/**
* Simple Parabolic PDE u' = del squared u
*
* With u = 0 on the boundaries of the unit cube. Subject to the initial
* condition u(0,x,y,z)=sin( PI x)sin( PI y)sin( PI z).
*/
void TestSimpleLinearParabolicSolver3DZeroDirich()
{
// read mesh on [0,1]x[0,1]x[0,1]
TrianglesMeshReader<3,3> mesh_reader("mesh/test/data/cube_136_elements");
TetrahedralMesh<3,3> mesh;
mesh.ConstructFromMeshReader(mesh_reader);
// Instantiate PDE object
HeatEquation<3> pde;
// Boundary conditions - zero dirichlet everywhere on boundary
BoundaryConditionsContainer<3,3,1> bcc;
bcc.DefineZeroDirichletOnMeshBoundary(&mesh);
// Solver
SimpleLinearParabolicSolver<3,3> solver(&mesh,&pde,&bcc);
/*
* Choose initial condition sin(x*pi)*sin(y*pi)*sin(z*pi) as
* this is an eigenfunction of the heat equation.
*/
std::vector<double> init_cond(mesh.GetNumNodes());
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double y = mesh.GetNode(i)->GetPoint()[1];
double z = mesh.GetNode(i)->GetPoint()[2];
init_cond[i] = sin(x*M_PI)*sin(y*M_PI)*sin(z*M_PI);
}
Vec initial_condition = PetscTools::CreateVec(init_cond);
double t_end = 0.1;
solver.SetTimes(0, t_end);
solver.SetTimeStep(0.001);
solver.SetInitialCondition(initial_condition);
Vec result = solver.Solve();
ReplicatableVector result_repl(result);
// Check solution is u = e^{-3*t*pi*pi} sin(x*pi)*sin(y*pi)*sin(z*pi), t=0.1
for (unsigned i=0; i<result_repl.GetSize(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double y = mesh.GetNode(i)->GetPoint()[1];
double z = mesh.GetNode(i)->GetPoint()[2];
double u = exp(-3*t_end*M_PI*M_PI)*sin(x*M_PI)*sin(y*M_PI)*sin(z*M_PI);
TS_ASSERT_DELTA(result_repl[i], u, 0.1);
}
PetscTools::Destroy(initial_condition);
PetscTools::Destroy(result);
}
void TestHeatEquationWithSourceWithCoupledOdeSystemIn1dWithZeroNeumann()
{
// Create mesh of domain [0,1]
TrianglesMeshReader<1,1> mesh_reader("mesh/test/data/1D_0_to_1_100_elements");
TetrahedralMesh<1,1> mesh;
mesh.ConstructFromMeshReader(mesh_reader);
// Create PDE system object
HeatEquationWithSourceForCoupledOdeSystem<1> pde;
// Define zero Neumann boundary conditions
BoundaryConditionsContainer<1,1,1> bcc;
ConstBoundaryCondition<1>* p_boundary_condition = new ConstBoundaryCondition<1>(0.0);
TetrahedralMesh<1,1>::BoundaryElementIterator iter = mesh.GetBoundaryElementIteratorBegin();
bcc.AddNeumannBoundaryCondition(*iter, p_boundary_condition);
iter = mesh.GetBoundaryElementIteratorEnd();
iter--;
bcc.AddNeumannBoundaryCondition(*iter, p_boundary_condition);
// Create the correct number of ODE systems
double a = 5.0;
std::vector<AbstractOdeSystemForCoupledPdeSystem*> ode_systems;
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
ode_systems.push_back(new OdeSystemForCoupledHeatEquationWithSource(a));
}
// Create PDE system solver
LinearParabolicPdeSystemWithCoupledOdeSystemSolver<1,1,1> solver(&mesh, &pde, &bcc, ode_systems);
// Test setting end time and timestep
TS_ASSERT_THROWS_THIS(solver.SetTimes(1.0, 0.0), "Start time has to be less than end time");
TS_ASSERT_THROWS_THIS(solver.SetTimeStep(0.0), "Time step has to be greater than zero");
// Set end time and timestep
double t_end = 0.1;
solver.SetTimes(0, t_end);
solver.SetTimeStep(0.001);
// Set initial condition u(x,0) = 1 + cos(pi*x)
std::vector<double> init_cond(mesh.GetNumNodes());
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
init_cond[i] = 1 + cos(M_PI*x);
}
Vec initial_condition = PetscTools::CreateVec(init_cond);
solver.SetInitialCondition(initial_condition);
// Solve PDE system and store result
Vec result = solver.Solve();
ReplicatableVector result_repl(result);
/*
* Test that solution is given by
*
* u(x,t) = 1 + (1 - exp(-a*t))/a + exp(-pi*pi*t)*cos(pi*x),
* v(x,t) = exp(-a*t),
*
* with t = t_end.
*/
for (unsigned i=0; i<result_repl.GetSize(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double u = 1 + (1 - exp(-a*t_end))/a + exp(-M_PI*M_PI*t_end)*cos(M_PI*x);
TS_ASSERT_DELTA(result_repl[i], u, 0.1);
double u_from_v = solver.GetOdeSystemAtNode(i)->rGetPdeSolution()[0];
TS_ASSERT_DELTA(result_repl[i], u_from_v, 0.1);
double v = exp(-a*t_end);
TS_ASSERT_DELTA(ode_systems[i]->rGetStateVariables()[0], v, 0.1);
}
// Test the method GetOdeSystemAtNode()
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
TS_ASSERT(solver.GetOdeSystemAtNode(i) != NULL);
TS_ASSERT_DELTA(static_cast<OdeSystemForCoupledHeatEquationWithSource*>(solver.GetOdeSystemAtNode(i))->GetA(), 5.0, 1e-6);
}
// Tidy up
PetscTools::Destroy(initial_condition);
PetscTools::Destroy(result);
}
void TestHeatEquationWithCoupledOdeSystemIn2dWithZeroDirichlet()
{
// Create mesh of the domain [0,1]x[0,1]
TrianglesMeshReader<2,2> mesh_reader("mesh/test/data/square_4096_elements");
TetrahedralMesh<2,2> mesh;
mesh.ConstructFromMeshReader(mesh_reader);
// Create PDE system object
HeatEquationForCoupledOdeSystem<2> pde;
// Define zero Dirichlet boundary conditions on entire boundary
BoundaryConditionsContainer<2,2,1> bcc;
bcc.DefineZeroDirichletOnMeshBoundary(&mesh);
// Create the correct number of ODE systems
double a = 5.0;
std::vector<AbstractOdeSystemForCoupledPdeSystem*> ode_systems;
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
ode_systems.push_back(new OdeSystemForCoupledHeatEquation(a));
}
// Create PDE system solver
LinearParabolicPdeSystemWithCoupledOdeSystemSolver<2,2,1> solver(&mesh, &pde, &bcc, ode_systems);
// Set end time and timestep
double t_end = 0.01;
solver.SetTimes(0, t_end);
solver.SetTimeStep(0.001);
/*
* Set initial condition
*
* u(x,y,0) = sin(pi*x)*sin(pi*y),
*
* which is an eigenfunction of the heat equation.
*/
std::vector<double> init_cond(mesh.GetNumNodes());
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double y = mesh.GetNode(i)->GetPoint()[1];
init_cond[i] = sin(M_PI*x)*sin(M_PI*y);
}
Vec initial_condition = PetscTools::CreateVec(init_cond);
solver.SetInitialCondition(initial_condition);
// Solve PDE system and store result
Vec result = solver.Solve();
ReplicatableVector result_repl(result);
/*
* Test that solution is given by
*
* u(x,y,t) = e^{-2*pi*pi*t} sin(pi*x)*sin(pi*y),
* v(x,y,t) = 1 + (1 - e^{-2*pi*pi*t})*sin(pi*x)*sin(pi*y)*a/(2*pi*pi),
*
* with t = t_end.
*/
for (unsigned i=0; i<result_repl.GetSize(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double y = mesh.GetNode(i)->GetPoint()[1];
double u = exp(-2*M_PI*M_PI*t_end)*sin(M_PI*x)*sin(M_PI*y);
double v = 1.0 + (a/(2*M_PI*M_PI))*(1 - exp(-2*M_PI*M_PI*t_end))*sin(M_PI*x)*sin(M_PI*y);
TS_ASSERT_DELTA(result_repl[i], u, 0.01);
TS_ASSERT_DELTA(ode_systems[i]->rGetStateVariables()[0], v, 0.01);
}
// Tidy up
PetscTools::Destroy(initial_condition);
PetscTools::Destroy(result);
}
void TestHeatEquationWithCoupledOdeSystemIn1dWithMixed()
{
// Create mesh of domain [0,1]
TrianglesMeshReader<1,1> mesh_reader("mesh/test/data/1D_0_to_1_100_elements");
TetrahedralMesh<1,1> mesh;
mesh.ConstructFromMeshReader(mesh_reader);
// Create PDE system object
HeatEquationForCoupledOdeSystem<1> pde;
// Define non-zero Neumann boundary condition at x=0
BoundaryConditionsContainer<1,1,1> bcc;
ConstBoundaryCondition<1>* p_boundary_condition = new ConstBoundaryCondition<1>(1.0);
TetrahedralMesh<1,1>::BoundaryElementIterator iter = mesh.GetBoundaryElementIteratorBegin();
bcc.AddNeumannBoundaryCondition(*iter, p_boundary_condition);
// Define zero Dirichlet boundary condition at x=1
ConstBoundaryCondition<1>* p_boundary_condition2 = new ConstBoundaryCondition<1>(0.0);
TetrahedralMesh<1,1>::BoundaryNodeIterator node_iter = mesh.GetBoundaryNodeIteratorEnd();
--node_iter;
bcc.AddDirichletBoundaryCondition(*node_iter, p_boundary_condition2);
// Create the correct number of ODE systems
double a = 5.0;
std::vector<AbstractOdeSystemForCoupledPdeSystem*> ode_systems;
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
ode_systems.push_back(new OdeSystemForCoupledHeatEquation(a));
}
// Create PDE system solver
LinearParabolicPdeSystemWithCoupledOdeSystemSolver<1,1,1> solver(&mesh, &pde, &bcc, ode_systems);
// Set end time and timestep
double t_end = 0.1;
solver.SetTimes(0, t_end);
solver.SetTimeStep(0.001);
// Set initial condition u(x,0) = 1 - x
std::vector<double> init_cond(mesh.GetNumNodes());
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
init_cond[i] = 1 - x;
}
Vec initial_condition = PetscTools::CreateVec(init_cond);
solver.SetInitialCondition(initial_condition);
// Solve PDE system and store result
Vec result = solver.Solve();
ReplicatableVector result_repl(result);
/*
* Test that solution is given by
*
* u(x,t) = 1 - x,
* v(x,t) = 1 + a*(1-x)*t,
*
* with t = t_end.
*/
for (unsigned i=0; i<result_repl.GetSize(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double u = 1 - x;
TS_ASSERT_DELTA(result_repl[i], u, 0.1);
double v = 1 + a*(1-x)*t_end;
TS_ASSERT_DELTA(ode_systems[i]->rGetStateVariables()[0], v, 0.1);
}
// Tidy up
PetscTools::Destroy(initial_condition);
PetscTools::Destroy(result);
}
void TestSolvingEllipticPde() throw(Exception)
{
/* First we declare a mesh reader which reads mesh data files of the 'Triangle'
* format. The path given is relative to the main Chaste directory. As we are in 2d,
* the reader will look for three datafiles, [name].nodes, [name].ele and [name].edge.
* Note that the first template argument here is the spatial dimension of the
* elements in the mesh ({{{ELEMENT_DIM}}}), and the second is the dimension of the nodes,
* i.e. the dimension of the space the mesh lives in ({{{SPACE_DIM}}}). Usually
* {{{ELEMENT_DIM}}} and {{{SPACE_DIM}}} will be equal. */
TrianglesMeshReader<2,2> mesh_reader("mesh/test/data/square_128_elements");
/* Now declare a tetrahedral mesh with the same dimensions... */
TetrahedralMesh<2,2> mesh;
/* ... and construct the mesh using the mesh reader. */
mesh.ConstructFromMeshReader(mesh_reader);
/* Next we instantiate an instance of our PDE we wish to solve. */
MyPde pde;
/* A set of boundary conditions are stored in a {{{BoundaryConditionsContainer}}}. The
* three template arguments are ELEMENT_DIM, SPACE_DIM and PROBLEM_DIM, the latter being
* the number of unknowns we are solving for. We have one unknown (ie u is a scalar, not
* a vector), so in this case {{{PROBLEM_DIM}}}=1. */
BoundaryConditionsContainer<2,2,1> bcc;
/* Defining the boundary conditions is the only particularly fiddly part of solving PDEs,
* unless they are very simple, such as u=0 on the boundary, which could be done
* as follows: */
//bcc.DefineZeroDirichletOnMeshBoundary(&mesh);
/* We want to specify u=0 on x=0 and y=0. To do this, we first create the boundary condition
* object saying what the value of the condition is at any particular point in space. Here
* we use the class `ConstBoundaryCondition`, a subclass of `AbstractBoundaryCondition` that
* yields the same constant value (0.0 here) everywhere it is used.
*
* Note that the object is allocated with `new`. The `BoundaryConditionsContainer` object deals
* with deleting its associated boundary condition objects. Note too that we could allocate a
* separate condition object for each boundary node, but using the same object where possible is
* more memory efficient.
*/
ConstBoundaryCondition<2>* p_zero_boundary_condition = new ConstBoundaryCondition<2>(0.0);
/* We then get a boundary node iterator from the mesh... */
TetrahedralMesh<2,2>::BoundaryNodeIterator iter = mesh.GetBoundaryNodeIteratorBegin();
/* ...and loop over the boundary nodes, getting the x and y values. */
while (iter < mesh.GetBoundaryNodeIteratorEnd())
{
double x = (*iter)->GetPoint()[0];
double y = (*iter)->GetPoint()[1];
/* If x=0 or y=0... */
if ((x==0) || (y==0))
{
/* ...associate the zero boundary condition created above with this boundary node
* ({{{*iter}}} being a pointer to a {{{Node<2>}}}).
*/
bcc.AddDirichletBoundaryCondition(*iter, p_zero_boundary_condition);
}
iter++;
}
/* Now we create Neumann boundary conditions for the ''surface elements'' on x=1 and y=1. Note that
* Dirichlet boundary conditions are defined on nodes, whereas Neumann boundary conditions are
* defined on surface elements. Note also that the natural boundary condition statement for this
* PDE is (D grad u).n = g(x) (where n is the outward-facing surface normal), and g(x) is a prescribed
* function, ''not'' something like du/dn=g(x). Hence the boundary condition we are specifying is
* (D grad u).n = 0.
*
* '''Important note for 1D:''' This means that if we were solving 2u,,xx,,=f(x) in 1D, and
* wanted to specify du/dx=1 on the LHS boundary, the Neumann boundary value we have to specify is
* -2, as n=-1 (outward facing normal) so (D gradu).n = -2 when du/dx=1.
*
* To define Neumann bcs, we reuse the zero boundary condition object defined above, but apply it
* at surface elements. We loop over these using another iterator provided by the mesh class.
*/
TetrahedralMesh<2,2>::BoundaryElementIterator surf_iter
= mesh.GetBoundaryElementIteratorBegin();
while (surf_iter != mesh.GetBoundaryElementIteratorEnd())
{
/* Get the x and y values of any node (here, the 0th) in the element. */
unsigned node_index = (*surf_iter)->GetNodeGlobalIndex(0);
double x = mesh.GetNode(node_index)->GetPoint()[0];
double y = mesh.GetNode(node_index)->GetPoint()[1];
/* If x=1 or y=1... */
if ( (fabs(x-1.0) < 1e-6) || (fabs(y-1.0) < 1e-6) )
{
/* ...associate the boundary condition with the surface element. */
bcc.AddNeumannBoundaryCondition(*surf_iter, p_zero_boundary_condition);
}
/* Finally increment the iterator. */
surf_iter++;
}
/* Next we define the solver of the PDE.
* To solve an {{{AbstractLinearEllipticPde}}} (which is the type of PDE {{{MyPde}}} is),
* we use a {{{SimpleLinearEllipticSolver}}}. The solver, again templated over
* {{{ELEMENT_DIM}}} and {{{SPACE_DIM}}}, needs to be given (pointers to) the mesh,
* pde and boundary conditions.
*/
SimpleLinearEllipticSolver<2,2> solver(&mesh, &pde, &bcc);
/* To solve, just call {{{Solve()}}}. A PETSc vector is returned. */
Vec result = solver.Solve();
/* It is a pain to access the individual components of a PETSc vector, even when running only on
* one process. A helper class called {{{ReplicatableVector}}} has been created. Create
* an instance of one of these, using the PETSc {{{Vec}}} as the data. The ''i''th
* component of {{{result}}} can now be obtained by simply doing {{{result_repl[i]}}}.
*/
ReplicatableVector result_repl(result);
/* Let us write out the solution to a file. To do this, create an
* {{{OutputFileHandler}}}, passing in the directory we want files written to.
* This is relative to the directory defined by the CHASTE_TEST_OUTPUT environment
* variable - usually `/tmp/$USER/testoutput`. Note by default the output directory
* passed in is emptied by this command. To avoid this, {{{false}}} can be passed in as a second
* parameter.
*/
OutputFileHandler output_file_handler("TestSolvingLinearPdeTutorial");
/* Create an {{{out_stream}}}, which is a stream to a particular file. An {{{out_stream}}}
* is a smart pointer to a `std::ofstream`. */
out_stream p_file = output_file_handler.OpenOutputFile("linear_solution.txt");
/* Loop over the entries of the solution. */
for (unsigned i=0; i<result_repl.GetSize(); i++)
{
/* Get the x and y-values of the node corresponding to this entry. The method
* {{{GetNode}}} on the mesh class returns a pointer to a {{{Node}}}. */
double x = mesh.GetNode(i)->rGetLocation()[0];
double y = mesh.GetNode(i)->rGetLocation()[1];
/* Get the computed solution at this node from the {{{ReplicatableVector}}}. */
double u = result_repl[i];
/* Finally, write x, y and u to the output file. The solution could then be
* visualised in (eg) matlab, using the commands:
* {{{sol=load('linear_solution.txt'); plot3(sol(:,1),sol(:,2),sol(:,3),'.');}}}*/
(*p_file) << x << " " << y << " " << u << "\n";
}
/* All PETSc {{{Vec}}}s should be destroyed when they are no longer needed, or you will have a memory leak. */
PetscTools::Destroy(result);
}
// test 2D problem - takes a long time to run.
// solution is incorrect to specified tolerance.
void xTestSimpleLinearParabolicSolver2DNeumannWithSmallTimeStepAndFineMesh()
{
// Create mesh from mesh reader
FemlabMeshReader<2,2> mesh_reader("mesh/test/data/",
"femlab_fine_square_nodes.dat",
"femlab_fine_square_elements.dat",
"femlab_fine_square_edges.dat");
TetrahedralMesh<2,2> mesh;
mesh.ConstructFromMeshReader(mesh_reader);
// Instantiate PDE object
HeatEquation<2> pde;
// Boundary conditions - zero dirichlet on boundary;
BoundaryConditionsContainer<2,2,1> bcc;
TetrahedralMesh<2,2>::BoundaryNodeIterator iter = mesh.GetBoundaryNodeIteratorBegin();
while (iter != mesh.GetBoundaryNodeIteratorEnd())
{
double x = (*iter)->GetPoint()[0];
double y = (*iter)->GetPoint()[1];
ConstBoundaryCondition<2>* p_dirichlet_boundary_condition =
new ConstBoundaryCondition<2>(x);
if (fabs(y) < 0.01)
{
bcc.AddDirichletBoundaryCondition(*iter, p_dirichlet_boundary_condition);
}
if (fabs(y - 1.0) < 0.01)
{
bcc.AddDirichletBoundaryCondition(*iter, p_dirichlet_boundary_condition);
}
if (fabs(x) < 0.01)
{
bcc.AddDirichletBoundaryCondition(*iter, p_dirichlet_boundary_condition);
}
iter++;
}
TetrahedralMesh<2,2>::BoundaryElementIterator surf_iter = mesh.GetBoundaryElementIteratorBegin();
ConstBoundaryCondition<2>* p_neumann_boundary_condition =
new ConstBoundaryCondition<2>(1.0);
while (surf_iter != mesh.GetBoundaryElementIteratorEnd())
{
int node = (*surf_iter)->GetNodeGlobalIndex(0);
double x = mesh.GetNode(node)->GetPoint()[0];
if (fabs(x - 1.0) < 0.01)
{
bcc.AddNeumannBoundaryCondition(*surf_iter, p_neumann_boundary_condition);
}
surf_iter++;
}
// Solver
SimpleLinearParabolicSolver<2,2> solver(&mesh,&pde,&bcc);
// Initial condition u(0,x,y) = sin(0.5*M_PI*x)*sin(M_PI*y)+x
std::vector<double> init_cond(mesh.GetNumNodes());
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double y = mesh.GetNode(i)->GetPoint()[1];
init_cond[i] = sin(0.5*M_PI*x)*sin(M_PI*y)+x;
}
Vec initial_condition = PetscTools::CreateVec(init_cond);
double t_end = 0.1;
solver.SetTimes(0, t_end);
solver.SetTimeStep(0.001);
solver.SetInitialCondition(initial_condition);
Vec result = solver.Solve();
ReplicatableVector result_repl(result);
// Check solution is u = e^{-5/4*M_PI*M_PI*t} sin(0.5*M_PI*x)*sin(M_PI*y)+x, t=0.1
for (unsigned i=0; i<result_repl.GetSize(); i++)
{
double x = mesh.GetNode(i)->GetPoint()[0];
double y = mesh.GetNode(i)->GetPoint()[1];
double u = exp((-5/4)*M_PI*M_PI*t_end) * sin(0.5*M_PI*x) * sin(M_PI*y) + x;
TS_ASSERT_DELTA(result_repl[i], u, 0.001);
}
PetscTools::Destroy(result);
PetscTools::Destroy(initial_condition);
}
