// CalcAtIp calculates volume data such as S and G at natural coordinate r
// Input:
// x[ndim][nverts+?] -- coordinates matrix of solid element
// ip -- integration point
// Output:
// S, DSdR, DxdR, DRdx, G, and J
func (o *Shape) CalcAtIp(x [][]float64, ip *Ipoint, derivs bool) (err error) {
// S and dSdR
o.Func(o.S, o.dSdR, ip.R, ip.S, ip.T, derivs)
if !derivs {
return
}
if o.Gndim == 1 {
// calculate Jvec3d == dxdR
for i := 0; i < len(x); i++ {
o.Jvec3d[i] = 0.0
for m := 0; m < o.Nverts; m++ {
o.Jvec3d[i] += x[i][m] * o.dSdR[m][0] // dxdR := x * dSdR
}
}
// calculate J = norm of Jvec3d
o.J = la.VecNorm(o.Jvec3d)
// calculate G
for m := 0; m < o.Nverts; m++ {
o.Gvec[m] = o.dSdR[m][0] / o.J
}
return
}
// dxdR := sum_n x * dSdR => dx_i/dR_j := sum_n x^n_i * dS^n/dR_j
for i := 0; i < len(x); i++ {
for j := 0; j < o.Gndim; j++ {
o.dxdR[i][j] = 0.0
for n := 0; n < o.Nverts; n++ {
o.dxdR[i][j] += x[i][n] * o.dSdR[n][j]
}
}
}
// dRdx := inv(dxdR)
o.J, err = la.MatInv(o.dRdx, o.dxdR, MINDET)
if err != nil {
return
}
// G == dSdx := dSdR * dRdx => dS^m/dR_i := sum_i dS^m/dR_i * dR_i/dx_j
la.MatMul(o.G, 1, o.dSdR, o.dRdx)
return
}
// InvMap computes the natural coordinates r, given the real coordinate y
// Input:
// y[ndim] -- are the 2D/3D point coordinates
// x[ndim][nverts+?] -- coordinates matrix of solid element
// Output:
// r[3] -- are the natural coordinates of given point
func (o *Shape) InvMap(r, y []float64, x [][]float64) (err error) {
// check
if o.Gndim == 1 {
return chk.Err("Inverse mapping is not implemented in 1D\n")
}
var δRnorm float64
e := make([]float64, o.Gndim) // residual
δr := make([]float64, o.Gndim) // corrector
r[0], r[1], r[2] = 0, 0, 0 // first trial
it := 0
derivs := true
for it = 0; it < INVMAP_NIT; it++ {
// shape functions and derivatives
o.Func(o.S, o.dSdR, r[0], r[1], r[2], derivs)
// residual: e = y - x * S
for i := 0; i < o.Gndim; i++ {
e[i] = y[i]
for j := 0; j < o.Nverts; j++ {
e[i] -= x[i][j] * o.S[j]
}
}
// Jmat == dxdR = x * dSdR;
for i := 0; i < len(x); i++ {
for j := 0; j < o.Gndim; j++ {
o.dxdR[i][j] = 0.0
for k := 0; k < o.Nverts; k++ {
o.dxdR[i][j] += x[i][k] * o.dSdR[k][j] // dxdR := x * dSdR
}
}
}
// Jimat == dRdx = Jmat.inverse();
o.J, err = la.MatInv(o.dRdx, o.dxdR, MINDET)
if err != nil {
return
}
// corrector: dR = Jimat * e
for i := 0; i < o.Gndim; i++ {
δr[i] = 0.0
for j := 0; j < o.Gndim; j++ {
δr[i] += o.dRdx[i][j] * e[j]
}
}
// converged?
δRnorm = 0.0
for i := 0; i < o.Gndim; i++ {
r[i] += δr[i]
δRnorm += δr[i] * δr[i]
// fix r outside range
if r[i] < -1.0 || r[i] > 1.0 {
if math.Abs(r[i]-(-1.0)) < INVMAP_TOL {
r[i] = -1.0
}
if math.Abs(r[i]-1.0) < INVMAP_TOL {
r[i] = 1.0
}
}
}
if math.Sqrt(δRnorm) < INVMAP_TOL {
break
}
}
// check
if it == INVMAP_NIT {
return
}
return
}