1889 * Solve
for the displacement
using a Newton-Raphson method. We
break this
1890 * function into the nonlinear
loop and the function that solves the
1891 * linearized Newton-Raphson step:
1896 * std::pair<unsigned int, double>
1901 * Solution retrieval as well as post-processing and writing
data to file :
1907 *
void output_results()
const;
1911 * Finally, some member variables that describe the current state: A
1912 * collection of the parameters used to describe the problem setup...
1915 * Parameters::AllParameters parameters;
1919 * ...the
volume of the reference configuration...
1922 *
double vol_reference;
1926 * ...and description of the geometry on which the problem is solved:
1933 * Also, keep track of the current time and the time spent evaluating
1942 * A storage
object for quadrature
point information. As opposed to
1943 * @ref step_18
"step-18", deal.II
's native quadrature point data manager is employed
1947 * CellDataStorage<typename Triangulation<dim>::cell_iterator,
1948 * PointHistory<dim>>
1949 * quadrature_point_history;
1953 * A description of the finite-element system including the displacement
1954 * polynomial degree, the degree-of-freedom handler, number of DoFs per
1955 * cell and the extractor objects used to retrieve information from the
1959 * const unsigned int degree;
1960 * const FESystem<dim> fe;
1961 * DoFHandler<dim> dof_handler;
1962 * const unsigned int dofs_per_cell;
1963 * const FEValuesExtractors::Vector u_fe;
1964 * const FEValuesExtractors::Scalar p_fe;
1965 * const FEValuesExtractors::Scalar J_fe;
1969 * Description of how the block-system is arranged. There are 3 blocks,
1970 * the first contains a vector DOF @f$\mathbf{u}@f$ while the other two
1971 * describe scalar DOFs, @f$\widetilde{p}@f$ and @f$\widetilde{J}@f$.
1974 * static const unsigned int n_blocks = 3;
1975 * static const unsigned int n_components = dim + 2;
1976 * static const unsigned int first_u_component = 0;
1977 * static const unsigned int p_component = dim;
1978 * static const unsigned int J_component = dim + 1;
1987 * std::vector<types::global_dof_index> dofs_per_block;
1988 * std::vector<types::global_dof_index> element_indices_u;
1989 * std::vector<types::global_dof_index> element_indices_p;
1990 * std::vector<types::global_dof_index> element_indices_J;
1994 * Rules for Gauss-quadrature on both the cell and faces. The number of
1995 * quadrature points on both cells and faces is recorded.
1998 * const QGauss<dim> qf_cell;
1999 * const QGauss<dim - 1> qf_face;
2000 * const unsigned int n_q_points;
2001 * const unsigned int n_q_points_f;
2005 * Objects that store the converged solution and right-hand side vectors,
2006 * as well as the tangent matrix. There is an AffineConstraints object used
2007 * to keep track of constraints. We make use of a sparsity pattern
2008 * designed for a block system.
2011 * AffineConstraints<double> constraints;
2012 * BlockSparsityPattern sparsity_pattern;
2013 * BlockSparseMatrix<double> tangent_matrix;
2014 * BlockVector<double> system_rhs;
2015 * BlockVector<double> solution_n;
2019 * Then define a number of variables to store norms and update norms and
2020 * normalization factors.
2039 * void normalize(const Errors &rhs)
2041 * if (rhs.norm != 0.0)
2051 * double norm, u, p, J;
2054 * Errors error_residual, error_residual_0, error_residual_norm, error_update,
2055 * error_update_0, error_update_norm;
2059 * Methods to calculate error measures
2062 * void get_error_residual(Errors &error_residual);
2064 * void get_error_update(const BlockVector<double> &newton_update,
2065 * Errors &error_update);
2067 * std::pair<double, double> get_error_dilation() const;
2071 * Compute the volume in the spatial configuration
2074 * double compute_vol_current() const;
2078 * Print information to screen in a pleasing way...
2081 * static void print_conv_header();
2083 * void print_conv_footer();
2089 * <a name="step_44-ImplementationofthecodeSolidcodeclass"></a>
2090 * <h3>Implementation of the <code>Solid</code> class</h3>
2095 * <a name="step_44-Publicinterface"></a>
2096 * <h4>Public interface</h4>
2100 * We initialize the Solid class using data extracted from the parameter file.
2103 * template <int dim>
2104 * Solid<dim>::Solid(const std::string &input_file)
2105 * : parameters(input_file)
2106 * , vol_reference(0.)
2107 * , triangulation(Triangulation<dim>::maximum_smoothing)
2108 * , time(parameters.end_time, parameters.delta_t)
2109 * , timer(std::cout, TimerOutput::summary, TimerOutput::wall_times)
2110 * , degree(parameters.poly_degree)
2114 * The Finite Element System is composed of dim continuous displacement
2115 * DOFs, and discontinuous pressure and dilatation DOFs. In an attempt to
2116 * satisfy the Babuska-Brezzi or LBB stability conditions (see Hughes
2117 * (2000)), we set up a @f$Q_n \times DGP_{n-1} \times DGP_{n-1}@f$
2118 * system. @f$Q_2 \times DGP_1 \times DGP_1@f$ elements satisfy this
2119 * condition, while @f$Q_1 \times DGP_0 \times DGP_0@f$ elements do
2120 * not. However, it has been shown that the latter demonstrate good
2121 * convergence characteristics nonetheless.
2124 * fe(FE_Q<dim>(parameters.poly_degree) ^ dim, // displacement
2125 * FE_DGP<dim>(parameters.poly_degree - 1), // pressure
2126 * FE_DGP<dim>(parameters.poly_degree - 1)) // dilatation
2127 * , dof_handler(triangulation)
2128 * , dofs_per_cell(fe.n_dofs_per_cell())
2129 * , u_fe(first_u_component)
2130 * , p_fe(p_component)
2131 * , J_fe(J_component)
2132 * , dofs_per_block(n_blocks)
2133 * , qf_cell(parameters.quad_order)
2134 * , qf_face(parameters.quad_order)
2135 * , n_q_points(qf_cell.size())
2136 * , n_q_points_f(qf_face.size())
2138 * Assert(dim == 2 || dim == 3,
2139 * ExcMessage("This problem only works in 2 or 3 space dimensions."));
2140 * determine_component_extractors();
2146 * In solving the quasi-static problem, the time becomes a loading parameter,
2147 * i.e. we increasing the loading linearly with time, making the two concepts
2148 * interchangeable. We choose to increment time linearly using a constant time
2153 * We start the function with preprocessing, setting the initial dilatation
2154 * values, and then output the initial grid before starting the simulation
2155 * proper with the first time (and loading)
2160 * Care must be taken (or at least some thought given) when imposing the
2161 * constraint @f$\widetilde{J}=1@f$ on the initial solution field. The constraint
2162 * corresponds to the determinant of the deformation gradient in the
2163 * undeformed configuration, which is the identity tensor. We use
2164 * FE_DGP bases to interpolate the dilatation field, thus we can't
2165 * simply set the corresponding dof to unity as they correspond to the
2166 * coefficients of a truncated Legendre polynomial.
2169 * indicating the hanging node constraints. We have
none in
this program
2170 * So we have to create a constraint
object. In its original state, constraint
2171 * objects are unsorted, and have to be
sorted (
using the
2173 * @ref step_21
"step-21" for more information. We only need to enforce the
initial condition
2174 * on the dilatation. In order to
do this, we make use of a
2176 * n_components to 1. This is exactly what we want. Have a look at its usage
2177 * in @ref step_20
"step-20" for more information.
2180 *
template <
int dim>
2181 *
void Solid<dim>::run()
2187 * constraints.
close();
2192 * dof_handler, constraints,
QGauss<dim>(degree + 2), J_mask, solution_n);
2199 * We then declare the incremental solution update @f$\varDelta
2200 * \mathbf{\Xi} \dealcoloneq \{\varDelta \mathbf{u},\varDelta \widetilde{p},
2201 * \varDelta \widetilde{J} \}@f$ and start the
loop over the time domain.
2205 * At the beginning, we reset the solution update
for this time step...
2209 *
while (time.current() < time.end())
2211 * solution_delta = 0.0;
2215 * ...solve the current time step and update total solution vector
2216 * @f$\mathbf{\Xi}_{\textrm{n}} = \mathbf{\Xi}_{\textrm{n-1}} +
2217 * \varDelta \mathbf{\Xi}@f$...
2220 * solve_nonlinear_timestep(solution_delta);
2221 * solution_n += solution_delta;
2225 * ...and plot the results before moving on happily to the next time
2238 * <a name=
"step_44-Privateinterface"></a>
2239 * <h3>Private interface</h3>
2244 * <a name=
"step_44-Threadingbuildingblocksstructures"></a>
2245 * <h4>Threading-building-blocks structures</h4>
2250 * We use the Threading Building Blocks library (TBB) to perform as many
2251 * computationally intensive distributed tasks as possible. In particular, we
2252 *
assemble the tangent
matrix and right hand side vector, the
static
2253 * condensation contributions, and update
data stored at the quadrature points
2254 *
using TBB. Our main tool
for this is the
WorkStream class (see the @ref
2255 * threads topic
for more information).
2259 * Firstly we deal with the tangent matrix and right-hand side assembly
2260 * structures. The PerTaskData
object stores local contributions to the global
2264 * template <int dim>
2265 *
struct Solid<dim>::PerTaskData_ASM
2269 * std::vector<types::global_dof_index> local_dof_indices;
2271 * PerTaskData_ASM(
const unsigned int dofs_per_cell)
2273 * , cell_rhs(dofs_per_cell)
2274 * , local_dof_indices(dofs_per_cell)
2287 * On the other hand, the ScratchData
object stores the larger objects such as
2288 * the shape-function
values array (<code>Nx</code>) and a shape function
2293 *
template <
int dim>
2294 *
struct Solid<dim>::ScratchData_ASM
2299 * std::vector<std::vector<double>> Nx;
2300 * std::vector<std::vector<Tensor<2, dim>>> grad_Nx;
2301 * std::vector<std::vector<SymmetricTensor<2, dim>>> symm_grad_Nx;
2308 * : fe_values(fe_cell, qf_cell, uf_cell)
2309 * , fe_face_values(fe_cell, qf_face, uf_face)
2310 * , Nx(qf_cell.size(), std::vector<double>(fe_cell.n_dofs_per_cell()))
2311 * , grad_Nx(qf_cell.size(),
2313 * , symm_grad_Nx(qf_cell.size(),
2315 * fe_cell.n_dofs_per_cell()))
2318 * ScratchData_ASM(
const ScratchData_ASM &rhs)
2319 * : fe_values(rhs.fe_values.get_fe(),
2320 * rhs.fe_values.get_quadrature(),
2321 * rhs.fe_values.get_update_flags())
2322 * , fe_face_values(rhs.fe_face_values.get_fe(),
2323 * rhs.fe_face_values.get_quadrature(),
2324 * rhs.fe_face_values.get_update_flags())
2326 * , grad_Nx(rhs.grad_Nx)
2327 * , symm_grad_Nx(rhs.symm_grad_Nx)
2332 *
const unsigned int n_q_points = Nx.size();
2333 *
const unsigned int n_dofs_per_cell = Nx[0].size();
2334 *
for (
unsigned int q_point = 0; q_point < n_q_points; ++q_point)
2336 *
Assert(Nx[q_point].
size() == n_dofs_per_cell, ExcInternalError());
2337 *
Assert(grad_Nx[q_point].
size() == n_dofs_per_cell,
2338 * ExcInternalError());
2339 *
Assert(symm_grad_Nx[q_point].
size() == n_dofs_per_cell,
2340 * ExcInternalError());
2341 *
for (
unsigned int k = 0; k < n_dofs_per_cell; ++k)
2343 * Nx[q_point][k] = 0.0;
2344 * grad_Nx[q_point][k] = 0.0;
2345 * symm_grad_Nx[q_point][k] = 0.0;
2354 * Then we define structures to
assemble the statically condensed tangent
2355 *
matrix. Recall that we wish to solve
for a displacement-based formulation.
2356 * We
do the condensation at the element
level as the @f$\widetilde{p}@f$ and
2357 * @f$\widetilde{J}@f$ fields are element-wise discontinuous. As these operations
2358 * are
matrix-based, we need to set up a number of matrices to store the local
2359 * contributions from a number of the tangent
matrix sub-blocks. We place
2360 * these in the PerTaskData
struct.
2364 * We choose not to reset any
data in the <code>reset()</code> function as the
2365 *
matrix extraction and replacement tools will take care of
this
2368 *
template <
int dim>
2369 *
struct Solid<dim>::PerTaskData_SC
2372 * std::vector<types::global_dof_index> local_dof_indices;
2384 * PerTaskData_SC(
const unsigned int dofs_per_cell,
2385 *
const unsigned int n_u,
2386 *
const unsigned int n_p,
2387 *
const unsigned int n_J)
2389 * , local_dof_indices(dofs_per_cell)
2390 * , k_orig(dofs_per_cell, dofs_per_cell)
2394 * , k_pJ_inv(n_p, n_J)
2395 * , k_bbar(n_u, n_u)
2408 * The ScratchData
object for the operations we wish to perform here is empty
2409 * since we need no temporary
data, but it still needs to be defined
for the
2410 * current implementation of TBB in deal.II. So we create a dummy
struct for
2414 *
template <
int dim>
2415 *
struct Solid<dim>::ScratchData_SC
2424 * And
finally we define the structures to assist with updating the quadrature
2425 *
point information. Similar to the SC assembly process, we
do not need the
2426 * PerTaskData object (since there is nothing to store here) but must define
2427 * one nonetheless. Note that
this is because
for the operation that we have
2428 * here -- updating the
data on quadrature points -- the operation is purely
2429 * local: the things we
do on every cell get consumed on every cell, without
2430 * any global aggregation operation as is usually the
case when
using the
2431 *
WorkStream class. The fact that we still have to define a per-task
data
2432 * structure points to the fact that the
WorkStream class may be ill-suited to
2433 *
this operation (we could, in principle simply create a
new task
using
2435 * it
this way anyway.
2436 * Furthermore, should there be different material models associated with a
2437 * quadrature
point, requiring varying levels of computational expense, then
2438 * the method used here could be advantageous.
2441 *
template <
int dim>
2442 *
struct Solid<dim>::PerTaskData_UQPH
2451 * The ScratchData
object will be used to store an alias
for the solution
2452 * vector so that we don
't have to copy this large data structure. We then
2453 * define a number of vectors to extract the solution values and gradients at
2454 * the quadrature points.
2457 * template <int dim>
2458 * struct Solid<dim>::ScratchData_UQPH
2460 * const BlockVector<double> &solution_total;
2462 * std::vector<Tensor<2, dim>> solution_grads_u_total;
2463 * std::vector<double> solution_values_p_total;
2464 * std::vector<double> solution_values_J_total;
2466 * FEValues<dim> fe_values;
2468 * ScratchData_UQPH(const FiniteElement<dim> &fe_cell,
2469 * const QGauss<dim> &qf_cell,
2470 * const UpdateFlags uf_cell,
2471 * const BlockVector<double> &solution_total)
2472 * : solution_total(solution_total)
2473 * , solution_grads_u_total(qf_cell.size())
2474 * , solution_values_p_total(qf_cell.size())
2475 * , solution_values_J_total(qf_cell.size())
2476 * , fe_values(fe_cell, qf_cell, uf_cell)
2479 * ScratchData_UQPH(const ScratchData_UQPH &rhs)
2480 * : solution_total(rhs.solution_total)
2481 * , solution_grads_u_total(rhs.solution_grads_u_total)
2482 * , solution_values_p_total(rhs.solution_values_p_total)
2483 * , solution_values_J_total(rhs.solution_values_J_total)
2484 * , fe_values(rhs.fe_values.get_fe(),
2485 * rhs.fe_values.get_quadrature(),
2486 * rhs.fe_values.get_update_flags())
2491 * const unsigned int n_q_points = solution_grads_u_total.size();
2492 * for (unsigned int q = 0; q < n_q_points; ++q)
2494 * solution_grads_u_total[q] = 0.0;
2495 * solution_values_p_total[q] = 0.0;
2496 * solution_values_J_total[q] = 0.0;
2505 * <a name="step_44-Solidmake_grid"></a>
2506 * <h4>Solid::make_grid</h4>
2510 * On to the first of the private member functions. Here we create the
2511 * triangulation of the domain, for which we choose the scaled cube with each
2512 * face given a boundary ID number. The grid must be refined at least once
2513 * for the indentation problem.
2517 * We then determine the volume of the reference configuration and print it
2521 * template <int dim>
2522 * void Solid<dim>::make_grid()
2524 * GridGenerator::hyper_rectangle(
2526 * (dim == 3 ? Point<dim>(0.0, 0.0, 0.0) : Point<dim>(0.0, 0.0)),
2527 * (dim == 3 ? Point<dim>(1.0, 1.0, 1.0) : Point<dim>(1.0, 1.0)),
2529 * GridTools::scale(parameters.scale, triangulation);
2530 * triangulation.refine_global(std::max(1U, parameters.global_refinement));
2532 * vol_reference = GridTools::volume(triangulation);
2533 * std::cout << "Grid:\n\t Reference volume: " << vol_reference << std::endl;
2537 * Since we wish to apply a Neumann BC to a patch on the top surface, we
2538 * must find the cell faces in this part of the domain and mark them with
2539 * a distinct boundary ID number. The faces we are looking for are on the
2540 * +y surface and will get boundary ID 6 (zero through five are already
2541 * used when creating the six faces of the cube domain):
2544 * for (const auto &cell : triangulation.active_cell_iterators())
2545 * for (const auto &face : cell->face_iterators())
2547 * if (face->at_boundary() == true &&
2548 * face->center()[1] == 1.0 * parameters.scale)
2552 * if (face->center()[0] < 0.5 * parameters.scale &&
2553 * face->center()[2] < 0.5 * parameters.scale)
2554 * face->set_boundary_id(6);
2558 * if (face->center()[0] < 0.5 * parameters.scale)
2559 * face->set_boundary_id(6);
2569 * <a name="step_44-Solidsystem_setup"></a>
2570 * <h4>Solid::system_setup</h4>
2574 * Next we describe how the FE system is setup. We first determine the number
2575 * of components per block. Since the displacement is a vector component, the
2576 * first dim components belong to it, while the next two describe scalar
2577 * pressure and dilatation DOFs.
2580 * template <int dim>
2581 * void Solid<dim>::system_setup()
2583 * timer.enter_subsection("Setup system");
2585 * std::vector<unsigned int> block_component(n_components,
2586 * u_dof); // Displacement
2587 * block_component[p_component] = p_dof; // Pressure
2588 * block_component[J_component] = J_dof; // Dilatation
2592 * The DOF handler is then initialized and we renumber the grid in an
2593 * efficient manner. We also record the number of DOFs per block.
2596 * dof_handler.distribute_dofs(fe);
2597 * DoFRenumbering::Cuthill_McKee(dof_handler);
2598 * DoFRenumbering::component_wise(dof_handler, block_component);
2601 * DoFTools::count_dofs_per_fe_block(dof_handler, block_component);
2603 * std::cout << "Triangulation:"
2604 * << "\n\t Number of active cells: "
2605 * << triangulation.n_active_cells()
2606 * << "\n\t Number of degrees of freedom: " << dof_handler.n_dofs()
2611 * Setup the sparsity pattern and tangent matrix
2614 * tangent_matrix.clear();
2616 * BlockDynamicSparsityPattern dsp(dofs_per_block, dofs_per_block);
2620 * The global system matrix initially has the following structure
2622 * \underbrace{\begin{bmatrix}
2623 * \mathsf{\mathbf{K}}_{uu} & \mathsf{\mathbf{K}}_{u\widetilde{p}} &
2625 * \\ \mathsf{\mathbf{K}}_{\widetilde{p}u} & \mathbf{0} &
2626 * \mathsf{\mathbf{K}}_{\widetilde{p}\widetilde{J}}
2627 * \\ \mathbf{0} & \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{p}} &
2628 * \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{J}}
2629 * \end{bmatrix}}_{\mathsf{\mathbf{K}}(\mathbf{\Xi}_{\textrm{i}})}
2630 * \underbrace{\begin{bmatrix}
2632 * \\ d \widetilde{\mathsf{\mathbf{p}}}
2633 * \\ d \widetilde{\mathsf{\mathbf{J}}}
2634 * \end{bmatrix}}_{d \mathbf{\Xi}}
2636 * \underbrace{\begin{bmatrix}
2637 * \mathsf{\mathbf{F}}_{u}(\mathbf{u}_{\textrm{i}})
2638 * \\ \mathsf{\mathbf{F}}_{\widetilde{p}}(\widetilde{p}_{\textrm{i}})
2639 * \\ \mathsf{\mathbf{F}}_{\widetilde{J}}(\widetilde{J}_{\textrm{i}})
2640 * \end{bmatrix}}_{ \mathsf{\mathbf{F}}(\mathbf{\Xi}_{\textrm{i}}) } \, .
2642 * We optimize the sparsity pattern to reflect this structure
2643 * and prevent unnecessary data creation for the right-diagonal
2647 * Table<2, DoFTools::Coupling> coupling(n_components, n_components);
2648 * for (unsigned int ii = 0; ii < n_components; ++ii)
2649 * for (unsigned int jj = 0; jj < n_components; ++jj)
2650 * if (((ii < p_component) && (jj == J_component)) ||
2651 * ((ii == J_component) && (jj < p_component)) ||
2652 * ((ii == p_component) && (jj == p_component)))
2653 * coupling[ii][jj] = DoFTools::none;
2655 * coupling[ii][jj] = DoFTools::always;
2656 * DoFTools::make_sparsity_pattern(
2657 * dof_handler, coupling, dsp, constraints, false);
2658 * sparsity_pattern.copy_from(dsp);
2661 * tangent_matrix.reinit(sparsity_pattern);
2665 * We then set up storage vectors
2668 * system_rhs.reinit(dofs_per_block);
2669 * solution_n.reinit(dofs_per_block);
2673 * ...and finally set up the quadrature
2679 * timer.leave_subsection();
2686 * <a name="step_44-Soliddetermine_component_extractors"></a>
2687 * <h4>Solid::determine_component_extractors</h4>
2688 * Next we compute some information from the FE system that describes which
2689 * local element DOFs are attached to which block component. This is used
2690 * later to extract sub-blocks from the global matrix.
2694 * In essence, all we need is for the FESystem object to indicate to which
2695 * block component a DOF on the reference cell is attached to. Currently, the
2696 * interpolation fields are setup such that 0 indicates a displacement DOF, 1
2697 * a pressure DOF and 2 a dilatation DOF.
2700 * template <int dim>
2701 * void Solid<dim>::determine_component_extractors()
2703 * element_indices_u.clear();
2704 * element_indices_p.clear();
2705 * element_indices_J.clear();
2707 * for (unsigned int k = 0; k < fe.n_dofs_per_cell(); ++k)
2709 * const unsigned int k_group = fe.system_to_base_index(k).first.first;
2710 * if (k_group == u_dof)
2711 * element_indices_u.push_back(k);
2712 * else if (k_group == p_dof)
2713 * element_indices_p.push_back(k);
2714 * else if (k_group == J_dof)
2715 * element_indices_J.push_back(k);
2718 * Assert(k_group <= J_dof, ExcInternalError());
2726 * <a name="step_44-Solidsetup_qph"></a>
2727 * <h4>Solid::setup_qph</h4>
2728 * The method used to store quadrature information is already described in
2729 * @ref step_18 "step-18". Here we implement a similar setup for a SMP machine.
2733 * Firstly the actual QPH data objects are created. This must be done only
2734 * once the grid is refined to its finest level.
2737 * template <int dim>
2738 * void Solid<dim>::setup_qph()
2740 * std::cout << " Setting up quadrature point data..." << std::endl;
2742 * quadrature_point_history.initialize(triangulation.begin_active(),
2743 * triangulation.end(),
2748 * Next we set up the initial quadrature point data.
2749 * Note that when the quadrature point data is retrieved,
2750 * it is returned as a vector of smart pointers.
2753 * for (const auto &cell : triangulation.active_cell_iterators())
2755 * const std::vector<std::shared_ptr<PointHistory<dim>>> lqph =
2756 * quadrature_point_history.get_data(cell);
2757 * Assert(lqph.size() == n_q_points, ExcInternalError());
2759 * for (unsigned int q_point = 0; q_point < n_q_points; ++q_point)
2760 * lqph[q_point]->setup_lqp(parameters);
2767 * <a name="step_44-Solidupdate_qph_incremental"></a>
2768 * <h4>Solid::update_qph_incremental</h4>
2769 * As the update of QP information occurs frequently and involves a number of
2770 * expensive operations, we define a multithreaded approach to distributing
2771 * the task across a number of CPU cores.
2775 * To start this, we first we need to obtain the total solution as it stands
2776 * at this Newton increment and then create the initial copy of the scratch
2777 * and copy data objects:
2780 * template <int dim>
2782 * Solid<dim>::update_qph_incremental(const BlockVector<double> &solution_delta)
2784 * timer.enter_subsection("Update QPH data");
2785 * std::cout << " UQPH " << std::flush;
2787 * const BlockVector<double> solution_total(
2788 * get_total_solution(solution_delta));
2790 * const UpdateFlags uf_UQPH(update_values | update_gradients);
2791 * PerTaskData_UQPH per_task_data_UQPH;
2792 * ScratchData_UQPH scratch_data_UQPH(fe, qf_cell, uf_UQPH, solution_total);
2796 * We then pass them and the one-cell update function to the WorkStream to
2800 * WorkStream::run(dof_handler.active_cell_iterators(),
2802 * &Solid::update_qph_incremental_one_cell,
2803 * &Solid::copy_local_to_global_UQPH,
2804 * scratch_data_UQPH,
2805 * per_task_data_UQPH);
2807 * timer.leave_subsection();
2813 * Now we describe how we extract data from the solution vector and pass it
2814 * along to each QP storage object for processing.
2817 * template <int dim>
2818 * void Solid<dim>::update_qph_incremental_one_cell(
2819 * const typename DoFHandler<dim>::active_cell_iterator &cell,
2820 * ScratchData_UQPH &scratch,
2821 * PerTaskData_UQPH & /*data*/)
2823 * const std::vector<std::shared_ptr<PointHistory<dim>>> lqph =
2824 * quadrature_point_history.get_data(cell);
2825 * Assert(lqph.size() == n_q_points, ExcInternalError());
2827 * Assert(scratch.solution_grads_u_total.size() == n_q_points,
2828 * ExcInternalError());
2829 * Assert(scratch.solution_values_p_total.size() == n_q_points,
2830 * ExcInternalError());
2831 * Assert(scratch.solution_values_J_total.size() == n_q_points,
2832 * ExcInternalError());
2838 * We first need to find the values and gradients at quadrature points
2839 * inside the current cell and then we update each local QP using the
2840 * displacement gradient and total pressure and dilatation solution
2844 * scratch.fe_values.reinit(cell);
2845 * scratch.fe_values[u_fe].get_function_gradients(
2846 * scratch.solution_total, scratch.solution_grads_u_total);
2847 * scratch.fe_values[p_fe].get_function_values(
2848 * scratch.solution_total, scratch.solution_values_p_total);
2849 * scratch.fe_values[J_fe].get_function_values(
2850 * scratch.solution_total, scratch.solution_values_J_total);
2852 * for (const unsigned int q_point :
2853 * scratch.fe_values.quadrature_point_indices())
2854 * lqph[q_point]->update_values(scratch.solution_grads_u_total[q_point],
2855 * scratch.solution_values_p_total[q_point],
2856 * scratch.solution_values_J_total[q_point]);
2863 * <a name="step_44-Solidsolve_nonlinear_timestep"></a>
2864 * <h4>Solid::solve_nonlinear_timestep</h4>
2868 * The next function is the driver method for the Newton-Raphson scheme. At
2869 * its top we create a new vector to store the current Newton update step,
2870 * reset the error storage objects and print solver header.
2873 * template <int dim>
2874 * void Solid<dim>::solve_nonlinear_timestep(BlockVector<double> &solution_delta)
2876 * std::cout << std::endl
2877 * << "Timestep " << time.get_timestep() << " @ " << time.current()
2878 * << 's
' << std::endl;
2880 * BlockVector<double> newton_update(dofs_per_block);
2882 * error_residual.reset();
2883 * error_residual_0.reset();
2884 * error_residual_norm.reset();
2885 * error_update.reset();
2886 * error_update_0.reset();
2887 * error_update_norm.reset();
2889 * print_conv_header();
2893 * We now perform a number of Newton iterations to iteratively solve the
2894 * nonlinear problem. Since the problem is fully nonlinear and we are
2895 * using a full Newton method, the data stored in the tangent matrix and
2896 * right-hand side vector is not reusable and must be cleared at each
2897 * Newton step. We then initially build the linear system and
2898 * check for convergence (and store this value in the first iteration).
2899 * The unconstrained DOFs of the rhs vector hold the out-of-balance
2900 * forces, and collectively determine whether or not the equilibrium
2901 * solution has been attained.
2905 * Although for this particular problem we could potentially construct the
2906 * RHS vector before assembling the system matrix, for the sake of
2907 * extensibility we choose not to do so. The benefit to assembling the RHS
2908 * vector and system matrix separately is that the latter is an expensive
2909 * operation and we can potentially avoid an extra assembly process by not
2910 * assembling the tangent matrix when convergence is attained. However, this
2911 * makes parallelizing the code using MPI more difficult. Furthermore, when
2912 * extending the problem to the transient case additional contributions to
2913 * the RHS may result from the time discretization and application of
2914 * constraints for the velocity and acceleration fields.
2917 * unsigned int newton_iteration = 0;
2918 * for (; newton_iteration < parameters.max_iterations_NR; ++newton_iteration)
2920 * std::cout << ' ' << std::setw(2) << newton_iteration << ' '
2925 * We construct the linear system, but hold off on solving it
2926 * (a step that should be significantly more expensive than assembly):
2929 * make_constraints(newton_iteration);
2930 * assemble_system();
2934 * We can now determine the normalized residual error and check for
2935 * solution convergence:
2938 * get_error_residual(error_residual);
2939 * if (newton_iteration == 0)
2940 * error_residual_0 = error_residual;
2942 * error_residual_norm = error_residual;
2943 * error_residual_norm.normalize(error_residual_0);
2945 * if (newton_iteration > 0 && error_update_norm.u <= parameters.tol_u &&
2946 * error_residual_norm.u <= parameters.tol_f)
2948 * std::cout << " CONVERGED! " << std::endl;
2949 * print_conv_footer();
2956 * If we have decided that we want to continue with the iteration, we
2957 * solve the linearized system:
2960 * const std::pair<unsigned int, double> lin_solver_output =
2961 * solve_linear_system(newton_update);
2965 * We can now determine the normalized Newton update error:
2968 * get_error_update(newton_update, error_update);
2969 * if (newton_iteration == 0)
2970 * error_update_0 = error_update;
2972 * error_update_norm = error_update;
2973 * error_update_norm.normalize(error_update_0);
2977 * Lastly, since we implicitly accept the solution step we can perform
2978 * the actual update of the solution increment for the current time
2979 * step, update all quadrature point information pertaining to
2980 * this new displacement and stress state and continue iterating:
2983 * solution_delta += newton_update;
2984 * update_qph_incremental(solution_delta);
2986 * std::cout << " | " << std::fixed << std::setprecision(3) << std::setw(7)
2987 * << std::scientific << lin_solver_output.first << " "
2988 * << lin_solver_output.second << " "
2989 * << error_residual_norm.norm << " " << error_residual_norm.u
2990 * << " " << error_residual_norm.p << " "
2991 * << error_residual_norm.J << " " << error_update_norm.norm
2992 * << " " << error_update_norm.u << " " << error_update_norm.p
2993 * << " " << error_update_norm.J << " " << std::endl;
2998 * At the end, if it turns out that we have in fact done more iterations
2999 * than the parameter file allowed, we raise an exception that can be
3000 * caught in the main() function. The call <code>AssertThrow(condition,
3001 * exc_object)</code> is in essence equivalent to <code>if (!cond) throw
3002 * exc_object;</code> but the former form fills certain fields in the
3003 * exception object that identify the location (filename and line number)
3004 * where the exception was raised to make it simpler to identify where the
3008 * AssertThrow(newton_iteration < parameters.max_iterations_NR,
3009 * ExcMessage("No convergence in nonlinear solver!"));
3016 * <a name="step_44-Solidprint_conv_headerandSolidprint_conv_footer"></a>
3017 * <h4>Solid::print_conv_header and Solid::print_conv_footer</h4>
3021 * This program prints out data in a nice table that is updated
3022 * on a per-iteration basis. The next two functions set up the table
3023 * header and footer:
3026 * template <int dim>
3027 * void Solid<dim>::print_conv_header()
3029 * static const unsigned int l_width = 150;
3031 * for (unsigned int i = 0; i < l_width; ++i)
3033 * std::cout << std::endl;
3035 * std::cout << " SOLVER STEP "
3036 * << " | LIN_IT LIN_RES RES_NORM "
3037 * << " RES_U RES_P RES_J NU_NORM "
3038 * << " NU_U NU_P NU_J " << std::endl;
3040 * for (unsigned int i = 0; i < l_width; ++i)
3042 * std::cout << std::endl;
3047 * template <int dim>
3048 * void Solid<dim>::print_conv_footer()
3050 * static const unsigned int l_width = 150;
3052 * for (unsigned int i = 0; i < l_width; ++i)
3054 * std::cout << std::endl;
3056 * const std::pair<double, double> error_dil = get_error_dilation();
3058 * std::cout << "Relative errors:" << std::endl
3059 * << "Displacement:\t" << error_update.u / error_update_0.u
3061 * << "Force: \t\t" << error_residual.u / error_residual_0.u
3063 * << "Dilatation:\t" << error_dil.first << std::endl
3064 * << "v / V_0:\t" << error_dil.second * vol_reference << " / "
3065 * << vol_reference << " = " << error_dil.second << std::endl;
3072 * <a name="step_44-Solidget_error_dilation"></a>
3073 * <h4>Solid::get_error_dilation</h4>
3077 * Calculate the volume of the domain in the spatial configuration
3080 * template <int dim>
3081 * double Solid<dim>::compute_vol_current() const
3083 * double vol_current = 0.0;
3085 * FEValues<dim> fe_values(fe, qf_cell, update_JxW_values);
3087 * for (const auto &cell : triangulation.active_cell_iterators())
3089 * fe_values.reinit(cell);
3093 * In contrast to that which was previously called for,
3094 * in this instance the quadrature point data is specifically
3095 * non-modifiable since we will only be accessing data.
3096 * We ensure that the right get_data function is called by
3097 * marking this update function as constant.
3100 * const std::vector<std::shared_ptr<const PointHistory<dim>>> lqph =
3101 * quadrature_point_history.get_data(cell);
3102 * Assert(lqph.size() == n_q_points, ExcInternalError());
3104 * for (const unsigned int q_point : fe_values.quadrature_point_indices())
3106 * const double det_F_qp = lqph[q_point]->get_det_F();
3107 * const double JxW = fe_values.JxW(q_point);
3109 * vol_current += det_F_qp * JxW;
3112 * Assert(vol_current > 0.0, ExcInternalError());
3113 * return vol_current;
3118 * Calculate how well the dilatation @f$\widetilde{J}@f$ agrees with @f$J
3119 * \dealcoloneq \textrm{det}\ \mathbf{F}@f$ from the @f$L^2@f$ error @f$ \bigl[
3120 * \int_{\Omega_0} {[ J - \widetilde{J}]}^{2}\textrm{d}V \bigr]^{1/2}@f$.
3121 * We also return the ratio of the current volume of the
3122 * domain to the reference volume. This is of interest for incompressible
3123 * media where we want to check how well the isochoric constraint has been
3127 * template <int dim>
3128 * std::pair<double, double> Solid<dim>::get_error_dilation() const
3130 * double dil_L2_error = 0.0;
3132 * FEValues<dim> fe_values(fe, qf_cell, update_JxW_values);
3134 * for (const auto &cell : triangulation.active_cell_iterators())
3136 * fe_values.reinit(cell);
3138 * const std::vector<std::shared_ptr<const PointHistory<dim>>> lqph =
3139 * quadrature_point_history.get_data(cell);
3140 * Assert(lqph.size() == n_q_points, ExcInternalError());
3142 * for (const unsigned int q_point : fe_values.quadrature_point_indices())
3144 * const double det_F_qp = lqph[q_point]->get_det_F();
3145 * const double J_tilde_qp = lqph[q_point]->get_J_tilde();
3146 * const double the_error_qp_squared =
3147 * Utilities::fixed_power<2>((det_F_qp - J_tilde_qp));
3148 * const double JxW = fe_values.JxW(q_point);
3150 * dil_L2_error += the_error_qp_squared * JxW;
3154 * return std::make_pair(std::sqrt(dil_L2_error),
3155 * compute_vol_current() / vol_reference);
3162 * <a name="step_44-Solidget_error_residual"></a>
3163 * <h4>Solid::get_error_residual</h4>
3167 * Determine the true residual error for the problem. That is, determine the
3168 * error in the residual for the unconstrained degrees of freedom. Note that
3169 * to do so, we need to ignore constrained DOFs by setting the residual in
3170 * these vector components to zero.
3173 * template <int dim>
3174 * void Solid<dim>::get_error_residual(Errors &error_residual)
3176 * BlockVector<double> error_res(dofs_per_block);
3178 * for (unsigned int i = 0; i < dof_handler.n_dofs(); ++i)
3179 * if (!constraints.is_constrained(i))
3180 * error_res(i) = system_rhs(i);
3182 * error_residual.norm = error_res.l2_norm();
3183 * error_residual.u = error_res.block(u_dof).l2_norm();
3184 * error_residual.p = error_res.block(p_dof).l2_norm();
3185 * error_residual.J = error_res.block(J_dof).l2_norm();
3192 * <a name="step_44-Solidget_error_update"></a>
3193 * <h4>Solid::get_error_update</h4>
3197 * Determine the true Newton update error for the problem
3200 * template <int dim>
3201 * void Solid<dim>::get_error_update(const BlockVector<double> &newton_update,
3202 * Errors &error_update)
3204 * BlockVector<double> error_ud(dofs_per_block);
3205 * for (unsigned int i = 0; i < dof_handler.n_dofs(); ++i)
3206 * if (!constraints.is_constrained(i))
3207 * error_ud(i) = newton_update(i);
3209 * error_update.norm = error_ud.l2_norm();
3210 * error_update.u = error_ud.block(u_dof).l2_norm();
3211 * error_update.p = error_ud.block(p_dof).l2_norm();
3212 * error_update.J = error_ud.block(J_dof).l2_norm();
3220 * <a name="step_44-Solidget_total_solution"></a>
3221 * <h4>Solid::get_total_solution</h4>
3225 * This function provides the total solution, which is valid at any Newton
3226 * step. This is required as, to reduce computational error, the total
3227 * solution is only updated at the end of the timestep.
3230 * template <int dim>
3231 * BlockVector<double> Solid<dim>::get_total_solution(
3232 * const BlockVector<double> &solution_delta) const
3234 * BlockVector<double> solution_total(solution_n);
3235 * solution_total += solution_delta;
3236 * return solution_total;
3243 * <a name="step_44-Solidassemble_system"></a>
3244 * <h4>Solid::assemble_system</h4>
3248 * Since we use TBB for assembly, we simply setup a copy of the
3249 * data structures required for the process and pass them, along
3250 * with the assembly functions to the WorkStream object for processing. Note
3251 * that we must ensure that the matrix and RHS vector are reset before any
3252 * assembly operations can occur. Furthermore, since we are describing a
3253 * problem with Neumann BCs, we will need the face normals and so must specify
3254 * this in the face update flags.
3257 * template <int dim>
3258 * void Solid<dim>::assemble_system()
3260 * timer.enter_subsection("Assemble system");
3261 * std::cout << " ASM_SYS " << std::flush;
3263 * tangent_matrix = 0.0;
3266 * const UpdateFlags uf_cell(update_values | update_gradients |
3267 * update_JxW_values);
3268 * const UpdateFlags uf_face(update_values | update_normal_vectors |
3269 * update_JxW_values);
3271 * PerTaskData_ASM per_task_data(dofs_per_cell);
3272 * ScratchData_ASM scratch_data(fe, qf_cell, uf_cell, qf_face, uf_face);
3276 * The syntax used here to pass data to the WorkStream class
3277 * is discussed in @ref step_13 "step-13".
3281 * dof_handler.active_cell_iterators(),
3282 * [this](const typename DoFHandler<dim>::active_cell_iterator &cell,
3283 * ScratchData_ASM &scratch,
3284 * PerTaskData_ASM &data) {
3285 * this->assemble_system_one_cell(cell, scratch, data);
3287 * [this](const PerTaskData_ASM &data) {
3288 * this->constraints.distribute_local_to_global(data.cell_matrix,
3290 * data.local_dof_indices,
3297 * timer.leave_subsection();
3302 * Of course, we still have to define how we assemble the tangent matrix
3303 * contribution for a single cell. We first need to reset and initialize some
3304 * of the scratch data structures and retrieve some basic information
3305 * regarding the DOF numbering on this cell. We can precalculate the cell
3306 * shape function values and gradients. Note that the shape function gradients
3307 * are defined with regard to the current configuration. That is
3308 * @f$\textrm{grad}\ \boldsymbol{\varphi} = \textrm{Grad}\ \boldsymbol{\varphi}
3309 * \ \mathbf{F}^{-1}@f$.
3312 * template <int dim>
3313 * void Solid<dim>::assemble_system_one_cell(
3314 * const typename DoFHandler<dim>::active_cell_iterator &cell,
3315 * ScratchData_ASM &scratch,
3316 * PerTaskData_ASM &data) const
3320 * scratch.fe_values.reinit(cell);
3321 * cell->get_dof_indices(data.local_dof_indices);
3323 * const std::vector<std::shared_ptr<const PointHistory<dim>>> lqph =
3324 * quadrature_point_history.get_data(cell);
3325 * Assert(lqph.size() == n_q_points, ExcInternalError());
3327 * for (const unsigned int q_point :
3328 * scratch.fe_values.quadrature_point_indices())
3330 * const Tensor<2, dim> F_inv = lqph[q_point]->get_F_inv();
3331 * for (const unsigned int k : scratch.fe_values.dof_indices())
3333 * const unsigned int k_group = fe.system_to_base_index(k).first.first;
3335 * if (k_group == u_dof)
3337 * scratch.grad_Nx[q_point][k] =
3338 * scratch.fe_values[u_fe].gradient(k, q_point) * F_inv;
3339 * scratch.symm_grad_Nx[q_point][k] =
3340 * symmetrize(scratch.grad_Nx[q_point][k]);
3342 * else if (k_group == p_dof)
3343 * scratch.Nx[q_point][k] =
3344 * scratch.fe_values[p_fe].value(k, q_point);
3345 * else if (k_group == J_dof)
3346 * scratch.Nx[q_point][k] =
3347 * scratch.fe_values[J_fe].value(k, q_point);
3349 * Assert(k_group <= J_dof, ExcInternalError());
3355 * Now we build the local cell @ref GlossStiffnessMatrix "stiffness matrix" and RHS vector. Since the
3356 * global and local system matrices are symmetric, we can exploit this
3357 * property by building only the lower half of the local matrix and copying
3358 * the values to the upper half. So we only assemble half of the
3359 * @f$\mathsf{\mathbf{k}}_{uu}@f$, @f$\mathsf{\mathbf{k}}_{\widetilde{p}
3360 * \widetilde{p}} = \mathbf{0}@f$, @f$\mathsf{\mathbf{k}}_{\widetilde{J}
3361 * \widetilde{J}}@f$ blocks, while the whole
3362 * @f$\mathsf{\mathbf{k}}_{\widetilde{p} \widetilde{J}}@f$,
3363 * @f$\mathsf{\mathbf{k}}_{u \widetilde{J}} = \mathbf{0}@f$,
3364 * @f$\mathsf{\mathbf{k}}_{u \widetilde{p}}@f$ blocks are built.
3368 * In doing so, we first extract some configuration dependent variables
3369 * from our quadrature history objects for the current quadrature point.
3372 * for (const unsigned int q_point :
3373 * scratch.fe_values.quadrature_point_indices())
3375 * const SymmetricTensor<2, dim> tau = lqph[q_point]->get_tau();
3376 * const Tensor<2, dim> tau_ns = lqph[q_point]->get_tau();
3377 * const SymmetricTensor<4, dim> Jc = lqph[q_point]->get_Jc();
3378 * const double det_F = lqph[q_point]->get_det_F();
3379 * const double p_tilde = lqph[q_point]->get_p_tilde();
3380 * const double J_tilde = lqph[q_point]->get_J_tilde();
3381 * const double dPsi_vol_dJ = lqph[q_point]->get_dPsi_vol_dJ();
3382 * const double d2Psi_vol_dJ2 = lqph[q_point]->get_d2Psi_vol_dJ2();
3383 * const SymmetricTensor<2, dim> &I =
3384 * Physics::Elasticity::StandardTensors<dim>::I;
3388 * These two tensors store some precomputed data. Their use will
3389 * explained shortly.
3392 * SymmetricTensor<2, dim> symm_grad_Nx_i_x_Jc;
3393 * Tensor<1, dim> grad_Nx_i_comp_i_x_tau;
3397 * Next we define some aliases to make the assembly process easier to
3401 * const std::vector<double> &N = scratch.Nx[q_point];
3402 * const std::vector<SymmetricTensor<2, dim>> &symm_grad_Nx =
3403 * scratch.symm_grad_Nx[q_point];
3404 * const std::vector<Tensor<2, dim>> &grad_Nx = scratch.grad_Nx[q_point];
3405 * const double JxW = scratch.fe_values.JxW(q_point);
3407 * for (const unsigned int i : scratch.fe_values.dof_indices())
3409 * const unsigned int component_i =
3410 * fe.system_to_component_index(i).first;
3411 * const unsigned int i_group = fe.system_to_base_index(i).first.first;
3415 * We first compute the contributions
3416 * from the internal forces. Note, by
3417 * definition of the rhs as the negative
3418 * of the residual, these contributions
3422 * if (i_group == u_dof)
3423 * data.cell_rhs(i) -= (symm_grad_Nx[i] * tau) * JxW;
3424 * else if (i_group == p_dof)
3425 * data.cell_rhs(i) -= N[i] * (det_F - J_tilde) * JxW;
3426 * else if (i_group == J_dof)
3427 * data.cell_rhs(i) -= N[i] * (dPsi_vol_dJ - p_tilde) * JxW;
3429 * Assert(i_group <= J_dof, ExcInternalError());
3433 * Before we go into the inner loop, we have one final chance to
3434 * introduce some optimizations. We've already taken into account
3435 * the symmetry of the system, and we can now precompute some
3436 * common terms that are repeatedly applied in the inner
loop.
3437 * We won
't be excessive here, but will rather focus on expensive
3438 * operations, namely those involving the rank-4 material stiffness
3439 * tensor and the rank-2 stress tensor.
3443 * What we may observe is that both of these tensors are contracted
3444 * with shape function gradients indexed on the "i" DoF. This
3445 * implies that this particular operation remains constant as we
3446 * loop over the "j" DoF. For that reason, we can extract this from
3447 * the inner loop and save the many operations that, for each
3448 * quadrature point and DoF index "i" and repeated over index "j"
3449 * are required to double contract a rank-2 symmetric tensor with a
3450 * rank-4 symmetric tensor, and a rank-1 tensor with a rank-2
3455 * At the loss of some readability, this small change will reduce
3456 * the assembly time of the symmetrized system by about half when
3457 * using the simulation default parameters, and becomes more
3458 * significant as the h-refinement level increases.
3461 * if (i_group == u_dof)
3463 * symm_grad_Nx_i_x_Jc = symm_grad_Nx[i] * Jc;
3464 * grad_Nx_i_comp_i_x_tau = grad_Nx[i][component_i] * tau_ns;
3469 * Now we're prepared to compute the tangent
matrix contributions:
3472 *
for (
const unsigned int j :
3473 * scratch.fe_values.dof_indices_ending_at(i))
3475 * const unsigned
int component_j =
3476 * fe.system_to_component_index(j).
first;
3477 *
const unsigned int j_group =
3478 * fe.system_to_base_index(j).first.first;
3482 * This is the @f$\mathsf{\mathbf{k}}_{uu}@f$
3483 * contribution. It comprises a material contribution, and a
3484 * geometrical stress contribution which is only added along
3485 * the local
matrix diagonals:
3488 *
if ((i_group == j_group) && (i_group == u_dof))
3492 * The material contribution:
3495 *
data.cell_matrix(i, j) += symm_grad_Nx_i_x_Jc *
3496 * symm_grad_Nx[j] * JxW;
3500 * The geometrical stress contribution:
3503 *
if (component_i == component_j)
3504 *
data.cell_matrix(i, j) +=
3505 * grad_Nx_i_comp_i_x_tau * grad_Nx[j][component_j] * JxW;
3509 * Next is the @f$\mathsf{\mathbf{k}}_{ \widetilde{p} u}@f$
3513 *
else if ((i_group == p_dof) && (j_group == u_dof))
3515 *
data.cell_matrix(i, j) += N[i] * det_F *
3516 * (symm_grad_Nx[j] * I) * JxW;
3520 * and lastly the @f$\mathsf{\mathbf{k}}_{ \widetilde{J}
3521 * \widetilde{p}}@f$ and @f$\mathsf{\mathbf{k}}_{ \widetilde{J}
3522 * \widetilde{J}}@f$ contributions:
3525 *
else if ((i_group == J_dof) && (j_group == p_dof))
3526 *
data.cell_matrix(i, j) -= N[i] * N[j] * JxW;
3527 *
else if ((i_group == j_group) && (i_group == J_dof))
3528 *
data.cell_matrix(i, j) += N[i] * d2Psi_vol_dJ2 * N[j] * JxW;
3530 *
Assert((i_group <= J_dof) && (j_group <= J_dof),
3531 * ExcInternalError());
3539 * cell face exists on a boundary on which a traction is applied and add
3540 * the contribution
if this is the
case.
3543 *
for (
const auto &face : cell->face_iterators())
3544 * if (face->at_boundary() && face->
boundary_id() == 6)
3546 * scratch.fe_face_values.
reinit(cell, face);
3548 *
for (
const unsigned int f_q_point :
3549 * scratch.fe_face_values.quadrature_point_indices())
3551 * const
Tensor<1, dim> &N =
3552 * scratch.fe_face_values.normal_vector(f_q_point);
3556 * Using the face normal at
this quadrature
point we specify the
3557 * traction in reference configuration. For
this problem, a
3558 * defined pressure is applied in the reference configuration.
3559 * The direction of the applied traction is assumed not to
3560 * evolve with the deformation of the domain. The traction is
3561 * defined
using the
first Piola-Kirchhoff stress is simply
3562 * @f$\mathbf{t} = \mathbf{P}\mathbf{N} = [p_0 \mathbf{I}]
3563 * \mathbf{N} = p_0 \mathbf{N}@f$ We use the time variable to
3564 * linearly ramp up the pressure load.
3568 * Note that the contributions to the right hand side vector we
3569 * compute here only exist in the displacement components of the
3573 *
static const double p0 =
3574 * -4.0 / (parameters.scale * parameters.scale);
3575 *
const double time_ramp = (time.current() / time.end());
3576 *
const double pressure = p0 * parameters.p_p0 * time_ramp;
3579 *
for (
const unsigned int i : scratch.fe_values.dof_indices())
3581 * const unsigned
int i_group =
3584 *
if (i_group == u_dof)
3586 *
const unsigned int component_i =
3587 * fe.system_to_component_index(i).first;
3589 * scratch.fe_face_values.shape_value(i, f_q_point);
3590 *
const double JxW = scratch.fe_face_values.JxW(f_q_point);
3592 *
data.cell_rhs(i) += (Ni * traction[component_i]) * JxW;
3600 * Finally, we need to
copy the lower half of the local
matrix into the
3604 *
for (
const unsigned int i : scratch.fe_values.dof_indices())
3605 * for (const unsigned
int j :
3606 * scratch.fe_values.dof_indices_starting_at(i + 1))
3615 * <a name=
"step_44-Solidmake_constraints"></a>
3616 * <h4>Solid::make_constraints</h4>
3617 * The constraints
for this problem are simple to describe.
3618 * In
this particular example, the boundary
values will be calculated
for
3619 * the two
first iterations of Newton
's algorithm. In general, one would
3620 * build non-homogeneous constraints in the zeroth iteration (that is, when
3621 * `apply_dirichlet_bc == true` in the code block that follows) and build
3622 * only the corresponding homogeneous constraints in the following step. While
3623 * the current example has only homogeneous constraints, previous experiences
3624 * have shown that a common error is forgetting to add the extra condition
3625 * when refactoring the code to specific uses. This could lead to errors that
3626 * are hard to debug. In this spirit, we choose to make the code more verbose
3627 * in terms of what operations are performed at each Newton step.
3630 * template <int dim>
3631 * void Solid<dim>::make_constraints(const int it_nr)
3635 * Since we (a) are dealing with an iterative Newton method, (b) are using
3636 * an incremental formulation for the displacement, and (c) apply the
3637 * constraints to the incremental displacement field, any non-homogeneous
3638 * constraints on the displacement update should only be specified at the
3639 * zeroth iteration. No subsequent contributions are to be made since the
3640 * constraints will be exactly satisfied after that iteration.
3643 * const bool apply_dirichlet_bc = (it_nr == 0);
3647 * Furthermore, after the first Newton iteration within a timestep, the
3648 * constraints remain the same and we do not need to modify or rebuild them
3649 * so long as we do not clear the @p constraints object.
3654 * std::cout << " --- " << std::flush;
3658 * std::cout << " CST " << std::flush;
3660 * if (apply_dirichlet_bc)
3664 * At the zeroth Newton iteration we wish to apply the full set of
3665 * non-homogeneous and homogeneous constraints that represent the
3666 * boundary conditions on the displacement increment. Since in general
3667 * the constraints may be different at each time step, we need to clear
3668 * the constraints matrix and completely rebuild it. An example case
3669 * would be if a surface is accelerating; in such a scenario the change
3670 * in displacement is non-constant between each time step.
3673 * constraints.clear();
3677 * The boundary conditions for the indentation problem in 3d are as
3678 * follows: On the -x, -y and -z faces (IDs 0,2,4) we set up a symmetry
3679 * condition to allow only planar movement while the +x and +z faces
3680 * (IDs 1,5) are traction free. In this contrived problem, part of the
3681 * +y face (ID 3) is set to have no motion in the x- and z-component.
3682 * Finally, as described earlier, the other part of the +y face has an
3683 * the applied pressure but is also constrained in the x- and
3688 * In the following, we will have to tell the function interpolation
3689 * boundary values which components of the solution vector should be
3690 * constrained (i.e., whether it's the x-, y-, z-displacements or
3691 * combinations thereof). This is done
using ComponentMask objects (see
3692 * @ref GlossComponentMask) which we can get from the finite element
if we
3693 * provide it with an extractor
object for the component we wish to
3694 * select. To
this end we
first set up such extractor objects and later
3695 * use it when generating the relevant component masks:
3709 * fe.component_mask(x_displacement));
3719 * fe.component_mask(y_displacement));
3734 * (fe.component_mask(x_displacement) |
3735 * fe.component_mask(z_displacement)));
3745 * fe.component_mask(z_displacement));
3756 * (fe.component_mask(x_displacement) |
3757 * fe.component_mask(z_displacement)));
3770 * (fe.component_mask(x_displacement)));
3780 * (fe.component_mask(x_displacement)));
3788 * As all Dirichlet constraints are fulfilled exactly after the zeroth
3789 * Newton iteration, we want to ensure that no further modification are
3790 * made to those entries. This implies that we want to convert
3791 * all non-homogeneous Dirichlet constraints into homogeneous ones.
3795 * In
this example the procedure to
do this is quite straightforward,
3796 * and in fact we can (and will) circumvent any unnecessary operations
3797 * when only homogeneous boundary conditions are applied.
3798 * In a more
general problem one should be mindful of hanging node
3799 * and periodic constraints, which may also introduce some
3800 * inhomogeneities. It might then be advantageous to keep disparate
3801 * objects
for the different
types of constraints, and
merge them
3802 * together once the homogeneous Dirichlet constraints have been
3806 *
if (constraints.has_inhomogeneities())
3810 * Since the
affine constraints were finalized at the previous
3811 * Newton iteration, they may not be modified directly. So
3812 * we need to
copy them to another temporary
object and make
3813 * modification there. Once we
're done, we'll transfer them
3814 * back to the main @p constraints
object.
3818 *
for (
unsigned int dof = 0; dof != dof_handler.n_dofs(); ++dof)
3819 *
if (homogeneous_constraints.is_inhomogeneously_constrained(dof))
3820 * homogeneous_constraints.set_inhomogeneity(dof, 0.0);
3822 * constraints.clear();
3823 * constraints.copy_from(homogeneous_constraints);
3827 * constraints.close();
3833 * <a name=
"step_44-Solidassemble_sc"></a>
3834 * <h4>Solid::assemble_sc</h4>
3835 * Solving the entire block system is a bit problematic as there are no
3836 * contributions to the @f$\mathsf{\mathbf{
K}}_{ \widetilde{J} \widetilde{J}}@f$
3837 * block, rendering it noninvertible (when
using an iterative solver).
3838 * Since the pressure and dilatation variables DOFs are discontinuous, we can
3839 * condense them out to form a smaller displacement-only system which
3840 * we will then solve and subsequently post-process to retrieve the
3841 * pressure and dilatation solutions.
3845 * The
static condensation process could be performed at a global
level but we
3846 * need the inverse of one of the blocks. However, since the pressure and
3847 * dilatation variables are discontinuous, the
static condensation (SC)
3848 * operation can also be done on a per-cell basis and we can produce the
3850 * @f$\mathsf{\mathbf{
K}}_{\widetilde{p}\widetilde{J}}@f$ block by inverting the
3851 * local blocks. We can again use TBB to
do this since each operation will be
3859 * \mathsf{\mathbf{
K}}_{\textrm{con}}
3860 * = \bigl[ \mathsf{\mathbf{
K}}_{uu} +
3861 * \overline{\overline{\mathsf{\mathbf{
K}}}}~ \bigr]
3863 * from each element
's contributions. These contributions are then added to
3864 * the global stiffness matrix. Given this description, the following two
3865 * functions should be clear:
3868 * template <int dim>
3869 * void Solid<dim>::assemble_sc()
3871 * timer.enter_subsection("Perform static condensation");
3872 * std::cout << " ASM_SC " << std::flush;
3874 * PerTaskData_SC per_task_data(dofs_per_cell,
3875 * element_indices_u.size(),
3876 * element_indices_p.size(),
3877 * element_indices_J.size());
3878 * ScratchData_SC scratch_data;
3880 * WorkStream::run(dof_handler.active_cell_iterators(),
3882 * &Solid::assemble_sc_one_cell,
3883 * &Solid::copy_local_to_global_sc,
3887 * timer.leave_subsection();
3891 * template <int dim>
3892 * void Solid<dim>::copy_local_to_global_sc(const PerTaskData_SC &data)
3894 * for (unsigned int i = 0; i < dofs_per_cell; ++i)
3895 * for (unsigned int j = 0; j < dofs_per_cell; ++j)
3896 * tangent_matrix.add(data.local_dof_indices[i],
3897 * data.local_dof_indices[j],
3898 * data.cell_matrix(i, j));
3904 * Now we describe the static condensation process. As per usual, we must
3905 * first find out which global numbers the degrees of freedom on this cell
3906 * have and reset some data structures:
3909 * template <int dim>
3910 * void Solid<dim>::assemble_sc_one_cell(
3911 * const typename DoFHandler<dim>::active_cell_iterator &cell,
3912 * ScratchData_SC &scratch,
3913 * PerTaskData_SC &data)
3917 * cell->get_dof_indices(data.local_dof_indices);
3921 * We now extract the contribution of the dofs associated with the current
3922 * cell to the global stiffness matrix. The discontinuous nature of the
3923 * @f$\widetilde{p}@f$ and @f$\widetilde{J}@f$ interpolations mean that their is
3924 * no coupling of the local contributions at the global level. This is not
3925 * the case with the @f$\mathbf{u}@f$ dof. In other words,
3926 * @f$\mathsf{\mathbf{k}}_{\widetilde{J} \widetilde{p}}@f$,
3927 * @f$\mathsf{\mathbf{k}}_{\widetilde{p} \widetilde{p}}@f$ and
3928 * @f$\mathsf{\mathbf{k}}_{\widetilde{J} \widetilde{p}}@f$, when extracted
3929 * from the global stiffness matrix are the element contributions. This
3930 * is not the case for @f$\mathsf{\mathbf{k}}_{uu}@f$.
3934 * Note: A lower-case symbol is used to denote element stiffness matrices.
3938 * Currently the matrix corresponding to
3939 * the dof associated with the current element
3940 * (denoted somewhat loosely as @f$\mathsf{\mathbf{k}}@f$)
3944 * \mathsf{\mathbf{k}}_{uu} & \mathsf{\mathbf{k}}_{u\widetilde{p}}
3946 * \\ \mathsf{\mathbf{k}}_{\widetilde{p}u} & \mathbf{0} &
3947 * \mathsf{\mathbf{k}}_{\widetilde{p}\widetilde{J}}
3948 * \\ \mathbf{0} & \mathsf{\mathbf{k}}_{\widetilde{J}\widetilde{p}} &
3949 * \mathsf{\mathbf{k}}_{\widetilde{J}\widetilde{J}} \end{bmatrix}
3954 * We now need to modify it such that it appear as
3957 * \mathsf{\mathbf{k}}_{\textrm{con}} &
3958 * \mathsf{\mathbf{k}}_{u\widetilde{p}} & \mathbf{0}
3959 * \\ \mathsf{\mathbf{k}}_{\widetilde{p}u} & \mathbf{0} &
3960 * \mathsf{\mathbf{k}}_{\widetilde{p}\widetilde{J}}^{-1}
3961 * \\ \mathbf{0} & \mathsf{\mathbf{k}}_{\widetilde{J}\widetilde{p}} &
3962 * \mathsf{\mathbf{k}}_{\widetilde{J}\widetilde{J}} \end{bmatrix}
3964 * with @f$\mathsf{\mathbf{k}}_{\textrm{con}} = \bigl[
3965 * \mathsf{\mathbf{k}}_{uu} +\overline{\overline{\mathsf{\mathbf{k}}}}~
3966 * \bigr]@f$ where @f$ \overline{\overline{\mathsf{\mathbf{k}}}}
3967 * \dealcoloneq \mathsf{\mathbf{k}}_{u\widetilde{p}}
3968 * \overline{\mathsf{\mathbf{k}}} \mathsf{\mathbf{k}}_{\widetilde{p}u}
3972 * \overline{\mathsf{\mathbf{k}}} =
3973 * \mathsf{\mathbf{k}}_{\widetilde{J}\widetilde{p}}^{-1}
3974 * \mathsf{\mathbf{k}}_{\widetilde{J}\widetilde{J}}
3975 * \mathsf{\mathbf{k}}_{\widetilde{p}\widetilde{J}}^{-1}
3980 * At this point, we need to take note of
3981 * the fact that global data already exists
3982 * in the @f$\mathsf{\mathbf{K}}_{uu}@f$,
3983 * @f$\mathsf{\mathbf{K}}_{\widetilde{p} \widetilde{J}}@f$
3985 * @f$\mathsf{\mathbf{K}}_{\widetilde{J} \widetilde{p}}@f$
3986 * sub-blocks. So if we are to modify them, we must account for the data
3987 * that is already there (i.e. simply add to it or remove it if
3988 * necessary). Since the copy_local_to_global operation is a "+="
3989 * operation, we need to take this into account
3993 * For the @f$\mathsf{\mathbf{K}}_{uu}@f$ block in particular, this means that
3994 * contributions have been added from the surrounding cells, so we need to
3995 * be careful when we manipulate this block. We can't just erase the
4000 * This is the strategy we will employ to get the sub-blocks we want:
4004 * - @f$ {\mathsf{\mathbf{k}}}_{\textrm{store}}@f$:
4005 * Since we don
't have access to @f$\mathsf{\mathbf{k}}_{uu}@f$,
4006 * but we know its contribution is added to
4007 * the global @f$\mathsf{\mathbf{K}}_{uu}@f$ matrix, we just want
4008 * to add the element wise
4009 * static-condensation @f$\overline{\overline{\mathsf{\mathbf{k}}}}@f$.
4013 * - @f$\mathsf{\mathbf{k}}^{-1}_{\widetilde{p} \widetilde{J}}@f$:
4014 * Similarly, @f$\mathsf{\mathbf{k}}_{\widetilde{p}
4015 * \widetilde{J}}@f$ exists in
4016 * the subblock. Since the copy
4017 * operation is a += operation, we
4018 * need to subtract the existing
4019 * @f$\mathsf{\mathbf{k}}_{\widetilde{p} \widetilde{J}}@f$
4020 * submatrix in addition to
4021 * "adding" that which we wish to
4026 * - @f$\mathsf{\mathbf{k}}^{-1}_{\widetilde{J} \widetilde{p}}@f$:
4027 * Since the global matrix
4028 * is symmetric, this block is the
4029 * same as the one above and we
4031 * @f$\mathsf{\mathbf{k}}^{-1}_{\widetilde{p} \widetilde{J}}@f$
4032 * as a substitute for this one.
4036 * We first extract element data from the
4037 * system matrix. So first we get the
4038 * entire subblock for the cell, then
4039 * extract @f$\mathsf{\mathbf{k}}@f$
4040 * for the dofs associated with
4041 * the current element
4044 * data.k_orig.extract_submatrix_from(tangent_matrix,
4045 * data.local_dof_indices,
4046 * data.local_dof_indices);
4049 * and next the local matrices for
4050 * @f$\mathsf{\mathbf{k}}_{ \widetilde{p} u}@f$
4051 * @f$\mathsf{\mathbf{k}}_{ \widetilde{p} \widetilde{J}}@f$
4053 * @f$\mathsf{\mathbf{k}}_{ \widetilde{J} \widetilde{J}}@f$:
4056 * data.k_pu.extract_submatrix_from(data.k_orig,
4057 * element_indices_p,
4058 * element_indices_u);
4059 * data.k_pJ.extract_submatrix_from(data.k_orig,
4060 * element_indices_p,
4061 * element_indices_J);
4062 * data.k_JJ.extract_submatrix_from(data.k_orig,
4063 * element_indices_J,
4064 * element_indices_J);
4068 * To get the inverse of @f$\mathsf{\mathbf{k}}_{\widetilde{p}
4069 * \widetilde{J}}@f$, we invert it directly. This operation is relatively
4070 * inexpensive since @f$\mathsf{\mathbf{k}}_{\widetilde{p} \widetilde{J}}@f$
4071 * since block-diagonal.
4074 * data.k_pJ_inv.invert(data.k_pJ);
4078 * Now we can make condensation terms to
4079 * add to the @f$\mathsf{\mathbf{k}}_{uu}@f$
4080 * block and put them in
4081 * the cell local matrix
4083 * \mathsf{\mathbf{A}}
4085 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{p} \widetilde{J}}
4086 * \mathsf{\mathbf{k}}_{\widetilde{p} u}
4090 * data.k_pJ_inv.mmult(data.A, data.k_pu);
4094 * \mathsf{\mathbf{B}}
4096 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{J} \widetilde{J}}
4097 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{p} \widetilde{J}}
4098 * \mathsf{\mathbf{k}}_{\widetilde{p} u}
4102 * data.k_JJ.mmult(data.B, data.A);
4106 * \mathsf{\mathbf{C}}
4108 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{J} \widetilde{p}}
4109 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{J} \widetilde{J}}
4110 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{p} \widetilde{J}}
4111 * \mathsf{\mathbf{k}}_{\widetilde{p} u}
4115 * data.k_pJ_inv.Tmmult(data.C, data.B);
4119 * \overline{\overline{\mathsf{\mathbf{k}}}}
4121 * \mathsf{\mathbf{k}}_{u \widetilde{p}}
4122 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{J} \widetilde{p}}
4123 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{J} \widetilde{J}}
4124 * \mathsf{\mathbf{k}}^{-1}_{\widetilde{p} \widetilde{J}}
4125 * \mathsf{\mathbf{k}}_{\widetilde{p} u}
4129 * data.k_pu.Tmmult(data.k_bbar, data.C);
4130 * data.k_bbar.scatter_matrix_to(element_indices_u,
4131 * element_indices_u,
4132 * data.cell_matrix);
4137 * @f$\mathsf{\mathbf{k}}^{-1}_{ \widetilde{p} \widetilde{J}}@f$
4139 * @f$\mathsf{\mathbf{k}}_{ \widetilde{p} \widetilde{J}}@f$
4140 * block for post-processing. Note again
4141 * that we need to remove the
4142 * contribution that already exists there.
4145 * data.k_pJ_inv.add(-1.0, data.k_pJ);
4146 * data.k_pJ_inv.scatter_matrix_to(element_indices_p,
4147 * element_indices_J,
4148 * data.cell_matrix);
4154 * <a name="step_44-Solidsolve_linear_system"></a>
4155 * <h4>Solid::solve_linear_system</h4>
4156 * We now have all of the necessary components to use one of two possible
4157 * methods to solve the linearised system. The first is to perform static
4158 * condensation on an element level, which requires some alterations
4159 * to the tangent matrix and RHS vector. Alternatively, the full block
4160 * system can be solved by performing condensation on a global level.
4161 * Below we implement both approaches.
4164 * template <int dim>
4165 * std::pair<unsigned int, double>
4166 * Solid<dim>::solve_linear_system(BlockVector<double> &newton_update)
4168 * unsigned int lin_it = 0;
4169 * double lin_res = 0.0;
4171 * if (parameters.use_static_condensation == true)
4175 * Firstly, here is the approach using the (permanent) augmentation of
4176 * the tangent matrix. For the following, recall that
4178 * \mathsf{\mathbf{K}}_{\textrm{store}}
4181 * \mathsf{\mathbf{K}}_{\textrm{con}} &
4182 * \mathsf{\mathbf{K}}_{u\widetilde{p}} & \mathbf{0}
4183 * \\ \mathsf{\mathbf{K}}_{\widetilde{p}u} & \mathbf{0} &
4184 * \mathsf{\mathbf{K}}_{\widetilde{p}\widetilde{J}}^{-1}
4186 * \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{p}} &
4187 * \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{J}} \end{bmatrix} \, .
4191 * d \widetilde{\mathsf{\mathbf{p}}}
4193 * \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{p}}^{-1}
4195 * \mathsf{\mathbf{F}}_{\widetilde{J}}
4197 * \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{J}}
4198 * d \widetilde{\mathsf{\mathbf{J}}} \bigr]
4199 * \\ d \widetilde{\mathsf{\mathbf{J}}}
4201 * \mathsf{\mathbf{K}}_{\widetilde{p}\widetilde{J}}^{-1}
4203 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4204 * - \mathsf{\mathbf{K}}_{\widetilde{p}u} d
4205 * \mathsf{\mathbf{u}} \bigr]
4206 * \\ \Rightarrow d \widetilde{\mathsf{\mathbf{p}}}
4207 * &= \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{p}}^{-1}
4208 * \mathsf{\mathbf{F}}_{\widetilde{J}}
4210 * \underbrace{\bigl[\mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{p}}^{-1}
4211 * \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{J}}
4212 * \mathsf{\mathbf{K}}_{\widetilde{p}\widetilde{J}}^{-1}\bigr]}_{\overline{\mathsf{\mathbf{K}}}}\bigl[
4213 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4214 * - \mathsf{\mathbf{K}}_{\widetilde{p}u} d
4215 * \mathsf{\mathbf{u}} \bigr]
4219 * \underbrace{\bigl[ \mathsf{\mathbf{K}}_{uu} +
4220 * \overline{\overline{\mathsf{\mathbf{K}}}}~ \bigr]
4221 * }_{\mathsf{\mathbf{K}}_{\textrm{con}}} d
4222 * \mathsf{\mathbf{u}}
4226 * \mathsf{\mathbf{F}}_{u}
4227 * - \mathsf{\mathbf{K}}_{u\widetilde{p}} \bigl[
4228 * \mathsf{\mathbf{K}}_{\widetilde{J}\widetilde{p}}^{-1}
4229 * \mathsf{\mathbf{F}}_{\widetilde{J}}
4231 * \overline{\mathsf{\mathbf{K}}}\mathsf{\mathbf{F}}_{\widetilde{p}}
4233 * \Bigr]}_{\mathsf{\mathbf{F}}_{\textrm{con}}}
4237 * \overline{\overline{\mathsf{\mathbf{K}}}} \dealcoloneq
4238 * \mathsf{\mathbf{K}}_{u\widetilde{p}}
4239 * \overline{\mathsf{\mathbf{K}}}
4240 * \mathsf{\mathbf{K}}_{\widetilde{p}u} \, .
4245 * At the top, we allocate two temporary vectors to help with the
4246 * static condensation, and variables to store the number of
4247 * linear solver iterations and the (hopefully converged) residual.
4250 * BlockVector<double> A(dofs_per_block);
4251 * BlockVector<double> B(dofs_per_block);
4256 * In the first step of this function, we solve for the incremental
4257 * displacement @f$d\mathbf{u}@f$. To this end, we perform static
4258 * condensation to make
4259 * @f$\mathsf{\mathbf{K}}_{\textrm{con}}
4260 * = \bigl[ \mathsf{\mathbf{K}}_{uu} +
4261 * \overline{\overline{\mathsf{\mathbf{K}}}}~ \bigr]@f$
4263 * @f$\mathsf{\mathbf{K}}^{-1}_{\widetilde{p} \widetilde{J}}@f$
4264 * in the original @f$\mathsf{\mathbf{K}}_{\widetilde{p} \widetilde{J}}@f$
4265 * block. That is, we make @f$\mathsf{\mathbf{K}}_{\textrm{store}}@f$.
4274 * \mathsf{\mathbf{A}}_{\widetilde{J}}
4276 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{p} \widetilde{J}}
4277 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4281 * tangent_matrix.block(p_dof, J_dof)
4282 * .vmult(A.block(J_dof), system_rhs.block(p_dof));
4286 * \mathsf{\mathbf{B}}_{\widetilde{J}}
4288 * \mathsf{\mathbf{K}}_{\widetilde{J} \widetilde{J}}
4289 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{p} \widetilde{J}}
4290 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4294 * tangent_matrix.block(J_dof, J_dof)
4295 * .vmult(B.block(J_dof), A.block(J_dof));
4299 * \mathsf{\mathbf{A}}_{\widetilde{J}}
4301 * \mathsf{\mathbf{F}}_{\widetilde{J}}
4303 * \mathsf{\mathbf{K}}_{\widetilde{J} \widetilde{J}}
4304 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{p} \widetilde{J}}
4305 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4309 * A.block(J_dof) = system_rhs.block(J_dof);
4310 * A.block(J_dof) -= B.block(J_dof);
4314 * \mathsf{\mathbf{A}}_{\widetilde{J}}
4316 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{J} \widetilde{p}}
4318 * \mathsf{\mathbf{F}}_{\widetilde{J}}
4320 * \mathsf{\mathbf{K}}_{\widetilde{J} \widetilde{J}}
4321 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{p} \widetilde{J}}
4322 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4327 * tangent_matrix.block(p_dof, J_dof)
4328 * .Tvmult(A.block(p_dof), A.block(J_dof));
4332 * \mathsf{\mathbf{A}}_{u}
4334 * \mathsf{\mathbf{K}}_{u \widetilde{p}}
4335 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{J} \widetilde{p}}
4337 * \mathsf{\mathbf{F}}_{\widetilde{J}}
4339 * \mathsf{\mathbf{K}}_{\widetilde{J} \widetilde{J}}
4340 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{p} \widetilde{J}}
4341 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4346 * tangent_matrix.block(u_dof, p_dof)
4347 * .vmult(A.block(u_dof), A.block(p_dof));
4351 * \mathsf{\mathbf{F}}_{\text{con}}
4353 * \mathsf{\mathbf{F}}_{u}
4355 * \mathsf{\mathbf{K}}_{u \widetilde{p}}
4356 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{J} \widetilde{p}}
4358 * \mathsf{\mathbf{F}}_{\widetilde{J}}
4360 * \mathsf{\mathbf{K}}_{\widetilde{J} \widetilde{J}}
4361 * \mathsf{\mathbf{K}}^{-1}_{\widetilde{p} \widetilde{J}}
4362 * \mathsf{\mathbf{F}}_{\widetilde{p}}
4367 * system_rhs.block(u_dof) -= A.block(u_dof);
4369 * timer.enter_subsection("Linear solver");
4370 * std::cout << " SLV " << std::flush;
4371 * if (parameters.type_lin == "CG")
4373 * const auto solver_its = static_cast<unsigned int>(
4374 * tangent_matrix.block(u_dof, u_dof).m() *
4375 * parameters.max_iterations_lin);
4376 * const double tol_sol =
4377 * parameters.tol_lin * system_rhs.block(u_dof).l2_norm();
4379 * SolverControl solver_control(solver_its, tol_sol);
4381 * GrowingVectorMemory<Vector<double>> GVM;
4382 * SolverCG<Vector<double>> solver_CG(solver_control, GVM);
4386 * We've chosen by
default a SSOR preconditioner as it appears to
4387 * provide the fastest solver convergence characteristics
for this
4388 * problem on a single-thread machine. However,
this might not be
4389 *
true for different problem sizes.
4393 * preconditioner(parameters.preconditioner_type,
4394 * parameters.preconditioner_relaxation);
4395 * preconditioner.use_matrix(tangent_matrix.block(u_dof, u_dof));
4397 * solver_CG.solve(tangent_matrix.block(u_dof, u_dof),
4398 * newton_update.block(u_dof),
4399 * system_rhs.block(u_dof),
4402 * lin_it = solver_control.last_step();
4403 * lin_res = solver_control.last_value();
4405 *
else if (parameters.type_lin ==
"Direct")
4409 * Otherwise
if the problem is small
4410 * enough, a direct solver can be
4415 * A_direct.
initialize(tangent_matrix.block(u_dof, u_dof));
4416 * A_direct.vmult(newton_update.block(u_dof),
4417 * system_rhs.block(u_dof));
4423 *
Assert(
false, ExcMessage(
"Linear solver type not implemented"));
4425 * timer.leave_subsection();
4430 * Now that we have the displacement update, distribute the constraints
4431 * back to the Newton update:
4434 * constraints.distribute(newton_update);
4436 * timer.enter_subsection(
"Linear solver postprocessing");
4437 * std::cout <<
" PP " << std::flush;
4441 * The next step after solving the displacement
4442 * problem is to post-process to get the
4443 * dilatation solution from the
4446 * d \widetilde{\mathsf{\mathbf{J}}}
4447 * = \mathsf{\mathbf{
K}}_{\widetilde{p}\widetilde{J}}^{-1} \bigl[
4448 * \mathsf{\mathbf{
F}}_{\widetilde{p}}
4449 * - \mathsf{\mathbf{
K}}_{\widetilde{p}u} d \mathsf{\mathbf{u}}
4458 * \mathsf{\mathbf{A}}_{\widetilde{p}}
4460 * \mathsf{\mathbf{
K}}_{\widetilde{p}u} d \mathsf{\mathbf{u}}
4464 * tangent_matrix.block(p_dof, u_dof)
4465 * .vmult(A.block(p_dof), newton_update.block(u_dof));
4469 * \mathsf{\mathbf{A}}_{\widetilde{p}}
4471 * -\mathsf{\mathbf{
K}}_{\widetilde{p}u} d \mathsf{\mathbf{u}}
4475 * A.block(p_dof) *= -1.0;
4479 * \mathsf{\mathbf{A}}_{\widetilde{p}}
4481 * \mathsf{\mathbf{
F}}_{\widetilde{p}}
4482 * -\mathsf{\mathbf{
K}}_{\widetilde{p}u} d \mathsf{\mathbf{u}}
4486 * A.block(p_dof) += system_rhs.block(p_dof);
4490 * d\mathsf{\mathbf{\widetilde{J}}}
4492 * \mathsf{\mathbf{
K}}^{-1}_{\widetilde{p}\widetilde{J}}
4494 * \mathsf{\mathbf{
F}}_{\widetilde{p}}
4495 * -\mathsf{\mathbf{
K}}_{\widetilde{p}u} d \mathsf{\mathbf{u}}
4500 * tangent_matrix.block(p_dof, J_dof)
4501 * .vmult(newton_update.block(J_dof), A.block(p_dof));
4506 * we ensure here that any Dirichlet constraints
4507 * are distributed on the updated solution:
4510 * constraints.distribute(newton_update);
4514 * Finally we solve
for the pressure
4515 * update with the substitution:
4517 * d \widetilde{\mathsf{\mathbf{p}}}
4519 * \mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{p}}^{-1}
4521 * \mathsf{\mathbf{
F}}_{\widetilde{J}}
4522 * - \mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{J}}
4523 * d \widetilde{\mathsf{\mathbf{J}}}
4532 * \mathsf{\mathbf{A}}_{\widetilde{J}}
4534 * \mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{J}}
4535 * d \widetilde{\mathsf{\mathbf{J}}}
4539 * tangent_matrix.block(J_dof, J_dof)
4540 * .vmult(A.block(J_dof), newton_update.block(J_dof));
4544 * \mathsf{\mathbf{A}}_{\widetilde{J}}
4546 * -\mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{J}}
4547 * d \widetilde{\mathsf{\mathbf{J}}}
4551 * A.block(J_dof) *= -1.0;
4555 * \mathsf{\mathbf{A}}_{\widetilde{J}}
4557 * \mathsf{\mathbf{
F}}_{\widetilde{J}}
4559 * \mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{J}}
4560 * d \widetilde{\mathsf{\mathbf{J}}}
4564 * A.block(J_dof) += system_rhs.block(J_dof);
4569 * d \widetilde{\mathsf{\mathbf{p}}}
4571 * \mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{p}}^{-1}
4573 * \mathsf{\mathbf{
F}}_{\widetilde{J}}
4574 * - \mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{J}}
4575 * d \widetilde{\mathsf{\mathbf{J}}}
4580 * tangent_matrix.block(p_dof, J_dof)
4581 * .Tvmult(newton_update.block(p_dof), A.block(J_dof));
4586 * We are now at the
end, so we distribute all
4587 * constrained dofs back to the Newton
4591 * constraints.distribute(newton_update);
4593 * timer.leave_subsection();
4597 * std::cout <<
" ------ " << std::flush;
4599 * timer.enter_subsection(
"Linear solver");
4600 * std::cout <<
" SLV " << std::flush;
4602 *
if (parameters.type_lin ==
"CG")
4606 * Manual condensation of the dilatation and pressure fields on
4607 * a local
level, and subsequent post-processing, took quite a
4608 * bit of effort to achieve. To recap, we had to produce the
4610 * @f$\mathsf{\mathbf{
K}}_{\widetilde{p}\widetilde{J}}^{-1}@f$, which
4611 * was permanently written into the global tangent
matrix. We then
4612 * permanently modified @f$\mathsf{\mathbf{
K}}_{uu}@f$ to produce
4613 * @f$\mathsf{\mathbf{
K}}_{\textrm{con}}@f$. This involved the
4614 * extraction and manipulation of local sub-blocks of the tangent
4615 *
matrix. After solving
for the displacement, the individual
4616 *
matrix-vector operations required to solve
for dilatation and
4617 * pressure were carefully implemented. Contrast these many sequence
4618 * of steps to the much simpler and transparent implementation
using
4623 * For ease of later use, we define some aliases
for
4624 * blocks in the RHS vector
4633 * ... and
for blocks in the Newton update vector.
4642 * We next define some linear operators
for the tangent
matrix
4643 * sub-blocks We will exploit the symmetry of the system, so not all
4644 * blocks are required.
4660 * We then construct a
LinearOperator that represents the inverse of
4662 * @f$\mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{p}}@f$. Since it is
4663 *
diagonal (or, when a higher order ansatz it used, nearly
4664 * diagonal), a Jacobi preconditioner is suitable.
4668 * preconditioner_K_Jp_inv(
"jacobi");
4669 * preconditioner_K_Jp_inv.use_matrix(
4670 * tangent_matrix.block(J_dof, p_dof));
4672 *
static_cast<unsigned int>(tangent_matrix.block(J_dof, p_dof).m() *
4673 * parameters.max_iterations_lin),
4675 * parameters.tol_lin);
4677 * solver_K_Jp_inv.
select(
"cg");
4678 * solver_K_Jp_inv.set_control(solver_control_K_Jp_inv);
4679 *
const auto K_Jp_inv =
4684 * Now we can construct that
transpose of
4685 * @f$\mathsf{\mathbf{
K}}_{\widetilde{J}\widetilde{p}}^{-1}@f$ and a
4686 * linear
operator that represents the condensed operations
4687 * @f$\overline{\mathsf{\mathbf{
K}}}@f$ and
4688 * @f$\overline{\overline{\mathsf{\mathbf{
K}}}}@f$ and the
final
4690 * @f$\mathsf{\mathbf{
K}}_{\textrm{con}}@f$.
4692 * here, but for clarity and the purpose of demonstrating the
4693 * similarities between the formulation and implementation of the
4694 * linear solution scheme, we will perform these operations
4699 * const auto K_pp_bar = K_Jp_inv * K_JJ * K_pJ_inv;
4700 * const auto K_uu_bar_bar = K_up * K_pp_bar * K_pu;
4701 * const auto K_uu_con = K_uu + K_uu_bar_bar;
4705 * Lastly, we define an operator for inverse of augmented stiffness
4706 * matrix, namely @f$\mathsf{\mathbf{
K}}_{\textrm{con}}^{-1}@f$. Note
4707 * that the preconditioner
for the augmented stiffness
matrix is
4708 * different to the
case when we use
static condensation. In
this
4709 * instance, the preconditioner is based on a non-modified
4710 * @f$\mathsf{\mathbf{
K}}_{uu}@f$,
while with the
first approach we
4711 * actually modified the entries of
this sub-block. However, since
4712 * @f$\mathsf{\mathbf{
K}}_{\textrm{con}}@f$ and
4713 * @f$\mathsf{\mathbf{
K}}_{uu}@f$ operate on the same space, it remains
4714 * adequate
for this problem.
4718 * preconditioner_K_con_inv(parameters.preconditioner_type,
4719 * parameters.preconditioner_relaxation);
4720 * preconditioner_K_con_inv.use_matrix(
4721 * tangent_matrix.block(u_dof, u_dof));
4723 *
static_cast<unsigned int>(tangent_matrix.block(u_dof, u_dof).m() *
4724 * parameters.max_iterations_lin),
4726 * parameters.tol_lin);
4728 * solver_K_con_inv.
select(
"cg");
4729 * solver_K_con_inv.set_control(solver_control_K_con_inv);
4730 *
const auto K_uu_con_inv =
4733 * preconditioner_K_con_inv);
4737 * Now we are in a position to solve
for the displacement field.
4738 * We can nest the linear operations, and the result is immediately
4739 * written to the Newton update vector.
4740 * It is clear that the implementation closely mimics the derivation
4741 * stated in the introduction.
4745 * K_uu_con_inv * (f_u - K_up * (K_Jp_inv * f_J - K_pp_bar * f_p));
4747 * timer.leave_subsection();
4751 * The operations need to post-process
for the dilatation and
4752 * pressure fields are just as easy to express.
4755 * timer.enter_subsection(
"Linear solver postprocessing");
4756 * std::cout <<
" PP " << std::flush;
4758 * d_J = K_pJ_inv * (f_p - K_pu * d_u);
4759 * d_p = K_Jp_inv * (f_J - K_JJ * d_J);
4761 * lin_it = solver_control_K_con_inv.last_step();
4762 * lin_res = solver_control_K_con_inv.last_value();
4764 *
else if (parameters.type_lin ==
"Direct")
4768 * Solve the full block system with
4769 * a direct solver. As it is relatively
4770 * robust, it may be immune to problem
4771 * arising from the presence of the zero
4772 * @f$\mathsf{\mathbf{
K}}_{ \widetilde{J} \widetilde{J}}@f$
4778 * A_direct.vmult(newton_update, system_rhs);
4783 * std::cout <<
" -- " << std::flush;
4786 *
Assert(
false, ExcMessage(
"Linear solver type not implemented"));
4788 * timer.leave_subsection();
4792 * Finally, we again ensure here that any Dirichlet
4793 * constraints are distributed on the updated solution:
4796 * constraints.distribute(newton_update);
4799 *
return std::make_pair(lin_it, lin_res);
4805 * <a name=
"step_44-Solidoutput_results"></a>
4806 * <h4>Solid::output_results</h4>
4807 * Here we present how the results are written to file to be viewed
4808 *
using ParaView or VisIt. The method is similar to that shown in previous
4809 * tutorials so will not be discussed in detail.
4813 * @note As of 2023, Visit 3.3.3 can still not deal with higher-order cells.
4814 * Rather, it simply reports that there is no
data to show. To view the
4815 * results of
this program with Visit, you will want to comment out the
4816 * line that sets `output_flags.write_higher_order_cells =
true;`. On the
4817 * other hand, Paraview is able to understand VTU files with higher order
4821 *
template <
int dim>
4822 *
void Solid<dim>::output_results() const
4825 * std::vector<DataComponentInterpretation::DataComponentInterpretation>
4826 * data_component_interpretation(
4828 * data_component_interpretation.push_back(
4830 * data_component_interpretation.push_back(
4833 * std::vector<std::string> solution_name(dim,
"displacement");
4834 * solution_name.emplace_back(
"pressure");
4835 * solution_name.emplace_back(
"dilatation");
4839 * output_flags.physical_units[
"displacement"] =
"m";
4840 * data_out.set_flags(output_flags);
4842 * data_out.attach_dof_handler(dof_handler);
4843 * data_out.add_data_vector(solution_n,
4846 * data_component_interpretation);
4850 * Since we are dealing with a large deformation problem, it would be nice
4852 * linked with the
DataOut class provides an interface through which this
4853 * can be achieved without physically moving the grid points in the
4855 * a temporary vector and then create the Eulerian mapping. We also
4856 * specify the polynomial degree to the
DataOut object in order to produce
4857 * a more refined output
data set when higher order polynomials are used.
4861 *
for (
unsigned int i = 0; i < soln.size(); ++i)
4862 * soln(i) = solution_n(i);
4864 * data_out.build_patches(q_mapping, degree);
4866 * std::ofstream output(
"solution-" + std::to_string(dim) +
"d-" +
4867 * std::to_string(time.get_timestep()) +
".vtu");
4868 * data_out.write_vtu(output);
4877 * <a name=
"step_44-Mainfunction"></a>
4878 * <h3>Main function</h3>
4879 * Lastly we provide the main driver function which appears
4880 * no different to the other tutorials.
4885 *
using namespace Step44;
4889 *
const unsigned int dim = 3;
4890 * Solid<dim> solid(
"parameters.prm");
4893 *
catch (std::exception &exc)
4895 * std::cerr << std::endl
4897 * <<
"----------------------------------------------------"
4899 * std::cerr <<
"Exception on processing: " << std::endl
4900 * << exc.what() << std::endl
4901 * <<
"Aborting!" << std::endl
4902 * <<
"----------------------------------------------------"
4909 * std::cerr << std::endl
4911 * <<
"----------------------------------------------------"
4913 * std::cerr <<
"Unknown exception!" << std::endl
4914 * <<
"Aborting!" << std::endl
4915 * <<
"----------------------------------------------------"
4923<a name=
"step_44-Results"></a><h1>Results</h1>
4926Firstly, we present a comparison of a series of 3-
d results with those
4927in the literature (see Reese et al (2000)) to demonstrate that the program works as expected.
4929We begin with a comparison of the convergence with mesh refinement for the @f$Q_1-DGPM_0-DGPM_0@f$ and
4930@f$Q_2-DGPM_1-DGPM_1@f$ formulations, as summarised in the figure below.
4931The vertical displacement of the midpoint of the upper surface of the block is used to assess convergence.
4932Both schemes demonstrate good convergence properties for varying values of the load parameter @f$p/p_0@f$.
4933The results agree with those in the literature.
4934The lower-order formulation typically overestimates the displacement for low levels of refinement,
4935while the higher-order interpolation scheme underestimates it, but be a lesser degree.
4936This benchmark, and a series of others not shown here, give us confidence that the code is working
4939<table align="center" class="tutorial" cellspacing="3" cellpadding="3">
4944 Convergence of the @f$Q_1-DGPM_0-DGPM_0@f$ formulation in 3-d.
4950 Convergence of the @f$Q_2-DGPM_1-DGPM_1@f$ formulation in 3-d.
4957A typical screen output generated by running the problem is shown below.
4958The particular case demonstrated is that of the @f$Q_2-DGPM_1-DGPM_1@f$ formulation.
4959It is clear that, using the Newton-Raphson method, quadratic convergence of the solution is obtained.
4960Solution convergence is achieved within 5 Newton increments for all time-steps.
4961The converged displacement's @f$L_2@f$-norm is several orders of magnitude less than the geometry scale.
4965 Reference volume: 1e-09
4967 Number of active cells: 64
4968 Number of degrees of freedom: 2699
4969 Setting up quadrature point
data...
4972___________________________________________________________________________________________________________________________________________________________
4973 SOLVER STEP | LIN_IT LIN_RES RES_NORM RES_U RES_P RES_J NU_NORM NU_U NU_P NU_J
4974___________________________________________________________________________________________________________________________________________________________
4975 0 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 786 2.118e-06 1.000e+00 1.000e+00 0.000e+00 0.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
4976 1 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 552 1.031e-03 8.563e-02 8.563e-02 9.200e-13 3.929e-08 1.060e-01 3.816e-02 1.060e-01 1.060e-01
4977 2 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 667 5.602e-06 2.482e-03 2.482e-03 3.373e-15 2.982e-10 2.936e-03 2.053e-04 2.936e-03 2.936e-03
4978 3 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 856 6.469e-10 2.129e-06 2.129e-06 2.245e-19 1.244e-13 1.887e-06 7.289e-07 1.887e-06 1.887e-06
4980___________________________________________________________________________________________________________________________________________________________
4982Displacement: 7.289e-07
4984Dilatation: 1.353e-07
4985v / V_0: 1.000e-09 / 1.000e-09 = 1.000e+00
4990Timestep 10 @ 1.000e+00s
4991___________________________________________________________________________________________________________________________________________________________
4992 SOLVER STEP | LIN_IT LIN_RES RES_NORM RES_U RES_P RES_J NU_NORM NU_U NU_P NU_J
4993___________________________________________________________________________________________________________________________________________________________
4994 0 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 874 2.358e-06 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00 1.000e+00
4995 1 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 658 2.942e-04 1.544e-01 1.544e-01 1.208e+13 1.855e+06 6.014e-02 7.398e-02 6.014e-02 6.014e-02
4996 2 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 790 2.206e-06 2.908e-03 2.908e-03 7.302e+10 2.067e+03 2.716e-03 1.433e-03 2.716e-03 2.717e-03
4997 3 ASM_R ASM_K CST ASM_SC SLV PP UQPH | 893 2.374e-09 1.919e-06 1.919e-06 4.527e+07 4.100e+00 1.672e-06 6.842e-07 1.672e-06 1.672e-06
4999___________________________________________________________________________________________________________________________________________________________
5001Displacement: 6.842e-07
5003Dilatation: 1.528e-06
5004v / V_0: 1.000e-09 / 1.000e-09 = 1.000e+00
5009Using the
Timer class, we can discern which parts of the code require the highest computational expense.
5010For a case with a large number of degrees-of-freedom (i.e. a high
level of refinement), a typical output of the
Timer is given below.
5011Much of the code in the tutorial has been developed based on the optimizations described,
5012discussed and demonstrated in @ref step_18 "step-18" and others.
5013With over 93% of the time being spent in the linear solver, it is obvious that it may be necessary
5014to invest in a better solver for large three-dimensional problems.
5015The SSOR preconditioner is not multithreaded but is effective for this class of solid problems.
5016It may be beneficial to investigate the use of another solver such as those available through the Trilinos library.
5020+---------------------------------------------+------------+------------+
5021| Total wallclock time elapsed since start | 9.874e+02s | |
5023| Section | no. calls | wall time | % of total |
5024+---------------------------------+-----------+------------+------------+
5025| Assemble system right-hand side | 53 | 1.727e+00s | 1.75e-01% |
5026| Assemble tangent matrix | 43 | 2.707e+01s | 2.74e+00% |
5027| Linear solver | 43 | 9.248e+02s | 9.37e+01% |
5028| Linear solver postprocessing | 43 | 2.743e-02s | 2.78e-03% |
5029| Perform static condensation | 43 | 1.437e+01s | 1.46e+00% |
5030| Setup system | 1 | 3.897e-01s | 3.95e-02% |
5031| Update QPH
data | 43 | 5.770e-01s | 5.84e-02% |
5032+---------------------------------+-----------+------------+------------+
5036We then used ParaView to visualize the results for two cases.
5037The
first was for the coarsest grid and the lowest-order interpolation method: @f$Q_1-DGPM_0-DGPM_0@f$.
5038The
second was on a refined grid using a @f$Q_2-DGPM_1-DGPM_1@f$ formulation.
5039The vertical component of the displacement, the pressure @f$\widetilde{p}@f$ and the dilatation @f$\widetilde{J}@f$ fields
5043For the
first case it is clear that the coarse spatial discretization coupled with large displacements leads to a low quality solution
5044(the loading ratio is @f$p/p_0=80@f$).
5045Additionally, the pressure difference between elements is very large.
5046The
constant pressure field on the element means that the large pressure
gradient is not captured.
5047However, it should be noted that locking, which would be present in a standard @f$Q_1@f$ displacement formulation does not arise
5048even in
this poorly discretised
case.
5049The
final vertical displacement of the tracked node on the top surface of the block is still within 12.5% of the converged solution.
5050The pressure solution is very coarse and has large jumps between adjacent cells.
5051It is clear that the
volume nearest to the applied traction undergoes compression
while the outer extents
5052of the domain are in a state of expansion.
5053The dilatation solution field and pressure field are clearly linked,
5054with
positive dilatation indicating regions of
positive pressure and
negative showing regions placed in compression.
5055As discussed in the Introduction, a compressive pressure has a
negative sign
5057This stems from the definition of the volumetric strain energy function
5058and is opposite to the physically realistic interpretation of pressure.
5061<table align=
"center" class=
"tutorial" cellspacing=
"3" cellpadding=
"3">
5064 <img src=
"https://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-displacement.png" alt=
"">
5066 Z-displacement solution
for the 3-
d problem.
5070 <img src=
"https://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-pressure.png" alt=
"">
5072 Discontinuous piece-wise
constant pressure field.
5076 <img src=
"https://www.dealii.org/images/steps/developer/step-44.Q1-P0_gr_1_p_ratio_80-dilatation.png" alt=
"">
5078 Discontinuous piece-wise
constant dilatation field.
5084Combining spatial refinement and a higher-order interpolation scheme results in a high-quality solution.
5085Three grid refinements coupled with a @f$Q_2-DGPM_1-DGPM_1@f$ formulation produces
5086a result that clearly captures the mechanics of the problem.
5087The deformation of the traction surface is well resolved.
5088We can now observe the actual extent of the applied traction, with the maximum force being applied
5089at the central
point of the surface causing the largest compression.
5090Even though very high strains are experienced in the domain,
5091especially at the boundary of the region of applied traction,
5092the solution remains accurate.
5093The pressure field is captured in far greater detail than before.
5094There is a clear distinction and transition between regions of compression and expansion,
5095and the linear approximation of the pressure field allows a refined visualization
5096of the pressure at the sub-element
scale.
5097It should however be noted that the pressure field remains discontinuous
5098and could be smoothed on a continuous grid
for the post-processing purposes.
5102<table align=
"center" class=
"tutorial" cellspacing=
"3" cellpadding=
"3">
5105 <img src=
"https://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-displacement.png" alt=
"">
5107 Z-displacement solution
for the 3-
d problem.
5111 <img src=
"https://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-pressure.png" alt=
"">
5113 Discontinuous linear pressure field.
5117 <img src=
"https://www.dealii.org/images/steps/developer/step-44.Q2-P1_gr_3_p_ratio_80-dilatation.png" alt=
"">
5119 Discontinuous linear dilatation field.
5125This brief analysis of the results demonstrates that the three-field formulation is effective
5126in circumventing volumetric locking
for highly-incompressible media.
5127The mixed formulation is able to accurately simulate the displacement of a
5128near-incompressible block under compression.
5129The command-line output indicates that the volumetric change under extreme compression resulted in
5130less than 0.01%
volume change
for a Poisson
's ratio of 0.4999.
5132In terms of run-time, the @f$Q_2-DGPM_1-DGPM_1@f$ formulation tends to be more computationally expensive
5133than the @f$Q_1-DGPM_0-DGPM_0@f$ for a similar number of degrees-of-freedom
5134(produced by adding an extra grid refinement level for the lower-order interpolation).
5135This is shown in the graph below for a batch of tests run consecutively on a single 4-core (8-thread) machine.
5136The increase in computational time for the higher-order method is likely due to
5137the increased band-width required for the higher-order elements.
5138As previously mentioned, the use of a better solver and preconditioner may mitigate the
5139expense of using a higher-order formulation.
5140It was observed that for the given problem using the multithreaded Jacobi preconditioner can reduce the
5141computational runtime by up to 72% (for the worst case being a higher-order formulation with a large number
5142of degrees-of-freedom) in comparison to the single-thread SSOR preconditioner.
5143However, it is the author's experience that the Jacobi method of preconditioning may not be suitable
for
5144some finite-strain problems involving alternative constitutive models.
5147<table align=
"center" class=
"tutorial" cellspacing=
"3" cellpadding=
"3">
5150 <img src=
"https://www.dealii.org/images/steps/developer/step-44.Normalised_runtime.png" alt=
"">
5152 Runtime on a 4-core machine, normalised against the lowest grid resolution @f$Q_1-DGPM_0-DGPM_0@f$ solution that utilised a SSOR preconditioner.
5159Lastly, results
for the displacement solution
for the 2-
d problem are showcased below
for
5160two different levels of grid refinement.
5161It is clear that due to the extra constraints imposed by simulating in 2-
d that the resulting
5162displacement field, although qualitatively similar, is different to that of the 3-
d case.
5165<table align=
"center" class=
"tutorial" cellspacing=
"3" cellpadding=
"3">
5168 <img src=
"https://www.dealii.org/images/steps/developer/step-44.2d-gr_2.png" alt=
"">
5170 Y-displacement solution in 2-
d for 2 global grid refinement levels.
5174 <img src=
"https://www.dealii.org/images/steps/developer/step-44.2d-gr_5.png" alt=
"">
5176 Y-displacement solution in 2-
d for 5 global grid refinement levels.
5182<a name=
"step-44-extensions"></a>
5183<a name=
"step_44-Possibilitiesforextensions"></a><h3>Possibilities
for extensions</h3>
5186There are a number of obvious extensions
for this work:
5188- Firstly, an additional constraint could be added to the free-energy
5189 function in order to enforce a high degree of incompressibility in
5190 materials. An additional Lagrange multiplier would be introduced,
5191 but
this could most easily be dealt with
using the principle of
5192 augmented Lagrange multipliers. This is demonstrated in <em>Simo and
5193 Taylor (1991) </em>.
5194- The constitutive relationship used in
this
5195 model is relatively basic. It may be beneficial to
split the material
5196 class into two separate classes, one dealing with the volumetric
5197 response and the other the isochoric response, and produce a generic
5198 materials class (i.
e. having abstract virtual
functions that derived
5199 classes have to implement) that would allow for the addition of more complex
5200 material models. Such models could include other hyperelastic
5201 materials, plasticity and viscoelastic materials and others.
5202- The program has been developed for solving problems on single-node
5203 multicore machines. With a little effort, the program could be
5204 extended to a large-
scale computing environment through the use of
5205 PETSc or Trilinos, using a similar technique to that demonstrated in
5206 @ref step_40
"step-40". This would mostly involve changes to the setup, assembly,
5207 <code>PointHistory</code> and linear solver routines.
5208- As this program assumes quasi-static equilibrium, extensions to
5209 include
dynamic effects would be necessary to study problems where
5210 inertial effects are important,
e.g. problems involving impact.
5211- Load and solution limiting procedures may be necessary for highly
5212 nonlinear problems. It is possible to add a linesearch algorithm to
5213 limit the step
size within a Newton increment to ensure optimum
5214 convergence. It may also be necessary to use a load limiting method,
5215 such as the Riks method, to solve unstable problems involving
5216 geometric instability such as buckling and snap-through.
5217- Many physical problems involve contact. It is possible to include
5218 the effect of frictional or frictionless contact between objects
5219 into this program. This would involve the addition of an extra term
5220 in the free-energy functional and therefore an addition to the
5221 assembly routine. One would also need to manage the contact problem
5222 (detection and stress calculations) itself. An alternative to
5223 additional penalty terms in the free-energy functional would be to
5224 use active set methods such as the one used in @ref step_41
"step-41".
5225- The complete condensation procedure using LinearOperators has been
5226 coded into the linear solver routine. This could also have been
5228 operator to condense out one or more of the fields in a more
5230- Finally, adaptive mesh refinement, as demonstrated in @ref step_6
"step-6" and
5231 @ref step_18
"step-18", could provide additional solution accuracy.
5234<a name=
"step_44-PlainProg"></a>
5235<h1> The plain program</h1>
5236@include
"step-44.cc"
void select(const std::string &name)
void initialize(const SparsityPattern &sparsity_pattern)
#define Assert(cond, exc)
LinearOperator< Range, Domain, Payload > linear_operator(const OperatorExemplar &, const Matrix &)
LinearOperator< Domain, Range, Payload > transpose_operator(const LinearOperator< Range, Domain, Payload > &op)
LinearOperator< Range_2, Domain_2, Payload > schur_complement(const LinearOperator< Domain_1, Range_1, Payload > &A_inv, const LinearOperator< Range_1, Domain_2, Payload > &B, const LinearOperator< Range_2, Domain_1, Payload > &C, const LinearOperator< Range_2, Domain_2, Payload > &D)
LinearOperator< Domain, Range, Payload > inverse_operator(const LinearOperator< Range, Domain, Payload > &op, Solver &solver, const Preconditioner &preconditioner)
void loop(IteratorType begin, std_cxx20::type_identity_t< IteratorType > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(std_cxx20::type_identity_t< DOFINFO > &, std_cxx20::type_identity_t< DOFINFO > &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, AssemblerType &assembler, const LoopControl &lctrl=LoopControl())
Task< RT > new_task(const std::function< RT()> &function)
std::vector< index_type > data
std::vector< value_type > split(const typename ::Triangulation< dim, spacedim >::cell_iterator &parent, const value_type parent_value)
CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt K
@ component_is_part_of_vector
Expression sign(const Expression &x)
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
void cell_matrix(FullMatrix< double > &M, const FEValuesBase< dim > &fe, const FEValuesBase< dim > &fetest, const ArrayView< const std::vector< double > > &velocity, const double factor=1.)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
SymmetricTensor< 2, dim, Number > C(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
Tensor< 2, dim, Number > F(const Tensor< 2, dim, Number > &Grad_u)
VectorType::value_type * end(VectorType &V)
bool check(const ConstraintKinds kind_in, const unsigned int dim)
DEAL_II_HOST constexpr TableIndices< 2 > merge(const TableIndices< 2 > &previous_indices, const unsigned int new_index, const unsigned int position)
void copy(const T *begin, const T *end, U *dest)
int(&) functions(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
bool write_higher_order_cells
1.9.8