/* == Sliding boundary conditions ==
*
* It is common to require a Dirichlet boundary condition where the displacement/position in one dimension
* is fixed, but the displacement/position in the others are free. This can be easily done when
* collecting the new locations for the fixed nodes, as shown in the following example. Here, we
* take a unit square, apply gravity downward, and suppose the Y=0 surface is like a frictionless boundary,
* so that, for the nodes on Y=0, we specify y=0 but leave x free (Here (X,Y)=old position, (x,y)=new position).
* (Note though that this wouldn't be enough to uniquely specify the final solution - an arbitrary
* translation in the Y direction could be added a solution to obtain another valid solution, so we
* fully fix the node at the origin.)
*/
void TestWithSlidingDirichletBoundaryConditions() throw(Exception)
{
QuadraticMesh<2> mesh;
mesh.ConstructRegularSlabMesh(0.1 /*stepsize*/, 1.0 /*width*/, 1.0 /*height*/);
ExponentialMaterialLaw<2> law(1.0, 0.5); // First parameter is 'a', second 'b', in W=a*exp(b(I1-3))
/* Create fixed nodes and locations... */
std::vector<unsigned> fixed_nodes;
std::vector<c_vector<double,2> > locations;
/* Fix node 0 (the node at the origin) */
fixed_nodes.push_back(0);
locations.push_back(zero_vector<double>(2));
/* For the rest, if the node is on the Y=0 surface.. */
for (unsigned i=1; i<mesh.GetNumNodes(); i++)
{
if ( fabs(mesh.GetNode(i)->rGetLocation()[1])<1e-6)
{
/* ..add it to the list of fixed nodes.. */
fixed_nodes.push_back(i);
/* ..and define y to be 0 but x is fixed */
c_vector<double,2> new_location;
new_location(0) = SolidMechanicsProblemDefinition<2>::FREE;
new_location(1) = 0.0;
locations.push_back(new_location);
}
}
/* Set the material law and fixed nodes, add some gravity, and solve */
SolidMechanicsProblemDefinition<2> problem_defn(mesh);
problem_defn.SetMaterialLaw(INCOMPRESSIBLE,&law);
problem_defn.SetFixedNodes(fixed_nodes, locations);
c_vector<double,2> gravity = zero_vector<double>(2);
gravity(1) = -0.5;
problem_defn.SetBodyForce(gravity);
IncompressibleNonlinearElasticitySolver<2> solver(mesh,
problem_defn,
"ElasticitySlidingBcsExample");
solver.Solve();
solver.CreateCmguiOutput();
/* Check the node at (1,0) has moved but has stayed on Y=0 */
TS_ASSERT_LESS_THAN(1.0, solver.rGetDeformedPosition()[10](0));
TS_ASSERT_DELTA(solver.rGetDeformedPosition()[10](1), 0.0, 1e-3);
}
void TestBarPressureOnUndersideCompressible() throw(Exception)
{
EXIT_IF_PARALLEL; // petsc's direct solve only runs one 1 proc - MUMPS may be the answer for direct solves in parallel
std::string base = "BarPressureOnUndersideCompressible";
std::stringstream output_dir;
/////////////////////////////////////
// Solve on coarse mesh..
/////////////////////////////////////
QuadraticMesh<3> mesh;
double h = 1.0;
mesh.ConstructRegularSlabMesh(h, 10.0, 1.0, 1.0);
output_dir << base << "h_" << h;
std::vector<double> solution;
unsigned num_iters = SolvePressureOnUndersideCompressible(mesh, output_dir.str(), solution, false, true);
TS_ASSERT_EQUALS(num_iters, 5u);
}
void TestReadMeshes(void) throw(Exception)
{
{
READER_2D reader("mesh/test/data/square_4_elements_gmsh.msh");
TetrahedralMesh<2,2> mesh;
mesh.ConstructFromMeshReader(reader);
TS_ASSERT_EQUALS(mesh.GetNumNodes(), 5u);
TS_ASSERT_EQUALS(mesh.GetNumElements(), 4u);
TS_ASSERT_EQUALS(mesh.GetNumBoundaryElements(), 4u);
}
{
READER_3D reader("mesh/test/data/simple_cube_gmsh.msh");
TetrahedralMesh<3,3> mesh;
mesh.ConstructFromMeshReader(reader);
TS_ASSERT_EQUALS(mesh.GetNumNodes(), 14u);
TS_ASSERT_EQUALS(mesh.GetNumElements(), 24u);
TS_ASSERT_EQUALS(mesh.GetNumBoundaryElements(), 24u);
}
{
READER_2D reader("mesh/test/data/quad_square_4_elements_gmsh.msh",2,2);
QuadraticMesh<2> mesh;
mesh.ConstructFromMeshReader(reader);
TS_ASSERT_EQUALS(mesh.GetNumNodes(), 13u);
TS_ASSERT_EQUALS(mesh.GetNumElements(), 4u);
TS_ASSERT_EQUALS(mesh.GetNumBoundaryElements(), 4u);
}
{
READER_3D reader("mesh/test/data/quad_cube_gmsh.msh",2,2);
QuadraticMesh<3> mesh;
mesh.ConstructFromMeshReader(reader);
TS_ASSERT_EQUALS(mesh.GetNumNodes(), 63u);
TS_ASSERT_EQUALS(mesh.GetNumElements(), 24u);
TS_ASSERT_EQUALS(mesh.GetNumBoundaryElements(), 24u);
}
}
void TestBarPressureOnUnderside() throw(Exception)
{
EXIT_IF_PARALLEL; // petsc's direct solve only runs one 1 proc - MUMPS may be the answer for direct solves in parallel
std::string base = "BarPressureOnUnderside";
std::stringstream output_dir;
/////////////////////////////////////
// Solve on coarse mesh..
/////////////////////////////////////
QuadraticMesh<3> mesh;
double h = 1.0;
mesh.ConstructRegularSlabMesh(h, 10.0, 1.0, 1.0);
output_dir << base << "h_" << h;
std::vector<double> solution;
unsigned num_iters = SolvePressureOnUnderside(mesh, output_dir.str(), solution, false, true);
TS_ASSERT_EQUALS(num_iters, 6u);
/////////////////////////////////////
// Solve on slightly finer mesh..
/////////////////////////////////////
h = 0.5;
QuadraticMesh<3> mesh2;
mesh2.ConstructRegularSlabMesh(h, 10.0, 1.0, 1.0);
std::vector<double> solution2;
InterpolateMechanicsSolutionToNewMesh<3>(mesh, solution, mesh2, solution2, INCOMPRESSIBLE);
output_dir.str("");
output_dir << base << "h_" << h;
num_iters = SolvePressureOnUnderside(mesh2, output_dir.str(), solution2, true, true);
TS_ASSERT_EQUALS(num_iters, 2u);
///////////////////////////////////////
// Could continue onto finer meshes..
///////////////////////////////////////
}
/* == Incompressible deformation: non-zero displacement boundary conditions, functional tractions ==
*
* We now consider a more complicated example. We prescribe particular new locations for the nodes
* on the Dirichlet boundary, and also show how to prescribe a traction that is given in functional form
* rather than prescribed for each boundary element.
*/
void TestIncompressibleProblemMoreComplicatedExample() throw(Exception)
{
/* Create a mesh */
QuadraticMesh<2> mesh;
mesh.ConstructRegularSlabMesh(0.1 /*stepsize*/, 1.0 /*width*/, 1.0 /*height*/);
/* Use a different material law this time, an exponential material law.
* The material law needs to inherit from `AbstractIncompressibleMaterialLaw`,
* and there are a few implemented, see `continuum_mechanics/src/problem/material_laws` */
ExponentialMaterialLaw<2> law(1.0, 0.5); // First parameter is 'a', second 'b', in W=a*exp(b(I1-3))
/* Now specify the fixed nodes, and their new locations. Create `std::vector`s for each. */
std::vector<unsigned> fixed_nodes;
std::vector<c_vector<double,2> > locations;
/* Loop over the mesh nodes */
for (unsigned i=0; i<mesh.GetNumNodes(); i++)
{
/* If the node is on the Y=0 surface (the LHS) */
if ( fabs(mesh.GetNode(i)->rGetLocation()[1])<1e-6)
{
/* Add it to the list of fixed nodes */
fixed_nodes.push_back(i);
/* and define a new position x=(X,0.1*X^2^) */
c_vector<double,2> new_location;
double X = mesh.GetNode(i)->rGetLocation()[0];
new_location(0) = X;
new_location(1) = 0.1*X*X;
locations.push_back(new_location);
}
}
/* Now collect all the boundary elements on the top surface, as before, except
* here we don't create the tractions for each element
*/
std::vector<BoundaryElement<1,2>*> boundary_elems;
for (TetrahedralMesh<2,2>::BoundaryElementIterator iter = mesh.GetBoundaryElementIteratorBegin();
iter != mesh.GetBoundaryElementIteratorEnd();
++iter)
{
/* If Y=1, have found a boundary element */
if (fabs((*iter)->CalculateCentroid()[1] - 1.0)<1e-6)
{
BoundaryElement<1,2>* p_element = *iter;
boundary_elems.push_back(p_element);
}
}
/* Create a problem definition object, and this time calling `SetFixedNodes`
* which takes in the new locations of the fixed nodes.
*/
SolidMechanicsProblemDefinition<2> problem_defn(mesh);
problem_defn.SetMaterialLaw(INCOMPRESSIBLE,&law);
problem_defn.SetFixedNodes(fixed_nodes, locations);
/* Now call `SetTractionBoundaryConditions`, which takes in a vector of
* boundary elements as in the previous test. However this time the second argument
* is a ''function pointer'' (just the name of the function) to a
* function returning traction in terms of position (and time [see below]).
* This function is defined above, before the tests. It has to take in a `c_vector` (position)
* and a double (time), and returns a `c_vector` (traction), and will only be called
* using points in the boundary elements being passed in. The function `MyTraction`
* (defined above, before the tests) above defines a horizontal traction (ie a shear stress, since it is
* applied to the top surface) which increases in magnitude across the object.
*/
problem_defn.SetTractionBoundaryConditions(boundary_elems, MyTraction);
/* Note: You can also call `problem_defn.SetBodyForce(MyBodyForce)`, passing in a function
* instead of a vector, although isn't really physically useful, it is only really useful
* for constructing problems with exact solutions.
*
* Create the solver as before */
IncompressibleNonlinearElasticitySolver<2> solver(mesh,
problem_defn,
"IncompressibleElasticityMoreComplicatedExample");
/* Call `Solve()` */
solver.Solve();
/* Another quick check */
TS_ASSERT_EQUALS(solver.GetNumNewtonIterations(), 6u);
/* Visualise as before.
*
* '''Advanced:''' Note that the function `MyTraction` takes in time, which it didn't use. In the above it would have been called
* with t=0. The current time can be set using `SetCurrentTime()`. The idea is that the user may want to solve a
* sequence of static problems with time-dependent tractions (say), for which they should allow `MyTraction` to
* depend on time, and put the solve inside a time-loop, for example:
*/
//for (double t=0; t<T; t+=dt)
//{
// solver.SetCurrentTime(t);
// solver.Solve();
//}
/* In this the current time would be passed through to `MyTraction`
*
* Create Cmgui output
*/
solver.CreateCmguiOutput();
/* This is just to check that nothing has been accidentally changed in this test */
TS_ASSERT_DELTA(solver.rGetDeformedPosition()[98](0), 1.4543, 1e-3);
TS_ASSERT_DELTA(solver.rGetDeformedPosition()[98](1), 0.5638, 1e-3);
}
/* == Compressible deformation, and other bits and pieces ==
*
* In this test, we will show the (very minor) changes required to solve a compressible nonlinear
* elasticity problem, we will describe and show how to specify 'pressure on deformed body'
* boundary conditions, we illustrate how a quadratic mesh can be generated using a linear mesh
* input files, and we also illustrate how `Solve()` can be called repeatedly, with loading
* changing between the solves.
*
* Note: for other examples of compressible solves, including problems with an exact solution, see the
* file `continuum_mechanics/test/TestCompressibleNonlinearElasticitySolver.hpp`
*/
void TestSolvingCompressibleProblem() throw (Exception)
{
/* All mechanics problems must take in quadratic meshes, but the mesh files for
* (standard) linear meshes in Triangles/Tetgen can be automatically converted
* to quadratic meshes, by simply doing the following. (The mesh loaded here is a disk
* centred at the origin with radius 1).
*/
QuadraticMesh<2> mesh;
TrianglesMeshReader<2,2> reader("mesh/test/data/disk_522_elements");
mesh.ConstructFromLinearMeshReader(reader);
/* Compressible problems require a compressible material law, ie one that
* inherits from `AbstractCompressibleMaterialLaw`. The `CompressibleMooneyRivlinMaterialLaw`
* is one such example; instantiate one of these
*/
CompressibleMooneyRivlinMaterialLaw<2> law(1.0, 0.5);
/* For this problem, we fix the nodes on the surface for which Y < -0.9 */
std::vector<unsigned> fixed_nodes;
for ( TetrahedralMesh<2,2>::BoundaryNodeIterator iter = mesh.GetBoundaryNodeIteratorBegin();
iter != mesh.GetBoundaryNodeIteratorEnd();
++iter)
{
double Y = (*iter)->rGetLocation()[1];
if(Y < -0.9)
{
fixed_nodes.push_back((*iter)->GetIndex());
}
}
/* We will (later) apply Neumann boundary conditions to surface elements which lie below Y=0,
* and these are collected here. (Minor, subtle, comment: we don't bother here checking Y>-0.9,
* so the surface elements collected here include the ones on the Dirichlet boundary. This doesn't
* matter as the Dirichlet boundary conditions to the nonlinear system essentially overwrite
* an Neumann-related effects).
*/
std::vector<BoundaryElement<1,2>*> boundary_elems;
for (TetrahedralMesh<2,2>::BoundaryElementIterator iter
= mesh.GetBoundaryElementIteratorBegin();
iter != mesh.GetBoundaryElementIteratorEnd();
++iter)
{
BoundaryElement<1,2>* p_element = *iter;
if(p_element->CalculateCentroid()[1]<0.0)
{
boundary_elems.push_back(p_element);
}
}
assert(boundary_elems.size()>0);
/* Create the problem definition class, and set the law again, this time
* stating that the law is compressible
*/
SolidMechanicsProblemDefinition<2> problem_defn(mesh);
problem_defn.SetMaterialLaw(COMPRESSIBLE,&law);
/* Set the fixed nodes and gravity */
problem_defn.SetZeroDisplacementNodes(fixed_nodes);
/* The elasticity solvers have two nonlinear solvers implemented, one hand-coded and one which uses PETSc's SNES
* solver. The latter is not the default but can be more robust (and will probably be the default in later
* versions). This is how it can be used. (This option can also be called if the compiled binary is run from
* the command line (see ChasteGuides/RunningBinariesFromCommandLine) using the option "-mech_use_snes").
*/
problem_defn.SetSolveUsingSnes();
/* This line tells the solver to output info about the nonlinear solve as it progresses, and can be used with
* or without the SNES option above. The corresponding command line option is "-mech_verbose"
*/
problem_defn.SetVerboseDuringSolve();
c_vector<double,2> gravity;
gravity(0) = 0;
gravity(1) = 0.1;
problem_defn.SetBodyForce(gravity);
/* Declare the compressible solver, which has the same interface as the incompressible
* one, and call `Solve()`
*/
CompressibleNonlinearElasticitySolver<2> solver(mesh,
problem_defn,
"CompressibleSolidMechanicsExample");
solver.Solve();
/* Now we call add additional boundary conditions, and call `Solve() again. Firstly: these
* Neumann conditions here are not specified traction boundary conditions (such BCs are specified
* on the undeformed body), but instead, the (more natural) specification of a pressure
* exactly in the ''normal direction on the deformed body''. We have to provide a set of boundary
* elements of the mesh, and a pressure to act on those elements. The solver will automatically
* compute the deformed normal directions on which the pressure acts. Note: with this type of
* BC, the ordering of the nodes on the boundary elements needs to be consistent, otherwise some
* normals will be computed to be inward and others outward. The solver will check this on the
* original mesh and throw an exception if this is not the case. (The required ordering is such that:
* in 2D, surface element nodes are ordered anticlockwise (looking at the whole mesh); in 3D, looking
* at any surface element from outside the mesh, the three nodes are ordered anticlockwise. (Triangle/tetgen
* automatically create boundary elements like this)).
*/
double external_pressure = -0.4; // negative sign => inward pressure
problem_defn.SetApplyNormalPressureOnDeformedSurface(boundary_elems, external_pressure);
/* Call `Solve()` again, so now solving with fixed nodes, gravity, and pressure. The solution from
* the previous solve will be used as the initial guess. Although at the moment the solution from the
* previous call to `Solve()` will be over-written, calling `Solve()` repeatedly may be useful for
* some problems: sometimes, Newton's method will fail to converge for given force/pressures etc, and it can
* be (very) helpful to ''increment'' the loading. For example, set the gravity to be (0,-9.81/3), solve,
* then set it to be (0,-2*9.81/3), solve again, and finally set it to be (0,-9.81) and solve for a
* final time
*/
solver.Solve();
solver.CreateCmguiOutput();
}
void TestAssembler2d() throw (Exception)
{
QuadraticMesh<2> mesh;
TrianglesMeshReader<2,2> mesh_reader("mesh/test/data/canonical_triangle_quadratic", 2, 2, false);
mesh.ConstructFromMeshReader(mesh_reader);
ContinuumMechanicsProblemDefinition<2> problem_defn(mesh);
double t1 = 2.6854233;
double t2 = 3.2574578;
// for the boundary element on the y=0 surface, create a traction
std::vector<BoundaryElement<1,2>*> boundary_elems;
std::vector<c_vector<double,2> > tractions;
c_vector<double,2> traction;
traction(0) = t1;
traction(1) = t2;
for (TetrahedralMesh<2,2>::BoundaryElementIterator iter
= mesh.GetBoundaryElementIteratorBegin();
iter != mesh.GetBoundaryElementIteratorEnd();
++iter)
{
if (fabs((*iter)->CalculateCentroid()[1])<1e-4)
{
BoundaryElement<1,2>* p_element = *iter;
boundary_elems.push_back(p_element);
tractions.push_back(traction);
}
}
assert(boundary_elems.size()==1);
problem_defn.SetTractionBoundaryConditions(boundary_elems, tractions);
ContinuumMechanicsNeumannBcsAssembler<2> assembler(&mesh, &problem_defn);
TS_ASSERT_THROWS_THIS(assembler.AssembleVector(), "Vector to be assembled has not been set");
Vec bad_sized_vec = PetscTools::CreateVec(2);
assembler.SetVectorToAssemble(bad_sized_vec, true);
TS_ASSERT_THROWS_THIS(assembler.AssembleVector(), "Vector provided to be assembled has size 2, not expected size of 18 ((dim+1)*num_nodes)");
Vec vec = PetscTools::CreateVec(3*mesh.GetNumNodes());
assembler.SetVectorToAssemble(vec, true);
assembler.AssembleVector();
ReplicatableVector vec_repl(vec);
// Note: on a 1d boundary element, intgl phi_i dx = 1/6 for the bases on the vertices
// and intgl phi_i dx = 4/6 for the basis at the interior node. (Here the integrals
// are over the canonical 1d element, [0,1], which is also the physical element for this
// mesh.
// node 0 is on the surface, and is a vertex
TS_ASSERT_DELTA(vec_repl[0], t1/6.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[1], t2/6.0, 1e-8);
// node 1 is on the surface, and is a vertex
TS_ASSERT_DELTA(vec_repl[3], t1/6.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[4], t2/6.0, 1e-8);
// nodes 2, 3, 4 are not on the surface
for(unsigned i=2; i<5; i++)
{
TS_ASSERT_DELTA(vec_repl[3*i], 0.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[3*i], 0.0, 1e-8);
}
// node 5 is on the surface, and is an interior node
TS_ASSERT_DELTA(vec_repl[15], 4*t1/6.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[16], 4*t2/6.0, 1e-8);
// the rest of the vector is the pressure block and should be zero.
TS_ASSERT_DELTA(vec_repl[2], 0.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[5], 0.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[8], 0.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[11], 0.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[14], 0.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[17], 0.0, 1e-8);
PetscTools::Destroy(vec);
PetscTools::Destroy(bad_sized_vec);
}
/*
* Test that the matrix is calculated correctly on the cannonical triangle.
* Tests against the analytical solution calculated by hand.
*/
void TestAssembler() throw(Exception)
{
QuadraticMesh<2> mesh;
TrianglesMeshReader<2,2> mesh_reader("mesh/test/data/canonical_triangle_quadratic", 2, 2, false);
mesh.ConstructFromMeshReader(mesh_reader);
double mu = 2.0;
c_vector<double,2> body_force;
double g1 = 1.34254;
double g2 = 75.3422;
body_force(0) = g1;
body_force(1) = g2;
StokesFlowProblemDefinition<2> problem_defn(mesh);
problem_defn.SetViscosity(mu);
problem_defn.SetBodyForce(body_force);
StokesFlowAssembler<2> assembler(&mesh, &problem_defn);
// The tests below test the assembler against hand-calculated variables for
// an OLD weak form (corresponding to different boundary conditions), not the
// current Stokes weak form. This factor converts the assembler to use the old
// weak form. See documentation for this variable for more details.
assembler.mScaleFactor = 0.0;
Vec vec = PetscTools::CreateVec(18);
Mat mat;
PetscTools::SetupMat(mat, 18, 18, 18);
assembler.SetVectorToAssemble(vec, true);
assembler.SetMatrixToAssemble(mat, true);
assembler.Assemble();
PetscMatTools::Finalise(mat);
double A[6][6] = {
{ 1.0, 1.0/6.0, 1.0/6.0, 0.0, -2.0/3.0, -2.0/3.0},
{ 1.0/6.0, 1.0/2.0, 0.0, 0.0, 0.0, -2.0/3.0},
{ 1.0/6.0, 0.0, 1.0/2.0, 0.0, -2.0/3.0, 0.0},
{ 0.0, 0.0, 0.0, 8.0/3.0, -4.0/3.0, -4.0/3.0},
{ -2.0/3.0, 0.0, -2.0/3.0, -4.0/3.0, 8.0/3.0, 0.0},
{ -2.0/3.0, -2.0/3.0, 0.0, -4.0/3.0, 0.0, 8.0/3.0}
};
double Bx[6][3] = {
{ -1.0/6.0, 0.0, 0.0},
{ 0.0, 1.0/6.0, 0.0},
{ 0.0, 0.0, 0.0},
{ 1.0/6.0, 1.0/6.0, 1.0/3.0},
{ -1.0/6.0, -1.0/6.0, -1.0/3.0},
{ 1.0/6.0, -1.0/6.0, 0.0},
};
double By[6][3] = {
{ -1.0/6.0, 0.0, 0.0},
{ 0.0, 0.0, 0.0},
{ 0.0, 0.0, 1.0/6.0},
{ 1.0/6.0, 1.0/3.0, 1.0/6.0},
{ 1.0/6.0, 0.0, -1.0/6.0},
{ -1.0/6.0, -1.0/3.0, -1.0/6.0},
};
c_matrix<double,18,18> exact_A = zero_matrix<double>(18);
// The diagonal 6x6 blocks
for (unsigned i=0; i<6; i++)
{
for (unsigned j=0; j<6; j++)
{
exact_A(3*i, 3*j) = mu*A[i][j];
exact_A(3*i+1,3*j+1) = mu*A[i][j];
}
}
// The 6x3 Blocks
for (unsigned i=0; i<6; i++)
{
for (unsigned j=0; j<3; j++)
{
exact_A(3*i,3*j+2) = -Bx[i][j];
exact_A(3*i+1,3*j+2) = -By[i][j];
//- as -Div(U)=0
exact_A(3*j+2,3*i) = -Bx[i][j];
exact_A(3*j+2,3*i+1) = -By[i][j];
}
}
int lo, hi;
MatGetOwnershipRange(mat, &lo, &hi);
for (unsigned i=lo; i<(unsigned)hi; i++)
{
for (unsigned j=0; j<18; j++)
{
TS_ASSERT_DELTA(PetscMatTools::GetElement(mat,i,j), exact_A(i,j), 1e-9);
}
}
ReplicatableVector vec_repl(vec);
// The first 6 entries in the vector correspond to nodes 0, 1, 2, i.e. the vertices.
// For these nodes, it can be shown that the integral of the corresponding
// basis function is zero, i.e. \intgl_{canonical element} \phi_i dV = 0.0 for i=0,1,2, phi_i the
// i-th QUADRATIC basis.
for(unsigned i=0; i<3; i++)
{
TS_ASSERT_DELTA(vec_repl[3*i], g1*0.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[3*i+1], g2*0.0, 1e-8);
}
// The next 6 entries in the vector correspond to nodes 3, 4, 5, i.e. the internal edges.
// For these nodes, it can be shown that the integral of the corresponding
// basis function is 1/6, i.e. \intgl_{canonical element} \phi_i dV = 1/6 for i=3,4,5, phi_i the
// i-th QUADRATIC basis.
for(unsigned i=3; i<6; i++)
{
TS_ASSERT_DELTA(vec_repl[3*i], g1/6.0, 1e-8);
TS_ASSERT_DELTA(vec_repl[3*i+1], g2/6.0, 1e-8);
}
// The pressure-block of the RHS vector should be zero.
TS_ASSERT_DELTA(vec_repl[2], 0.0, 1e-9);
TS_ASSERT_DELTA(vec_repl[5], 0.0, 1e-9);
TS_ASSERT_DELTA(vec_repl[8], 0.0, 1e-9);
TS_ASSERT_DELTA(vec_repl[11], 0.0, 1e-9);
TS_ASSERT_DELTA(vec_repl[14], 0.0, 1e-9);
TS_ASSERT_DELTA(vec_repl[17], 0.0, 1e-9);
// Replace the body force with a functional body force (see MyBodyForce) above, and
// assemble the vector again. This bit isn't so much to test the vector, but
// to test the physical location being integrated at is interpolated correctly
// and passed into the force function - see asserts in MyBodyForce.
mesh.Translate(0.5, 0.8);
Vec vec2 = PetscTools::CreateVec(18);
assembler.SetVectorToAssemble(vec2, true);
problem_defn.SetBodyForce(MyBodyForce);
assembler.Assemble();
ReplicatableVector vec_repl2(vec2);
for(unsigned i=3; i<6; i++)
{
TS_ASSERT_DELTA(vec_repl2[3*i], 10.0/6.0, 1e-8);
TS_ASSERT_DELTA(vec_repl2[3*i+1], 20.0/6.0, 1e-8);
}
PetscTools::Destroy(vec);
PetscTools::Destroy(vec2);
PetscTools::Destroy(mat);
}
/* == 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() throw(Exception)
{
/* 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);
}
/* 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() throw(Exception)
{
/* 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.
* (`QuadraticMesh`s 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";
/* HOW_TO_TAG Continuum mechanics
* Visualise nonlinear elasticity problems solutions, including visualisng strains
*/
/* 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);
}
VoltageInterpolaterOntoMechanicsMesh<DIM>::VoltageInterpolaterOntoMechanicsMesh(
TetrahedralMesh<DIM,DIM>& rElectricsMesh,
QuadraticMesh<DIM>& rMechanicsMesh,
std::vector<std::string>& rVariableNames,
std::string directory,
std::string inputFileNamePrefix)
{
// Read the data from the HDF5 file
Hdf5DataReader reader(directory,inputFileNamePrefix);
unsigned num_timesteps = reader.GetUnlimitedDimensionValues().size();
// set up the elements and weights for the coarse nodes in the fine mesh
FineCoarseMeshPair<DIM> mesh_pair(rElectricsMesh, rMechanicsMesh);
mesh_pair.SetUpBoxesOnFineMesh();
mesh_pair.ComputeFineElementsAndWeightsForCoarseNodes(true);
assert(mesh_pair.rGetElementsAndWeights().size()==rMechanicsMesh.GetNumNodes());
// create and setup a writer
Hdf5DataWriter* p_writer = new Hdf5DataWriter(*rMechanicsMesh.GetDistributedVectorFactory(),
directory,
"voltage_mechanics_mesh",
false, //don't clean
false);
std::vector<int> columns_id;
for (unsigned var_index = 0; var_index < rVariableNames.size(); var_index++)
{
std::string var_name = rVariableNames[var_index];
columns_id.push_back( p_writer->DefineVariable(var_name,"mV") );
}
p_writer->DefineUnlimitedDimension("Time","msecs", num_timesteps);
p_writer->DefineFixedDimension( rMechanicsMesh.GetNumNodes() );
p_writer->EndDefineMode();
assert(columns_id.size() == rVariableNames.size());
// set up a vector to read into
DistributedVectorFactory factory(rElectricsMesh.GetNumNodes());
Vec voltage = factory.CreateVec();
std::vector<double> interpolated_voltages(rMechanicsMesh.GetNumNodes());
Vec voltage_coarse = NULL;
for(unsigned time_step=0; time_step<num_timesteps; time_step++)
{
for (unsigned var_index = 0; var_index < rVariableNames.size(); var_index++)
{
std::string var_name = rVariableNames[var_index];
// read
reader.GetVariableOverNodes(voltage, var_name, time_step);
ReplicatableVector voltage_repl(voltage);
// interpolate
for(unsigned i=0; i<mesh_pair.rGetElementsAndWeights().size(); i++)
{
double interpolated_voltage = 0;
Element<DIM,DIM>& element = *(rElectricsMesh.GetElement(mesh_pair.rGetElementsAndWeights()[i].ElementNum));
for(unsigned node_index = 0; node_index<element.GetNumNodes(); node_index++)
{
unsigned global_node_index = element.GetNodeGlobalIndex(node_index);
interpolated_voltage += voltage_repl[global_node_index]*mesh_pair.rGetElementsAndWeights()[i].Weights(node_index);
}
interpolated_voltages[i] = interpolated_voltage;
}
if(voltage_coarse!=NULL)
{
PetscTools::Destroy(voltage_coarse);
}
voltage_coarse = PetscTools::CreateVec(interpolated_voltages);
// write
p_writer->PutVector(columns_id[var_index], voltage_coarse);
}
p_writer->PutUnlimitedVariable(time_step);
p_writer->AdvanceAlongUnlimitedDimension();
}
if(voltage_coarse!=NULL)
{
PetscTools::Destroy(voltage);
PetscTools::Destroy(voltage_coarse);
}
// delete to flush
delete p_writer;
// Convert the new data to CMGUI format.
// alter the directory in HeartConfig as that is where Hdf5ToCmguiConverter decides
// where to output
std::string config_directory = HeartConfig::Instance()->GetOutputDirectory();
HeartConfig::Instance()->SetOutputDirectory(directory);
Hdf5ToCmguiConverter<DIM,DIM> converter(FileFinder(directory, RelativeTo::ChasteTestOutput),
"voltage_mechanics_mesh",
&rMechanicsMesh,
false);
HeartConfig::Instance()->SetOutputDirectory(config_directory);
}
void InterpolateMechanicsSolutionToNewMesh(QuadraticMesh<3>& rCoarseMesh, std::vector<double>& rCoarseSolution,
QuadraticMesh<3>& rFineMesh, std::vector<double>& rFineSolution,
CompressibilityType compressibilityType)
{
unsigned NUM_UNKNOWNS = (compressibilityType==INCOMPRESSIBLE ? DIM+1 : DIM);
if(rCoarseSolution.size() != rCoarseMesh.GetNumNodes()*NUM_UNKNOWNS)
{
EXCEPTION("rCoarseSolution not correct size");
}
if(rFineSolution.size() != rFineMesh.GetNumNodes()*NUM_UNKNOWNS)
{
rFineSolution.resize(rFineMesh.GetNumNodes()*NUM_UNKNOWNS);
}
c_vector<double, (DIM+1)*(DIM+2)/2> quad_basis;
for(unsigned i=0; i<rFineMesh.GetNumNodes(); i++)
{
// find containing elements and weights in coarse mesh
ChastePoint<DIM> point = rFineMesh.GetNode(i)->GetPoint();
unsigned coarse_element_index = rCoarseMesh.GetContainingElementIndex(point,
false);
Element<DIM,DIM>* p_coarse_element = rCoarseMesh.GetElement(coarse_element_index);
c_vector<double,DIM+1> weight = p_coarse_element->CalculateInterpolationWeights(point);
c_vector<double,DIM> xi;
xi(0) = weight(1);
xi(1) = weight(2);
if(DIM==3)
{
xi(2) = weight(3);
}
QuadraticBasisFunction<DIM>::ComputeBasisFunctions(xi, quad_basis);
// interpolate (u,p) (don't do anything for p if compressible)
c_vector<double,DIM+1> fine_solution = zero_vector<double>(DIM+1);
for(unsigned elem_node_index=0; elem_node_index<(DIM+1)*(DIM+2)/2; elem_node_index++)
{
unsigned coarse_node = p_coarse_element->GetNodeGlobalIndex(elem_node_index);
c_vector<double,DIM+1> coarse_solution_at_node;
for(unsigned j=0; j<DIM; j++)
{
coarse_solution_at_node(j) = rCoarseSolution[NUM_UNKNOWNS*coarse_node + j];
}
if(compressibilityType==INCOMPRESSIBLE)
{
coarse_solution_at_node(DIM) = rCoarseSolution[NUM_UNKNOWNS*coarse_node + DIM];
}
fine_solution += coarse_solution_at_node*quad_basis(elem_node_index);
}
for(unsigned j=0; j<DIM; j++)
{
rFineSolution[NUM_UNKNOWNS*i + j] = fine_solution(j);
}
if(compressibilityType==INCOMPRESSIBLE)
{
rFineSolution[NUM_UNKNOWNS*i + DIM] = fine_solution(DIM);
// Whilst the returned p from a solve is defined properly at all nodes, during the solve linear basis functions are
// used for p and therefore p not computed explicitly at internal nodes, and the solver solves for p=0 at these internal
// nodes. (After the solve, p is interpolated from vertices to internal nodes)
if(rFineMesh.GetNode(i)->IsInternal())
{
rFineSolution[NUM_UNKNOWNS*i + DIM] = 0.0;
}
}
}
}
// Similar to above but uses a compressible material
unsigned SolvePressureOnUndersideCompressible(QuadraticMesh<3>& rMesh, std::string outputDirectory,
std::vector<double>& rSolution,
bool useSolutionAsGuess,
double scaleFactor = 1.0)
{
CompressibleExponentialLaw<3> law;
std::vector<unsigned> fixed_nodes = NonlinearElasticityTools<3>::GetNodesByComponentValue(rMesh, 0, 0.0);
std::vector<BoundaryElement<2,3>*> boundary_elems;
double pressure = 0.001;
for (TetrahedralMesh<3,3>::BoundaryElementIterator iter
= rMesh.GetBoundaryElementIteratorBegin();
iter != rMesh.GetBoundaryElementIteratorEnd();
++iter)
{
BoundaryElement<2,3>* p_element = *iter;
double Z = p_element->CalculateCentroid()[2];
if(fabs(Z)<1e-6)
{
boundary_elems.push_back(p_element);
}
}
SolidMechanicsProblemDefinition<3> problem_defn(rMesh);
problem_defn.SetMaterialLaw(COMPRESSIBLE,&law);
problem_defn.SetZeroDisplacementNodes(fixed_nodes);
problem_defn.SetApplyNormalPressureOnDeformedSurface(boundary_elems, pressure*scaleFactor);
problem_defn.SetVerboseDuringSolve();
CompressibleNonlinearElasticitySolver<3> solver(rMesh,problem_defn,outputDirectory);
solver.SetWriteOutputEachNewtonIteration();
// cover the SetTakeFullFirstNewtonStep() method, and the SetUsePetscDirectSolve() method
solver.SetTakeFullFirstNewtonStep();
solver.SetUsePetscDirectSolve();
if(useSolutionAsGuess)
{
if(solver.rGetCurrentSolution().size()!=rSolution.size())
{
EXCEPTION("Badly-sized input");
}
for(unsigned i=0; i<rSolution.size(); i++)
{
solver.rGetCurrentSolution()[i] = rSolution[i];
}
}
if(scaleFactor < 1.0)
{
try
{
solver.Solve();
}
catch (Exception& e)
{
// not final Solve, so don't quit
WARNING(e.GetMessage());
}
}
else
{
solver.Solve();
solver.CreateCmguiOutput();
VtkNonlinearElasticitySolutionWriter<3> vtk_writer(solver);
vtk_writer.SetWriteElementWiseStrains(DEFORMATION_TENSOR_C);
vtk_writer.Write();
}
rSolution.clear();
rSolution.resize(solver.rGetCurrentSolution().size());
for(unsigned i=0; i<rSolution.size(); i++)
{
rSolution[i] = solver.rGetCurrentSolution()[i];
}
return solver.GetNumNewtonIterations();
}