746 * sparsity_pattern.copy_from(dsp);
748 * system_matrix.reinit(sparsity_pattern);
755 * <a name=
"ElasticProblemassemble_system"></a>
756 * <h4>ElasticProblem::assemble_system</h4>
760 * The big changes in
this program are in the creation of
matrix and right
761 * hand side, since they are problem-dependent. We will go through that
762 * process step-by-step, since it is a bit more complicated than in previous
767 * The
first parts of
this function are the same as before, however: setting
768 * up a suitable quadrature formula, initializing an
FEValues object for the
769 * (vector-valued) finite element we use as well as the quadrature
object,
770 * and declaring a number of auxiliary arrays. In addition, we declare the
771 * ever same two abbreviations: <code>n_q_points</code> and
772 * <code>dofs_per_cell</code>. The number of degrees of freedom per cell we
773 * now obviously ask from the composed finite element rather than from the
774 * underlying
scalar Q1 element. Here, it is <code>dim</code> times the
775 * number of degrees of freedom per cell of the Q1 element, though this is
776 * not explicit knowledge we need to care about:
780 * void ElasticProblem<dim>::assemble_system()
782 *
QGauss<dim> quadrature_formula(fe.degree + 1);
785 * quadrature_formula,
789 *
const unsigned int dofs_per_cell = fe.n_dofs_per_cell();
790 *
const unsigned int n_q_points = quadrature_formula.size();
795 * std::vector<types::global_dof_index> local_dof_indices(dofs_per_cell);
799 * As was shown in previous examples as well, we need a place where to
800 * store the
values of the coefficients at all the quadrature points on a
801 * cell. In the present situation, we have two coefficients,
lambda and
805 * std::vector<double> lambda_values(n_q_points);
806 * std::vector<double> mu_values(n_q_points);
810 * Well, we could as well have omitted the above two arrays since we will
811 * use
constant coefficients
for both
lambda and mu, which can be declared
813 *
value 1.0. Although we could omit the respective factors in the
814 * assemblage of the
matrix, we use them here
for purpose of
823 * right_hand_side just once per cell to make things simpler.
826 * std::vector<Tensor<1, dim>> rhs_values(n_q_points);
830 * Now we can
begin with the
loop over all cells:
833 *
for (
const auto &cell : dof_handler.active_cell_iterators())
838 * fe_values.reinit(cell);
842 * Next we get the
values of the coefficients at the quadrature
843 * points. Likewise
for the right hand side:
846 *
lambda.value_list(fe_values.get_quadrature_points(), lambda_values);
847 * mu.value_list(fe_values.get_quadrature_points(), mu_values);
848 * right_hand_side(fe_values.get_quadrature_points(), rhs_values);
852 * Then
assemble the entries of the local @ref GlossStiffnessMatrix
"stiffness matrix" and right
853 * hand side vector. This follows almost one-to-one the pattern
854 * described in the introduction of
this example. One of the few
855 * comments in place is that we can compute the number
856 * <code>comp(i)</code>, i.e. the
index of the only
nonzero vector
857 * component of shape function <code>i</code>
using the
858 * <code>fe.system_to_component_index(i).first</code> function
call
863 * (By accessing the <code>
first</code> variable of the
return value
864 * of the <code>system_to_component_index</code> function, you might
865 * already have guessed that there is more in it. In fact, the
866 * function returns a <code>std::pair@<
unsigned int,
unsigned
867 *
int@></code>, of which the
first element is <code>comp(i)</code>
868 * and the
second is the value <code>base(i)</code> also noted in the
869 * introduction, i.e. the
index of
this shape function within all the
871 * i.e. <code>base(i)</code> in the diction of the introduction. This
872 * is not a number that we are usually interested in, however.)
880 *
for (
const unsigned int i : fe_values.dof_indices())
882 * const unsigned
int component_i =
883 * fe.system_to_component_index(i).
first;
885 *
for (
const unsigned int j : fe_values.dof_indices())
887 * const unsigned
int component_j =
888 * fe.system_to_component_index(j).
first;
890 *
for (
const unsigned int q_point :
891 * fe_values.quadrature_point_indices())
896 * The
first term is @f$(\
lambda \partial_i u_i, \partial_j
897 * v_j) + (\mu \partial_i u_j, \partial_j v_i)@f$. Note
898 * that <code>shape_grad(i,q_point)</code> returns the
900 * shape function at quadrature
point q_point. The
901 * component <code>comp(i)</code> of the
gradient, which
902 * is the derivative of this only
nonzero vector
903 * component of the i-th shape function with respect to
904 * the comp(i)th coordinate is accessed by the appended
909 * (fe_values.shape_grad(i, q_point)[component_i] *
910 * fe_values.shape_grad(j, q_point)[component_j] *
911 * lambda_values[q_point])
913 * (fe_values.shape_grad(i, q_point)[component_j] *
914 * fe_values.shape_grad(j, q_point)[component_i] *
915 * mu_values[q_point])
919 * The
second term is @f$(\mu \nabla u_i, \nabla
920 * v_j)@f$. We need not access a specific component of
921 * the
gradient, since we only have to compute the
923 * overloaded version of <tt>operator*</tt> takes
924 * care, as in previous examples.
928 * Note that by using the <tt>?:</tt> operator, we only
929 * do this if <tt>component_i</tt> equals
930 * <tt>component_j</tt>, otherwise a zero is added
931 * (which will be optimized away by the compiler).
934 * ((component_i == component_j) ?
935 * (fe_values.shape_grad(i, q_point) *
936 * fe_values.shape_grad(j, q_point) *
937 * mu_values[q_point]) :
940 * fe_values.JxW(q_point);
947 * Assembling the right hand side is also just as discussed in the
951 *
for (
const unsigned int i : fe_values.dof_indices())
953 * const unsigned
int component_i =
954 * fe.system_to_component_index(i).
first;
956 *
for (
const unsigned int q_point :
957 * fe_values.quadrature_point_indices())
958 * cell_rhs(i) += fe_values.shape_value(i, q_point) *
959 * rhs_values[q_point][component_i] *
960 * fe_values.JxW(q_point);
965 * The transfer from local degrees of freedom into the global
matrix
966 * and right hand side vector does not depend on the equation under
967 * consideration, and is thus the same as in all previous
971 * cell->get_dof_indices(local_dof_indices);
972 * constraints.distribute_local_to_global(
973 * cell_matrix, cell_rhs, local_dof_indices, system_matrix, system_rhs);
982 * <a name=
"ElasticProblemsolve"></a>
983 * <h4>ElasticProblem::solve</h4>
987 * The solver does not care about where the system of equations comes, as
989 * requirements
for the use of the CG solver), which the system indeed
990 * is. Therefore, we need not change anything.
994 *
void ElasticProblem<dim>::solve()
1000 * preconditioner.
initialize(system_matrix, 1.2);
1002 * cg.solve(system_matrix, solution, system_rhs, preconditioner);
1004 * constraints.distribute(solution);
1011 * <a name=
"ElasticProblemrefine_grid"></a>
1012 * <h4>ElasticProblem::refine_grid</h4>
1016 * The function that does the refinement of the grid is the same as in the
1017 * @ref step_6
"step-6" example. The quadrature formula is adapted to the linear elements
1018 * again. Note that the error estimator by
default adds up the estimated
1019 * obtained from all components of the finite element solution, i.e., it
1020 * uses the displacement in all directions with the same weight. If we would
1021 * like the grid to be adapted to the x-displacement only, we could pass the
1022 * function an additional parameter which tells it to
do so and
do not
1023 * consider the displacements in all other directions
for the error
1024 * indicators. However,
for the current problem, it seems appropriate to
1025 * consider all displacement components with
equal weight.
1028 *
template <
int dim>
1029 *
void ElasticProblem<dim>::refine_grid()
1037 * estimated_error_per_cell);
1040 * estimated_error_per_cell,
1051 * <a name=
"ElasticProblemoutput_results"></a>
1052 * <h4>ElasticProblem::output_results</h4>
1056 * The output happens mostly as has been shown in previous examples
1057 * already. The only difference is that the solution function is vector
1058 * valued. The
DataOut class takes care of this automatically, but we have
1059 * to give each component of the solution vector a different name.
1063 * To
do this, the DataOut::add_vector() function wants a vector of
1064 * strings. Since the number of components is the same as the number
1065 * of dimensions we are working in, we use the <code>switch</code>
1070 * We note that some graphics programs have restriction on what
1071 * characters are allowed in the names of variables. deal.II therefore
1072 * supports only the minimal subset of these characters that is supported
1073 * by all programs. Basically, these are letters,
numbers, underscores,
1074 * and some other characters, but in particular no whitespace and
1075 * minus/hyphen. The library will throw an exception otherwise, at least
1080 * After listing the 1d, 2d, and 3d case, it is good style to let the
1081 * program die if we run upon a case which we did not consider. Remember
1082 * that the
Assert macro generates an exception if the condition in the
1083 *
first parameter is not satisfied. Of course, the condition
1084 * <code>false</code> can never be satisfied, so the program will always
1085 * abort whenever it gets to the default statement:
1088 * template <
int dim>
1089 *
void ElasticProblem<dim>::output_results(const
unsigned int cycle) const
1094 * std::vector<std::string> solution_names;
1098 * solution_names.emplace_back(
"displacement");
1101 * solution_names.emplace_back(
"x_displacement");
1102 * solution_names.emplace_back(
"y_displacement");
1105 * solution_names.emplace_back(
"x_displacement");
1106 * solution_names.emplace_back(
"y_displacement");
1107 * solution_names.emplace_back(
"z_displacement");
1110 *
Assert(
false, ExcNotImplemented());
1115 * After setting up the names
for the different components of the
1116 * solution vector, we can add the solution vector to the list of
1117 * data vectors scheduled
for output. Note that the following
1118 * function takes a vector of strings as
second argument, whereas
1119 * the one which we have used in all previous examples accepted a
1120 *
string there. (In fact, the function we had used before would
1121 * convert the single
string into a vector with only one element
1122 * and forwards that to the other function.)
1125 * data_out.add_data_vector(solution, solution_names);
1126 * data_out.build_patches();
1128 * std::ofstream output(
"solution-" + std::to_string(cycle) +
".vtk");
1129 * data_out.write_vtk(output);
1137 * <a name=
"ElasticProblemrun"></a>
1138 * <h4>ElasticProblem::run</h4>
1142 * The <code>
run</code> function does the same things as in @ref step_6
"step-6",
for
1143 * example. This time, we use the square [-1,1]^
d as domain, and we
refine
1144 * it globally four times before starting the
first iteration.
1148 * The reason
for refining is a bit accidental: we use the
QGauss
1149 * quadrature formula with two points in each direction
for integration of the
1150 * right hand side; that means that there are four quadrature points on each
1151 * cell (in 2d). If we only
refine the
initial grid once globally, then there
1152 * will be only four quadrature points in each direction on the
1153 * domain. However, the right hand side function was chosen to be rather
1154 * localized and in that
case, by pure chance, it happens that all quadrature
1155 * points lie at points where the right hand side function is zero (in
1156 * mathematical terms, the quadrature points happen to be at points outside
1157 * the <i>support</i> of the right hand side function). The right hand side
1158 * vector computed with quadrature will then contain only zeroes (even though
1159 * it would of course be nonzero
if we had computed the right hand side vector
1160 * exactly
using the integral) and the solution of the system of
1161 * equations is the zero vector, i.e., a finite element function that is zero
1162 * everywhere. In a sense, we
1163 * should not be surprised that
this is happening since we have chosen
1164 * an
initial grid that is totally unsuitable
for the problem at hand.
1168 * The unfortunate thing is that
if the discrete solution is
constant, then
1170 *
for each cell as well, and the
call to
1171 * Triangulation::refine_and_coarsen_fixed_number() will not flag any cells
1172 * for refinement (why should it if the indicated error is zero for each
1173 * cell?). The grid in the next iteration will therefore consist of four
1174 * cells only as well, and the same problem occurs again.
1178 * The conclusion needs to be: while of course we will not choose the
1179 * initial grid to be well-suited for the accurate solution of the problem,
1180 * we must at least choose it such that it has the chance to capture the
1181 * important features of the solution. In this case, it needs to be able to
1182 * see the right hand side. Thus, we refine globally four times. (Any larger
1183 * number of global refinement steps would of course also work.)
1186 * template <
int dim>
1187 *
void ElasticProblem<dim>::run()
1189 *
for (
unsigned int cycle = 0; cycle < 8; ++cycle)
1191 * std::cout <<
"Cycle " << cycle <<
':' << std::endl;
1201 * std::cout <<
" Number of active cells: "
1206 * std::cout <<
" Number of degrees of freedom: " << dof_handler.n_dofs()
1209 * assemble_system();
1211 * output_results(cycle);
1219 * <a name=
"Thecodemaincodefunction"></a>
1220 * <h3>The <code>main</code> function</h3>
1224 * After closing the <code>Step8</code>
namespace in the last line above, the
1225 * following is the main function of the program and is again exactly like in
1226 * @ref step_6
"step-6" (apart from the changed
class names, of course).
1233 * Step8::ElasticProblem<2> elastic_problem_2d;
1234 * elastic_problem_2d.run();
1236 *
catch (std::exception &exc)
1238 * std::cerr << std::endl
1240 * <<
"----------------------------------------------------"
1242 * std::cerr <<
"Exception on processing: " << std::endl
1243 * << exc.what() << std::endl
1244 * <<
"Aborting!" << std::endl
1245 * <<
"----------------------------------------------------"
1252 * std::cerr << std::endl
1254 * <<
"----------------------------------------------------"
1256 * std::cerr <<
"Unknown exception!" << std::endl
1257 * <<
"Aborting!" << std::endl
1258 * <<
"----------------------------------------------------"
1266<a name=
"Results"></a><h1>Results</h1>
1270There is not much to be said about the results of
this program, other than
1271that they look nice. All images were made
using VisIt from the
1272output files that the program wrote to disk. The
first two pictures show
1273the @f$x@f$- and @f$y@f$-displacements as
scalar components:
1278<img src=
"https://www.dealii.org/images/steps/developer/step-8.x.png" alt=
"">
1281<img src=
"https://www.dealii.org/images/steps/developer/step-8.y.png" alt=
"">
1287You can clearly see the sources of @f$x@f$-displacement around @f$x=0.5@f$ and
1288@f$x=-0.5@f$, and of @f$y@f$-displacement at the origin.
1290What one frequently would like to
do is to show the displacement as a vector
1291field, i.e., vectors that
for each point illustrate the direction and magnitude
1292of displacement. Unfortunately, that
's a bit more involved. To understand why
1293this is so, remember that we have just defined our finite element as a
1294collection of two components (in <code>dim=2</code> dimensions). Nowhere have
1295we said that this is not just a pressure and a concentration (two scalar
1296quantities) but that the two components actually are the parts of a
1297vector-valued quantity, namely the displacement. Absent this knowledge, the
1298DataOut class assumes that all individual variables we print are separate
1299scalars, and VisIt and Paraview then faithfully assume that this is indeed what it is. In
1300other words, once we have written the data as scalars, there is nothing in
1301these programs that allows us to paste these two scalar fields back together as a
1302vector field. Where we would have to attack this problem is at the root,
1303namely in <code>ElasticProblem::output_results()</code>. We won't
do so here but
1304instead refer the reader to the @ref step_22
"step-22" program where we show how to
do this
1305for a more
general situation. That said, we couldn
't help generating the data
1306anyway that would show how this would look if implemented as discussed in
1307@ref step_22 "step-22". The vector field then looks like this (VisIt and Paraview
1308randomly select a few
1309hundred vertices from which to draw the vectors; drawing them from each
1310individual vertex would make the picture unreadable):
1312<img src="https://www.dealii.org/images/steps/developer/step-8.vectors.png" alt="">
1315We note that one may have intuitively expected the
1316solution to be symmetric about the @f$x@f$- and @f$y@f$-axes since the @f$x@f$- and
1317@f$y@f$-forces are symmetric with respect to these axes. However, the force
1318considered as a vector is not symmetric and consequently neither is
1322<a name="PlainProg"></a>
1323<h1> The plain program</h1>
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
static void estimate(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const Quadrature< dim - 1 > &quadrature, const std::map< types::boundary_id, const Function< spacedim, typename InputVector::value_type > * > &neumann_bc, const InputVector &solution, Vector< float > &error, const ComponentMask &component_mask=ComponentMask(), const Function< spacedim > *coefficients=nullptr, const unsigned int n_threads=numbers::invalid_unsigned_int, const types::subdomain_id subdomain_id=numbers::invalid_subdomain_id, const types::material_id material_id=numbers::invalid_material_id, const Strategy strategy=cell_diameter_over_24)
void initialize(const MatrixType &A, const AdditionalData ¶meters=AdditionalData())
#define Assert(cond, exc)
void loop(ITERATOR begin, std_cxx20::type_identity_t< ITERATOR > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, ASSEMBLER &assembler, const LoopControl &lctrl=LoopControl())
@ update_values
Shape function values.
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
void hyper_cube(Triangulation< dim, spacedim > &tria, const double left=0., const double right=1., const bool colorize=false)
void refine(Triangulation< dim, spacedim > &tria, const Vector< Number > &criteria, const double threshold, const unsigned int max_to_mark=numbers::invalid_unsigned_int)
void refine_and_coarsen_fixed_number(Triangulation< dim, spacedim > &triangulation, const Vector< Number > &criteria, const double top_fraction_of_cells, const double bottom_fraction_of_cells, const unsigned int max_n_cells=std::numeric_limits< unsigned int >::max())
@ matrix
Contents is actually a matrix.
@ symmetric
Matrix is symmetric.
@ 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 > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
VectorType::value_type * begin(VectorType &V)
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
int(&) functions(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation