int main(int argc, char** argv)
{
try
{
const double pi = 4.0*atan(1.0);
double lambda = 1.25*pi*pi;
int nx = 32;
int nt = 10;
double tFinal = 1.0/lambda;
Sundance::setOption("nx", nx, "Number of elements");
Sundance::setOption("nt", nt, "Number of timesteps");
Sundance::setOption("tFinal", tFinal, "Final time");
Sundance::init(&argc, &argv);
/* Creation of vector type */
VectorType<double> vecType = new EpetraVectorType();
/* Set up mesh */
MeshType meshType = new BasicSimplicialMeshType();
MeshSource meshSrc = new PartitionedRectangleMesher(
0.0, 1.0, nx,
0.0, 1.0, nx,
meshType);
Mesh mesh = meshSrc.getMesh();
/*
* Specification of cell filters
*/
CellFilter interior = new MaximalCellFilter();
CellFilter edges = new DimensionalCellFilter(1);
CellFilter west = edges.coordSubset(0, 0.0);
CellFilter east = edges.coordSubset(0, 1.0);
CellFilter south = edges.coordSubset(1, 0.0);
CellFilter north = edges.coordSubset(1, 1.0);
/* set up test and unknown functions */
BasisFamily basis = new Lagrange(1);
Expr u = new UnknownFunction(basis, "u");
Expr v = new TestFunction(basis, "v");
/* set up differential operators */
Expr grad = gradient(2);
Expr x = new CoordExpr(0);
Expr y = new CoordExpr(1);
Expr t = new Sundance::Parameter(0.0);
Expr tPrev = new Sundance::Parameter(0.0);
DiscreteSpace discSpace(mesh, basis, vecType);
Expr uExact = cos(0.5*pi*y)*sin(pi*x)*exp(-lambda*t);
L2Projector proj(discSpace, uExact);
Expr uPrev = proj.project();
/*
* We need a quadrature rule for doing the integrations
*/
QuadratureFamily quad = new GaussianQuadrature(2);
double deltaT = tFinal/nt;
Expr gWest = -pi*exp(-lambda*t)*cos(0.5*pi*y);
Expr gWestPrev = -pi*exp(-lambda*tPrev)*cos(0.5*pi*y);
/* Create the weak form */
Expr eqn = Integral(interior, v*(u-uPrev)/deltaT
+ 0.5*(grad*v)*(grad*u + grad*uPrev), quad)
+ Integral(west, -0.5*v*(gWest+gWestPrev), quad);
Expr bc = EssentialBC(east + north, v*u, quad);
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
LinearSolver<double> solver
= LinearSolverBuilder::createSolver("amesos.xml");
FieldWriter w0 = new VTKWriter("TransientHeat2D-0");
w0.addMesh(mesh);
w0.addField("T", new ExprFieldWrapper(uPrev[0]));
w0.write();
for (int i=0; i<nt; i++)
{
t.setParameterValue((i+1)*deltaT);
tPrev.setParameterValue(i*deltaT);
Out::root() << "t=" << (i+1)*deltaT << endl;
Expr uNext = prob.solve(solver);
ostringstream oss;
oss << "TransientHeat2D-" << i+1;
FieldWriter w = new VTKWriter(oss.str());
w.addMesh(mesh);
w.addField("T", new ExprFieldWrapper(uNext[0]));
w.write();
updateDiscreteFunction(uNext, uPrev);
}
double err = L2Norm(mesh, interior, uExact-uPrev, quad);
Out::root() << "error norm=" << err << endl;
double h = 1.0/(nx-1.0);
double tol = 0.1*(pow(h,2.0) + pow(lambda*deltaT, 2.0));
Out::root() << "tol=" << tol << endl;
Sundance::passFailTest(err, tol);
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
return Sundance::testStatus();
}
int main(int argc, char** argv)
{
try
{
Sundance::init(&argc, &argv);
/* We will do our linear algebra using Epetra */
VectorType<double> vecType = new EpetraVectorType();
/* Create a mesh. It will be of type BasisSimplicialMesh, and will
* be built using a PartitionedLineMesher. */
MeshType meshType = new BasicSimplicialMeshType();
int nx = 32;
MeshSource mesher = new PartitionedRectangleMesher(
0.0, 1.0, nx,
0.0, 1.0, nx,
meshType);
Mesh mesh = mesher.getMesh();
/* Create a cell filter that will identify the maximal cells
* in the interior of the domain */
CellFilter interior = new MaximalCellFilter();
/* Make cell filters for the east and west boundaries */
CellFilter edges = new DimensionalCellFilter(1);
CellFilter west = edges.coordSubset(0, 0.0);
CellFilter east = edges.coordSubset(0, 1.0);
/* Create unknown and test functions */
Expr u = new UnknownFunction(new Lagrange(1), "u");
Expr v = new TestFunction(new Lagrange(1), "v");
/* Create differential operator and coordinate function */
Expr x = new CoordExpr(0);
Expr grad = gradient(1);
/* We need a quadrature rule for doing the integrations */
QuadratureFamily quad = new GaussianQuadrature(4);
/* Define the parameter */
Expr xi = new Sundance::Parameter(0.0);
/* Construct a forcing term to provide an exact solution for
* validation purposes. This will involve the parameter. */
Expr uEx = x*(1.0-x)*(1.0+xi*exp(x));
Expr f = -(-20 - exp(x)*xi*(1 + 32*x + 10*x*x +
exp(x)*(-1 + 2*x*(2 + x))*xi))/10.0;
/* Define the weak form, using the parameter expression. This weak form
* can be used for all parameter values. */
Expr eqn = Integral(interior,
(1.0 + 0.1*xi*exp(x))*(grad*v)*(grad*u) - f*v, quad);
/* Define the Dirichlet BC */
Expr h = new CellDiameterExpr();
Expr bc = EssentialBC(east + west, v*u/h, quad);
/* We can now set up the linear problem. This can be reused
* for different parameter values. */
LinearProblem prob(mesh, eqn, bc, v, u, vecType);
/* make a projector for the exact solution. Just like the
* problem, this can be reused for different parameter values. */
DiscreteSpace ds(mesh, new Lagrange(1), vecType);
L2Projector proj(ds, uEx);
/* Get the solver and declare variables for the results */
LinearSolver<double> solver = LinearSolverBuilder::createSolver("aztec-ml.xml");
Expr soln;
SolverState<double> state;
/* Set up the sweep from xi=0 to xi=xiMax in nSteps steps. */
int nSteps = 10;
double xiMax = 2.0;
/* Make an array in which to keep the observed errors */
Array<double> err(nSteps);
/* Do the sweep */
for (int n=0; n<nSteps; n++)
{
/* Update the parameter value */
double xiVal = xiMax*n/(nSteps - 1.0);
xi.setParameterValue(xiVal);
Out::root() << "step n=" << n << " of " << nSteps << " xi=" << xiVal;
/* Solve the problem. The updated parameter value is automatically used. */
state = prob.solve(solver, soln);
TEUCHOS_TEST_FOR_EXCEPTION(state.finalState() != SolveConverged,
std::runtime_error,
"solve failed!");
/* Project the exact solution onto a discrrete space for viz. The updated
* parameter value is automatically used. */
Expr uEx0 = proj.project();
/* Write the approximate and exact solutions for viz */
FieldWriter w = new VTKWriter("ParameterSweep-" + Teuchos::toString(n));
w.addMesh(mesh);
w.addField("u", new ExprFieldWrapper(soln[0]));
w.addField("uEx", new ExprFieldWrapper(uEx0[0]));
w.write();
/* Compute the L2 norm of the error */
err[n] = L2Norm(mesh, interior, soln-uEx, quad);
Out::root() << " L2 error = " << err[n] << endl;
}
/* The errors are O(h^2), so use that to set a tolerance */
double hVal = 1.0/(nx-1.0);
double fudge = 2.0;
double tol = fudge*hVal*hVal;
/* Find the max error over all parameter values */
double maxErr = *std::max_element(err.begin(), err.end());
/* Check the error */
Sundance::passFailTest(maxErr, tol);
}
catch(std::exception& e)
{
Sundance::handleException(e);
}
Sundance::finalize();
return Sundance::testStatus();
}
