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polynomialBasis.cpp 16.47 KiB
// Gmsh - Copyright (C) 1997-2016 C. Geuzaine, J.-F. Remacle
//
// See the LICENSE.txt file for license information. Please report all
// bugs and problems to the public mailing list <gmsh@onelab.info>.
//
// Contributor(s):
// Koen Hillewaert
// Jonathan Lambrechts
//
#include <stdlib.h>
#include <cmath>
#include "GmshDefines.h"
#include "GmshMessage.h"
#include "polynomialBasis.h"
#include "GaussIntegration.h"
#include "pointsGenerators.h"
#include <limits>
namespace {
fullMatrix<double> generateLagrangeMonomialCoefficients
(const fullMatrix<double>& monomial, const fullMatrix<double>& point)
{
if(monomial.size1() != point.size1() || monomial.size2() != point.size2()){
Msg::Fatal("Wrong sizes for Lagrange coefficients generation %d %d -- %d %d",
monomial.size1(), point.size1(),
monomial.size2(), point.size2() );
return fullMatrix<double>(1, 1);
}
int ndofs = monomial.size1();
int dim = monomial.size2();
fullMatrix<double> Vandermonde(ndofs, ndofs);
for (int i = 0; i < ndofs; i++) {
for (int j = 0; j < ndofs; j++) {
double dd = 1.;
for (int k = 0; k < dim; k++) dd *= pow_int(point(j, k), monomial(i, k));
Vandermonde(i, j) = dd;
}
}
fullMatrix<double> coefficient(ndofs, ndofs);
Vandermonde.invert(coefficient);
return coefficient;
}
/*
void printClosure(polynomialBasis::clCont &fullClosure,
std::vector<int> &closureRef,
polynomialBasis::clCont &closures)
{
for (unsigned int i = 0; i < closures.size(); i++) {
printf("%3i (%2i): ", i, closureRef[i]);
if(closureRef[i]==-1){
printf("\n");
continue;
}
for (unsigned int j = 0; j < fullClosure[i].size(); j++) {
printf("%2i ", fullClosure[i][j]);
}
printf (" -- ");
for (unsigned int j = 0; j < closures[closureRef[i]].size(); j++) {
std::string equalSign = "-";
if (fullClosure[i][closures[closureRef[i]][j]] != closures[i][j])
equalSign = "#";
printf("%2i%s%2i ", fullClosure[i][closures[closureRef[i]][j]],equalSign.c_str(),
closures[i][j]);
} printf("\n");
}
}
*/
}
polynomialBasis::polynomialBasis(int tag) : nodalBasis(tag)
{
switch (parentType) {
case TYPE_PNT :
monomials = gmshGenerateMonomialsLine(0);
break;
case TYPE_LIN :
monomials = gmshGenerateMonomialsLine(order);
break;
case TYPE_TRI :
monomials = gmshGenerateMonomialsTriangle(order, serendip);
break;
case TYPE_QUA :
monomials = serendip ? gmshGenerateMonomialsQuadSerendipity(order) :
gmshGenerateMonomialsQuadrangle(order);
break;
case TYPE_TET :
monomials = gmshGenerateMonomialsTetrahedron(order, serendip);
break;
case TYPE_PRI :
monomials = serendip ? gmshGenerateMonomialsPrismSerendipity(order) :
gmshGenerateMonomialsPrism(order);
break;
case TYPE_HEX :
monomials = serendip ? gmshGenerateMonomialsHexaSerendipity(order) :
gmshGenerateMonomialsHexahedron(order);
break;
}
coefficients = generateLagrangeMonomialCoefficients(monomials, points);
}
polynomialBasis::~polynomialBasis()
{
}
void polynomialBasis::evaluateMonomials(double u, double v, double w, double p[]) const
{
for (int j = 0; j < monomials.size1(); ++j) {
p[j] = 1.;
switch (dimension) {
case 3 : p[j] *= pow_int(w, monomials(j, 2));
case 2 : p[j] *= pow_int(v, monomials(j, 1));
case 1 : p[j] *= pow_int(u, monomials(j, 0));
default :
break;
}
}
}
void polynomialBasis::f(double u, double v, double w, double *sf) const
{
double p[1256];
evaluateMonomials(u, v, w, p);
for (int i = 0; i < coefficients.size1(); i++) {
sf[i] = 0.0;
for (int j = 0; j < coefficients.size2(); j++) {
sf[i] += coefficients(i, j) * p[j];
}
}
}
void polynomialBasis::f(const fullMatrix<double> &coord, fullMatrix<double> &sf) const
{ double p[1256];
sf.resize (coord.size1(), coefficients.size1());
for (int iPoint = 0; iPoint < coord.size1(); iPoint++) {
evaluateMonomials(coord(iPoint, 0), coord.size2() > 1 ? coord(iPoint, 1) : 0,
coord.size2() > 2 ? coord(iPoint, 2) : 0, p);
for (int i = 0; i < coefficients.size1(); i++) {
sf(iPoint,i) = 0.;
for (int j = 0; j < coefficients.size2(); j++)
sf(iPoint,i) += coefficients(i, j) * p[j];
}
}
}
void polynomialBasis::df(const fullMatrix<double> &coord, fullMatrix<double> &dfm) const
{
double dfv[1256][3];
dfm.resize (coefficients.size1(), coord.size1() * 3, false);
int dimCoord = coord.size2();
for (int iPoint=0; iPoint< coord.size1(); iPoint++) {
df(coord(iPoint, 0), dimCoord > 1 ? coord(iPoint, 1) : 0.,
dimCoord > 2 ? coord(iPoint, 2) : 0., dfv);
for (int i = 0; i < coefficients.size1(); i++) {
dfm(i, iPoint * 3 + 0) = dfv[i][0];
dfm(i, iPoint * 3 + 1) = dfv[i][1];
dfm(i, iPoint * 3 + 2) = dfv[i][2];
}
}
}
void polynomialBasis::df(double u, double v, double w, double grads[][3]) const
{
switch (monomials.size2()) {
case 1:
for (int i = 0; i < coefficients.size1(); i++){
grads[i][0] = 0;
grads[i][1] = 0;
grads[i][2] = 0;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 0)
grads[i][0] += coefficients(i, j) *
pow_int(u, (monomials(j, 0) - 1)) * monomials(j, 0);
}
}
break;
case 2:
for (int i = 0; i < coefficients.size1(); i++){
grads[i][0] = 0;
grads[i][1] = 0;
grads[i][2] = 0;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 0)
grads[i][0] += coefficients(i, j) *
pow_int(u, (monomials(j, 0) - 1)) * monomials(j, 0) *
pow_int(v, monomials(j, 1));
if (monomials(j, 1) > 0)
grads[i][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, (monomials(j, 1) - 1)) * monomials(j, 1);
}
}
break;
case 3:
for (int i = 0; i < coefficients.size1(); i++){
grads[i][0] = 0;
grads[i][1] = 0;
grads[i][2] = 0;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 0)
grads[i][0] += coefficients(i, j) *
pow_int(u, (monomials(j, 0) - 1)) * monomials(j, 0) * pow_int(v, monomials(j, 1)) *
pow_int(w, monomials(j, 2));
if (monomials(j, 1) > 0)
grads[i][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, (monomials(j, 1) - 1)) * monomials(j, 1) *
pow_int(w, monomials(j, 2));
if (monomials(j, 2) > 0)
grads[i][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1)) *
pow_int(w, (monomials(j, 2) - 1)) * monomials(j, 2);
}
}
break;
}
}
void polynomialBasis::ddf(double u, double v, double w, double hess[][3][3]) const
{
switch (monomials.size2()) {
case 1:
for (int i = 0; i < coefficients.size1(); i++){
hess[i][0][0] = hess[i][0][1] = hess[i][0][2] = 0;
hess[i][1][0] = hess[i][1][1] = hess[i][1][2] = 0;
hess[i][2][0] = hess[i][2][1] = hess[i][2][2] = 0;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 1) // second derivative !=0
hess[i][0][0] += coefficients(i, j) * pow_int(u, (monomials(j, 0) - 2)) *
monomials(j, 0) * (monomials(j, 0) - 1);
}
}
break;
case 2:
for (int i = 0; i < coefficients.size1(); i++){
hess[i][0][0] = hess[i][0][1] = hess[i][0][2] = 0;
hess[i][1][0] = hess[i][1][1] = hess[i][1][2] = 0;
hess[i][2][0] = hess[i][2][1] = hess[i][2][2] = 0;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 1) // second derivative !=0
hess[i][0][0] += coefficients(i, j) * pow_int(u, (monomials(j, 0) - 2)) *
monomials(j, 0) * (monomials(j, 0) - 1) * pow_int(v, monomials(j, 1));
if ((monomials(j, 1) > 0) && (monomials(j, 0) > 0))
hess[i][0][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 1) * monomials(j, 0) *
pow_int(v, monomials(j, 1) - 1) * monomials(j, 1);
if (monomials(j, 1) > 1)
hess[i][1][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1) - 2) * monomials(j, 1) * (monomials(j, 1) - 1);
}
hess[i][1][0] = hess[i][0][1];
}
break;
case 3:
for (int i = 0; i < coefficients.size1(); i++){
hess[i][0][0] = hess[i][0][1] = hess[i][0][2] = 0;
hess[i][1][0] = hess[i][1][1] = hess[i][1][2] = 0;
hess[i][2][0] = hess[i][2][1] = hess[i][2][2] = 0;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 1)
hess[i][0][0] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 2) * monomials(j, 0) * (monomials(j, 0)-1) *
pow_int(v, monomials(j, 1)) *
pow_int(w, monomials(j, 2));
if ((monomials(j, 0) > 0) && (monomials(j, 1) > 0))
hess[i][0][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 1) * monomials(j, 0) * pow_int(v, monomials(j, 1) - 1) * monomials(j, 1) *
pow_int(w, monomials(j, 2));
if ((monomials(j, 0) > 0) && (monomials(j, 2) > 0))
hess[i][0][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 1) * monomials(j, 0) *
pow_int(v, monomials(j, 1)) *
pow_int(w, monomials(j, 2) - 1) * monomials(j, 2);
if (monomials(j, 1) > 1)
hess[i][1][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1) - 2) * monomials(j, 1) * (monomials(j, 1)-1) *
pow_int(w, monomials(j, 2));
if ((monomials(j, 1) > 0) && (monomials(j, 2) > 0))
hess[i][1][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1) - 1) * monomials(j, 1) *
pow_int(w, monomials(j, 2) - 1) * monomials(j, 2);
if (monomials(j, 2) > 1)
hess[i][2][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1)) *
pow_int(w, monomials(j, 2) - 2) * monomials(j, 2) * (monomials(j, 2) - 1);
}
hess[i][1][0] = hess[i][0][1];
hess[i][2][0] = hess[i][0][2];
hess[i][2][1] = hess[i][1][2];
}
break;
}
}
void polynomialBasis::dddf(double u, double v, double w, double third[][3][3][3]) const
{
switch (monomials.size2()) {
case 1:
for (int i = 0; i < coefficients.size1(); i++){
for (int p=0; p<3; p++)
for (int q=0; q<3; q++)
for (int r=0; r<3; r++)
third[i][p][q][r] = 0.;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 2) // third derivative !=0
third[i][0][0][0] += coefficients(i, j) * pow_int(u, (monomials(j, 0) - 3)) *
monomials(j, 0) * (monomials(j, 0) - 1)*(monomials(j, 0) - 2);
}
}
break;
case 2:
for (int i = 0; i < coefficients.size1(); i++){
for (int p=0; p<3; p++)
for (int q=0; q<3; q++)
for (int r=0; r<3; r++)
third[i][p][q][r] = 0.;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 2) // second derivative !=0
third[i][0][0][0] += coefficients(i, j) *
pow_int(u, (monomials(j, 0) - 3))*monomials(j, 0) *
(monomials(j, 0) - 1) *(monomials(j, 0) - 2) *
pow_int(v, monomials(j, 1));
if ((monomials(j, 0) > 1) && (monomials(j, 1) > 0))
third[i][0][0][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 2) * monomials(j, 0)*(monomials(j, 0) - 1) *
pow_int(v, monomials(j, 1) - 1) * monomials(j, 1);
if ((monomials(j, 0) > 0) && (monomials(j, 1) > 1))
third[i][0][1][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 1) *monomials(j, 0)*
pow_int(v, monomials(j, 1) - 2) * monomials(j, 1) * (monomials(j, 1) - 1);
if (monomials(j, 1) > 2)
third[i][1][1][1] += coefficients(i, j) * pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1) - 3) * monomials(j, 1) *
(monomials(j, 1) - 1) * (monomials(j, 1) - 2);
}
third[i][0][1][0] = third[i][0][0][1];
third[i][1][0][0] = third[i][0][0][1];
third[i][1][0][1] = third[i][0][1][1];
third[i][1][1][0] = third[i][0][1][1];
}
break;
case 3:
for (int i = 0; i < coefficients.size1(); i++){
for (int p=0; p<3; p++)
for (int q=0; q<3; q++)
for (int r=0; r<3; r++)
third[i][p][q][r] = 0.;
for(int j = 0; j < coefficients.size2(); j++){
if (monomials(j, 0) > 2) // second derivative !=0
third[i][0][0][0] += coefficients(i, j) *
pow_int(u, (monomials(j, 0) - 3)) *monomials(j, 0) *
(monomials(j, 0) - 1) * (monomials(j, 0) - 2) *
pow_int(v, monomials(j, 1))*
pow_int(w, monomials(j, 2));
if ((monomials(j, 0) > 1) && (monomials(j, 1) > 0))
third[i][0][0][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 2) * monomials(j, 0)*(monomials(j, 0) - 1) *
pow_int(v, monomials(j, 1) - 1) * monomials(j, 1)*
pow_int(w, monomials(j, 2));
if ((monomials(j, 0) > 1) && (monomials(j, 2) > 0))
third[i][0][0][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 2) * monomials(j, 0)*(monomials(j, 0) - 1) *
pow_int(v, monomials(j, 1)) *
pow_int(w, monomials(j, 2) - 1)* monomials(j, 2);
if ((monomials(j, 0) > 0) && (monomials(j, 1) > 1))
third[i][0][1][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 1) *monomials(j, 0)*
pow_int(v, monomials(j, 1) - 2) * monomials(j, 1) * (monomials(j, 1) - 1)*
pow_int(w, monomials(j, 2));
if ((monomials(j, 0) > 0) && (monomials(j, 1) > 0)&& (monomials(j, 2) > 0))
third[i][0][1][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 1) *monomials(j, 0)*
pow_int(v, monomials(j, 1) - 1) *monomials(j, 1) *
pow_int(w, monomials(j, 2) - 1) *monomials(j, 2);
if ((monomials(j, 0) > 0) && (monomials(j, 2) > 1))
third[i][0][2][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0) - 1) *monomials(j, 0)*
pow_int(v, monomials(j, 1))*
pow_int(w, monomials(j, 2) - 2) * monomials(j, 2) * (monomials(j, 2) - 1);
if ((monomials(j, 1) > 2))
third[i][1][1][1] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1)-3) * monomials(j, 1) *
(monomials(j, 1) - 1) * (monomials(j, 1) - 2)*
pow_int(w, monomials(j, 2));
if ((monomials(j, 1) > 1) && (monomials(j, 2) > 0))
third[i][1][1][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1)-2) * monomials(j, 1) * (monomials(j, 1) - 1)*
pow_int(w, monomials(j, 2) - 1) *monomials(j, 2) ;
if ((monomials(j, 1) > 0) && (monomials(j, 2) > 1))
third[i][1][2][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1)-1) *monomials(j, 1)* pow_int(w, monomials(j, 2) - 2) * monomials(j, 2) * (monomials(j, 2) - 1) ;
if ((monomials(j, 2) > 2))
third[i][2][2][2] += coefficients(i, j) *
pow_int(u, monomials(j, 0)) *
pow_int(v, monomials(j, 1))*
pow_int(w, monomials(j, 2) - 3) * monomials(j, 2) *
(monomials(j, 2) - 1) * (monomials(j, 2) - 2);
}
third[i][0][1][0] = third[i][0][0][1];
third[i][1][0][0] = third[i][0][0][1];
third[i][2][0][0] = third[i][0][0][2];
third[i][0][2][0] = third[i][0][0][2];
third[i][1][0][1] = third[i][0][1][1];
third[i][1][1][0] = third[i][0][1][1];
third[i][0][2][1] = third[i][0][1][2];
third[i][1][0][2] = third[i][0][1][2];
third[i][1][2][0] = third[i][0][1][2];
third[i][2][1][0] = third[i][0][1][2];
third[i][2][0][1] = third[i][0][1][2];
third[i][2][2][0] = third[i][0][2][2];
third[i][2][0][2] = third[i][0][2][2];
third[i][1][2][1] = third[i][1][1][2];
third[i][2][1][1] = third[i][1][1][2];
third[i][2][2][1] = third[i][1][2][2];
third[i][2][1][2] = third[i][1][2][2];
}
break;
}
}