// CalcD computes D = dσ_new/dε_new consistent with StressUpdate
func (o *VonMises) CalcD(D [][]float64, s *State, firstIt bool) (err error) {
// set first Δγ
if firstIt {
s.Dgam = 0
}
// elastic
if !s.Loading {
return o.SmallElasticity.CalcD(D, s)
}
// elastoplastic => consistent stiffness
σ := s.Sig
Δγ := s.Dgam
p, q := tsr.M_p(σ), tsr.M_q(σ)
qtr := q + Δγ*3.0*o.G
m := 1.0 - Δγ*3.0*o.G/qtr
nstr := tsr.SQ2by3 * qtr // norm(str)
for i := 0; i < o.Nsig; i++ {
o.ten[i] = (σ[i] + p*tsr.Im[i]) / (m * nstr) // ten := unit(str) = snew / (m * nstr)
}
hp := 3.0*o.G + o.H
a1 := o.K
b2 := 6.0 * o.G * o.G * (Δγ/qtr - 1.0/hp)
for i := 0; i < o.Nsig; i++ {
for j := 0; j < o.Nsig; j++ {
D[i][j] = 2.0*o.G*m*tsr.Psd[i][j] + a1*tsr.Im[i]*tsr.Im[j] + b2*o.ten[i]*o.ten[j]
}
}
return
}
// Update updates stresses for given strains
func (o *DruckerPrager) Update(s *State, ε, Δε []float64, eid, ipid int, time float64) (err error) {
// set flags
s.Loading = false // => not elastoplastic
s.ApexReturn = false // => not return-to-apex
s.Dgam = 0 // Δγ := 0
// accessors
σ := s.Sig
α0 := &s.Alp[0]
// copy of α0 at beginning of step
α0ini := *α0
// trial stress
var devΔε_i float64
trΔε := Δε[0] + Δε[1] + Δε[2]
for i := 0; i < o.Nsig; i++ {
devΔε_i = Δε[i] - trΔε*tsr.Im[i]/3.0
o.ten[i] = σ[i] + o.K*trΔε*tsr.Im[i] + 2.0*o.G*devΔε_i // ten := σtr
}
ptr, qtr := tsr.M_p(o.ten), tsr.M_q(o.ten)
// trial yield function
ftr := qtr - o.M*ptr - o.qy0 - o.H*(*α0)
// elastic update
if ftr <= 0.0 {
copy(σ, o.ten) // σ := ten = σtr
return
}
// elastoplastic update
var str_i float64
hp := 3.0*o.G + o.K*o.M*o.Mb + o.H
s.Dgam = ftr / hp
*α0 += s.Dgam
pnew := ptr + s.Dgam*o.K*o.Mb
m := 1.0 - s.Dgam*3.0*o.G/qtr
for i := 0; i < o.Nsig; i++ {
str_i = o.ten[i] + ptr*tsr.Im[i]
σ[i] = m*str_i - pnew*tsr.Im[i]
}
s.Loading = true
// check for apex singularity
acone := qtr - s.Dgam*3.0*o.G
if acone < 0 {
s.Dgam = (-o.M*ptr - o.qy0 - o.H*α0ini) / (3.0*o.K*o.M + o.H)
*α0 = α0ini + s.Dgam
pnew = ptr + s.Dgam*3.0*o.K
for i := 0; i < o.Nsig; i++ {
σ[i] = -pnew * tsr.Im[i]
}
s.ApexReturn = true
}
return
}
// CalcD computes D = dσ_new/dε_new consistent with StressUpdate
func (o *DruckerPrager) CalcD(D [][]float64, s *State, firstIt bool) (err error) {
// set first Δγ
if firstIt {
s.Dgam = 0
}
// elastic
if !s.Loading {
return o.SmallElasticity.CalcD(D, s)
}
// return to apex
if s.ApexReturn {
a1 := o.K * o.H / (3.0*o.K*o.M + o.H)
for i := 0; i < o.Nsig; i++ {
for j := 0; j < o.Nsig; j++ {
D[i][j] = a1 * tsr.Im[i] * tsr.Im[j]
}
}
return
}
// elastoplastic => consistent stiffness
σ := s.Sig
Δγ := s.Dgam
p, q := tsr.M_p(σ), tsr.M_q(σ)
qtr := q + Δγ*3.0*o.G
m := 1.0 - Δγ*3.0*o.G/qtr
nstr := tsr.SQ2by3 * qtr // norm(str)
for i := 0; i < o.Nsig; i++ {
o.ten[i] = (σ[i] + p*tsr.Im[i]) / (m * nstr) // ten := unit(str) = snew / (m * nstr)
}
hp := 3.0*o.G + o.K*o.M*o.Mb + o.H
a1 := o.K - o.K*o.K*o.Mb*o.M/hp
a2 := -2.0 * o.G * o.K * o.Mb * tsr.SQ3by2 / hp
b1 := -tsr.SQ6 * o.G * o.M * o.K / hp
b2 := 6.0 * o.G * o.G * (Δγ/qtr - 1.0/hp)
for i := 0; i < o.Nsig; i++ {
for j := 0; j < o.Nsig; j++ {
D[i][j] = 2.0*o.G*m*tsr.Psd[i][j] +
a1*tsr.Im[i]*tsr.Im[j] +
a2*tsr.Im[i]*o.ten[j] +
b1*o.ten[i]*tsr.Im[j] +
b2*o.ten[i]*o.ten[j]
}
}
return
}
// Update updates stresses for given strains
func (o *VonMises) Update(s *State, ε, Δε []float64, eid, ipid int, time float64) (err error) {
// set flags
s.Loading = false // => not elastoplastic
s.ApexReturn = false // => not return-to-apex
s.Dgam = 0 // Δγ := 0
// accessors
σ := s.Sig
α0 := &s.Alp[0]
// trial stress
var devΔε_i float64
trΔε := Δε[0] + Δε[1] + Δε[2]
for i := 0; i < o.Nsig; i++ {
devΔε_i = Δε[i] - trΔε*tsr.Im[i]/3.0
o.ten[i] = σ[i] + o.K*trΔε*tsr.Im[i] + 2.0*o.G*devΔε_i // ten := σtr
}
ptr, qtr := tsr.M_p(o.ten), tsr.M_q(o.ten)
// trial yield function
ftr := qtr - o.qy0 - o.H*(*α0)
// elastic update
if ftr <= 0.0 {
copy(σ, o.ten) // σ := ten = σtr
return
}
// elastoplastic update
var str_i float64
hp := 3.0*o.G + o.H
s.Dgam = ftr / hp
*α0 += s.Dgam
pnew := ptr
m := 1.0 - s.Dgam*3.0*o.G/qtr
for i := 0; i < o.Nsig; i++ {
str_i = o.ten[i] + ptr*tsr.Im[i]
σ[i] = m*str_i - pnew*tsr.Im[i]
}
s.Loading = true
return
}
func (o *Plotter) Plot_p_q(x, y []float64, res []*State, sts [][]float64, last bool) {
// stress path
nr := len(res)
k := nr - 1
var xmi, xma, ymi, yma float64
for i := 0; i < nr; i++ {
x[i], y[i] = o.P[i], o.Q[i]
if o.Multq {
mult := fun.Sign(o.W[i])
y[i] *= mult
}
if o.UseOct {
x[i] *= tsr.SQ3
y[i] *= tsr.SQ2by3
}
if i == 0 {
xmi, xma = x[i], x[i]
ymi, yma = y[i], y[i]
} else {
xmi = min(xmi, x[i])
xma = max(xma, x[i])
ymi = min(ymi, y[i])
yma = max(yma, y[i])
}
if o.SMPon {
x[i], y[i], _ = tsr.M_pq_smp(res[i].Sig, o.SMPa, o.SMPb, o.SMPβ, o.SMPϵ)
}
}
plt.Plot(x, y, io.Sf("'r.', ls='%s', clip_on=0, color='%s', marker='%s', label=r'%s'", o.Ls, o.Clr, o.Mrk, o.Lbl))
plt.PlotOne(x[0], y[0], io.Sf("'bo', clip_on=0, color='%s', marker='%s', ms=%d", o.SpClr, o.SpMrk, o.SpMs))
plt.PlotOne(x[k], y[k], io.Sf("'bs', clip_on=0, color='%s', marker='%s', ms=%d", o.SpClr, o.EpMrk, o.EpMs))
// yield surface
if o.WithYs && o.m != nil {
mx, my := 1.0, 1.0
if o.UseOct {
mx, my = tsr.SQ3, tsr.SQ2by3
}
if o.UsePmin {
xmi = min(xmi, o.Pmin*mx)
}
if o.UsePmax {
xma = max(xma, o.Pmax*mx)
yma = max(yma, o.Pmax*my)
}
xmi, xma, ymi, yma = o.fix_range(xmi, xmi, xma, ymi, yma)
if o.PqLims != nil {
xmi, xma, ymi, yma = o.PqLims[0], o.PqLims[1], o.PqLims[2], o.PqLims[3]
}
//io.Pforan("xmi,xma ymi,yma = %v,%v %v,%v\n", xmi,xma, ymi,yma)
dx := (xma - xmi) / float64(o.NptsPq-1)
dy := (yma - ymi) / float64(o.NptsPq-1)
xx := la.MatAlloc(o.NptsPq, o.NptsPq)
yy := la.MatAlloc(o.NptsPq, o.NptsPq)
za := la.MatAlloc(o.NptsPq, o.NptsPq)
zb := la.MatAlloc(o.NptsPq, o.NptsPq)
var p, q, σa, σb, σc, λ0, λ1, λ2 float64
v := NewState(len(res[0].Sig), len(res[0].Alp), false, len(res[0].EpsE) > 0)
for k := 0; k < nr; k++ {
copy(v.Alp, res[k].Alp)
v.Dgam = res[k].Dgam
for i := 0; i < o.NptsPq; i++ {
for j := 0; j < o.NptsPq; j++ {
xx[i][j] = xmi + float64(i)*dx
yy[i][j] = ymi + float64(j)*dy
p, q = xx[i][j], yy[i][j]
if o.UseOct {
p /= tsr.SQ3
q /= tsr.SQ2by3
}
σa, σb, σc = tsr.PQW2O(p, q, o.W[k])
λ0, λ1, λ2 = tsr.O2L(σa, σb, σc)
v.Sig[0], v.Sig[1], v.Sig[2] = λ0, λ1, λ2
ys := o.m.YieldFuncs(v)
za[i][j] = ys[0]
if o.nsurf > 1 {
zb[i][j] = ys[1]
}
if o.SMPon {
xx[i][j], yy[i][j], _ = tsr.M_pq_smp(v.Sig, o.SMPa, o.SMPb, o.SMPβ, o.SMPϵ)
}
}
}
plt.ContourSimple(xx, yy, za, io.Sf("colors=['%s'], levels=[0], linestyles=['%s'], linewidths=[%g], clip_on=0", o.YsClr0, o.YsLs0, o.YsLw0)+o.ArgsYs)
if o.nsurf > 1 {
plt.ContourSimple(xx, yy, zb, io.Sf("colors=['%s'], levels=[0], linestyles=['%s'], linewidths=[%g], clip_on=0", o.YsClr1, o.YsLs1, o.YsLw1)+o.ArgsYs)
}
}
}
// predictor-corrector
if len(o.PreCor) > 1 {
var p, q, pnew, qnew float64
for i := 1; i < len(o.PreCor); i++ {
p = tsr.M_p(o.PreCor[i-1])
q = tsr.M_q(o.PreCor[i-1])
pnew = tsr.M_p(o.PreCor[i])
qnew = tsr.M_q(o.PreCor[i])
if o.UseOct {
p *= tsr.SQ3
pnew *= tsr.SQ3
q *= tsr.SQ2by3
qnew *= tsr.SQ2by3
}
if o.SMPon {
p, q, _ = tsr.M_pq_smp(o.PreCor[i-1], o.SMPa, o.SMPb, o.SMPβ, o.SMPϵ)
pnew, qnew, _ = tsr.M_pq_smp(o.PreCor[i], o.SMPa, o.SMPb, o.SMPβ, o.SMPϵ)
}
if math.Abs(pnew-p) > 1e-10 || math.Abs(qnew-q) > 1e-10 {
plt.Arrow(p, q, pnew, qnew, io.Sf("sc=%d, fc='%s', ec='%s'", o.ArrWid, o.ClrPC, o.ClrPC))
}
}
}
// settings
if last {
plt.Equal()
xl, yl := "$p_{cam}$", "$q_{cam}$"
if o.UseOct {
xl, yl = "$p_{oct}$", "$q_{oct}$"
}
if o.SMPon {
xl, yl = "$p_{smp}$", "$q_{smp}$"
}
if o.AxLblX != "" {
xl = o.AxLblX
}
if o.AxLblY != "" {
yl = o.AxLblY
}
plt.Gll(xl, yl, "leg_out=1, leg_ncol=4, leg_hlen=1.5")
if lims, ok := o.Lims["p,q"]; ok {
plt.AxisLims(lims)
}
if lims, ok := o.Lims["p,q,ys"]; ok {
plt.AxisLims(lims)
}
}
}
// YieldFs computes the yield functions
func (o DruckerPrager) YieldFuncs(s *State) []float64 {
p, q := tsr.M_p(s.Sig), tsr.M_q(s.Sig)
α0 := s.Alp[0]
return []float64{q - o.M*p - o.qy0 - o.H*α0}
}
// YieldFs computes the yield functions
func (o VonMises) YieldFuncs(s *State) []float64 {
q := tsr.M_q(s.Sig)
α0 := s.Alp[0]
return []float64{q - o.qy0 - o.H*α0}
}