/* == 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);
}
/* == 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);
}
/* 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);
}
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;
}
}
}
}