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netgen/libsrc/meshing/curvedelems.cpp
Joachim Schoeberl 91d5c9888d ARRAY -> Array
2009-01-25 12:35:25 +00:00

3168 lines
78 KiB
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#include <mystdlib.h>
#include "meshing.hpp"
#include "../general/autodiff.hpp"
namespace netgen
{
// bool rational = true;
static void ComputeGaussRule (int n, Array<double> & xi, Array<double> & wi)
{
xi.SetSize (n);
wi.SetSize (n);
int m = (n+1)/2;
double p1, p2, p3;
double pp, z, z1;
for (int i = 1; i <= m; i++)
{
z = cos ( M_PI * (i - 0.25) / (n + 0.5));
while(1)
{
p1 = 1; p2 = 0;
for (int j = 1; j <= n; j++)
{
p3 = p2; p2 = p1;
p1 = ((2 * j - 1) * z * p2 - (j - 1) * p3) / j;
}
// p1 is legendre polynomial
pp = n * (z*p1-p2) / (z*z - 1);
z1 = z;
z = z1-p1/pp;
if (fabs (z - z1) < 1e-14) break;
}
xi[i-1] = 0.5 * (1 - z);
xi[n-i] = 0.5 * (1 + z);
wi[i-1] = wi[n-i] = 1.0 / ( (1 - z * z) * pp * pp);
}
}
// compute edge bubbles up to order n, x \in (-1, 1)
static void CalcEdgeShape (int n, double x, double * shape)
{
double p1 = x, p2 = -1, p3 = 0;
for (int j=2; j<=n; j++)
{
p3=p2; p2=p1;
p1=( (2*j-3) * x * p2 - (j-3) * p3) / j;
shape[j-2] = p1;
}
}
static void CalcEdgeDx (int n, double x, double * dshape)
{
double p1 = x, p2 = -1, p3 = 0;
double p1dx = 1, p2dx = 0, p3dx = 0;
for (int j=2; j<=n; j++)
{
p3=p2; p2=p1;
p3dx = p2dx; p2dx = p1dx;
p1=( (2*j-3) * x * p2 - (j-3) * p3) / j;
p1dx = ( (2*j-3) * (x * p2dx + p2) - (j-3) * p3dx) / j;
dshape[j-2] = p1dx;
}
}
static void CalcEdgeShapeDx (int n, double x, double * shape, double * dshape)
{
double p1 = x, p2 = -1, p3 = 0;
double p1dx = 1, p2dx = 0, p3dx = 0;
for (int j=2; j<=n; j++)
{
p3=p2; p2=p1;
p3dx = p2dx; p2dx = p1dx;
p1=( (2*j-3) * x * p2 - (j-3) * p3) / j;
p1dx = ( (2*j-3) * (x * p2dx + p2) - (j-3) * p3dx) / j;
shape[j-2] = p1;
dshape[j-2] = p1dx;
}
}
// compute L_i(x/t) * t^i
static void CalcScaledEdgeShape (int n, double x, double t, double * shape)
{
static bool init = false;
static double coefs[100][2];
if (!init)
{
for (int j = 0; j < 100; j++)
{
coefs[j][0] = double(2*j+1)/(j+2);
coefs[j][1] = -double(j-1)/(j+2);
}
init = true;
}
double p1 = x, p2 = -1, p3 = 0;
double tt = t*t;
for (int j=0; j<=n-2; j++)
{
p3=p2; p2=p1;
p1= coefs[j][0] * x * p2 + coefs[j][1] * tt*p3;
// p1=( (2*j+1) * x * p2 - t*t*(j-1) * p3) / (j+2);
shape[j] = p1;
}
}
template <int DIST>
static void CalcScaledEdgeShapeDxDt (int n, double x, double t, double * dshape)
{
double p1 = x, p2 = -1, p3 = 0;
double p1dx = 1, p1dt = 0;
double p2dx = 0, p2dt = 0;
double p3dx = 0, p3dt = 0;
for (int j=0; j<=n-2; j++)
{
p3=p2; p3dx=p2dx; p3dt = p2dt;
p2=p1; p2dx=p1dx; p2dt = p1dt;
p1 = ( (2*j+1) * x * p2 - t*t*(j-1) * p3) / (j+2);
p1dx = ( (2*j+1) * (x * p2dx + p2) - t*t*(j-1) * p3dx) / (j+2);
p1dt = ( (2*j+1) * x * p2dt - (j-1)* (t*t*p3dt+2*t*p3)) / (j+2);
// shape[j] = p1;
dshape[DIST*j ] = p1dx;
dshape[DIST*j+1] = p1dt;
}
}
template <class Tx, class Tres>
static void LegendrePolynomial (int n, Tx x, Tres * values)
{
switch (n)
{
case 0:
values[0] = 1;
break;
case 1:
values[0] = 1;
values[1] = x;
break;
default:
if (n < 0) return;
Tx p1 = 1.0, p2 = 0.0, p3;
values[0] = 1.0;
for (int j=1; j<=n; j++)
{
p3 = p2; p2 = p1;
p1 = ((2.0*j-1.0)*x*p2 - (j-1.0)*p3) / j;
values[j] = p1;
}
}
}
template <class Tx, class Tt, class Tres>
static void ScaledLegendrePolynomial (int n, Tx x, Tt t, Tres * values)
{
switch (n)
{
case 0:
values[0] = 1.0;
break;
case 1:
values[0] = 1.0;
values[1] = x;
break;
default:
if (n < 0) return;
Tx p1 = 1.0, p2 = 0.0, p3;
values[0] = 1.0;
for (int j=1; j<=n; j++)
{
p3 = p2; p2 = p1;
p1 = ((2.0*j-1.0)*x*p2 - t*t*(j-1.0)*p3) / j;
values[j] = p1;
}
}
}
template <class S, class T>
inline void JacobiPolynomial (int n, S x, double alpha, double beta, T * values)
{
S p1 = 1.0, p2 = 0.0, p3;
if (n >= 0)
p2 = values[0] = 1.0;
if (n >= 1)
p1 = values[1] = 0.5 * (2*(alpha+1)+(alpha+beta+2)*(x-1));
for (int i = 1; i < n; i++)
{
p3 = p2; p2=p1;
p1 =
1.0 / ( 2 * (i+1) * (i+alpha+beta+1) * (2*i+alpha+beta) ) *
(
( (2*i+alpha+beta+1)*(alpha*alpha-beta*beta) +
(2*i+alpha+beta)*(2*i+alpha+beta+1)*(2*i+alpha+beta+2) * x)
* p2
- 2*(i+alpha)*(i+beta) * (2*i+alpha+beta+2) * p3
);
values[i+1] = p1;
}
}
template <class S, class St, class T>
inline void ScaledJacobiPolynomial (int n, S x, St t, double alpha, double beta, T * values)
{
/*
S p1 = 1.0, p2 = 0.0, p3;
if (n >= 0) values[0] = 1.0;
*/
S p1 = 1.0, p2 = 0.0, p3;
if (n >= 0)
p2 = values[0] = 1.0;
if (n >= 1)
p1 = values[1] = 0.5 * (2*(alpha+1)*t+(alpha+beta+2)*(x-t));
for (int i=1; i < n; i++)
{
p3 = p2; p2=p1;
p1 =
1.0 / ( 2 * (i+1) * (i+alpha+beta+1) * (2*i+alpha+beta) ) *
(
( (2*i+alpha+beta+1)*(alpha*alpha-beta*beta) * t +
(2*i+alpha+beta)*(2*i+alpha+beta+1)*(2*i+alpha+beta+2) * x)
* p2
- 2*(i+alpha)*(i+beta) * (2*i+alpha+beta+2) * t * t * p3
);
values[i+1] = p1;
}
}
// compute face bubbles up to order n, 0 < y, y-x < 1, x+y < 1
template <class Tx, class Ty, class Ts>
static void CalcTrigShape (int n, Tx x, Ty y, Ts * shape)
{
if (n < 3) return;
Tx hx[50], hy[50*50];
// ScaledLegendrePolynomial (n-3, 2*x-1, 1-y, hx);
ScaledJacobiPolynomial (n-3, x, 1-y, 2, 2, hx);
// LegendrePolynomial (n-3, 2*y-1, hy);
for (int ix = 0; ix <= n-3; ix++)
// JacobiPolynomial (n-3, 2*y-1, 0, 0, hy+50*ix);
JacobiPolynomial (n-3, 2*y-1, 2*ix+5, 2, hy+50*ix);
int ii = 0;
Tx bub = (1+x-y)*y*(1-x-y);
for (int iy = 0; iy <= n-3; iy++)
for (int ix = 0; ix <= n-3-iy; ix++)
shape[ii++] = bub * hx[ix]*hy[iy+50*ix];
}
static void CalcTrigShapeDxDy (int n, double x, double y, double * dshape)
{
if (n < 3) return;
AutoDiff<2> adx(x, 0);
AutoDiff<2> ady(y, 1);
AutoDiff<2> res[2000];
CalcTrigShape (n, adx, ady, &res[0]);
int ndof = (n-1)*(n-2)/2;
for (int i = 0; i < ndof; i++)
{
dshape[2*i] = res[i].DValue(0);
dshape[2*i+1] = res[i].DValue(1);
}
/*
if (n < 3) return;
int ndof = (n-1)*(n-2)/2;
double h1[1000], h2[1000];
double eps = 1e-4;
CalcTrigShape (n, x+eps, y, h1);
CalcTrigShape (n, x-eps, y, h2);
for (int i = 0; i < ndof; i++)
dshape[2*i] = (h1[i]-h2[i])/(2*eps);
CalcTrigShape (n, x, y+eps, h1);
CalcTrigShape (n, x, y-eps, h2);
for (int i = 0; i < ndof; i++)
dshape[2*i+1] = (h1[i]-h2[i])/(2*eps);
*/
}
// compute face bubbles up to order n, 0 < y, y-x < 1, x+y < 1
template <class Tx, class Ty, class Tt, class Tr>
static void CalcScaledTrigShape (int n, Tx x, Ty y, Tt t, Tr * shape)
{
if (n < 3) return;
Tx hx[50], hy[50*50];
// ScaledLegendrePolynomial (n-3, (2*x-1), t-y, hx);
ScaledJacobiPolynomial (n-3, x, t-y, 2, 2, hx);
// ScaledLegendrePolynomial (n-3, (2*y-1), t, hy);
for (int ix = 0; ix <= n-3; ix++)
ScaledJacobiPolynomial (n-3, 2*y-1, t, 2*ix+5, 2, hy+50*ix);
int ii = 0;
Tx bub = (t+x-y)*y*(t-x-y);
for (int iy = 0; iy <= n-3; iy++)
for (int ix = 0; ix <= n-3-iy; ix++)
shape[ii++] = bub * hx[ix]*hy[iy+50*ix];
}
// compute face bubbles up to order n, 0 < y, y-x < 1, x+y < 1
static void CalcScaledTrigShapeDxDyDt (int n, double x, double y, double t, double * dshape)
{
if (n < 3) return;
AutoDiff<3> adx(x, 0);
AutoDiff<3> ady(y, 1);
AutoDiff<3> adt(t, 2);
AutoDiff<3> res[2000];
CalcScaledTrigShape (n, adx, ady, adt, &res[0]);
double dshape1[6000];
int ndof = (n-1)*(n-2)/2;
for (int i = 0; i < ndof; i++)
{
dshape[3*i] = res[i].DValue(0);
dshape[3*i+1] = res[i].DValue(1);
dshape[3*i+2] = res[i].DValue(2);
}
/*
if (n < 3) return;
double hvl[1000], hvr[1000];
int nd = (n-1)*(n-2)/2;
double eps = 1e-6;
CalcScaledTrigShape (n, x-eps, y, t, hvl);
CalcScaledTrigShape (n, x+eps, y, t, hvr);
for (int i = 0; i < nd; i++)
dshape[3*i] = (hvr[i]-hvl[i])/(2*eps);
CalcScaledTrigShape (n, x, y-eps, t, hvl);
CalcScaledTrigShape (n, x, y+eps, t, hvr);
for (int i = 0; i < nd; i++)
dshape[3*i+1] = (hvr[i]-hvl[i])/(2*eps);
CalcScaledTrigShape (n, x, y, t-eps, hvl);
CalcScaledTrigShape (n, x, y, t+eps, hvr);
for (int i = 0; i < nd; i++)
dshape[3*i+2] = (hvr[i]-hvl[i])/(2*eps);
*/
/*
for (int i = 0; i < 3*nd; i++)
if (fabs (dshape[i]-dshape1[i]) > 1e-8 * fabs(dshape[i]) + 1e-6)
{
cerr
cerr << "big difference: " << dshape1[i] << " != " << dshape[i] << endl;
}
*/
}
CurvedElements :: CurvedElements (const Mesh & amesh)
: mesh (amesh)
{
order = 1;
rational = 0;
ishighorder = 0;
}
CurvedElements :: ~CurvedElements()
{
;
}
void CurvedElements :: BuildCurvedElements(Refinement * ref, int aorder,
bool arational)
{
order = aorder;
ishighorder = 0;
if (mesh.coarsemesh)
{
mesh.coarsemesh->GetCurvedElements().BuildCurvedElements (ref, aorder, arational);
order = aorder;
rational = arational;
ishighorder = (order > 1);
return;
}
PrintMessage (1, "Curve elements, order = ", aorder);
if (rational) PrintMessage (1, "curved elements with rational splines");
const_cast<Mesh&> (mesh).UpdateTopology();
const MeshTopology & top = mesh.GetTopology();
rational = arational;
Array<int> edgenrs;
edgeorder.SetSize (top.GetNEdges());
faceorder.SetSize (top.GetNFaces());
edgeorder = 1;
faceorder = 1;
if (rational)
{
edgeweight.SetSize (top.GetNEdges());
edgeweight = 1.0;
}
if (aorder <= 1)
{
for (ElementIndex ei = 0; ei < mesh.GetNE(); ei++)
if (mesh[ei].GetType() == TET10)
ishighorder = 1;
return;
}
if (rational) aorder = 2;
if (mesh.GetDimension() == 3)
for (SurfaceElementIndex i = 0; i < mesh.GetNSE(); i++)
{
top.GetSurfaceElementEdges (i+1, edgenrs);
for (int j = 0; j < edgenrs.Size(); j++)
edgeorder[edgenrs[j]-1] = aorder;
faceorder[top.GetSurfaceElementFace (i+1)-1] = aorder;
}
for (SegmentIndex i = 0; i < mesh.GetNSeg(); i++)
edgeorder[top.GetSegmentEdge (i+1)-1] = aorder;
if (rational)
{
edgeorder = 2;
faceorder = 1;
}
edgecoeffsindex.SetSize (top.GetNEdges()+1);
int nd = 0;
for (int i = 0; i < top.GetNEdges(); i++)
{
edgecoeffsindex[i] = nd;
nd += max (0, edgeorder[i]-1);
}
edgecoeffsindex[top.GetNEdges()] = nd;
edgecoeffs.SetSize (nd);
edgecoeffs = Vec<3> (0,0,0);
facecoeffsindex.SetSize (top.GetNFaces()+1);
nd = 0;
for (int i = 0; i < top.GetNFaces(); i++)
{
facecoeffsindex[i] = nd;
if (top.GetFaceType(i+1) == TRIG)
nd += max (0, (faceorder[i]-1)*(faceorder[i]-2)/2);
else
nd += max (0, sqr(faceorder[i]-1));
}
facecoeffsindex[top.GetNFaces()] = nd;
facecoeffs.SetSize (nd);
facecoeffs = Vec<3> (0,0,0);
if (!ref || aorder <= 1)
{
order = aorder;
return;
}
Array<double> xi, weight;
ComputeGaussRule (aorder+4, xi, weight); // on (0,1)
PrintMessage (3, "Curving edges");
if (mesh.GetDimension() == 3 || rational)
for (SurfaceElementIndex i = 0; i < mesh.GetNSE(); i++)
{
SetThreadPercent(double(i)/mesh.GetNSE()*100.);
const Element2d & el = mesh[i];
top.GetSurfaceElementEdges (i+1, edgenrs);
for (int j = 0; j < edgenrs.Size(); j++)
edgenrs[j]--;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (el.GetType());
for (int i2 = 0; i2 < edgenrs.Size(); i2++)
{
PointIndex pi1 = edges[i2][0]-1;
PointIndex pi2 = edges[i2][1]-1;
bool swap = el[pi1] > el[pi2];
Point<3> p1 = mesh[el[pi1]];
Point<3> p2 = mesh[el[pi2]];
int order1 = edgeorder[edgenrs[i2]];
int ndof = max (0, order1-1);
if (rational && order1 >= 2)
{
Point<3> pm = Center (p1, p2);
int surfnr = mesh.GetFaceDescriptor(el.GetIndex()).SurfNr();
Vec<3> n1 = ref -> GetNormal (p1, surfnr, el.GeomInfoPi(edges[i2][0]));
Vec<3> n2 = ref -> GetNormal (p2, surfnr, el.GeomInfoPi(edges[i2][1]));
// p3 = pm + alpha1 n1 + alpha2 n2
Mat<2> mat, inv;
Vec<2> rhs, sol;
mat(0,0) = n1*n1;
mat(0,1) = mat(1,0) = n1*n2;
mat(1,1) = n2*n2;
rhs(0) = n1 * (p1-pm);
rhs(1) = n2 * (p2-pm);
Point<3> p3;
if (fabs (Det (mat)) > 1e-10)
{
CalcInverse (mat, inv);
sol = inv * rhs;
p3 = pm + sol(0) * n1 + sol(1) * n2;
}
else
p3 = pm;
edgecoeffs[edgecoeffsindex[edgenrs[i2]]] = Vec<3> (p3);
double wold = 1, w = 1, dw = 0.1;
double dold = 1e99;
while (fabs (dw) > 1e-12)
{
Vec<3> v05 = 0.25 * Vec<3> (p1) + 0.5*w* Vec<3>(p3) + 0.25 * Vec<3> (p2);
v05 /= 1 + (w-1) * 0.5;
Point<3> p05 (v05), pp05(v05);
ref -> ProjectToSurface (pp05, surfnr, el.GeomInfoPi(edges[i2][0]));
double d = Dist (pp05, p05);
if (d < dold)
{
dold = d;
wold = w;
w += dw;
}
else
{
dw *= -0.7;
w = wold + dw;
}
}
edgeweight[edgenrs[i2]] = w;
continue;
}
Vector shape(ndof);
DenseMatrix mat(ndof, ndof), inv(ndof, ndof),
rhs(ndof, 3), sol(ndof, 3);
rhs = 0.0;
mat = 0.0;
for (int j = 0; j < xi.Size(); j++)
{
Point<3> p;
Point<3> pp;
PointGeomInfo ppgi;
if (swap)
{
p = p1 + xi[j] * (p2-p1);
ref -> PointBetween (p1, p2, xi[j],
mesh.GetFaceDescriptor(el.GetIndex()).SurfNr(),
el.GeomInfoPi(edges[i2][0]),
el.GeomInfoPi(edges[i2][1]),
pp, ppgi);
}
else
{
p = p2 + xi[j] * (p1-p2);
ref -> PointBetween (p2, p1, xi[j],
mesh.GetFaceDescriptor(el.GetIndex()).SurfNr(),
el.GeomInfoPi(edges[i2][1]),
el.GeomInfoPi(edges[i2][0]),
pp, ppgi);
}
Vec<3> dist = pp - p;
CalcEdgeShape (order1, 2*xi[j]-1, &shape(0));
for (int k = 0; k < ndof; k++)
for (int l = 0; l < ndof; l++)
mat(k,l) += weight[j] * shape(k) * shape(l);
for (int k = 0; k < ndof; k++)
for (int l = 0; l < 3; l++)
rhs(k,l) += weight[j] * shape(k) * dist(l);
}
CalcInverse (mat, inv);
Mult (inv, rhs, sol);
int first = edgecoeffsindex[edgenrs[i2]];
for (int j = 0; j < ndof; j++)
for (int k = 0; k < 3; k++)
edgecoeffs[first+j](k) = sol(j,k);
}
}
for (SegmentIndex i = 0; i < mesh.GetNSeg(); i++)
{
SetThreadPercent(double(i)/mesh.GetNSeg()*100.);
const Segment & seg = mesh[i];
PointIndex pi1 = mesh[i].p1;
PointIndex pi2 = mesh[i].p2;
bool swap = (pi1 > pi2);
Point<3> p1 = mesh[pi1];
Point<3> p2 = mesh[pi2];
int segnr = top.GetSegmentEdge (i+1)-1;
int order1 = edgeorder[segnr];
int ndof = max (0, order1-1);
if (rational)
{
Vec<3> tau1 = ref -> GetTangent (p1, seg.surfnr2, seg.surfnr1, seg.epgeominfo[0]);
Vec<3> tau2 = ref -> GetTangent (p2, seg.surfnr2, seg.surfnr1, seg.epgeominfo[1]);
// p1 + alpha1 tau1 = p2 + alpha2 tau2;
Mat<3,2> mat;
Mat<2,3> inv;
Vec<3> rhs;
Vec<2> sol;
for (int j = 0; j < 3; j++)
{
mat(j,0) = tau1(j);
mat(j,1) = -tau2(j);
rhs(j) = p2(j)-p1(j);
}
CalcInverse (mat, inv);
sol = inv * rhs;
Point<3> p3 = p1+sol(0) * tau1;
edgecoeffs[edgecoeffsindex[segnr]] = Vec<3> (p3);
// double dopt = 1e99;
// double wopt = 0;
// for (double w = 0; w <= 2; w += 0.0001)
// {
// Vec<3> v05 = 0.25 * Vec<3> (p1) + 0.5*w* Vec<3>(p3) + 0.25 * Vec<3> (p2);
// v05 /= 1 + (w-1) * 0.5;
// Point<3> p05 (v05), pp05(v05);
// ref -> ProjectToEdge (pp05, seg.surfnr1, seg.surfnr2, seg.epgeominfo[0]);
// double d = Dist (pp05, p05);
// if (d < dopt)
// {
// wopt = w;
// dopt = d;
// }
// }
double wold = 1, w = 1, dw = 0.1;
double dold = 1e99;
while (fabs (dw) > 1e-12)
{
Vec<3> v05 = 0.25 * Vec<3> (p1) + 0.5*w* Vec<3>(p3) + 0.25 * Vec<3> (p2);
v05 /= 1 + (w-1) * 0.5;
Point<3> p05 (v05), pp05(v05);
ref -> ProjectToEdge (pp05, seg.surfnr1, seg.surfnr2, seg.epgeominfo[0]);
double d = Dist (pp05, p05);
if (d < dold)
{
dold = d;
wold = w;
w += dw;
}
else
{
dw *= -0.7;
w = wold + dw;
}
// *testout << "w = " << w << ", dw = " << dw << endl;
}
// cout << "wopt = " << w << ", dopt = " << dold << endl;
edgeweight[segnr] = w;
// cout << "p1 = " << p1 << ", tau1 = " << tau1 << ", alpha1 = " << sol(0) << endl;
// cout << "p2 = " << p2 << ", tau2 = " << tau2 << ", alpha2 = " << -sol(1) << endl;
// cout << "p+alpha tau = " << p1 + sol(0) * tau1
// << " =?= " << p2 +sol(1) * tau2 << endl;
}
else
{
Vector shape(ndof);
DenseMatrix mat(ndof, ndof), inv(ndof, ndof),
rhs(ndof, 3), sol(ndof, 3);
rhs = 0.0;
mat = 0.0;
for (int j = 0; j < xi.Size(); j++)
{
Point<3> p;
Point<3> pp;
EdgePointGeomInfo ppgi;
if (swap)
{
p = p1 + xi[j] * (p2-p1);
ref -> PointBetween (p1, p2, xi[j],
seg.surfnr2, seg.surfnr1,
seg.epgeominfo[0], seg.epgeominfo[1],
pp, ppgi);
}
else
{
p = p2 + xi[j] * (p1-p2);
ref -> PointBetween (p2, p1, xi[j],
seg.surfnr2, seg.surfnr1,
seg.epgeominfo[1], seg.epgeominfo[0],
pp, ppgi);
}
Vec<3> dist = pp - p;
CalcEdgeShape (order1, 2*xi[j]-1, &shape(0));
for (int k = 0; k < ndof; k++)
for (int l = 0; l < ndof; l++)
mat(k,l) += weight[j] * shape(k) * shape(l);
for (int k = 0; k < ndof; k++)
for (int l = 0; l < 3; l++)
rhs(k,l) += weight[j] * shape(k) * dist(l);
}
CalcInverse (mat, inv);
Mult (inv, rhs, sol);
int first = edgecoeffsindex[segnr];
for (int j = 0; j < ndof; j++)
for (int k = 0; k < 3; k++)
edgecoeffs[first+j](k) = sol(j,k);
}
}
PrintMessage (3, "Curving faces");
if (mesh.GetDimension() == 3)
for (SurfaceElementIndex i = 0; i < mesh.GetNSE(); i++)
{
SetThreadPercent(double(i)/mesh.GetNSE()*100.);
const Element2d & el = mesh[i];
int facenr = top.GetSurfaceElementFace (i+1)-1;
if (el.GetType() == TRIG && order >= 3)
{
int fnums[] = { 0, 1, 2 };
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
if (el[fnums[1]] > el[fnums[2]]) swap (fnums[1], fnums[2]);
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
int order1 = faceorder[facenr];
int ndof = max (0, (order1-1)*(order1-2)/2);
Vector shape(ndof);
DenseMatrix mat(ndof, ndof), inv(ndof, ndof),
rhs(ndof, 3), sol(ndof, 3);
rhs = 0.0;
mat = 0.0;
for (int jx = 0; jx < xi.Size(); jx++)
for (int jy = 0; jy < xi.Size(); jy++)
{
double y = xi[jy];
double x = (1-y) * xi[jx];
double lami[] = { x, y, 1-x-y };
double wi = weight[jx]*weight[jy]*(1-y);
Point<2> xii (x, y);
Point<3> p1, p2;
CalcSurfaceTransformation (xii, i, p1);
p2 = p1;
ref -> ProjectToSurface (p2, mesh.GetFaceDescriptor(el.GetIndex()).SurfNr());
Vec<3> dist = p2-p1;
CalcTrigShape (order1, lami[fnums[1]]-lami[fnums[0]],
1-lami[fnums[1]]-lami[fnums[0]], &shape(0));
for (int k = 0; k < ndof; k++)
for (int l = 0; l < ndof; l++)
mat(k,l) += wi * shape(k) * shape(l);
for (int k = 0; k < ndof; k++)
for (int l = 0; l < 3; l++)
rhs(k,l) += wi * shape(k) * dist(l);
}
// *testout << "mat = " << endl << mat << endl;
// CalcInverse (mat, inv);
// Mult (inv, rhs, sol);
for (int i = 0; i < ndof; i++)
for (int j = 0; j < 3; j++)
sol(i,j) = rhs(i,j) / mat(i,i); // Orthogonal basis !
int first = facecoeffsindex[facenr];
for (int j = 0; j < ndof; j++)
for (int k = 0; k < 3; k++)
facecoeffs[first+j](k) = sol(j,k);
}
}
PrintMessage (3, "Complete");
// compress edge and face tables
int newbase = 0;
for (int i = 0; i < edgeorder.Size(); i++)
{
bool curved = 0;
int oldbase = edgecoeffsindex[i];
nd = edgecoeffsindex[i+1] - edgecoeffsindex[i];
for (int j = 0; j < nd; j++)
if (edgecoeffs[oldbase+j].Length() > 1e-12)
curved = 1;
if (rational) curved = 1;
if (curved && newbase != oldbase)
for (int j = 0; j < nd; j++)
edgecoeffs[newbase+j] = edgecoeffs[oldbase+j];
edgecoeffsindex[i] = newbase;
if (!curved) edgeorder[i] = 1;
if (curved) newbase += nd;
}
edgecoeffsindex.Last() = newbase;
newbase = 0;
for (int i = 0; i < faceorder.Size(); i++)
{
bool curved = 0;
int oldbase = facecoeffsindex[i];
nd = facecoeffsindex[i+1] - facecoeffsindex[i];
for (int j = 0; j < nd; j++)
if (facecoeffs[oldbase+j].Length() > 1e-12)
curved = 1;
if (curved && newbase != oldbase)
for (int j = 0; j < nd; j++)
facecoeffs[newbase+j] = facecoeffs[oldbase+j];
facecoeffsindex[i] = newbase;
if (!curved) faceorder[i] = 1;
if (curved) newbase += nd;
}
facecoeffsindex.Last() = newbase;
ishighorder = (order > 1);
// (*testout) << "edgecoeffs = " << endl << edgecoeffs << endl;
// (*testout) << "facecoeffs = " << endl << facecoeffs << endl;
}
// *********************** Transform edges *****************************
bool CurvedElements :: IsSegmentCurved (SegmentIndex elnr) const
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
return mesh.coarsemesh->GetCurvedElements().IsSegmentCurved (hpref_el.coarse_elnr);
}
SegmentInfo info;
info.elnr = elnr;
info.order = order;
info.ndof = info.nv = 2;
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
info.edgenr = top.GetSegmentEdge (elnr+1)-1;
info.ndof += edgeorder[info.edgenr]-1;
}
return (info.ndof > info.nv);
}
void CurvedElements ::
CalcSegmentTransformation (double xi, SegmentIndex elnr,
Point<3> * x, Vec<3> * dxdxi, bool * curved)
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
// xi umrechnen
double lami[2] = { xi, 1-xi };
double dlami[2] = { 1, -1 };
double coarse_xi = 0;
double trans = 0;
for (int i = 0; i < 2; i++)
{
coarse_xi += hpref_el.param[i][0] * lami[i];
trans += hpref_el.param[i][0] * dlami[i];
}
mesh.coarsemesh->GetCurvedElements().CalcSegmentTransformation (coarse_xi, hpref_el.coarse_elnr, x, dxdxi, curved);
if (dxdxi) *dxdxi *= trans;
return;
}
Vector shapes, dshapes;
Array<Vec<3> > coefs;
SegmentInfo info;
info.elnr = elnr;
info.order = order;
info.ndof = info.nv = 2;
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
info.edgenr = top.GetSegmentEdge (elnr+1)-1;
info.ndof += edgeorder[info.edgenr]-1;
}
CalcElementShapes (info, xi, shapes);
GetCoefficients (info, coefs);
*x = 0;
for (int i = 0; i < shapes.Size(); i++)
*x += shapes(i) * coefs[i];
if (dxdxi)
{
CalcElementDShapes (info, xi, dshapes);
*dxdxi = 0;
for (int i = 0; i < shapes.Size(); i++)
for (int j = 0; j < 3; j++)
(*dxdxi)(j) += dshapes(i) * coefs[i](j);
}
if (curved)
*curved = (info.order > 1);
// cout << "Segment, |x| = " << Abs2(Vec<3> (*x) ) << endl;
}
void CurvedElements ::
CalcElementShapes (SegmentInfo & info, double xi, Vector & shapes) const
{
if (rational && info.order == 2)
{
shapes.SetSize(3);
double w = edgeweight[info.edgenr];
shapes(0) = xi*xi;
shapes(1) = (1-xi)*(1-xi);
shapes(2) = 2*w*xi*(1-xi);
shapes *= 1.0 / (1 + (w-1) *2*xi*(1-xi));
return;
}
shapes.SetSize(info.ndof);
shapes(0) = xi;
shapes(1) = 1-xi;
if (info.order >= 2)
{
if (mesh[info.elnr].p1 > mesh[info.elnr].p2)
xi = 1-xi;
CalcEdgeShape (edgeorder[info.edgenr], 2*xi-1, &shapes(2));
}
}
void CurvedElements ::
CalcElementDShapes (SegmentInfo & info, double xi, Vector & dshapes) const
{
if (rational && info.order == 2)
{
dshapes.SetSize(3);
double wi = edgeweight[info.edgenr];
double shapes[3];
shapes[0] = xi*xi;
shapes[1] = (1-xi)*(1-xi);
shapes[2] = 2*wi*xi*(1-xi);
double w = 1 + (wi-1) *2*xi*(1-xi);
double dw = (wi-1) * (2 - 4*xi);
dshapes(0) = 2*xi;
dshapes(1) = 2*(xi-1);
dshapes(2) = 2*wi*(1-2*xi);
for (int j = 0;j < 3; j++)
dshapes(j) = dshapes(j) / w - shapes[j] * dw / (w*w);
return;
}
dshapes.SetSize(info.ndof);
dshapes = 0;
dshapes(0) = 1;
dshapes(1) = -1;
// int order = edgeorder[info.edgenr];
if (info.order >= 2)
{
double fac = 2;
if (mesh[info.elnr].p1 > mesh[info.elnr].p2)
{
xi = 1-xi;
fac *= -1;
}
CalcEdgeDx (edgeorder[info.edgenr], 2*xi-1, &dshapes(2));
for (int i = 2; i < dshapes.Size(); i++)
dshapes(i) *= fac;
}
// ??? not implemented ????
}
void CurvedElements ::
GetCoefficients (SegmentInfo & info, Array<Vec<3> > & coefs) const
{
const Segment & el = mesh[info.elnr];
coefs.SetSize(info.ndof);
coefs[0] = Vec<3> (mesh[el.p1]);
coefs[1] = Vec<3> (mesh[el.p2]);
if (info.order >= 2)
{
int first = edgecoeffsindex[info.edgenr];
int next = edgecoeffsindex[info.edgenr+1];
for (int i = 0; i < next-first; i++)
coefs[i+2] = edgecoeffs[first+i];
}
}
// ********************** Transform surface elements *******************
bool CurvedElements :: IsSurfaceElementCurved (SurfaceElementIndex elnr) const
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
return mesh.coarsemesh->GetCurvedElements().IsSurfaceElementCurved (hpref_el.coarse_elnr);
}
const Element2d & el = mesh[elnr];
ELEMENT_TYPE type = el.GetType();
SurfaceElementInfo info;
info.elnr = elnr;
info.order = order;
switch (type)
{
case TRIG : info.nv = 3; break;
case QUAD : info.nv = 4; break;
case TRIG6: return true;
default:
cerr << "undef element in CalcSurfaceTrafo" << endl;
}
info.ndof = info.nv;
// info.ndof = info.nv = ( (type == TRIG) || (type == TRIG6) ) ? 3 : 4;
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
top.GetSurfaceElementEdges (elnr+1, info.edgenrs);
for (int i = 0; i < info.edgenrs.Size(); i++)
info.edgenrs[i]--;
info.facenr = top.GetSurfaceElementFace (elnr+1)-1;
for (int i = 0; i < info.edgenrs.Size(); i++)
info.ndof += edgecoeffsindex[info.edgenrs[i]+1] - edgecoeffsindex[info.edgenrs[i]];
info.ndof += facecoeffsindex[info.facenr+1] - facecoeffsindex[info.facenr];
}
return (info.ndof > info.nv);
}
void CurvedElements ::
CalcSurfaceTransformation (Point<2> xi, SurfaceElementIndex elnr,
Point<3> * x, Mat<3,2> * dxdxi, bool * curved)
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
// xi umrechnen
double lami[4];
FlatVector vlami(4, lami);
vlami = 0;
mesh[elnr].GetShapeNew (xi, vlami);
Mat<2,2> trans;
Mat<3,2> dxdxic;
if (dxdxi)
{
MatrixFixWidth<2> dlami(4);
dlami = 0;
mesh[elnr].GetDShapeNew (xi, dlami);
trans = 0;
for (int k = 0; k < 2; k++)
for (int l = 0; l < 2; l++)
for (int i = 0; i < hpref_el.np; i++)
trans(l,k) += hpref_el.param[i][l] * dlami(i, k);
}
Point<2> coarse_xi(0,0);
for (int i = 0; i < hpref_el.np; i++)
for (int j = 0; j < 2; j++)
coarse_xi(j) += hpref_el.param[i][j] * lami[i];
mesh.coarsemesh->GetCurvedElements().CalcSurfaceTransformation (coarse_xi, hpref_el.coarse_elnr, x, &dxdxic, curved);
if (dxdxi)
*dxdxi = dxdxic * trans;
return;
}
Vector shapes;
DenseMatrix dshapes;
Array<Vec<3> > coefs;
const Element2d & el = mesh[elnr];
ELEMENT_TYPE type = el.GetType();
SurfaceElementInfo info;
info.elnr = elnr;
info.order = order;
switch (type)
{
case TRIG : info.nv = 3; break;
case QUAD : info.nv = 4; break;
case TRIG6: info.nv = 6; break;
default:
cerr << "undef element in CalcSurfaceTrafo" << endl;
}
info.ndof = info.nv;
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
top.GetSurfaceElementEdges (elnr+1, info.edgenrs);
for (int i = 0; i < info.edgenrs.Size(); i++)
info.edgenrs[i]--;
info.facenr = top.GetSurfaceElementFace (elnr+1)-1;
bool firsttry = true;
bool problem = false;
while(firsttry || problem)
{
problem = false;
for (int i = 0; !problem && i < info.edgenrs.Size(); i++)
{
if(info.edgenrs[i]+1 >= edgecoeffsindex.Size())
problem = true;
else
info.ndof += edgecoeffsindex[info.edgenrs[i]+1] - edgecoeffsindex[info.edgenrs[i]];
}
if(info.facenr+1 >= facecoeffsindex.Size())
problem = true;
else
info.ndof += facecoeffsindex[info.facenr+1] - facecoeffsindex[info.facenr];
if(problem && !firsttry)
throw NgException("something wrong with curved elements");
if(problem)
BuildCurvedElements(NULL,order,rational);
firsttry = false;
}
}
CalcElementShapes (info, xi, shapes);
GetCoefficients (info, coefs);
*x = 0;
for (int i = 0; i < coefs.Size(); i++)
*x += shapes(i) * coefs[i];
if (dxdxi)
{
CalcElementDShapes (info, xi, dshapes);
*dxdxi = 0;
for (int i = 0; i < coefs.Size(); i++)
for (int j = 0; j < 3; j++)
for (int k = 0; k < 2; k++)
(*dxdxi)(j,k) += dshapes(i,k) * coefs[i](j);
}
if (curved)
*curved = (info.ndof > info.nv);
}
void CurvedElements ::
CalcElementShapes (SurfaceElementInfo & info, const Point<2> & xi, Vector & shapes) const
{
const Element2d & el = mesh[info.elnr];
shapes.SetSize(info.ndof);
shapes = 0;
if (rational && info.order >= 2)
{
shapes.SetSize(6);
double w = 1;
double lami[3] = { xi(0), xi(1), 1-xi(0)-xi(1) };
for (int j = 0; j < 3; j++)
shapes(j) = lami[j] * lami[j];
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TRIG);
for (int j = 0; j < 3; j++)
{
double wi = edgeweight[info.edgenrs[j]];
shapes(j+3) = 2 * wi * lami[edges[j][0]-1] * lami[edges[j][1]-1];
w += (wi-1) * 2 * lami[edges[j][0]-1] * lami[edges[j][1]-1];
}
shapes *= 1.0 / w;
return;
}
switch (el.GetType())
{
case TRIG:
{
shapes(0) = xi(0);
shapes(1) = xi(1);
shapes(2) = 1-xi(0)-xi(1);
if (info.order == 1) return;
int ii = 3;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TRIG);
for (int i = 0; i < 3; i++)
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = edges[i][0]-1, vi2 = edges[i][1]-1;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcScaledEdgeShape (eorder, shapes(vi1)-shapes(vi2), shapes(vi1)+shapes(vi2), &shapes(ii));
ii += eorder-1;
}
}
int forder = faceorder[info.facenr];
if (forder >= 3)
{
int fnums[] = { 0, 1, 2 };
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
if (el[fnums[1]] > el[fnums[2]]) swap (fnums[1], fnums[2]);
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
CalcTrigShape (forder,
shapes(fnums[1])-shapes(fnums[0]),
1-shapes(fnums[1])-shapes(fnums[0]), &shapes(ii));
}
break;
}
case TRIG6:
{
if (shapes.Size() == 3)
{
shapes(0) = xi(0);
shapes(1) = xi(1);
shapes(2) = 1-xi(0)-xi(1);
}
else
{
double x = xi(0);
double y = xi(1);
double lam3 = 1-x-y;
shapes(0) = x * (2*x-1);
shapes(1) = y * (2*y-1);
shapes(2) = lam3 * (2*lam3-1);
shapes(3) = 4 * y * lam3;
shapes(4) = 4 * x * lam3;
shapes(5) = 4 * x * y;
}
break;
}
case QUAD:
{
shapes(0) = (1-xi(0))*(1-xi(1));
shapes(1) = xi(0) *(1-xi(1));
shapes(2) = xi(0) * xi(1) ;
shapes(3) = (1-xi(0))* xi(1) ;
if (info.order == 1) return;
double mu[4] = {
1 - xi(0) + 1 - xi(1),
xi(0) + 1 - xi(1),
xi(0) + xi(1),
1 - xi(0) + xi(1),
};
int ii = 4;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (QUAD);
for (int i = 0; i < 4; i++)
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = edges[i][0]-1, vi2 = edges[i][1]-1;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcEdgeShape (eorder, mu[vi1]-mu[vi2], &shapes(ii));
double lame = shapes(vi1)+shapes(vi2);
for (int j = 0; j < order-1; j++)
shapes(ii+j) *= lame;
ii += eorder-1;
}
}
for (int i = ii; i < info.ndof; i++)
shapes(i) = 0;
break;
}
};
}
void CurvedElements ::
CalcElementDShapes (SurfaceElementInfo & info, const Point<2> & xi, DenseMatrix & dshapes) const
{
const Element2d & el = mesh[info.elnr];
ELEMENT_TYPE type = el.GetType();
double lami[4];
dshapes.SetSize(info.ndof,2);
dshapes = 0;
// *testout << "calcelementdshapes, info.ndof = " << info.ndof << endl;
if (rational && info.order >= 2)
{
double w = 1;
double dw[2] = { 0, 0 };
lami[0] = xi(0); lami[1] = xi(1); lami[2] = 1-xi(0)-xi(1);
double dlami[3][2] = { { 1, 0 }, { 0, 1 }, { -1, -1 }};
double shapes[6];
for (int j = 0; j < 3; j++)
{
shapes[j] = lami[j] * lami[j];
dshapes(j,0) = 2 * lami[j] * dlami[j][0];
dshapes(j,1) = 2 * lami[j] * dlami[j][1];
}
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TRIG);
for (int j = 0; j < 3; j++)
{
double wi = edgeweight[info.edgenrs[j]];
shapes[j+3] = 2 * wi * lami[edges[j][0]-1] * lami[edges[j][1]-1];
for (int k = 0; k < 2; k++)
dshapes(j+3,k) = 2*wi* (lami[edges[j][0]-1] * dlami[edges[j][1]-1][k] +
lami[edges[j][1]-1] * dlami[edges[j][0]-1][k]);
w += (wi-1) * 2 * lami[edges[j][0]-1] * lami[edges[j][1]-1];
for (int k = 0; k < 2; k++)
dw[k] += 2*(wi-1) * (lami[edges[j][0]-1] * dlami[edges[j][1]-1][k] +
lami[edges[j][1]-1] * dlami[edges[j][0]-1][k]);
}
// shapes *= 1.0 / w;
dshapes *= 1.0 / w;
for (int i = 0; i < 6; i++)
for (int j = 0; j < 2; j++)
dshapes(i,j) -= shapes[i] * dw[j] / (w*w);
return;
}
switch (type)
{
case TRIG:
{
dshapes(0,0) = 1;
dshapes(1,1) = 1;
dshapes(2,0) = -1;
dshapes(2,1) = -1;
if (info.order == 1) return;
// *testout << "info.order = " << info.order << endl;
lami[0] = xi(0);
lami[1] = xi(1);
lami[2] = 1-xi(0)-xi(1);
int ii = 3;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TRIG);
for (int i = 0; i < 3; i++)
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = edges[i][0]-1, vi2 = edges[i][1]-1;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcScaledEdgeShapeDxDt<2> (eorder, lami[vi1]-lami[vi2], lami[vi1]+lami[vi2], &dshapes(ii,0));
Mat<2,2> trans;
for (int j = 0; j < 2; j++)
{
trans(0,j) = dshapes(vi1,j)-dshapes(vi2,j);
trans(1,j) = dshapes(vi1,j)+dshapes(vi2,j);
}
for (int j = 0; j < eorder-1; j++)
{
double ddx = dshapes(ii+j,0);
double ddt = dshapes(ii+j,1);
dshapes(ii+j,0) = ddx * trans(0,0) + ddt * trans(1,0);
dshapes(ii+j,1) = ddx * trans(0,1) + ddt * trans(1,1);
}
ii += eorder-1;
}
}
int forder = faceorder[info.facenr];
// *testout << "forder = " << forder << endl;
if (forder >= 3)
{
int fnums[] = { 0, 1, 2 };
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
if (el[fnums[1]] > el[fnums[2]]) swap (fnums[1], fnums[2]);
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
CalcTrigShapeDxDy (forder,
lami[fnums[1]]-lami[fnums[0]],
1-lami[fnums[1]]-lami[fnums[0]], &dshapes(ii,0));
int nd = (forder-1)*(forder-2)/2;
Mat<2,2> trans;
for (int j = 0; j < 2; j++)
{
trans(0,j) = dshapes(fnums[1],j)-dshapes(fnums[0],j);
trans(1,j) = -dshapes(fnums[1],j)-dshapes(fnums[0],j);
}
for (int j = 0; j < nd; j++)
{
double ddx = dshapes(ii+j,0);
double ddt = dshapes(ii+j,1);
dshapes(ii+j,0) = ddx * trans(0,0) + ddt * trans(1,0);
dshapes(ii+j,1) = ddx * trans(0,1) + ddt * trans(1,1);
}
}
break;
}
case TRIG6:
{
if (dshapes.Height() == 3)
{
dshapes = 0.0;
dshapes(0,0) = 1;
dshapes(1,1) = 1;
dshapes(2,0) = -1;
dshapes(2,1) = -1;
}
else
{
AutoDiff<2> x(xi(0), 0);
AutoDiff<2> y(xi(1), 1);
AutoDiff<2> lam3 = 1-x-y;
AutoDiff<2> shapes[6];
shapes[0] = x * (2*x-1);
shapes[1] = y * (2*y-1);
shapes[2] = lam3 * (2*lam3-1);
shapes[3] = 4 * y * lam3;
shapes[4] = 4 * x * lam3;
shapes[5] = 4 * x * y;
for (int i = 0; i < 6; i++)
{
dshapes(i,0) = shapes[i].DValue(0);
dshapes(i,1) = shapes[i].DValue(1);
}
}
break;
}
case QUAD:
{
dshapes(0,0) = -(1-xi(1));
dshapes(0,1) = -(1-xi(0));
dshapes(1,0) = (1-xi(1));
dshapes(1,1) = -xi(0);
dshapes(2,0) = xi(1);
dshapes(2,1) = xi(0);
dshapes(3,0) = -xi(1);
dshapes(3,1) = (1-xi(0));
if (info.order == 1) return;
double shapes[4] = {
(1-xi(0))*(1-xi(1)),
xi(0) *(1-xi(1)),
xi(0) * xi(1) ,
(1-xi(0))* xi(1)
};
double mu[4] = {
1 - xi(0) + 1 - xi(1),
xi(0) + 1 - xi(1),
xi(0) + xi(1),
1 - xi(0) + xi(1),
};
double dmu[4][2] = {
{ -1, -1 },
{ 1, -1 },
{ 1, 1 },
{ -1, 1 } };
// double hshapes[20], hdshapes[20];
ArrayMem<double, 20> hshapes(order+1), hdshapes(order+1);
int ii = 4;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (QUAD);
for (int i = 0; i < 4; i++)
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = edges[i][0]-1, vi2 = edges[i][1]-1;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcEdgeShapeDx (eorder, mu[vi1]-mu[vi2], &hshapes[0], &hdshapes[0]);
double lame = shapes[vi1]+shapes[vi2];
double dlame[2] = {
dshapes(vi1, 0) + dshapes(vi2, 0),
dshapes(vi1, 1) + dshapes(vi2, 1) };
for (int j = 0; j < eorder-1; j++)
for (int k = 0; k < 2; k++)
dshapes(ii+j, k) =
lame * hdshapes[j] * (dmu[vi1][k]-dmu[vi2][k])
+ dlame[k] * hshapes[j];
ii += eorder-1;
}
}
/*
*testout << "quad, dshape = " << endl << dshapes << endl;
for (int i = 0; i < 2; i++)
{
Point<2> xil = xi, xir = xi;
Vector shapesl(dshapes.Height()), shapesr(dshapes.Height());
xil(i) -= 1e-6;
xir(i) += 1e-6;
CalcElementShapes (info, xil, shapesl);
CalcElementShapes (info, xir, shapesr);
for (int j = 0; j < dshapes.Height(); j++)
dshapes(j,i) = 1.0 / 2e-6 * (shapesr(j)-shapesl(j));
}
*testout << "quad, num dshape = " << endl << dshapes << endl;
*/
break;
}
};
}
void CurvedElements ::
GetCoefficients (SurfaceElementInfo & info, Array<Vec<3> > & coefs) const
{
const Element2d & el = mesh[info.elnr];
coefs.SetSize (info.ndof);
// coefs = Vec<3> (0,0,0);
for (int i = 0; i < info.nv; i++)
coefs[i] = Vec<3> (mesh[el[i]]);
if (info.order == 1) return;
int ii = info.nv;
for (int i = 0; i < info.edgenrs.Size(); i++)
{
int first = edgecoeffsindex[info.edgenrs[i]];
int next = edgecoeffsindex[info.edgenrs[i]+1];
for (int j = first; j < next; j++, ii++)
coefs[ii] = edgecoeffs[j];
}
int first = facecoeffsindex[info.facenr];
int next = facecoeffsindex[info.facenr+1];
for (int j = first; j < next; j++, ii++)
coefs[ii] = facecoeffs[j];
}
// ********************** Transform volume elements *******************
bool CurvedElements :: IsElementCurved (ElementIndex elnr) const
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
return mesh.coarsemesh->GetCurvedElements().IsElementCurved (hpref_el.coarse_elnr);
}
const Element & el = mesh[elnr];
ELEMENT_TYPE type = el.GetType();
ElementInfo info;
info.elnr = elnr;
info.order = order;
info.ndof = info.nv = MeshTopology::GetNPoints (type);
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
info.nedges = top.GetElementEdges (elnr+1, info.edgenrs, 0);
for (int i = 0; i < info.nedges; i++)
info.edgenrs[i]--;
info.nfaces = top.GetElementFaces (elnr+1, info.facenrs, 0);
for (int i = 0; i < info.nfaces; i++)
info.facenrs[i]--;
for (int i = 0; i < info.nedges; i++)
info.ndof += edgecoeffsindex[info.edgenrs[i]+1] - edgecoeffsindex[info.edgenrs[i]];
for (int i = 0; i < info.nfaces; i++)
info.ndof += facecoeffsindex[info.facenrs[i]+1] - facecoeffsindex[info.facenrs[i]];
}
return (info.ndof > info.nv);
}
void CurvedElements ::
CalcElementTransformation (Point<3> xi, ElementIndex elnr,
Point<3> * x, Mat<3,3> * dxdxi, // bool * curved,
void * buffer, bool valid)
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
// xi umrechnen
double lami[8];
FlatVector vlami(8, lami);
vlami = 0;
mesh[elnr].GetShapeNew (xi, vlami);
Mat<3,3> trans, dxdxic;
if (dxdxi)
{
MatrixFixWidth<3> dlami(8);
dlami = 0;
mesh[elnr].GetDShapeNew (xi, dlami);
trans = 0;
for (int k = 0; k < 3; k++)
for (int l = 0; l < 3; l++)
for (int i = 0; i < hpref_el.np; i++)
trans(l,k) += hpref_el.param[i][l] * dlami(i, k);
}
Point<3> coarse_xi(0,0,0);
for (int i = 0; i < hpref_el.np; i++)
for (int j = 0; j < 3; j++)
coarse_xi(j) += hpref_el.param[i][j] * lami[i];
mesh.coarsemesh->GetCurvedElements().CalcElementTransformation (coarse_xi, hpref_el.coarse_elnr, x, &dxdxic /* , curved */);
if (dxdxi)
*dxdxi = dxdxic * trans;
return;
}
Vector shapes;
MatrixFixWidth<3> dshapes;
const Element & el = mesh[elnr];
ELEMENT_TYPE type = el.GetType();
ElementInfo hinfo;
ElementInfo & info = (buffer) ? *static_cast<ElementInfo*> (buffer) : hinfo;
if (!valid)
{
info.elnr = elnr;
info.order = order;
info.ndof = info.nv = MeshTopology::GetNPoints (type);
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
info.nedges = top.GetElementEdges (elnr+1, info.edgenrs, 0);
for (int i = 0; i < info.nedges; i++)
info.edgenrs[i]--;
info.nfaces = top.GetElementFaces (elnr+1, info.facenrs, 0);
for (int i = 0; i < info.nfaces; i++)
info.facenrs[i]--;
for (int i = 0; i < info.nedges; i++)
info.ndof += edgecoeffsindex[info.edgenrs[i]+1] - edgecoeffsindex[info.edgenrs[i]];
for (int i = 0; i < info.nfaces; i++)
info.ndof += facecoeffsindex[info.facenrs[i]+1] - facecoeffsindex[info.facenrs[i]];
}
}
CalcElementShapes (info, xi, shapes);
Vec<3> * coefs = (info.ndof <= 10) ?
&info.hcoefs[0] : new Vec<3> [info.ndof];
if (info.ndof > 10 || !valid)
GetCoefficients (info, coefs);
if (x)
{
*x = 0;
for (int i = 0; i < shapes.Size(); i++)
*x += shapes(i) * coefs[i];
}
if (dxdxi)
{
if (valid && info.order == 1 && info.nv == 4) // a linear tet
{
*dxdxi = info.hdxdxi;
}
else
{
CalcElementDShapes (info, xi, dshapes);
*dxdxi = 0;
for (int i = 0; i < shapes.Size(); i++)
for (int j = 0; j < 3; j++)
for (int k = 0; k < 3; k++)
(*dxdxi)(j,k) += dshapes(i,k) * coefs[i](j);
info.hdxdxi = *dxdxi;
}
}
// *testout << "curved_elements, dshapes = " << endl << dshapes << endl;
// if (curved) *curved = (info.ndof > info.nv);
if (info.ndof > 10) delete [] coefs;
}
void CurvedElements :: CalcElementShapes (ElementInfo & info, const Point<3> & xi, Vector & shapes) const
{
const Element & el = mesh[info.elnr];
if (rational && info.order >= 2)
{
shapes.SetSize(10);
double w = 1;
double lami[4] = { xi(0), xi(1), xi(2), 1-xi(0)-xi(1)-xi(2) };
for (int j = 0; j < 4; j++)
shapes(j) = lami[j] * lami[j];
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TET);
for (int j = 0; j < 6; j++)
{
double wi = edgeweight[info.edgenrs[j]];
shapes(j+4) = 2 * wi * lami[edges[j][0]-1] * lami[edges[j][1]-1];
w += (wi-1) * 2 * lami[edges[j][0]-1] * lami[edges[j][1]-1];
}
shapes *= 1.0 / w;
return;
}
shapes.SetSize(info.ndof);
switch (el.GetType())
{
case TET:
{
shapes(0) = xi(0);
shapes(1) = xi(1);
shapes(2) = xi(2);
shapes(3) = 1-xi(0)-xi(1)-xi(2);
if (info.order == 1) return;
int ii = 4;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TET);
for (int i = 0; i < 6; i++)
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = edges[i][0]-1, vi2 = edges[i][1]-1;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcScaledEdgeShape (eorder, shapes(vi1)-shapes(vi2), shapes(vi1)+shapes(vi2), &shapes(ii));
ii += eorder-1;
}
}
const ELEMENT_FACE * faces = MeshTopology::GetFaces (TET);
for (int i = 0; i < 4; i++)
{
int forder = faceorder[info.facenrs[i]];
if (forder >= 3)
{
int fnums[] = { faces[i][0]-1, faces[i][1]-1, faces[i][2]-1 };
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
if (el[fnums[1]] > el[fnums[2]]) swap (fnums[1], fnums[2]);
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
CalcScaledTrigShape (forder,
shapes(fnums[1])-shapes(fnums[0]), shapes(fnums[2]),
shapes(fnums[0])+shapes(fnums[1])+shapes(fnums[2]), &shapes(ii));
ii += (forder-1)*(forder-2)/2;
// CalcScaledEdgeShape (forder, shapes(vi1)-shapes(vi2), shapes(vi1)+shapes(vi2), &shapes(ii));
// ii += forder-1;
}
}
break;
}
case TET10:
{
double x = xi(0);
double y = xi(1);
double z = xi(2);
double lam4 = 1 - x - y - z;
/*
shapes(0) = xi(0);
shapes(1) = xi(1);
shapes(2) = xi(2);
shapes(3) = 1-xi(0)-xi(1)-xi(2);
*/
shapes(0) = 2 * x * x - x;
shapes(1) = 2 * y * y - y;
shapes(2) = 2 * z * z - z;
shapes(3) = 2 * lam4 * lam4 - lam4;
shapes(4) = 4 * x * y;
shapes(5) = 4 * x * z;
shapes(6) = 4 * x * lam4;
shapes(7) = 4 * y * z;
shapes(8) = 4 * y * lam4;
shapes(9) = 4 * z * lam4;
break;
}
case PRISM:
{
double lami[6] = { xi(0), xi(1), 1-xi(0)-xi(1), xi(0), xi(1), 1-xi(0)-xi(1) };
double lamiz[6] = { 1-xi(2), 1-xi(2), 1-xi(2), xi(2), xi(2), xi(2) };
for (int i = 0; i < 6; i++)
shapes(i) = lami[i%3] * ( (i < 3) ? (1-xi(2)) : xi(2) );
for (int i = 6; i < info.ndof; i++)
shapes(i) = 0;
if (info.order == 1) return;
int ii = 6;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (PRISM);
for (int i = 0; i < 6; i++) // horizontal edges
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = (edges[i][0]-1) % 3, vi2 = (edges[i][1]-1) % 3;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcScaledEdgeShape (eorder, lami[vi1]-lami[vi2], lami[vi1]+lami[vi2], &shapes(ii));
double facz = (i < 3) ? (1-xi(2)) : xi(2);
for (int j = 0; j < eorder-1; j++)
shapes(ii+j) *= facz;
ii += eorder-1;
}
}
for (int i = 6; i < 9; i++) // vertical edges
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = (edges[i][0]-1), vi2 = (edges[i][1]-1);
if (el[vi1] > el[vi2]) swap (vi1, vi2);
double bubz = lamiz[vi1]*lamiz[vi2];
double polyz = lamiz[vi1] - lamiz[vi2];
double bubxy = lami[(vi1)%3];
for (int j = 0; j < eorder-1; j++)
{
shapes(ii+j) = bubxy * bubz;
bubz *= polyz;
}
ii += eorder-1;
}
}
// FACE SHAPES
const ELEMENT_FACE * faces = MeshTopology::GetFaces (PRISM);
for (int i = 0; i < 2; i++)
{
int forder = faceorder[info.facenrs[i]];
if ( forder < 3 ) continue;
int fav[3] = { faces[i][0]-1, faces[i][1]-1, faces[i][2]-1 };
if(el[fav[0]] > el[fav[1]]) swap(fav[0],fav[1]);
if(el[fav[1]] > el[fav[2]]) swap(fav[1],fav[2]);
if(el[fav[0]] > el[fav[1]]) swap(fav[0],fav[1]);
CalcTrigShape (forder,
lami[fav[2]]-lami[fav[1]], lami[fav[0]],
&shapes(ii));
int ndf = (forder+1)*(forder+2)/2 - 3 - 3*(forder-1);
for ( int j = 0; j < ndf; j++ )
shapes(ii+j) *= lamiz[fav[1]];
ii += ndf;
}
break;
}
case PYRAMID:
{
shapes = 0.0;
double x = xi(0);
double y = xi(1);
double z = xi(2);
if (z == 1.) z = 1-1e-10;
shapes[0] = (1-z-x)*(1-z-y) / (1-z);
shapes[1] = x*(1-z-y) / (1-z);
shapes[2] = x*y / (1-z);
shapes[3] = (1-z-x)*y / (1-z);
shapes[4] = z;
if (info.order == 1) return;
int ii = 5;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (PYRAMID);
for (int i = 0; i < 4; i++) // horizontal edges
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = (edges[i][0]-1), vi2 = (edges[i][1]-1);
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcScaledEdgeShape (eorder, shapes[vi1]-shapes[vi2], shapes[vi1]+shapes[vi2], &shapes(ii));
double fac = (shapes[vi1]+shapes[vi2]) / (1-z);
for (int j = 0; j < eorder-1; j++)
shapes(ii+j) *= fac;
ii += eorder-1;
}
}
break;
}
case HEX:
{
shapes = 0.0;
double x = xi(0);
double y = xi(1);
double z = xi(2);
shapes[0] = (1-x)*(1-y)*(1-z);
shapes[1] = x *(1-y)*(1-z);
shapes[2] = x * y *(1-z);
shapes[3] = (1-x)* y *(1-z);
shapes[4] = (1-x)*(1-y)*(z);
shapes[5] = x *(1-y)*(z);
shapes[6] = x * y *(z);
shapes[7] = (1-x)* y *(z);
break;
}
};
}
void CurvedElements ::
CalcElementDShapes (ElementInfo & info, const Point<3> & xi, MatrixFixWidth<3> & dshapes) const
{
const Element & el = mesh[info.elnr];
dshapes.SetSize(info.ndof);
dshapes = 0.0;
if (rational && info.order >= 2)
{
double w = 1;
double dw[3] = { 0, 0, 0 };
double lami[4] = { xi(0), xi(1), xi(2), 1-xi(0)-xi(1)-xi(2) };
double dlami[4][3] = { { 1, 0, 0 }, { 0, 1, 0 }, { 0, 0, 1 }, { -1, -1, -1 }};
double shapes[10];
for (int j = 0; j < 4; j++)
{
shapes[j] = lami[j] * lami[j];
dshapes(j,0) = 2 * lami[j] * dlami[j][0];
dshapes(j,1) = 2 * lami[j] * dlami[j][1];
dshapes(j,2) = 2 * lami[j] * dlami[j][2];
}
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TET);
for (int j = 0; j < 6; j++)
{
double wi = edgeweight[info.edgenrs[j]];
shapes[j+4] = 2 * wi * lami[edges[j][0]-1] * lami[edges[j][1]-1];
for (int k = 0; k < 3; k++)
dshapes(j+4,k) = 2*wi* (lami[edges[j][0]-1] * dlami[edges[j][1]-1][k] +
lami[edges[j][1]-1] * dlami[edges[j][0]-1][k]);
w += (wi-1) * 2 * lami[edges[j][0]-1] * lami[edges[j][1]-1];
for (int k = 0; k < 3; k++)
dw[k] += 2*(wi-1) * (lami[edges[j][0]-1] * dlami[edges[j][1]-1][k] +
lami[edges[j][1]-1] * dlami[edges[j][0]-1][k]);
}
// shapes *= 1.0 / w;
dshapes *= 1.0 / w;
for (int i = 0; i < 10; i++)
for (int j = 0; j < 3; j++)
dshapes(i,j) -= shapes[i] * dw[j] / (w*w);
return;
}
switch (el.GetType())
{
case TET:
{
dshapes(0,0) = 1;
dshapes(1,1) = 1;
dshapes(2,2) = 1;
dshapes(3,0) = -1;
dshapes(3,1) = -1;
dshapes(3,2) = -1;
if (info.order == 1) return;
double lami[] = { xi(0), xi(1), xi(2), 1-xi(0)-xi(1)-xi(2) };
int ii = 4;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (TET);
for (int i = 0; i < 6; i++)
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = edges[i][0]-1, vi2 = edges[i][1]-1;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
CalcScaledEdgeShapeDxDt<3> (eorder, lami[vi1]-lami[vi2], lami[vi1]+lami[vi2], &dshapes(ii,0));
Mat<2,3> trans;
for (int j = 0; j < 3; j++)
{
trans(0,j) = dshapes(vi1,j)-dshapes(vi2,j);
trans(1,j) = dshapes(vi1,j)+dshapes(vi2,j);
}
for (int j = 0; j < order-1; j++)
{
double ddx = dshapes(ii+j,0);
double ddt = dshapes(ii+j,1);
dshapes(ii+j,0) = ddx * trans(0,0) + ddt * trans(1,0);
dshapes(ii+j,1) = ddx * trans(0,1) + ddt * trans(1,1);
dshapes(ii+j,2) = ddx * trans(0,2) + ddt * trans(1,2);
}
ii += eorder-1;
}
}
const ELEMENT_FACE * faces = MeshTopology::GetFaces (TET);
for (int i = 0; i < 4; i++)
{
int forder = faceorder[info.facenrs[i]];
if (forder >= 3)
{
int fnums[] = { faces[i][0]-1, faces[i][1]-1, faces[i][2]-1 };
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
if (el[fnums[1]] > el[fnums[2]]) swap (fnums[1], fnums[2]);
if (el[fnums[0]] > el[fnums[1]]) swap (fnums[0], fnums[1]);
CalcScaledTrigShapeDxDyDt (forder,
lami[fnums[1]]-lami[fnums[0]],
lami[fnums[2]], lami[fnums[0]]+lami[fnums[1]]+lami[fnums[2]],
&dshapes(ii,0));
Mat<3,3> trans;
for (int j = 0; j < 3; j++)
{
trans(0,j) = dshapes(fnums[1],j)-dshapes(fnums[0],j);
trans(1,j) = dshapes(fnums[2],j);
trans(2,j) = dshapes(fnums[0],j)+dshapes(fnums[1],j)+dshapes(fnums[2],j);
}
int nfd = (forder-1)*(forder-2)/2;
for (int j = 0; j < nfd; j++)
{
double ddx = dshapes(ii+j,0);
double ddy = dshapes(ii+j,1);
double ddt = dshapes(ii+j,2);
dshapes(ii+j,0) = ddx * trans(0,0) + ddy * trans(1,0) + ddt * trans(2,0);
dshapes(ii+j,1) = ddx * trans(0,1) + ddy * trans(1,1) + ddt * trans(2,1);
dshapes(ii+j,2) = ddx * trans(0,2) + ddy * trans(1,2) + ddt * trans(2,2);
}
ii += nfd;
}
}
break;
}
case TET10:
{
if (dshapes.Height() == 4)
{
dshapes = 0.0;
dshapes(0,0) = 1;
dshapes(1,1) = 1;
dshapes(2,2) = 1;
dshapes(3,0) = -1;
dshapes(3,1) = -1;
dshapes(3,2) = -1;
}
else
{
AutoDiff<3> x(xi(0), 0);
AutoDiff<3> y(xi(1), 1);
AutoDiff<3> z(xi(2), 2);
AutoDiff<3> lam4 = 1-x-y-z;
AutoDiff<3> shapes[10];
shapes[0] = 2 * x * x - x;
shapes[1] = 2 * y * y - y;
shapes[2] = 2 * z * z - z;
shapes[3] = 2 * lam4 * lam4 - lam4;
shapes[4] = 4 * x * y;
shapes[5] = 4 * x * z;
shapes[6] = 4 * x * lam4;
shapes[7] = 4 * y * z;
shapes[8] = 4 * y * lam4;
shapes[9] = 4 * z * lam4;
for (int i = 0; i < 10; i++)
{
dshapes(i,0) = shapes[i].DValue(0);
dshapes(i,1) = shapes[i].DValue(1);
dshapes(i,2) = shapes[i].DValue(2);
}
}
break;
break;
}
case PRISM:
{
double lami[6] = { xi(0), xi(1), 1-xi(0)-xi(1), xi(0), xi(1), 1-xi(0)-xi(1) };
double lamiz[6] = { 1-xi(2), 1-xi(2), 1-xi(2), xi(2), xi(2), xi(2) };
double dlamiz[6] = { -1, -1, -1, 1, 1, 1 };
double dlami[6][2] =
{ { 1, 0, },
{ 0, 1, },
{ -1, -1 },
{ 1, 0, },
{ 0, 1, },
{ -1, -1 } };
for (int i = 0; i < 6; i++)
{
// shapes(i) = lami[i%3] * ( (i < 3) ? (1-xi(2)) : xi(2) );
dshapes(i,0) = dlami[i%3][0] * ( (i < 3) ? (1-xi(2)) : xi(2) );
dshapes(i,1) = dlami[i%3][1] * ( (i < 3) ? (1-xi(2)) : xi(2) );
dshapes(i,2) = lami[i%3] * ( (i < 3) ? -1 : 1 );
}
int ii = 6;
if (info.order == 1) return;
const ELEMENT_EDGE * edges = MeshTopology::GetEdges (PRISM);
for (int i = 0; i < 6; i++) // horizontal edges
{
int order = edgeorder[info.edgenrs[i]];
if (order >= 2)
{
int vi1 = (edges[i][0]-1) % 3, vi2 = (edges[i][1]-1) % 3;
if (el[vi1] > el[vi2]) swap (vi1, vi2);
Vector shapei(order+1);
CalcScaledEdgeShapeDxDt<3> (order, lami[vi1]-lami[vi2], lami[vi1]+lami[vi2], &dshapes(ii,0) );
CalcScaledEdgeShape(order, lami[vi1]-lami[vi2], lami[vi1]+lami[vi2], &shapei(0) );
Mat<2,2> trans;
for (int j = 0; j < 2; j++)
{
trans(0,j) = dlami[vi1][j]-dlami[vi2][j];
trans(1,j) = dlami[vi1][j]+dlami[vi2][j];
}
for (int j = 0; j < order-1; j++)
{
double ddx = dshapes(ii+j,0);
double ddt = dshapes(ii+j,1);
dshapes(ii+j,0) = ddx * trans(0,0) + ddt * trans(1,0);
dshapes(ii+j,1) = ddx * trans(0,1) + ddt * trans(1,1);
}
double facz = (i < 3) ? (1-xi(2)) : xi(2);
double dfacz = (i < 3) ? (-1) : 1;
for (int j = 0; j < order-1; j++)
{
dshapes(ii+j,0) *= facz;
dshapes(ii+j,1) *= facz;
dshapes(ii+j,2) = shapei(j) * dfacz;
}
ii += order-1;
}
}
for (int i = 6; i < 9; i++) // vertical edges
{
int eorder = edgeorder[info.edgenrs[i]];
if (eorder >= 2)
{
int vi1 = (edges[i][0]-1), vi2 = (edges[i][1]-1);
if (el[vi1] > el[vi2]) swap (vi1, vi2);
double bubz = lamiz[vi1] * lamiz[vi2];
double dbubz = dlamiz[vi1]*lamiz[vi2] + lamiz[vi1]*dlamiz[vi2];
double polyz = lamiz[vi1] - lamiz[vi2];
double dpolyz = dlamiz[vi1] - dlamiz[vi2];
double bubxy = lami[(vi1)%3];
double dbubxydx = dlami[(vi1)%3][0];
double dbubxydy = dlami[(vi1)%3][1];
for (int j = 0; j < eorder-1; j++)
{
dshapes(ii+j,0) = dbubxydx * bubz;
dshapes(ii+j,1) = dbubxydy * bubz;
dshapes(ii+j,2) = bubxy * dbubz;
dbubz = bubz * dpolyz + dbubz * polyz;
bubz *= polyz;
}
ii += eorder-1;
}
}
if (info.order == 2) return;
// FACE SHAPES
const ELEMENT_FACE * faces = MeshTopology::GetFaces (PRISM);
for (int i = 0; i < 2; i++)
{
int forder = faceorder[info.facenrs[i]];
if ( forder < 3 ) continue;
int ndf = (forder+1)*(forder+2)/2 - 3 - 3*(forder-1);
int fav[3] = { faces[i][0]-1, faces[i][1]-1, faces[i][2]-1 };
if(el[fav[0]] > el[fav[1]]) swap(fav[0],fav[1]);
if(el[fav[1]] > el[fav[2]]) swap(fav[1],fav[2]);
if(el[fav[0]] > el[fav[1]]) swap(fav[0],fav[1]);
MatrixFixWidth<2> dshapei(ndf);
Vector shapei(ndf);
CalcTrigShapeDxDy (forder,
lami[fav[2]]-lami[fav[1]], lami[fav[0]],
&dshapei(0,0));
CalcTrigShape (forder, lami[fav[2]]-lami[fav[1]], lami[fav[0]],
&shapei(0));
Mat<2,2> trans;
for (int j = 0; j < 2; j++)
{
trans(0,j) = dlami[fav[2]][j]-dlami[fav[1]][j];
trans(1,j) = dlami[fav[0]][j];
}
for (int j = 0; j < order-1; j++)
{
double ddx = dshapes(ii+j,0);
double ddt = dshapes(ii+j,1);
dshapes(ii+j,0) = ddx * trans(0,0) + ddt * trans(1,0);
dshapes(ii+j,1) = ddx * trans(0,1) + ddt * trans(1,1);
}
for ( int j = 0; j < ndf; j++ )
{
dshapes(ii+j,0) *= lamiz[fav[1]];
dshapes(ii+j,1) *= lamiz[fav[1]];
dshapes(ii+j,2) = shapei(j) * dlamiz[fav[1]];
}
ii += ndf;
}
break;
}
case PYRAMID:
{
dshapes = 0.0;
double x = xi(0);
double y = xi(1);
double z = xi(2);
if (z == 1.) z = 1-1e-10;
double z1 = 1-z;
double z2 = z1*z1;
dshapes(0,0) = -(z1-y)/z1;
dshapes(0,1) = -(z1-x)/z1;
dshapes(0,2) = ((x+y+2*z-2)*z1+(z1-y)*(z1-x))/z2;
dshapes(1,0) = (z1-y)/z1;
dshapes(1,1) = -x/z1;
dshapes(1,2) = (-x*z1+x*(z1-y))/z2;
dshapes(2,0) = y/z1;
dshapes(2,1) = x/z1;
dshapes(2,2) = x*y/z2;
dshapes(3,0) = -y/z1;
dshapes(3,1) = (z1-x)/z1;
dshapes(3,2) = (-y*z1+y*(z1-x))/z2;
dshapes(4,0) = 0;
dshapes(4,1) = 0;
dshapes(4,2) = 1;
/* old:
vdshape[0] = Vec<3>( -(z1-y)/z1, -(z1-x)/z1, ((x+y+2*z-2)*z1+(z1-y)*(z1-x))/z2 );
vdshape[1] = Vec<3>( (z1-y)/z1, -x/z1, (-x*z1+x*(z1-y))/z2 );
vdshape[2] = Vec<3>( y/z1, x/z1, x*y/z2 );
vdshape[3] = Vec<3>( -y/z1, (z1-x)/z1, (-y*z1+y*(z1-x))/z2 );
vdshape[4] = Vec<3>( 0, 0, 1 );
*/
break;
}
case HEX:
{
dshapes = 0.0;
double x = xi(0);
double y = xi(1);
double z = xi(2);
// shapes[0] = (1-x)*(1-y)*(1-z);
dshapes(0,0) = - (1-y)*(1-z);
dshapes(0,1) = (1-x) * (-1) * (1-z);
dshapes(0,2) = (1-x) * (1-y) * (-1);
// shapes[1] = x *(1-y)*(1-z);
dshapes(1,0) = (1-y)*(1-z);
dshapes(1,1) = -x * (1-z);
dshapes(1,2) = -x * (1-y);
// shapes[2] = x * y *(1-z);
dshapes(2,0) = y * (1-z);
dshapes(2,1) = x * (1-z);
dshapes(2,2) = -x * y;
// shapes[3] = (1-x)* y *(1-z);
dshapes(3,0) = -y * (1-z);
dshapes(3,1) = (1-x) * (1-z);
dshapes(3,2) = -(1-x) * y;
// shapes[4] = (1-x)*(1-y)*z;
dshapes(4,0) = - (1-y)*z;
dshapes(4,1) = (1-x) * (-1) * z;
dshapes(4,2) = (1-x) * (1-y) * 1;
// shapes[5] = x *(1-y)*z;
dshapes(5,0) = (1-y)*z;
dshapes(5,1) = -x * z;
dshapes(5,2) = x * (1-y);
// shapes[6] = x * y *z;
dshapes(6,0) = y * z;
dshapes(6,1) = x * z;
dshapes(6,2) = x * y;
// shapes[7] = (1-x)* y *z;
dshapes(7,0) = -y * z;
dshapes(7,1) = (1-x) * z;
dshapes(7,2) = (1-x) * y;
break;
}
}
/*
DenseMatrix dshapes2 (info.ndof, 3);
Vector shapesl(info.ndof);
Vector shapesr(info.ndof);
double eps = 1e-6;
for (int i = 0; i < 3; i++)
{
Point<3> xl = xi;
Point<3> xr = xi;
xl(i) -= eps;
xr(i) += eps;
CalcElementShapes (info, xl, shapesl);
CalcElementShapes (info, xr, shapesr);
for (int j = 0; j < info.ndof; j++)
dshapes2(j,i) = (shapesr(j)-shapesl(j)) / (2*eps);
}
(*testout) << "dshapes = " << endl << dshapes << endl;
(*testout) << "dshapes2 = " << endl << dshapes2 << endl;
dshapes2 -= dshapes;
(*testout) << "diff = " << endl << dshapes2 << endl;
*/
}
void CurvedElements ::
GetCoefficients (ElementInfo & info, Vec<3> * coefs) const
{
// cout << "getcoeffs, info.elnr = " << info.elnr << endl;
// cout << "getcoeffs, info.nv = " << info.nv << endl;
const Element & el = mesh[info.elnr];
/*
coefs.SetSize (info.ndof);
coefs = Vec<3> (0,0,0);
*/
/*
for (int i = 0; i < info.ndof; i++)
coefs[i] = Vec<3> (0,0,0);
*/
for (int i = 0; i < info.nv; i++)
coefs[i] = Vec<3> (mesh[el[i]]);
if (info.order == 1) return;
int ii = info.nv;
for (int i = 0; i < info.nedges; i++)
{
int first = edgecoeffsindex[info.edgenrs[i]];
int next = edgecoeffsindex[info.edgenrs[i]+1];
for (int j = first; j < next; j++, ii++)
coefs[ii] = edgecoeffs[j];
}
for (int i = 0; i < info.nfaces; i++)
{
int first = facecoeffsindex[info.facenrs[i]];
int next = facecoeffsindex[info.facenrs[i]+1];
for (int j = first; j < next; j++, ii++)
coefs[ii] = facecoeffs[j];
}
}
void CurvedElements ::
CalcMultiPointSegmentTransformation (Array<double> * xi, SegmentIndex segnr,
Array<Point<3> > * x,
Array<Vec<3> > * dxdxi)
{
;
}
void CurvedElements ::
CalcMultiPointSurfaceTransformation (Array< Point<2> > * xi, SurfaceElementIndex elnr,
Array< Point<3> > * x,
Array< Mat<3,2> > * dxdxi)
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
// xi umrechnen
double lami[4];
FlatVector vlami(4, lami);
ArrayMem<Point<2>, 50> coarse_xi (xi->Size());
for (int pi = 0; pi < xi->Size(); pi++)
{
vlami = 0;
mesh[elnr].GetShapeNew ( (*xi)[pi], vlami);
Point<2> cxi(0,0);
for (int i = 0; i < hpref_el.np; i++)
for (int j = 0; j < 2; j++)
cxi(j) += hpref_el.param[i][j] * lami[i];
coarse_xi[pi] = cxi;
}
mesh.coarsemesh->GetCurvedElements().
CalcMultiPointSurfaceTransformation (&coarse_xi, hpref_el.coarse_elnr, x, dxdxi);
Mat<2,2> trans;
Mat<3,2> dxdxic;
if (dxdxi)
{
MatrixFixWidth<2> dlami(4);
dlami = 0;
for (int pi = 0; pi < xi->Size(); pi++)
{
mesh[elnr].GetDShapeNew ( (*xi)[pi], dlami);
trans = 0;
for (int k = 0; k < 2; k++)
for (int l = 0; l < 2; l++)
for (int i = 0; i < hpref_el.np; i++)
trans(l,k) += hpref_el.param[i][l] * dlami(i, k);
dxdxic = (*dxdxi)[pi];
(*dxdxi)[pi] = dxdxic * trans;
}
}
return;
}
Vector shapes;
DenseMatrix dshapes;
Array<Vec<3> > coefs;
const Element2d & el = mesh[elnr];
ELEMENT_TYPE type = el.GetType();
SurfaceElementInfo info;
info.elnr = elnr;
info.order = order;
switch (type)
{
case TRIG : info.nv = 3; break;
case QUAD : info.nv = 4; break;
case TRIG6: info.nv = 6; break;
default:
cerr << "undef element in CalcMultPointSurfaceTrao" << endl;
}
info.ndof = info.nv;
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
top.GetSurfaceElementEdges (elnr+1, info.edgenrs);
for (int i = 0; i < info.edgenrs.Size(); i++)
info.edgenrs[i]--;
info.facenr = top.GetSurfaceElementFace (elnr+1)-1;
for (int i = 0; i < info.edgenrs.Size(); i++)
info.ndof += edgecoeffsindex[info.edgenrs[i]+1] - edgecoeffsindex[info.edgenrs[i]];
info.ndof += facecoeffsindex[info.facenr+1] - facecoeffsindex[info.facenr];
}
GetCoefficients (info, coefs);
if (x)
{
for (int j = 0; j < xi->Size(); j++)
{
CalcElementShapes (info, (*xi)[j], shapes);
Point<3> val(0,0,0);
for (int i = 0; i < coefs.Size(); i++)
val += shapes(i) * coefs[i];
(*x)[j] = val;
}
}
if (dxdxi)
{
for (int ip = 0; ip < xi->Size(); ip++)
{
CalcElementDShapes (info, (*xi)[ip], dshapes);
/*
(*dxdxi)[ip] = 0;
for (int i = 0; i < coefs.Size(); i++)
for (int j = 0; j < 3; j++)
for (int k = 0; k < 2; k++)
(*dxdxi)[ip](j,k) += dshapes(i,k) * coefs[i](j);
*/
Mat<3,2> ds;
ds = 0.0;
for (int i = 0; i < coefs.Size(); i++)
for (int j = 0; j < 3; j++)
for (int k = 0; k < 2; k++)
ds(j,k) += dshapes(i,k) * coefs[i](j);
(*dxdxi)[ip] = ds;
}
}
}
void CurvedElements ::
CalcMultiPointElementTransformation (Array< Point<3> > * xi, ElementIndex elnr,
Array< Point<3> > * x,
Array< Mat<3,3> > * dxdxi)
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
// xi umrechnen
double lami[8];
FlatVector vlami(8, lami);
ArrayMem<Point<3>, 50> coarse_xi (xi->Size());
for (int pi = 0; pi < xi->Size(); pi++)
{
vlami = 0;
mesh[elnr].GetShapeNew ( (*xi)[pi], vlami);
Point<3> cxi(0,0,0);
for (int i = 0; i < hpref_el.np; i++)
for (int j = 0; j < 3; j++)
cxi(j) += hpref_el.param[i][j] * lami[i];
coarse_xi[pi] = cxi;
}
mesh.coarsemesh->GetCurvedElements().
CalcMultiPointElementTransformation (&coarse_xi, hpref_el.coarse_elnr, x, dxdxi);
Mat<3,3> trans, dxdxic;
if (dxdxi)
{
MatrixFixWidth<3> dlami(8);
dlami = 0;
for (int pi = 0; pi < xi->Size(); pi++)
{
mesh[elnr].GetDShapeNew ( (*xi)[pi], dlami);
trans = 0;
for (int k = 0; k < 3; k++)
for (int l = 0; l < 3; l++)
for (int i = 0; i < hpref_el.np; i++)
trans(l,k) += hpref_el.param[i][l] * dlami(i, k);
dxdxic = (*dxdxi)[pi];
(*dxdxi)[pi] = dxdxic * trans;
}
}
return;
}
Vector shapes;
MatrixFixWidth<3> dshapes;
const Element & el = mesh[elnr];
ELEMENT_TYPE type = el.GetType();
ElementInfo info;
info.elnr = elnr;
info.order = order;
info.ndof = info.nv = MeshTopology::GetNPoints (type);
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
info.nedges = top.GetElementEdges (elnr+1, info.edgenrs, 0);
for (int i = 0; i < info.nedges; i++)
info.edgenrs[i]--;
info.nfaces = top.GetElementFaces (elnr+1, info.facenrs, 0);
for (int i = 0; i < info.nfaces; i++)
info.facenrs[i]--;
for (int i = 0; i < info.nedges; i++)
info.ndof += edgecoeffsindex[info.edgenrs[i]+1] - edgecoeffsindex[info.edgenrs[i]];
for (int i = 0; i < info.nfaces; i++)
info.ndof += facecoeffsindex[info.facenrs[i]+1] - facecoeffsindex[info.facenrs[i]];
// info.ndof += facecoeffsindex[info.facenr+1] - facecoeffsindex[info.facenr];
}
Array<Vec<3> > coefs(info.ndof);
GetCoefficients (info, &coefs[0]);
if (x)
{
for (int j = 0; j < xi->Size(); j++)
{
CalcElementShapes (info, (*xi)[j], shapes);
(*x)[j] = 0;
for (int i = 0; i < coefs.Size(); i++)
(*x)[j] += shapes(i) * coefs[i];
}
}
if (dxdxi)
{
for (int ip = 0; ip < xi->Size(); ip++)
{
CalcElementDShapes (info, (*xi)[ip], dshapes);
/*
(*dxdxi)[ip] = 0;
for (int i = 0; i < coefs.Size(); i++)
for (int j = 0; j < 3; j++)
for (int k = 0; k < 3; k++)
(*dxdxi)[ip](j,k) += dshapes(i,k) * coefs[i](j);
*/
Mat<3,3> ds;
ds = 0;
for (int i = 0; i < coefs.Size(); i++)
for (int j = 0; j < 3; j++)
for (int k = 0; k < 3; k++)
ds(j,k) += dshapes(i,k) * coefs[i](j);
(*dxdxi)[ip] = ds;
}
}
}
void CurvedElements ::
CalcMultiPointElementTransformation (ElementIndex elnr, int n,
const double * xi, int sxi,
double * x, int sx,
double * dxdxi, int sdxdxi)
{
if (mesh.coarsemesh)
{
const HPRefElement & hpref_el =
(*mesh.hpelements) [mesh[elnr].hp_elnr];
// xi umrechnen
double lami[8];
FlatVector vlami(8, lami);
ArrayMem<double, 100> coarse_xi (3*n);
for (int pi = 0; pi < n; pi++)
{
vlami = 0;
Point<3> pxi;
for (int j = 0; j < 3; j++)
pxi(j) = xi[pi*sxi+j];
mesh[elnr].GetShapeNew ( pxi, vlami);
Point<3> cxi(0,0,0);
for (int i = 0; i < hpref_el.np; i++)
for (int j = 0; j < 3; j++)
cxi(j) += hpref_el.param[i][j] * lami[i];
for (int j = 0; j < 3; j++)
coarse_xi[3*pi+j] = cxi(j);
}
mesh.coarsemesh->GetCurvedElements().
CalcMultiPointElementTransformation (hpref_el.coarse_elnr, n,
&coarse_xi[0], 3,
x, sx,
dxdxi, sdxdxi);
Mat<3,3> trans, dxdxic;
if (dxdxi)
{
MatrixFixWidth<3> dlami(8);
dlami = 0;
for (int pi = 0; pi < n; pi++)
{
Point<3> pxi;
for (int j = 0; j < 3; j++)
pxi(j) = xi[pi*sxi+j];
mesh[elnr].GetDShapeNew (pxi, dlami);
trans = 0;
for (int k = 0; k < 3; k++)
for (int l = 0; l < 3; l++)
for (int i = 0; i < hpref_el.np; i++)
trans(l,k) += hpref_el.param[i][l] * dlami(i, k);
Mat<3> mat_dxdxic, mat_dxdxi;
for (int j = 0; j < 3; j++)
for (int k = 0; k < 3; k++)
mat_dxdxic(j,k) = dxdxi[pi*sdxdxi+3*j+k];
mat_dxdxi = mat_dxdxic * trans;
for (int j = 0; j < 3; j++)
for (int k = 0; k < 3; k++)
dxdxi[pi*sdxdxi+3*j+k] = mat_dxdxi(j,k);
// dxdxic = (*dxdxi)[pi];
// (*dxdxi)[pi] = dxdxic * trans;
}
}
return;
}
Vector shapes;
MatrixFixWidth<3> dshapes;
const Element & el = mesh[elnr];
ELEMENT_TYPE type = el.GetType();
ElementInfo info;
info.elnr = elnr;
info.order = order;
info.ndof = info.nv = MeshTopology::GetNPoints (type);
if (info.order > 1)
{
const MeshTopology & top = mesh.GetTopology();
info.nedges = top.GetElementEdges (elnr+1, info.edgenrs, 0);
for (int i = 0; i < info.nedges; i++)
info.edgenrs[i]--;
info.nfaces = top.GetElementFaces (elnr+1, info.facenrs, 0);
for (int i = 0; i < info.nfaces; i++)
info.facenrs[i]--;
for (int i = 0; i < info.nedges; i++)
info.ndof += edgecoeffsindex[info.edgenrs[i]+1] - edgecoeffsindex[info.edgenrs[i]];
for (int i = 0; i < info.nfaces; i++)
info.ndof += facecoeffsindex[info.facenrs[i]+1] - facecoeffsindex[info.facenrs[i]];
// info.ndof += facecoeffsindex[info.facenr+1] - facecoeffsindex[info.facenr];
}
Array<Vec<3> > coefs(info.ndof);
GetCoefficients (info, &coefs[0]);
if (x)
{
for (int j = 0; j < n; j++)
{
Point<3> xij, xj;
for (int k = 0; k < 3; k++)
xij(k) = xi[j*sxi+k];
CalcElementShapes (info, xij, shapes);
xj = 0;
for (int i = 0; i < coefs.Size(); i++)
xj += shapes(i) * coefs[i];
for (int k = 0; k < 3; k++)
x[j*sx+k] = xj(k);
}
}
if (dxdxi)
{
for (int ip = 0; ip < n; ip++)
{
Point<3> xij;
for (int k = 0; k < 3; k++)
xij(k) = xi[ip*sxi+k];
CalcElementDShapes (info, xij, dshapes);
Mat<3> dxdxij;
dxdxij = 0.0;
for (int i = 0; i < coefs.Size(); i++)
for (int j = 0; j < 3; j++)
for (int k = 0; k < 3; k++)
dxdxij(j,k) += dshapes(i,k) * coefs[i](j);
for (int j = 0; j < 3; j++)
for (int k = 0; k < 3; k++)
dxdxi[ip*sdxdxi+3*j+k] = dxdxij(j,k);
}
}
}
};
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