Solving Nonlinear Pdes

This tutorial is automatically generated from TestSolvingNonlinearPdesTutorial.hpp at revision 71ac325b969f. Note that the code is given in full at the bottom of the page.

An example showing how to solve a nonlinear elliptic PDE. Also includes function-based boundary conditions.

In this tutorial we show how Chaste can be used to solve nonlinear elliptic PDEs. We will solve the PDE $\nabla .(u \nabla u) + 1 = 0$, on a square domain, with boundary conditions $u=0$ on $y=0$, and Neumann boundary conditions: $(u \nabla u).\mathbf{n} = 0$ on $x=0$ and $x=1$; and $(u \nabla u).\mathbf{n} = y$ on $y=1$.

For nonlinear PDEs, the finite element equations are of the form $\mathbf{F}(\mathbf{U})=0$, where $\mathbf{U}=(U_1 , \dots , U_N)$ is a vector of the unknowns at each node, and $\mathbf{F}$ is some non-linear vector valued function. To solve this, a nonlinear solver is required. Chaste can solve this with Newton’s method, or (default) use PETSc’s nonlinear solvers. Solvers of such nonlinear problems usually require the Jacobian of the problem, i.e. the matrix $A = \partial F/ \partial U$, or at least an approximation of the Jacobian.

The following header files need to be included, as in the linear PDEs tutorial.

#include <cxxtest/TestSuite.h>
#include "UblasIncludes.hpp"
#include "TrianglesMeshReader.hpp"
#include "TetrahedralMesh.hpp"
#include "BoundaryConditionsContainer.hpp"
#include "ConstBoundaryCondition.hpp"
#include "OutputFileHandler.hpp"
#include "PetscSetupAndFinalize.hpp"

This is the solver for nonlinear elliptic PDEs.

#include "SimpleNonlinearEllipticSolver.hpp"

In this test we also show how to define Neumman boundary conditions which depend on spatial location, for which the following class is needed.

#include "FunctionalBoundaryCondition.hpp"

We will choose to use the Chaste Newton solver rather than PETSc’s nonlinear solver.

#include "SimpleNewtonNonlinearSolver.hpp"

As in the linear PDEs tutorial, we have to define the PDE class we want to solve (assuming one has not already been created). Nonlinear elliptic PDEs should inherit from AbstractNonlinearEllipticPde, which has five pure methods which have to be implemented in this concrete class. Here, we define the PDE $\nabla .(u \nabla u) + 1 = 0$.

class MyNonlinearPde : public AbstractNonlinearEllipticPde<2>
{
public:

The first is the part of the source term that is independent of $u$.

    double ComputeLinearSourceTerm(const ChastePoint<2>& rX)
    {
        return 1.0;
    }

The second is the part of the source term that is dependent on $u$.

    double ComputeNonlinearSourceTerm(const ChastePoint<2>& rX, double u)
    {
        return 0.0;
    }

The third is the diffusion tensor, which unlike in the linear case can be dependent on $u$. The diffusion tensor should be symmetric and positive definite.

    c_matrix<double,2,2> ComputeDiffusionTerm(const ChastePoint<2>& rX, double u)
    {
        return identity_matrix<double>(2)*u;
    }

We also need to provide the derivatives with respect to $u$ of the last two methods, so that the Jacobian matrix can be assembled. The derivative of the nonlinear source term is

    double ComputeNonlinearSourceTermPrime(const ChastePoint<2>& , double )
    {
        return 0.0;
    }

And the derivative of the diffusion tensor is just the identity matrix.

    c_matrix<double,2,2> ComputeDiffusionTermPrime(const ChastePoint<2>& rX, double u)
    {
        return identity_matrix<double>(2);
    }
};

We also need to define a (global) function that will become the Neumman boundary conditions, via the FunctionalBoundaryCondition class (see below). This function is $f(x,y) = y$.

double MyNeummanFunction(const ChastePoint<2>& rX)
{
    return rX[1];
}

Next, we define the test suite, as before.

class TestSolvingNonlinearPdesTutorial : public CxxTest::TestSuite
{
public:

Define a particular test.

    void TestSolvingNonlinearEllipticPde()
    {

As usual, first create a mesh.

        TrianglesMeshReader<2,2> mesh_reader("mesh/test/data/square_128_elements");
        TetrahedralMesh<2,2> mesh;
        mesh.ConstructFromMeshReader(mesh_reader);

Next, instantiate the PDE to be solved.

        MyNonlinearPde pde;

Then we have to define the boundary conditions. First, the Dirichlet boundary condition, $u=0$ on $x=0$, using the boundary node iterator.

        BoundaryConditionsContainer<2,2,1> bcc;
        ConstBoundaryCondition<2>* p_zero_bc = new ConstBoundaryCondition<2>(0.0);
        for (TetrahedralMesh<2,2>::BoundaryNodeIterator node_iter = mesh.GetBoundaryNodeIteratorBegin();
             node_iter != mesh.GetBoundaryNodeIteratorEnd();
             node_iter++)
        {
            if (fabs((*node_iter)->GetPoint()[1]) < 1e-12)
            {
                bcc.AddDirichletBoundaryCondition(*node_iter, p_zero_bc);
            }
        }

And then the Neumman conditions. Neumann boundary condition are defined on surface elements, and for this problem, the Neumman boundary value depends on the position in space, so we make use of the FunctionalBoundaryCondition object, which contains a pointer to a function, and just returns the value of that function for the required point when the GetValue method is called.

        FunctionalBoundaryCondition<2>* p_functional_bc = new FunctionalBoundaryCondition<2>(&MyNeummanFunction);

Loop over surface elements.

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

Get the $y$ value of any node (here, the zero-th).

            double y = (*elt_iter)->GetNodeLocation(0,1);

If y=1…

            if (fabs(y-1.0) < 1e-12)
            {

… then associate the functional boundary condition, $(D \nabla u).n = y$, with the surface element…

                bcc.AddNeumannBoundaryCondition(*elt_iter, p_functional_bc);
            }
            else
            {

…else associate the zero boundary condition (i.e. zero flux) with this element.

                bcc.AddNeumannBoundaryCondition(*elt_iter, p_zero_bc);
            }
        }

Note that in the above loop, the zero Neumman boundary condition was applied to all surface elements for which $y \neq 1$, which included the Dirichlet surface $y=0$. This is OK, as Dirichlet boundary conditions are applied to the finite element matrix after Neumman boundary conditions, where the appropriate rows in the matrix are overwritten.

This is the solver for solving nonlinear problems, which, as usual, takes in the mesh, the PDE, and the boundary conditions.

        SimpleNonlinearEllipticSolver<2,2> solver(&mesh, &pde, &bcc);

The solver also needs to be given an initial guess, which will be a PETSc vector. We can make use of a helper method to create it.

        Vec initial_guess = PetscTools::CreateAndSetVec(mesh.GetNumNodes(), 0.25);

Optional: To use Chaste’s Newton solver to solve nonlinear vector equations that are assembled, rather than the default PETSc nonlinear solvers, we can do the following:

        SimpleNewtonNonlinearSolver newton_solver;
        solver.SetNonlinearSolver(&newton_solver);

Optional: We can also manually set tolerances, and whether to print statistics, with this nonlinear vector equation solver

        newton_solver.SetTolerance(1e-10);
        newton_solver.SetWriteStats();

Now call Solve, passing in the initial guess

        Vec answer = solver.Solve(initial_guess);

Note that we could have got the solver to not use an analytical Jacobian and use a numerically-calculated Jacobian instead, by passing in false as a second parameter:

        //Vec answer = solver.Solve(initial_guess, false);

Once solved, we can check the obtained solution against the analytical solution.

        ReplicatableVector answer_repl(answer);
        for (unsigned i=0; i<answer_repl.GetSize(); i++)
        {
            double y = mesh.GetNode(i)->GetPoint()[1];
            double exact_u = sqrt(y*(4-y));
            TS_ASSERT_DELTA(answer_repl[i], exact_u, 0.15);
        }

Finally, we have to remember to destroy the PETSc Vecs.

        PetscTools::Destroy(initial_guess);
        PetscTools::Destroy(answer);
    }
};

Full code

#include <cxxtest/TestSuite.h>
#include "UblasIncludes.hpp"
#include "TrianglesMeshReader.hpp"
#include "TetrahedralMesh.hpp"
#include "BoundaryConditionsContainer.hpp"
#include "ConstBoundaryCondition.hpp"
#include "OutputFileHandler.hpp"
#include "PetscSetupAndFinalize.hpp"
#include "SimpleNonlinearEllipticSolver.hpp"
#include "FunctionalBoundaryCondition.hpp"
#include "SimpleNewtonNonlinearSolver.hpp"

class MyNonlinearPde : public AbstractNonlinearEllipticPde<2>
{
public:

    double ComputeLinearSourceTerm(const ChastePoint<2>& rX)
    {
        return 1.0;
    }

    double ComputeNonlinearSourceTerm(const ChastePoint<2>& rX, double u)
    {
        return 0.0;
    }

    c_matrix<double,2,2> ComputeDiffusionTerm(const ChastePoint<2>& rX, double u)
    {
        return identity_matrix<double>(2)*u;
    }

    double ComputeNonlinearSourceTermPrime(const ChastePoint<2>& , double )
    {
        return 0.0;
    }

    c_matrix<double,2,2> ComputeDiffusionTermPrime(const ChastePoint<2>& rX, double u)
    {
        return identity_matrix<double>(2);
    }
};

double MyNeummanFunction(const ChastePoint<2>& rX)
{
    return rX[1];
}

class TestSolvingNonlinearPdesTutorial : public CxxTest::TestSuite
{
public:
    void TestSolvingNonlinearEllipticPde()
    {
        TrianglesMeshReader<2,2> mesh_reader("mesh/test/data/square_128_elements");
        TetrahedralMesh<2,2> mesh;
        mesh.ConstructFromMeshReader(mesh_reader);

        MyNonlinearPde pde;

        BoundaryConditionsContainer<2,2,1> bcc;
        ConstBoundaryCondition<2>* p_zero_bc = new ConstBoundaryCondition<2>(0.0);
        for (TetrahedralMesh<2,2>::BoundaryNodeIterator node_iter = mesh.GetBoundaryNodeIteratorBegin();
             node_iter != mesh.GetBoundaryNodeIteratorEnd();
             node_iter++)
        {
            if (fabs((*node_iter)->GetPoint()[1]) < 1e-12)
            {
                bcc.AddDirichletBoundaryCondition(*node_iter, p_zero_bc);
            }
        }

        FunctionalBoundaryCondition<2>* p_functional_bc = new FunctionalBoundaryCondition<2>(&MyNeummanFunction);
        for (TetrahedralMesh<2,2>::BoundaryElementIterator elt_iter = mesh.GetBoundaryElementIteratorBegin();
             elt_iter != mesh.GetBoundaryElementIteratorEnd();
             elt_iter++)
        {
            double y = (*elt_iter)->GetNodeLocation(0,1);
            if (fabs(y-1.0) < 1e-12)
            {
                bcc.AddNeumannBoundaryCondition(*elt_iter, p_functional_bc);
            }
            else
            {
                bcc.AddNeumannBoundaryCondition(*elt_iter, p_zero_bc);
            }
        }
        SimpleNonlinearEllipticSolver<2,2> solver(&mesh, &pde, &bcc);

        Vec initial_guess = PetscTools::CreateAndSetVec(mesh.GetNumNodes(), 0.25);

        SimpleNewtonNonlinearSolver newton_solver;
        solver.SetNonlinearSolver(&newton_solver);
        newton_solver.SetTolerance(1e-10);
        newton_solver.SetWriteStats();

        Vec answer = solver.Solve(initial_guess);

        //Vec answer = solver.Solve(initial_guess, false);

        ReplicatableVector answer_repl(answer);
        for (unsigned i=0; i<answer_repl.GetSize(); i++)
        {
            double y = mesh.GetNode(i)->GetPoint()[1];
            double exact_u = sqrt(y*(4-y));
            TS_ASSERT_DELTA(answer_repl[i], exact_u, 0.15);
        }

        PetscTools::Destroy(initial_guess);
        PetscTools::Destroy(answer);
    }
};

Solving More Elasticity Problems

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Solving Nonlinear Pdes

An example showing how to solve a nonlinear elliptic PDE. Also includes function-based boundary conditions.#

Full code#

An example showing how to solve a nonlinear elliptic PDE. Also includes function-based boundary conditions.

Full code