Basis_HDIV_TRI_In_FEM<Scalar,ArrayScalar>::Basis_HDIV_TRI_In_FEM( const int n ,
const EPointType pointType ):
Phis( n ),
coeffs( (n+1)*(n+2) , n*(n+2) )
{
const int N = n*(n+2);
this -> basisCardinality_ = N;
this -> basisDegree_ = n;
this -> basisCellTopology_ = shards::CellTopology(shards::getCellTopologyData<shards::Triangle<3> >() );
this -> basisType_ = BASIS_FEM_FIAT;
this -> basisCoordinates_ = COORDINATES_CARTESIAN;
this -> basisTagsAreSet_ = false;
const int littleN = n*(n+1); // dim of (P_{n-1})^2 -- smaller space
const int bigN = (n+1)*(n+2); // dim of (P_{n})^2 -- larger space
const int scalarSmallestN = (n-1)*n / 2;
const int scalarLittleN = littleN/2;
const int scalarBigN = bigN/2;
// first, need to project the basis for RT space onto the
// orthogonal basis of degree n
// get coefficients of PkHx
Teuchos::SerialDenseMatrix<int,Scalar> V1(bigN, N);
// basis for the space is
// { (phi_i,0) }_{i=0}^{scalarLittleN-1} ,
// { (0,phi_i) }_{i=0}^{scalarLittleN-1} ,
// { (x,y) . phi_i}_{i=scalarLittleN}^{scalarBigN-1}
// columns of V1 are expansion of this basis in terms of the basis
// for P_{n}^2
// these two loops get the first two sets of basis functions
for (int i=0;i<scalarLittleN;i++) {
V1(i,i) = 1.0;
V1(scalarBigN+i,scalarLittleN+i) = 1.0;
}
// now I need to integrate { (x,y) phi } against the big basis
// first, get a cubature rule.
CubatureDirectTriDefault<Scalar,ArrayScalar > myCub( 2 * n );
ArrayScalar cubPoints( myCub.getNumPoints() , 2 );
ArrayScalar cubWeights( myCub.getNumPoints() );
myCub.getCubature( cubPoints , cubWeights );
// tabulate the scalar orthonormal basis at cubature points
ArrayScalar phisAtCubPoints( scalarBigN , myCub.getNumPoints() );
Phis.getValues( phisAtCubPoints , cubPoints , OPERATOR_VALUE );
// now do the integration
for (int i=0;i<n;i++) {
for (int j=0;j<scalarBigN;j++) { // int (x,y) phi_i \cdot (phi_j,0)
V1(j,littleN+i) = 0.0;
for (int k=0;k<myCub.getNumPoints();k++) {
V1(j,littleN+i) +=
cubWeights(k) * cubPoints(k,0)
* phisAtCubPoints(scalarSmallestN+i,k)
* phisAtCubPoints(j,k);
}
}
for (int j=0;j<scalarBigN;j++) { // int (x,y) phi_i \cdot (0,phi_j)
V1(j+scalarBigN,littleN+i) = 0.0;
for (int k=0;k<myCub.getNumPoints();k++) {
V1(j+scalarBigN,littleN+i) +=
cubWeights(k) * cubPoints(k,1)
* phisAtCubPoints(scalarSmallestN+i,k)
* phisAtCubPoints(j,k);
}
}
}
//std::cout << V1 << "\n";
// next, apply the RT nodes (rows) to the basis for (P_n)^2 (columns)
Teuchos::SerialDenseMatrix<int,Scalar> V2(N , bigN);
// first 3 * degree nodes are normals at each edge
// get the points on the line
ArrayScalar linePts( n , 1 );
if (pointType == POINTTYPE_WARPBLEND) {
CubatureDirectLineGauss<Scalar> edgeRule( n );
ArrayScalar edgeCubWts( n );
edgeRule.getCubature( linePts , edgeCubWts );
}
else if (pointType == POINTTYPE_EQUISPACED ) {
shards::CellTopology linetop(shards::getCellTopologyData<shards::Line<2> >() );
PointTools::getLattice<Scalar,ArrayScalar >( linePts ,
linetop ,
n+1 , 1 ,
POINTTYPE_EQUISPACED );
}
// holds the image of the line points
ArrayScalar edgePts( n , 2 );
ArrayScalar phisAtEdgePoints( scalarBigN , n );
// these are scaled by the appropriate edge lengths.
const Scalar nx[] = {0.0,1.0,-1.0};
const Scalar ny[] = {-1.0,1.0,0.0};
for (int i=0;i<3;i++) { // loop over edges
CellTools<Scalar>::mapToReferenceSubcell( edgePts ,
linePts ,
1 ,
i ,
this->basisCellTopology_ );
Phis.getValues( phisAtEdgePoints , edgePts , OPERATOR_VALUE );
// loop over points (rows of V2)
for (int j=0;j<n;j++) {
// loop over orthonormal basis functions (columns of V2)
for (int k=0;k<scalarBigN;k++) {
V2(n*i+j,k) = nx[i] * phisAtEdgePoints(k,j);
V2(n*i+j,k+scalarBigN) = ny[i] * phisAtEdgePoints(k,j);
}
}
}
// next map the points to each edge
// remaining nodes are divided into two pieces: point value of x
// components and point values of y components. These are
// evaluated at the interior of a lattice of degree + 1, For then
// the degree == 1 space corresponds classicaly to RT0 and so gets
// no internal nodes, and degree == 2 corresponds to RT1 and needs
// one internal node per vector component.
const int numInternalPoints = PointTools::getLatticeSize( this->getBaseCellTopology() ,
n + 1 ,
1 );
if (numInternalPoints > 0) {
ArrayScalar internalPoints( numInternalPoints , 2 );
PointTools::getLattice<Scalar,ArrayScalar >( internalPoints ,
this->getBaseCellTopology() ,
n + 1 ,
1 ,
pointType );
ArrayScalar phisAtInternalPoints( scalarBigN , numInternalPoints );
Phis.getValues( phisAtInternalPoints , internalPoints , OPERATOR_VALUE );
// copy values into right positions of V2
for (int i=0;i<numInternalPoints;i++) {
for (int j=0;j<scalarBigN;j++) {
// x component
V2(3*n+i,j) = phisAtInternalPoints(j,i);
// y component
V2(3*n+numInternalPoints+i,scalarBigN+j) = phisAtInternalPoints(j,i);
}
}
}
// std::cout << "Nodes on big basis\n";
// std::cout << V2 << "\n";
// std::cout << "End nodes\n";
Teuchos::SerialDenseMatrix<int,Scalar> Vsdm( N , N );
// multiply V2 * V1 --> V
Vsdm.multiply( Teuchos::NO_TRANS , Teuchos::NO_TRANS , 1.0 , V2 , V1 , 0.0 );
// std::cout << "Vandermonde:\n";
// std::cout << Vsdm << "\n";
// std::cout << "End Vandermonde\n";
Teuchos::SerialDenseSolver<int,Scalar> solver;
solver.setMatrix( rcp( &Vsdm , false ) );
solver.invert( );
Teuchos::SerialDenseMatrix<int,Scalar> Csdm( bigN , N );
Csdm.multiply( Teuchos::NO_TRANS , Teuchos::NO_TRANS , 1.0 , V1 , Vsdm , 0.0 );
// std::cout << Csdm << "\n";
for (int i=0;i<bigN;i++) {
for (int j=0;j<N;j++) {
coeffs(i,j) = Csdm(i,j);
}
}
}
Basis_HCURL_TRI_In_FEM<Scalar,ArrayScalar>::Basis_HCURL_TRI_In_FEM( const int n ,
const EPointType pointType ):
Phis_( n ),
coeffs_( (n+1)*(n+2) , n*(n+2) )
{
const int N = n*(n+2);
this -> basisCardinality_ = N;
this -> basisDegree_ = n;
this -> basisCellTopology_ = shards::CellTopology(shards::getCellTopologyData<shards::Triangle<3> >() );
this -> basisType_ = BASIS_FEM_FIAT;
this -> basisCoordinates_ = COORDINATES_CARTESIAN;
this -> basisTagsAreSet_ = false;
const int littleN = n*(n+1); // dim of (P_{n-1})^2 -- smaller space
const int bigN = (n+1)*(n+2); // dim of (P_{n})^2 -- larger space
const int scalarSmallestN = (n-1)*n / 2;
const int scalarLittleN = littleN/2;
const int scalarBigN = bigN/2;
// first, need to project the basis for Nedelec space onto the
// orthogonal basis of degree n
// get coefficients of PkHx
Teuchos::SerialDenseMatrix<int,Scalar> V1(bigN, N);
// basis for the space is
// { (phi_i,0) }_{i=0}^{scalarLittleN-1} ,
// { (0,phi_i) }_{i=0}^{scalarLittleN-1} ,
// { (x,y) \times phi_i}_{i=scalarLittleN}^{scalarBigN-1}
// { (x,y) \times phi = (y phi , -x \phi)
// columns of V1 are expansion of this basis in terms of the basis
// for P_{n}^2
// these two loops get the first two sets of basis functions
for (int i=0;i<scalarLittleN;i++) {
V1(i,i) = 1.0;
V1(scalarBigN+i,scalarLittleN+i) = 1.0;
}
// now I need to integrate { (x,y) \times phi } against the big basis
// first, get a cubature rule.
CubatureDirectTriDefault<Scalar,ArrayScalar > myCub( 2 * n );
ArrayScalar cubPoints( myCub.getNumPoints() , 2 );
ArrayScalar cubWeights( myCub.getNumPoints() );
myCub.getCubature( cubPoints , cubWeights );
// tabulate the scalar orthonormal basis at cubature points
ArrayScalar phisAtCubPoints( scalarBigN , myCub.getNumPoints() );
Phis_.getValues( phisAtCubPoints , cubPoints , OPERATOR_VALUE );
// now do the integration
for (int i=0;i<n;i++) {
for (int j=0;j<scalarBigN;j++) { // int (x,y) phi_i \cdot (phi_j,0)
V1(j,littleN+i) = 0.0;
for (int k=0;k<myCub.getNumPoints();k++) {
V1(j,littleN+i) -=
cubWeights(k) * cubPoints(k,1)
* phisAtCubPoints(scalarSmallestN+i,k)
* phisAtCubPoints(j,k);
}
}
for (int j=0;j<scalarBigN;j++) { // int (x,y) phi_i \cdot (0,phi_j)
V1(j+scalarBigN,littleN+i) = 0.0;
for (int k=0;k<myCub.getNumPoints();k++) {
V1(j+scalarBigN,littleN+i) +=
cubWeights(k) * cubPoints(k,0)
* phisAtCubPoints(scalarSmallestN+i,k)
* phisAtCubPoints(j,k);
}
}
}
//std::cout << V1 << "\n";
// next, apply the RT nodes (rows) to the basis for (P_n)^2 (columns)
Teuchos::SerialDenseMatrix<int,Scalar> V2(N , bigN);
// first 3 * degree nodes are normals at each edge
// get the points on the line
ArrayScalar linePts( n , 1 );
if (pointType == POINTTYPE_WARPBLEND) {
CubatureDirectLineGauss<Scalar> edgeRule( 2*n - 1 );
ArrayScalar edgeCubWts( n );
edgeRule.getCubature( linePts , edgeCubWts );
}
else if (pointType == POINTTYPE_EQUISPACED ) {
shards::CellTopology linetop(shards::getCellTopologyData<shards::Line<2> >() );
PointTools::getLattice<Scalar,ArrayScalar >( linePts ,
linetop ,
n+1 , 1 ,
POINTTYPE_EQUISPACED );
}
ArrayScalar edgePts( n , 2 );
ArrayScalar phisAtEdgePoints( scalarBigN , n );
ArrayScalar edgeTan(2);
for (int i=0;i<3;i++) { // loop over edges
CellTools<Scalar>::getReferenceEdgeTangent( edgeTan ,
i ,
this->basisCellTopology_ );
/* multiply by 2.0 to account for a Jacobian in Pavel's definition */
for (int j=0;j<2;j++) {
edgeTan(j) *= 2.0;
}
CellTools<Scalar>::mapToReferenceSubcell( edgePts ,
linePts ,
1 ,
i ,
this->basisCellTopology_ );
Phis_.getValues( phisAtEdgePoints , edgePts , OPERATOR_VALUE );
// loop over points (rows of V2)
for (int j=0;j<n;j++) {
// loop over orthonormal basis functions (columns of V2)
for (int k=0;k<scalarBigN;k++) {
V2(n*i+j,k) = edgeTan(0) * phisAtEdgePoints(k,j);
V2(n*i+j,k+scalarBigN) = edgeTan(1) * phisAtEdgePoints(k,j);
}
}
}
// remaining nodes are x- and y- components at internal points, if n > 1
// this code is exactly the same as it is for HDIV
const int numInternalPoints = PointTools::getLatticeSize( this->getBaseCellTopology() ,
n + 1 ,
1 );
if (numInternalPoints > 0) {
ArrayScalar internalPoints( numInternalPoints , 2 );
PointTools::getLattice<Scalar,ArrayScalar >( internalPoints ,
this->getBaseCellTopology() ,
n + 1 ,
1 ,
pointType );
ArrayScalar phisAtInternalPoints( scalarBigN , numInternalPoints );
Phis_.getValues( phisAtInternalPoints , internalPoints , OPERATOR_VALUE );
// copy values into right positions of V2
for (int i=0;i<numInternalPoints;i++) {
for (int j=0;j<scalarBigN;j++) {
// x component
V2(3*n+i,j) = phisAtInternalPoints(j,i);
// y component
V2(3*n+numInternalPoints+i,scalarBigN+j) = phisAtInternalPoints(j,i);
}
}
}
// std::cout << "Nodes on big basis\n";
// std::cout << V2 << "\n";
// std::cout << "End nodes\n";
Teuchos::SerialDenseMatrix<int,Scalar> Vsdm( N , N );
// multiply V2 * V1 --> V
Vsdm.multiply( Teuchos::NO_TRANS , Teuchos::NO_TRANS , 1.0 , V2 , V1 , 0.0 );
// std::cout << "Vandermonde:\n";
// std::cout << Vsdm << "\n";
// std::cout << "End Vandermonde\n";
Teuchos::SerialDenseSolver<int,Scalar> solver;
solver.setMatrix( rcp( &Vsdm , false ) );
solver.invert( );
Teuchos::SerialDenseMatrix<int,Scalar> Csdm( bigN , N );
Csdm.multiply( Teuchos::NO_TRANS , Teuchos::NO_TRANS , 1.0 , V1 , Vsdm , 0.0 );
// std::cout << Csdm << "\n";
for (int i=0;i<bigN;i++) {
for (int j=0;j<N;j++) {
coeffs_(i,j) = Csdm(i,j);
}
}
initializeTags();
this->basisTagsAreSet_ = true;
}
