// Calc_c computes the elastic/plastic transition zone
// TODO: check what's 'c' exactly
func (o *PressCylin) Calc_c(P float64) float64 {
var nls num.NlSolver
defer nls.Clean()
o.P_fx = P
Res := []float64{(o.a + o.b) / 2.0} // initial values
nls.Init(1, o.fx_fun, nil, o.dfdx_fun, true, false, nil)
nls.Solve(Res, true)
return Res[0]
}
// CalcEps0 computes initial strains from stresses (s.Sig)
// Note: initial strains are elastic strains => εe_ini := ε0
func (o *HyperElast1) CalcEps0(s *State) {
s0 := make([]float64, o.Nsig)
s0no, p0, q0 := tsr.M_devσ(s0, s.Sig)
ev0 := -p0 / o.K0
ed0 := q0 / (3.0 * o.G0)
if !o.le {
var nls num.NlSolver
nls.Init(2, func(fx, x []float64) error { // ev=x[0], ed=x[1]
εv, εd := x[0], x[1]
p, q := o.Calc_pq(εv, εd)
fx[0] = p - p0
fx[1] = q - q0
return nil
}, nil, func(J [][]float64, x []float64) (err error) {
εv, εd := x[0], x[1]
pv := o.pa * math.Exp(-o.a*εv)
Dvv := o.a * (1.0 + 1.5*o.a*o.κb*εd*εd) * pv
Dvd := -3.0 * o.a * o.κb * εd * pv
Ddd := 3.0 * (o.G0 + o.κb*pv)
J[0][0] = -Dvv
J[0][1] = -Dvd
J[1][0] = Dvd
J[1][1] = Ddd
return nil
}, true, false, map[string]float64{"lSearch": 0})
x := []float64{ev0, ed0}
nls.SetTols(1e-10, 1e-10, 1e-14, num.EPS)
//nls.ChkConv = false
//nls.CheckJ(x, 1e-6, true, false)
silent := true
err := nls.Solve(x, silent)
if err != nil {
chk.Panic("HyperElast1: CalcEps0: non-linear solver failed:\n%v", err)
}
ev0, ed0 = x[0], x[1]
}
if s0no > 0 {
for i := 0; i < o.Nsig; i++ {
s.EpsE[i] = ev0*tsr.Im[i]/3.0 + tsr.SQ3by2*ed0*(s0[i]/s0no)
}
return
}
for i := 0; i < o.Nsig; i++ {
s.EpsE[i] = ev0 * tsr.Im[i] / 3.0
}
}
// Run computes β starting witn an initial guess
func (o *ReliabFORM) Run(βtrial float64, verbose bool, args ...interface{}) (β float64, μ, σ, x []float64) {
// initial random variables
β = βtrial
nx := len(o.μ)
μ = make([]float64, nx) // mean values (equivalent normal value)
σ = make([]float64, nx) // deviation values (equivalent normal value)
x = make([]float64, nx) // current vector of random variables defining min(β)
for i := 0; i < nx; i++ {
μ[i] = o.μ[i]
σ[i] = o.σ[i]
x[i] = o.μ[i]
}
// lognormal distribution structure
var lnd DistLogNormal
// has lognormal random variable?
haslrv := false
for _, found := range o.lrv {
if found {
haslrv = true
break
}
}
// function to compute β with x-constant
// gβ(β) = g(μ - β・A・σ) = 0
var err error
gβfcn := func(fy, y []float64) error {
βtmp := y[0]
for i := 0; i < nx; i++ {
o.xtmp[i] = μ[i] - βtmp*o.α[i]*σ[i]
}
fy[0], err = o.gfcn(o.xtmp, args)
if err != nil {
chk.Panic("cannot compute gfcn(%v):\n%v", o.xtmp, err)
}
return nil
}
// derivative of gβ w.r.t β
hβfcn := func(dfdy [][]float64, y []float64) error {
βtmp := y[0]
for i := 0; i < nx; i++ {
o.xtmp[i] = μ[i] - βtmp*o.α[i]*σ[i]
}
err = o.hfcn(o.dgdx, o.xtmp, args)
if err != nil {
chk.Panic("cannot compute hfcn(%v):\n%v", o.xtmp, err)
}
dfdy[0][0] = 0
for i := 0; i < nx; i++ {
dfdy[0][0] -= o.dgdx[i] * o.α[i] * σ[i]
}
return nil
}
// nonlinear solver with y[0] = β
// solving: gβ(β) = g(μ - β・A・σ) = 0
var nls num.NlSolver
nls.Init(1, gβfcn, nil, hβfcn, true, false, nil)
defer nls.Clean()
// message
if verbose {
io.Pf("\n%s", io.StrThickLine(60))
}
// plotting
plot := o.PlotFnk != ""
if nx != 2 {
plot = false
}
if plot {
if o.PlotNp < 3 {
o.PlotNp = 41
}
var umin, umax, vmin, vmax float64
if o.PlotCf < 1 {
o.PlotCf = 2
}
if len(o.PlotUrange) == 0 {
umin, umax = μ[0]-o.PlotCf*μ[0], μ[0]+o.PlotCf*μ[0]
vmin, vmax = μ[1]-o.PlotCf*μ[1], μ[1]+o.PlotCf*μ[1]
} else {
chk.IntAssert(len(o.PlotUrange), 2)
chk.IntAssert(len(o.PlotVrange), 2)
umin, umax = o.PlotUrange[0], o.PlotUrange[1]
vmin, vmax = o.PlotVrange[0], o.PlotVrange[1]
}
o.PlotU, o.PlotV = utl.MeshGrid2D(umin, umax, vmin, vmax, o.PlotNp, o.PlotNp)
o.PlotZ = la.MatAlloc(o.PlotNp, o.PlotNp)
plt.SetForEps(0.8, 300)
for i := 0; i < o.PlotNp; i++ {
for j := 0; j < o.PlotNp; j++ {
o.xtmp[0] = o.PlotU[i][j]
o.xtmp[1] = o.PlotV[i][j]
o.PlotZ[i][j], err = o.gfcn(o.xtmp, args)
if err != nil {
chk.Panic("cannot compute gfcn(%v):\n%v", x, err)
}
}
}
plt.Contour(o.PlotU, o.PlotV, o.PlotZ, "")
plt.ContourSimple(o.PlotU, o.PlotV, o.PlotZ, true, 8, "levels=[0], colors=['yellow']")
plt.PlotOne(x[0], x[1], "'ro', label='initial'")
}
// iterations to find β
var dat VarData
B := []float64{β}
itB := 0
for itB = 0; itB < o.NmaxItB; itB++ {
// message
if verbose {
gx, err := o.gfcn(x, args)
if err != nil {
chk.Panic("cannot compute gfcn(%v):\n%v", x, err)
}
io.Pf("%s itB=%d β=%g g=%g\n", io.StrThinLine(60), itB, β, gx)
}
// plot
if plot {
plt.PlotOne(x[0], x[1], "'r.'")
}
// compute direction cosines
itA := 0
for itA = 0; itA < o.NmaxItA; itA++ {
// has lognormal random variable (lrv)
if haslrv {
// find equivalent normal mean and std deviation for lognormal variables
for i := 0; i < nx; i++ {
if o.lrv[i] {
// set distribution
dat.M, dat.S = o.μ[i], o.σ[i]
lnd.Init(&dat)
// update μ and σ
fx := lnd.Pdf(x[i])
Φinvx := (math.Log(x[i]) - lnd.M) / lnd.S
φx := math.Exp(-Φinvx*Φinvx/2.0) / math.Sqrt2 / math.SqrtPi
σ[i] = φx / fx
μ[i] = x[i] - Φinvx*σ[i]
}
}
}
// compute direction cosines
err = o.hfcn(o.dgdx, x, args)
if err != nil {
chk.Panic("cannot compute hfcn(%v):\n%v", x, err)
}
den := 0.0
for i := 0; i < nx; i++ {
den += math.Pow(o.dgdx[i]*σ[i], 2.0)
}
den = math.Sqrt(den)
αerr := 0.0 // difference on α
for i := 0; i < nx; i++ {
αnew := o.dgdx[i] * σ[i] / den
αerr += math.Pow(αnew-o.α[i], 2.0)
o.α[i] = αnew
}
αerr = math.Sqrt(αerr)
// message
if verbose {
io.Pf(" itA=%d\n", itA)
io.Pf("%12s%12s%12s%12s\n", "x", "μ", "σ", "α")
for i := 0; i < nx; i++ {
io.Pf("%12.3f%12.3f%12.3f%12.3f\n", x[i], μ[i], σ[i], o.α[i])
}
}
// update x-star
for i := 0; i < nx; i++ {
x[i] = μ[i] - β*o.α[i]*σ[i]
}
// check convergence on α
if itA > 1 && αerr < o.TolA {
if verbose {
io.Pfgrey(". . . converged on α with αerr=%g . . .\n", αerr)
}
break
}
}
// failed to converge on α
if itA == o.NmaxItA {
chk.Panic("failed to convege on α")
}
// compute new β
B[0] = β
nls.Solve(B, o.NlsSilent)
βerr := math.Abs(B[0] - β)
β = B[0]
if o.NlsCheckJ {
nls.CheckJ(B, o.NlsCheckJtol, true, false)
}
// update x-star
for i := 0; i < nx; i++ {
x[i] = μ[i] - β*o.α[i]*σ[i]
}
// check convergence on β
if βerr < o.TolB {
if verbose {
io.Pfgrey2(". . . converged on β with βerr=%g . . .\n", βerr)
}
break
}
}
// failed to converge on β
if itB == o.NmaxItB {
chk.Panic("failed to converge on β")
}
// message
if verbose {
gx, err := o.gfcn(x, args)
if err != nil {
chk.Panic("cannot compute gfcn(%v):\n%v", x, err)
}
io.Pfgreen("x = %v\n", x)
io.Pfgreen("g = %v\n", gx)
io.PfGreen("β = %v\n", β)
}
// plot
if plot {
plt.Gll("$x_0$", "$x_1$", "")
plt.Cross("")
plt.SaveD("/tmp/gosl", "fig_form_"+o.PlotFnk+".eps")
}
return
}
func main() {
PI := math.Pi
yf := func(x float64) float64 {
return 1.0 - math.Sqrt(x) - math.Sin(10.0*math.Pi*x)*x
}
dydx := func(x float64) float64 {
return -math.Sin(10.0*PI*x) - 10.0*PI*x*math.Cos(10.0*PI*x) - 1.0/(2.0*math.Sqrt(x))
}
var nlsDY num.NlSolver
nlsDY.Init(1, func(fx, x []float64) error {
fx[0] = dydx(x[0])
return nil
}, nil, nil, false, true, nil)
defer nlsDY.Clean()
X := []float64{0.09, 0.25, 0.45, 0.65, 0.85}
Y := make([]float64, len(X))
for i, x := range X {
// find min
xx := []float64{x}
err := nlsDY.Solve(xx, true)
if err != nil {
io.PfRed("dydx nls failed:\n%v", err)
return
}
X[i] = xx[0]
Y[i] = yf(X[i])
}
// find next point along horizontal line
Xnext := []float64{0.2, 0.4, 0.6, 0.8}
Ynext := make([]float64, len(Xnext))
for i, xnext := range Xnext {
var nlsX num.NlSolver
nlsX.Init(1, func(fx, x []float64) error {
fx[0] = Y[i] - yf(x[0])
return nil
}, nil, nil, false, true, nil)
defer nlsX.Clean()
xx := []float64{xnext}
err := nlsX.Solve(xx, true)
if err != nil {
io.PfRed("dydx nls failed:\n%v", err)
return
}
Xnext[i] = xx[0]
Ynext[i] = yf(Xnext[i])
}
// auxiliary points
XX := []float64{
0, X[0],
Xnext[0], X[1],
Xnext[1], X[2],
Xnext[2], X[3],
Xnext[3], X[4],
}
YY := []float64{
1, Y[0],
Ynext[0], Y[1],
Ynext[1], Y[2],
Ynext[2], Y[3],
Ynext[3], Y[4],
}
io.Pforan("XX = %.3f\n", XX)
io.Pforan("YY = %.3f\n", YY)
// find arc-length
arclen := 0.0
for i := 0; i < len(XX); i += 2 {
a := XX[i]
b := XX[i+1]
if i == 0 {
a += 1e-7
}
var quad num.Simp
quad.Init(func(x float64) float64 {
return math.Sqrt(1.0 + math.Pow(dydx(x), 2.0))
}, a, b, 1e-4)
res, err := quad.Integrate()
if err != nil {
io.PfRed("quad failed:\n%v", err)
return
}
arclen += res
io.Pf("int(...) from %.15f to %.15f = %g\n", a, b, res)
}
io.Pforan("arclen = %v\n", arclen)
np := 201
xx := utl.LinSpace(0, 1, np)
yy := make([]float64, np)
for i := 0; i < np; i++ {
yy[i] = yf(xx[i])
}
plt.Plot(xx, yy, "'b-', clip_on=0")
for i, x := range X {
plt.PlotOne(x, Y[i], "'r|', mew=2, ms=30, clip_on=0")
}
for i, x := range Xnext {
plt.PlotOne(x, Ynext[i], "'r_', mew=2, ms=30, clip_on=0")
}
for i := 0; i < len(XX); i += 2 {
x0, y0 := XX[i], YY[i]
x1, y1 := XX[i+1], YY[i+1]
plt.Arrow(x0, y0, x1, y1, "")
}
plt.SetXnticks(11)
plt.Gll("x", "y", "")
plt.SaveD("/tmp/goga", "calcZDT3pts.eps")
}