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vtkLagrangeTriangle.cxx
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vtkLagrangeTriangle.cxx
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/*=========================================================================
Program: Visualization Toolkit
Module: vtkLagrangeTriangle.cxx
Copyright (c) Ken Martin, Will Schroeder, Bill Lorensen
All rights reserved.
See Copyright.txt or http://www.kitware.com/Copyright.htm for details.
This software is distributed WITHOUT ANY WARRANTY; without even
the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
PURPOSE. See the above copyright notice for more information.
=========================================================================*/
#include "vtkLagrangeTriangle.h"
#include "vtkCellArray.h"
#include "vtkCellData.h"
#include "vtkDoubleArray.h"
#include "vtkIncrementalPointLocator.h"
#include "vtkLagrangeCurve.h"
#include "vtkLine.h"
#include "vtkMath.h"
#include "vtkObjectFactory.h"
#include "vtkPointData.h"
#include "vtkPoints.h"
#include "vtkTriangle.h"
#define ENABLE_CACHING
#define SEVEN_POINT_TRIANGLE
vtkStandardNewMacro(vtkLagrangeTriangle);
//----------------------------------------------------------------------------
vtkLagrangeTriangle::vtkLagrangeTriangle() : ParametricCoordinates(nullptr)
{
this->Order = 0;
this->Edge = vtkLagrangeCurve::New();
this->Face = vtkTriangle::New();
this->Scalars = vtkDoubleArray::New();
this->Scalars->SetNumberOfTuples(3);
this->Points->SetNumberOfPoints(3);
this->PointIds->SetNumberOfIds(3);
for (int i = 0; i < 3; i++)
{
this->Points->SetPoint(i, 0.0, 0.0, 0.0);
this->PointIds->SetId(i,0);
}
}
//----------------------------------------------------------------------------
vtkLagrangeTriangle::~vtkLagrangeTriangle()
{
delete [] this->ParametricCoordinates;
this->Edge->Delete();
this->Face->Delete();
this->Scalars->Delete();
}
//----------------------------------------------------------------------------
vtkCell *vtkLagrangeTriangle::GetEdge(int edgeId)
{
this->Edge->PointIds->Reset();
this->Edge->Points->Reset();
vtkIdType order = this->GetOrder();
vtkIdType bindex[3] = {0,0,0};
bindex[(edgeId+2)%3] = order;
for (vtkIdType i=0;i<=order;i++)
{
vtkIdType triangleIndex = this->ToIndex(bindex);
// The ordering for points in vtkLagrangeCurve are: first point, then last
// point, and then the remaining points in sequence. This loop iterates over
// the edge in sequence starting with the first point. The following value
// maps from this iteration loop to the edge's ordering.
vtkIdType edgeIndex = (i == 0 ? 0 : (i == order ? 1 : i + 1));
this->Edge->GetPointIds()->InsertId(edgeIndex, this->PointIds->GetId(triangleIndex));
this->Edge->GetPoints()->InsertPoint(edgeIndex, this->Points->GetPoint(triangleIndex));
bindex[(edgeId+2)%3]--;
bindex[edgeId]++;
}
return this->Edge;
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::Initialize()
{
vtkIdType order = this->ComputeOrder();
if (this->Order != order)
{
// Reset our caches
this->Order = order;
this->NumberOfSubtriangles = this->ComputeNumberOfSubtriangles();
#ifdef ENABLE_CACHING
for (vtkIdType i = 0; i < this->GetPointIds()->GetNumberOfIds(); i++)
{
this->BarycentricIndexMap[3*i] = -1;
}
// we sacrifice memory for efficiency here
vtkIdType nIndexMap = (this->Order+1)*(this->Order+1);
for (vtkIdType i = 0; i < nIndexMap; i++)
{
this->IndexMap[i] = -1;
}
vtkIdType nSubtriangles = this->GetNumberOfSubtriangles();
for (vtkIdType i = 0; i < nSubtriangles; i++)
{
this->SubtriangleIndexMap[9*i] = -1;
}
#endif
}
}
//----------------------------------------------------------------------------
vtkIdType vtkLagrangeTriangle::ComputeNumberOfSubtriangles()
{
#ifdef SEVEN_POINT_TRIANGLE
if (this->Points->GetNumberOfPoints() == 7)
{
return 6;
}
#endif
vtkIdType order = this->GetOrder();
return order*order;
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::SubtriangleBarycentricPointIndices(
vtkIdType cellIndex, vtkIdType (&pointBIndices)[3][3])
{
assert(cellIndex < this->GetNumberOfSubtriangles());
#ifdef SEVEN_POINT_TRIANGLE
if (this->Points->GetNumberOfPoints() == 7)
{
pointBIndices[0][0] = cellIndex;
if (cellIndex < 3)
{
pointBIndices[1][0] = (cellIndex+3)%6;
}
else
{
pointBIndices[1][0] = (cellIndex+1)%3;
}
pointBIndices[2][0] = 6;
return;
}
#endif
#ifdef ENABLE_CACHING
vtkIdType cellIndexStart = 9*cellIndex;
if (this->SubtriangleIndexMap[cellIndexStart] == -1)
#endif
{
vtkIdType order = this->GetOrder();
if (order == 1)
{
pointBIndices[0][0] = 0; pointBIndices[0][1] = 0; pointBIndices[0][2] = 1;
pointBIndices[1][0] = 1; pointBIndices[1][1] = 0; pointBIndices[1][2] = 0;
pointBIndices[2][0] = 0; pointBIndices[2][1] = 1; pointBIndices[2][2] = 0;
}
else
{
vtkIdType nRightSideUp = order*(order+1)/2;
if (cellIndex < nRightSideUp)
{
// there are nRightSideUp subtriangles whose orientation is the same as
// the parent triangle. We traverse them here.
vtkLagrangeTriangle::BarycentricIndex(cellIndex, pointBIndices[0],
order - 1);
pointBIndices[0][2] += 1;
pointBIndices[1][0] = pointBIndices[0][0] + 1;
pointBIndices[1][1] = pointBIndices[0][1];
pointBIndices[1][2] = pointBIndices[0][2] - 1;
pointBIndices[2][0] = pointBIndices[0][0];
pointBIndices[2][1] = pointBIndices[0][1] + 1;
pointBIndices[2][2] = pointBIndices[0][2] - 1;
}
else
{
// the remaining subtriangles are inverted with respect to the parent
// triangle. We traverse them here.
if (order == 2)
{
pointBIndices[0][0]=1; pointBIndices[0][1]=1; pointBIndices[0][2]=0;
pointBIndices[1][0]=0; pointBIndices[1][1]=1; pointBIndices[1][2]=1;
pointBIndices[2][0]=1; pointBIndices[2][1]=0; pointBIndices[2][2]=1;
}
else
{
vtkLagrangeTriangle::BarycentricIndex(cellIndex - nRightSideUp,
pointBIndices[1], order - 2);
pointBIndices[1][1] += 1;
pointBIndices[1][2] += 1;
pointBIndices[2][0] = pointBIndices[1][0] + 1;
pointBIndices[2][1] = pointBIndices[1][1] - 1;
pointBIndices[2][2] = pointBIndices[1][2];
pointBIndices[0][0] = pointBIndices[1][0] + 1;
pointBIndices[0][1] = pointBIndices[1][1];
pointBIndices[0][2] = pointBIndices[1][2] - 1;
}
}
}
#ifdef ENABLE_CACHING
for (vtkIdType i=0;i<3;i++)
{
for (vtkIdType j=0;j<3;j++)
{
this->SubtriangleIndexMap[cellIndexStart + 3*i + j]=pointBIndices[i][j];
}
}
#endif
}
#ifdef ENABLE_CACHING
else
{
for (vtkIdType i=0;i<3;i++)
{
for (vtkIdType j=0;j<3;j++)
{
pointBIndices[i][j]=this->SubtriangleIndexMap[cellIndexStart + 3*i + j];
}
}
}
#endif
}
//----------------------------------------------------------------------------
int vtkLagrangeTriangle::CellBoundary(int vtkNotUsed(subId), const double pcoords[3],
vtkIdList *pts)
{
double t1=pcoords[0]-pcoords[1];
double t2=0.5*(1.0-pcoords[0])-pcoords[1];
double t3=2.0*pcoords[0]+pcoords[1]-1.0;
pts->SetNumberOfIds(2);
// compare against three lines in parametric space that divide element
// into three pieces
if ( t1 >= 0.0 && t2 >= 0.0 )
{
pts->SetId(0,this->PointIds->GetId(0));
pts->SetId(1,this->PointIds->GetId(1));
}
else if ( t2 < 0.0 && t3 >= 0.0 )
{
pts->SetId(0,this->PointIds->GetId(1));
pts->SetId(1,this->PointIds->GetId(2));
}
else //( t1 < 0.0 && t3 < 0.0 )
{
pts->SetId(0,this->PointIds->GetId(2));
pts->SetId(1,this->PointIds->GetId(0));
}
if ( pcoords[0] < 0.0 || pcoords[1] < 0.0 ||
pcoords[0] > 1.0 || pcoords[1] > 1.0 ||
(1.0 - pcoords[0] - pcoords[1]) < 0.0 )
{
return 0;
}
else
{
return 1;
}
}
//----------------------------------------------------------------------------
int vtkLagrangeTriangle::EvaluatePosition(const double x[3], double closestPoint[3],
int& subId, double pcoords[3],
double& minDist2, double weights[])
{
double pc[3], dist2, tempWeights[3], closest[3];
double pcoordsMin[3] = {0., 0., 0.};
int returnStatus=0, status, ignoreId;
vtkIdType minBIndices[3][3], bindices[3][3], pointIndices[3];
vtkIdType order = this->GetOrder();
vtkIdType numberOfSubtriangles = this->GetNumberOfSubtriangles();
minDist2 = VTK_DOUBLE_MAX;
for (vtkIdType subCellId = 0; subCellId < numberOfSubtriangles; subCellId++)
{
this->SubtriangleBarycentricPointIndices(subCellId, bindices);
for (vtkIdType i=0;i<3;i++)
{
pointIndices[i] = this->ToIndex(bindices[i]);
this->Face->Points->SetPoint(i, this->Points->GetPoint(pointIndices[i]));
}
status = this->Face->EvaluatePosition(x, closest, ignoreId, pc, dist2,
tempWeights);
if ( status != -1 && dist2 < minDist2 )
{
returnStatus = status;
minDist2 = dist2;
subId = subCellId;
pcoordsMin[0] = pc[0];
pcoordsMin[1] = pc[1];
for (vtkIdType i=0;i<3;i++)
{
for (vtkIdType j=0;j<3;j++)
{
minBIndices[i][j] = bindices[i][j];
}
}
}
}
// adjust parametric coordinates
if ( returnStatus != -1 )
{
for (vtkIdType i=0;i<3;i++)
{
pcoords[i] =
(i < 2 ? (minBIndices[0][i] +
pcoordsMin[0]*(minBIndices[1][i] - minBIndices[0][i]) +
pcoordsMin[1]*(minBIndices[2][i] - minBIndices[0][i]))/order
: 0.);
}
if(closestPoint!=nullptr)
{
// Compute both closestPoint and weights
this->EvaluateLocation(subId,pcoords,closestPoint,weights);
}
else
{
// Compute weights only
this->InterpolateFunctions(pcoords,weights);
}
}
return returnStatus;
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::EvaluateLocation(int& vtkNotUsed(subId),
const double pcoords[3], double x[3],
double *weights)
{
x[0] = x[1] = x[2] = 0.;
this->InterpolateFunctions(pcoords,weights);
double p[3];
vtkIdType nPoints = this->GetPoints()->GetNumberOfPoints();
for (vtkIdType idx = 0; idx < nPoints; idx++)
{
this->Points->GetPoint(idx,p);
for (vtkIdType jdx = 0; jdx < 3; jdx++)
{
x[jdx] += p[jdx]*weights[idx];
}
}
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::Contour(double value,
vtkDataArray* cellScalars,
vtkIncrementalPointLocator* locator,
vtkCellArray *verts,
vtkCellArray* lines,
vtkCellArray* polys,
vtkPointData* inPd,
vtkPointData* outPd,
vtkCellData* inCd,
vtkIdType cellId,
vtkCellData* outCd)
{
vtkIdType bindices[3][3];
vtkIdType numberOfSubtriangles = this->GetNumberOfSubtriangles();
for (vtkIdType subCellId = 0; subCellId < numberOfSubtriangles; subCellId++)
{
this->SubtriangleBarycentricPointIndices(subCellId, bindices);
for (vtkIdType i=0;i<3;i++)
{
vtkIdType pointIndex = this->ToIndex(bindices[i]);
this->Face->Points->SetPoint(i, this->Points->GetPoint(pointIndex));
if ( outPd )
{
this->Face->PointIds->SetId(i, this->PointIds->GetId(pointIndex));
}
this->Scalars->SetTuple(i, cellScalars->GetTuple(pointIndex));
}
this->Face->Contour(value, this->Scalars, locator, verts,
lines, polys, inPd, outPd, inCd, cellId, outCd);
}
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::Clip(double value,
vtkDataArray* cellScalars,
vtkIncrementalPointLocator* locator,
vtkCellArray* polys,
vtkPointData* inPd,
vtkPointData* outPd,
vtkCellData* inCd,
vtkIdType cellId,
vtkCellData* outCd,
int insideOut)
{
vtkIdType bindices[3][3];
vtkIdType numberOfSubtriangles = this->GetNumberOfSubtriangles();
for (vtkIdType subCellId = 0; subCellId < numberOfSubtriangles; subCellId++)
{
this->SubtriangleBarycentricPointIndices(subCellId, bindices);
for (vtkIdType i=0;i<3;i++)
{
vtkIdType pointIndex = this->ToIndex(bindices[i]);
this->Face->Points->SetPoint(i, this->Points->GetPoint(pointIndex));
if ( outPd )
{
this->Face->PointIds->SetId(i, this->PointIds->GetId(pointIndex));
}
this->Scalars->SetTuple(i, cellScalars->GetTuple(pointIndex));
}
this->Face->Clip(value, this->Scalars, locator, polys, inPd, outPd,
inCd, cellId, outCd, insideOut);
}
}
//----------------------------------------------------------------------------
int vtkLagrangeTriangle::IntersectWithLine(const double* p1,
const double* p2,
double tol,
double& t,
double* x,
double* pcoords,
int& subId)
{
vtkIdType bindices[3][3];
vtkIdType order = this->GetOrder();
vtkIdType numberOfSubtriangles = this->GetNumberOfSubtriangles();
int subTest;
t = VTK_DOUBLE_MAX;
double tTmp;
double xMin[3], pcoordsMin[3];
for (vtkIdType subCellId = 0; subCellId < numberOfSubtriangles; subCellId++)
{
this->SubtriangleBarycentricPointIndices(subCellId, bindices);
for (vtkIdType i=0;i<3;i++)
{
vtkIdType pointIndex = this->ToIndex(bindices[i]);
this->Face->Points->SetPoint(i, this->Points->GetPoint(pointIndex));
}
if (this->Face->IntersectWithLine(p1, p2, tol, tTmp, xMin, pcoordsMin,
subTest) && tTmp < t)
{
for (vtkIdType i=0;i<3;i++)
{
x[i] = xMin[i];
pcoords[i] = (i < 2 ?
(bindices[0][i] +
pcoordsMin[0]*(bindices[1][i] - bindices[0][i]) +
pcoordsMin[1]*(bindices[2][i] - bindices[0][i]))/order
: 0.);
}
t = tTmp;
}
}
subId = 0;
return (t == VTK_DOUBLE_MAX ? 0 : 1);
}
//----------------------------------------------------------------------------
int vtkLagrangeTriangle::Triangulate(int vtkNotUsed(index), vtkIdList *ptIds,
vtkPoints *pts)
{
pts->Reset();
ptIds->Reset();
#ifdef SEVEN_POINT_TRIANGLE
if (this->Points->GetNumberOfPoints() == 7)
{
static const int edgeOrder[7] = { 0, 3, 1, 4, 2, 5, 0 };
pts->SetNumberOfPoints(18);
ptIds->SetNumberOfIds(18);
vtkIdType pointId = 0;
for (vtkIdType i=0;i<6;i++)
{
ptIds->SetId(pointId,this->PointIds->GetId(edgeOrder[i]));
pts->SetPoint(pointId,this->Points->GetPoint(edgeOrder[i]));
pointId++;
ptIds->SetId(pointId,this->PointIds->GetId(edgeOrder[i+1]));
pts->SetPoint(pointId,this->Points->GetPoint(edgeOrder[i+1]));
pointId++;
ptIds->SetId(pointId,this->PointIds->GetId(6));
pts->SetPoint(pointId,this->Points->GetPoint(6));
pointId++;
}
return 1;
}
#endif
vtkIdType bindices[3][3];
vtkIdType numberOfSubtriangles = this->GetNumberOfSubtriangles();
pts->SetNumberOfPoints(3*numberOfSubtriangles);
ptIds->SetNumberOfIds(3*numberOfSubtriangles);
for (vtkIdType subCellId = 0; subCellId < numberOfSubtriangles; subCellId++)
{
this->SubtriangleBarycentricPointIndices(subCellId, bindices);
for (vtkIdType i=0;i<3;i++)
{
vtkIdType pointIndex = this->ToIndex(bindices[i]);
ptIds->SetId(3*subCellId+i,this->PointIds->GetId(pointIndex));
pts->SetPoint(3*subCellId+i,this->Points->GetPoint(pointIndex));
}
}
return 1;
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::JacobianInverse(const double pcoords[3], double**inverse,
double* derivs)
{
// Given parametric coordinates compute inverse Jacobian transformation
// matrix. Returns 9 elements of 3x3 inverse Jacobian plus interpolation
// function derivatives.
int i, j, k;
double *m[3], m0[3], m1[3], m2[3];
double x[3];
vtkIdType numberOfPoints = this->Points->GetNumberOfPoints();
// compute interpolation function derivatives
this->InterpolateDerivs(pcoords, derivs);
// create Jacobian matrix
m[0] = m0; m[1] = m1; m[2] = m2;
for (i=0; i < 3; i++) //initialize matrix
{
m0[i] = m1[i] = m2[i] = 0.0;
}
for (j=0; j < numberOfPoints; j++)
{
this->Points->GetPoint(j, x);
for (i=0; i < 3; i++)
{
for (k=0; k < this->GetCellDimension(); k++)
{
m[k][i] += x[i] * derivs[numberOfPoints*k + j];
}
}
}
// Compute third row vector in transposed Jacobian and normalize it, so that
// Jacobian determinant stays the same.
if (this->GetCellDimension() == 2)
{
vtkMath::Cross(m0,m1,m2);
}
if ( vtkMath::Normalize(m2) == 0.0 || !vtkMath::InvertMatrix(m,inverse,3))
{
vtkErrorMacro(<<"Jacobian inverse not found");
return;
}
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::Derivatives(int vtkNotUsed(subId),
const double pcoords[3],
const double* values,
int dim,
double *derivs)
{
double *jI[3], j0[3], j1[3], j2[3];
double fDs[(VTK_LAGRANGE_TRIANGLE_MAX_ORDER + 1) *
(VTK_LAGRANGE_TRIANGLE_MAX_ORDER + 2)];
double sum[3];
int i, j, k;
vtkIdType numberOfPoints = this->Points->GetNumberOfPoints();
// compute inverse Jacobian and interpolation function derivatives
jI[0] = j0; jI[1] = j1; jI[2] = j2;
this->JacobianInverse(pcoords, jI, fDs);
// now compute derivates of values provided
for (k=0; k < dim; k++) //loop over values per vertex
{
sum[0] = sum[1] = sum[2] = 0.0;
for ( i=0; i < numberOfPoints; i++) //loop over interp. function derivatives
{
sum[0] += fDs[i] * values[dim*i + k];
sum[1] += fDs[numberOfPoints + i] * values[dim*i + k];
}
for (j=0; j < 3; j++) //loop over derivative directions
{
derivs[3*k + j] = 0.;
for (i=0; i < this->GetCellDimension(); i++)
{
derivs[3*k + j] += sum[i]*jI[j][i];
}
}
}
}
//----------------------------------------------------------------------------
#ifdef SEVEN_POINT_TRIANGLE
namespace
{
double SevenPointTriangleCoords[7*3] = {0.,0.,0.,
1.,0.,0.,
0.,1.,0.,
.5,0.,0.,
.5,.5,0.,
0.,.5,0.,
1./3.,1./3.,0.};
}
#endif
double* vtkLagrangeTriangle::GetParametricCoords()
{
#ifdef SEVEN_POINT_TRIANGLE
if (this->Points->GetNumberOfPoints() == 7)
{
return SevenPointTriangleCoords;
}
#endif
if (!this->ParametricCoordinates)
{
vtkIdType order = this->GetOrder();
vtkIdType nPoints = (order + 1)*(order + 2)/2;
this->ParametricCoordinates = new double[3*nPoints];
ComputeParametricCoords(this->ParametricCoordinates,order);
}
return this->ParametricCoordinates;
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::ComputeParametricCoords(double* coords,
vtkIdType order)
{
double max = static_cast<double>(order);
double min = 0.;
vtkIdType pIdx = 0;
double p[3];
vtkIdType ord;
for (ord = order; ord > 0; ord-=3)
{
// add the vertex points
for (vtkIdType dim=0;dim<3;dim++)
{
coords[pIdx + dim] = min/order;
coords[pIdx + (dim+1)%3] = min/order;
coords[pIdx + (dim+2)%3] = max/order;
pIdx += 3;
}
// add the edge points
if (ord > 1)
{
for (vtkIdType dim=0;dim<3;dim++)
{
p[dim] = p[(dim+1)%3] = min/order;
p[(dim+2)%3] = max/order;
for (vtkIdType i=0;i<ord-1;i++)
{
p[dim] += 1./order;
p[(dim+2)%3] -= 1./order;
for (vtkIdType j=0;j<3;j++)
{
coords[pIdx + j] = p[j];
}
pIdx += 3;
}
}
}
max -= 2.;
min += 1.;
}
if (ord == 0)
{
coords[pIdx] = min/order;
coords[pIdx + 1] = min/order;
coords[pIdx + 2] = min/order;
pIdx += 3;
}
// For parametric coordinates, we project the barycentric coordinates
// onto the z=0 plane
for (vtkIdType i=2;i<pIdx;i+=3)
{
coords[i] = 0.;
}
}
//----------------------------------------------------------------------------
int vtkLagrangeTriangle::GetParametricCenter(double pcoords[3])
{
pcoords[0] = pcoords[1] = 1./3.;
pcoords[2] = 0.;
return 0;
}
//----------------------------------------------------------------------------
double vtkLagrangeTriangle::GetParametricDistance(const double pcoords[3])
{
int i;
double pDist, pDistMax=0.0;
double pc[3];
pc[0] = pcoords[0];
pc[1] = pcoords[1];
pc[2] = 1.0 - pcoords[0] - pcoords[1];
for (i=0; i<3; i++)
{
if ( pc[i] < 0.0 )
{
pDist = -pc[i];
}
else if ( pc[i] > 1.0 )
{
pDist = pc[i] - 1.0;
}
else //inside the cell in the parametric direction
{
pDist = 0.0;
}
if ( pDist > pDistMax )
{
pDistMax = pDist;
}
}
return pDistMax;
}
//----------------------------------------------------------------------------
double vtkLagrangeTriangle::eta(vtkIdType n, vtkIdType chi, double sigma)
{
double result = 1.;
for (vtkIdType i=1;i<=chi;i++)
{
result *= (n*sigma - i + 1.)/i;
}
return result;
}
//----------------------------------------------------------------------------
double vtkLagrangeTriangle::d_eta(vtkIdType n, vtkIdType chi, double sigma)
{
if (chi == 0)
{
return 0.;
}
else
{
double chi_d = static_cast<double>(chi);
return (n/chi_d*eta(n,chi-1,sigma) +
(n*sigma-chi_d+1.)/chi_d*d_eta(n,chi-1,sigma));
}
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::InterpolateFunctions(const double pcoords[3],
double* weights)
{
// Adapted from P. Silvester, "High-Order Polynomial Triangular Finite
// Elements for Potential Problems". Int. J. Engng Sci. Vol. 7, pp. 849-861.
// Pergamon Press, 1969. The generic method is valid for all orders, but we
// unroll the first two orders to reduce computational cost.
double tau[3] = {pcoords[0], pcoords[1], 1. - pcoords[0] - pcoords[1]};
vtkIdType n = this->GetOrder();
if (n == 1)
{
// for the linear case, we simply return the parametric coordinates, rotated
// into the parametric frame (e.g. barycentric tau_2 = parametric x).
weights[0] = tau[2];
weights[1] = tau[0];
weights[2] = tau[1];
}
else if (n == 2)
{
#ifdef SEVEN_POINT_TRIANGLE
if (this->GetPoints()->GetNumberOfPoints() == 7)
{
double rs = tau[0]*tau[1];
double rt = tau[0]*tau[2];
double st = tau[1]*tau[2];
double rst = rs*tau[2];
weights[0] = tau[2] + 3.0*rst - 2.0*rt - 2.0*st;
weights[1] = tau[0] + 3.0*rst - 2.0*rt - 2.0*rs;
weights[2] = tau[1] + 3.0*rst - 2.0*rs - 2.0*st;
weights[3] = 4.0*rt - 12.0*rst;
weights[4] = 4.0*rs - 12.0*rst;
weights[5] = 4.0*st - 12.0*rst;
weights[6] = 27.0*rst;
return;
}
#endif
weights[0] = tau[2]*(2.0*tau[2] - 1.0);
weights[1] = tau[0]*(2.0*tau[0] - 1.0);
weights[2] = tau[1]*(2.0*tau[1] - 1.0);
weights[3] = 4.0 * tau[0] * tau[2];
weights[4] = 4.0 * tau[0] * tau[1];
weights[5] = 4.0 * tau[1] * tau[2];
}
else
{
vtkIdType nPoints = this->GetPoints()->GetNumberOfPoints();
for (vtkIdType idx=0; idx < nPoints; idx++)
{
weights[idx] = 1.;
vtkIdType lambda[3];
this->ToBarycentricIndex(idx,lambda);
for (vtkIdType dim=0;dim<3;dim++)
{
weights[idx] *= eta(n, lambda[dim], tau[dim]);
}
}
}
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::InterpolateDerivs(const double pcoords[3],
double* derivs)
{
// Analytic differentiation of the triangle shape functions, as defined in
// P. Silvester, "High-Order Polynomial Triangular Finite Elements for
// Potential Problems". Int. J. Engng Sci. Vol. 7, pp. 849-861. Pergamon
// Press, 1969. The generic method is valid for all orders, but we unroll the
// first two orders to reduce computational cost.
double tau[3] = {pcoords[0], pcoords[1], 1. - pcoords[0] - pcoords[1]};
vtkIdType n = this->GetOrder();
if (n == 1)
{
derivs[0] = -1;
derivs[1] = 1;
derivs[2] = 0;
derivs[3] = -1;
derivs[4] = 0;
derivs[5] = 1;
}
else if (n == 2)
{
#ifdef SEVEN_POINT_TRIANGLE
if (this->GetPoints()->GetNumberOfPoints() == 7)
{
double tmr = tau[2] - tau[0];
double tms = tau[2] - tau[1];
derivs[0] = -1.0 + 3.0*tau[1]*tmr - 2.0*tmr + 2.0*tau[1];
derivs[1] = 1.0 + 3.0*tau[1]*tmr - 2.0*tmr - 2.0*tau[1];
derivs[2] = 3.0*tau[1]*tmr;
derivs[3] = 4.0*tmr - 12.0*tau[1]*tmr;
derivs[4] = 4.0*tau[1] - 12.0*tau[1]*tmr;
derivs[5] = -4.0*tau[1] - 12.0*tau[1]*tmr;
derivs[6] = 27.0*tau[1]*tmr;
derivs[7] = -1.0 + 3.0*tau[0]*tms - 2.0*tms + 2.0*tau[0];
derivs[8] = 3.0*tau[0]*tms;
derivs[9] = 1.0 + 3.0*tau[0]*tms - 2.0*tms - 2.0*tau[0];
derivs[10] = -4.0*tau[0] - 12.0*tau[0]*tms;
derivs[11] = 4.0*tau[0] - 12.0*tau[0]*tms;
derivs[12] = 4.0*tms - 12.0*tau[0]*tms;
derivs[13] = 27.0*tau[0]*tms;
return;
}
#endif
derivs[0] = 1.0 - 4.0*tau[2];
derivs[1] = 4.0*tau[0] - 1.0;
derivs[2] = 0.0;
derivs[3] = 4.0*(tau[2] - tau[0]);
derivs[4] = 4.0*tau[1];
derivs[5] = -4.0*tau[1];
derivs[6] = 1.0 - 4.0*tau[2];
derivs[7] = 0.0;
derivs[8] = 4.0*tau[1] - 1.0;
derivs[9] = -4.0*tau[0];
derivs[10] = 4.0*tau[0];
derivs[11] = 4.0*(tau[2] - tau[1]);
}
else
{
vtkIdType nPoints = this->GetPoints()->GetNumberOfPoints();
for (vtkIdType idx = 0; idx < nPoints; idx++)
{
vtkIdType lambda[3];
this->ToBarycentricIndex(idx,lambda);
double eta_alpha = eta(n,lambda[0],tau[0]);
double eta_beta = eta(n,lambda[1],tau[1]);
double eta_gamma = eta(n,lambda[2],tau[2]);
double d_eta_alpha = d_eta(n,lambda[0],tau[0]);
double d_eta_beta = d_eta(n,lambda[1],tau[1]);
double d_eta_gamma = d_eta(n,lambda[2],tau[2]);
double d_f_d_tau1 = (d_eta_alpha*eta_beta*eta_gamma -
eta_alpha*eta_beta*d_eta_gamma);
double d_f_d_tau2 = (eta_alpha*d_eta_beta*eta_gamma -
eta_alpha*eta_beta*d_eta_gamma);
derivs[idx] = d_f_d_tau1;
derivs[nPoints + idx] = d_f_d_tau2;
}
}
}
//----------------------------------------------------------------------------
vtkIdType vtkLagrangeTriangle::ComputeOrder()
{
// when order = n, # points = (n+1)*(n+2)/2
return (sqrt(8*this->GetPoints()->GetNumberOfPoints() + 1) - 3)/2;
}
//----------------------------------------------------------------------------
void vtkLagrangeTriangle::ToBarycentricIndex(vtkIdType index,
vtkIdType* bindex)
{
#ifdef ENABLE_CACHING
if (this->BarycentricIndexMap[3*index] == -1)
{
vtkLagrangeTriangle::BarycentricIndex(index,
&this->BarycentricIndexMap[3*index],
this->GetOrder());
}
for (vtkIdType i=0;i<3;i++)
{
bindex[i] = this->BarycentricIndexMap[3*index + i];
}
#else
return vtkLagrangeTriangle::BarycentricIndex(index,bindex,this->GetOrder());
#endif
}
//----------------------------------------------------------------------------
vtkIdType vtkLagrangeTriangle::ToIndex(const vtkIdType* bindex)
{
#ifdef SEVEN_POINT_TRIANGLE
if (this->Points->GetNumberOfPoints() == 7)
{
return bindex[0];
}
#endif
#ifdef ENABLE_CACHING
vtkIdType cacheIdx = ((this->Order+1)*bindex[0] + bindex[1]);
if (this->IndexMap[cacheIdx] == -1)
{
this->IndexMap[cacheIdx] =
vtkLagrangeTriangle::Index(bindex, this->GetOrder());
}
return this->IndexMap[cacheIdx];
#else
return vtkLagrangeTriangle::Index(bindex, this->GetOrder());
#endif
}