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c577ee0f43500a2fade34e62cc55cacb44be0840
mfem/fem/fe/fe_h1.cpp
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Veselin Dobrev 8acd5cd3a2 Fix a size bug in thread-safe mode in H1_TriangleElement::CalcHessian. 
Fix use-after-delete bug in nurbs_ex10p.cpp.

Use relative tolerance in the "Collocated Derivative Kernels" unit test
to resolve failures in some setups with the original absolute tolerance.
2026-02-13 10:29:22 -08:00

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// Copyright (c) 2010-2025, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
// H1 Finite Element classes
#include "fe_h1.hpp"
namespace mfem
{
using namespace std;
H1_SegmentElement::H1_SegmentElement(const int p, const int btype)
: NodalTensorFiniteElement(1, p, VerifyClosed(btype), H1_DOF_MAP)
{
const real_t *cp = poly1d.ClosedPoints(p, b_type);
#ifndef MFEM_THREAD_SAFE
shape_x.SetSize(p+1);
dshape_x.SetSize(p+1);
d2shape_x.SetSize(p+1);
#endif
Nodes.IntPoint(0).x = cp[0];
Nodes.IntPoint(1).x = cp[p];
for (int i = 1; i < p; i++)
{
Nodes.IntPoint(i+1).x = cp[i];
}
}
void H1_SegmentElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1);
#endif
basis1d.Eval(ip.x, shape_x);
shape(0) = shape_x(0);
shape(1) = shape_x(p);
for (int i = 1; i < p; i++)
{
shape(i+1) = shape_x(i);
}
}
void H1_SegmentElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), dshape_x(p+1);
#endif
basis1d.Eval(ip.x, shape_x, dshape_x);
dshape(0,0) = dshape_x(0);
dshape(1,0) = dshape_x(p);
for (int i = 1; i < p; i++)
{
dshape(i+1,0) = dshape_x(i);
}
}
void H1_SegmentElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &Hessian) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), dshape_x(p+1), d2shape_x(p+1);
#endif
basis1d.Eval(ip.x, shape_x, dshape_x, d2shape_x);
Hessian(0,0) = d2shape_x(0);
Hessian(1,0) = d2shape_x(p);
for (int i = 1; i < p; i++)
{
Hessian(i+1,0) = d2shape_x(i);
}
}
void H1_SegmentElement::ProjectDelta(int vertex, Vector &dofs) const
{
const int p = order;
const real_t *cp = poly1d.ClosedPoints(p, b_type);
switch (vertex)
{
case 0:
dofs(0) = poly1d.CalcDelta(p, (1.0 - cp[0]));
dofs(1) = poly1d.CalcDelta(p, (1.0 - cp[p]));
for (int i = 1; i < p; i++)
{
dofs(i+1) = poly1d.CalcDelta(p, (1.0 - cp[i]));
}
break;
case 1:
dofs(0) = poly1d.CalcDelta(p, cp[0]);
dofs(1) = poly1d.CalcDelta(p, cp[p]);
for (int i = 1; i < p; i++)
{
dofs(i+1) = poly1d.CalcDelta(p, cp[i]);
}
break;
}
}
H1_QuadrilateralElement::H1_QuadrilateralElement(const int p, const int btype)
: NodalTensorFiniteElement(2, p, VerifyClosed(btype), H1_DOF_MAP)
{
const real_t *cp = poly1d.ClosedPoints(p, b_type);
#ifndef MFEM_THREAD_SAFE
const int p1 = p + 1;
shape_x.SetSize(p1);
shape_y.SetSize(p1);
dshape_x.SetSize(p1);
dshape_y.SetSize(p1);
d2shape_x.SetSize(p1);
d2shape_y.SetSize(p1);
#endif
int o = 0;
for (int j = 0; j <= p; j++)
{
for (int i = 0; i <= p; i++)
{
Nodes.IntPoint(dof_map[o++]).Set2(cp[i], cp[j]);
}
}
}
void H1_QuadrilateralElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1);
#endif
basis1d.Eval(ip.x, shape_x);
basis1d.Eval(ip.y, shape_y);
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
shape(dof_map[o++]) = shape_x(i)*shape_y(j);
}
}
void H1_QuadrilateralElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1), dshape_x(p+1), dshape_y(p+1);
#endif
basis1d.Eval(ip.x, shape_x, dshape_x);
basis1d.Eval(ip.y, shape_y, dshape_y);
for (int o = 0, j = 0; j <= p; j++)
{
for (int i = 0; i <= p; i++)
{
dshape(dof_map[o],0) = dshape_x(i)* shape_y(j);
dshape(dof_map[o],1) = shape_x(i)*dshape_y(j); o++;
}
}
}
void H1_QuadrilateralElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &Hessian) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1), dshape_x(p+1), dshape_y(p+1),
d2shape_x(p+1), d2shape_y(p+1);
#endif
basis1d.Eval(ip.x, shape_x, dshape_x, d2shape_x);
basis1d.Eval(ip.y, shape_y, dshape_y, d2shape_y);
for (int o = 0, j = 0; j <= p; j++)
{
for (int i = 0; i <= p; i++)
{
Hessian(dof_map[o],0) = d2shape_x(i)* shape_y(j);
Hessian(dof_map[o],1) = dshape_x(i)* dshape_y(j);
Hessian(dof_map[o],2) = shape_x(i)*d2shape_y(j); o++;
}
}
}
void H1_QuadrilateralElement::ProjectDelta(int vertex, Vector &dofs) const
{
const int p = order;
const real_t *cp = poly1d.ClosedPoints(p, b_type);
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1);
#endif
for (int i = 0; i <= p; i++)
{
shape_x(i) = poly1d.CalcDelta(p, (1.0 - cp[i]));
shape_y(i) = poly1d.CalcDelta(p, cp[i]);
}
switch (vertex)
{
case 0:
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_x(i)*shape_x(j);
}
break;
case 1:
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_y(i)*shape_x(j);
}
break;
case 2:
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_y(i)*shape_y(j);
}
break;
case 3:
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_x(i)*shape_y(j);
}
break;
}
}
H1_HexahedronElement::H1_HexahedronElement(const int p, const int btype)
: NodalTensorFiniteElement(3, p, VerifyClosed(btype), H1_DOF_MAP)
{
const real_t *cp = poly1d.ClosedPoints(p, b_type);
#ifndef MFEM_THREAD_SAFE
const int p1 = p + 1;
shape_x.SetSize(p1);
shape_y.SetSize(p1);
shape_z.SetSize(p1);
dshape_x.SetSize(p1);
dshape_y.SetSize(p1);
dshape_z.SetSize(p1);
d2shape_x.SetSize(p1);
d2shape_y.SetSize(p1);
d2shape_z.SetSize(p1);
#endif
int o = 0;
for (int k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
Nodes.IntPoint(dof_map[o++]).Set3(cp[i], cp[j], cp[k]);
}
}
void H1_HexahedronElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1), shape_z(p+1);
#endif
basis1d.Eval(ip.x, shape_x);
basis1d.Eval(ip.y, shape_y);
basis1d.Eval(ip.z, shape_z);
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
shape(dof_map[o++]) = shape_x(i)*shape_y(j)*shape_z(k);
}
}
void H1_HexahedronElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1), shape_z(p+1);
Vector dshape_x(p+1), dshape_y(p+1), dshape_z(p+1);
#endif
basis1d.Eval(ip.x, shape_x, dshape_x);
basis1d.Eval(ip.y, shape_y, dshape_y);
basis1d.Eval(ip.z, shape_z, dshape_z);
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dshape(dof_map[o],0) = dshape_x(i)* shape_y(j)* shape_z(k);
dshape(dof_map[o],1) = shape_x(i)*dshape_y(j)* shape_z(k);
dshape(dof_map[o],2) = shape_x(i)* shape_y(j)*dshape_z(k); o++;
}
}
void H1_HexahedronElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &Hessian) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1), shape_z(p+1);
Vector dshape_x(p+1), dshape_y(p+1), dshape_z(p+1);
Vector d2shape_x(p+1), d2shape_y(p+1), d2shape_z(p+1);
#endif
basis1d.Eval(ip.x, shape_x, dshape_x, d2shape_x);
basis1d.Eval(ip.y, shape_y, dshape_y, d2shape_y);
basis1d.Eval(ip.z, shape_z, dshape_z, d2shape_z);
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
Hessian(dof_map[o],0) = d2shape_x(i)* shape_y(j)* shape_z(k);
Hessian(dof_map[o],1) = dshape_x(i)* dshape_y(j)* shape_z(k);
Hessian(dof_map[o],2) = dshape_x(i)* shape_y(j)* dshape_z(k);
Hessian(dof_map[o],3) = shape_x(i)*d2shape_y(j)* shape_z(k);
Hessian(dof_map[o],4) = shape_x(i)* dshape_y(j)* dshape_z(k);
Hessian(dof_map[o],5) = shape_x(i)* shape_y(j)*d2shape_z(k);
o++;
}
}
void H1_HexahedronElement::ProjectDelta(int vertex, Vector &dofs) const
{
const int p = order;
const real_t *cp = poly1d.ClosedPoints(p,b_type);
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p+1), shape_y(p+1);
#endif
for (int i = 0; i <= p; i++)
{
shape_x(i) = poly1d.CalcDelta(p, (1.0 - cp[i]));
shape_y(i) = poly1d.CalcDelta(p, cp[i]);
}
switch (vertex)
{
case 0:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_x(i)*shape_x(j)*shape_x(k);
}
break;
case 1:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_y(i)*shape_x(j)*shape_x(k);
}
break;
case 2:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_y(i)*shape_y(j)*shape_x(k);
}
break;
case 3:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_x(i)*shape_y(j)*shape_x(k);
}
break;
case 4:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_x(i)*shape_x(j)*shape_y(k);
}
break;
case 5:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_y(i)*shape_x(j)*shape_y(k);
}
break;
case 6:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_y(i)*shape_y(j)*shape_y(k);
}
break;
case 7:
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j <= p; j++)
for (int i = 0; i <= p; i++)
{
dofs(dof_map[o++]) = shape_x(i)*shape_y(j)*shape_y(k);
}
break;
}
}
H1_TriangleElement::H1_TriangleElement(const int p, const int btype)
: NodalFiniteElement(2, Geometry::TRIANGLE, ((p + 1)*(p + 2))/2, p,
FunctionSpace::Pk)
{
const real_t *cp = poly1d.ClosedPoints(p, VerifyNodal(VerifyClosed(btype)));
#ifndef MFEM_THREAD_SAFE
shape_x.SetSize(p + 1);
shape_y.SetSize(p + 1);
shape_l.SetSize(p + 1);
dshape_x.SetSize(p + 1);
dshape_y.SetSize(p + 1);
dshape_l.SetSize(p + 1);
ddshape_x.SetSize(p + 1);
ddshape_y.SetSize(p + 1);
ddshape_l.SetSize(p + 1);
u.SetSize(dof);
du.SetSize(dof, dim);
ddu.SetSize(dof, (dim * (dim + 1)) / 2 );
#else
Vector shape_x(p + 1), shape_y(p + 1), shape_l(p + 1);
#endif
int p2p3 = 2*p + 3;
auto idx = [p2p3](int i, int j) { return ((p2p3-j)*j)/2+i; };
lex_ordering.SetSize(dof);
// vertices
lex_ordering[idx(0,0)] = 0;
Nodes.IntPoint(0).Set2(cp[0], cp[0]);
lex_ordering[idx(p,0)] = 1;
Nodes.IntPoint(1).Set2(cp[p], cp[0]);
lex_ordering[idx(0,p)] = 2;
Nodes.IntPoint(2).Set2(cp[0], cp[p]);
// edges
int o = 3;
for (int i = 1; i < p; i++)
{
lex_ordering[idx(i,0)] = o;
Nodes.IntPoint(o++).Set2(cp[i], cp[0]);
}
for (int i = 1; i < p; i++)
{
lex_ordering[idx(p-i,i)] = o;
Nodes.IntPoint(o++).Set2(cp[p-i], cp[i]);
}
for (int i = 1; i < p; i++)
{
lex_ordering[idx(0,p-i)] = o;
Nodes.IntPoint(o++).Set2(cp[0], cp[p-i]);
}
// interior
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++)
{
const real_t w = cp[i] + cp[j] + cp[p-i-j];
lex_ordering[idx(i,j)] = o;
Nodes.IntPoint(o++).Set2(cp[i]/w, cp[j]/w);
}
DenseMatrix T(dof);
for (int k = 0; k < dof; k++)
{
IntegrationPoint &ip = Nodes.IntPoint(k);
poly1d.CalcBasis(p, ip.x, shape_x);
poly1d.CalcBasis(p, ip.y, shape_y);
poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l);
o = 0;
for (int j = 0; j <= p; j++)
for (int i = 0; i + j <= p; i++)
{
T(o++, k) = shape_x(i)*shape_y(j)*shape_l(p-i-j);
}
}
Ti.Factor(T);
// mfem::out << "H1_TriangleElement(" << p << ") : "; Ti.TestInversion();
}
void H1_TriangleElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p + 1), shape_y(p + 1), shape_l(p + 1), u(dof);
#endif
poly1d.CalcBasis(p, ip.x, shape_x);
poly1d.CalcBasis(p, ip.y, shape_y);
poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l);
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i + j <= p; i++)
{
u(o++) = shape_x(i)*shape_y(j)*shape_l(p-i-j);
}
Ti.Mult(u, shape);
}
void H1_TriangleElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p + 1), shape_y(p + 1), shape_l(p + 1);
Vector dshape_x(p + 1), dshape_y(p + 1), dshape_l(p + 1);
DenseMatrix du(dof, dim);
#endif
poly1d.CalcBasis(p, ip.x, shape_x, dshape_x);
poly1d.CalcBasis(p, ip.y, shape_y, dshape_y);
poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l, dshape_l);
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i + j <= p; i++)
{
int k = p - i - j;
du(o,0) = ((dshape_x(i)* shape_l(k)) -
( shape_x(i)*dshape_l(k)))*shape_y(j);
du(o,1) = ((dshape_y(j)* shape_l(k)) -
( shape_y(j)*dshape_l(k)))*shape_x(i);
o++;
}
Ti.Mult(du, dshape);
}
void H1_TriangleElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &ddshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p + 1), shape_y(p + 1), shape_l(p + 1);
Vector dshape_x(p + 1), dshape_y(p + 1), dshape_l(p + 1);
Vector ddshape_x(p + 1), ddshape_y(p + 1), ddshape_l(p + 1);
DenseMatrix ddu(dof, (dim*(dim+1))/2);
#endif
poly1d.CalcBasis(p, ip.x, shape_x, dshape_x, ddshape_x);
poly1d.CalcBasis(p, ip.y, shape_y, dshape_y, ddshape_y);
poly1d.CalcBasis(p, 1. - ip.x - ip.y, shape_l, dshape_l, ddshape_l);
for (int o = 0, j = 0; j <= p; j++)
for (int i = 0; i + j <= p; i++)
{
int k = p - i - j;
// u_xx, u_xy, u_yy
ddu(o,0) = ((ddshape_x(i) * shape_l(k)) - 2. * (dshape_x(i) * dshape_l(k)) +
(shape_x(i) * ddshape_l(k))) * shape_y(j);
ddu(o,1) = (((shape_x(i) * ddshape_l(k)) - dshape_x(i) * dshape_l(k)) * shape_y(
j)) + (((dshape_x(i) * shape_l(k)) - (shape_x(i) * dshape_l(k))) * dshape_y(j));
ddu(o,2) = ((ddshape_y(j) * shape_l(k)) - 2. * (dshape_y(j) * dshape_l(k)) +
(shape_y(j) * ddshape_l(k))) * shape_x(i);
o++;
}
Ti.Mult(ddu, ddshape);
}
H1_TetrahedronElement::H1_TetrahedronElement(const int p, const int btype)
: NodalFiniteElement(3, Geometry::TETRAHEDRON, ((p + 1)*(p + 2)*(p + 3))/6,
p, FunctionSpace::Pk)
{
const real_t *cp = poly1d.ClosedPoints(p, VerifyNodal(VerifyClosed(btype)));
#ifndef MFEM_THREAD_SAFE
shape_x.SetSize(p + 1);
shape_y.SetSize(p + 1);
shape_z.SetSize(p + 1);
shape_l.SetSize(p + 1);
dshape_x.SetSize(p + 1);
dshape_y.SetSize(p + 1);
dshape_z.SetSize(p + 1);
dshape_l.SetSize(p + 1);
ddshape_x.SetSize(p + 1);
ddshape_y.SetSize(p + 1);
ddshape_z.SetSize(p + 1);
ddshape_l.SetSize(p + 1);
u.SetSize(dof);
du.SetSize(dof, dim);
ddu.SetSize(dof, (dim * (dim + 1)) / 2);
#else
Vector shape_x(p + 1), shape_y(p + 1), shape_z(p + 1), shape_l(p + 1);
#endif
auto tri = [](int k) { return (k*(k + 1))/2; };
auto tet = [](int k) { return (k*(k + 1)*(k + 2))/6; };
int ndof = tet(p+1);
auto idx = [tri, tet, p, ndof](int i, int j, int k)
{
return ndof - tet(p - k) - tri(p + 1 - k - j) + i;
};
lex_ordering.SetSize(dof);
// vertices
lex_ordering[idx(0,0,0)] = 0;
Nodes.IntPoint(0).Set3(cp[0], cp[0], cp[0]);
lex_ordering[idx(p,0,0)] = 1;
Nodes.IntPoint(1).Set3(cp[p], cp[0], cp[0]);
lex_ordering[idx(0,p,0)] = 2;
Nodes.IntPoint(2).Set3(cp[0], cp[p], cp[0]);
lex_ordering[idx(0,0,p)] = 3;
Nodes.IntPoint(3).Set3(cp[0], cp[0], cp[p]);
// edges (see Tetrahedron::edges in mesh/tetrahedron.cpp)
int o = 4;
for (int i = 1; i < p; i++) // (0,1)
{
lex_ordering[idx(i,0,0)] = o;
Nodes.IntPoint(o++).Set3(cp[i], cp[0], cp[0]);
}
for (int i = 1; i < p; i++) // (0,2)
{
lex_ordering[idx(0,i,0)] = o;
Nodes.IntPoint(o++).Set3(cp[0], cp[i], cp[0]);
}
for (int i = 1; i < p; i++) // (0,3)
{
lex_ordering[idx(0,0,i)] = o;
Nodes.IntPoint(o++).Set3(cp[0], cp[0], cp[i]);
}
for (int i = 1; i < p; i++) // (1,2)
{
lex_ordering[idx(p-i,i,0)] = o;
Nodes.IntPoint(o++).Set3(cp[p-i], cp[i], cp[0]);
}
for (int i = 1; i < p; i++) // (1,3)
{
lex_ordering[idx(p-i,0,i)] = o;
Nodes.IntPoint(o++).Set3(cp[p-i], cp[0], cp[i]);
}
for (int i = 1; i < p; i++) // (2,3)
{
lex_ordering[idx(0,p-i,i)] = o;
Nodes.IntPoint(o++).Set3(cp[0], cp[p-i], cp[i]);
}
// faces (see Mesh::GenerateFaces in mesh/mesh.cpp)
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (1,2,3)
{
lex_ordering[idx(p-i-j,i,j)] = o;
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[p-i-j]/w, cp[i]/w, cp[j]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (0,3,2)
{
lex_ordering[idx(0,j,i)] = o;
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[0], cp[j]/w, cp[i]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (0,1,3)
{
lex_ordering[idx(i,0,j)] = o;
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[i]/w, cp[0], cp[j]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (0,2,1)
{
lex_ordering[idx(j,i,0)] = o;
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[j]/w, cp[i]/w, cp[0]);
}
// interior
for (int k = 1; k < p; k++)
for (int j = 1; j + k < p; j++)
for (int i = 1; i + j + k < p; i++)
{
lex_ordering[idx(i,j,k)] = o;
real_t w = cp[i] + cp[j] + cp[k] + cp[p-i-j-k];
Nodes.IntPoint(o++).Set3(cp[i]/w, cp[j]/w, cp[k]/w);
}
DenseMatrix T(dof);
for (int m = 0; m < dof; m++)
{
IntegrationPoint &ip = Nodes.IntPoint(m);
poly1d.CalcBasis(p, ip.x, shape_x);
poly1d.CalcBasis(p, ip.y, shape_y);
poly1d.CalcBasis(p, ip.z, shape_z);
poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l);
o = 0;
for (int k = 0; k <= p; k++)
for (int j = 0; j + k <= p; j++)
for (int i = 0; i + j + k <= p; i++)
{
T(o++, m) = shape_x(i)*shape_y(j)*shape_z(k)*shape_l(p-i-j-k);
}
}
Ti.Factor(T);
// mfem::out << "H1_TetrahedronElement(" << p << ") : "; Ti.TestInversion();
}
void H1_TetrahedronElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p + 1), shape_y(p + 1), shape_z(p + 1), shape_l(p + 1);
Vector u(dof);
#endif
poly1d.CalcBasis(p, ip.x, shape_x);
poly1d.CalcBasis(p, ip.y, shape_y);
poly1d.CalcBasis(p, ip.z, shape_z);
poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l);
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j + k <= p; j++)
for (int i = 0; i + j + k <= p; i++)
{
u(o++) = shape_x(i)*shape_y(j)*shape_z(k)*shape_l(p-i-j-k);
}
Ti.Mult(u, shape);
}
void H1_TetrahedronElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p + 1), shape_y(p + 1), shape_z(p + 1), shape_l(p + 1);
Vector dshape_x(p + 1), dshape_y(p + 1), dshape_z(p + 1), dshape_l(p + 1);
DenseMatrix du(dof, dim);
#endif
poly1d.CalcBasis(p, ip.x, shape_x, dshape_x);
poly1d.CalcBasis(p, ip.y, shape_y, dshape_y);
poly1d.CalcBasis(p, ip.z, shape_z, dshape_z);
poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l, dshape_l);
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j + k <= p; j++)
for (int i = 0; i + j + k <= p; i++)
{
int l = p - i - j - k;
du(o,0) = ((dshape_x(i)* shape_l(l)) -
( shape_x(i)*dshape_l(l)))*shape_y(j)*shape_z(k);
du(o,1) = ((dshape_y(j)* shape_l(l)) -
( shape_y(j)*dshape_l(l)))*shape_x(i)*shape_z(k);
du(o,2) = ((dshape_z(k)* shape_l(l)) -
( shape_z(k)*dshape_l(l)))*shape_x(i)*shape_y(j);
o++;
}
Ti.Mult(du, dshape);
}
void H1_TetrahedronElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &ddshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(p + 1), shape_y(p + 1), shape_z(p + 1), shape_l(p + 1);
Vector dshape_x(p + 1), dshape_y(p + 1), dshape_z(p + 1), dshape_l(p + 1);
Vector ddshape_x(p + 1), ddshape_y(p + 1), ddshape_z(p + 1), ddshape_l(p + 1);
DenseMatrix ddu(dof, ((dim + 1) * dim) / 2);
#endif
poly1d.CalcBasis(p, ip.x, shape_x, dshape_x, ddshape_x);
poly1d.CalcBasis(p, ip.y, shape_y, dshape_y, ddshape_y);
poly1d.CalcBasis(p, ip.z, shape_z, dshape_z, ddshape_z);
poly1d.CalcBasis(p, 1. - ip.x - ip.y - ip.z, shape_l, dshape_l, ddshape_l);
for (int o = 0, k = 0; k <= p; k++)
for (int j = 0; j + k <= p; j++)
for (int i = 0; i + j + k <= p; i++)
{
// u_xx, u_xy, u_xz, u_yy, u_yz, u_zz
int l = p - i - j - k;
ddu(o,0) = ((ddshape_x(i) * shape_l(l)) - 2. * (dshape_x(i) * dshape_l(l)) +
(shape_x(i) * ddshape_l(l))) * shape_y(j) * shape_z(k);
ddu(o,1) = ((dshape_y(j) * ((dshape_x(i) * shape_l(l)) -
(shape_x(i) * dshape_l(l)))) +
(shape_y(j) * ((ddshape_l(l) * shape_x(i)) -
(dshape_x(i) * dshape_l(l)))))* shape_z(k);
ddu(o,2) = ((dshape_z(k) * ((dshape_x(i) * shape_l(l)) -
(shape_x(i) * dshape_l(l)))) +
(shape_z(k) * ((ddshape_l(l) * shape_x(i)) -
(dshape_x(i) * dshape_l(l)))))* shape_y(j);
ddu(o,3) = ((ddshape_y(j) * shape_l(l)) - 2. * (dshape_y(j) * dshape_l(l)) +
(shape_y(j) * ddshape_l(l))) * shape_x(i) * shape_z(k);
ddu(o,4) = ((dshape_z(k) * ((dshape_y(j) * shape_l(l)) -
(shape_y(j)*dshape_l(l))) ) +
(shape_z(k)* ((ddshape_l(l)*shape_y(j)) -
(dshape_y(j) * dshape_l(l)) ) ) )* shape_x(i);
ddu(o,5) = ((ddshape_z(k) * shape_l(l)) - 2. * (dshape_z(k) * dshape_l(l)) +
(shape_z(k) * ddshape_l(l))) * shape_y(j) * shape_x(i);
o++;
}
Ti.Mult(ddu, ddshape);
}
// TODO: use a FunctionSpace specific to wedges instead of Qk.
H1_WedgeElement::H1_WedgeElement(const int p,
const int btype)
: NodalFiniteElement(3, Geometry::PRISM, ((p + 1)*(p + 1)*(p + 2))/2,
p, FunctionSpace::Qk),
TriangleFE(p, btype),
SegmentFE(p, btype)
{
#ifndef MFEM_THREAD_SAFE
t_shape.SetSize(TriangleFE.GetDof());
s_shape.SetSize(SegmentFE.GetDof());
t_dshape.SetSize(TriangleFE.GetDof(), 2);
s_dshape.SetSize(SegmentFE.GetDof(), 1);
#endif
t_dof.SetSize(dof);
s_dof.SetSize(dof);
int p2p3 = 2*p + 3, ntri = ((p + 1)*(p + 2))/2;
auto idx = [p2p3,ntri](int i, int j, int k)
{
return k*ntri + ((p2p3-j)*j)/2+i;
};
lex_ordering.SetSize(dof);
int o = 0;
// Nodal DoFs
lex_ordering[idx(0,0,0)] = o++;
lex_ordering[idx(p,0,0)] = o++;
lex_ordering[idx(0,p,0)] = o++;
lex_ordering[idx(0,0,p)] = o++;
lex_ordering[idx(p,0,p)] = o++;
lex_ordering[idx(0,p,p)] = o++;
t_dof[0] = 0; s_dof[0] = 0;
t_dof[1] = 1; s_dof[1] = 0;
t_dof[2] = 2; s_dof[2] = 0;
t_dof[3] = 0; s_dof[3] = 1;
t_dof[4] = 1; s_dof[4] = 1;
t_dof[5] = 2; s_dof[5] = 1;
// Edge DoFs
int k = 0;
int ne = p-1;
for (int i=1; i<p; i++)
{
lex_ordering[idx(i,0,0)] = o + 0*ne + k;
lex_ordering[idx(p-i,i,0)] = o + 1*ne + k;
lex_ordering[idx(0,p-i,0)] = o + 2*ne + k;
lex_ordering[idx(i,0,p)] = o + 3*ne + k;
lex_ordering[idx(p-i,i,p)] = o + 4*ne + k;
lex_ordering[idx(0,p-i,p)] = o + 5*ne + k;
lex_ordering[idx(0,0,i)] = o + 6*ne + k;
lex_ordering[idx(p,0,i)] = o + 7*ne + k;
lex_ordering[idx(0,p,i)] = o + 8*ne + k;
t_dof[5 + 0 * ne + i] = 2 + 0 * ne + i; s_dof[5 + 0 * ne + i] = 0;
t_dof[5 + 1 * ne + i] = 2 + 1 * ne + i; s_dof[5 + 1 * ne + i] = 0;
t_dof[5 + 2 * ne + i] = 2 + 2 * ne + i; s_dof[5 + 2 * ne + i] = 0;
t_dof[5 + 3 * ne + i] = 2 + 0 * ne + i; s_dof[5 + 3 * ne + i] = 1;
t_dof[5 + 4 * ne + i] = 2 + 1 * ne + i; s_dof[5 + 4 * ne + i] = 1;
t_dof[5 + 5 * ne + i] = 2 + 2 * ne + i; s_dof[5 + 5 * ne + i] = 1;
t_dof[5 + 6 * ne + i] = 0; s_dof[5 + 6 * ne + i] = i + 1;
t_dof[5 + 7 * ne + i] = 1; s_dof[5 + 7 * ne + i] = i + 1;
t_dof[5 + 8 * ne + i] = 2; s_dof[5 + 8 * ne + i] = i + 1;
++k;
}
o += 9*ne;
// Triangular Face DoFs
k=0;
int nt = (p-1)*(p-2)/2;
for (int j=1; j<p; j++)
{
for (int i=1; i<p-j; i++)
{
int l = j - p + (((2 * p - 1) - i) * i) / 2;
lex_ordering[idx(i,j,0)] = o+l;
lex_ordering[idx(i,j,p)] = o+nt+k;
t_dof[6 + 9 * ne + k] = 3 * p + l; s_dof[6 + 9 * ne + k] = 0;
t_dof[6 + 9 * ne + nt + k] = 3 * p + k; s_dof[6 + 9 * ne + nt + k] = 1;
k++;
}
}
o += 2*nt;
// Quadrilateral Face DoFs
k=0;
int nq = (p-1)*(p-1);
for (int j=1; j<p; j++)
{
for (int i=1; i<p; i++)
{
lex_ordering[idx(i,0,j)] = o+k;
lex_ordering[idx(p-i,i,j)] = o+nq+k;
lex_ordering[idx(0,p-i,j)] = o+2*nq+k;
t_dof[6 + 9 * ne + 2 * nt + 0 * nq + k] = 2 + 0 * ne + i;
t_dof[6 + 9 * ne + 2 * nt + 1 * nq + k] = 2 + 1 * ne + i;
t_dof[6 + 9 * ne + 2 * nt + 2 * nq + k] = 2 + 2 * ne + i;
s_dof[6 + 9 * ne + 2 * nt + 0 * nq + k] = 1 + j;
s_dof[6 + 9 * ne + 2 * nt + 1 * nq + k] = 1 + j;
s_dof[6 + 9 * ne + 2 * nt + 2 * nq + k] = 1 + j;
k++;
}
}
o += 3*nq;
// Interior DoFs
int m=0;
for (k=1; k<p; k++)
{
int l=0;
for (int j=1; j<p; j++)
{
for (int i=1; i+j<p; i++)
{
lex_ordering[idx(i,j,k)] = o++;
t_dof[6 + 9 * ne + 2 * nt + 3 * nq + m] = 3 * p + l;
s_dof[6 + 9 * ne + 2 * nt + 3 * nq + m] = 1 + k;
l++; m++;
}
}
}
// Define Nodes
const IntegrationRule & t_Nodes = TriangleFE.GetNodes();
const IntegrationRule & s_Nodes = SegmentFE.GetNodes();
for (int i=0; i<dof; i++)
{
Nodes.IntPoint(i).x = t_Nodes.IntPoint(t_dof[i]).x;
Nodes.IntPoint(i).y = t_Nodes.IntPoint(t_dof[i]).y;
Nodes.IntPoint(i).z = s_Nodes.IntPoint(s_dof[i]).x;
}
}
void H1_WedgeElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
#ifdef MFEM_THREAD_SAFE
Vector t_shape(TriangleFE.GetDof());
Vector s_shape(SegmentFE.GetDof());
#endif
IntegrationPoint ipz; ipz.x = ip.z; ipz.y = 0.0; ipz.z = 0.0;
TriangleFE.CalcShape(ip, t_shape);
SegmentFE.CalcShape(ipz, s_shape);
for (int i=0; i<dof; i++)
{
shape[i] = t_shape[t_dof[i]] * s_shape[s_dof[i]];
}
}
void H1_WedgeElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
#ifdef MFEM_THREAD_SAFE
Vector t_shape(TriangleFE.GetDof());
DenseMatrix t_dshape(TriangleFE.GetDof(), 2);
Vector s_shape(SegmentFE.GetDof());
DenseMatrix s_dshape(SegmentFE.GetDof(), 1);
#endif
IntegrationPoint ipz; ipz.x = ip.z; ipz.y = 0.0; ipz.z = 0.0;
TriangleFE.CalcShape(ip, t_shape);
TriangleFE.CalcDShape(ip, t_dshape);
SegmentFE.CalcShape(ipz, s_shape);
SegmentFE.CalcDShape(ipz, s_dshape);
for (int i=0; i<dof; i++)
{
dshape(i, 0) = t_dshape(t_dof[i],0) * s_shape[s_dof[i]];
dshape(i, 1) = t_dshape(t_dof[i],1) * s_shape[s_dof[i]];
dshape(i, 2) = t_shape[t_dof[i]] * s_dshape(s_dof[i],0);
}
}
H1_FuentesPyramidElement::H1_FuentesPyramidElement(const int p, const int btype)
: NodalFiniteElement(3, Geometry::PYRAMID,
p * (p * p + 3) + 1, // Fuentes et al
p, FunctionSpace::Uk)
{
zmax = 0.0;
const real_t *cp = poly1d.ClosedPoints(p, VerifyNodal(VerifyClosed(btype)));
#ifndef MFEM_THREAD_SAFE
tmp_i.SetSize(p + 1);
tmp1_ij.SetSize(p + 1, p + 1);
tmp2_ij.SetSize(p + 1, dim);
tmp_ijk.SetSize(p + 1, p + 1, dim);
tmp_u.SetSize(dof);
tmp_du.SetSize(dof, dim);
#else
Vector tmp_i(p + 1);
DenseMatrix tmp1_ij(p + 1, p + 1);
#endif
// vertices
Nodes.IntPoint(0).Set3(cp[0], cp[0], cp[0]);
Nodes.IntPoint(1).Set3(cp[p], cp[0], cp[0]);
Nodes.IntPoint(2).Set3(cp[p], cp[p], cp[0]);
Nodes.IntPoint(3).Set3(cp[0], cp[p], cp[0]);
Nodes.IntPoint(4).Set3(cp[0], cp[0], cp[p]);
// edges
int o = 5;
for (int i = 1; i < p; i++) // (0,1)
{
Nodes.IntPoint(o++).Set3(cp[i], cp[0], cp[0]);
}
for (int i = 1; i < p; i++) // (1,2)
{
Nodes.IntPoint(o++).Set3(cp[p], cp[i], cp[0]);
}
for (int i = 1; i < p; i++) // (3,2)
{
Nodes.IntPoint(o++).Set3(cp[i], cp[p], cp[0]);
}
for (int i = 1; i < p; i++) // (0,3)
{
Nodes.IntPoint(o++).Set3(cp[0], cp[i], cp[0]);
}
for (int i = 1; i < p; i++) // (0,4)
{
Nodes.IntPoint(o++).Set3(cp[0], cp[0], cp[i]);
}
for (int i = 1; i < p; i++) // (1,4)
{
Nodes.IntPoint(o++).Set3(cp[p-i], cp[0], cp[i]);
}
for (int i = 1; i < p; i++) // (2,4)
{
Nodes.IntPoint(o++).Set3(cp[p-i], cp[p-i], cp[i]);
}
for (int i = 1; i < p; i++) // (3,4)
{
Nodes.IntPoint(o++).Set3(cp[0], cp[p-i], cp[i]);
}
// quadrilateral face
for (int j = 1; j < p; j++)
{
for (int i = 1; i < p; i++)
{
Nodes.IntPoint(o++).Set3(cp[i], cp[p-j], cp[0]);
}
}
// triangular faces
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (0,1,4)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[i]/w, cp[0], cp[j]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (1,2,4)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3((cp[i] + cp[p-i-j])/w, cp[i]/w, cp[j]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (2,3,4)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[p-i-j]/w, (cp[i] + cp[p-i-j])/w, cp[j]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (3,0,4)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[0], cp[p-i-j]/w, cp[j]/w);
}
// Points based on Fuentes' interior bubbles
for (int k = 1; k < p; k++)
{
for (int j = 1; j < p; j++)
{
for (int i = 1; i < p; i++)
{
Nodes.IntPoint(o++).Set3(cp[i] * (1.0 - cp[k]),
cp[j] * (1.0 - cp[k]),
cp[k]);
}
}
}
MFEM_ASSERT(o == dof,
"Number of nodes does not match the "
"number of degrees of freedom");
DenseMatrix T(dof);
for (int m = 0; m < dof; m++)
{
const IntegrationPoint &ip = Nodes.IntPoint(m);
Vector col(T.GetColumn(m), dof);
calcBasis(order, ip, tmp_i, tmp1_ij, col);
}
Ti.Factor(T);
}
void H1_FuentesPyramidElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector tmp_i(p + 1);
Vector tmp_u(dof);
DenseMatrix tmp1_ij(p + 1, p + 1);
#endif
calcBasis(p, ip, tmp_i, tmp1_ij, tmp_u);
Ti.Mult(tmp_u, shape);
}
void H1_FuentesPyramidElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector tmp_i(p + 1);
DenseMatrix tmp1_ij(p + 1, p + 1);
DenseMatrix tmp2_ij(p + 1, dim);
DenseTensor tmp_ijk(p + 1, p + 1, dim);
DenseMatrix tmp_du(dof, dim);
#endif
calcGradBasis(p, ip, tmp_i, tmp2_ij, tmp1_ij, tmp_ijk, tmp_du);
Ti.Mult(tmp_du, dshape);
}
void H1_FuentesPyramidElement::CalcRawShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector tmp_i(p + 1);
DenseMatrix tmp1_ij(p + 1, p + 1);
#endif
calcBasis(p, ip, tmp_i, tmp1_ij, shape);
}
void H1_FuentesPyramidElement::CalcRawDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector tmp_i(p + 1);
DenseMatrix tmp1_ij(p + 1, p + 1);
DenseMatrix tmp2_ij(p + 1, dim);
DenseTensor tmp_ijk(p + 1, p + 1, dim);
#endif
calcGradBasis(p, ip, tmp_i, tmp2_ij, tmp1_ij, tmp_ijk, dshape);
}
void H1_FuentesPyramidElement::calcBasis(const int p,
const IntegrationPoint &ip,
Vector &phi_i, DenseMatrix &phi_ij,
Vector &u) const
{
real_t x = ip.x;
real_t y = ip.y;
real_t z = ip.z;
Vector xy({x,y});
zmax = std::max(z, zmax);
real_t mu;
int o = 0;
// Vertices
u[0] = lam1(x, y, z);
u[1] = lam2(x, y, z);
u[2] = lam3(x, y, z);
u[3] = lam4(x, y, z);
u[4] = lam5(x, y, z);
o += 5;
// Mixed edges (base edges)
if (CheckZ(z) && p >= 2)
{
// (a,b) = (1,2), c = 0
phi_E(p, nu01(z, xy, 1), phi_i);
mu = mu0(z, xy, 2);
for (int i = 2; i <= p; i++, o++)
{
u[o] = mu * phi_i[i];
}
// (a,b) = (1,2), c = 1
mu = mu1(z, xy, 2);
for (int i = 2; i <= p; i++, o++)
{
u[o] = mu * phi_i[i];
}
// (a,b) = (2,1), c = 0
phi_E(p, nu01(z, xy, 2), phi_i);
mu = mu0(z, xy, 1);
for (int i = 2; i <= p; i++, o++)
{
u[o] = mu * phi_i[i];
}
// (a,b) = (2,1), c = 1
mu = mu1(z, xy, 1);
for (int i = 2; i <= p; i++, o++)
{
u[o] = mu * phi_i[i];
}
}
else
{
for (int i = 0; i < 4 * (p - 1); i++, o++)
{
u[o] = 0.0;
}
}
// Triangle edges (upright edges)
if (p >= 2)
{
phi_E(p, lam15(x, y, z), phi_i);
for (int i = 2; i<= p; i++, o++)
{
u[o] = phi_i[i];
}
phi_E(p, lam25(x, y, z), phi_i);
for (int i = 2; i<= p; i++, o++)
{
u[o] = phi_i[i];
}
phi_E(p, lam35(x, y, z), phi_i);
for (int i = 2; i<= p; i++, o++)
{
u[o] = phi_i[i];
}
phi_E(p, lam45(x, y, z), phi_i);
for (int i = 2; i<= p; i++, o++)
{
u[o] = phi_i[i];
}
}
// Quadrilateral face
if (CheckZ(z) && p >= 2)
{
phi_Q(p, mu01(z, xy, 1), mu01(z, xy, 2), phi_ij);
mu = mu0(z);
for (int j = 2; j <= p; j++)
{
for (int i = 2; i <= p; i++, o++)
{
u[o] = mu * phi_ij(i,j);
}
}
}
else
{
for (int j = 2; j <= p; j++)
{
for (int i = 2; i <= p; i++, o++)
{
u[o] = 0.0;
}
}
}
// Triangular faces
if (CheckZ(z) && p >= 3)
{
// (a,b) = (1,2), c = 0
phi_T(p, nu012(z, xy, 1), phi_ij);
mu = mu0(z, xy, 2);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
{
u[o] = mu * phi_ij(i,j);
}
// (a,b) = (1,2), c = 1
mu = mu1(z, xy, 2);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
{
u[o] = mu * phi_ij(i,j);
}
// (a,b) = (2,1), c = 0
phi_T(p, nu012(z, xy, 2), phi_ij);
mu = mu0(z, xy, 1);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
{
u[o] = mu * phi_ij(i,j);
}
// (a,b) = (2,1), c = 1
mu = mu1(z, xy, 1);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
{
u[o] = mu * phi_ij(i,j);
}
}
else
{
for (int i = 0; i < 2 * (p - 1) * (p - 2); i++, o++)
{
u[o] = 0.0;
}
}
// Interior
if (CheckZ(z) && p >= 2)
{
phi_Q(p, mu01(z, xy, 1), mu01(z, xy, 2), phi_ij);
phi_E(p, mu01(z), phi_i);
for (int k = 2; k <= p; k++)
{
for (int j = 2; j <= p; j++)
{
for (int i = 2; i <= p; i++, o++)
{
u[o] = phi_ij(i,j) * phi_i(k);
}
}
}
}
else
{
for (int i = 0; i < (p - 1) * (p - 1) * (p - 1); i++, o++)
{
u[o]= 0.0;
}
}
}
void H1_FuentesPyramidElement::calcGradBasis(const int p,
const IntegrationPoint &ip,
Vector &phi_i,
DenseMatrix &dphi_i,
DenseMatrix &phi_ij,
DenseTensor &dphi_ij,
DenseMatrix &du) const
{
real_t x = ip.x;
real_t y = ip.y;
real_t z = ip.z;
Vector xy({x,y});
zmax = std::max(z, zmax);
real_t mu;
Vector dmu(3);
Vector dlam(3);
int o = 0;
// Vertices
dlam = grad_lam1(x, y, z);
for (int d=0; d<3; d++) { du(0, d) = dlam(d); }
dlam = grad_lam2(x, y, z);
for (int d=0; d<3; d++) { du(1, d) = dlam(d); }
dlam = grad_lam3(x, y, z);
for (int d=0; d<3; d++) { du(2, d) = dlam(d); }
dlam = grad_lam4(x, y, z);
for (int d=0; d<3; d++) { du(3, d) = dlam(d); }
dlam = grad_lam5(x, y, z);
for (int d=0; d<3; d++) { du(4, d) = dlam(d); }
o += 5;
// Mixed edges (base edges)
if (CheckZ(z) && p >= 2)
{
// (a,b) = (1,2), c = 0
phi_E(p, nu01(z, xy, 1), grad_nu01(z, xy, 1), phi_i, dphi_i);
mu = mu0(z, xy, 2);
dmu = grad_mu0(z, xy, 2);;
for (int i = 2; i <= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_i[i] + mu * dphi_i(i, d);
}
// (a,b) = (1,2), c = 1
mu = mu1(z, xy, 2);
dmu = grad_mu1(z, xy, 2);;
for (int i = 2; i <= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_i[i] + mu * dphi_i(i, d);
}
// (a,b) = (2,1), c = 0
phi_E(p, nu01(z, xy, 2), grad_nu01(z, xy, 2), phi_i, dphi_i);
mu = mu0(z, xy, 1);
dmu = grad_mu0(z, xy, 1);;
for (int i = 2; i <= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_i[i] + mu * dphi_i(i, d);
}
// (a,b) = (2,1), c = 1
mu = mu1(z, xy, 1);
dmu = grad_mu1(z, xy, 1);;
for (int i = 2; i <= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_i[i] + mu * dphi_i(i, d);
}
}
else
{
for (int i = 0; i < 4 * (p - 1); i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = 0.0;
}
}
// Triangle edges (upright edges)
if (p >= 2)
{
phi_E(p, lam15(x, y, z), grad_lam15(x,y,z), phi_i, dphi_i);
for (int i = 2; i<= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dphi_i(i, d);
}
phi_E(p, lam25(x, y, z), grad_lam25(x, y, z), phi_i, dphi_i);
for (int i = 2; i<= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dphi_i(i, d);
}
phi_E(p, lam35(x, y, z), grad_lam35(x, y, z), phi_i, dphi_i);
for (int i = 2; i<= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dphi_i(i, d);
}
phi_E(p, lam45(x, y, z), grad_lam45(x, y, z), phi_i, dphi_i);
for (int i = 2; i<= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dphi_i(i, d);
}
}
// Quadrilateral face
if (CheckZ(z) && p >= 2)
{
phi_Q(p, mu01(z, xy, 1), grad_mu01(z, xy, 1),
mu01(z, xy, 2), grad_mu01(z, xy, 2), phi_ij, dphi_ij);
mu = mu0(z);
dmu = grad_mu0(z);
for (int j = 2; j <= p; j++)
for (int i = 2; i <= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_ij(i, j) + mu * dphi_ij(i, j, d);
}
}
else
{
for (int j = 2; j <= p; j++)
for (int i = 2; i <= p; i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = 0.0;
}
}
// Triangular faces
if (CheckZ(z) && p >= 3)
{
// (a,b) = (1,2), c = 0
phi_T(p, nu012(z, xy, 1), grad_nu012(z, xy, 1), phi_ij, dphi_ij);
mu = mu0(z, xy, 2);
dmu = grad_mu0(z, xy, 2);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_ij(i, j) + mu * dphi_ij(i, j, d);
}
// (a,b) = (1,2), c = 1
mu = mu1(z, xy, 2);
dmu = grad_mu1(z, xy, 2);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_ij(i, j) + mu * dphi_ij(i, j, d);
}
// (a,b) = (2,1), c = 0
phi_T(p, nu012(z, xy, 2), grad_nu012(z, xy, 2), phi_ij, dphi_ij);
mu = mu0(z, xy, 1);
dmu = grad_mu0(z, xy, 1);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_ij(i, j) + mu * dphi_ij(i, j, d);
}
// (a,b) = (2,1), c = 1
mu = mu1(z, xy, 1);
dmu = grad_mu1(z, xy, 1);
for (int i = 2; i < p; i++)
for (int j = 1; i + j <= p; j++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = dmu(d) * phi_ij(i, j) + mu * dphi_ij(i, j, d);
}
}
else
{
for (int i = 0; i < 2 * (p - 1) * (p - 2); i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = 0.0;
}
}
// Interior
if (CheckZ(z) && p >= 2)
{
phi_Q(p, mu01(z, xy, 1), grad_mu01(z, xy, 1),
mu01(z, xy, 2), grad_mu01(z, xy, 2), phi_ij, dphi_ij);
phi_E(p, mu01(z), grad_mu01(z), phi_i, dphi_i);
for (int k = 2; k <= p; k++)
for (int j = 2; j <= p; j++)
for (int i = 2; i <= p; i++, o++)
for (int d=0; d<3; d++)
du(o, d) = dphi_ij(i, j, d) * phi_i(k) +
phi_ij(i, j) * dphi_i(k, d);
}
else
{
for (int i = 0; i < (p - 1) * (p - 1) * (p - 1); i++, o++)
for (int d=0; d<3; d++)
{
du(o, d) = 0.0;
}
}
}
H1_BergotPyramidElement::H1_BergotPyramidElement(const int p, const int btype)
: NodalFiniteElement(3, Geometry::PYRAMID,
(p + 1) * (p + 2) * (2 * p + 3) / 6, // Bergot (JSC)
p, FunctionSpace::Uk)
{
const real_t *cp = poly1d.ClosedPoints(p, VerifyNodal(VerifyClosed(btype)));
#ifndef MFEM_THREAD_SAFE
shape_x.SetSize(p + 1);
shape_y.SetSize(p + 1);
shape_z.SetSize(p + 1);
dshape_x.SetSize(p + 1);
dshape_y.SetSize(p + 1);
dshape_z.SetSize(p + 1);
dshape_z_dt.SetSize(p + 1);
ddshape_x.SetSize(p + 1);
ddshape_y.SetSize(p + 1);
ddshape_z.SetSize(p + 1);
u.SetSize(dof);
du.SetSize(dof, dim);
ddu.SetSize(dof, (dim * (dim + 1)) / 2);
#else
Vector shape_x(p + 1), shape_y(p + 1), shape_z(p + 1);
#endif
// vertices
Nodes.IntPoint(0).Set3(cp[0], cp[0], cp[0]);
Nodes.IntPoint(1).Set3(cp[p], cp[0], cp[0]);
Nodes.IntPoint(2).Set3(cp[p], cp[p], cp[0]);
Nodes.IntPoint(3).Set3(cp[0], cp[p], cp[0]);
Nodes.IntPoint(4).Set3(cp[0], cp[0], cp[p]);
// edges
int o = 5;
for (int i = 1; i < p; i++) // (0,1)
{
Nodes.IntPoint(o++).Set3(cp[i], cp[0], cp[0]);
}
for (int i = 1; i < p; i++) // (1,2)
{
Nodes.IntPoint(o++).Set3(cp[p], cp[i], cp[0]);
}
for (int i = 1; i < p; i++) // (3,2)
{
Nodes.IntPoint(o++).Set3(cp[i], cp[p], cp[0]);
}
for (int i = 1; i < p; i++) // (0,3)
{
Nodes.IntPoint(o++).Set3(cp[0], cp[i], cp[0]);
}
for (int i = 1; i < p; i++) // (0,4)
{
Nodes.IntPoint(o++).Set3(cp[0], cp[0], cp[i]);
}
for (int i = 1; i < p; i++) // (1,4)
{
Nodes.IntPoint(o++).Set3(cp[p-i], cp[0], cp[i]);
}
for (int i = 1; i < p; i++) // (2,4)
{
Nodes.IntPoint(o++).Set3(cp[p-i], cp[p-i], cp[i]);
}
for (int i = 1; i < p; i++) // (3,4)
{
Nodes.IntPoint(o++).Set3(cp[0], cp[p-i], cp[i]);
}
// quadrilateral face
for (int j = 1; j < p; j++)
{
for (int i = 1; i < p; i++)
{
Nodes.IntPoint(o++).Set3(cp[i], cp[j], cp[0]);
}
}
// triangular faces
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (0,1,4)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[i]/w, cp[0], cp[j]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (1,2,4)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(1.0 - cp[j]/w, cp[i]/w, cp[j]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (3,4,2)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[j]/w, 1.0 - cp[i]/w, cp[i]/w);
}
for (int j = 1; j < p; j++)
for (int i = 1; i + j < p; i++) // (0,4,3)
{
real_t w = cp[i] + cp[j] + cp[p-i-j];
Nodes.IntPoint(o++).Set3(cp[0], cp[j]/w, cp[i]/w);
}
// interior
for (int k = 1; k < p - 1; k++)
{
for (int j = 1; j < p - k; j++)
{
real_t wjk = cp[j] + cp[k] + cp[p-j-k];
for (int i = 1; i < p - k; i++)
{
real_t wik = cp[i] + cp[k] + cp[p-i-k];
real_t w = wik * wjk * cp[p-k];
Nodes.IntPoint(o++).Set3(cp[i] * (cp[j] + cp[p-j-k]) / w,
cp[j] * (cp[i] + cp[p-i-k]) / w,
cp[k] * cp[p-k] / w);
}
}
}
MFEM_ASSERT(o == dof,
"Number of nodes does not match the "
"number of degrees of freedom");
DenseMatrix T(dof);
for (int m = 0; m < dof; m++)
{
const IntegrationPoint &ip = Nodes.IntPoint(m);
real_t x = (ip.z < 1.0) ? (ip.x / (1.0 - ip.z)) : 0.0;
real_t y = (ip.z < 1.0) ? (ip.y / (1.0 - ip.z)) : 0.0;
real_t z = ip.z;
poly1d.CalcLegendre(p, x, shape_x.GetData());
poly1d.CalcLegendre(p, y, shape_y.GetData());
o = 0;
for (int i = 0; i <= p; i++)
{
for (int j = 0; j <= p; j++)
{
int maxij = std::max(i, j);
FuentesPyramid::CalcScaledJacobi(p-maxij, 2.0 * (maxij + 1.0),
z, 1.0, shape_z);
for (int k = 0; k <= p - maxij; k++)
{
T(o++, m) = shape_x(i) * shape_y(j) * shape_z(k) *
pow(1.0 - ip.z, maxij);
}
}
}
}
Ti.Factor(T);
}
void H1_BergotPyramidElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
Vector shape_x(order+1);
Vector shape_y(order+1);
Vector shape_z(order+1);
Vector u(dof);
#endif
real_t x = (ip.z < 1.0) ? (ip.x / (1.0 - ip.z)) : 0.0;
real_t y = (ip.z < 1.0) ? (ip.y / (1.0 - ip.z)) : 0.0;
real_t z = ip.z;
poly1d.CalcLegendre(p, x, shape_x.GetData());
poly1d.CalcLegendre(p, y, shape_y.GetData());
int o = 0;
for (int i = 0; i <= p; i++)
for (int j = 0; j <= p; j++)
{
int maxij = std::max(i, j);
FuentesPyramid::CalcScaledJacobi(p-maxij, 2.0 * (maxij + 1.0), z, 1.0,
shape_z);
for (int k = 0; k <= p - maxij; k++)
u[o++] = shape_x(i) * shape_y(j) * shape_z(k) *
pow(1.0 - ip.z, maxij);
}
Ti.Mult(u, shape);
}
void H1_BergotPyramidElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const int p = order;
#ifdef MFEM_THREAD_SAFE
DenseMatrix du(dof, dim);
Vector shape_x(order+1);
Vector shape_y(order+1);
Vector shape_z(order+1);
Vector dshape_x(order+1);
Vector dshape_y(order+1);
Vector dshape_z(order+1);
Vector dshape_z_dt(order+1);
#endif
real_t x = (ip.z < 1.0) ? (ip.x / (1.0 - ip.z)) : 0.0;
real_t y = (ip.z < 1.0) ? (ip.y / (1.0 - ip.z)) : 0.0;
real_t z = ip.z;
poly1d.CalcLegendre(p, x, shape_x.GetData(), dshape_x.GetData());
poly1d.CalcLegendre(p, y, shape_y.GetData(), dshape_y.GetData());
int o = 0;
for (int i = 0; i <= p; i++)
for (int j = 0; j <= p; j++)
{
int maxij = std::max(i, j);
FuentesPyramid::CalcScaledJacobi(p-maxij, 2.0 * (maxij + 1.0), z, 1.0,
shape_z, dshape_z, dshape_z_dt);
for (int k = 0; k <= p - maxij; k++, o++)
{
du(o,0) = dshape_x(i) * shape_y(j) * shape_z(k) *
pow(1.0 - ip.z, maxij - 1);
du(o,1) = shape_x(i) * dshape_y(j) * shape_z(k) *
pow(1.0 - ip.z, maxij - 1);
du(o,2) = shape_x(i) * shape_y(j) * dshape_z(k) *
pow(1.0 - ip.z, maxij) +
(ip.x * dshape_x(i) * shape_y(j) +
ip.y * shape_x(i) * dshape_y(j)) *
shape_z(k) * pow(1.0 - ip.z, maxij - 2) -
maxij * shape_x(i) * shape_y(j) * shape_z(k) *
pow(1.0 - ip.z, maxij - 1);
}
}
Ti.Mult(du, dshape);
}
}
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