HexNodeBasis::HexNodeBasis(size_t order){
// Reference Space //
refSpace = new HexReferenceSpace;
nRefSpace = getReferenceSpace().getNReferenceSpace();
const vector<vector<vector<size_t> > >&
edgeIdx = refSpace->getEdgeNodeIndex();
const vector<vector<vector<size_t> > >&
faceIdx = refSpace->getFaceNodeIndex();
// Set Basis Type //
this->order = order;
type = 0;
dim = 3;
nVertex = 8;
nEdge = 12 * (order - 1);
nFace = 6 * (order - 1) * (order - 1);
nCell = (order - 1) * (order - 1) * (order - 1);
nFunction = nVertex + nEdge + nFace + nCell;
// Legendre Polynomial //
Polynomial* legendre = new Polynomial[order];
Legendre::integrated(legendre, order);
// Lagrange & Lifting //
const Polynomial lagrange[8] =
{
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 1, 0)) *
(Polynomial(1, 0, 0, 1)))
};
const Polynomial lifting[8] =
{
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 1))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 1, 0)) +
(Polynomial(1, 0, 0, 1)))
};
// Basis //
basis = new Polynomial**[nRefSpace];
for(size_t s = 0; s < nRefSpace; s++)
basis[s] = new Polynomial*[nFunction];
// Vertex Based //
for(size_t s = 0; s < nRefSpace; s++){
basis[s][0] = new Polynomial(lagrange[0]);
basis[s][1] = new Polynomial(lagrange[1]);
basis[s][2] = new Polynomial(lagrange[2]);
basis[s][3] = new Polynomial(lagrange[3]);
basis[s][4] = new Polynomial(lagrange[4]);
basis[s][5] = new Polynomial(lagrange[5]);
basis[s][6] = new Polynomial(lagrange[6]);
basis[s][7] = new Polynomial(lagrange[7]);
}
// Edge Based //
for(size_t s = 0; s < nRefSpace; s++){
size_t i = nVertex;
for(size_t e = 0; e < 12; e++){
for(size_t l = 1; l < order; l++){
basis[s][i] =
new Polynomial(legendre[l].compose(lifting[edgeIdx[s][e][1]] -
lifting[edgeIdx[s][e][0]])
*
(lagrange[edgeIdx[s][e][0]] +
lagrange[edgeIdx[s][e][1]]));
i++;
}
}
}
// Face Based //
for(size_t s = 0; s < nRefSpace; s++){
size_t i = nVertex + nEdge;
for(size_t f = 0; f < 6; f++){
for(size_t l1 = 1; l1 < order; l1++){
for(size_t l2 = 1; l2 < order; l2++){
Polynomial sum =
lagrange[faceIdx[s][f][0]] +
lagrange[faceIdx[s][f][1]] +
lagrange[faceIdx[s][f][2]] +
lagrange[faceIdx[s][f][3]];
basis[s][i] =
new Polynomial(legendre[l1].compose(lifting[faceIdx[s][f][0]] -
lifting[faceIdx[s][f][1]]) *
legendre[l2].compose(lifting[faceIdx[s][f][0]] -
lifting[faceIdx[s][f][3]]) *
sum);
i++;
}
}
}
}
// Cell Based //
Polynomial px = Polynomial(2, 1, 0, 0);
Polynomial py = Polynomial(2, 0, 1, 0);
Polynomial pz = Polynomial(2, 0, 0, 1);
px = px - Polynomial(1, 0, 0, 0);
py = py - Polynomial(1, 0, 0, 0);
pz = pz - Polynomial(1, 0, 0, 0);
for(size_t s = 0; s < nRefSpace; s++){
size_t i = nVertex + nEdge + nFace;
for(size_t l1 = 1; l1 < order; l1++){
for(size_t l2 = 1; l2 < order; l2++){
for(size_t l3 = 1; l3 < order; l3++){
basis[s][i] =
new Polynomial(legendre[l1].compose(px) *
legendre[l2].compose(py) *
legendre[l3].compose(pz));
i++;
}
}
}
}
// Mapping to Gmsh Quad //
// x = (u + 1) / 2
// y = (v + 1) / 2
//
// (x, y) = Zaglmayr Ref Quad
// (u, v) = Gmsh Ref Quad
Polynomial mapX(Polynomial(0.5, 1, 0, 0) +
Polynomial(0.5, 0, 0, 0));
Polynomial mapY(Polynomial(0.5, 0, 1, 0) +
Polynomial(0.5, 0, 0, 0));
Polynomial mapZ(Polynomial(0.5, 0, 0, 1) +
Polynomial(0.5, 0, 0, 0));
for(size_t s = 0; s < nRefSpace; s++){
for(size_t i = 0; i < nFunction; i++){
Polynomial* tmp;
tmp = basis[s][i];
basis[s][i] = new Polynomial(tmp->compose(mapX, mapY, mapZ));
delete tmp;
}
}
// Free Temporary Sapce //
delete[] legendre;
}
QuadNodeBasis::QuadNodeBasis(size_t order){
// Set Basis Type //
this->order = order;
type = TYPE_QUA;
dim = 2;
nVertex = 4;
nEdge = 4 * (order - 1);
nFace = (order - 1) * (order - 1);
nCell = 0;
nFunction = nVertex + nEdge + nFace + nCell;
// Reference Space //
const size_t nOrientation = ReferenceSpaceManager::getNOrientation(type);
const vector<vector<vector<size_t> > >&
edgeIdx = ReferenceSpaceManager::getEdgeNodeIndex(type);
const vector<vector<vector<size_t> > >&
faceIdx = ReferenceSpaceManager::getFaceNodeIndex(type);
// Legendre Polynomial //
Polynomial* legendre = new Polynomial[order];
Legendre::integrated(legendre, order);
// Lagrange & Lifting //
const Polynomial lagrange[4] =
{
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0))),
Polynomial((Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0))),
Polynomial((Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 1, 0))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) *
(Polynomial(1, 0, 1, 0)))
};
const Polynomial lifting[4] =
{
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0))),
Polynomial((Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 0, 0) - Polynomial(1, 0, 1, 0))),
Polynomial((Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 1, 0))),
Polynomial((Polynomial(1, 0, 0, 0) - Polynomial(1, 1, 0, 0)) +
(Polynomial(1, 0, 1, 0)))
};
// Basis //
basis = new Polynomial**[nOrientation];
for(size_t s = 0; s < nOrientation; s++)
basis[s] = new Polynomial*[nFunction];
// Vertex Based //
for(size_t s = 0; s < nOrientation; s++){
basis[s][0] = new Polynomial(lagrange[0]);
basis[s][1] = new Polynomial(lagrange[1]);
basis[s][2] = new Polynomial(lagrange[2]);
basis[s][3] = new Polynomial(lagrange[3]);
}
// Edge Based //
for(size_t s = 0; s < nOrientation; s++){
size_t i = nVertex;
for(size_t e = 0; e < 4; e++){
for(size_t l = 1; l < order; l++){
basis[s][i] =
new Polynomial(legendre[l].compose(lifting[edgeIdx[s][e][1]] -
lifting[edgeIdx[s][e][0]])
*
(lagrange[edgeIdx[s][e][0]] +
lagrange[edgeIdx[s][e][1]]));
i++;
}
}
}
// Face Based //
// NB: We use (*(*faceIdx[s])[f])[]
// where f = 0, because triangles
// have only ONE face: the face '0'
for(size_t s = 0; s < nOrientation; s++){
size_t i = nVertex + nEdge;
for(size_t l1 = 1; l1 < order; l1++){
for(size_t l2 = 1; l2 < order; l2++){
basis[s][i] =
new Polynomial(legendre[l1].compose(lifting[faceIdx[s][0][0]] -
lifting[faceIdx[s][0][1]])
*
legendre[l2].compose(lifting[faceIdx[s][0][0]] -
lifting[faceIdx[s][0][3]]));
i++;
}
}
}
// Mapping to Gmsh Quad //
// x = (u + 1) / 2
// y = (v + 1) / 2
//
// (x, y) = Zaglmayr Ref Quad
// (u, v) = Gmsh Ref Quad
Polynomial mapX(Polynomial(0.5, 1, 0, 0) +
Polynomial(0.5, 0, 0, 0));
Polynomial mapY(Polynomial(0.5, 0, 1, 0) +
Polynomial(0.5, 0, 0, 0));
for(size_t s = 0; s < nOrientation; s++){
for(size_t i = 0; i < nFunction; i++){
Polynomial* tmp;
tmp = basis[s][i];
basis[s][i] = new Polynomial(tmp->compose(mapX, mapY));
delete tmp;
}
}
// Free Temporary Sapce //
delete[] legendre;
}