// spectral_decomp computes the spectral decomposition of b := F*tr(F) tensor
func (o *Ogden) b_and_spectral_decomp(F [][]float64) (err error) {
// determinant of F
o.J, err = tsr.Inv(o.Fi, F)
if err != nil {
return
}
// left Cauchy-Green tensor
tsr.LeftCauchyGreenDef(o.b, F)
// eigenvalues and eigenprojectors
tsr.Ten2Man(o.bm, o.b)
err = tsr.M_EigenValsProjsNum(o.P, o.λ, o.bm)
if err != nil {
return
}
o.λ[0] = math.Sqrt(o.λ[0])
o.λ[1] = math.Sqrt(o.λ[1])
o.λ[2] = math.Sqrt(o.λ[2])
return
}
// CalcD computes algorithmic tangent operator
func (o *PrincStrainsUp) CalcD(D [][]float64, s *State) (err error) {
// elastic response
if !s.Loading {
o.Mdl.ElastD(D, s)
return
}
// eigenvalues/projectors of trial elastic strain
err = tsr.M_EigenValsProjsNum(o.P, o.Lεetr, s.EpsTr)
if err != nil {
return
}
// derivatives of eigenprojectors w.r.t trial elastic strains
err = tsr.M_EigenProjsDerivAuto(o.dPdT, s.EpsTr, o.Lεetr, o.P)
if err != nil {
io.Pforan("EpsTr = %v\n", s.EpsTr)
io.Pforan("Lεetr = %v\n", o.Lεetr)
la.PrintMat("P", o.P, "%10g", false)
return
}
// eigenvalues of strains
err = tsr.M_EigenValsNum(o.Lεe, s.EpsE)
if err != nil {
return
}
// compute Lσ, De and Jacobian
o.Mdl.E_CalcSig(o.Lσ, o.Lεe)
err = o.Mdl.L_SecondDerivs(o.N, o.Nb, o.A, o.h, o.Mb, o.a, o.b, o.c, o.Lσ, s.Alp)
if err != nil {
return err
}
o.Mdl.E_CalcDe(o.De, o.Lεe)
o.calcJafterDerivs(o.J, o.Lεe, s.Alp, s.Dgam)
// invert Jacobian => Ji
err = la.MatInvG(o.Ji, o.J, 1e-10)
if err != nil {
return
}
// compute De and Dt = De * Ji
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
o.Dt[i][j] = 0
for k := 0; k < 3; k++ {
o.Dt[i][j] += o.De[i][k] * o.Ji[k][j]
}
}
}
// compute D
for i := 0; i < o.Nsig; i++ {
for j := 0; j < o.Nsig; j++ {
D[i][j] = 0.0
for k := 0; k < 3; k++ {
for l := 0; l < 3; l++ {
D[i][j] += o.Dt[k][l] * o.P[k][i] * o.P[l][j]
}
D[i][j] += o.Lσ[k] * o.dPdT[k][i][j]
}
}
}
return
}
// Update updates state
func (o *PrincStrainsUp) Update(s *State, ε, Δε []float64, eid, ipid int, time float64) (err error) {
// debugging
if o.DbgOn {
o.dbg_init(s, ε, Δε, eid, ipid)
}
// trial strains
for i := 0; i < o.Nsig; i++ {
s.EpsE[i] += Δε[i]
}
copy(s.EpsTr, s.EpsE)
// eigenvalues/projectors of trial elastic strain
err = tsr.M_EigenValsProjsNum(o.P, o.Lεetr, s.EpsTr)
if err != nil {
return
}
// trial stresses
o.Mdl.E_CalcSig(o.Lσ, o.Lεetr)
// debugging
if o.DbgOn {
defer o.dbg_end(s, ε, eid, ipid)() // TODO: fix this
}
// check loading condition => elastic update?
ftr := o.Mdl.L_YieldFunc(o.Lσ, s.Alp)
if ftr <= o.Fzero {
s.Dgam = 0
s.Loading = false
for i := 0; i < o.Nsig; i++ {
s.Sig[i] = o.Lσ[0]*o.P[0][i] + o.Lσ[1]*o.P[1][i] + o.Lσ[2]*o.P[2][i]
}
return
}
// initial values
for i := 0; i < 3; i++ {
o.x[i] = o.Lεetr[i]
}
for i := 0; i < o.Nalp; i++ {
o.αn[i] = s.Alp[i]
o.x[3+i] = s.Alp[i]
}
o.x[3+o.Nalp] = 0 // Δγ
// check Jacobian
if o.ChkJac {
var cnd float64
cnd, err = o.nls.CheckJ(o.x, o.ChkJacTol, true, o.ChkSilent)
io.Pfred("before: cnd(J) = %v\n", cnd)
}
// modify b
bsmp := o.Mdl.Get_bsmp()
if bsmp > 0 && o.Nbsmp > 1 {
o.Mdl.Set_bsmp(0)
defer func() { o.Mdl.Set_bsmp(bsmp) }()
δb := bsmp / float64(o.Nbsmp-1)
for i := 0; i < o.Nbsmp; i++ {
b := float64(i) * δb
err = o.do_solve(b, eid, ipid, time)
if err != nil {
return
}
}
} else {
err = o.do_solve(bsmp, eid, ipid, time)
if err != nil {
return
}
}
// check Jacobian again
if o.ChkJac {
var cnd float64
cnd, err = o.nls.CheckJ(o.x, o.ChkJacTol, true, o.ChkSilent)
io.Pfred("after: cnd(J) = %v\n", cnd)
if err != nil {
return
}
}
// set new state
εe, α, Δγ := o.x[:3], o.x[3:3+o.Nalp], o.x[3+o.Nalp]
o.Mdl.E_CalcSig(o.Lσ, εe)
for i := 0; i < o.Nsig; i++ {
s.Sig[i] = o.Lσ[0]*o.P[0][i] + o.Lσ[1]*o.P[1][i] + o.Lσ[2]*o.P[2][i]
s.EpsE[i] = εe[0]*o.P[0][i] + εe[1]*o.P[1][i] + εe[2]*o.P[2][i]
}
copy(s.Alp, α)
s.Dgam = Δγ
s.Loading = true
return
}