Solving Elasticity Problems

This tutorial is automatically generated from TestSolvingElasticityProblemsTutorial.hpp at revision 4d762407bc1f. Note that the code is given in full at the bottom of the page.

Solving solid mechanics problems

In this tutorial we show how Chaste can be used to solve solid mechanics problems. We assume the reader has some familiarity with solid mechanics problems (the equations of nonlinear elasticity are given in the PDF on equations and finite element implementations (see ChasteGuides –> Miscellaneous information)). It is also best to have had a look at the solving linear PDEs tutorials.

In brief, there several facets to solid mechanics models:

Time-dependent problems versus static problems
Linear elasticity versus nonlinear elasticity
Compressible versus incompressible materials
The type of material behaviour (elastic, visco-elastic, etc..)
Specification of geometry, material law, body force, displacement boundary conditions, and traction boundary conditions

The solvers currently implemented are STATIC (time-independent) and use NONLINEAR ELASTICITY. There are solvers for both INCOMPRESSIBLE and COMPRESSIBLE deformations. The material behaviour is assumed to be ELASTIC (stress is just a function of strain, not strain-rate etc), and in particular HYPER-ELASTIC (stress is a function of strain via a ‘strain energy function’, for which stress is obtained by differentiating the strain energy function with respect to strain).

To solve a mechanics problem we need to

Choose the solver (compressible or incompressible)
Specify the geometry (ie the mesh)
Specify the material law (ie the strain-energy function) (incompressible or compressible as appropriate).
Specify the BODY FORCE – this is a force density acting throughout the body (eg. acceleration due to gravity), and also the mass density.
Specify some DISPLACEMENT BOUNDARY CONDITIONS – some part of the boundary must have the displacement specified on it
Specify TRACTION BOUNDARY CONDITIONS (if non-zero) on the rest of the boundary – tractions are forces per unit area applied the rest of the surface of the deformable object.

VERY IMPORTANT NOTE: For incompressible problems, make sure you read the comment about HYPRE below before going to 3D or refining meshes in these tests.

Another note: mechanics problems are not currently implemented to scale in parallel yet.

As always we include this first class as a test suite

#include <cxxtest/TestSuite.h>

On some systems there is a clash between Boost Ublas includes and PETSc. This can be resolved by making sure that Chaste’s interface to the Boost libraries are included as early as possible.

#include "UblasCustomFunctions.hpp"

The incompressible solver is called IncompressibleNonlinearElasticitySolver

#include "IncompressibleNonlinearElasticitySolver.hpp"

The simplest incompressible material law is the Mooney-Rivlin material law (of which Neo-Hookean laws are a subset)

#include "MooneyRivlinMaterialLaw.hpp"

Another incompressible material law

#include "ExponentialMaterialLaw.hpp"

This is a useful helper class

#include "NonlinearElasticityTools.hpp"

For visualising results in Paraview

#include "VtkNonlinearElasticitySolutionWriter.hpp"

As before: PetscSetupAndFinalize.hpp must be included in every test that uses PETSc. Note that it cannot be included in the source code.

#include "PetscSetupAndFinalize.hpp"

Simple incompressible deformation: 2D shape hanging under gravity

class TestSolvingElasticityProblemsTutorial : public CxxTest::TestSuite
{
public:

In the first test we use INCOMPRESSIBLE nonlinear elasticity. For incompressible elasticity, there is a constraint on the deformation, which results in a pressure field (a Lagrange multiplier) which is solved for together with the deformation.

All the mechanics solvers solve for the deformation using the finite element method with QUADRATIC basis functions for the deformation. This necessitates the use of a QuadraticMesh - such meshes have extra nodes that aren’t vertices of elements, in this case midway along each edge. (The displacement is solved for at each node in the mesh (including internal [non-vertex] nodes), whereas the pressure is only solved for at each vertex - in FEM terms, quadratic interpolation for displacement, linear interpolation for pressure, which is required for stability. The pressure at internal nodes is computed by linear interpolation).

    void TestSimpleIncompressibleProblem()
    {

First, define the geometry. This should be specified using the QuadraticMesh class, which inherits from TetrahedralMesh and has mostly the same interface. Here we define a 0.8 by 1 rectangle, with elements 0.1 wide. (QuadraticMeshs can also be read in using TrianglesMeshReader; see next tutorial/rest of code base for examples of this).

        QuadraticMesh<2> mesh;
        mesh.ConstructRegularSlabMesh(0.1 /*stepsize*/, 0.8 /*width*/, 1.0 /*height*/);

We use a Mooney-Rivlin material law, which applies to isotropic materials and has two parameters. Restricted to 2D however, it only has one parameter, which can be thought of as the total stiffness. We declare a Mooney-Rivlin law, setting the parameter to 1.

        MooneyRivlinMaterialLaw<2> law(1.0);

Next, the body force density. In realistic problems this will either be acceleration due to gravity (ie b=(0,-9.81)) or zero if the effect of gravity can be neglected. In this problem we apply a gravity-like downward force.

        c_vector<double,2> body_force;
        body_force(0) =  0.0;
        body_force(1) = -2.0;

Two types of boundary condition are required: displacement and traction. As with the other PDE solvers, the displacement (Dirichlet) boundary conditions are specified at nodes, whereas traction (Neumann) boundary conditions are specified on boundary elements.

In this test we apply displacement boundary conditions on one surface of the mesh, the upper (Y=1.0) surface. We are going to specify zero-displacement at these nodes. We do not specify any traction boundary conditions, which means that (effectively) zero-traction boundary conditions (ie zero pressures) are applied on the three other surfaces.

We need to get a std::vector of all the node indices that we want to fix. The NonlinearElasticityTools has a static method for helping do this: the following gets all the nodes for which Y=1.0. The second argument (the ‘1’) indicates Y . (So, for example, GetNodesByComponentValue(mesh, 0, 10) would get the nodes on X=10).

        std::vector<unsigned> fixed_nodes = NonlinearElasticityTools<2>::GetNodesByComponentValue(mesh, 1, 1.0);

Before creating the solver we create a SolidMechanicsProblemDefinition object, which contains everything that defines the problem: mesh, material law, body force, the fixed nodes and their locations, any traction boundary conditions, and the density (which multiplies the body force, otherwise isn’t used).

        SolidMechanicsProblemDefinition<2> problem_defn(mesh);

Set the material problem on the problem definition object, saying that the problem, and the material law, is incompressible. All material law files can be found in continuum_mechanics/src/problem/material_laws.

        problem_defn.SetMaterialLaw(INCOMPRESSIBLE,&law);

Set the fixed nodes, choosing zero displacement for these nodes (see later for how to provide locations for the fixed nodes).

        problem_defn.SetZeroDisplacementNodes(fixed_nodes);

Set the body force and the density. (Note that the second line isn’t technically needed, as internally the density is initialised to 1)

        problem_defn.SetBodyForce(body_force);
        problem_defn.SetDensity(1.0);

Now we create the (incompressible) solver, passing in the mesh, problem definition and output directory

        IncompressibleNonlinearElasticitySolver<2> solver(mesh,
                                                          problem_defn,
                                                          "SimpleIncompressibleElasticityTutorial");

.. and to compute the solution, just call Solve()

        solver.Solve();

Visualisation. Go to the folder SimpleIncompressibleElasticityTutorial in your test-output directory. There should be 2 files, initial.nodes and solution.nodes. These are the original nodal positions and the deformed positions. Each file has two columns, the x and y locations of each node. To visualise the solution in say Matlab or Octave, you could do: x=load('solution.nodes'); plot(x(:,1),x(:,2),'k*'). For Cmgui output, see below.

To get the actual solution from the solver, use these two methods. Note that the first gets the deformed position (ie the new location, not the displacement). They are both of size num_total_nodes.

        std::vector<c_vector<double,2> >& r_deformed_positions = solver.rGetDeformedPosition();
        std::vector<double>& r_pressures = solver.rGetPressures();

Let us obtain the values of the new position, and the pressure, at the bottom right corner node.

        unsigned node_index = 8;
        assert( fabs(mesh.GetNode(node_index)->rGetLocation()[0] - 0.8) < 1e-6); // check that X=0.8, ie that we have the correct node,
        assert( fabs(mesh.GetNode(node_index)->rGetLocation()[1] - 0.0) < 1e-6); // check that Y=0.0, ie that we have the correct node,
        std::cout << "New position: " << r_deformed_positions[node_index](0) << " " << r_deformed_positions[node_index](1) << "\n";
        std::cout << "Pressure: " << r_pressures[node_index] << "\n";

One visualiser is Cmgui. This method can be used to convert all the output files to Cmgui format. They are placed in [OUTPUT_DIRECTORY]/cmgui. A script is created to easily load the data: in a terminal cd to this directory and call cmgui LoadSolutions.com. (In this directory, the initial position is given by solution_0.exnode, the deformed by solution_1.exnode).

        solver.CreateCmguiOutput();

The recommended visualiser is Paraview, for which Chaste must be installed with VTK. With paraview, strains (and in the future stresses) can be visualised on the undeformed/deformed geometry). We can create VTK output using the VtkNonlinearElasticitySolutionWriter class. The undeformed geometry, solution displacement, and pressure (if incompressible problem) are written to file, and below we also choose to write the deformation tensor C for each element.

        VtkNonlinearElasticitySolutionWriter<2> vtk_writer(solver);
        vtk_writer.SetWriteElementWiseStrains(DEFORMATION_TENSOR_C); // other options are DEFORMATION_GRADIENT_F and LAGRANGE_STRAIN_E
        vtk_writer.Write();

These are just to check that nothing has been accidentally changed in this test. Newton’s method (with damping) was used to solve the nonlinear problem, and we check that 4 iterations were needed to converge.

        TS_ASSERT_DELTA(r_deformed_positions[node_index](0),  0.7980, 1e-3);
        TS_ASSERT_DELTA(r_deformed_positions[node_index](1), -0.1129, 1e-3);
        TS_ASSERT_EQUALS(solver.GetNumNewtonIterations(), 4u);
    }

Exercise: convert to a compressible solver and compare the resultant deformations. The next tutorial describes how to solve for a compressible deformation, but the changes are essentially trivial: IncompressibleNonlinearElasticitySolver needs to be changed to CompressibleNonlinearElasticitySolver, the line problem_defn.SetMaterialLaw(..) needs changing, and the material law itself should be of type AbstractCompressibleMaterialLaw. An example is CompressibleMooneyRivlinMaterialLaw. Also solver.rGetPressures() doesn’t exist (or make sense) when the solver is an CompressibleNonlinearElasticitySolver.

Incompressible deformation: 2D shape hanging under gravity with a balancing traction

We now repeat the above test but include a traction on the bottom surface (Y=0). We apply this in the inward direction so that is counters (somewhat) the effect of gravity. We also show how stresses and strains can be written to file.

    void TestIncompressibleProblemWithTractions()
    {

All of this is exactly as above

        QuadraticMesh<2> mesh;
        mesh.ConstructRegularSlabMesh(0.1 /*stepsize*/, 0.8 /*width*/, 1.0 /*height*/);

        MooneyRivlinMaterialLaw<2> law(1.0);

        c_vector<double,2> body_force;
        body_force(0) =  0.0;
        body_force(1) = -2.0;

        std::vector<unsigned> fixed_nodes = NonlinearElasticityTools<2>::GetNodesByComponentValue(mesh, 1, 1.0);

Now the traction boundary conditions. We need to collect all the boundary elements on the surface which we want to apply non-zero tractions, put them in a std::vector, and create a corresponding std::vector of the tractions for each of the boundary elements. Note that the each traction is a 2D vector with dimensions of pressure.

First, declare the data structures:

        std::vector<BoundaryElement<1,2>*> boundary_elems;
        std::vector<c_vector<double,2> > tractions;

Create a constant traction

        c_vector<double,2> traction;
        traction(0) = 0;
        traction(1) = 1.0; // this choice of sign corresponds to an inward force (if applied to the bottom surface)

Loop over boundary elements

        for (TetrahedralMesh<2,2>::BoundaryElementIterator iter = mesh.GetBoundaryElementIteratorBegin();
             iter != mesh.GetBoundaryElementIteratorEnd();
             ++iter)
        {

If the centre of the element has Y value of 0.0, it is on the surface we need

            if (fabs((*iter)->CalculateCentroid()[1] - 0.0) < 1e-6)
            {

Put the boundary element and the constant traction into the stores.

                BoundaryElement<1,2>* p_element = *iter;
                boundary_elems.push_back(p_element);
                tractions.push_back(traction);
            }
        }

A quick check

        assert(boundary_elems.size() == 8u);

Now create the problem definition object, setting the material law, fixed nodes and body force as before (this time not calling SetDensity(), so using the default density of 1.0, and also calling a method for setting tractions, which takes in the boundary elements and tractions for each of those elements.

        SolidMechanicsProblemDefinition<2> problem_defn(mesh);
        problem_defn.SetMaterialLaw(INCOMPRESSIBLE,&law);
        problem_defn.SetZeroDisplacementNodes(fixed_nodes);
        problem_defn.SetBodyForce(body_force);
        problem_defn.SetTractionBoundaryConditions(boundary_elems, tractions);

Create solver as before

        IncompressibleNonlinearElasticitySolver<2> solver(mesh,
                                                          problem_defn,
                                                          "IncompressibleElasticityWithTractionsTutorial");

In this test we also output the stress and strain. For the former, we have to tell the solver to store the stresses that are computed during the solve.

        solver.SetComputeAverageStressPerElementDuringSolve();

Call Solve()

        solver.Solve();

If VTK output is written (discussed above) strains can be visualised. Alternatively, we can create text files for strains and stresses by doing the following.

Write the final deformation gradients to file. The i-th line of this file provides the deformation gradient F, written as ‘F(0,0) F(0,1) F(1,0) F(1,1)’, evaluated at the centroid of the i-th element. The first variable can also be DEFORMATION_TENSOR_C or LAGRANGE_STRAIN_E to write C or E. The second parameter is the file name.

        solver.WriteCurrentStrains(DEFORMATION_GRADIENT_F,"deformation_grad");

Since we called SetComputeAverageStressPerElementDuringSolve, we can write the stresses to file too. However, note that for each element this is not the stress evaluated at the centroid, but the mean average of the stresses evaluated at the quadrature points - for technical cardiac electromechanics reasons, it is difficult to define the stress at non-quadrature points.

        solver.WriteCurrentAverageElementStresses("2nd_PK_stress");

Another quick check

        TS_ASSERT_EQUALS(solver.GetNumNewtonIterations(), 4u);

Visualise as before by going to the output directory and doing x=load('solution.nodes'); plot(x(:,1),x(:,2),'m*') in Matlab/octave, or by using Cmgui. The effect of the traction should be clear (especially when compared to the results of the first test).

Create Cmgui output

        solver.CreateCmguiOutput();

This is just to check that nothing has been accidentally changed in this test

        TS_ASSERT_DELTA(solver.rGetDeformedPosition()[8](0), 0.8561, 1e-3);
        TS_ASSERT_DELTA(solver.rGetDeformedPosition()[8](1), 0.0310, 1e-3);
    }
};

More examples are given in the next tutorial

IMPORTANT: Using HYPRE

Mechanics solves being nonlinear are expensive, so it is recommended you also use CMAKE_BUILD_TYPE=Release when running with CMake on larger problems.

When running incompressible problems in 3D, or with more elements, it is vital to also change the linear solver to use HYPRE, an algebraic multigrid solver. Without HYRPE, the linear solve (i) may become very very slow; or (ii) may not converge, in which case the nonlinear solve will (probably) not converge. HYPRE is (currently) not a pre-requisite for installing Chaste, hence this is not (currently) the default linear solver for incompressible mechanics problems, although this will change in the future.

HYPRE should be considered a pre-requisite for large incompressible mechanics problems.

To use HYPRE, you need to have PETSc installed with HYPRE. However, if you followed installation instructions for Chaste 2.1 or later, you probably do already have PETSc installed with HYPRE.

To switch on HYPRE, open the file continuum_mechanics/src/solver/AbstractNonlinearElasticitySolver and uncomment the line #define MECH_USE_HYPRE near the top of the file (currently: line 57).

Note: PETSc unfortunately doesn’t quit if you try to use HYPRE without it being installed, but it spew lots of error messages.

Full code

#include <cxxtest/TestSuite.h>
#include "UblasCustomFunctions.hpp"
#include "IncompressibleNonlinearElasticitySolver.hpp"
#include "MooneyRivlinMaterialLaw.hpp"
#include "ExponentialMaterialLaw.hpp"
#include "NonlinearElasticityTools.hpp"
#include "VtkNonlinearElasticitySolutionWriter.hpp"
#include "PetscSetupAndFinalize.hpp"

class TestSolvingElasticityProblemsTutorial : public CxxTest::TestSuite
{
public:
    void TestSimpleIncompressibleProblem()
    {
        QuadraticMesh<2> mesh;
        mesh.ConstructRegularSlabMesh(0.1 /*stepsize*/, 0.8 /*width*/, 1.0 /*height*/);

        MooneyRivlinMaterialLaw<2> law(1.0);

        c_vector<double,2> body_force;
        body_force(0) =  0.0;
        body_force(1) = -2.0;

        std::vector<unsigned> fixed_nodes = NonlinearElasticityTools<2>::GetNodesByComponentValue(mesh, 1, 1.0);

        SolidMechanicsProblemDefinition<2> problem_defn(mesh);

        problem_defn.SetMaterialLaw(INCOMPRESSIBLE,&law);

        problem_defn.SetZeroDisplacementNodes(fixed_nodes);
        problem_defn.SetBodyForce(body_force);
        problem_defn.SetDensity(1.0);

        IncompressibleNonlinearElasticitySolver<2> solver(mesh,
                                                          problem_defn,
                                                          "SimpleIncompressibleElasticityTutorial");

        solver.Solve();

        std::vector<c_vector<double,2> >& r_deformed_positions = solver.rGetDeformedPosition();
        std::vector<double>& r_pressures = solver.rGetPressures();
        unsigned node_index = 8;
        assert( fabs(mesh.GetNode(node_index)->rGetLocation()[0] - 0.8) < 1e-6); // check that X=0.8, ie that we have the correct node,
        assert( fabs(mesh.GetNode(node_index)->rGetLocation()[1] - 0.0) < 1e-6); // check that Y=0.0, ie that we have the correct node,
        std::cout << "New position: " << r_deformed_positions[node_index](0) << " " << r_deformed_positions[node_index](1) << "\n";
        std::cout << "Pressure: " << r_pressures[node_index] << "\n";

        solver.CreateCmguiOutput();

        VtkNonlinearElasticitySolutionWriter<2> vtk_writer(solver);
        vtk_writer.SetWriteElementWiseStrains(DEFORMATION_TENSOR_C); // other options are DEFORMATION_GRADIENT_F and LAGRANGE_STRAIN_E
        vtk_writer.Write();

        TS_ASSERT_DELTA(r_deformed_positions[node_index](0),  0.7980, 1e-3);
        TS_ASSERT_DELTA(r_deformed_positions[node_index](1), -0.1129, 1e-3);
        TS_ASSERT_EQUALS(solver.GetNumNewtonIterations(), 4u);
    }
    void TestIncompressibleProblemWithTractions()
    {
        QuadraticMesh<2> mesh;
        mesh.ConstructRegularSlabMesh(0.1 /*stepsize*/, 0.8 /*width*/, 1.0 /*height*/);

        MooneyRivlinMaterialLaw<2> law(1.0);

        c_vector<double,2> body_force;
        body_force(0) =  0.0;
        body_force(1) = -2.0;

        std::vector<unsigned> fixed_nodes = NonlinearElasticityTools<2>::GetNodesByComponentValue(mesh, 1, 1.0);

        std::vector<BoundaryElement<1,2>*> boundary_elems;
        std::vector<c_vector<double,2> > tractions;
        c_vector<double,2> traction;
        traction(0) = 0;
        traction(1) = 1.0; // this choice of sign corresponds to an inward force (if applied to the bottom surface)
        for (TetrahedralMesh<2,2>::BoundaryElementIterator iter = mesh.GetBoundaryElementIteratorBegin();
             iter != mesh.GetBoundaryElementIteratorEnd();
             ++iter)
        {
            if (fabs((*iter)->CalculateCentroid()[1] - 0.0) < 1e-6)
            {
                BoundaryElement<1,2>* p_element = *iter;
                boundary_elems.push_back(p_element);
                tractions.push_back(traction);
            }
        }
        assert(boundary_elems.size() == 8u);

        SolidMechanicsProblemDefinition<2> problem_defn(mesh);
        problem_defn.SetMaterialLaw(INCOMPRESSIBLE,&law);
        problem_defn.SetZeroDisplacementNodes(fixed_nodes);
        problem_defn.SetBodyForce(body_force);
        problem_defn.SetTractionBoundaryConditions(boundary_elems, tractions);

        IncompressibleNonlinearElasticitySolver<2> solver(mesh,
                                                          problem_defn,
                                                          "IncompressibleElasticityWithTractionsTutorial");

        solver.SetComputeAverageStressPerElementDuringSolve();

        solver.Solve();

        solver.WriteCurrentStrains(DEFORMATION_GRADIENT_F,"deformation_grad");

        solver.WriteCurrentAverageElementStresses("2nd_PK_stress");

        TS_ASSERT_EQUALS(solver.GetNumNewtonIterations(), 4u);

        solver.CreateCmguiOutput();

        TS_ASSERT_DELTA(solver.rGetDeformedPosition()[8](0), 0.8561, 1e-3);
        TS_ASSERT_DELTA(solver.rGetDeformedPosition()[8](1), 0.0310, 1e-3);
    }
};

Solving a Cardiac Problem

Solving Linear Parabolic Pde Systems With Coupled Ode Systems

Solving Elasticity Problems

Solving solid mechanics problems#

Simple incompressible deformation: 2D shape hanging under gravity#

Incompressible deformation: 2D shape hanging under gravity with a balancing traction#

IMPORTANT: Using HYPRE#

Full code#

Solving solid mechanics problems

Simple incompressible deformation: 2D shape hanging under gravity

Incompressible deformation: 2D shape hanging under gravity with a balancing traction

IMPORTANT: Using HYPRE

Full code