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mfem/fem/fe/fe_fixed_order.cpp
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Ido Akkerman 9a47796ea3 Add trivia hessians
2025-12-02 16:59:06 +01:00

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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.
// Fixed Order Finite Element classes
#include "fe_fixed_order.hpp"
#include "../coefficient.hpp"
namespace mfem
{
using namespace std;
PointFiniteElement::PointFiniteElement()
: NodalFiniteElement(0, Geometry::POINT, 1, 0)
{
lex_ordering.SetSize(1);
lex_ordering[0] = 0;
Nodes.IntPoint(0).x = 0.0;
}
void PointFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.;
}
void PointFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
// dshape is (1 x 0) - nothing to compute
}
Linear1DFiniteElement::Linear1DFiniteElement()
: NodalFiniteElement(1, Geometry::SEGMENT, 2, 1)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(1).x = 1.0;
}
void Linear1DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1. - ip.x;
shape(1) = ip.x;
}
void Linear1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = -1.;
dshape(1,0) = 1.;
}
void Linear1DFiniteElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &h) const
{
h = 0.0;
}
Linear2DFiniteElement::Linear2DFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 3, 1)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
}
void Linear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1. - ip.x - ip.y;
shape(1) = ip.x;
shape(2) = ip.y;
}
void Linear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = -1.; dshape(0,1) = -1.;
dshape(1,0) = 1.; dshape(1,1) = 0.;
dshape(2,0) = 0.; dshape(2,1) = 1.;
}
void Linear2DFiniteElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &h) const
{
h = 0.0;
}
BiLinear2DFiniteElement::BiLinear2DFiniteElement()
: NodalFiniteElement(2, Geometry::SQUARE, 4, 1, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 1.0;
}
void BiLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = (1. - ip.x) * (1. - ip.y) ;
shape(1) = ip.x * (1. - ip.y) ;
shape(2) = ip.x * ip.y ;
shape(3) = (1. - ip.x) * ip.y ;
}
void BiLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = -1. + ip.y; dshape(0,1) = -1. + ip.x ;
dshape(1,0) = 1. - ip.y; dshape(1,1) = -ip.x ;
dshape(2,0) = ip.y ; dshape(2,1) = ip.x ;
dshape(3,0) = -ip.y ; dshape(3,1) = 1. - ip.x ;
}
void BiLinear2DFiniteElement::CalcHessian(
const IntegrationPoint &ip, DenseMatrix &h) const
{
h(0,0) = 0.; h(0,1) = 1.; h(0,2) = 0.;
h(1,0) = 0.; h(1,1) = -1.; h(1,2) = 0.;
h(2,0) = 0.; h(2,1) = 1.; h(2,2) = 0.;
h(3,0) = 0.; h(3,1) = -1.; h(3,2) = 0.;
}
GaussLinear2DFiniteElement::GaussLinear2DFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 3, 1, FunctionSpace::Pk)
{
Nodes.IntPoint(0).x = 1./6.;
Nodes.IntPoint(0).y = 1./6.;
Nodes.IntPoint(1).x = 2./3.;
Nodes.IntPoint(1).y = 1./6.;
Nodes.IntPoint(2).x = 1./6.;
Nodes.IntPoint(2).y = 2./3.;
}
void GaussLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const real_t x = ip.x, y = ip.y;
shape(0) = 5./3. - 2. * (x + y);
shape(1) = 2. * (x - 1./6.);
shape(2) = 2. * (y - 1./6.);
}
void GaussLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = -2.; dshape(0,1) = -2.;
dshape(1,0) = 2.; dshape(1,1) = 0.;
dshape(2,0) = 0.; dshape(2,1) = 2.;
}
void GaussLinear2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
dofs(vertex) = 2./3.;
dofs((vertex+1)%3) = 1./6.;
dofs((vertex+2)%3) = 1./6.;
}
// 0.5-0.5/sqrt(3) and 0.5+0.5/sqrt(3)
const real_t GaussBiLinear2DFiniteElement::p[] =
{ 0.2113248654051871177454256, 0.7886751345948128822545744 };
GaussBiLinear2DFiniteElement::GaussBiLinear2DFiniteElement()
: NodalFiniteElement(2, Geometry::SQUARE, 4, 1, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = p[0];
Nodes.IntPoint(0).y = p[0];
Nodes.IntPoint(1).x = p[1];
Nodes.IntPoint(1).y = p[0];
Nodes.IntPoint(2).x = p[1];
Nodes.IntPoint(2).y = p[1];
Nodes.IntPoint(3).x = p[0];
Nodes.IntPoint(3).y = p[1];
}
void GaussBiLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const real_t x = ip.x, y = ip.y;
shape(0) = 3. * (p[1] - x) * (p[1] - y);
shape(1) = 3. * (x - p[0]) * (p[1] - y);
shape(2) = 3. * (x - p[0]) * (y - p[0]);
shape(3) = 3. * (p[1] - x) * (y - p[0]);
}
void GaussBiLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const real_t x = ip.x, y = ip.y;
dshape(0,0) = 3. * (y - p[1]); dshape(0,1) = 3. * (x - p[1]);
dshape(1,0) = 3. * (p[1] - y); dshape(1,1) = 3. * (p[0] - x);
dshape(2,0) = 3. * (y - p[0]); dshape(2,1) = 3. * (x - p[0]);
dshape(3,0) = 3. * (p[0] - y); dshape(3,1) = 3. * (p[1] - x);
}
void GaussBiLinear2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
#if 1
dofs(vertex) = p[1]*p[1];
dofs((vertex+1)%4) = p[0]*p[1];
dofs((vertex+2)%4) = p[0]*p[0];
dofs((vertex+3)%4) = p[0]*p[1];
#else
dofs = 1.0;
#endif
}
P1OnQuadFiniteElement::P1OnQuadFiniteElement()
: NodalFiniteElement(2, Geometry::SQUARE, 3, 1, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
}
void P1OnQuadFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1. - ip.x - ip.y;
shape(1) = ip.x;
shape(2) = ip.y;
}
void P1OnQuadFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = -1.; dshape(0,1) = -1.;
dshape(1,0) = 1.; dshape(1,1) = 0.;
dshape(2,0) = 0.; dshape(2,1) = 1.;
}
Quad1DFiniteElement::Quad1DFiniteElement()
: NodalFiniteElement(1, Geometry::SEGMENT, 3, 2)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(2).x = 0.5;
}
void Quad1DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x;
real_t l1 = 1.0 - x, l2 = x, l3 = 2. * x - 1.;
shape(0) = l1 * (-l3);
shape(1) = l2 * l3;
shape(2) = 4. * l1 * l2;
}
void Quad1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x;
dshape(0,0) = 4. * x - 3.;
dshape(1,0) = 4. * x - 1.;
dshape(2,0) = 4. - 8. * x;
}
Quad2DFiniteElement::Quad2DFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 6, 2)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.5;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.5;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 0.5;
}
void Quad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x, y = ip.y;
real_t l1 = 1.-x-y, l2 = x, l3 = y;
shape(0) = l1 * (2. * l1 - 1.);
shape(1) = l2 * (2. * l2 - 1.);
shape(2) = l3 * (2. * l3 - 1.);
shape(3) = 4. * l1 * l2;
shape(4) = 4. * l2 * l3;
shape(5) = 4. * l3 * l1;
}
void Quad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x, y = ip.y;
dshape(0,0) =
dshape(0,1) = 4. * (x + y) - 3.;
dshape(1,0) = 4. * x - 1.;
dshape(1,1) = 0.;
dshape(2,0) = 0.;
dshape(2,1) = 4. * y - 1.;
dshape(3,0) = -4. * (2. * x + y - 1.);
dshape(3,1) = -4. * x;
dshape(4,0) = 4. * y;
dshape(4,1) = 4. * x;
dshape(5,0) = -4. * y;
dshape(5,1) = -4. * (x + 2. * y - 1.);
}
void Quad2DFiniteElement::CalcHessian (const IntegrationPoint &ip,
DenseMatrix &h) const
{
h(0,0) = 4.;
h(0,1) = 4.;
h(0,2) = 4.;
h(1,0) = 4.;
h(1,1) = 0.;
h(1,2) = 0.;
h(2,0) = 0.;
h(2,1) = 0.;
h(2,2) = 4.;
h(3,0) = -8.;
h(3,1) = -4.;
h(3,2) = 0.;
h(4,0) = 0.;
h(4,1) = 4.;
h(4,2) = 0.;
h(5,0) = 0.;
h(5,1) = -4.;
h(5,2) = -8.;
}
void Quad2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
#if 0
dofs = 1.;
#else
dofs = 0.;
dofs(vertex) = 1.;
switch (vertex)
{
case 0: dofs(3) = 0.25; dofs(5) = 0.25; break;
case 1: dofs(3) = 0.25; dofs(4) = 0.25; break;
case 2: dofs(4) = 0.25; dofs(5) = 0.25; break;
}
#endif
}
const real_t GaussQuad2DFiniteElement::p[] =
{ 0.0915762135097707434595714634022015, 0.445948490915964886318329253883051 };
GaussQuad2DFiniteElement::GaussQuad2DFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 6, 2), A(6), D(6,2), pol(6)
{
Nodes.IntPoint(0).x = p[0];
Nodes.IntPoint(0).y = p[0];
Nodes.IntPoint(1).x = 1. - 2. * p[0];
Nodes.IntPoint(1).y = p[0];
Nodes.IntPoint(2).x = p[0];
Nodes.IntPoint(2).y = 1. - 2. * p[0];
Nodes.IntPoint(3).x = p[1];
Nodes.IntPoint(3).y = p[1];
Nodes.IntPoint(4).x = 1. - 2. * p[1];
Nodes.IntPoint(4).y = p[1];
Nodes.IntPoint(5).x = p[1];
Nodes.IntPoint(5).y = 1. - 2. * p[1];
for (int i = 0; i < 6; i++)
{
const real_t x = Nodes.IntPoint(i).x, y = Nodes.IntPoint(i).y;
A(0,i) = 1.;
A(1,i) = x;
A(2,i) = y;
A(3,i) = x * x;
A(4,i) = x * y;
A(5,i) = y * y;
}
A.Invert();
}
void GaussQuad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const real_t x = ip.x, y = ip.y;
pol(0) = 1.;
pol(1) = x;
pol(2) = y;
pol(3) = x * x;
pol(4) = x * y;
pol(5) = y * y;
A.Mult(pol, shape);
}
void GaussQuad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const real_t x = ip.x, y = ip.y;
D(0,0) = 0.; D(0,1) = 0.;
D(1,0) = 1.; D(1,1) = 0.;
D(2,0) = 0.; D(2,1) = 1.;
D(3,0) = 2. * x; D(3,1) = 0.;
D(4,0) = y; D(4,1) = x;
D(5,0) = 0.; D(5,1) = 2. * y;
Mult(A, D, dshape);
}
BiQuad2DFiniteElement::BiQuad2DFiniteElement()
: NodalFiniteElement(2, Geometry::SQUARE, 9, 2, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 1.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(5).x = 1.0;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(6).x = 0.5;
Nodes.IntPoint(6).y = 1.0;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(8).x = 0.5;
Nodes.IntPoint(8).y = 0.5;
}
void BiQuad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x, y = ip.y;
real_t l1x, l2x, l3x, l1y, l2y, l3y;
l1x = (x - 1.) * (2. * x - 1);
l2x = 4. * x * (1. - x);
l3x = x * (2. * x - 1.);
l1y = (y - 1.) * (2. * y - 1);
l2y = 4. * y * (1. - y);
l3y = y * (2. * y - 1.);
shape(0) = l1x * l1y;
shape(4) = l2x * l1y;
shape(1) = l3x * l1y;
shape(7) = l1x * l2y;
shape(8) = l2x * l2y;
shape(5) = l3x * l2y;
shape(3) = l1x * l3y;
shape(6) = l2x * l3y;
shape(2) = l3x * l3y;
}
void BiQuad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x, y = ip.y;
real_t l1x, l2x, l3x, l1y, l2y, l3y;
real_t d1x, d2x, d3x, d1y, d2y, d3y;
l1x = (x - 1.) * (2. * x - 1);
l2x = 4. * x * (1. - x);
l3x = x * (2. * x - 1.);
l1y = (y - 1.) * (2. * y - 1);
l2y = 4. * y * (1. - y);
l3y = y * (2. * y - 1.);
d1x = 4. * x - 3.;
d2x = 4. - 8. * x;
d3x = 4. * x - 1.;
d1y = 4. * y - 3.;
d2y = 4. - 8. * y;
d3y = 4. * y - 1.;
dshape(0,0) = d1x * l1y;
dshape(0,1) = l1x * d1y;
dshape(4,0) = d2x * l1y;
dshape(4,1) = l2x * d1y;
dshape(1,0) = d3x * l1y;
dshape(1,1) = l3x * d1y;
dshape(7,0) = d1x * l2y;
dshape(7,1) = l1x * d2y;
dshape(8,0) = d2x * l2y;
dshape(8,1) = l2x * d2y;
dshape(5,0) = d3x * l2y;
dshape(5,1) = l3x * d2y;
dshape(3,0) = d1x * l3y;
dshape(3,1) = l1x * d3y;
dshape(6,0) = d2x * l3y;
dshape(6,1) = l2x * d3y;
dshape(2,0) = d3x * l3y;
dshape(2,1) = l3x * d3y;
}
void BiQuad2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
#if 0
dofs = 1.;
#else
dofs = 0.;
dofs(vertex) = 1.;
switch (vertex)
{
case 0: dofs(4) = 0.25; dofs(7) = 0.25; break;
case 1: dofs(4) = 0.25; dofs(5) = 0.25; break;
case 2: dofs(5) = 0.25; dofs(6) = 0.25; break;
case 3: dofs(6) = 0.25; dofs(7) = 0.25; break;
}
dofs(8) = 1./16.;
#endif
}
GaussBiQuad2DFiniteElement::GaussBiQuad2DFiniteElement()
: NodalFiniteElement(2, Geometry::SQUARE, 9, 2, FunctionSpace::Qk)
{
const real_t p1 = 0.5*(1.-sqrt(3./5.));
Nodes.IntPoint(0).x = p1;
Nodes.IntPoint(0).y = p1;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = p1;
Nodes.IntPoint(1).x = 1.-p1;
Nodes.IntPoint(1).y = p1;
Nodes.IntPoint(7).x = p1;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(8).x = 0.5;
Nodes.IntPoint(8).y = 0.5;
Nodes.IntPoint(5).x = 1.-p1;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(3).x = p1;
Nodes.IntPoint(3).y = 1.-p1;
Nodes.IntPoint(6).x = 0.5;
Nodes.IntPoint(6).y = 1.-p1;
Nodes.IntPoint(2).x = 1.-p1;
Nodes.IntPoint(2).y = 1.-p1;
}
void GaussBiQuad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const real_t a = sqrt(5./3.);
const real_t p1 = 0.5*(1.-sqrt(3./5.));
real_t x = a*(ip.x-p1), y = a*(ip.y-p1);
real_t l1x, l2x, l3x, l1y, l2y, l3y;
l1x = (x - 1.) * (2. * x - 1);
l2x = 4. * x * (1. - x);
l3x = x * (2. * x - 1.);
l1y = (y - 1.) * (2. * y - 1);
l2y = 4. * y * (1. - y);
l3y = y * (2. * y - 1.);
shape(0) = l1x * l1y;
shape(4) = l2x * l1y;
shape(1) = l3x * l1y;
shape(7) = l1x * l2y;
shape(8) = l2x * l2y;
shape(5) = l3x * l2y;
shape(3) = l1x * l3y;
shape(6) = l2x * l3y;
shape(2) = l3x * l3y;
}
void GaussBiQuad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const real_t a = sqrt(5./3.);
const real_t p1 = 0.5*(1.-sqrt(3./5.));
real_t x = a*(ip.x-p1), y = a*(ip.y-p1);
real_t l1x, l2x, l3x, l1y, l2y, l3y;
real_t d1x, d2x, d3x, d1y, d2y, d3y;
l1x = (x - 1.) * (2. * x - 1);
l2x = 4. * x * (1. - x);
l3x = x * (2. * x - 1.);
l1y = (y - 1.) * (2. * y - 1);
l2y = 4. * y * (1. - y);
l3y = y * (2. * y - 1.);
d1x = a * (4. * x - 3.);
d2x = a * (4. - 8. * x);
d3x = a * (4. * x - 1.);
d1y = a * (4. * y - 3.);
d2y = a * (4. - 8. * y);
d3y = a * (4. * y - 1.);
dshape(0,0) = d1x * l1y;
dshape(0,1) = l1x * d1y;
dshape(4,0) = d2x * l1y;
dshape(4,1) = l2x * d1y;
dshape(1,0) = d3x * l1y;
dshape(1,1) = l3x * d1y;
dshape(7,0) = d1x * l2y;
dshape(7,1) = l1x * d2y;
dshape(8,0) = d2x * l2y;
dshape(8,1) = l2x * d2y;
dshape(5,0) = d3x * l2y;
dshape(5,1) = l3x * d2y;
dshape(3,0) = d1x * l3y;
dshape(3,1) = l1x * d3y;
dshape(6,0) = d2x * l3y;
dshape(6,1) = l2x * d3y;
dshape(2,0) = d3x * l3y;
dshape(2,1) = l3x * d3y;
}
BiCubic2DFiniteElement::BiCubic2DFiniteElement()
: NodalFiniteElement (2, Geometry::SQUARE, 16, 3, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.;
Nodes.IntPoint(0).y = 0.;
Nodes.IntPoint(1).x = 1.;
Nodes.IntPoint(1).y = 0.;
Nodes.IntPoint(2).x = 1.;
Nodes.IntPoint(2).y = 1.;
Nodes.IntPoint(3).x = 0.;
Nodes.IntPoint(3).y = 1.;
Nodes.IntPoint(4).x = 1./3.;
Nodes.IntPoint(4).y = 0.;
Nodes.IntPoint(5).x = 2./3.;
Nodes.IntPoint(5).y = 0.;
Nodes.IntPoint(6).x = 1.;
Nodes.IntPoint(6).y = 1./3.;
Nodes.IntPoint(7).x = 1.;
Nodes.IntPoint(7).y = 2./3.;
Nodes.IntPoint(8).x = 2./3.;
Nodes.IntPoint(8).y = 1.;
Nodes.IntPoint(9).x = 1./3.;
Nodes.IntPoint(9).y = 1.;
Nodes.IntPoint(10).x = 0.;
Nodes.IntPoint(10).y = 2./3.;
Nodes.IntPoint(11).x = 0.;
Nodes.IntPoint(11).y = 1./3.;
Nodes.IntPoint(12).x = 1./3.;
Nodes.IntPoint(12).y = 1./3.;
Nodes.IntPoint(13).x = 2./3.;
Nodes.IntPoint(13).y = 1./3.;
Nodes.IntPoint(14).x = 1./3.;
Nodes.IntPoint(14).y = 2./3.;
Nodes.IntPoint(15).x = 2./3.;
Nodes.IntPoint(15).y = 2./3.;
}
void BiCubic2DFiniteElement::CalcShape(
const IntegrationPoint &ip, Vector &shape) const
{
real_t x = ip.x, y = ip.y;
real_t w1x, w2x, w3x, w1y, w2y, w3y;
real_t l0x, l1x, l2x, l3x, l0y, l1y, l2y, l3y;
w1x = x - 1./3.; w2x = x - 2./3.; w3x = x - 1.;
w1y = y - 1./3.; w2y = y - 2./3.; w3y = y - 1.;
l0x = (- 4.5) * w1x * w2x * w3x;
l1x = ( 13.5) * x * w2x * w3x;
l2x = (-13.5) * x * w1x * w3x;
l3x = ( 4.5) * x * w1x * w2x;
l0y = (- 4.5) * w1y * w2y * w3y;
l1y = ( 13.5) * y * w2y * w3y;
l2y = (-13.5) * y * w1y * w3y;
l3y = ( 4.5) * y * w1y * w2y;
shape(0) = l0x * l0y;
shape(1) = l3x * l0y;
shape(2) = l3x * l3y;
shape(3) = l0x * l3y;
shape(4) = l1x * l0y;
shape(5) = l2x * l0y;
shape(6) = l3x * l1y;
shape(7) = l3x * l2y;
shape(8) = l2x * l3y;
shape(9) = l1x * l3y;
shape(10) = l0x * l2y;
shape(11) = l0x * l1y;
shape(12) = l1x * l1y;
shape(13) = l2x * l1y;
shape(14) = l1x * l2y;
shape(15) = l2x * l2y;
}
void BiCubic2DFiniteElement::CalcDShape(
const IntegrationPoint &ip, DenseMatrix &dshape) const
{
real_t x = ip.x, y = ip.y;
real_t w1x, w2x, w3x, w1y, w2y, w3y;
real_t l0x, l1x, l2x, l3x, l0y, l1y, l2y, l3y;
real_t d0x, d1x, d2x, d3x, d0y, d1y, d2y, d3y;
w1x = x - 1./3.; w2x = x - 2./3.; w3x = x - 1.;
w1y = y - 1./3.; w2y = y - 2./3.; w3y = y - 1.;
l0x = (- 4.5) * w1x * w2x * w3x;
l1x = ( 13.5) * x * w2x * w3x;
l2x = (-13.5) * x * w1x * w3x;
l3x = ( 4.5) * x * w1x * w2x;
l0y = (- 4.5) * w1y * w2y * w3y;
l1y = ( 13.5) * y * w2y * w3y;
l2y = (-13.5) * y * w1y * w3y;
l3y = ( 4.5) * y * w1y * w2y;
d0x = -5.5 + ( 18. - 13.5 * x) * x;
d1x = 9. + (-45. + 40.5 * x) * x;
d2x = -4.5 + ( 36. - 40.5 * x) * x;
d3x = 1. + (- 9. + 13.5 * x) * x;
d0y = -5.5 + ( 18. - 13.5 * y) * y;
d1y = 9. + (-45. + 40.5 * y) * y;
d2y = -4.5 + ( 36. - 40.5 * y) * y;
d3y = 1. + (- 9. + 13.5 * y) * y;
dshape( 0,0) = d0x * l0y; dshape( 0,1) = l0x * d0y;
dshape( 1,0) = d3x * l0y; dshape( 1,1) = l3x * d0y;
dshape( 2,0) = d3x * l3y; dshape( 2,1) = l3x * d3y;
dshape( 3,0) = d0x * l3y; dshape( 3,1) = l0x * d3y;
dshape( 4,0) = d1x * l0y; dshape( 4,1) = l1x * d0y;
dshape( 5,0) = d2x * l0y; dshape( 5,1) = l2x * d0y;
dshape( 6,0) = d3x * l1y; dshape( 6,1) = l3x * d1y;
dshape( 7,0) = d3x * l2y; dshape( 7,1) = l3x * d2y;
dshape( 8,0) = d2x * l3y; dshape( 8,1) = l2x * d3y;
dshape( 9,0) = d1x * l3y; dshape( 9,1) = l1x * d3y;
dshape(10,0) = d0x * l2y; dshape(10,1) = l0x * d2y;
dshape(11,0) = d0x * l1y; dshape(11,1) = l0x * d1y;
dshape(12,0) = d1x * l1y; dshape(12,1) = l1x * d1y;
dshape(13,0) = d2x * l1y; dshape(13,1) = l2x * d1y;
dshape(14,0) = d1x * l2y; dshape(14,1) = l1x * d2y;
dshape(15,0) = d2x * l2y; dshape(15,1) = l2x * d2y;
}
void BiCubic2DFiniteElement::CalcHessian(
const IntegrationPoint &ip, DenseMatrix &h) const
{
real_t x = ip.x, y = ip.y;
real_t w1x, w2x, w3x, w1y, w2y, w3y;
real_t l0x, l1x, l2x, l3x, l0y, l1y, l2y, l3y;
real_t d0x, d1x, d2x, d3x, d0y, d1y, d2y, d3y;
real_t h0x, h1x, h2x, h3x, h0y, h1y, h2y, h3y;
w1x = x - 1./3.; w2x = x - 2./3.; w3x = x - 1.;
w1y = y - 1./3.; w2y = y - 2./3.; w3y = y - 1.;
l0x = (- 4.5) * w1x * w2x * w3x;
l1x = ( 13.5) * x * w2x * w3x;
l2x = (-13.5) * x * w1x * w3x;
l3x = ( 4.5) * x * w1x * w2x;
l0y = (- 4.5) * w1y * w2y * w3y;
l1y = ( 13.5) * y * w2y * w3y;
l2y = (-13.5) * y * w1y * w3y;
l3y = ( 4.5) * y * w1y * w2y;
d0x = -5.5 + ( 18. - 13.5 * x) * x;
d1x = 9. + (-45. + 40.5 * x) * x;
d2x = -4.5 + ( 36. - 40.5 * x) * x;
d3x = 1. + (- 9. + 13.5 * x) * x;
d0y = -5.5 + ( 18. - 13.5 * y) * y;
d1y = 9. + (-45. + 40.5 * y) * y;
d2y = -4.5 + ( 36. - 40.5 * y) * y;
d3y = 1. + (- 9. + 13.5 * y) * y;
h0x = -27. * x + 18.;
h1x = 81. * x - 45.;
h2x = -81. * x + 36.;
h3x = 27. * x - 9.;
h0y = -27. * y + 18.;
h1y = 81. * y - 45.;
h2y = -81. * y + 36.;
h3y = 27. * y - 9.;
h( 0,0) = h0x * l0y; h( 0,1) = d0x * d0y; h( 0,2) = l0x * h0y;
h( 1,0) = h3x * l0y; h( 1,1) = d3x * d0y; h( 1,2) = l3x * h0y;
h( 2,0) = h3x * l3y; h( 2,1) = d3x * d3y; h( 2,2) = l3x * h3y;
h( 3,0) = h0x * l3y; h( 3,1) = d0x * d3y; h( 3,2) = l0x * h3y;
h( 4,0) = h1x * l0y; h( 4,1) = d1x * d0y; h( 4,2) = l1x * h0y;
h( 5,0) = h2x * l0y; h( 5,1) = d2x * d0y; h( 5,2) = l2x * h0y;
h( 6,0) = h3x * l1y; h( 6,1) = d3x * d1y; h( 6,2) = l3x * h1y;
h( 7,0) = h3x * l2y; h( 7,1) = d3x * d2y; h( 7,2) = l3x * h2y;
h( 8,0) = h2x * l3y; h( 8,1) = d2x * d3y; h( 8,2) = l2x * h3y;
h( 9,0) = h1x * l3y; h( 9,1) = d1x * d3y; h( 9,2) = l1x * h3y;
h(10,0) = h0x * l2y; h(10,1) = d0x * d2y; h(10,2) = l0x * h2y;
h(11,0) = h0x * l1y; h(11,1) = d0x * d1y; h(11,2) = l0x * h1y;
h(12,0) = h1x * l1y; h(12,1) = d1x * d1y; h(12,2) = l1x * h1y;
h(13,0) = h2x * l1y; h(13,1) = d2x * d1y; h(13,2) = l2x * h1y;
h(14,0) = h1x * l2y; h(14,1) = d1x * d2y; h(14,2) = l1x * h2y;
h(15,0) = h2x * l2y; h(15,1) = d2x * d2y; h(15,2) = l2x * h2y;
}
Cubic1DFiniteElement::Cubic1DFiniteElement()
: NodalFiniteElement(1, Geometry::SEGMENT, 4, 3)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(2).x = 0.33333333333333333333;
Nodes.IntPoint(3).x = 0.66666666666666666667;
}
void Cubic1DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x;
real_t l1 = x,
l2 = (1.0-x),
l3 = (0.33333333333333333333-x),
l4 = (0.66666666666666666667-x);
shape(0) = 4.5 * l2 * l3 * l4;
shape(1) = 4.5 * l1 * l3 * l4;
shape(2) = 13.5 * l1 * l2 * l4;
shape(3) = -13.5 * l1 * l2 * l3;
}
void Cubic1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x;
dshape(0,0) = -5.5 + x * (18. - 13.5 * x);
dshape(1,0) = 1. - x * (9. - 13.5 * x);
dshape(2,0) = 9. - x * (45. - 40.5 * x);
dshape(3,0) = -4.5 + x * (36. - 40.5 * x);
}
Cubic2DFiniteElement::Cubic2DFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 10, 3)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.33333333333333333333;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(4).x = 0.66666666666666666667;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(5).x = 0.66666666666666666667;
Nodes.IntPoint(5).y = 0.33333333333333333333;
Nodes.IntPoint(6).x = 0.33333333333333333333;
Nodes.IntPoint(6).y = 0.66666666666666666667;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 0.66666666666666666667;
Nodes.IntPoint(8).x = 0.0;
Nodes.IntPoint(8).y = 0.33333333333333333333;
Nodes.IntPoint(9).x = 0.33333333333333333333;
Nodes.IntPoint(9).y = 0.33333333333333333333;
}
void Cubic2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x, y = ip.y;
real_t l1 = (-1. + x + y),
lx = (-1. + 3.*x),
ly = (-1. + 3.*y);
shape(0) = -0.5*l1*(3.*l1 + 1.)*(3.*l1 + 2.);
shape(1) = 0.5*x*(lx - 1.)*lx;
shape(2) = 0.5*y*(-1. + ly)*ly;
shape(3) = 4.5*x*l1*(3.*l1 + 1.);
shape(4) = -4.5*x*lx*l1;
shape(5) = 4.5*x*lx*y;
shape(6) = 4.5*x*y*ly;
shape(7) = -4.5*y*l1*ly;
shape(8) = 4.5*y*l1*(1. + 3.*l1);
shape(9) = -27.*x*y*l1;
}
void Cubic2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x, y = ip.y;
dshape(0,0) = 0.5*(-11. + 36.*y - 9.*(x*(-4. + 3.*x) + 6.*x*y + 3.*y*y));
dshape(1,0) = 1. + 4.5*x*(-2. + 3.*x);
dshape(2,0) = 0.;
dshape(3,0) = 4.5*(2. + 9.*x*x - 5.*y + 3.*y*y + 2.*x*(-5. + 6.*y));
dshape(4,0) = -4.5*(1. - 1.*y + x*(-8. + 9.*x + 6.*y));
dshape(5,0) = 4.5*(-1. + 6.*x)*y;
dshape(6,0) = 4.5*y*(-1. + 3.*y);
dshape(7,0) = 4.5*(1. - 3.*y)*y;
dshape(8,0) = 4.5*y*(-5. + 6.*x + 6.*y);
dshape(9,0) = -27.*y*(-1. + 2.*x + y);
dshape(0,1) = 0.5*(-11. + 36.*y - 9.*(x*(-4. + 3.*x) + 6.*x*y + 3.*y*y));
dshape(1,1) = 0.;
dshape(2,1) = 1. + 4.5*y*(-2. + 3.*y);
dshape(3,1) = 4.5*x*(-5. + 6.*x + 6.*y);
dshape(4,1) = 4.5*(1. - 3.*x)*x;
dshape(5,1) = 4.5*x*(-1. + 3.*x);
dshape(6,1) = 4.5*x*(-1. + 6.*y);
dshape(7,1) = -4.5*(1. + x*(-1. + 6.*y) + y*(-8. + 9.*y));
dshape(8,1) = 4.5*(2. + 3.*x*x + y*(-10. + 9.*y) + x*(-5. + 12.*y));
dshape(9,1) = -27.*x*(-1. + x + 2.*y);
}
void Cubic2DFiniteElement::CalcHessian (const IntegrationPoint &ip,
DenseMatrix &h) const
{
real_t x = ip.x, y = ip.y;
h(0,0) = 18.-27.*(x+y);
h(0,1) = 18.-27.*(x+y);
h(0,2) = 18.-27.*(x+y);
h(1,0) = -9.+27.*x;
h(1,1) = 0.;
h(1,2) = 0.;
h(2,0) = 0.;
h(2,1) = 0.;
h(2,2) = -9.+27.*y;
h(3,0) = -45.+81.*x+54.*y;
h(3,1) = -22.5+54.*x+27.*y;
h(3,2) = 27.*x;
h(4,0) = 36.-81.*x-27.*y;
h(4,1) = 4.5-27.*x;
h(4,2) = 0.;
h(5,0) = 27.*y;
h(5,1) = -4.5+27.*x;
h(5,2) = 0.;
h(6,0) = 0.;
h(6,1) = -4.5+27.*y;
h(6,2) = 27.*x;
h(7,0) = 0.;
h(7,1) = 4.5-27.*y;
h(7,2) = 36.-27.*x-81.*y;
h(8,0) = 27.*y;
h(8,1) = -22.5+27.*x+54.*y;
h(8,2) = -45.+54.*x+81.*y;
h(9,0) = -54.*y;
h(9,1) = 27.-54.*(x+y);
h(9,2) = -54.*x;
}
Cubic3DFiniteElement::Cubic3DFiniteElement()
: NodalFiniteElement(3, Geometry::TETRAHEDRON, 20, 3)
{
Nodes.IntPoint(0).x = 0;
Nodes.IntPoint(0).y = 0;
Nodes.IntPoint(0).z = 0;
Nodes.IntPoint(1).x = 1.;
Nodes.IntPoint(1).y = 0;
Nodes.IntPoint(1).z = 0;
Nodes.IntPoint(2).x = 0;
Nodes.IntPoint(2).y = 1.;
Nodes.IntPoint(2).z = 0;
Nodes.IntPoint(3).x = 0;
Nodes.IntPoint(3).y = 0;
Nodes.IntPoint(3).z = 1.;
Nodes.IntPoint(4).x = 0.3333333333333333333333333333;
Nodes.IntPoint(4).y = 0;
Nodes.IntPoint(4).z = 0;
Nodes.IntPoint(5).x = 0.6666666666666666666666666667;
Nodes.IntPoint(5).y = 0;
Nodes.IntPoint(5).z = 0;
Nodes.IntPoint(6).x = 0;
Nodes.IntPoint(6).y = 0.3333333333333333333333333333;
Nodes.IntPoint(6).z = 0;
Nodes.IntPoint(7).x = 0;
Nodes.IntPoint(7).y = 0.6666666666666666666666666667;
Nodes.IntPoint(7).z = 0;
Nodes.IntPoint(8).x = 0;
Nodes.IntPoint(8).y = 0;
Nodes.IntPoint(8).z = 0.3333333333333333333333333333;
Nodes.IntPoint(9).x = 0;
Nodes.IntPoint(9).y = 0;
Nodes.IntPoint(9).z = 0.6666666666666666666666666667;
Nodes.IntPoint(10).x = 0.6666666666666666666666666667;
Nodes.IntPoint(10).y = 0.3333333333333333333333333333;
Nodes.IntPoint(10).z = 0;
Nodes.IntPoint(11).x = 0.3333333333333333333333333333;
Nodes.IntPoint(11).y = 0.6666666666666666666666666667;
Nodes.IntPoint(11).z = 0;
Nodes.IntPoint(12).x = 0.6666666666666666666666666667;
Nodes.IntPoint(12).y = 0;
Nodes.IntPoint(12).z = 0.3333333333333333333333333333;
Nodes.IntPoint(13).x = 0.3333333333333333333333333333;
Nodes.IntPoint(13).y = 0;
Nodes.IntPoint(13).z = 0.6666666666666666666666666667;
Nodes.IntPoint(14).x = 0;
Nodes.IntPoint(14).y = 0.6666666666666666666666666667;
Nodes.IntPoint(14).z = 0.3333333333333333333333333333;
Nodes.IntPoint(15).x = 0;
Nodes.IntPoint(15).y = 0.3333333333333333333333333333;
Nodes.IntPoint(15).z = 0.6666666666666666666666666667;
Nodes.IntPoint(16).x = 0.3333333333333333333333333333;
Nodes.IntPoint(16).y = 0.3333333333333333333333333333;
Nodes.IntPoint(16).z = 0.3333333333333333333333333333;
Nodes.IntPoint(17).x = 0;
Nodes.IntPoint(17).y = 0.3333333333333333333333333333;
Nodes.IntPoint(17).z = 0.3333333333333333333333333333;
Nodes.IntPoint(18).x = 0.3333333333333333333333333333;
Nodes.IntPoint(18).y = 0;
Nodes.IntPoint(18).z = 0.3333333333333333333333333333;
Nodes.IntPoint(19).x = 0.3333333333333333333333333333;
Nodes.IntPoint(19).y = 0.3333333333333333333333333333;
Nodes.IntPoint(19).z = 0;
}
void Cubic3DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
shape(0) = -((-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z)*
(-1 + 3*x + 3*y + 3*z))/2.;
shape(4) = (9*x*(-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z))/2.;
shape(5) = (-9*x*(-1 + 3*x)*(-1 + x + y + z))/2.;
shape(1) = (x*(2 + 9*(-1 + x)*x))/2.;
shape(6) = (9*y*(-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z))/2.;
shape(19) = -27*x*y*(-1 + x + y + z);
shape(10) = (9*x*(-1 + 3*x)*y)/2.;
shape(7) = (-9*y*(-1 + 3*y)*(-1 + x + y + z))/2.;
shape(11) = (9*x*y*(-1 + 3*y))/2.;
shape(2) = (y*(2 + 9*(-1 + y)*y))/2.;
shape(8) = (9*z*(-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z))/2.;
shape(18) = -27*x*z*(-1 + x + y + z);
shape(12) = (9*x*(-1 + 3*x)*z)/2.;
shape(17) = -27*y*z*(-1 + x + y + z);
shape(16) = 27*x*y*z;
shape(14) = (9*y*(-1 + 3*y)*z)/2.;
shape(9) = (-9*z*(-1 + x + y + z)*(-1 + 3*z))/2.;
shape(13) = (9*x*z*(-1 + 3*z))/2.;
shape(15) = (9*y*z*(-1 + 3*z))/2.;
shape(3) = (z*(2 + 9*(-1 + z)*z))/2.;
}
void Cubic3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
dshape(0,0) = (-11 + 36*y + 36*z - 9*(3*pow(x,2) + 3*pow(y + z,2) +
x*(-4 + 6*y + 6*z)))/2.;
dshape(0,1) = (-11 + 36*y + 36*z - 9*(3*pow(x,2) + 3*pow(y + z,2) +
x*(-4 + 6*y + 6*z)))/2.;
dshape(0,2) = (-11 + 36*y + 36*z - 9*(3*pow(x,2) + 3*pow(y + z,2) +
x*(-4 + 6*y + 6*z)))/2.;
dshape(4,0) = (9*(9*pow(x,2) + (-1 + y + z)*(-2 + 3*y + 3*z) +
2*x*(-5 + 6*y + 6*z)))/2.;
dshape(4,1) = (9*x*(-5 + 6*x + 6*y + 6*z))/2.;
dshape(4,2) = (9*x*(-5 + 6*x + 6*y + 6*z))/2.;
dshape(5,0) = (-9*(1 - y - z + x*(-8 + 9*x + 6*y + 6*z)))/2.;
dshape(5,1) = (9*(1 - 3*x)*x)/2.;
dshape(5,2) = (9*(1 - 3*x)*x)/2.;
dshape(1,0) = 1 + (9*x*(-2 + 3*x))/2.;
dshape(1,1) = 0;
dshape(1,2) = 0;
dshape(6,0) = (9*y*(-5 + 6*x + 6*y + 6*z))/2.;
dshape(6,1) = (9*(2 + 3*pow(x,2) - 10*y - 5*z + 3*(y + z)*(3*y + z) +
x*(-5 + 12*y + 6*z)))/2.;
dshape(6,2) = (9*y*(-5 + 6*x + 6*y + 6*z))/2.;
dshape(19,0) = -27*y*(-1 + 2*x + y + z);
dshape(19,1) = -27*x*(-1 + x + 2*y + z);
dshape(19,2) = -27*x*y;
dshape(10,0) = (9*(-1 + 6*x)*y)/2.;
dshape(10,1) = (9*x*(-1 + 3*x))/2.;
dshape(10,2) = 0;
dshape(7,0) = (9*(1 - 3*y)*y)/2.;
dshape(7,1) = (-9*(1 + x*(-1 + 6*y) - z + y*(-8 + 9*y + 6*z)))/2.;
dshape(7,2) = (9*(1 - 3*y)*y)/2.;
dshape(11,0) = (9*y*(-1 + 3*y))/2.;
dshape(11,1) = (9*x*(-1 + 6*y))/2.;
dshape(11,2) = 0;
dshape(2,0) = 0;
dshape(2,1) = 1 + (9*y*(-2 + 3*y))/2.;
dshape(2,2) = 0;
dshape(8,0) = (9*z*(-5 + 6*x + 6*y + 6*z))/2.;
dshape(8,1) = (9*z*(-5 + 6*x + 6*y + 6*z))/2.;
dshape(8,2) = (9*(2 + 3*pow(x,2) - 5*y - 10*z + 3*(y + z)*(y + 3*z) +
x*(-5 + 6*y + 12*z)))/2.;
dshape(18,0) = -27*z*(-1 + 2*x + y + z);
dshape(18,1) = -27*x*z;
dshape(18,2) = -27*x*(-1 + x + y + 2*z);
dshape(12,0) = (9*(-1 + 6*x)*z)/2.;
dshape(12,1) = 0;
dshape(12,2) = (9*x*(-1 + 3*x))/2.;
dshape(17,0) = -27*y*z;
dshape(17,1) = -27*z*(-1 + x + 2*y + z);
dshape(17,2) = -27*y*(-1 + x + y + 2*z);
dshape(16,0) = 27*y*z;
dshape(16,1) = 27*x*z;
dshape(16,2) = 27*x*y;
dshape(14,0) = 0;
dshape(14,1) = (9*(-1 + 6*y)*z)/2.;
dshape(14,2) = (9*y*(-1 + 3*y))/2.;
dshape(9,0) = (9*(1 - 3*z)*z)/2.;
dshape(9,1) = (9*(1 - 3*z)*z)/2.;
dshape(9,2) = (9*(-1 + x + y + 8*z - 6*(x + y)*z - 9*pow(z,2)))/2.;
dshape(13,0) = (9*z*(-1 + 3*z))/2.;
dshape(13,1) = 0;
dshape(13,2) = (9*x*(-1 + 6*z))/2.;
dshape(15,0) = 0;
dshape(15,1) = (9*z*(-1 + 3*z))/2.;
dshape(15,2) = (9*y*(-1 + 6*z))/2.;
dshape(3,0) = 0;
dshape(3,1) = 0;
dshape(3,2) = 1 + (9*z*(-2 + 3*z))/2.;
}
P0TriangleFiniteElement::P0TriangleFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 1, 0)
{
Nodes.IntPoint(0).x = 0.333333333333333333;
Nodes.IntPoint(0).y = 0.333333333333333333;
}
void P0TriangleFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0;
}
void P0TriangleFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0;
dshape(0,1) = 0.0;
}
P0QuadFiniteElement::P0QuadFiniteElement()
: NodalFiniteElement(2, Geometry::SQUARE, 1, 0, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.5;
}
void P0QuadFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0;
}
void P0QuadFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0;
dshape(0,1) = 0.0;
}
Linear3DFiniteElement::Linear3DFiniteElement()
: NodalFiniteElement(3, Geometry::TETRAHEDRON, 4, 1)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(3).z = 1.0;
}
void Linear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1. - ip.x - ip.y - ip.z;
shape(1) = ip.x;
shape(2) = ip.y;
shape(3) = ip.z;
}
void Linear3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
if (dshape.Height() == 4)
{
real_t *A = &dshape(0,0);
A[0] = -1.; A[4] = -1.; A[8] = -1.;
A[1] = 1.; A[5] = 0.; A[9] = 0.;
A[2] = 0.; A[6] = 1.; A[10] = 0.;
A[3] = 0.; A[7] = 0.; A[11] = 1.;
}
else
{
dshape(0,0) = -1.; dshape(0,1) = -1.; dshape(0,2) = -1.;
dshape(1,0) = 1.; dshape(1,1) = 0.; dshape(1,2) = 0.;
dshape(2,0) = 0.; dshape(2,1) = 1.; dshape(2,2) = 0.;
dshape(3,0) = 0.; dshape(3,1) = 0.; dshape(3,2) = 1.;
}
}
void Linear3DFiniteElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &h) const
{
h = 0.0;
}
void Linear3DFiniteElement::GetFaceDofs (int face, int **dofs, int *ndofs)
const
{
static int face_dofs[4][3] = {{1, 2, 3}, {0, 2, 3}, {0, 1, 3}, {0, 1, 2}};
*ndofs = 3;
*dofs = face_dofs[face];
}
// TODO: use a FunctionSpace specific to wedges instead of Qk.
LinearWedgeFiniteElement::LinearWedgeFiniteElement()
: NodalFiniteElement(3, Geometry::PRISM, 6, 1, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(3).z = 1.0;
Nodes.IntPoint(4).x = 1.0;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 1.0;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 1.0;
Nodes.IntPoint(5).z = 1.0;
}
void LinearWedgeFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = (1. - ip.x - ip.y) * (1. - ip.z);
shape(1) = ip.x * (1. - ip.z);
shape(2) = ip.y * (1. - ip.z);
shape(3) = (1. - ip.x - ip.y) * ip.z;
shape(4) = ip.x * ip.z;
shape(5) = ip.y * ip.z;
}
void LinearWedgeFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = -1. + ip.z;
dshape(0,1) = -1. + ip.z;
dshape(0,2) = -1. + ip.x + ip.y;
dshape(1,0) = 1. - ip.z;
dshape(1,1) = 0.;
dshape(1,2) = -ip.x;
dshape(2,0) = 0.;
dshape(2,1) = 1. - ip.z;
dshape(2,2) = -ip.y;
dshape(3,0) = -ip.z;
dshape(3,1) = -ip.z;
dshape(3,2) = 1. - ip.x - ip.y;
dshape(4,0) = ip.z;
dshape(4,1) = 0.;
dshape(4,2) = ip.x;
dshape(5,0) = 0.;
dshape(5,1) = ip.z;
dshape(5,2) = ip.y;
}
void LinearWedgeFiniteElement::GetFaceDofs (int face, int **dofs, int *ndofs)
const
{
static int face_dofs[5][4] =
{{0, 2, 1, -1}, {3, 4, 5, -1}, {0, 1, 4, 3}, {1, 2, 5, 4}, {2, 0, 3, 5}};
*ndofs = (face < 2) ? 3 : 4;
*dofs = face_dofs[face];
}
LinearPyramidFiniteElement::LinearPyramidFiniteElement()
: NodalFiniteElement(3, Geometry::PYRAMID, 5, 1, FunctionSpace::Uk)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 1.0;
Nodes.IntPoint(3).z = 0.0;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 1.0;
}
void LinearPyramidFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
real_t ox = 1.-x-z, oy = 1.-y-z, oz = 1.-z;
real_t tol = 1e-6;
if (oz <= tol)
{
// We must return the limit of the basis functions as z->1. In order to
// remain inside the pyramid in this limit the x and y coordinates must
// be approaching 0. The resulting limiting basis function values are:
shape(0) = 0.;
shape(1) = 0.;
shape(2) = 0.;
shape(3) = 0.;
shape(4) = 1.;
return;
}
real_t ozi = 1. / oz;
shape(0) = ox * oy * ozi;
shape(1) = x * oy * ozi;
shape(2) = x * y * ozi;
shape(3) = ox * y * ozi;
shape(4) = z;
}
void LinearPyramidFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
real_t ox = 1.-x-z, oy = 1.-y-z, oz = 1.-z;
real_t tol = 1e-6;
if (oz <= tol)
{
// At the apex of the pyramid the gradients of the basis functions are
// multivalued and depend on the direction from which the limit is taken.
// The following values correspond to the average of the gradients taken
// over all possible directions approaching the apex of the pyramid from
// within its interior.
dshape(0,0) = - 0.5;
dshape(0,1) = - 0.5;
dshape(0,2) = - 0.75;
dshape(1,0) = 0.5;
dshape(1,1) = - 0.5;
dshape(1,2) = - 0.25;
dshape(2,0) = 0.5;
dshape(2,1) = 0.5;
dshape(2,2) = 0.25;
dshape(3,0) = - 0.5;
dshape(3,1) = 0.5;
dshape(3,2) = - 0.25;
dshape(4,0) = 0.;
dshape(4,1) = 0.;
dshape(4,2) = 1.;
return;
}
real_t ozi = 1. / oz;
dshape(0,0) = - oy * ozi;
dshape(0,1) = - ox * ozi;
dshape(0,2) = x * y * ozi * ozi - 1.;
dshape(1,0) = oy * ozi;
dshape(1,1) = - x * ozi;
dshape(1,2) = - x * y * ozi * ozi;
dshape(2,0) = y * ozi;
dshape(2,1) = x * ozi;
dshape(2,2) = x * y * ozi * ozi;
dshape(3,0) = - y * ozi;
dshape(3,1) = ox * ozi;
dshape(3,2) = - x * y * ozi * ozi;
dshape(4,0) = 0.;
dshape(4,1) = 0.;
dshape(4,2) = 1.;
}
void LinearPyramidFiniteElement::GetFaceDofs (int face, int **dofs, int *ndofs)
const
{
static int face_dofs[5][4] =
{{3, 2, 1, 0}, {0, 1, 4, -1}, {1, 2, 4, -1}, {2, 3, 4, -1}, {3, 0, 4, -1}};
*ndofs = (face < 1) ? 4 : 3;
*dofs = face_dofs[face];
}
Quadratic3DFiniteElement::Quadratic3DFiniteElement()
: NodalFiniteElement(3, Geometry::TETRAHEDRON, 10, 2)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(3).z = 1.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 0.0;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(5).z = 0.0;
Nodes.IntPoint(6).x = 0.0;
Nodes.IntPoint(6).y = 0.0;
Nodes.IntPoint(6).z = 0.5;
Nodes.IntPoint(7).x = 0.5;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(7).z = 0.0;
Nodes.IntPoint(8).x = 0.5;
Nodes.IntPoint(8).y = 0.0;
Nodes.IntPoint(8).z = 0.5;
Nodes.IntPoint(9).x = 0.0;
Nodes.IntPoint(9).y = 0.5;
Nodes.IntPoint(9).z = 0.5;
}
void Quadratic3DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t L0, L1, L2, L3;
L0 = 1. - ip.x - ip.y - ip.z;
L1 = ip.x;
L2 = ip.y;
L3 = ip.z;
shape(0) = L0 * ( 2.0 * L0 - 1.0 );
shape(1) = L1 * ( 2.0 * L1 - 1.0 );
shape(2) = L2 * ( 2.0 * L2 - 1.0 );
shape(3) = L3 * ( 2.0 * L3 - 1.0 );
shape(4) = 4.0 * L0 * L1;
shape(5) = 4.0 * L0 * L2;
shape(6) = 4.0 * L0 * L3;
shape(7) = 4.0 * L1 * L2;
shape(8) = 4.0 * L1 * L3;
shape(9) = 4.0 * L2 * L3;
}
void Quadratic3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x, y, z, L0;
x = ip.x;
y = ip.y;
z = ip.z;
L0 = 1.0 - x - y - z;
dshape(0,0) = dshape(0,1) = dshape(0,2) = 1.0 - 4.0 * L0;
dshape(1,0) = -1.0 + 4.0 * x; dshape(1,1) = 0.0; dshape(1,2) = 0.0;
dshape(2,0) = 0.0; dshape(2,1) = -1.0 + 4.0 * y; dshape(2,2) = 0.0;
dshape(3,0) = dshape(3,1) = 0.0; dshape(3,2) = -1.0 + 4.0 * z;
dshape(4,0) = 4.0 * (L0 - x); dshape(4,1) = dshape(4,2) = -4.0 * x;
dshape(5,0) = dshape(5,2) = -4.0 * y; dshape(5,1) = 4.0 * (L0 - y);
dshape(6,0) = dshape(6,1) = -4.0 * z; dshape(6,2) = 4.0 * (L0 - z);
dshape(7,0) = 4.0 * y; dshape(7,1) = 4.0 * x; dshape(7,2) = 0.0;
dshape(8,0) = 4.0 * z; dshape(8,1) = 0.0; dshape(8,2) = 4.0 * x;
dshape(9,0) = 0.0; dshape(9,1) = 4.0 * z; dshape(9,2) = 4.0 * y;
}
TriLinear3DFiniteElement::TriLinear3DFiniteElement()
: NodalFiniteElement(3, Geometry::CUBE, 8, 1, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 1.0;
Nodes.IntPoint(3).z = 0.0;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 1.0;
Nodes.IntPoint(5).x = 1.0;
Nodes.IntPoint(5).y = 0.0;
Nodes.IntPoint(5).z = 1.0;
Nodes.IntPoint(6).x = 1.0;
Nodes.IntPoint(6).y = 1.0;
Nodes.IntPoint(6).z = 1.0;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 1.0;
Nodes.IntPoint(7).z = 1.0;
}
void TriLinear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
real_t ox = 1.-x, oy = 1.-y, oz = 1.-z;
shape(0) = ox * oy * oz;
shape(1) = x * oy * oz;
shape(2) = x * y * oz;
shape(3) = ox * y * oz;
shape(4) = ox * oy * z;
shape(5) = x * oy * z;
shape(6) = x * y * z;
shape(7) = ox * y * z;
}
void TriLinear3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
real_t ox = 1.-x, oy = 1.-y, oz = 1.-z;
dshape(0,0) = - oy * oz;
dshape(0,1) = - ox * oz;
dshape(0,2) = - ox * oy;
dshape(1,0) = oy * oz;
dshape(1,1) = - x * oz;
dshape(1,2) = - x * oy;
dshape(2,0) = y * oz;
dshape(2,1) = x * oz;
dshape(2,2) = - x * y;
dshape(3,0) = - y * oz;
dshape(3,1) = ox * oz;
dshape(3,2) = - ox * y;
dshape(4,0) = - oy * z;
dshape(4,1) = - ox * z;
dshape(4,2) = ox * oy;
dshape(5,0) = oy * z;
dshape(5,1) = - x * z;
dshape(5,2) = x * oy;
dshape(6,0) = y * z;
dshape(6,1) = x * z;
dshape(6,2) = x * y;
dshape(7,0) = - y * z;
dshape(7,1) = ox * z;
dshape(7,2) = ox * y;
}
void TriLinear3DFiniteElement::CalcHessian(const IntegrationPoint &ip,
DenseMatrix &h) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
real_t ox = 1.-x, oy = 1.-y, oz = 1.-z;
h(0,0) = 0.; h(0,1) = oz; h(0,2) = oy;
h(0,3) = 0.; h(0,4) = ox; h(0,5) = 0.;
h(1,0) = 0.; h(1,1) = -oz; h(1,2) = -oy;
h(1,3) = 0.; h(1,4) = x; h(1,5) = 0.;
h(2,0) = 0.; h(2,1) = oz; h(2,2) = -y;
h(2,3) = 0.; h(2,4) = -x; h(2,5) = 0.;
h(3,0) = 0.; h(3,1) = -oz; h(3,2) = y;
h(3,3) = 0.; h(3,4) = -ox; h(3,5) = 0.;
h(4,0) = 0.; h(4,1) = z; h(4,2) = -oy;
h(4,3) = 0.; h(4,4) = -ox; h(4,5) = 0.;
h(5,0) = 0.; h(5,1) = -z; h(5,2) = oy;
h(5,3) = 0.; h(5,4) = -x; h(5,5) = 0.;
h(6,0) = 0.; h(6,1) = z; h(6,2) = y;
h(6,3) = 0.; h(6,4) = x; h(6,5) = 0.;
h(7,0) = 0.; h(7,1) = -z; h(7,2) = -y;
h(7,3) = 0.; h(7,4) = ox; h(7,5) = 0.;
}
P0SegmentFiniteElement::P0SegmentFiniteElement(int Ord)
: NodalFiniteElement(1, Geometry::SEGMENT, 1, Ord) // default Ord = 0
{
Nodes.IntPoint(0).x = 0.5;
}
void P0SegmentFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0;
}
void P0SegmentFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0;
}
CrouzeixRaviartFiniteElement::CrouzeixRaviartFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 3, 1)
{
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 0.5;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 0.5;
}
void CrouzeixRaviartFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0 - 2.0 * ip.y;
shape(1) = -1.0 + 2.0 * ( ip.x + ip.y );
shape(2) = 1.0 - 2.0 * ip.x;
}
void CrouzeixRaviartFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0; dshape(0,1) = -2.0;
dshape(1,0) = 2.0; dshape(1,1) = 2.0;
dshape(2,0) = -2.0; dshape(2,1) = 0.0;
}
CrouzeixRaviartQuadFiniteElement::CrouzeixRaviartQuadFiniteElement()
// the FunctionSpace should be rotated (45 degrees) Q_1
// i.e. the span of { 1, x, y, x^2 - y^2 }
: NodalFiniteElement(2, Geometry::SQUARE, 4, 2, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(2).x = 0.5;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.5;
}
void CrouzeixRaviartQuadFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const real_t l1 = ip.x+ip.y-0.5, l2 = 1.-l1, l3 = ip.x-ip.y+0.5, l4 = 1.-l3;
shape(0) = l2 * l3;
shape(1) = l1 * l3;
shape(2) = l1 * l4;
shape(3) = l2 * l4;
}
void CrouzeixRaviartQuadFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const real_t x2 = 2.*ip.x, y2 = 2.*ip.y;
dshape(0,0) = 1. - x2; dshape(0,1) = -2. + y2;
dshape(1,0) = x2; dshape(1,1) = 1. - y2;
dshape(2,0) = 1. - x2; dshape(2,1) = y2;
dshape(3,0) = -2. + x2; dshape(3,1) = 1. - y2;
}
RT0TriangleFiniteElement::RT0TriangleFiniteElement()
: VectorFiniteElement(2, Geometry::TRIANGLE, 3, 1, H_DIV)
{
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 0.5;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 0.5;
}
void RT0TriangleFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y;
shape(0,0) = x;
shape(0,1) = y - 1.;
shape(1,0) = x;
shape(1,1) = y;
shape(2,0) = x - 1.;
shape(2,1) = y;
}
void RT0TriangleFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
divshape(0) = 2.;
divshape(1) = 2.;
divshape(2) = 2.;
}
const real_t RT0TriangleFiniteElement::nk[3][2] =
{ {0, -1}, {1, 1}, {-1, 0} };
void RT0TriangleFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 3; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 3; j++)
{
real_t d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT0TriangleFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[2];
Vector xk (vk, 2);
for (k = 0; k < 3; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
for (j = 0; j < 3; j++)
if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT0TriangleFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[2];
Vector xk (vk, 2);
for (int k = 0; k < 3; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
}
}
RT0QuadFiniteElement::RT0QuadFiniteElement()
: VectorFiniteElement(2, Geometry::SQUARE, 4, 1, H_DIV,
FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(2).x = 0.5;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.5;
}
void RT0QuadFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y;
shape(0,0) = 0;
shape(0,1) = y - 1.;
shape(1,0) = x;
shape(1,1) = 0;
shape(2,0) = 0;
shape(2,1) = y;
shape(3,0) = x - 1.;
shape(3,1) = 0;
}
void RT0QuadFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
divshape(0) = 1.;
divshape(1) = 1.;
divshape(2) = 1.;
divshape(3) = 1.;
}
const real_t RT0QuadFiniteElement::nk[4][2] =
{ {0, -1}, {1, 0}, {0, 1}, {-1, 0} };
void RT0QuadFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 4; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 4; j++)
{
real_t d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT0QuadFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear (more to have embedding?)
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[2];
Vector xk (vk, 2);
for (k = 0; k < 4; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
for (j = 0; j < 4; j++)
if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT0QuadFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[2];
Vector xk (vk, 2);
for (int k = 0; k < 4; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
}
}
RT1TriangleFiniteElement::RT1TriangleFiniteElement()
: VectorFiniteElement(2, Geometry::TRIANGLE, 8, 2, H_DIV)
{
Nodes.IntPoint(0).x = 0.33333333333333333333;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 0.66666666666666666667;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 0.66666666666666666667;
Nodes.IntPoint(2).y = 0.33333333333333333333;
Nodes.IntPoint(3).x = 0.33333333333333333333;
Nodes.IntPoint(3).y = 0.66666666666666666667;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.66666666666666666667;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 0.33333333333333333333;
Nodes.IntPoint(6).x = 0.33333333333333333333;
Nodes.IntPoint(6).y = 0.33333333333333333333;
Nodes.IntPoint(7).x = 0.33333333333333333333;
Nodes.IntPoint(7).y = 0.33333333333333333333;
}
void RT1TriangleFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y;
shape(0,0) = -2 * x * (-1 + x + 2 * y);
shape(0,1) = -2 * (-1 + y) * (-1 + x + 2 * y);
shape(1,0) = 2 * x * (x - y);
shape(1,1) = 2 * (x - y) * (-1 + y);
shape(2,0) = 2 * x * (-1 + 2 * x + y);
shape(2,1) = 2 * y * (-1 + 2 * x + y);
shape(3,0) = 2 * x * (-1 + x + 2 * y);
shape(3,1) = 2 * y * (-1 + x + 2 * y);
shape(4,0) = -2 * (-1 + x) * (x - y);
shape(4,1) = 2 * y * (-x + y);
shape(5,0) = -2 * (-1 + x) * (-1 + 2 * x + y);
shape(5,1) = -2 * y * (-1 + 2 * x + y);
shape(6,0) = -3 * x * (-2 + 2 * x + y);
shape(6,1) = -3 * y * (-1 + 2 * x + y);
shape(7,0) = -3 * x * (-1 + x + 2 * y);
shape(7,1) = -3 * y * (-2 + x + 2 * y);
}
void RT1TriangleFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
real_t x = ip.x, y = ip.y;
divshape(0) = -2 * (-4 + 3 * x + 6 * y);
divshape(1) = 2 + 6 * x - 6 * y;
divshape(2) = -4 + 12 * x + 6 * y;
divshape(3) = -4 + 6 * x + 12 * y;
divshape(4) = 2 - 6 * x + 6 * y;
divshape(5) = -2 * (-4 + 6 * x + 3 * y);
divshape(6) = -9 * (-1 + 2 * x + y);
divshape(7) = -9 * (-1 + x + 2 * y);
}
const real_t RT1TriangleFiniteElement::nk[8][2] =
{
{ 0,-1}, { 0,-1},
{ 1, 1}, { 1, 1},
{-1, 0}, {-1, 0},
{ 1, 0}, { 0, 1}
};
void RT1TriangleFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 8; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 8; j++)
{
real_t d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT1QuadFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear (more to have embedding?)
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[2];
Vector xk (vk, 2);
for (k = 0; k < 8; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
for (j = 0; j < 8; j++)
if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT1TriangleFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
real_t vk[2];
Vector xk (vk, 2);
for (int k = 0; k < 8; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
dofs(k) *= 0.5;
}
}
RT1QuadFiniteElement::RT1QuadFiniteElement()
: VectorFiniteElement(2, Geometry::SQUARE, 12, 2, H_DIV,
FunctionSpace::Qk)
{
// y = 0
Nodes.IntPoint(0).x = 1./3.;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 2./3.;
Nodes.IntPoint(1).y = 0.0;
// x = 1
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 1./3.;
Nodes.IntPoint(3).x = 1.0;
Nodes.IntPoint(3).y = 2./3.;
// y = 1
Nodes.IntPoint(4).x = 2./3.;
Nodes.IntPoint(4).y = 1.0;
Nodes.IntPoint(5).x = 1./3.;
Nodes.IntPoint(5).y = 1.0;
// x = 0
Nodes.IntPoint(6).x = 0.0;
Nodes.IntPoint(6).y = 2./3.;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 1./3.;
// x = 0.5 (interior)
Nodes.IntPoint(8).x = 0.5;
Nodes.IntPoint(8).y = 1./3.;
Nodes.IntPoint(9).x = 0.5;
Nodes.IntPoint(9).y = 2./3.;
// y = 0.5 (interior)
Nodes.IntPoint(10).x = 1./3.;
Nodes.IntPoint(10).y = 0.5;
Nodes.IntPoint(11).x = 2./3.;
Nodes.IntPoint(11).y = 0.5;
}
void RT1QuadFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y;
// y = 0
shape(0,0) = 0;
shape(0,1) = -( 1. - 3.*y + 2.*y*y)*( 2. - 3.*x);
shape(1,0) = 0;
shape(1,1) = -( 1. - 3.*y + 2.*y*y)*(-1. + 3.*x);
// x = 1
shape(2,0) = (-x + 2.*x*x)*( 2. - 3.*y);
shape(2,1) = 0;
shape(3,0) = (-x + 2.*x*x)*(-1. + 3.*y);
shape(3,1) = 0;
// y = 1
shape(4,0) = 0;
shape(4,1) = (-y + 2.*y*y)*(-1. + 3.*x);
shape(5,0) = 0;
shape(5,1) = (-y + 2.*y*y)*( 2. - 3.*x);
// x = 0
shape(6,0) = -(1. - 3.*x + 2.*x*x)*(-1. + 3.*y);
shape(6,1) = 0;
shape(7,0) = -(1. - 3.*x + 2.*x*x)*( 2. - 3.*y);
shape(7,1) = 0;
// x = 0.5 (interior)
shape(8,0) = (4.*x - 4.*x*x)*( 2. - 3.*y);
shape(8,1) = 0;
shape(9,0) = (4.*x - 4.*x*x)*(-1. + 3.*y);
shape(9,1) = 0;
// y = 0.5 (interior)
shape(10,0) = 0;
shape(10,1) = (4.*y - 4.*y*y)*( 2. - 3.*x);
shape(11,0) = 0;
shape(11,1) = (4.*y - 4.*y*y)*(-1. + 3.*x);
}
void RT1QuadFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
real_t x = ip.x, y = ip.y;
divshape(0) = -(-3. + 4.*y)*( 2. - 3.*x);
divshape(1) = -(-3. + 4.*y)*(-1. + 3.*x);
divshape(2) = (-1. + 4.*x)*( 2. - 3.*y);
divshape(3) = (-1. + 4.*x)*(-1. + 3.*y);
divshape(4) = (-1. + 4.*y)*(-1. + 3.*x);
divshape(5) = (-1. + 4.*y)*( 2. - 3.*x);
divshape(6) = -(-3. + 4.*x)*(-1. + 3.*y);
divshape(7) = -(-3. + 4.*x)*( 2. - 3.*y);
divshape(8) = ( 4. - 8.*x)*( 2. - 3.*y);
divshape(9) = ( 4. - 8.*x)*(-1. + 3.*y);
divshape(10) = ( 4. - 8.*y)*( 2. - 3.*x);
divshape(11) = ( 4. - 8.*y)*(-1. + 3.*x);
}
const real_t RT1QuadFiniteElement::nk[12][2] =
{
// y = 0
{0,-1}, {0,-1},
// X = 1
{1, 0}, {1, 0},
// y = 1
{0, 1}, {0, 1},
// x = 0
{-1,0}, {-1,0},
// x = 0.5 (interior)
{1, 0}, {1, 0},
// y = 0.5 (interior)
{0, 1}, {0, 1}
};
void RT1QuadFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 12; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 12; j++)
{
real_t d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT1QuadFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear (more to have embedding?)
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[2];
Vector xk (vk, 2);
for (k = 0; k < 12; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
for (j = 0; j < 12; j++)
if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT1QuadFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
real_t vk[2];
Vector xk (vk, 2);
for (int k = 0; k < 12; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
}
}
const real_t RT2TriangleFiniteElement::M[15][15] =
{
// *INDENT-OFF*
{
0, -5.3237900077244501311, 5.3237900077244501311, 16.647580015448900262,
0, 24.442740046346700787, -16.647580015448900262, -12.,
-19.118950038622250656, -47.237900077244501311, 0, -34.414110069520051180,
12., 30.590320061795601049, 15.295160030897800524
},
{
0, 1.5, -1.5, -15., 0, 2.625, 15., 15., -4.125, 30., 0, -14.625, -15.,
-15., 10.5
},
{
0, -0.67620999227554986889, 0.67620999227554986889, 7.3524199845510997378,
0, -3.4427400463467007866, -7.3524199845510997378, -12.,
4.1189500386222506555, -0.76209992275549868892, 0, 7.4141100695200511800,
12., -6.5903200617956010489, -3.2951600308978005244
},
{
0, 0, 1.5, 0, 0, 1.5, -11.471370023173350393, 0, 2.4713700231733503933,
-11.471370023173350393, 0, 2.4713700231733503933, 15.295160030897800524,
0, -3.2951600308978005244
},
{
0, 0, 4.875, 0, 0, 4.875, -16.875, 0, -16.875, -16.875, 0, -16.875, 10.5,
36., 10.5
},
{
0, 0, 1.5, 0, 0, 1.5, 2.4713700231733503933, 0, -11.471370023173350393,
2.4713700231733503933, 0, -11.471370023173350393, -3.2951600308978005244,
0, 15.295160030897800524
},
{
-0.67620999227554986889, 0, -3.4427400463467007866, 0,
7.3524199845510997378, 0.67620999227554986889, 7.4141100695200511800, 0,
-0.76209992275549868892, 4.1189500386222506555, -12.,
-7.3524199845510997378, -3.2951600308978005244, -6.5903200617956010489,
12.
},
{
1.5, 0, 2.625, 0, -15., -1.5, -14.625, 0, 30., -4.125, 15., 15., 10.5,
-15., -15.
},
{
-5.3237900077244501311, 0, 24.442740046346700787, 0, 16.647580015448900262,
5.3237900077244501311, -34.414110069520051180, 0, -47.237900077244501311,
-19.118950038622250656, -12., -16.647580015448900262, 15.295160030897800524,
30.590320061795601049, 12.
},
{ 0, 0, 18., 0, 0, 6., -42., 0, -30., -26., 0, -14., 24., 32., 8.},
{ 0, 0, 6., 0, 0, 18., -14., 0, -26., -30., 0, -42., 8., 32., 24.},
{ 0, 0, -6., 0, 0, -4., 30., 0, 4., 22., 0, 4., -24., -16., 0},
{ 0, 0, -4., 0, 0, -8., 20., 0, 8., 36., 0, 8., -16., -32., 0},
{ 0, 0, -8., 0, 0, -4., 8., 0, 36., 8., 0, 20., 0, -32., -16.},
{ 0, 0, -4., 0, 0, -6., 4., 0, 22., 4., 0, 30., 0, -16., -24.}
// *INDENT-ON*
};
RT2TriangleFiniteElement::RT2TriangleFiniteElement()
: VectorFiniteElement(2, Geometry::TRIANGLE, 15, 3, H_DIV)
{
const real_t p = 0.11270166537925831148;
Nodes.IntPoint(0).x = p;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 0.5;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 1.-p;
Nodes.IntPoint(2).y = 0.0;
Nodes.IntPoint(3).x = 1.-p;
Nodes.IntPoint(3).y = p;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.5;
Nodes.IntPoint(5).x = p;
Nodes.IntPoint(5).y = 1.-p;
Nodes.IntPoint(6).x = 0.0;
Nodes.IntPoint(6).y = 1.-p;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(8).x = 0.0;
Nodes.IntPoint(8).y = p;
Nodes.IntPoint(9).x = 0.25;
Nodes.IntPoint(9).y = 0.25;
Nodes.IntPoint(10).x = 0.25;
Nodes.IntPoint(10).y = 0.25;
Nodes.IntPoint(11).x = 0.5;
Nodes.IntPoint(11).y = 0.25;
Nodes.IntPoint(12).x = 0.5;
Nodes.IntPoint(12).y = 0.25;
Nodes.IntPoint(13).x = 0.25;
Nodes.IntPoint(13).y = 0.5;
Nodes.IntPoint(14).x = 0.25;
Nodes.IntPoint(14).y = 0.5;
}
void RT2TriangleFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y;
real_t Bx[15] = {1., 0., x, 0., y, 0., x*x, 0., x*y, 0., y*y, 0., x*x*x,
x*x*y, x*y*y
};
real_t By[15] = {0., 1., 0., x, 0., y, 0., x*x, 0., x*y, 0., y*y,
x*x*y, x*y*y, y*y*y
};
for (int i = 0; i < 15; i++)
{
real_t cx = 0.0, cy = 0.0;
for (int j = 0; j < 15; j++)
{
cx += M[i][j] * Bx[j];
cy += M[i][j] * By[j];
}
shape(i,0) = cx;
shape(i,1) = cy;
}
}
void RT2TriangleFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
real_t x = ip.x, y = ip.y;
constexpr real_t f2 = 2.0;
constexpr real_t f4 = 4.0;
real_t DivB[15] = {0., 0., 1., 0., 0., 1., f2*x, 0., y, x, 0., f2*y,
f4*x*x, f4*x*y, f4*y*y
};
for (int i = 0; i < 15; i++)
{
real_t div = 0.0;
for (int j = 0; j < 15; j++)
{
div += M[i][j] * DivB[j];
}
divshape(i) = div;
}
}
const real_t RT2QuadFiniteElement::pt[4] = {0.,1./3.,2./3.,1.};
const real_t RT2QuadFiniteElement::dpt[3] = {0.25,0.5,0.75};
RT2QuadFiniteElement::RT2QuadFiniteElement()
: VectorFiniteElement(2, Geometry::SQUARE, 24, 3, H_DIV,
FunctionSpace::Qk)
{
// y = 0 (pt[0])
Nodes.IntPoint(0).x = dpt[0]; Nodes.IntPoint(0).y = pt[0];
Nodes.IntPoint(1).x = dpt[1]; Nodes.IntPoint(1).y = pt[0];
Nodes.IntPoint(2).x = dpt[2]; Nodes.IntPoint(2).y = pt[0];
// x = 1 (pt[3])
Nodes.IntPoint(3).x = pt[3]; Nodes.IntPoint(3).y = dpt[0];
Nodes.IntPoint(4).x = pt[3]; Nodes.IntPoint(4).y = dpt[1];
Nodes.IntPoint(5).x = pt[3]; Nodes.IntPoint(5).y = dpt[2];
// y = 1 (pt[3])
Nodes.IntPoint(6).x = dpt[2]; Nodes.IntPoint(6).y = pt[3];
Nodes.IntPoint(7).x = dpt[1]; Nodes.IntPoint(7).y = pt[3];
Nodes.IntPoint(8).x = dpt[0]; Nodes.IntPoint(8).y = pt[3];
// x = 0 (pt[0])
Nodes.IntPoint(9).x = pt[0]; Nodes.IntPoint(9).y = dpt[2];
Nodes.IntPoint(10).x = pt[0]; Nodes.IntPoint(10).y = dpt[1];
Nodes.IntPoint(11).x = pt[0]; Nodes.IntPoint(11).y = dpt[0];
// x = pt[1] (interior)
Nodes.IntPoint(12).x = pt[1]; Nodes.IntPoint(12).y = dpt[0];
Nodes.IntPoint(13).x = pt[1]; Nodes.IntPoint(13).y = dpt[1];
Nodes.IntPoint(14).x = pt[1]; Nodes.IntPoint(14).y = dpt[2];
// x = pt[2] (interior)
Nodes.IntPoint(15).x = pt[2]; Nodes.IntPoint(15).y = dpt[0];
Nodes.IntPoint(16).x = pt[2]; Nodes.IntPoint(16).y = dpt[1];
Nodes.IntPoint(17).x = pt[2]; Nodes.IntPoint(17).y = dpt[2];
// y = pt[1] (interior)
Nodes.IntPoint(18).x = dpt[0]; Nodes.IntPoint(18).y = pt[1];
Nodes.IntPoint(19).x = dpt[1]; Nodes.IntPoint(19).y = pt[1];
Nodes.IntPoint(20).x = dpt[2]; Nodes.IntPoint(20).y = pt[1];
// y = pt[2] (interior)
Nodes.IntPoint(21).x = dpt[0]; Nodes.IntPoint(21).y = pt[2];
Nodes.IntPoint(22).x = dpt[1]; Nodes.IntPoint(22).y = pt[2];
Nodes.IntPoint(23).x = dpt[2]; Nodes.IntPoint(23).y = pt[2];
}
void RT2QuadFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y;
real_t ax0 = pt[0] - x;
real_t ax1 = pt[1] - x;
real_t ax2 = pt[2] - x;
real_t ax3 = pt[3] - x;
real_t by0 = dpt[0] - y;
real_t by1 = dpt[1] - y;
real_t by2 = dpt[2] - y;
real_t ay0 = pt[0] - y;
real_t ay1 = pt[1] - y;
real_t ay2 = pt[2] - y;
real_t ay3 = pt[3] - y;
real_t bx0 = dpt[0] - x;
real_t bx1 = dpt[1] - x;
real_t bx2 = dpt[2] - x;
real_t A01 = pt[0] - pt[1];
real_t A02 = pt[0] - pt[2];
real_t A12 = pt[1] - pt[2];
real_t A03 = pt[0] - pt[3];
real_t A13 = pt[1] - pt[3];
real_t A23 = pt[2] - pt[3];
real_t B01 = dpt[0] - dpt[1];
real_t B02 = dpt[0] - dpt[2];
real_t B12 = dpt[1] - dpt[2];
real_t tx0 = (bx1*bx2)/(B01*B02);
real_t tx1 = -(bx0*bx2)/(B01*B12);
real_t tx2 = (bx0*bx1)/(B02*B12);
real_t ty0 = (by1*by2)/(B01*B02);
real_t ty1 = -(by0*by2)/(B01*B12);
real_t ty2 = (by0*by1)/(B02*B12);
// y = 0 (p[0])
shape(0, 0) = 0;
shape(0, 1) = (ay1*ay2*ay3)/(A01*A02*A03)*tx0;
shape(1, 0) = 0;
shape(1, 1) = (ay1*ay2*ay3)/(A01*A02*A03)*tx1;
shape(2, 0) = 0;
shape(2, 1) = (ay1*ay2*ay3)/(A01*A02*A03)*tx2;
// x = 1 (p[3])
shape(3, 0) = (ax0*ax1*ax2)/(A03*A13*A23)*ty0;
shape(3, 1) = 0;
shape(4, 0) = (ax0*ax1*ax2)/(A03*A13*A23)*ty1;
shape(4, 1) = 0;
shape(5, 0) = (ax0*ax1*ax2)/(A03*A13*A23)*ty2;
shape(5, 1) = 0;
// y = 1 (p[3])
shape(6, 0) = 0;
shape(6, 1) = (ay0*ay1*ay2)/(A03*A13*A23)*tx2;
shape(7, 0) = 0;
shape(7, 1) = (ay0*ay1*ay2)/(A03*A13*A23)*tx1;
shape(8, 0) = 0;
shape(8, 1) = (ay0*ay1*ay2)/(A03*A13*A23)*tx0;
// x = 0 (p[0])
shape(9, 0) = (ax1*ax2*ax3)/(A01*A02*A03)*ty2;
shape(9, 1) = 0;
shape(10, 0) = (ax1*ax2*ax3)/(A01*A02*A03)*ty1;
shape(10, 1) = 0;
shape(11, 0) = (ax1*ax2*ax3)/(A01*A02*A03)*ty0;
shape(11, 1) = 0;
// x = p[1] (interior)
shape(12, 0) = (ax0*ax2*ax3)/(A01*A12*A13)*ty0;
shape(12, 1) = 0;
shape(13, 0) = (ax0*ax2*ax3)/(A01*A12*A13)*ty1;
shape(13, 1) = 0;
shape(14, 0) = (ax0*ax2*ax3)/(A01*A12*A13)*ty2;
shape(14, 1) = 0;
// x = p[2] (interior)
shape(15, 0) = -(ax0*ax1*ax3)/(A02*A12*A23)*ty0;
shape(15, 1) = 0;
shape(16, 0) = -(ax0*ax1*ax3)/(A02*A12*A23)*ty1;
shape(16, 1) = 0;
shape(17, 0) = -(ax0*ax1*ax3)/(A02*A12*A23)*ty2;
shape(17, 1) = 0;
// y = p[1] (interior)
shape(18, 0) = 0;
shape(18, 1) = (ay0*ay2*ay3)/(A01*A12*A13)*tx0;
shape(19, 0) = 0;
shape(19, 1) = (ay0*ay2*ay3)/(A01*A12*A13)*tx1;
shape(20, 0) = 0;
shape(20, 1) = (ay0*ay2*ay3)/(A01*A12*A13)*tx2;
// y = p[2] (interior)
shape(21, 0) = 0;
shape(21, 1) = -(ay0*ay1*ay3)/(A02*A12*A23)*tx0;
shape(22, 0) = 0;
shape(22, 1) = -(ay0*ay1*ay3)/(A02*A12*A23)*tx1;
shape(23, 0) = 0;
shape(23, 1) = -(ay0*ay1*ay3)/(A02*A12*A23)*tx2;
}
void RT2QuadFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
real_t x = ip.x, y = ip.y;
real_t a01 = pt[0]*pt[1];
real_t a02 = pt[0]*pt[2];
real_t a12 = pt[1]*pt[2];
real_t a03 = pt[0]*pt[3];
real_t a13 = pt[1]*pt[3];
real_t a23 = pt[2]*pt[3];
real_t bx0 = dpt[0] - x;
real_t bx1 = dpt[1] - x;
real_t bx2 = dpt[2] - x;
real_t by0 = dpt[0] - y;
real_t by1 = dpt[1] - y;
real_t by2 = dpt[2] - y;
real_t A01 = pt[0] - pt[1];
real_t A02 = pt[0] - pt[2];
real_t A12 = pt[1] - pt[2];
real_t A03 = pt[0] - pt[3];
real_t A13 = pt[1] - pt[3];
real_t A23 = pt[2] - pt[3];
real_t A012 = pt[0] + pt[1] + pt[2];
real_t A013 = pt[0] + pt[1] + pt[3];
real_t A023 = pt[0] + pt[2] + pt[3];
real_t A123 = pt[1] + pt[2] + pt[3];
real_t B01 = dpt[0] - dpt[1];
real_t B02 = dpt[0] - dpt[2];
real_t B12 = dpt[1] - dpt[2];
real_t tx0 = (bx1*bx2)/(B01*B02);
real_t tx1 = -(bx0*bx2)/(B01*B12);
real_t tx2 = (bx0*bx1)/(B02*B12);
real_t ty0 = (by1*by2)/(B01*B02);
real_t ty1 = -(by0*by2)/(B01*B12);
real_t ty2 = (by0*by1)/(B02*B12);
// y = 0 (p[0])
divshape(0) = -(a12 + a13 + a23 - 2.*A123*y + 3.*y*y)/(A01*A02*A03)*tx0;
divshape(1) = -(a12 + a13 + a23 - 2.*A123*y + 3.*y*y)/(A01*A02*A03)*tx1;
divshape(2) = -(a12 + a13 + a23 - 2.*A123*y + 3.*y*y)/(A01*A02*A03)*tx2;
// x = 1 (p[3])
divshape(3) = -(a01 + a02 + a12 - 2.*A012*x + 3.*x*x)/(A03*A13*A23)*ty0;
divshape(4) = -(a01 + a02 + a12 - 2.*A012*x + 3.*x*x)/(A03*A13*A23)*ty1;
divshape(5) = -(a01 + a02 + a12 - 2.*A012*x + 3.*x*x)/(A03*A13*A23)*ty2;
// y = 1 (p[3])
divshape(6) = -(a01 + a02 + a12 - 2.*A012*y + 3.*y*y)/(A03*A13*A23)*tx2;
divshape(7) = -(a01 + a02 + a12 - 2.*A012*y + 3.*y*y)/(A03*A13*A23)*tx1;
divshape(8) = -(a01 + a02 + a12 - 2.*A012*y + 3.*y*y)/(A03*A13*A23)*tx0;
// x = 0 (p[0])
divshape(9) = -(a12 + a13 + a23 - 2.*A123*x + 3.*x*x)/(A01*A02*A03)*ty2;
divshape(10) = -(a12 + a13 + a23 - 2.*A123*x + 3.*x*x)/(A01*A02*A03)*ty1;
divshape(11) = -(a12 + a13 + a23 - 2.*A123*x + 3.*x*x)/(A01*A02*A03)*ty0;
// x = p[1] (interior)
divshape(12) = -(a02 + a03 + a23 - 2.*A023*x + 3.*x*x)/(A01*A12*A13)*ty0;
divshape(13) = -(a02 + a03 + a23 - 2.*A023*x + 3.*x*x)/(A01*A12*A13)*ty1;
divshape(14) = -(a02 + a03 + a23 - 2.*A023*x + 3.*x*x)/(A01*A12*A13)*ty2;
// x = p[2] (interior)
divshape(15) = (a01 + a03 + a13 - 2.*A013*x + 3.*x*x)/(A02*A12*A23)*ty0;
divshape(16) = (a01 + a03 + a13 - 2.*A013*x + 3.*x*x)/(A02*A12*A23)*ty1;
divshape(17) = (a01 + a03 + a13 - 2.*A013*x + 3.*x*x)/(A02*A12*A23)*ty2;
// y = p[1] (interior)
divshape(18) = -(a02 + a03 + a23 - 2.*A023*y + 3.*y*y)/(A01*A12*A13)*tx0;
divshape(19) = -(a02 + a03 + a23 - 2.*A023*y + 3.*y*y)/(A01*A12*A13)*tx1;
divshape(20) = -(a02 + a03 + a23 - 2.*A023*y + 3.*y*y)/(A01*A12*A13)*tx2;
// y = p[2] (interior)
divshape(21) = (a01 + a03 + a13 - 2.*A013*y + 3.*y*y)/(A02*A12*A23)*tx0;
divshape(22) = (a01 + a03 + a13 - 2.*A013*y + 3.*y*y)/(A02*A12*A23)*tx1;
divshape(23) = (a01 + a03 + a13 - 2.*A013*y + 3.*y*y)/(A02*A12*A23)*tx2;
}
const real_t RT2QuadFiniteElement::nk[24][2] =
{
// y = 0
{0,-1}, {0,-1}, {0,-1},
// x = 1
{1, 0}, {1, 0}, {1, 0},
// y = 1
{0, 1}, {0, 1}, {0, 1},
// x = 0
{-1,0}, {-1,0}, {-1,0},
// x = p[1] (interior)
{1, 0}, {1, 0}, {1, 0},
// x = p[2] (interior)
{1, 0}, {1, 0}, {1, 0},
// y = p[1] (interior)
{0, 1}, {0, 1}, {0, 1},
// y = p[1] (interior)
{0, 1}, {0, 1}, {0, 1}
};
void RT2QuadFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 24; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 24; j++)
{
real_t d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT2QuadFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear (more to have embedding?)
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[2];
Vector xk (vk, 2);
for (k = 0; k < 24; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
for (j = 0; j < 24; j++)
if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT2QuadFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
real_t vk[2];
Vector xk (vk, 2);
for (int k = 0; k < 24; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
}
}
P1SegmentFiniteElement::P1SegmentFiniteElement()
: NodalFiniteElement(1, Geometry::SEGMENT, 2, 1)
{
Nodes.IntPoint(0).x = 0.33333333333333333333;
Nodes.IntPoint(1).x = 0.66666666666666666667;
}
void P1SegmentFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x;
shape(0) = 2. - 3. * x;
shape(1) = 3. * x - 1.;
}
void P1SegmentFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = -3.;
dshape(1,0) = 3.;
}
P2SegmentFiniteElement::P2SegmentFiniteElement()
: NodalFiniteElement(1, Geometry::SEGMENT, 3, 2)
{
const real_t p = 0.11270166537925831148;
Nodes.IntPoint(0).x = p;
Nodes.IntPoint(1).x = 0.5;
Nodes.IntPoint(2).x = 1.-p;
}
void P2SegmentFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
const real_t p = 0.11270166537925831148;
const real_t w = 1./((1-2*p)*(1-2*p));
real_t x = ip.x;
shape(0) = (2*x-1)*(x-1+p)*w;
shape(1) = 4*(x-1+p)*(p-x)*w;
shape(2) = (2*x-1)*(x-p)*w;
}
void P2SegmentFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const real_t p = 0.11270166537925831148;
const real_t w = 1./((1-2*p)*(1-2*p));
real_t x = ip.x;
dshape(0,0) = (-3+4*x+2*p)*w;
dshape(1,0) = (4-8*x)*w;
dshape(2,0) = (-1+4*x-2*p)*w;
}
Lagrange1DFiniteElement::Lagrange1DFiniteElement(int degree)
: NodalFiniteElement(1, Geometry::SEGMENT, degree+1, degree)
{
int i, m = degree;
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(1).x = 1.0;
for (i = 1; i < m; i++)
{
Nodes.IntPoint(i+1).x = real_t(i) / m;
}
rwk.SetSize(degree+1);
#ifndef MFEM_THREAD_SAFE
rxxk.SetSize(degree+1);
#endif
rwk(0) = 1.0;
for (i = 1; i <= m; i++)
{
rwk(i) = rwk(i-1) * ( (real_t)(m) / (real_t)(i) );
}
for (i = 0; i < m/2+1; i++)
{
rwk(m-i) = ( rwk(i) *= rwk(m-i) );
}
for (i = m-1; i >= 0; i -= 2)
{
rwk(i) = -rwk(i);
}
}
void Lagrange1DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t w, wk, x = ip.x;
int i, k, m = GetOrder();
#ifdef MFEM_THREAD_SAFE
Vector rxxk(m+1);
#endif
k = (int) floor ( m * x + 0.5 );
k = k > m ? m : k < 0 ? 0 : k; // clamp k to [0,m]
wk = 1.0;
for (i = 0; i <= m; i++)
if (i != k)
{
wk *= ( rxxk(i) = x - (real_t)(i) / m );
}
w = wk * ( rxxk(k) = x - (real_t)(k) / m );
if (k != 0)
{
shape(0) = w * rwk(0) / rxxk(0);
}
else
{
shape(0) = wk * rwk(0);
}
if (k != m)
{
shape(1) = w * rwk(m) / rxxk(m);
}
else
{
shape(1) = wk * rwk(k);
}
for (i = 1; i < m; i++)
if (i != k)
{
shape(i+1) = w * rwk(i) / rxxk(i);
}
else
{
shape(k+1) = wk * rwk(k);
}
}
void Lagrange1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t s, srx, w, wk, x = ip.x;
int i, k, m = GetOrder();
#ifdef MFEM_THREAD_SAFE
Vector rxxk(m+1);
#endif
k = (int) floor ( m * x + 0.5 );
k = k > m ? m : k < 0 ? 0 : k; // clamp k to [0,m]
wk = 1.0;
for (i = 0; i <= m; i++)
if (i != k)
{
wk *= ( rxxk(i) = x - (real_t)(i) / m );
}
w = wk * ( rxxk(k) = x - (real_t)(k) / m );
for (i = 0; i <= m; i++)
{
rxxk(i) = 1.0 / rxxk(i);
}
srx = 0.0;
for (i = 0; i <= m; i++)
if (i != k)
{
srx += rxxk(i);
}
s = w * srx + wk;
if (k != 0)
{
dshape(0,0) = (s - w * rxxk(0)) * rwk(0) * rxxk(0);
}
else
{
dshape(0,0) = wk * srx * rwk(0);
}
if (k != m)
{
dshape(1,0) = (s - w * rxxk(m)) * rwk(m) * rxxk(m);
}
else
{
dshape(1,0) = wk * srx * rwk(k);
}
for (i = 1; i < m; i++)
if (i != k)
{
dshape(i+1,0) = (s - w * rxxk(i)) * rwk(i) * rxxk(i);
}
else
{
dshape(k+1,0) = wk * srx * rwk(k);
}
}
P1TetNonConfFiniteElement::P1TetNonConfFiniteElement()
: NodalFiniteElement(3, Geometry::TETRAHEDRON, 4, 1)
{
Nodes.IntPoint(0).x = 0.33333333333333333333;
Nodes.IntPoint(0).y = 0.33333333333333333333;
Nodes.IntPoint(0).z = 0.33333333333333333333;
Nodes.IntPoint(1).x = 0.0;
Nodes.IntPoint(1).y = 0.33333333333333333333;
Nodes.IntPoint(1).z = 0.33333333333333333333;
Nodes.IntPoint(2).x = 0.33333333333333333333;
Nodes.IntPoint(2).y = 0.0;
Nodes.IntPoint(2).z = 0.33333333333333333333;
Nodes.IntPoint(3).x = 0.33333333333333333333;
Nodes.IntPoint(3).y = 0.33333333333333333333;
Nodes.IntPoint(3).z = 0.0;
}
void P1TetNonConfFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t L0, L1, L2, L3;
L1 = ip.x; L2 = ip.y; L3 = ip.z; L0 = 1.0 - L1 - L2 - L3;
shape(0) = 1.0 - 3.0 * L0;
shape(1) = 1.0 - 3.0 * L1;
shape(2) = 1.0 - 3.0 * L2;
shape(3) = 1.0 - 3.0 * L3;
}
void P1TetNonConfFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 3.0; dshape(0,1) = 3.0; dshape(0,2) = 3.0;
dshape(1,0) = -3.0; dshape(1,1) = 0.0; dshape(1,2) = 0.0;
dshape(2,0) = 0.0; dshape(2,1) = -3.0; dshape(2,2) = 0.0;
dshape(3,0) = 0.0; dshape(3,1) = 0.0; dshape(3,2) = -3.0;
}
P0TetFiniteElement::P0TetFiniteElement()
: NodalFiniteElement(3, Geometry::TETRAHEDRON, 1, 0)
{
Nodes.IntPoint(0).x = 0.25;
Nodes.IntPoint(0).y = 0.25;
Nodes.IntPoint(0).z = 0.25;
}
void P0TetFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0;
}
void P0TetFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0; dshape(0,1) = 0.0; dshape(0,2) = 0.0;
}
P0HexFiniteElement::P0HexFiniteElement()
: NodalFiniteElement(3, Geometry::CUBE, 1, 0, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.5;
Nodes.IntPoint(0).z = 0.5;
}
void P0HexFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0;
}
void P0HexFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0; dshape(0,1) = 0.0; dshape(0,2) = 0.0;
}
P0WdgFiniteElement::P0WdgFiniteElement()
: NodalFiniteElement(3, Geometry::PRISM, 1, 0, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 1.0 / 3.0;
Nodes.IntPoint(0).y = 1.0 / 3.0;
Nodes.IntPoint(0).z = 0.5;
}
void P0WdgFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0;
}
void P0WdgFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0; dshape(0,1) = 0.0; dshape(0,2) = 0.0;
}
P0PyrFiniteElement::P0PyrFiniteElement()
: NodalFiniteElement(3, Geometry::PYRAMID, 1, 0, FunctionSpace::Uk)
{
Nodes.IntPoint(0).x = 0.375;
Nodes.IntPoint(0).y = 0.375;
Nodes.IntPoint(0).z = 0.25;
}
void P0PyrFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
shape(0) = 1.0;
}
void P0PyrFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
dshape(0,0) = 0.0; dshape(0,1) = 0.0; dshape(0,2) = 0.0;
}
LagrangeHexFiniteElement::LagrangeHexFiniteElement (int degree)
: NodalFiniteElement(3, Geometry::CUBE, (degree+1)*(degree+1)*(degree+1),
degree, FunctionSpace::Qk)
{
if (degree == 2)
{
I = new int[dof];
J = new int[dof];
K = new int[dof];
// nodes
I[ 0] = 0; J[ 0] = 0; K[ 0] = 0;
I[ 1] = 1; J[ 1] = 0; K[ 1] = 0;
I[ 2] = 1; J[ 2] = 1; K[ 2] = 0;
I[ 3] = 0; J[ 3] = 1; K[ 3] = 0;
I[ 4] = 0; J[ 4] = 0; K[ 4] = 1;
I[ 5] = 1; J[ 5] = 0; K[ 5] = 1;
I[ 6] = 1; J[ 6] = 1; K[ 6] = 1;
I[ 7] = 0; J[ 7] = 1; K[ 7] = 1;
// edges
I[ 8] = 2; J[ 8] = 0; K[ 8] = 0;
I[ 9] = 1; J[ 9] = 2; K[ 9] = 0;
I[10] = 2; J[10] = 1; K[10] = 0;
I[11] = 0; J[11] = 2; K[11] = 0;
I[12] = 2; J[12] = 0; K[12] = 1;
I[13] = 1; J[13] = 2; K[13] = 1;
I[14] = 2; J[14] = 1; K[14] = 1;
I[15] = 0; J[15] = 2; K[15] = 1;
I[16] = 0; J[16] = 0; K[16] = 2;
I[17] = 1; J[17] = 0; K[17] = 2;
I[18] = 1; J[18] = 1; K[18] = 2;
I[19] = 0; J[19] = 1; K[19] = 2;
// faces
I[20] = 2; J[20] = 2; K[20] = 0;
I[21] = 2; J[21] = 0; K[21] = 2;
I[22] = 1; J[22] = 2; K[22] = 2;
I[23] = 2; J[23] = 1; K[23] = 2;
I[24] = 0; J[24] = 2; K[24] = 2;
I[25] = 2; J[25] = 2; K[25] = 1;
// element
I[26] = 2; J[26] = 2; K[26] = 2;
}
else if (degree == 3)
{
I = new int[dof];
J = new int[dof];
K = new int[dof];
// nodes
I[ 0] = 0; J[ 0] = 0; K[ 0] = 0;
I[ 1] = 1; J[ 1] = 0; K[ 1] = 0;
I[ 2] = 1; J[ 2] = 1; K[ 2] = 0;
I[ 3] = 0; J[ 3] = 1; K[ 3] = 0;
I[ 4] = 0; J[ 4] = 0; K[ 4] = 1;
I[ 5] = 1; J[ 5] = 0; K[ 5] = 1;
I[ 6] = 1; J[ 6] = 1; K[ 6] = 1;
I[ 7] = 0; J[ 7] = 1; K[ 7] = 1;
// edges
I[ 8] = 2; J[ 8] = 0; K[ 8] = 0;
I[ 9] = 3; J[ 9] = 0; K[ 9] = 0;
I[10] = 1; J[10] = 2; K[10] = 0;
I[11] = 1; J[11] = 3; K[11] = 0;
I[12] = 2; J[12] = 1; K[12] = 0;
I[13] = 3; J[13] = 1; K[13] = 0;
I[14] = 0; J[14] = 2; K[14] = 0;
I[15] = 0; J[15] = 3; K[15] = 0;
I[16] = 2; J[16] = 0; K[16] = 1;
I[17] = 3; J[17] = 0; K[17] = 1;
I[18] = 1; J[18] = 2; K[18] = 1;
I[19] = 1; J[19] = 3; K[19] = 1;
I[20] = 2; J[20] = 1; K[20] = 1;
I[21] = 3; J[21] = 1; K[21] = 1;
I[22] = 0; J[22] = 2; K[22] = 1;
I[23] = 0; J[23] = 3; K[23] = 1;
I[24] = 0; J[24] = 0; K[24] = 2;
I[25] = 0; J[25] = 0; K[25] = 3;
I[26] = 1; J[26] = 0; K[26] = 2;
I[27] = 1; J[27] = 0; K[27] = 3;
I[28] = 1; J[28] = 1; K[28] = 2;
I[29] = 1; J[29] = 1; K[29] = 3;
I[30] = 0; J[30] = 1; K[30] = 2;
I[31] = 0; J[31] = 1; K[31] = 3;
// faces
I[32] = 2; J[32] = 3; K[32] = 0;
I[33] = 3; J[33] = 3; K[33] = 0;
I[34] = 2; J[34] = 2; K[34] = 0;
I[35] = 3; J[35] = 2; K[35] = 0;
I[36] = 2; J[36] = 0; K[36] = 2;
I[37] = 3; J[37] = 0; K[37] = 2;
I[38] = 2; J[38] = 0; K[38] = 3;
I[39] = 3; J[39] = 0; K[39] = 3;
I[40] = 1; J[40] = 2; K[40] = 2;
I[41] = 1; J[41] = 3; K[41] = 2;
I[42] = 1; J[42] = 2; K[42] = 3;
I[43] = 1; J[43] = 3; K[43] = 3;
I[44] = 3; J[44] = 1; K[44] = 2;
I[45] = 2; J[45] = 1; K[45] = 2;
I[46] = 3; J[46] = 1; K[46] = 3;
I[47] = 2; J[47] = 1; K[47] = 3;
I[48] = 0; J[48] = 3; K[48] = 2;
I[49] = 0; J[49] = 2; K[49] = 2;
I[50] = 0; J[50] = 3; K[50] = 3;
I[51] = 0; J[51] = 2; K[51] = 3;
I[52] = 2; J[52] = 2; K[52] = 1;
I[53] = 3; J[53] = 2; K[53] = 1;
I[54] = 2; J[54] = 3; K[54] = 1;
I[55] = 3; J[55] = 3; K[55] = 1;
// element
I[56] = 2; J[56] = 2; K[56] = 2;
I[57] = 3; J[57] = 2; K[57] = 2;
I[58] = 3; J[58] = 3; K[58] = 2;
I[59] = 2; J[59] = 3; K[59] = 2;
I[60] = 2; J[60] = 2; K[60] = 3;
I[61] = 3; J[61] = 2; K[61] = 3;
I[62] = 3; J[62] = 3; K[62] = 3;
I[63] = 2; J[63] = 3; K[63] = 3;
}
else
{
mfem_error ("LagrangeHexFiniteElement::LagrangeHexFiniteElement");
}
fe1d = new Lagrange1DFiniteElement(degree);
dof1d = fe1d -> GetDof();
#ifndef MFEM_THREAD_SAFE
shape1dx.SetSize(dof1d);
shape1dy.SetSize(dof1d);
shape1dz.SetSize(dof1d);
dshape1dx.SetSize(dof1d,1);
dshape1dy.SetSize(dof1d,1);
dshape1dz.SetSize(dof1d,1);
#endif
for (int n = 0; n < dof; n++)
{
Nodes.IntPoint(n).x = fe1d -> GetNodes().IntPoint(I[n]).x;
Nodes.IntPoint(n).y = fe1d -> GetNodes().IntPoint(J[n]).x;
Nodes.IntPoint(n).z = fe1d -> GetNodes().IntPoint(K[n]).x;
}
}
void LagrangeHexFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
IntegrationPoint ipy, ipz;
ipy.x = ip.y;
ipz.x = ip.z;
#ifdef MFEM_THREAD_SAFE
Vector shape1dx(dof1d), shape1dy(dof1d), shape1dz(dof1d);
#endif
fe1d -> CalcShape(ip, shape1dx);
fe1d -> CalcShape(ipy, shape1dy);
fe1d -> CalcShape(ipz, shape1dz);
for (int n = 0; n < dof; n++)
{
shape(n) = shape1dx(I[n]) * shape1dy(J[n]) * shape1dz(K[n]);
}
}
void LagrangeHexFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
IntegrationPoint ipy, ipz;
ipy.x = ip.y;
ipz.x = ip.z;
#ifdef MFEM_THREAD_SAFE
Vector shape1dx(dof1d), shape1dy(dof1d), shape1dz(dof1d);
DenseMatrix dshape1dx(dof1d,1), dshape1dy(dof1d,1), dshape1dz(dof1d,1);
#endif
fe1d -> CalcShape(ip, shape1dx);
fe1d -> CalcShape(ipy, shape1dy);
fe1d -> CalcShape(ipz, shape1dz);
fe1d -> CalcDShape(ip, dshape1dx);
fe1d -> CalcDShape(ipy, dshape1dy);
fe1d -> CalcDShape(ipz, dshape1dz);
for (int n = 0; n < dof; n++)
{
dshape(n,0) = dshape1dx(I[n],0) * shape1dy(J[n]) * shape1dz(K[n]);
dshape(n,1) = shape1dx(I[n]) * dshape1dy(J[n],0) * shape1dz(K[n]);
dshape(n,2) = shape1dx(I[n]) * shape1dy(J[n]) * dshape1dz(K[n],0);
}
}
LagrangeHexFiniteElement::~LagrangeHexFiniteElement ()
{
delete fe1d;
delete [] I;
delete [] J;
delete [] K;
}
RefinedLinear1DFiniteElement::RefinedLinear1DFiniteElement()
: NodalFiniteElement(1, Geometry::SEGMENT, 3, 4)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(2).x = 0.5;
}
void RefinedLinear1DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = ip.x;
if (x <= 0.5)
{
shape(0) = 1.0 - 2.0 * x;
shape(1) = 0.0;
shape(2) = 2.0 * x;
}
else
{
shape(0) = 0.0;
shape(1) = 2.0 * x - 1.0;
shape(2) = 2.0 - 2.0 * x;
}
}
void RefinedLinear1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
real_t x = ip.x;
if (x <= 0.5)
{
dshape(0,0) = - 2.0;
dshape(1,0) = 0.0;
dshape(2,0) = 2.0;
}
else
{
dshape(0,0) = 0.0;
dshape(1,0) = 2.0;
dshape(2,0) = - 2.0;
}
}
RefinedLinear2DFiniteElement::RefinedLinear2DFiniteElement()
: NodalFiniteElement(2, Geometry::TRIANGLE, 6, 5)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.5;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.5;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 0.5;
}
void RefinedLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
int i;
real_t L0, L1, L2;
L0 = 2.0 * ( 1. - ip.x - ip.y );
L1 = 2.0 * ( ip.x );
L2 = 2.0 * ( ip.y );
// The reference triangle is split in 4 triangles as follows:
//
// T0 - 0,3,5
// T1 - 1,3,4
// T2 - 2,4,5
// T3 - 3,4,5
for (i = 0; i < 6; i++)
{
shape(i) = 0.0;
}
if (L0 >= 1.0) // T0
{
shape(0) = L0 - 1.0;
shape(3) = L1;
shape(5) = L2;
}
else if (L1 >= 1.0) // T1
{
shape(3) = L0;
shape(1) = L1 - 1.0;
shape(4) = L2;
}
else if (L2 >= 1.0) // T2
{
shape(5) = L0;
shape(4) = L1;
shape(2) = L2 - 1.0;
}
else // T3
{
shape(3) = 1.0 - L2;
shape(4) = 1.0 - L0;
shape(5) = 1.0 - L1;
}
}
void RefinedLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
int i,j;
real_t L0, L1, L2;
L0 = 2.0 * ( 1. - ip.x - ip.y );
L1 = 2.0 * ( ip.x );
L2 = 2.0 * ( ip.y );
real_t DL0[2], DL1[2], DL2[2];
DL0[0] = -2.0; DL0[1] = -2.0;
DL1[0] = 2.0; DL1[1] = 0.0;
DL2[0] = 0.0; DL2[1] = 2.0;
for (i = 0; i < 6; i++)
for (j = 0; j < 2; j++)
{
dshape(i,j) = 0.0;
}
if (L0 >= 1.0) // T0
{
for (j = 0; j < 2; j++)
{
dshape(0,j) = DL0[j];
dshape(3,j) = DL1[j];
dshape(5,j) = DL2[j];
}
}
else if (L1 >= 1.0) // T1
{
for (j = 0; j < 2; j++)
{
dshape(3,j) = DL0[j];
dshape(1,j) = DL1[j];
dshape(4,j) = DL2[j];
}
}
else if (L2 >= 1.0) // T2
{
for (j = 0; j < 2; j++)
{
dshape(5,j) = DL0[j];
dshape(4,j) = DL1[j];
dshape(2,j) = DL2[j];
}
}
else // T3
{
for (j = 0; j < 2; j++)
{
dshape(3,j) = - DL2[j];
dshape(4,j) = - DL0[j];
dshape(5,j) = - DL1[j];
}
}
}
RefinedLinear3DFiniteElement::RefinedLinear3DFiniteElement()
: NodalFiniteElement(3, Geometry::TETRAHEDRON, 10, 4)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(3).z = 1.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 0.0;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(5).z = 0.0;
Nodes.IntPoint(6).x = 0.0;
Nodes.IntPoint(6).y = 0.0;
Nodes.IntPoint(6).z = 0.5;
Nodes.IntPoint(7).x = 0.5;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(7).z = 0.0;
Nodes.IntPoint(8).x = 0.5;
Nodes.IntPoint(8).y = 0.0;
Nodes.IntPoint(8).z = 0.5;
Nodes.IntPoint(9).x = 0.0;
Nodes.IntPoint(9).y = 0.5;
Nodes.IntPoint(9).z = 0.5;
}
void RefinedLinear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
int i;
real_t L0, L1, L2, L3, L4, L5;
L0 = 2.0 * ( 1. - ip.x - ip.y - ip.z );
L1 = 2.0 * ( ip.x );
L2 = 2.0 * ( ip.y );
L3 = 2.0 * ( ip.z );
L4 = 2.0 * ( ip.x + ip.y );
L5 = 2.0 * ( ip.y + ip.z );
// The reference tetrahedron is split in 8 tetrahedra as follows:
//
// T0 - 0,4,5,6
// T1 - 1,4,7,8
// T2 - 2,5,7,9
// T3 - 3,6,8,9
// T4 - 4,5,6,8
// T5 - 4,5,7,8
// T6 - 5,6,8,9
// T7 - 5,7,8,9
for (i = 0; i < 10; i++)
{
shape(i) = 0.0;
}
if (L0 >= 1.0) // T0
{
shape(0) = L0 - 1.0;
shape(4) = L1;
shape(5) = L2;
shape(6) = L3;
}
else if (L1 >= 1.0) // T1
{
shape(4) = L0;
shape(1) = L1 - 1.0;
shape(7) = L2;
shape(8) = L3;
}
else if (L2 >= 1.0) // T2
{
shape(5) = L0;
shape(7) = L1;
shape(2) = L2 - 1.0;
shape(9) = L3;
}
else if (L3 >= 1.0) // T3
{
shape(6) = L0;
shape(8) = L1;
shape(9) = L2;
shape(3) = L3 - 1.0;
}
else if ((L4 <= 1.0) && (L5 <= 1.0)) // T4
{
shape(4) = 1.0 - L5;
shape(5) = L2;
shape(6) = 1.0 - L4;
shape(8) = 1.0 - L0;
}
else if ((L4 >= 1.0) && (L5 <= 1.0)) // T5
{
shape(4) = 1.0 - L5;
shape(5) = 1.0 - L1;
shape(7) = L4 - 1.0;
shape(8) = L3;
}
else if ((L4 <= 1.0) && (L5 >= 1.0)) // T6
{
shape(5) = 1.0 - L3;
shape(6) = 1.0 - L4;
shape(8) = L1;
shape(9) = L5 - 1.0;
}
else if ((L4 >= 1.0) && (L5 >= 1.0)) // T7
{
shape(5) = L0;
shape(7) = L4 - 1.0;
shape(8) = 1.0 - L2;
shape(9) = L5 - 1.0;
}
}
void RefinedLinear3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
int i,j;
real_t L0, L1, L2, L3, L4, L5;
L0 = 2.0 * ( 1. - ip.x - ip.y - ip.z );
L1 = 2.0 * ( ip.x );
L2 = 2.0 * ( ip.y );
L3 = 2.0 * ( ip.z );
L4 = 2.0 * ( ip.x + ip.y );
L5 = 2.0 * ( ip.y + ip.z );
real_t DL0[3], DL1[3], DL2[3], DL3[3], DL4[3], DL5[3];
DL0[0] = -2.0; DL0[1] = -2.0; DL0[2] = -2.0;
DL1[0] = 2.0; DL1[1] = 0.0; DL1[2] = 0.0;
DL2[0] = 0.0; DL2[1] = 2.0; DL2[2] = 0.0;
DL3[0] = 0.0; DL3[1] = 0.0; DL3[2] = 2.0;
DL4[0] = 2.0; DL4[1] = 2.0; DL4[2] = 0.0;
DL5[0] = 0.0; DL5[1] = 2.0; DL5[2] = 2.0;
for (i = 0; i < 10; i++)
for (j = 0; j < 3; j++)
{
dshape(i,j) = 0.0;
}
if (L0 >= 1.0) // T0
{
for (j = 0; j < 3; j++)
{
dshape(0,j) = DL0[j];
dshape(4,j) = DL1[j];
dshape(5,j) = DL2[j];
dshape(6,j) = DL3[j];
}
}
else if (L1 >= 1.0) // T1
{
for (j = 0; j < 3; j++)
{
dshape(4,j) = DL0[j];
dshape(1,j) = DL1[j];
dshape(7,j) = DL2[j];
dshape(8,j) = DL3[j];
}
}
else if (L2 >= 1.0) // T2
{
for (j = 0; j < 3; j++)
{
dshape(5,j) = DL0[j];
dshape(7,j) = DL1[j];
dshape(2,j) = DL2[j];
dshape(9,j) = DL3[j];
}
}
else if (L3 >= 1.0) // T3
{
for (j = 0; j < 3; j++)
{
dshape(6,j) = DL0[j];
dshape(8,j) = DL1[j];
dshape(9,j) = DL2[j];
dshape(3,j) = DL3[j];
}
}
else if ((L4 <= 1.0) && (L5 <= 1.0)) // T4
{
for (j = 0; j < 3; j++)
{
dshape(4,j) = - DL5[j];
dshape(5,j) = DL2[j];
dshape(6,j) = - DL4[j];
dshape(8,j) = - DL0[j];
}
}
else if ((L4 >= 1.0) && (L5 <= 1.0)) // T5
{
for (j = 0; j < 3; j++)
{
dshape(4,j) = - DL5[j];
dshape(5,j) = - DL1[j];
dshape(7,j) = DL4[j];
dshape(8,j) = DL3[j];
}
}
else if ((L4 <= 1.0) && (L5 >= 1.0)) // T6
{
for (j = 0; j < 3; j++)
{
dshape(5,j) = - DL3[j];
dshape(6,j) = - DL4[j];
dshape(8,j) = DL1[j];
dshape(9,j) = DL5[j];
}
}
else if ((L4 >= 1.0) && (L5 >= 1.0)) // T7
{
for (j = 0; j < 3; j++)
{
dshape(5,j) = DL0[j];
dshape(7,j) = DL4[j];
dshape(8,j) = - DL2[j];
dshape(9,j) = DL5[j];
}
}
}
RefinedBiLinear2DFiniteElement::RefinedBiLinear2DFiniteElement()
: NodalFiniteElement(2, Geometry::SQUARE, 9, 1, FunctionSpace::rQk)
{
Nodes.IntPoint(0).x = 0.0;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 1.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(5).x = 1.0;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(6).x = 0.5;
Nodes.IntPoint(6).y = 1.0;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(8).x = 0.5;
Nodes.IntPoint(8).y = 0.5;
}
void RefinedBiLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
int i;
real_t x = ip.x, y = ip.y;
real_t Lx, Ly;
Lx = 2.0 * ( 1. - x );
Ly = 2.0 * ( 1. - y );
// The reference square is split in 4 squares as follows:
//
// T0 - 0,4,7,8
// T1 - 1,4,5,8
// T2 - 2,5,6,8
// T3 - 3,6,7,8
for (i = 0; i < 9; i++)
{
shape(i) = 0.0;
}
if ((x <= 0.5) && (y <= 0.5)) // T0
{
shape(0) = (Lx - 1.0) * (Ly - 1.0);
shape(4) = (2.0 - Lx) * (Ly - 1.0);
shape(8) = (2.0 - Lx) * (2.0 - Ly);
shape(7) = (Lx - 1.0) * (2.0 - Ly);
}
else if ((x >= 0.5) && (y <= 0.5)) // T1
{
shape(4) = Lx * (Ly - 1.0);
shape(1) = (1.0 - Lx) * (Ly - 1.0);
shape(5) = (1.0 - Lx) * (2.0 - Ly);
shape(8) = Lx * (2.0 - Ly);
}
else if ((x >= 0.5) && (y >= 0.5)) // T2
{
shape(8) = Lx * Ly ;
shape(5) = (1.0 - Lx) * Ly ;
shape(2) = (1.0 - Lx) * (1.0 - Ly);
shape(6) = Lx * (1.0 - Ly);
}
else if ((x <= 0.5) && (y >= 0.5)) // T3
{
shape(7) = (Lx - 1.0) * Ly ;
shape(8) = (2.0 - Lx) * Ly ;
shape(6) = (2.0 - Lx) * (1.0 - Ly);
shape(3) = (Lx - 1.0) * (1.0 - Ly);
}
}
void RefinedBiLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
int i,j;
real_t x = ip.x, y = ip.y;
real_t Lx, Ly;
Lx = 2.0 * ( 1. - x );
Ly = 2.0 * ( 1. - y );
for (i = 0; i < 9; i++)
for (j = 0; j < 2; j++)
{
dshape(i,j) = 0.0;
}
if ((x <= 0.5) && (y <= 0.5)) // T0
{
dshape(0,0) = 2.0 * (1.0 - Ly);
dshape(0,1) = 2.0 * (1.0 - Lx);
dshape(4,0) = 2.0 * (Ly - 1.0);
dshape(4,1) = -2.0 * (2.0 - Lx);
dshape(8,0) = 2.0 * (2.0 - Ly);
dshape(8,1) = 2.0 * (2.0 - Lx);
dshape(7,0) = -2.0 * (2.0 - Ly);
dshape(7,0) = 2.0 * (Lx - 1.0);
}
else if ((x >= 0.5) && (y <= 0.5)) // T1
{
dshape(4,0) = -2.0 * (Ly - 1.0);
dshape(4,1) = -2.0 * Lx;
dshape(1,0) = 2.0 * (Ly - 1.0);
dshape(1,1) = -2.0 * (1.0 - Lx);
dshape(5,0) = 2.0 * (2.0 - Ly);
dshape(5,1) = 2.0 * (1.0 - Lx);
dshape(8,0) = -2.0 * (2.0 - Ly);
dshape(8,1) = 2.0 * Lx;
}
else if ((x >= 0.5) && (y >= 0.5)) // T2
{
dshape(8,0) = -2.0 * Ly;
dshape(8,1) = -2.0 * Lx;
dshape(5,0) = 2.0 * Ly;
dshape(5,1) = -2.0 * (1.0 - Lx);
dshape(2,0) = 2.0 * (1.0 - Ly);
dshape(2,1) = 2.0 * (1.0 - Lx);
dshape(6,0) = -2.0 * (1.0 - Ly);
dshape(6,1) = 2.0 * Lx;
}
else if ((x <= 0.5) && (y >= 0.5)) // T3
{
dshape(7,0) = -2.0 * Ly;
dshape(7,1) = -2.0 * (Lx - 1.0);
dshape(8,0) = 2.0 * Ly ;
dshape(8,1) = -2.0 * (2.0 - Lx);
dshape(6,0) = 2.0 * (1.0 - Ly);
dshape(6,1) = 2.0 * (2.0 - Lx);
dshape(3,0) = -2.0 * (1.0 - Ly);
dshape(3,1) = 2.0 * (Lx - 1.0);
}
}
RefinedTriLinear3DFiniteElement::RefinedTriLinear3DFiniteElement()
: NodalFiniteElement(3, Geometry::CUBE, 27, 2, FunctionSpace::rQk)
{
real_t I[27];
real_t J[27];
real_t K[27];
// nodes
I[ 0] = 0.0; J[ 0] = 0.0; K[ 0] = 0.0;
I[ 1] = 1.0; J[ 1] = 0.0; K[ 1] = 0.0;
I[ 2] = 1.0; J[ 2] = 1.0; K[ 2] = 0.0;
I[ 3] = 0.0; J[ 3] = 1.0; K[ 3] = 0.0;
I[ 4] = 0.0; J[ 4] = 0.0; K[ 4] = 1.0;
I[ 5] = 1.0; J[ 5] = 0.0; K[ 5] = 1.0;
I[ 6] = 1.0; J[ 6] = 1.0; K[ 6] = 1.0;
I[ 7] = 0.0; J[ 7] = 1.0; K[ 7] = 1.0;
// edges
I[ 8] = 0.5; J[ 8] = 0.0; K[ 8] = 0.0;
I[ 9] = 1.0; J[ 9] = 0.5; K[ 9] = 0.0;
I[10] = 0.5; J[10] = 1.0; K[10] = 0.0;
I[11] = 0.0; J[11] = 0.5; K[11] = 0.0;
I[12] = 0.5; J[12] = 0.0; K[12] = 1.0;
I[13] = 1.0; J[13] = 0.5; K[13] = 1.0;
I[14] = 0.5; J[14] = 1.0; K[14] = 1.0;
I[15] = 0.0; J[15] = 0.5; K[15] = 1.0;
I[16] = 0.0; J[16] = 0.0; K[16] = 0.5;
I[17] = 1.0; J[17] = 0.0; K[17] = 0.5;
I[18] = 1.0; J[18] = 1.0; K[18] = 0.5;
I[19] = 0.0; J[19] = 1.0; K[19] = 0.5;
// faces
I[20] = 0.5; J[20] = 0.5; K[20] = 0.0;
I[21] = 0.5; J[21] = 0.0; K[21] = 0.5;
I[22] = 1.0; J[22] = 0.5; K[22] = 0.5;
I[23] = 0.5; J[23] = 1.0; K[23] = 0.5;
I[24] = 0.0; J[24] = 0.5; K[24] = 0.5;
I[25] = 0.5; J[25] = 0.5; K[25] = 1.0;
// element
I[26] = 0.5; J[26] = 0.5; K[26] = 0.5;
for (int n = 0; n < 27; n++)
{
Nodes.IntPoint(n).x = I[n];
Nodes.IntPoint(n).y = J[n];
Nodes.IntPoint(n).z = K[n];
}
}
void RefinedTriLinear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
int i, N[8];
real_t Lx, Ly, Lz;
real_t x = ip.x, y = ip.y, z = ip.z;
for (i = 0; i < 27; i++)
{
shape(i) = 0.0;
}
if ((x <= 0.5) && (y <= 0.5) && (z <= 0.5)) // T0
{
Lx = 1.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 0;
N[1] = 8;
N[2] = 20;
N[3] = 11;
N[4] = 16;
N[5] = 21;
N[6] = 26;
N[7] = 24;
}
else if ((x >= 0.5) && (y <= 0.5) && (z <= 0.5)) // T1
{
Lx = 2.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 8;
N[1] = 1;
N[2] = 9;
N[3] = 20;
N[4] = 21;
N[5] = 17;
N[6] = 22;
N[7] = 26;
}
else if ((x <= 0.5) && (y >= 0.5) && (z <= 0.5)) // T2
{
Lx = 2.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 20;
N[1] = 9;
N[2] = 2;
N[3] = 10;
N[4] = 26;
N[5] = 22;
N[6] = 18;
N[7] = 23;
}
else if ((x >= 0.5) && (y >= 0.5) && (z <= 0.5)) // T3
{
Lx = 1.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 11;
N[1] = 20;
N[2] = 10;
N[3] = 3;
N[4] = 24;
N[5] = 26;
N[6] = 23;
N[7] = 19;
}
else if ((x <= 0.5) && (y <= 0.5) && (z >= 0.5)) // T4
{
Lx = 1.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 16;
N[1] = 21;
N[2] = 26;
N[3] = 24;
N[4] = 4;
N[5] = 12;
N[6] = 25;
N[7] = 15;
}
else if ((x >= 0.5) && (y <= 0.5) && (z >= 0.5)) // T5
{
Lx = 2.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 21;
N[1] = 17;
N[2] = 22;
N[3] = 26;
N[4] = 12;
N[5] = 5;
N[6] = 13;
N[7] = 25;
}
else if ((x <= 0.5) && (y >= 0.5) && (z >= 0.5)) // T6
{
Lx = 2.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 26;
N[1] = 22;
N[2] = 18;
N[3] = 23;
N[4] = 25;
N[5] = 13;
N[6] = 6;
N[7] = 14;
}
else // T7
{
Lx = 1.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 24;
N[1] = 26;
N[2] = 23;
N[3] = 19;
N[4] = 15;
N[5] = 25;
N[6] = 14;
N[7] = 7;
}
shape(N[0]) = Lx * Ly * Lz;
shape(N[1]) = (1 - Lx) * Ly * Lz;
shape(N[2]) = (1 - Lx) * (1 - Ly) * Lz;
shape(N[3]) = Lx * (1 - Ly) * Lz;
shape(N[4]) = Lx * Ly * (1 - Lz);
shape(N[5]) = (1 - Lx) * Ly * (1 - Lz);
shape(N[6]) = (1 - Lx) * (1 - Ly) * (1 - Lz);
shape(N[7]) = Lx * (1 - Ly) * (1 - Lz);
}
void RefinedTriLinear3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
int i, j, N[8];
real_t Lx, Ly, Lz;
real_t x = ip.x, y = ip.y, z = ip.z;
for (i = 0; i < 27; i++)
for (j = 0; j < 3; j++)
{
dshape(i,j) = 0.0;
}
if ((x <= 0.5) && (y <= 0.5) && (z <= 0.5)) // T0
{
Lx = 1.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 0;
N[1] = 8;
N[2] = 20;
N[3] = 11;
N[4] = 16;
N[5] = 21;
N[6] = 26;
N[7] = 24;
}
else if ((x >= 0.5) && (y <= 0.5) && (z <= 0.5)) // T1
{
Lx = 2.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 8;
N[1] = 1;
N[2] = 9;
N[3] = 20;
N[4] = 21;
N[5] = 17;
N[6] = 22;
N[7] = 26;
}
else if ((x <= 0.5) && (y >= 0.5) && (z <= 0.5)) // T2
{
Lx = 2.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 20;
N[1] = 9;
N[2] = 2;
N[3] = 10;
N[4] = 26;
N[5] = 22;
N[6] = 18;
N[7] = 23;
}
else if ((x >= 0.5) && (y >= 0.5) && (z <= 0.5)) // T3
{
Lx = 1.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 1.0 - 2.0 * z;
N[0] = 11;
N[1] = 20;
N[2] = 10;
N[3] = 3;
N[4] = 24;
N[5] = 26;
N[6] = 23;
N[7] = 19;
}
else if ((x <= 0.5) && (y <= 0.5) && (z >= 0.5)) // T4
{
Lx = 1.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 16;
N[1] = 21;
N[2] = 26;
N[3] = 24;
N[4] = 4;
N[5] = 12;
N[6] = 25;
N[7] = 15;
}
else if ((x >= 0.5) && (y <= 0.5) && (z >= 0.5)) // T5
{
Lx = 2.0 - 2.0 * x;
Ly = 1.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 21;
N[1] = 17;
N[2] = 22;
N[3] = 26;
N[4] = 12;
N[5] = 5;
N[6] = 13;
N[7] = 25;
}
else if ((x <= 0.5) && (y >= 0.5) && (z >= 0.5)) // T6
{
Lx = 2.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 26;
N[1] = 22;
N[2] = 18;
N[3] = 23;
N[4] = 25;
N[5] = 13;
N[6] = 6;
N[7] = 14;
}
else // T7
{
Lx = 1.0 - 2.0 * x;
Ly = 2.0 - 2.0 * y;
Lz = 2.0 - 2.0 * z;
N[0] = 24;
N[1] = 26;
N[2] = 23;
N[3] = 19;
N[4] = 15;
N[5] = 25;
N[6] = 14;
N[7] = 7;
}
dshape(N[0],0) = -2.0 * Ly * Lz ;
dshape(N[0],1) = -2.0 * Lx * Lz ;
dshape(N[0],2) = -2.0 * Lx * Ly ;
dshape(N[1],0) = 2.0 * Ly * Lz ;
dshape(N[1],1) = -2.0 * (1 - Lx) * Lz ;
dshape(N[1],2) = -2.0 * (1 - Lx) * Ly ;
dshape(N[2],0) = 2.0 * (1 - Ly) * Lz ;
dshape(N[2],1) = 2.0 * (1 - Lx) * Lz ;
dshape(N[2],2) = -2.0 * (1 - Lx) * (1 - Ly);
dshape(N[3],0) = -2.0 * (1 - Ly) * Lz ;
dshape(N[3],1) = 2.0 * Lx * Lz ;
dshape(N[3],2) = -2.0 * Lx * (1 - Ly);
dshape(N[4],0) = -2.0 * Ly * (1 - Lz);
dshape(N[4],1) = -2.0 * Lx * (1 - Lz);
dshape(N[4],2) = 2.0 * Lx * Ly ;
dshape(N[5],0) = 2.0 * Ly * (1 - Lz);
dshape(N[5],1) = -2.0 * (1 - Lx) * (1 - Lz);
dshape(N[5],2) = 2.0 * (1 - Lx) * Ly ;
dshape(N[6],0) = 2.0 * (1 - Ly) * (1 - Lz);
dshape(N[6],1) = 2.0 * (1 - Lx) * (1 - Lz);
dshape(N[6],2) = 2.0 * (1 - Lx) * (1 - Ly);
dshape(N[7],0) = -2.0 * (1 - Ly) * (1 - Lz);
dshape(N[7],1) = 2.0 * Lx * (1 - Lz);
dshape(N[7],2) = 2.0 * Lx * (1 - Ly);
}
Nedelec1HexFiniteElement::Nedelec1HexFiniteElement()
: VectorFiniteElement(3, Geometry::CUBE, 12, 1, H_CURL,
FunctionSpace::Qk)
{
// not real nodes ...
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.5;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.5;
Nodes.IntPoint(3).z = 0.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 1.0;
Nodes.IntPoint(5).x = 1.0;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(5).z = 1.0;
Nodes.IntPoint(6).x = 0.5;
Nodes.IntPoint(6).y = 1.0;
Nodes.IntPoint(6).z = 1.0;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(7).z = 1.0;
Nodes.IntPoint(8).x = 0.0;
Nodes.IntPoint(8).y = 0.0;
Nodes.IntPoint(8).z = 0.5;
Nodes.IntPoint(9).x = 1.0;
Nodes.IntPoint(9).y = 0.0;
Nodes.IntPoint(9).z = 0.5;
Nodes.IntPoint(10).x= 1.0;
Nodes.IntPoint(10).y= 1.0;
Nodes.IntPoint(10).z= 0.5;
Nodes.IntPoint(11).x= 0.0;
Nodes.IntPoint(11).y= 1.0;
Nodes.IntPoint(11).z= 0.5;
}
void Nedelec1HexFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
shape(0,0) = (1. - y) * (1. - z);
shape(0,1) = 0.;
shape(0,2) = 0.;
shape(2,0) = y * (1. - z);
shape(2,1) = 0.;
shape(2,2) = 0.;
shape(4,0) = z * (1. - y);
shape(4,1) = 0.;
shape(4,2) = 0.;
shape(6,0) = y * z;
shape(6,1) = 0.;
shape(6,2) = 0.;
shape(1,0) = 0.;
shape(1,1) = x * (1. - z);
shape(1,2) = 0.;
shape(3,0) = 0.;
shape(3,1) = (1. - x) * (1. - z);
shape(3,2) = 0.;
shape(5,0) = 0.;
shape(5,1) = x * z;
shape(5,2) = 0.;
shape(7,0) = 0.;
shape(7,1) = (1. - x) * z;
shape(7,2) = 0.;
shape(8,0) = 0.;
shape(8,1) = 0.;
shape(8,2) = (1. - x) * (1. - y);
shape(9,0) = 0.;
shape(9,1) = 0.;
shape(9,2) = x * (1. - y);
shape(10,0) = 0.;
shape(10,1) = 0.;
shape(10,2) = x * y;
shape(11,0) = 0.;
shape(11,1) = 0.;
shape(11,2) = y * (1. - x);
}
void Nedelec1HexFiniteElement::CalcCurlShape(const IntegrationPoint &ip,
DenseMatrix &curl_shape)
const
{
real_t x = ip.x, y = ip.y, z = ip.z;
curl_shape(0,0) = 0.;
curl_shape(0,1) = y - 1.;
curl_shape(0,2) = 1. - z;
curl_shape(2,0) = 0.;
curl_shape(2,1) = -y;
curl_shape(2,2) = z - 1.;
curl_shape(4,0) = 0;
curl_shape(4,1) = 1. - y;
curl_shape(4,2) = z;
curl_shape(6,0) = 0.;
curl_shape(6,1) = y;
curl_shape(6,2) = -z;
curl_shape(1,0) = x;
curl_shape(1,1) = 0.;
curl_shape(1,2) = 1. - z;
curl_shape(3,0) = 1. - x;
curl_shape(3,1) = 0.;
curl_shape(3,2) = z - 1.;
curl_shape(5,0) = -x;
curl_shape(5,1) = 0.;
curl_shape(5,2) = z;
curl_shape(7,0) = x - 1.;
curl_shape(7,1) = 0.;
curl_shape(7,2) = -z;
curl_shape(8,0) = x - 1.;
curl_shape(8,1) = 1. - y;
curl_shape(8,2) = 0.;
curl_shape(9,0) = -x;
curl_shape(9,1) = y - 1.;
curl_shape(9,2) = 0;
curl_shape(10,0) = x;
curl_shape(10,1) = -y;
curl_shape(10,2) = 0.;
curl_shape(11,0) = 1. - x;
curl_shape(11,1) = y;
curl_shape(11,2) = 0.;
}
const real_t Nedelec1HexFiniteElement::tk[12][3] =
{
{1,0,0}, {0,1,0}, {1,0,0}, {0,1,0},
{1,0,0}, {0,1,0}, {1,0,0}, {0,1,0},
{0,0,1}, {0,0,1}, {0,0,1}, {0,0,1}
};
void Nedelec1HexFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < dof; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < dof; j++)
{
real_t d = ( vshape(j,0)*tk[k][0] + vshape(j,1)*tk[k][1] +
vshape(j,2)*tk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "Nedelec1HexFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear (more to have embedding?)
const DenseMatrix &J = Trans.Jacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < dof; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = J tk
vk[0] = J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2];
vk[1] = J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2];
vk[2] = J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2];
for (j = 0; j < dof; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void Nedelec1HexFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < dof; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
const DenseMatrix &J = Trans.Jacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t J tk
dofs(k) =
vk[0] * ( J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2] ) +
vk[1] * ( J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2] ) +
vk[2] * ( J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2] );
}
}
void Nedelec1HexFiniteElement::ProjectGrad(const FiniteElement &fe,
ElementTransformation &Trans,
DenseMatrix &grad) const
{
DenseMatrix dshape(fe.GetDof(), 3);
Vector grad_k(fe.GetDof());
grad.SetSize(dof, fe.GetDof());
for (int k = 0; k < dof; k++)
{
fe.CalcDShape(Nodes.IntPoint(k), dshape);
dshape.Mult(tk[k], grad_k);
for (int j = 0; j < grad_k.Size(); j++)
{
grad(k,j) = (fabs(grad_k(j)) < 1e-12) ? 0.0 : grad_k(j);
}
}
}
Nedelec1TetFiniteElement::Nedelec1TetFiniteElement()
: VectorFiniteElement(3, Geometry::TETRAHEDRON, 6, 1, H_CURL)
{
// not real nodes ...
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 0.0;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 0.0;
Nodes.IntPoint(2).z = 0.5;
Nodes.IntPoint(3).x = 0.5;
Nodes.IntPoint(3).y = 0.5;
Nodes.IntPoint(3).z = 0.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 0.5;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(5).z = 0.5;
}
void Nedelec1TetFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
shape(0,0) = 1. - y - z;
shape(0,1) = x;
shape(0,2) = x;
shape(1,0) = y;
shape(1,1) = 1. - x - z;
shape(1,2) = y;
shape(2,0) = z;
shape(2,1) = z;
shape(2,2) = 1. - x - y;
shape(3,0) = -y;
shape(3,1) = x;
shape(3,2) = 0.;
shape(4,0) = -z;
shape(4,1) = 0.;
shape(4,2) = x;
shape(5,0) = 0.;
shape(5,1) = -z;
shape(5,2) = y;
}
void Nedelec1TetFiniteElement::CalcCurlShape(const IntegrationPoint &ip,
DenseMatrix &curl_shape)
const
{
curl_shape(0,0) = 0.;
curl_shape(0,1) = -2.;
curl_shape(0,2) = 2.;
curl_shape(1,0) = 2.;
curl_shape(1,1) = 0.;
curl_shape(1,2) = -2.;
curl_shape(2,0) = -2.;
curl_shape(2,1) = 2.;
curl_shape(2,2) = 0.;
curl_shape(3,0) = 0.;
curl_shape(3,1) = 0.;
curl_shape(3,2) = 2.;
curl_shape(4,0) = 0.;
curl_shape(4,1) = -2.;
curl_shape(4,2) = 0.;
curl_shape(5,0) = 2.;
curl_shape(5,1) = 0.;
curl_shape(5,2) = 0.;
}
const real_t Nedelec1TetFiniteElement::tk[6][3] =
{{1,0,0}, {0,1,0}, {0,0,1}, {-1,1,0}, {-1,0,1}, {0,-1,1}};
void Nedelec1TetFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < dof; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < dof; j++)
{
real_t d = ( vshape(j,0)*tk[k][0] + vshape(j,1)*tk[k][1] +
vshape(j,2)*tk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "Nedelec1TetFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
const DenseMatrix &J = Trans.Jacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < dof; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = J tk
vk[0] = J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2];
vk[1] = J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2];
vk[2] = J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2];
for (j = 0; j < dof; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void Nedelec1TetFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < dof; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
const DenseMatrix &J = Trans.Jacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t J tk
dofs(k) =
vk[0] * ( J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2] ) +
vk[1] * ( J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2] ) +
vk[2] * ( J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2] );
}
}
void Nedelec1TetFiniteElement::ProjectGrad(const FiniteElement &fe,
ElementTransformation &Trans,
DenseMatrix &grad) const
{
DenseMatrix dshape(fe.GetDof(), 3);
Vector grad_k(fe.GetDof());
grad.SetSize(dof, fe.GetDof());
for (int k = 0; k < dof; k++)
{
fe.CalcDShape(Nodes.IntPoint(k), dshape);
dshape.Mult(tk[k], grad_k);
for (int j = 0; j < grad_k.Size(); j++)
{
grad(k,j) = (fabs(grad_k(j)) < 1e-12) ? 0.0 : grad_k(j);
}
}
}
Nedelec1WdgFiniteElement::Nedelec1WdgFiniteElement()
: VectorFiniteElement(3, Geometry::PRISM, 9, 1, H_CURL, FunctionSpace::Qk)
{
// not real nodes ...
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 0.5;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.0;
Nodes.IntPoint(2).y = 0.5;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.5;
Nodes.IntPoint(3).y = 0.0;
Nodes.IntPoint(3).z = 1.0;
Nodes.IntPoint(4).x = 0.5;
Nodes.IntPoint(4).y = 0.5;
Nodes.IntPoint(4).z = 1.0;
Nodes.IntPoint(5).x = 0.0;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(5).z = 1.0;
Nodes.IntPoint(6).x = 0.0;
Nodes.IntPoint(6).y = 0.0;
Nodes.IntPoint(6).z = 0.5;
Nodes.IntPoint(7).x = 1.0;
Nodes.IntPoint(7).y = 0.0;
Nodes.IntPoint(7).z = 0.5;
Nodes.IntPoint(8).x = 0.0;
Nodes.IntPoint(8).y = 1.0;
Nodes.IntPoint(8).z = 0.5;
}
void Nedelec1WdgFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
shape(0,0) = (1. - y) * (1. - z);
shape(0,1) = x * (1. - z);
shape(0,2) = 0.;
shape(1,0) = - y * (1. - z);
shape(1,1) = x * (1. - z);
shape(1,2) = 0.;
shape(2,0) = - y * (1. - z);
shape(2,1) = - (1. - x) * (1. - z);
shape(2,2) = 0.;
shape(3,0) = (1. - y) * z;
shape(3,1) = x * z;
shape(3,2) = 0.;
shape(4,0) = - y * z;
shape(4,1) = x * z;
shape(4,2) = 0.;
shape(5,0) = - y * z;
shape(5,1) = - (1. - x) * z;
shape(5,2) = 0.;
shape(6,0) = 0.;
shape(6,1) = 0.;
shape(6,2) = 1. - x - y;
shape(7,0) = 0.;
shape(7,1) = 0.;
shape(7,2) = x;
shape(8,0) = 0.;
shape(8,1) = 0.;
shape(8,2) = y;
}
void Nedelec1WdgFiniteElement::CalcCurlShape(const IntegrationPoint &ip,
DenseMatrix &curl_shape)
const
{
real_t x = ip.x, y = ip.y, z2 = 2. * ip.z;
curl_shape(0,0) = x;
curl_shape(0,1) = - 1. + y;
curl_shape(0,2) = 2. - z2;
curl_shape(1,0) = x;
curl_shape(1,1) = y;
curl_shape(1,2) = 2. - z2;
curl_shape(2,0) = - 1. + x;
curl_shape(2,1) = y;
curl_shape(2,2) = 2. - z2;
curl_shape(3,0) = - x;
curl_shape(3,1) = 1. - y;
curl_shape(3,2) = z2;
curl_shape(4,0) = - x;
curl_shape(4,1) = - y;
curl_shape(4,2) = z2;
curl_shape(5,0) = 1. - x;
curl_shape(5,1) = - y;
curl_shape(5,2) = z2;
curl_shape(6,0) = - 1.;
curl_shape(6,1) = 1.;
curl_shape(6,2) = 0.;
curl_shape(7,0) = 0.;
curl_shape(7,1) = - 1.;
curl_shape(7,2) = 0.;
curl_shape(8,0) = 1.;
curl_shape(8,1) = 0.;
curl_shape(8,2) = 0.;
}
const real_t Nedelec1WdgFiniteElement::tk[9][3] =
{
{1,0,0}, {-1,1,0}, {0,-1,0}, {1,0,0}, {-1,1,0}, {0,-1,0},
{0,0,1}, {0,0,1}, {0,0,1}
};
void Nedelec1WdgFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < dof; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < dof; j++)
{
real_t d = ( vshape(j,0)*tk[k][0] + vshape(j,1)*tk[k][1] +
vshape(j,2)*tk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "Nedelec1WdgFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
const DenseMatrix &J = Trans.Jacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < dof; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = J tk
vk[0] = J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2];
vk[1] = J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2];
vk[2] = J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2];
for (j = 0; j < dof; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void Nedelec1WdgFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < dof; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
const DenseMatrix &J = Trans.Jacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t J tk
dofs(k) =
vk[0] * ( J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2] ) +
vk[1] * ( J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2] ) +
vk[2] * ( J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2] );
}
}
void Nedelec1WdgFiniteElement::ProjectGrad(const FiniteElement &fe,
ElementTransformation &Trans,
DenseMatrix &grad) const
{
DenseMatrix dshape(fe.GetDof(), 3);
Vector grad_k(fe.GetDof());
grad.SetSize(dof, fe.GetDof());
for (int k = 0; k < dof; k++)
{
fe.CalcDShape(Nodes.IntPoint(k), dshape);
dshape.Mult(tk[k], grad_k);
for (int j = 0; j < grad_k.Size(); j++)
{
grad(k,j) = (fabs(grad_k(j)) < 1e-12) ? 0.0 : grad_k(j);
}
}
}
Nedelec1PyrFiniteElement::Nedelec1PyrFiniteElement()
: VectorFiniteElement(3, Geometry::PYRAMID, 8, 1, H_CURL, FunctionSpace::Uk)
{
// not real nodes ...
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.0;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 1.0;
Nodes.IntPoint(1).y = 0.5;
Nodes.IntPoint(1).z = 0.0;
Nodes.IntPoint(2).x = 0.5;
Nodes.IntPoint(2).y = 1.0;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 0.0;
Nodes.IntPoint(3).y = 0.5;
Nodes.IntPoint(3).z = 0.0;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 0.5;
Nodes.IntPoint(5).x = 0.5;
Nodes.IntPoint(5).y = 0.0;
Nodes.IntPoint(5).z = 0.5;
Nodes.IntPoint(6).x = 0.5;
Nodes.IntPoint(6).y = 0.5;
Nodes.IntPoint(6).z = 0.5;
Nodes.IntPoint(7).x = 0.0;
Nodes.IntPoint(7).y = 0.5;
Nodes.IntPoint(7).z = 0.5;
}
void Nedelec1PyrFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z, z2 = 2. * ip.z;
real_t ox = 1. - x - z, oy = 1. - y - z, oz = 1. - z;
real_t tol = 1e-6;
if (oz <= tol)
{
// We must return the limit of the basis functions as z->1. In order to
// remain inside the pyramid in this limit the x and y coordinates must
// be approaching 0. Unfortunately we obtain different limits if we
// approach (0,0,1) from different directions. The values provided below
// are the limits as x->(1-z)/2 and y->(1-z)/2 i.e. along the line from
// the center of the base of the pyramid towards the apex. The resulting
// limiting basis function values are:
shape(0,0) = 0.;
shape(0,1) = 0.;
shape(0,2) = 0.;
shape(1,0) = 0.;
shape(1,1) = 0.;
shape(1,2) = 0.;
shape(2,0) = 0.;
shape(2,1) = 0.;
shape(2,2) = 0.;
shape(3,0) = 0.;
shape(3,1) = 0.;
shape(3,2) = 0.;
shape(4,0) = 0.5;
shape(4,1) = 0.5;
shape(4,2) = 0.75;
shape(5,0) = - 0.5;
shape(5,1) = 0.5;
shape(5,2) = 0.25;
shape(6,0) = - 0.5;
shape(6,1) = - 0.5;
shape(6,2) = - 0.25;
shape(7,0) = 0.5;
shape(7,1) = - 0.5;
shape(7,2) = 0.25;
return;
}
real_t ozi = 1.0 / oz;
shape(0,0) = oy;
shape(0,1) = 0.;
shape(0,2) = x * oy * ozi;
shape(1,0) = 0.;
shape(1,1) = x;
shape(1,2) = x * y * ozi;
shape(2,0) = y;
shape(2,1) = 0.;
shape(2,2) = x * y * ozi;
shape(3,0) = 0.;
shape(3,1) = ox;
shape(3,2) = ox * y * ozi;
shape(4,0) = oy * z * ozi;
shape(4,1) = ox * z * ozi;
shape(4,2) = 1. - x - y + x * y * (1. - z2) * ozi * ozi;
shape(5,0) = - oy * z * ozi;
shape(5,1) = x * z * ozi;
shape(5,2) = x * (1. - y * (1. - z2) * ozi * ozi);
shape(6,0) = - y * z * ozi;
shape(6,1) = - x * z * ozi;
shape(6,2) = x * y * (1. - z2) * ozi * ozi;
shape(7,0) = y * z * ozi;
shape(7,1) = - ox * z * ozi;
shape(7,2) = y * (1. - x * (1. - z2) * ozi * ozi);
}
void Nedelec1PyrFiniteElement::CalcCurlShape(const IntegrationPoint &ip,
DenseMatrix &curl_shape)
const
{
real_t x = ip.x, y = ip.y, z = ip.z, z2 = 2. * z;
real_t ox = 1. - x - z, oy = 1. - y - z, oz = 1. - z;
real_t tol = 1e-6;
if (oz <= tol)
{
// We must return the limit of the basis function derivatives as z->1.
// In order to remain inside the pyramid in this limit the x and y
// coordinates must be approaching 0. The resulting limiting basis
// function values are:
curl_shape(0,0) = - 0.5;
curl_shape(0,1) = - 1.5;
curl_shape(0,2) = 1.;
curl_shape(1,0) = 0.5;
curl_shape(1,1) = - 0.5;
curl_shape(1,2) = 1.;
curl_shape(2,0) = 0.5;
curl_shape(2,1) = - 0.5;
curl_shape(2,2) = - 1.;
curl_shape(3,0) = 1.5;
curl_shape(3,1) = 0.5;
curl_shape(3,2) = - 1.;
curl_shape(4,0) = - 1.;
curl_shape(4,1) = 1.;
curl_shape(4,2) = 0.;
curl_shape(5,0) = - 1.;
curl_shape(5,1) = - 1.;
curl_shape(5,2) = 0.;
curl_shape(6,0) = 1.;
curl_shape(6,1) = - 1.;
curl_shape(6,2) = 0.;
curl_shape(7,0) = 1.;
curl_shape(7,1) = 1.;
curl_shape(7,2) = 0.;
return;
}
real_t ozi = 1. / oz;
curl_shape(0,0) = - x * ozi;
curl_shape(0,1) = - 2. + y * ozi;
curl_shape(0,2) = 1.;
curl_shape(1,0) = x * ozi;
curl_shape(1,1) = - y * ozi;
curl_shape(1,2) = 1.;
curl_shape(2,0) = x * ozi;
curl_shape(2,1) = - y * ozi;
curl_shape(2,2) = - 1.;
curl_shape(3,0) = (2. - x - z2) * ozi;
curl_shape(3,1) = y * ozi;
curl_shape(3,2) = - 1.;
curl_shape(4,0) = - 2. * ox * ozi;
curl_shape(4,1) = 2. * oy * ozi;
curl_shape(4,2) = 0.;
curl_shape(5,0) = - 2. * x * ozi;
curl_shape(5,1) = - 2. * oy * ozi;
curl_shape(5,2) = 0.;
curl_shape(6,0) = 2. * x * ozi;
curl_shape(6,1) = - 2. * y * ozi;
curl_shape(6,2) = 0.;
curl_shape(7,0) = 2. * ox * ozi;
curl_shape(7,1) = 2. * y * ozi;
curl_shape(7,2) = 0.;
}
const real_t Nedelec1PyrFiniteElement::tk[8][3] =
{{1,0,0}, {0,1,0}, {1,0,0}, {0,1,0}, {0,0,1}, {-1,0,1}, {-1,-1,1}, {0,-1,1}};
void Nedelec1PyrFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < dof; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < dof; j++)
{
real_t d = ( vshape(j,0)*tk[k][0] + vshape(j,1)*tk[k][1] +
vshape(j,2)*tk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "Nedelec1PyrFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
const DenseMatrix &J = Trans.Jacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < dof; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = J tk
vk[0] = J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2];
vk[1] = J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2];
vk[2] = J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2];
for (j = 0; j < dof; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void Nedelec1PyrFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < dof; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
const DenseMatrix &J = Trans.Jacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t J tk
dofs(k) =
vk[0] * ( J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2] ) +
vk[1] * ( J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2] ) +
vk[2] * ( J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2] );
}
}
void Nedelec1PyrFiniteElement::ProjectGrad(const FiniteElement &fe,
ElementTransformation &Trans,
DenseMatrix &grad) const
{
DenseMatrix dshape(fe.GetDof(), 3);
Vector grad_k(fe.GetDof());
grad.SetSize(dof, fe.GetDof());
for (int k = 0; k < dof; k++)
{
fe.CalcDShape(Nodes.IntPoint(k), dshape);
dshape.Mult(tk[k], grad_k);
for (int j = 0; j < grad_k.Size(); j++)
{
grad(k,j) = (fabs(grad_k(j)) < 1e-12) ? 0.0 : grad_k(j);
}
}
}
Nedelec2PyrFiniteElement::Nedelec2PyrFiniteElement()
: VectorFiniteElement(3, Geometry::PYRAMID, 28, 2, H_CURL, FunctionSpace::Uk)
{
const real_t *eop = poly1d.OpenPoints(2 - 1);
const real_t fop = 1. / 3.;
// not real nodes ...
Nodes.IntPoint(0).Set3(eop[0], 0., 0.);
Nodes.IntPoint(1).Set3(eop[1], 0., 0.);
Nodes.IntPoint(2).Set3(1.0, eop[0], 0.);
Nodes.IntPoint(3).Set3(1.0, eop[1], 0.);
Nodes.IntPoint(4).Set3(eop[0], 1.0, 0.);
Nodes.IntPoint(5).Set3(eop[1], 1.0, 0.);
Nodes.IntPoint(6).Set3(0., eop[0], 0.);
Nodes.IntPoint(7).Set3(0., eop[1], 0.);
Nodes.IntPoint(8).Set3(0., 0., eop[0]);
Nodes.IntPoint(9).Set3(0., 0., eop[1]);
Nodes.IntPoint(10).Set3(eop[1], 0., eop[0]);
Nodes.IntPoint(11).Set3(eop[0], 0., eop[1]);
Nodes.IntPoint(12).Set3(eop[1], eop[1], eop[0]);
Nodes.IntPoint(13).Set3(eop[0], eop[0], eop[1]);
Nodes.IntPoint(14).Set3(0., eop[1], eop[0]);
Nodes.IntPoint(15).Set3(0., eop[0], eop[1]);
Nodes.IntPoint(16).Set3(eop[0], 0.5, 0.);
Nodes.IntPoint(17).Set3(eop[1], 0.5, 0.);
Nodes.IntPoint(18).Set3(0.5, eop[0], 0.);
Nodes.IntPoint(19).Set3(0.5, eop[1], 0.);
Nodes.IntPoint(20).Set3(fop, 0., fop);
Nodes.IntPoint(21).Set3(fop, 0., fop);
Nodes.IntPoint(22).Set3(2.*fop, fop, fop);
Nodes.IntPoint(23).Set3(2.*fop, fop, fop);
Nodes.IntPoint(24).Set3(fop, 2.*fop, fop);
Nodes.IntPoint(25).Set3(fop, 2.*fop, fop);
Nodes.IntPoint(26).Set3(0., fop, fop);
Nodes.IntPoint(27).Set3(0., fop, fop);
{
int n = 28;
DenseMatrix I(n,n);
DenseMatrix vecs(n,3);
I = 0.0;
for (int i=0; i<n; i++)
{
CalcVShape(Nodes.IntPoint(i), vecs);
for (int j=0; j<n; j++)
{
I(j,i) = vecs(j,0)*tk[i][0]+vecs(j,1)*tk[i][1]+vecs(j,2)*tk[i][2];
}
}
}
}
void Nedelec2PyrFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
shape = 0.0;
const real_t one = 1.0;
const real_t x = ip.x, y = ip.y, z = ip.z;
const real_t ox = one - x - z, oy = one - y - z, oz = one - z;
const real_t sq3 = sqrt(3.0);
const real_t tol = 1e-6;
if (oz <= tol)
{
// We must return the limit of the basis functions as z->1. In order to
// remain inside the pyramid in this limit the x and y coordinates must
// be approaching 0. The resulting limiting basis function values are:
shape(0,0) = 0.;
shape(0,1) = 0.;
shape(0,2) = 0.;
shape(1,0) = 0.;
shape(1,1) = 0.;
shape(1,2) = 0.;
shape(2,0) = 0.;
shape(2,1) = 0.;
shape(2,2) = 0.;
shape(3,0) = 0.;
shape(3,1) = 0.;
shape(3,2) = 0.;
shape(4,0) = 0.;
shape(4,1) = 0.;
shape(4,2) = 0.;
shape(5,0) = 0.;
shape(5,1) = 0.;
shape(5,2) = 0.;
shape(6,0) = 0.;
shape(6,1) = 0.;
shape(6,2) = 0.;
shape(7,0) = 0.;
shape(7,1) = 0.;
shape(7,2) = 0.;
return;
}
const real_t ozi = one / oz;
const real_t me0120[3] = {oy, 0., x * oy * ozi};
const real_t me1120[3] = {(x - ox) * oy, 0., (x - ox) * x * oy * ozi};
const real_t me0121[3] = {y, 0., x * y * ozi};
const real_t me1121[3] = {(x - ox) * y, 0., (x - ox) * x * y * ozi};
const real_t me0210[3] = {0., ox, ox * y * ozi};
const real_t me1210[3] = {0., ox * (y - oy), ox * y * (y - oy) * ozi};
const real_t me0211[3] = {0., x, x * y * ozi};
const real_t me1211[3] = {0., x * (y - oy), x * y * (y - oy) * ozi};
const real_t te01[3] = {oy * z * ozi, ox * z * ozi,
(ox * oy + (x * oy + ox * y) * z) * ozi * ozi
};
const real_t te11[3] = {oy * z * (z * oz - ox * oy) * ozi * ozi,
ox * z * (z * oz - ox * oy) * ozi * ozi,
(ox * oy + z * (x * oy + ox * y)) *
(z * oz - ox * oy) * ozi * ozi * ozi
};
const real_t te02[3] = {-oy * z * ozi, x * z * ozi,
x * (y * z + oy * oz) * ozi * ozi
};
const real_t te12[3] = {oy * z * (x * oy - z * oz) * ozi * ozi,
-x * z * (x * oy - z * oz) * ozi * ozi,
-x * (y * z + oy * oz) * (x * oy - z * oz)
* ozi * ozi * ozi
};
const real_t te03[3] = {-y * z * ozi, -x * z * ozi,
x * y * (one - 2_r * z) * ozi * ozi
};
const real_t te13[3] = {y * z * (x * y - z * oz) * ozi * ozi,
x * z * (x * y - z * oz) * ozi * ozi,
-x * y * (one - 2_r * z) * (x * y - z * oz)
* ozi * ozi * ozi
};
const real_t te04[3] = {y * z * ozi, -ox * z * ozi,
y * (x * z + ox * oz) * ozi * ozi
};
const real_t te14[3] = {-y * z * (ox * y - z * oz) * ozi * ozi,
ox * z * (ox * y - z * oz) * ozi * ozi,
-y * (x * z + ox * oz) * (ox * y - z * oz)
* ozi * ozi * ozi
};
const real_t qI02[3] = {-y * oy * ozi, 0., -x * y * oy * ozi * ozi};
const real_t qI12[3] = {-(x - ox) * y * oy * ozi * ozi, 0.,
-(x - ox) * x * y * oy * ozi * ozi * ozi
};
const real_t qII02[3] = {0., -x * ox * ozi, -x * y * ox * ozi * ozi};
const real_t qII12[3] = {0., -x * ox * (y - oy) * ozi * ozi,
-x * ox * y * (y - oy) * ozi * ozi * ozi
};
const real_t tI120[3] = {oy * z, 0., x * oy * z * ozi};
const real_t tI121[3] = {y * z, 0., x * y * z * ozi};
const real_t tI210[3] = {0., ox * z, ox * y * z * ozi};
const real_t tI211[3] = {0., x * z, x * y * z * ozi};
const real_t tII120[3] = {-ox * oy * z * ozi, 0., x * ox * oy * ozi};
const real_t tII121[3] = {-ox * y * z * ozi, 0., x * ox * y * ozi};
const real_t tII210[3] = {0., -ox * oy * z * ozi, ox * y * oy * ozi};
const real_t tII211[3] = {0., -x * oy * z * ozi, x * y * oy * ozi};
// Edge 0,1
for (int d=0; d<3; d++)
{
shape(0,d) = 0.5 * me0120[d] + qI02[d]
- sq3 * (0.5 * me1120[d] + qI12[d]) - 1.5 * tI120[d];
}
for (int d=0; d<3; d++)
{
shape(1,d) = 0.5 * me0120[d] + qI02[d]
+ sq3 * (0.5 * me1120[d] + qI12[d]) - 1.5 * tI120[d];
}
// Edge 1,2
for (int d=0; d<3; d++)
{
shape(2,d) = 0.5 * me0211[d] + qII02[d]
- sq3 * (0.5 * me1211[d] + qII12[d]) - 1.5 * tI211[d];
}
for (int d=0; d<3; d++)
{
shape(3,d) = 0.5 * me0211[d] + qII02[d]
+ sq3 * (0.5 * me1211[d] + qII12[d]) - 1.5 * tI211[d];
}
// Edge 3,2
for (int d=0; d<3; d++)
{
shape(4,d) = 0.5 * me0121[d] + qI02[d]
- sq3 * (0.5 * me1121[d] + qI12[d]) - 1.5 * tI121[d];
}
for (int d=0; d<3; d++)
{
shape(5,d) = 0.5 * me0121[d] + qI02[d]
+ sq3 * (0.5 * me1121[d] + qI12[d]) - 1.5 * tI121[d];
}
// Edge 0,3
for (int d=0; d<3; d++)
{
shape(6,d) = 0.5 * me0210[d] + qII02[d]
- sq3 * (0.5 * me1210[d] + qII12[d]) - 1.5 * tI210[d];
}
for (int d=0; d<3; d++)
{
shape(7,d) = 0.5 * me0210[d] + qII02[d]
+ sq3 * (0.5 * me1210[d] + qII12[d]) - 1.5 * tI210[d];
}
// Edge 0,4
for (int d=0; d<3; d++)
{
shape(8,d) = 0.5 * te01[d] - sq3 * 0.5 * te11[d]
- 1.5 * (tI120[d] + tII120[d] + tI210[d] + tII210[d]);
}
for (int d=0; d<3; d++)
{
shape(9,d) = 0.5 * te01[d] + sq3 * 0.5 * te11[d]
- 1.5 * (tI120[d] + tII120[d] + tI210[d] + tII210[d]);
}
// Edge 1,4
for (int d=0; d<3; d++)
{
shape(10,d) = 0.5 * te02[d] - sq3 * 0.5 * te12[d]
- 1.5 * (tII120[d] + tI211[d] + tII211[d]);
}
for (int d=0; d<3; d++)
{
shape(11,d) = 0.5 * te02[d] + sq3 * 0.5 * te12[d]
- 1.5 * (tII120[d] + tI211[d] + tII211[d]);
}
// Edge 2,4
for (int d=0; d<3; d++)
{
shape(12,d) = 0.5 * te03[d] - sq3 * 0.5 * te13[d]
- 1.5 * (tII211[d] + tII121[d]);
}
for (int d=0; d<3; d++)
{
shape(13,d) = 0.5 * te03[d] + sq3 * 0.5 * te13[d]
- 1.5 * (tII211[d] + tII121[d]);
}
// Edge 3,4
for (int d=0; d<3; d++)
{
shape(14,d) = 0.5 * te04[d] - sq3 * 0.5 * te14[d]
- 1.5 * (tI121[d] + tII121[d] + tII210[d]);
}
for (int d=0; d<3; d++)
{
shape(15,d) = 0.5 * te04[d] + sq3 * 0.5 * te14[d]
- 1.5 * (tI121[d] + tII121[d] + tII210[d]);
}
// Quadrilateral face
for (int d=0; d<3; d++)
{
shape(16,d) = -2. * qI02[d] + 2. * sq3 * qI12[d];
}
for (int d=0; d<3; d++)
{
shape(17,d) = -2. * qI02[d] - 2. * sq3 * qI12[d];
}
for (int d=0; d<3; d++)
{
shape(18,d) = 2. * qII02[d] - 2. * sq3 * qII12[d];
}
for (int d=0; d<3; d++)
{
shape(19,d) = 2. * qII02[d] + 2. * sq3 * qII12[d];
}
// Triangular face 0,1,4
for (int d=0; d<3; d++)
{
shape(20,d) = 3. * tI120[d] - 3. * tII120[d];
}
for (int d=0; d<3; d++)
{
shape(21,d) = 3. * tI120[d] + 6. * tII120[d];
}
// Triangular face 1,2,4
for (int d=0; d<3; d++)
{
shape(22,d) = 3. * tI211[d] - 3. * tII211[d];
}
for (int d=0; d<3; d++)
{
shape(23,d) = 3. * tI211[d] + 6. * tII211[d];
}
// Triangular face 2,3,4
for (int d=0; d<3; d++)
{
shape(24,d) = -6. * tI121[d] - 3. * tII121[d];
}
for (int d=0; d<3; d++)
{
shape(25,d) = 3. * tI121[d] + 6. * tII121[d];
}
// Triangular face 3,0,4
for (int d=0; d<3; d++)
{
shape(26,d) = -6. * tI210[d] - 3. * tII210[d];
}
for (int d=0; d<3; d++)
{
shape(27,d) = 3. * tI210[d] + 6. * tII210[d];
}
}
void Nedelec2PyrFiniteElement::CalcCurlShape(const IntegrationPoint &ip,
DenseMatrix &curl_shape)
const
{
const real_t one = 1.0;
const real_t x = ip.x, y = ip.y, z = ip.z, z2 = 2. * z;
const real_t ox = one - x - z, oy = one - y - z, oz = one - z;
const real_t tol = 1e-6;
if (oz <= tol)
{
// We must return the limit of the basis function derivatives as z->1.
// In order to remain inside the pyramid in this limit the x and y
// coordinates must be approaching 0. The resulting limiting basis
// function values are:
curl_shape(0,0) = 0.;
curl_shape(0,1) = - 2.;
curl_shape(0,2) = 1.;
curl_shape(1,0) = 0.;
curl_shape(1,1) = 0.;
curl_shape(1,2) = 1.;
curl_shape(2,0) = 0.;
curl_shape(2,1) = 0.;
curl_shape(2,2) = - 1.;
curl_shape(3,0) = 2.;
curl_shape(3,1) = 0.;
curl_shape(3,2) = - 1.;
curl_shape(4,0) = - 2.;
curl_shape(4,1) = 2.;
curl_shape(4,2) = 0.;
curl_shape(5,0) = 0.;
curl_shape(5,1) = - 2.;
curl_shape(5,2) = 0.;
curl_shape(6,0) = 0.;
curl_shape(6,1) = 0.;
curl_shape(6,2) = 0.;
curl_shape(7,0) = 2.;
curl_shape(7,1) = 0.;
curl_shape(7,2) = 0.;
return;
}
real_t ozi = one / oz;
curl_shape(0,0) = - x * ozi;
curl_shape(0,1) = - 2. + y * ozi;
curl_shape(0,2) = 1.;
curl_shape(1,0) = x * ozi;
curl_shape(1,1) = - y * ozi;
curl_shape(1,2) = 1.;
curl_shape(2,0) = x * ozi;
curl_shape(2,1) = - y * ozi;
curl_shape(2,2) = - 1.;
curl_shape(3,0) = (2. - x - z2) * ozi;
curl_shape(3,1) = y * ozi;
curl_shape(3,2) = - 1.;
curl_shape(4,0) = - 2. * ox * ozi;
curl_shape(4,1) = 2. * oy * ozi;
curl_shape(4,2) = 0.;
curl_shape(5,0) = - 2. * x * ozi;
curl_shape(5,1) = - 2. * oy * ozi;
curl_shape(5,2) = 0.;
curl_shape(6,0) = 2. * x * ozi;
curl_shape(6,1) = - 2. * y * ozi;
curl_shape(6,2) = 0.;
curl_shape(7,0) = 2. * ox * ozi;
curl_shape(7,1) = 2. * y * ozi;
curl_shape(7,2) = 0.;
}
const real_t Nedelec2PyrFiniteElement::tk[28][3] =
{
{1,0,0}, {1,0,0}, {0,1,0}, {0,1,0},
{1,0,0}, {1,0,0}, {0,1,0}, {0,1,0},
{0,0,1}, {0,0,1}, {-1,0,1}, {-1,0,1},
{-1,-1,1}, {-1,-1,1}, {0,-1,1}, {0,-1,1},
{1,0,0}, {1,0,0}, {0,-1,0}, {0,-1,0},
{1,0,0}, {0,0,1}, {0,1,0}, {-1,0,1},
{-1,0,0}, {-1,-1,1}, {0,-1,0}, {0,-1,1}
};
void Nedelec2PyrFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < dof; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < dof; j++)
{
real_t d = ( vshape(j,0)*tk[k][0] + vshape(j,1)*tk[k][1] +
vshape(j,2)*tk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "Nedelec1PyrFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
const DenseMatrix &J = Trans.Jacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < dof; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = J tk
vk[0] = J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2];
vk[1] = J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2];
vk[2] = J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2];
for (j = 0; j < dof; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void Nedelec2PyrFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < dof; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
const DenseMatrix &J = Trans.Jacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t J tk
dofs(k) =
vk[0] * ( J(0,0)*tk[k][0]+J(0,1)*tk[k][1]+J(0,2)*tk[k][2] ) +
vk[1] * ( J(1,0)*tk[k][0]+J(1,1)*tk[k][1]+J(1,2)*tk[k][2] ) +
vk[2] * ( J(2,0)*tk[k][0]+J(2,1)*tk[k][1]+J(2,2)*tk[k][2] );
}
}
void Nedelec2PyrFiniteElement::ProjectGrad(const FiniteElement &fe,
ElementTransformation &Trans,
DenseMatrix &grad) const
{
DenseMatrix dshape(fe.GetDof(), 3);
Vector grad_k(fe.GetDof());
grad.SetSize(dof, fe.GetDof());
for (int k = 0; k < dof; k++)
{
fe.CalcDShape(Nodes.IntPoint(k), dshape);
dshape.Mult(tk[k], grad_k);
for (int j = 0; j < grad_k.Size(); j++)
{
grad(k,j) = (fabs(grad_k(j)) < 1e-12) ? 0.0 : grad_k(j);
}
}
}
RT0HexFiniteElement::RT0HexFiniteElement()
: VectorFiniteElement(3, Geometry::CUBE, 6, 1, H_DIV, FunctionSpace::Qk)
{
// not real nodes ...
// z = 0, y = 0, x = 1, y = 1, x = 0, z = 1
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.5;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 0.5;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.5;
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 0.5;
Nodes.IntPoint(2).z = 0.5;
Nodes.IntPoint(3).x = 0.5;
Nodes.IntPoint(3).y = 1.0;
Nodes.IntPoint(3).z = 0.5;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.5;
Nodes.IntPoint(4).z = 0.5;
Nodes.IntPoint(5).x = 0.5;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(5).z = 1.0;
}
void RT0HexFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
// z = 0
shape(0,0) = 0.;
shape(0,1) = 0.;
shape(0,2) = z - 1.;
// y = 0
shape(1,0) = 0.;
shape(1,1) = y - 1.;
shape(1,2) = 0.;
// x = 1
shape(2,0) = x;
shape(2,1) = 0.;
shape(2,2) = 0.;
// y = 1
shape(3,0) = 0.;
shape(3,1) = y;
shape(3,2) = 0.;
// x = 0
shape(4,0) = x - 1.;
shape(4,1) = 0.;
shape(4,2) = 0.;
// z = 1
shape(5,0) = 0.;
shape(5,1) = 0.;
shape(5,2) = z;
}
void RT0HexFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
divshape(0) = 1.;
divshape(1) = 1.;
divshape(2) = 1.;
divshape(3) = 1.;
divshape(4) = 1.;
divshape(5) = 1.;
}
const real_t RT0HexFiniteElement::nk[6][3] =
{{0,0,-1}, {0,-1,0}, {1,0,0}, {0,1,0}, {-1,0,0}, {0,0,1}};
void RT0HexFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 6; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 6; j++)
{
real_t d = ( vshape(j,0)*nk[k][0] + vshape(j,1)*nk[k][1] +
vshape(j,2)*nk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT0HexFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < 6; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2];
vk[2] = Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2];
for (j = 0; j < 6; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT0HexFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < 6; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) =
vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2] ) +
vk[2] * ( Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2] );
}
}
RT1HexFiniteElement::RT1HexFiniteElement()
: VectorFiniteElement(3, Geometry::CUBE, 36, 2, H_DIV,
FunctionSpace::Qk)
{
// z = 0
Nodes.IntPoint(2).x = 1./3.;
Nodes.IntPoint(2).y = 1./3.;
Nodes.IntPoint(2).z = 0.0;
Nodes.IntPoint(3).x = 2./3.;
Nodes.IntPoint(3).y = 1./3.;
Nodes.IntPoint(3).z = 0.0;
Nodes.IntPoint(0).x = 1./3.;
Nodes.IntPoint(0).y = 2./3.;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 2./3.;
Nodes.IntPoint(1).y = 2./3.;
Nodes.IntPoint(1).z = 0.0;
// y = 0
Nodes.IntPoint(4).x = 1./3.;
Nodes.IntPoint(4).y = 0.0;
Nodes.IntPoint(4).z = 1./3.;
Nodes.IntPoint(5).x = 2./3.;
Nodes.IntPoint(5).y = 0.0;
Nodes.IntPoint(5).z = 1./3.;
Nodes.IntPoint(6).x = 1./3.;
Nodes.IntPoint(6).y = 0.0;
Nodes.IntPoint(6).z = 2./3.;
Nodes.IntPoint(7).x = 2./3.;
Nodes.IntPoint(7).y = 0.0;
Nodes.IntPoint(7).z = 2./3.;
// x = 1
Nodes.IntPoint(8).x = 1.0;
Nodes.IntPoint(8).y = 1./3.;
Nodes.IntPoint(8).z = 1./3.;
Nodes.IntPoint(9).x = 1.0;
Nodes.IntPoint(9).y = 2./3.;
Nodes.IntPoint(9).z = 1./3.;
Nodes.IntPoint(10).x = 1.0;
Nodes.IntPoint(10).y = 1./3.;
Nodes.IntPoint(10).z = 2./3.;
Nodes.IntPoint(11).x = 1.0;
Nodes.IntPoint(11).y = 2./3.;
Nodes.IntPoint(11).z = 2./3.;
// y = 1
Nodes.IntPoint(13).x = 1./3.;
Nodes.IntPoint(13).y = 1.0;
Nodes.IntPoint(13).z = 1./3.;
Nodes.IntPoint(12).x = 2./3.;
Nodes.IntPoint(12).y = 1.0;
Nodes.IntPoint(12).z = 1./3.;
Nodes.IntPoint(15).x = 1./3.;
Nodes.IntPoint(15).y = 1.0;
Nodes.IntPoint(15).z = 2./3.;
Nodes.IntPoint(14).x = 2./3.;
Nodes.IntPoint(14).y = 1.0;
Nodes.IntPoint(14).z = 2./3.;
// x = 0
Nodes.IntPoint(17).x = 0.0;
Nodes.IntPoint(17).y = 1./3.;
Nodes.IntPoint(17).z = 1./3.;
Nodes.IntPoint(16).x = 0.0;
Nodes.IntPoint(16).y = 2./3.;
Nodes.IntPoint(16).z = 1./3.;
Nodes.IntPoint(19).x = 0.0;
Nodes.IntPoint(19).y = 1./3.;
Nodes.IntPoint(19).z = 2./3.;
Nodes.IntPoint(18).x = 0.0;
Nodes.IntPoint(18).y = 2./3.;
Nodes.IntPoint(18).z = 2./3.;
// z = 1
Nodes.IntPoint(20).x = 1./3.;
Nodes.IntPoint(20).y = 1./3.;
Nodes.IntPoint(20).z = 1.0;
Nodes.IntPoint(21).x = 2./3.;
Nodes.IntPoint(21).y = 1./3.;
Nodes.IntPoint(21).z = 1.0;
Nodes.IntPoint(22).x = 1./3.;
Nodes.IntPoint(22).y = 2./3.;
Nodes.IntPoint(22).z = 1.0;
Nodes.IntPoint(23).x = 2./3.;
Nodes.IntPoint(23).y = 2./3.;
Nodes.IntPoint(23).z = 1.0;
// x = 0.5 (interior)
Nodes.IntPoint(24).x = 0.5;
Nodes.IntPoint(24).y = 1./3.;
Nodes.IntPoint(24).z = 1./3.;
Nodes.IntPoint(25).x = 0.5;
Nodes.IntPoint(25).y = 1./3.;
Nodes.IntPoint(25).z = 2./3.;
Nodes.IntPoint(26).x = 0.5;
Nodes.IntPoint(26).y = 2./3.;
Nodes.IntPoint(26).z = 1./3.;
Nodes.IntPoint(27).x = 0.5;
Nodes.IntPoint(27).y = 2./3.;
Nodes.IntPoint(27).z = 2./3.;
// y = 0.5 (interior)
Nodes.IntPoint(28).x = 1./3.;
Nodes.IntPoint(28).y = 0.5;
Nodes.IntPoint(28).z = 1./3.;
Nodes.IntPoint(29).x = 1./3.;
Nodes.IntPoint(29).y = 0.5;
Nodes.IntPoint(29).z = 2./3.;
Nodes.IntPoint(30).x = 2./3.;
Nodes.IntPoint(30).y = 0.5;
Nodes.IntPoint(30).z = 1./3.;
Nodes.IntPoint(31).x = 2./3.;
Nodes.IntPoint(31).y = 0.5;
Nodes.IntPoint(31).z = 2./3.;
// z = 0.5 (interior)
Nodes.IntPoint(32).x = 1./3.;
Nodes.IntPoint(32).y = 1./3.;
Nodes.IntPoint(32).z = 0.5;
Nodes.IntPoint(33).x = 1./3.;
Nodes.IntPoint(33).y = 2./3.;
Nodes.IntPoint(33).z = 0.5;
Nodes.IntPoint(34).x = 2./3.;
Nodes.IntPoint(34).y = 1./3.;
Nodes.IntPoint(34).z = 0.5;
Nodes.IntPoint(35).x = 2./3.;
Nodes.IntPoint(35).y = 2./3.;
Nodes.IntPoint(35).z = 0.5;
}
void RT1HexFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
// z = 0
shape(2,0) = 0.;
shape(2,1) = 0.;
shape(2,2) = -(1. - 3.*z + 2.*z*z)*( 2. - 3.*x)*( 2. - 3.*y);
shape(3,0) = 0.;
shape(3,1) = 0.;
shape(3,2) = -(1. - 3.*z + 2.*z*z)*(-1. + 3.*x)*( 2. - 3.*y);
shape(0,0) = 0.;
shape(0,1) = 0.;
shape(0,2) = -(1. - 3.*z + 2.*z*z)*( 2. - 3.*x)*(-1. + 3.*y);
shape(1,0) = 0.;
shape(1,1) = 0.;
shape(1,2) = -(1. - 3.*z + 2.*z*z)*(-1. + 3.*x)*(-1. + 3.*y);
// y = 0
shape(4,0) = 0.;
shape(4,1) = -(1. - 3.*y + 2.*y*y)*( 2. - 3.*x)*( 2. - 3.*z);
shape(4,2) = 0.;
shape(5,0) = 0.;
shape(5,1) = -(1. - 3.*y + 2.*y*y)*(-1. + 3.*x)*( 2. - 3.*z);
shape(5,2) = 0.;
shape(6,0) = 0.;
shape(6,1) = -(1. - 3.*y + 2.*y*y)*( 2. - 3.*x)*(-1. + 3.*z);
shape(6,2) = 0.;
shape(7,0) = 0.;
shape(7,1) = -(1. - 3.*y + 2.*y*y)*(-1. + 3.*x)*(-1. + 3.*z);
shape(7,2) = 0.;
// x = 1
shape(8,0) = (-x + 2.*x*x)*( 2. - 3.*y)*( 2. - 3.*z);
shape(8,1) = 0.;
shape(8,2) = 0.;
shape(9,0) = (-x + 2.*x*x)*(-1. + 3.*y)*( 2. - 3.*z);
shape(9,1) = 0.;
shape(9,2) = 0.;
shape(10,0) = (-x + 2.*x*x)*( 2. - 3.*y)*(-1. + 3.*z);
shape(10,1) = 0.;
shape(10,2) = 0.;
shape(11,0) = (-x + 2.*x*x)*(-1. + 3.*y)*(-1. + 3.*z);
shape(11,1) = 0.;
shape(11,2) = 0.;
// y = 1
shape(13,0) = 0.;
shape(13,1) = (-y + 2.*y*y)*( 2. - 3.*x)*( 2. - 3.*z);
shape(13,2) = 0.;
shape(12,0) = 0.;
shape(12,1) = (-y + 2.*y*y)*(-1. + 3.*x)*( 2. - 3.*z);
shape(12,2) = 0.;
shape(15,0) = 0.;
shape(15,1) = (-y + 2.*y*y)*( 2. - 3.*x)*(-1. + 3.*z);
shape(15,2) = 0.;
shape(14,0) = 0.;
shape(14,1) = (-y + 2.*y*y)*(-1. + 3.*x)*(-1. + 3.*z);
shape(14,2) = 0.;
// x = 0
shape(17,0) = -(1. - 3.*x + 2.*x*x)*( 2. - 3.*y)*( 2. - 3.*z);
shape(17,1) = 0.;
shape(17,2) = 0.;
shape(16,0) = -(1. - 3.*x + 2.*x*x)*(-1. + 3.*y)*( 2. - 3.*z);
shape(16,1) = 0.;
shape(16,2) = 0.;
shape(19,0) = -(1. - 3.*x + 2.*x*x)*( 2. - 3.*y)*(-1. + 3.*z);
shape(19,1) = 0.;
shape(19,2) = 0.;
shape(18,0) = -(1. - 3.*x + 2.*x*x)*(-1. + 3.*y)*(-1. + 3.*z);
shape(18,1) = 0.;
shape(18,2) = 0.;
// z = 1
shape(20,0) = 0.;
shape(20,1) = 0.;
shape(20,2) = (-z + 2.*z*z)*( 2. - 3.*x)*( 2. - 3.*y);
shape(21,0) = 0.;
shape(21,1) = 0.;
shape(21,2) = (-z + 2.*z*z)*(-1. + 3.*x)*( 2. - 3.*y);
shape(22,0) = 0.;
shape(22,1) = 0.;
shape(22,2) = (-z + 2.*z*z)*( 2. - 3.*x)*(-1. + 3.*y);
shape(23,0) = 0.;
shape(23,1) = 0.;
shape(23,2) = (-z + 2.*z*z)*(-1. + 3.*x)*(-1. + 3.*y);
// x = 0.5 (interior)
shape(24,0) = (4.*x - 4.*x*x)*( 2. - 3.*y)*( 2. - 3.*z);
shape(24,1) = 0.;
shape(24,2) = 0.;
shape(25,0) = (4.*x - 4.*x*x)*( 2. - 3.*y)*(-1. + 3.*z);
shape(25,1) = 0.;
shape(25,2) = 0.;
shape(26,0) = (4.*x - 4.*x*x)*(-1. + 3.*y)*( 2. - 3.*z);
shape(26,1) = 0.;
shape(26,2) = 0.;
shape(27,0) = (4.*x - 4.*x*x)*(-1. + 3.*y)*(-1. + 3.*z);
shape(27,1) = 0.;
shape(27,2) = 0.;
// y = 0.5 (interior)
shape(28,0) = 0.;
shape(28,1) = (4.*y - 4.*y*y)*( 2. - 3.*x)*( 2. - 3.*z);
shape(28,2) = 0.;
shape(29,0) = 0.;
shape(29,1) = (4.*y - 4.*y*y)*( 2. - 3.*x)*(-1. + 3.*z);
shape(29,2) = 0.;
shape(30,0) = 0.;
shape(30,1) = (4.*y - 4.*y*y)*(-1. + 3.*x)*( 2. - 3.*z);
shape(30,2) = 0.;
shape(31,0) = 0.;
shape(31,1) = (4.*y - 4.*y*y)*(-1. + 3.*x)*(-1. + 3.*z);
shape(31,2) = 0.;
// z = 0.5 (interior)
shape(32,0) = 0.;
shape(32,1) = 0.;
shape(32,2) = (4.*z - 4.*z*z)*( 2. - 3.*x)*( 2. - 3.*y);
shape(33,0) = 0.;
shape(33,1) = 0.;
shape(33,2) = (4.*z - 4.*z*z)*( 2. - 3.*x)*(-1. + 3.*y);
shape(34,0) = 0.;
shape(34,1) = 0.;
shape(34,2) = (4.*z - 4.*z*z)*(-1. + 3.*x)*( 2. - 3.*y);
shape(35,0) = 0.;
shape(35,1) = 0.;
shape(35,2) = (4.*z - 4.*z*z)*(-1. + 3.*x)*(-1. + 3.*y);
}
void RT1HexFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
real_t x = ip.x, y = ip.y, z = ip.z;
// z = 0
divshape(2) = -(-3. + 4.*z)*( 2. - 3.*x)*( 2. - 3.*y);
divshape(3) = -(-3. + 4.*z)*(-1. + 3.*x)*( 2. - 3.*y);
divshape(0) = -(-3. + 4.*z)*( 2. - 3.*x)*(-1. + 3.*y);
divshape(1) = -(-3. + 4.*z)*(-1. + 3.*x)*(-1. + 3.*y);
// y = 0
divshape(4) = -(-3. + 4.*y)*( 2. - 3.*x)*( 2. - 3.*z);
divshape(5) = -(-3. + 4.*y)*(-1. + 3.*x)*( 2. - 3.*z);
divshape(6) = -(-3. + 4.*y)*( 2. - 3.*x)*(-1. + 3.*z);
divshape(7) = -(-3. + 4.*y)*(-1. + 3.*x)*(-1. + 3.*z);
// x = 1
divshape(8) = (-1. + 4.*x)*( 2. - 3.*y)*( 2. - 3.*z);
divshape(9) = (-1. + 4.*x)*(-1. + 3.*y)*( 2. - 3.*z);
divshape(10) = (-1. + 4.*x)*( 2. - 3.*y)*(-1. + 3.*z);
divshape(11) = (-1. + 4.*x)*(-1. + 3.*y)*(-1. + 3.*z);
// y = 1
divshape(13) = (-1. + 4.*y)*( 2. - 3.*x)*( 2. - 3.*z);
divshape(12) = (-1. + 4.*y)*(-1. + 3.*x)*( 2. - 3.*z);
divshape(15) = (-1. + 4.*y)*( 2. - 3.*x)*(-1. + 3.*z);
divshape(14) = (-1. + 4.*y)*(-1. + 3.*x)*(-1. + 3.*z);
// x = 0
divshape(17) = -(-3. + 4.*x)*( 2. - 3.*y)*( 2. - 3.*z);
divshape(16) = -(-3. + 4.*x)*(-1. + 3.*y)*( 2. - 3.*z);
divshape(19) = -(-3. + 4.*x)*( 2. - 3.*y)*(-1. + 3.*z);
divshape(18) = -(-3. + 4.*x)*(-1. + 3.*y)*(-1. + 3.*z);
// z = 1
divshape(20) = (-1. + 4.*z)*( 2. - 3.*x)*( 2. - 3.*y);
divshape(21) = (-1. + 4.*z)*(-1. + 3.*x)*( 2. - 3.*y);
divshape(22) = (-1. + 4.*z)*( 2. - 3.*x)*(-1. + 3.*y);
divshape(23) = (-1. + 4.*z)*(-1. + 3.*x)*(-1. + 3.*y);
// x = 0.5 (interior)
divshape(24) = ( 4. - 8.*x)*( 2. - 3.*y)*( 2. - 3.*z);
divshape(25) = ( 4. - 8.*x)*( 2. - 3.*y)*(-1. + 3.*z);
divshape(26) = ( 4. - 8.*x)*(-1. + 3.*y)*( 2. - 3.*z);
divshape(27) = ( 4. - 8.*x)*(-1. + 3.*y)*(-1. + 3.*z);
// y = 0.5 (interior)
divshape(28) = ( 4. - 8.*y)*( 2. - 3.*x)*( 2. - 3.*z);
divshape(29) = ( 4. - 8.*y)*( 2. - 3.*x)*(-1. + 3.*z);
divshape(30) = ( 4. - 8.*y)*(-1. + 3.*x)*( 2. - 3.*z);
divshape(31) = ( 4. - 8.*y)*(-1. + 3.*x)*(-1. + 3.*z);
// z = 0.5 (interior)
divshape(32) = ( 4. - 8.*z)*( 2. - 3.*x)*( 2. - 3.*y);
divshape(33) = ( 4. - 8.*z)*( 2. - 3.*x)*(-1. + 3.*y);
divshape(34) = ( 4. - 8.*z)*(-1. + 3.*x)*( 2. - 3.*y);
divshape(35) = ( 4. - 8.*z)*(-1. + 3.*x)*(-1. + 3.*y);
}
const real_t RT1HexFiniteElement::nk[36][3] =
{
{0, 0,-1}, {0, 0,-1}, {0, 0,-1}, {0, 0,-1},
{0,-1, 0}, {0,-1, 0}, {0,-1, 0}, {0,-1, 0},
{1, 0, 0}, {1, 0, 0}, {1, 0, 0}, {1, 0, 0},
{0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0},
{-1,0, 0}, {-1,0, 0}, {-1,0, 0}, {-1,0, 0},
{0, 0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1},
{1, 0, 0}, {1, 0, 0}, {1, 0, 0}, {1, 0, 0},
{0, 1, 0}, {0, 1, 0}, {0, 1, 0}, {0, 1, 0},
{0, 0, 1}, {0, 0, 1}, {0, 0, 1}, {0, 0, 1}
};
void RT1HexFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 36; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 36; j++)
{
real_t d = ( vshape(j,0)*nk[k][0] + vshape(j,1)*nk[k][1] +
vshape(j,2)*nk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT0HexFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < 36; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2];
vk[2] = Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2];
for (j = 0; j < 36; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT1HexFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < 36; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) =
vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2] ) +
vk[2] * ( Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2] );
}
}
RT0TetFiniteElement::RT0TetFiniteElement()
: VectorFiniteElement(3, Geometry::TETRAHEDRON, 4, 1, H_DIV)
{
// not real nodes ...
Nodes.IntPoint(0).x = 0.33333333333333333333;
Nodes.IntPoint(0).y = 0.33333333333333333333;
Nodes.IntPoint(0).z = 0.33333333333333333333;
Nodes.IntPoint(1).x = 0.0;
Nodes.IntPoint(1).y = 0.33333333333333333333;
Nodes.IntPoint(1).z = 0.33333333333333333333;
Nodes.IntPoint(2).x = 0.33333333333333333333;
Nodes.IntPoint(2).y = 0.0;
Nodes.IntPoint(2).z = 0.33333333333333333333;
Nodes.IntPoint(3).x = 0.33333333333333333333;
Nodes.IntPoint(3).y = 0.33333333333333333333;
Nodes.IntPoint(3).z = 0.0;
}
void RT0TetFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x2 = 2.0*ip.x, y2 = 2.0*ip.y, z2 = 2.0*ip.z;
shape(0,0) = x2;
shape(0,1) = y2;
shape(0,2) = z2;
shape(1,0) = x2 - 2.0;
shape(1,1) = y2;
shape(1,2) = z2;
shape(2,0) = x2;
shape(2,1) = y2 - 2.0;
shape(2,2) = z2;
shape(3,0) = x2;
shape(3,1) = y2;
shape(3,2) = z2 - 2.0;
}
void RT0TetFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
divshape(0) = 6.0;
divshape(1) = 6.0;
divshape(2) = 6.0;
divshape(3) = 6.0;
}
const real_t RT0TetFiniteElement::nk[4][3] =
{{.5,.5,.5}, {-.5,0,0}, {0,-.5,0}, {0,0,-.5}};
void RT0TetFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < 4; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < 4; j++)
{
real_t d = ( vshape(j,0)*nk[k][0] + vshape(j,1)*nk[k][1] +
vshape(j,2)*nk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT0TetFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < 4; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2];
vk[2] = Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2];
for (j = 0; j < 4; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT0TetFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < 4; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) =
vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2] ) +
vk[2] * ( Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2] );
}
}
RT0WdgFiniteElement::RT0WdgFiniteElement()
: VectorFiniteElement(3, Geometry::PRISM, 5, 1, H_DIV, FunctionSpace::Pk)
{
// not real nodes ...
Nodes.IntPoint(0).x = 0.33333333333333333333;
Nodes.IntPoint(0).y = 0.33333333333333333333;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 0.33333333333333333333;
Nodes.IntPoint(1).y = 0.33333333333333333333;
Nodes.IntPoint(1).z = 1.0;
Nodes.IntPoint(2).x = 0.5;
Nodes.IntPoint(2).y = 0.0;
Nodes.IntPoint(2).z = 0.5;
Nodes.IntPoint(3).x = 0.5;
Nodes.IntPoint(3).y = 0.5;
Nodes.IntPoint(3).z = 0.5;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.5;
Nodes.IntPoint(4).z = 0.5;
}
void RT0WdgFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z2 = 2.0*ip.z;
shape(0,0) = 0.0;
shape(0,1) = 0.0;
shape(0,2) = z2 - 2.0;
shape(1,0) = 0.0;
shape(1,1) = 0.0;
shape(1,2) = z2;
shape(2,0) = x;
shape(2,1) = y - 1.0;
shape(2,2) = 0.0;
shape(3,0) = x;
shape(3,1) = y;
shape(3,2) = 0.0;
shape(4,0) = x - 1.0;
shape(4,1) = y;
shape(4,2) = 0.0;
}
void RT0WdgFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
divshape(0) = 2.0;
divshape(1) = 2.0;
divshape(2) = 2.0;
divshape(3) = 2.0;
divshape(4) = 2.0;
}
const real_t RT0WdgFiniteElement::nk[5][3] =
{{0.,0.,-.5}, {0.,0.,.5}, {0,-1.,0}, {1.,1.,0}, {-1.,0,0}};
void RT0WdgFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < dof; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < dof; j++)
{
real_t d = ( vshape(j,0)*nk[k][0] + vshape(j,1)*nk[k][1] +
vshape(j,2)*nk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT0WdgFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < dof; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2];
vk[2] = Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2];
for (j = 0; j < dof; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT0WdgFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < 5; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) =
vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2] ) +
vk[2] * ( Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2] );
}
}
void RT0WdgFiniteElement::ProjectCurl(const FiniteElement &fe,
ElementTransformation &Trans,
DenseMatrix &curl) const
{
DenseMatrix curl_shape(fe.GetDof(), dim);
Vector curl_k(fe.GetDof());
curl.SetSize(dof, fe.GetDof());
for (int k = 0; k < dof; k++)
{
fe.CalcCurlShape(Nodes.IntPoint(k), curl_shape);
curl_shape.Mult(nk[k], curl_k);
for (int j = 0; j < curl_k.Size(); j++)
{
curl(k,j) = (fabs(curl_k(j)) < 1e-12) ? 0.0 : curl_k(j);
}
}
}
RT0PyrFiniteElement::RT0PyrFiniteElement(bool rt0tets)
: VectorFiniteElement(3, Geometry::PYRAMID, 5, 1, H_DIV, FunctionSpace::Uk),
rt0(rt0tets)
{
// not real nodes ...
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.5;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 0.33333333333333333333;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.33333333333333333333;
Nodes.IntPoint(2).x = 0.66666666666666666667;
Nodes.IntPoint(2).y = 0.33333333333333333333;
Nodes.IntPoint(2).z = 0.33333333333333333333;
Nodes.IntPoint(3).x = 0.33333333333333333333;
Nodes.IntPoint(3).y = 0.66666666666666666667;
Nodes.IntPoint(3).z = 0.33333333333333333333;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.33333333333333333333;
Nodes.IntPoint(4).z = 0.33333333333333333333;
}
void RT0PyrFiniteElement::CalcVShape(const IntegrationPoint &ip,
DenseMatrix &shape) const
{
real_t x = ip.x, y = ip.y, z = ip.z, oz = 1.0 - z;
real_t x2 = 2.0*ip.x, y2 = 2.0*ip.y, z2 = 2.0*ip.z;
real_t tol = 1e-6;
if (oz <= tol)
{
// We must return the limit of the basis functions as z->1. In order to
// remain inside the pyramid in this limit the x and y coordinates must
// be approaching 0. Unfortunately we obtain different limits if we
// approach (0,0,1) from different directions. The values provided below
// are the limits of the average over the square cross section
// [0,epsilon]x[0,epsilon] at the height z = 1-epsilon as epsilon -> 0.
shape(0,0) = 0.0;
shape(0,1) = 0.0;
shape(0,2) = 0.0;
shape(1,0) = - 0.5;
shape(1,1) = - 1.5;
shape(1,2) = 1.;
shape(2,0) = 0.5;
shape(2,1) = - 0.5;
shape(2,2) = 1.0;
shape(3,0) = - 0.5;
shape(3,1) = 0.5;
shape(3,2) = 1.0;
shape(4,0) = - 1.5;
shape(4,1) = - 0.5;
shape(4,2) = 1.0;
if (!rt0)
{
for (int i=1; i<5; i++)
for (int j=0; j<3; j++)
{
shape(i, j) *= 0.5;
}
}
return;
}
real_t ozi = 1.0 / oz;
shape(0,0) = x;
shape(0,1) = y;
shape(0,2) = z - 1.;
shape(1,0) = - x * z * ozi;
shape(1,1) = (y2 + z2 - y * z - 2.0) * ozi;
shape(1,2) = z;
shape(2,0) = x * (2.0 - z) * ozi;
shape(2,1) = - y * z * ozi;
shape(2,2) = z;
shape(3,0) = - x * z * ozi;
shape(3,1) = y * (2.0 - z) * ozi;
shape(3,2) = z;
shape(4,0) = (x2 + z2 - x * z - 2.0) * ozi;
shape(4,1) = - y * z * ozi;
shape(4,2) = z;
if (!rt0)
{
for (int i=1; i<5; i++)
for (int j=0; j<3; j++)
{
shape(i, j) *= 0.5;
}
}
}
void RT0PyrFiniteElement::CalcDivShape(const IntegrationPoint &ip,
Vector &divshape) const
{
divshape(0) = 3.0;
divshape(1) = 3.0;
divshape(2) = 3.0;
divshape(3) = 3.0;
divshape(4) = 3.0;
if (!rt0)
{
for (int i=1; i<5; i++)
{
divshape(i) *= 0.5;
}
}
}
const real_t RT0PyrFiniteElement::nk[5][3] =
{{0.,0.,-1}, {0,-1,0}, {1,0,1}, {0,1,1}, {-1,0,0}};
void RT0PyrFiniteElement::GetLocalInterpolation (
ElementTransformation &Trans, DenseMatrix &I) const
{
int k, j;
#ifdef MFEM_THREAD_SAFE
DenseMatrix vshape(dof, dim);
#endif
#ifdef MFEM_DEBUG
for (k = 0; k < dof; k++)
{
CalcVShape (Nodes.IntPoint(k), vshape);
for (j = 0; j < dof; j++)
{
real_t d = ( vshape(j,0)*nk[k][0] + vshape(j,1)*nk[k][1] +
vshape(j,2)*nk[k][2] );
if (j == k) { d -= 1.0; }
if (fabs(d) > 1.0e-12)
{
mfem::err << "RT0PyrFiniteElement::GetLocalInterpolation (...)\n"
" k = " << k << ", j = " << j << ", d = " << d << endl;
mfem_error();
}
}
}
#endif
IntegrationPoint ip;
ip.x = ip.y = ip.z = 0.0;
Trans.SetIntPoint (&ip);
// Trans must be linear
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
real_t vk[3];
Vector xk (vk, 3);
for (k = 0; k < dof; k++)
{
Trans.Transform (Nodes.IntPoint (k), xk);
ip.x = vk[0]; ip.y = vk[1]; ip.z = vk[2];
CalcVShape (ip, vshape);
// vk = |J| J^{-t} nk
vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2];
vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2];
vk[2] = Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2];
for (j = 0; j < dof; j++)
if (fabs (I(k,j) = (vshape(j,0)*vk[0]+vshape(j,1)*vk[1]+
vshape(j,2)*vk[2])) < 1.0e-12)
{
I(k,j) = 0.0;
}
}
}
void RT0PyrFiniteElement::Project (
VectorCoefficient &vc, ElementTransformation &Trans,
Vector &dofs) const
{
real_t vk[3];
Vector xk (vk, 3);
for (int k = 0; k < dof; k++)
{
Trans.SetIntPoint (&Nodes.IntPoint (k));
// set Jinv = |J| J^{-t} = adj(J)^t
const DenseMatrix &Jinv = Trans.TransposeAdjugateJacobian();
vc.Eval (xk, Trans, Nodes.IntPoint (k));
// xk^t |J| J^{-t} nk
dofs(k) =
vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1]+Jinv(0,2)*nk[k][2] ) +
vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1]+Jinv(1,2)*nk[k][2] ) +
vk[2] * ( Jinv(2,0)*nk[k][0]+Jinv(2,1)*nk[k][1]+Jinv(2,2)*nk[k][2] );
if (!rt0 && k > 0) { dofs[k] *= 2.0; }
}
}
void RT0PyrFiniteElement::ProjectCurl(const FiniteElement &fe,
ElementTransformation &Trans,
DenseMatrix &curl) const
{
DenseMatrix curl_shape(fe.GetDof(), dim);
Vector curl_k(fe.GetDof());
curl.SetSize(dof, fe.GetDof());
for (int k = 0; k < dof; k++)
{
fe.CalcCurlShape(Nodes.IntPoint(k), curl_shape);
curl_shape.Mult(nk[k], curl_k);
if (!rt0 && k > 0) { curl_k *= 2.0; }
for (int j = 0; j < curl_k.Size(); j++)
{
curl(k,j) = (fabs(curl_k(j)) < 1e-12) ? 0.0 : curl_k(j);
}
}
}
RotTriLinearHexFiniteElement::RotTriLinearHexFiniteElement()
: NodalFiniteElement(3, Geometry::CUBE, 6, 2, FunctionSpace::Qk)
{
Nodes.IntPoint(0).x = 0.5;
Nodes.IntPoint(0).y = 0.5;
Nodes.IntPoint(0).z = 0.0;
Nodes.IntPoint(1).x = 0.5;
Nodes.IntPoint(1).y = 0.0;
Nodes.IntPoint(1).z = 0.5;
Nodes.IntPoint(2).x = 1.0;
Nodes.IntPoint(2).y = 0.5;
Nodes.IntPoint(2).z = 0.5;
Nodes.IntPoint(3).x = 0.5;
Nodes.IntPoint(3).y = 1.0;
Nodes.IntPoint(3).z = 0.5;
Nodes.IntPoint(4).x = 0.0;
Nodes.IntPoint(4).y = 0.5;
Nodes.IntPoint(4).z = 0.5;
Nodes.IntPoint(5).x = 0.5;
Nodes.IntPoint(5).y = 0.5;
Nodes.IntPoint(5).z = 1.0;
}
void RotTriLinearHexFiniteElement::CalcShape(const IntegrationPoint &ip,
Vector &shape) const
{
real_t x = 2. * ip.x - 1.;
real_t y = 2. * ip.y - 1.;
real_t z = 2. * ip.z - 1.;
real_t f5 = x * x - y * y;
real_t f6 = y * y - z * z;
shape(0) = (1./6.) * (1. - 3. * z - f5 - 2. * f6);
shape(1) = (1./6.) * (1. - 3. * y - f5 + f6);
shape(2) = (1./6.) * (1. + 3. * x + 2. * f5 + f6);
shape(3) = (1./6.) * (1. + 3. * y - f5 + f6);
shape(4) = (1./6.) * (1. - 3. * x + 2. * f5 + f6);
shape(5) = (1./6.) * (1. + 3. * z - f5 - 2. * f6);
}
void RotTriLinearHexFiniteElement::CalcDShape(const IntegrationPoint &ip,
DenseMatrix &dshape) const
{
const real_t a = 2./3.;
real_t xt = a * (1. - 2. * ip.x);
real_t yt = a * (1. - 2. * ip.y);
real_t zt = a * (1. - 2. * ip.z);
dshape(0,0) = xt;
dshape(0,1) = yt;
dshape(0,2) = -1. - 2. * zt;
dshape(1,0) = xt;
dshape(1,1) = -1. - 2. * yt;
dshape(1,2) = zt;
dshape(2,0) = 1. - 2. * xt;
dshape(2,1) = yt;
dshape(2,2) = zt;
dshape(3,0) = xt;
dshape(3,1) = 1. - 2. * yt;
dshape(3,2) = zt;
dshape(4,0) = -1. - 2. * xt;
dshape(4,1) = yt;
dshape(4,2) = zt;
dshape(5,0) = xt;
dshape(5,1) = yt;
dshape(5,2) = 1. - 2. * zt;
}
}
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