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Hörl, Maximilian authored
dg.hh 14.18 KiB
#ifndef DUNE_MMDG_DG_HH
#define DUNE_MMDG_DG_HH
#include <cmath>
#include <dune/common/dynmatrix.hh>
#include <dune/common/dynvector.hh>
#include <dune/grid/io/file/vtk.hh>
#include <dune/geometry/quadraturerules.hh>
#include <dune/mmdg/nonconformingp1vtkfunction.hh>
template<class GridView, class Mapper, class Problem>
class DG
{
public:
using Scalar = typename GridView::ctype;
static constexpr int dim = GridView::dimension;
using Matrix = Dune::DynamicMatrix<Scalar>; //NOTE: what is an appropriate sparse matrix type? -> BCRS
using Vector = Dune::DynamicVector<Scalar>;
using VTKFunction = Dune::VTKFunction<GridView>;
using P1Function = Dune::NonconformingP1VTKFunction<GridView,
Dune::DynamicVector<double>>;
//constructor
DG (const GridView& gridView, const Mapper& mapper,
const Problem& problem) :
gridView_(gridView), mapper_(mapper), problem_(problem),
dof((1 + dim) * gridView.size(0))
{
A = Matrix(dof, dof, 0.0); //initialize stiffness matrix
b = Vector(dof, 0.0); //initialize load vector
d = Vector(dof, 0.0); //initialize solution vector
}
const void operator() (const Scalar K, const Scalar mu)
{
//assemble stiffness matrix A and load vector b
assembleSLE(K, mu);
//NOTE: what would be an appropiate solver here?
A.solve(d, b);
//write solution to a vtk file
writeVTKoutput();
}
const GridView& gridView_;
const Mapper& mapper_; //element mapper for gridView_
const Problem& problem_; //the DG problem to be solved
const int dof; //degrees of freedom
private:
//assemble stiffness matrix A and load vector b
void assembleSLE (const Scalar K, const Scalar mu)
{
//we use the basis
// phi_elem,0 (x) = indicator(elem);
// phi_elem,i (x) = x[i]*indicator(elem);
//for elem in elements(gridView) and i = 1,...,dim
for (const auto& elem : elements(gridView_))
{
const int elemIdx = mapper_.index(elem);
const auto& geo = elem.geometry();
const double elemVol = geo.volume();
const auto& center = geo.center();
//in the system of linear equations (SLE) Ad = b,
//the index elemIdxSLE refers to the basis function phi_elem,0
//and the indices elemIdxSLE + i + 1, i = 1,...,dim, refer to the
//basis function phi_elem,i
const int elemIdxSLE = (dim + 1)*elemIdx;
/* //TODO: can be done outside of the loop?
const Dune::QuadratureRule<double,dim>& rule =
Dune::QuadratureRules<double,dim>::rule(geo.type(), problem_.quadratureOrder());
Dune::FieldVector<double, dim+1> update(0.0);
//NOTE: how are quadrature rules in Dune applied correctly?
for (const auto& ip : rule)
{
const auto& qp = ip.position();
//NOTE: is the volume of the element taken into account
//automatically?
const double weight = ip.weight();
//quadrature for int_elem q*phi_elem,0 dV
b[elemIdxSLE] += weight * problem_.q(qp);
//quadrature for int_elem q*phi_elem,i dV
for (int i = 0; i < dim; i++)
{
b[elemIdxSLE + i + 1] += weight * qp[i] * problem_.q(qp);
}
}
*/
//NOTE: makeshift solution for source term q = -1
b[elemIdxSLE] += -elemVol;
for (int i = 0; i < dim; i++)
{
b[elemIdxSLE + i + 1] += -elemVol * center[i];
}
for (int i = 0; i < dim; i++)
{
//exact evaluation of
// int_elem K*grad(phi_elem,i)*grad(phi_elem,i) dV
A[elemIdxSLE + i + 1][elemIdxSLE + i + 1] += K * elemVol;
}
//iterate over all intersection with the boundary of elem
for (const auto& intersct : intersections(gridView_, elem))
{
const auto& normal = intersct.centerUnitOuterNormal();
const auto& intersctGeo = intersct.geometry();
const double intersctVol = intersctGeo.volume();
const auto& intersctCenter = intersctGeo.center();
//TODO: quadrature rule cannot be used for dim = 1!
// const Dune::QuadratureRule<double,dim-1>& secondOrderRule =
// Dune::QuadratureRules<double,dim-1>::rule(
// intersct.type(), 2);
//storage for multiply used integral values,
//note that only the lower diagonal and diagonal entries of
//quadraticIntregrals are used
Dune::FieldVector<Scalar, dim> linearIntegrals(0.0);
Dune::FieldMatrix<Scalar, dim, dim> quadraticIntregrals(0.0);
//NOTE: is there a better type for a symmetric matrix?
for (int i = 0; i < dim; i++)
{
//use midpoint rule for exact evaluation of
// int_intersct x_i ds
linearIntegrals[i] = intersctVol * intersctCenter[i];
for (int j = 0; j <= i; j++) {
const auto& leftCorner = intersctGeo.corner(0);
const auto& rightCorner = intersctGeo.corner(1);
quadraticIntregrals[i][j] = intersctVol / 3 *
( leftCorner[i] * leftCorner[j] + rightCorner[i] * rightCorner[j]
+ 0.5 * (leftCorner[i] * rightCorner[j] +
leftCorner[j] * rightCorner[i]) );
/*
//use second order quadrature rule for exact evaluation of
// int_intersct x_i*x_j ds
//NOTE: use Simpson's rule instead manually?
for (const auto& ip : secondOrderRule)
{
const auto& qp = ip.position();
quadraticIntregrals[i][j] += //NOTE: volume necessary?
ip.weight() * qp[i] * qp[j] * intersctVol;
}
*/
//NOTE: probably unnecessary
quadraticIntregrals[j][i] = quadraticIntregrals[i][j];
}
}
//exact evaluation of
// int_intersct mu*jump(phi_elem,0)*jump(phi_elem,0) ds
A[elemIdxSLE][elemIdxSLE] += mu * intersctVol;
if (intersct.neighbor()) //intersct has neighboring element
{
//index of the neighboring element
const int neighborIdx = mapper_.index(intersct.outside());
const int neighborIdxSLE = (dim + 1)*neighborIdx;
for (int i = 0; i < dim; i++)
{ //we use the relations
// int_intersct mu*jump(phi_elem,0)*jump(phi_elem,i) ds
// = mu * int_intersct x_i ds
//and
// int_intersct avg(K*grad(phi_elem,i))*jump(phi_elem,0) ds
// = 0.5 * K * normal[i] * vol(intersct)
A[elemIdxSLE + i + 1][elemIdxSLE] +=
mu * linearIntegrals[i] - 0.5 * K * normal[i] * intersctVol;
for (int j = 0; j <= i; j++)
{
//we use the relations
// int_intersct mu*jump(phi_elem,i)*jump(phi_elem,j) ds
// = mu * int_intersct x_i*x_j ds
//and
// int_intersct avg(K*grad(phi_elem,i))*jump(phi_elem,j) ds
// = 0.5 * K * normal[i] * int_intersct x_j ds
A[elemIdxSLE + i + 1][elemIdxSLE + j + 1]
+= mu * quadraticIntregrals[i][j]
- 0.5 * K * (normal[i] * linearIntegrals[j]
+ normal[j] * linearIntegrals[i]);
}
}
if (neighborIdx > elemIdx)
{ //make sure that each facet is considered only once
continue;
}
//exact evaluation of
// int_intersct mu*jump(phi_elem,0)*jump(phi_neighbor,0) ds
A[elemIdxSLE][neighborIdxSLE] += -mu * intersctVol;
//stiffness matrix A is symmetric
A[neighborIdxSLE][elemIdxSLE] += A[elemIdxSLE][neighborIdxSLE];
for (int i = 0; i < dim; i++)
{
//we use the relations
// int_intersct mu*jump(phi_elem,i)
// *jump(phi_neighbor,0) ds
// = -mu*int_intersct x_i ds
//and
// int_intersct avg(K*grad(phi_neighbor,i))
// *jump(phi_elem,0) ds
// = 0.5 * K * normal[i] * vol(intersct)
A[elemIdxSLE + i + 1][neighborIdxSLE] +=
-mu * linearIntegrals[i] + 0.5 * K * normal[i] * intersctVol;
//we use the relations
// int_intersct mu*jump(phi_elem,0)
// *jump(phi_neighbor,i) ds
// = -mu*int_intersct x_i ds
//and
// int_intersct avg(K*grad(phi_neighbor,i))
// *jump(phi_elem,0) ds
// = 0.5 * K * normal[i] * vol(intersct)
A[elemIdxSLE][neighborIdxSLE + i + 1] +=
-mu * linearIntegrals[i] - 0.5 * K * normal[i] * intersctVol;
//stiffness matrix A is symmetric
A[neighborIdxSLE][elemIdxSLE + i + 1] +=
A[elemIdxSLE + i + 1][neighborIdxSLE];
A[neighborIdxSLE + i + 1][elemIdxSLE] +=
A[elemIdxSLE][neighborIdxSLE + i + 1];
for (int j = 0; j <= i; j++)
{
//we use the relations
// int_intersct mu*jump(phi_elem,i)
// *jump(phi_neighbor,j) ds
// = -mu*int_intersct x_i*x_j ds
//and
// int_intersct avg(K*grad(phi_neighbor,j))
// *jump(phi_elem,i) ds
// = 0.5 * K * normal[j] * int_intersct x_i ds
//as well as
// int_intersct avg(K*grad(phi_elem,i))
// *jump(phi_neighbor,j) ds
// = -0.5 * K * normal[i] * int_intersct x_j ds
A[elemIdxSLE + i + 1][neighborIdxSLE + j + 1] +=
-mu * quadraticIntregrals[i][j]
- 0.5 * K * (normal[j] * linearIntegrals[i]
- normal[i] * linearIntegrals[j]);
//stiffness matrix A is symmetric
A[neighborIdxSLE + j + 1][elemIdxSLE + i + 1] +=
A[elemIdxSLE + i + 1][neighborIdxSLE + j + 1];
if (i != j)
{
// int_intersct mu*jump(phi_elem,j)
// *jump(phi_neighbor,i) ds
// = -mu*int_intersct x_i*x_j ds
//and
// int_intersct avg(K*grad(phi_neighbor,i))
// *jump(phi_elem,j) ds
// = 0.5 * K * normal[i] * int_intersct x_j ds
//as well as
// int_intersct avg(K*grad(phi_elem,j))
// *jump(phi_neighbor,i) ds
// = -0.5 * K * normal[j] * int_intersct x_i ds
A[elemIdxSLE + j + 1][neighborIdxSLE + i + 1] +=
-mu * quadraticIntregrals[i][j]
- 0.5 * K * (normal[i] * linearIntegrals[j] - normal[j] * linearIntegrals[i]);
//stiffness matrix A is symmetric
A[neighborIdxSLE + i + 1][elemIdxSLE + j + 1] +=
A[elemIdxSLE + j + 1][neighborIdxSLE + i + 1];
}
}
}
}
else //boundary facet
{
for (int i = 0; i < dim; i++)
{ //we use the relations
// int_intersct mu*jump(phi_elem,0)*jump(phi_elem,i) ds
// = mu * int_intersct x_i ds
//and for boundary facets
// int_intersct avg(K*grad(phi_elem,i))
// *jump(phi_elem,0) ds
// = K * normal[i] * vol(intersct)
A[elemIdxSLE + i + 1][elemIdxSLE] +=
mu * linearIntegrals[i] - 0.5 * K * normal[i] * intersctVol;
for (int j = 0; j <= i; j++)
{
//we use the relations
// int_intersct mu*jump(phi_elem,i)*jump(phi_elem,j) ds
// = mu * int_intersct x_i*x_j ds
//and for boundary facets
// int_intersct avg(K*grad(phi_elem,i))
// *jump(phi_elem,j) ds
// = 0.5 * K * normal[i] * int_intersct x_j ds
A[elemIdxSLE + i + 1][elemIdxSLE + j + 1] +=
mu * quadraticIntregrals[i][j]
- 0.5 * K * (normal[i] * linearIntegrals[j]
+ normal[j] * linearIntegrals[i]);
}
}
}
}
//stiffness matrix A is symmetric
for (int i = 0; i < dim; i++)
{
A[elemIdxSLE][elemIdxSLE + i + 1] = A[elemIdxSLE + i + 1][elemIdxSLE];
for(int j = 0; j < i; j++)
{
A[elemIdxSLE + j + 1][elemIdxSLE + i + 1] =
A[elemIdxSLE + i + 1][elemIdxSLE + j + 1];
}
}
}
//NOTE: check if A is symmetric
for (int i = 0; i < dof; i++)
for (int j = 0; j < i; j++)
/*assert*/if(std::abs(A[i][j] - A[j][i]) >
std::numeric_limits<Scalar>::epsilon())
std::cout << i << ", " << j << std::endl;
}
//writes the solution to a vtk file
void writeVTKoutput () const
{
//storage for pressure data, for each element we store the
//pressure at the corners of the element, the VTKFunction will
//create the output using a linear interpolation for each element
Dune::DynamicVector<Scalar> pressure(dof, 0.0);
Dune::DynamicVector<Scalar> exactPressure(dof, 0.0);
for (const auto& elem : elements(gridView_))
{
const int elemIdxSLE = (dim + 1)*mapper_.index(elem);
const auto& geo = elem.geometry();
for (int k = 0; k < geo.corners(); k++)
{
if (problem_.hasExactSolution())
{
exactPressure[elemIdxSLE + k] = problem_.exactSolution(geo.corner(k));
}
//contribution of the basis function
// phi_elem,0 (x) = indicator(elem);
//at the kth corner of elem
pressure[elemIdxSLE + k] = d[elemIdxSLE];
for (int i = 0; i < dim; i++)
{
//contribution of the basis function
// phi_elem,i (x) = x[i]*indicator(elem);
//at the kth corner of elem
pressure[elemIdxSLE + k] += d[elemIdxSLE + i + 1] * geo.corner(k)[i];
}
}
}
//create output directory if necessary
mkdir("data", S_IRWXU | S_IRWXG | S_IROTH | S_IXOTH);
Dune::VTKWriter<GridView> vtkWriter(gridView_, Dune::VTK::nonconforming);
vtkWriter.addVertexData( std::shared_ptr<const VTKFunction>(
new P1Function(gridView_, pressure, "pressure")));
if (problem_.hasExactSolution())
{
vtkWriter.addVertexData( std::shared_ptr<const VTKFunction>(
new P1Function(gridView_, exactPressure, "exactPressure")));
}
vtkWriter.pwrite("pressureData", "", "data");
}
Matrix A; //stiffness matrix
Vector b; //load vector
Vector d; //solution vector
};
#endif