->parameters.load_type == "displacement")
 *               {
 *                   const std::vector<double> value = get_dirichlet_load(4,0);
 *                   const FEValuesExtractors::Scalar direction(this->x_displacement);
 *   
 *                   VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                              4,
 *                                                              Functions::ConstantFunction<dim>(value[0],this->n_components),
 *                                                              constraints,
 *                                                              this->fe.component_mask(direction));
 *   
 *                  VectorTools::interpolate_boundary_values( this->dof_handler_ref,
 *                                                             4,
 *                                                             Functions::ZeroFunction<dim>(this->n_components),
 *                                                             constraints,
 *                                                             (this->fe.component_mask(this->y_displacement) |
 *                                                              this->fe.component_mask(this->z_displacement) ));
 *               }
 *             }
 *   
 *             virtual Tensor<1,dim>
 *             get_neumann_traction (const types::boundary_id &boundary_id,
 *                                   const Point<dim>         &pt,
 *                                   const Tensor<1,dim>      &N) const override
 *             {
 *                 if (this->parameters.load_type == "pressure")
 *                 {
 *                   if (boundary_id ==  4)
 *                   {
 *                       const double final_load   = this->parameters.load;
 *                       const double current_time = this->time.get_current();
 *                       const double final_time   = this->time.get_end();
 *                       const double num_cycles   = 3.0;
 *                       const Tensor<1,3> axis ({0.0,1.0,0.0});
 *                       const double angle = numbers::PI;
 *                       static const Tensor< 2, dim, double> R(Physics::Transformations::Rotations::rotation_matrix_3d(axis,angle));
 *   
 *                       return  (final_load * (std::sin(2.0*(numbers::PI)*num_cycles*current_time/final_time)) * (R * N));
 *                   }
 *                 }
 *   
 *                 (void)pt;
 *   
 *                 return Tensor<1,dim>();
 *               }
 *   
 *             virtual types::boundary_id
 *             get_reaction_boundary_id_for_output() const override
 *             {
 *                 return 4;
 *             }
 *   
 *             virtual std::vector<double>
 *             get_dirichlet_load(const types::boundary_id   &boundary_id,
 *                                const int                  &direction) const override
 *             {
 *                   std::vector<double> displ_incr (dim, 0.0);
 *   
 *                   if ( (boundary_id == 4) && (direction == 0) )
 *                   {
 *                       const double final_displ  = this->parameters.load;
 *                       const double current_time = this->time.get_current();
 *                       const double final_time   = this->time.get_end();
 *                       const double delta_time   = this->time.get_delta_t();
 *                       const double num_cycles   = 3.0;
 *                       double current_displ = 0.0;
 *                       double previous_displ = 0.0;
 *   
 *                       if (this->parameters.num_cycle_sets == 1)
 *                       {
 *                           current_displ  = final_displ * (std::sin(2.0*(numbers::PI)*num_cycles*current_time/final_time));
 *                           previous_displ = final_displ * (std::sin(2.0*(numbers::PI)*num_cycles*(current_time-delta_time)/final_time));
 *                       }
 *                       else
 *                       {
 *                           AssertThrow(false, ExcMessage("Problem type not defined. Budday shear experiments implemented only for one set of cycles."));
 *                       }
 *                       displ_incr[0] = current_displ - previous_displ;
 *                   }
 *                   return displ_incr;
 *             }
 *       };
 *   
 *   }
 *   
 * @endcode
 * 
 * 
 * <a name="nonlinear-poro-viscoelasticity.cc-Mainfunction"></a> 
 * <h3>Main function</h3>
 * Lastly we provide the main driver function which is similar to the other tutorials.
 * 
 * @code
 *   int main (int argc, char *argv[])
 *   {
 *     using namespace dealii;
 *     using namespace NonLinearPoroViscoElasticity;
 *   
 *     const unsigned int n_tbb_processes = 1;
 *     Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, n_tbb_processes);
 *   
 *     try
 *       {
 *         Parameters::AllParameters parameters ("parameters.prm");
 *         if (parameters.geom_type == "Ehlers_tube_step_load")
 *         {
 *           VerificationEhlers1999StepLoad<3> solid_3d(parameters);
 *           solid_3d.run();
 *         }
 *         else if (parameters.geom_type == "Ehlers_tube_increase_load")
 *         {
 *           VerificationEhlers1999IncreaseLoad<3> solid_3d(parameters);
 *           solid_3d.run();
 *         }
 *         else if (parameters.geom_type == "Ehlers_cube_consolidation")
 *         {
 *           VerificationEhlers1999CubeConsolidation<3> solid_3d(parameters);
 *           solid_3d.run();
 *         }
 *         else if (parameters.geom_type == "Franceschini_consolidation")
 *         {
 *           Franceschini2006Consolidation<3> solid_3d(parameters);
 *           solid_3d.run();
 *         }
 *         else if (parameters.geom_type == "Budday_cube_tension_compression")
 *         {
 *           BrainBudday2017CubeTensionCompression<3> solid_3d(parameters);
 *           solid_3d.run();
 *         }
 *         else if (parameters.geom_type == "Budday_cube_tension_compression_fully_fixed")
 *         {
 *           BrainBudday2017CubeTensionCompressionFullyFixed<3> solid_3d(parameters);
 *           solid_3d.run();
 *         }
 *         else if (parameters.geom_type == "Budday_cube_shear_fully_fixed")
 *         {
 *           BrainBudday2017CubeShearFullyFixed<3> solid_3d(parameters);
 *           solid_3d.run();
 *         }
 *         else
 *         {
 *           AssertThrow(false, ExcMessage("Problem type not defined. Current setting: " + parameters.geom_type));
 *         }
 *   
 *       }
 *     catch (std::exception &exc)
 *       {
 *         if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
 *         {
 *             std::cerr << std::endl << std::endl
 *                       << "----------------------------------------------------"
 *                       << std::endl;
 *             std::cerr << "Exception on processing: " << std::endl << exc.what()
 *                       << std::endl << "Aborting!" << std::endl
 *                       << "----------------------------------------------------"
 *                       << std::endl;
 *   
 *             return 1;
 *         }
 *       }
 *     catch (...)
 *       {
 *         if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
 *         {
 *             std::cerr << std::endl << std::endl
 *                       << "----------------------------------------------------"
 *                       << std::endl;
 *             std::cerr << "Unknown exception!" << std::endl << "Aborting!"
 *                       << std::endl
 *                       << "----------------------------------------------------"
 *                       << std::endl;
 *             return 1;
 *         }
 *       }
 *     return 0;
 *   }
 * @endcode
 
 
*/