// CheckDSdR checks dSdR derivatives of shape structures
func CheckDSdR(tst *testing.T, shape *Shape, r []float64, tol float64, verbose bool) {
// auxiliary
r_tmp := make([]float64, len(r))
S_tmp := make([]float64, shape.Nverts)
// analytical
shape.Func(shape.S, shape.DSdR, r, true, -1)
// numerical
for n := 0; n < shape.Nverts; n++ {
for i := 0; i < shape.Gndim; i++ {
dSndRi, _ := num.DerivCentral(func(t float64, args ...interface{}) (Sn float64) {
copy(r_tmp, r)
r_tmp[i] = t
shape.Func(S_tmp, nil, r_tmp, false, -1)
Sn = S_tmp[n]
return
}, r[i], 1e-1)
if verbose {
io.Pfgrey2(" dS%ddR%d @ %5.2f = %v (num: %v)\n", n, i, r, shape.DSdR[n][i], dSndRi)
}
if math.Abs(shape.DSdR[n][i]-dSndRi) > tol {
tst.Errorf("nurbs dS%ddR%d failed with err = %g\n", n, i, math.Abs(shape.DSdR[n][i]-dSndRi))
return
}
}
}
}
// CheckDSdx checks G=dSdx derivatives of shape structures
func CheckDSdx(tst *testing.T, shape *Shape, xmat [][]float64, x []float64, tol float64, verbose bool) {
// find r corresponding to x
r := make([]float64, 3)
err := shape.InvMap(r, x, xmat)
if err != nil {
tst.Errorf("InvMap failed:\n%v", err)
return
}
// analytical
err = shape.CalcAtIp(xmat, r, true)
if err != nil {
tst.Errorf("CalcAtIp failed:\n%v", err)
return
}
// numerical
x_tmp := make([]float64, len(x))
for n := 0; n < shape.Nverts; n++ {
for i := 0; i < shape.Gndim; i++ {
dSnDxi, _ := num.DerivCentral(func(t float64, args ...interface{}) (Sn float64) {
copy(x_tmp, x)
x_tmp[i] = t
err = shape.InvMap(r, x_tmp, xmat)
if err != nil {
tst.Errorf("InvMap failed:\n%v", err)
return
}
err = shape.CalcAtIp(xmat, r, false)
if err != nil {
tst.Errorf("CalcAtIp failed:\n%v", err)
return
}
Sn = shape.S[n]
return
}, x[i], 1e-1)
if verbose {
io.Pfgrey2(" dS%dDx%d @ %5.2f = %v (num: %v)\n", n, i, x, shape.G[n][i], dSnDxi)
}
if math.Abs(shape.G[n][i]-dSnDxi) > tol {
tst.Errorf("nurbs dS%dDx%d failed with err = %g\n", n, i, math.Abs(shape.G[n][i]-dSnDxi))
return
}
}
}
}
// checkDerivs checks dSdR derivatives of shape structures
func checkDerivs(tst *testing.T, shape string, r []float64, tol float64, verbose bool) {
// information
fcn := Functions[shape]
ndim := GeomNdim[shape]
nverts := NumVerts[shape]
// allocate slices
S := make([]float64, nverts)
dSdR := utl.DblsAlloc(nverts, ndim)
// auxiliary
r_tmp := make([]float64, len(r))
S_tmp := make([]float64, nverts)
// analytical
fcn(S, dSdR, r, true)
// numerical
for n := 0; n < nverts; n++ {
for i := 0; i < ndim; i++ {
dSndRi, _ := num.DerivCentral(func(t float64, args ...interface{}) (Sn float64) {
copy(r_tmp, r)
r_tmp[i] = t
fcn(S_tmp, nil, r_tmp, false)
Sn = S_tmp[n]
return
}, r[i], 1e-1)
if verbose {
io.Pfgrey2(" dS%ddR%d @ %5.2f = %v (num: %v)\n", n, i, r, dSdR[n][i], dSndRi)
}
if math.Abs(dSdR[n][i]-dSndRi) > tol {
tst.Errorf("nurbs dS%ddR%d failed with err = %g\n", n, i, math.Abs(dSdR[n][i]-dSndRi))
return
}
}
}
}
// testing_compare_results_u compares results with u-formulation
func TestingCompareResultsU(tst *testing.T, simfname, cmpfname string, tolK, tolu, tols float64, skipK, verbose bool) {
// only root can run this test
if !Global.Root {
return
}
// read summary
sum := ReadSum(Global.Dirout, Global.Fnkey)
if sum == nil {
tst.Error("cannot read summary file for simulation=%q\n", simfname)
return
}
// allocate domain
distr := false
d := NewDomain(Global.Sim.Regions[0], distr)
if !d.SetStage(0, Global.Sim.Stages[0], distr) {
tst.Errorf("TestingCompareResultsU: SetStage failed\n")
return
}
// read file
buf, err := io.ReadFile(cmpfname)
if err != nil {
tst.Errorf("TestingCompareResultsU: ReadFile failed\n")
return
}
// unmarshal json
var cmp_set T_results_set
err = json.Unmarshal(buf, &cmp_set)
if err != nil {
tst.Errorf("TestingCompareResultsU: Unmarshal failed\n")
return
}
// run comparisons
dmult := 1.0
for idx, cmp := range cmp_set {
// displacements multiplier
if idx == 0 && math.Abs(cmp.DispMult) > 1e-10 {
dmult = cmp.DispMult
}
// time index
tidx := idx + 1
if verbose {
io.PfYel("\n\ntidx = %d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n", tidx)
}
// load gofem results
if !d.In(sum, tidx, true) {
tst.Errorf("TestingCompareResultsU: reading of results failed\n")
return
}
if verbose {
io.Pfyel("time = %v\n", d.Sol.T)
}
// check K matrices
if !skipK {
if verbose {
io.Pfgreen(". . . checking K matrices . . .\n")
}
for eid, Ksg := range cmp.Kmats {
if e, ok := d.Elems[eid].(*ElemU); ok {
if !e.AddToKb(d.Kb, d.Sol, true) {
tst.Errorf("TestingCompareResultsU: AddToKb failed\n")
return
}
chk.Matrix(tst, io.Sf("K%d", eid), tolK, e.K, Ksg)
}
}
}
// check displacements
if verbose {
io.Pfgreen(". . . checking displacements . . .\n")
}
for nid, usg := range cmp.Disp {
ix := d.Vid2node[nid].Dofs[0].Eq
iy := d.Vid2node[nid].Dofs[1].Eq
chk.AnaNum(tst, "ux", tolu, d.Sol.Y[ix], usg[0]*dmult, verbose)
chk.AnaNum(tst, "uy", tolu, d.Sol.Y[iy], usg[1]*dmult, verbose)
if len(usg) == 3 {
iz := d.Vid2node[nid].Dofs[2].Eq
chk.AnaNum(tst, "uz", tolu, d.Sol.Y[iz], usg[2]*dmult, verbose)
}
}
// check stresses
if true {
if verbose {
io.Pfgreen(". . . checking stresses . . .\n")
}
for eid, sig := range cmp.Sigmas {
if verbose {
io.Pforan("eid = %d\n", eid)
}
if e, ok := d.Cid2elem[eid].(*ElemU); ok {
for ip, val := range sig {
if verbose {
io.Pfgrey2("ip = %d\n", ip)
}
σ := e.States[ip].Sig
if len(val) == 6 {
chk.AnaNum(tst, "sx ", tols, σ[0], val[0], verbose)
chk.AnaNum(tst, "sy ", tols, σ[1], val[1], verbose)
} else {
chk.AnaNum(tst, "sx ", tols, σ[0], val[0], verbose)
chk.AnaNum(tst, "sy ", tols, σ[1], val[1], verbose)
chk.AnaNum(tst, "sxy", tols, σ[3]/SQ2, val[2], verbose)
if len(val) > 3 { // sx, sy, sxy, sz
chk.AnaNum(tst, "sz ", tols, σ[2], val[3], verbose)
}
}
}
}
}
}
}
}
// 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 Test_shape01(tst *testing.T) {
//utl.Tsilent = false
chk.PrintTitle("Test shape01")
for name, shape := range factory {
io.Pfyel("--------------------------------- %-6s---------------------------------\n", name)
// check S
tol := 1e-17
errS := 0.0
if name == "tri10" {
tol = 1e-14
}
for n := 0; n < shape.Nverts; n++ {
rst := []float64{0, 0, 0}
for i := 0; i < shape.Gndim; i++ {
rst[i] = shape.NatCoords[i][n]
}
shape.Func(shape.S, shape.dSdR, rst[0], rst[1], rst[2], false)
io.Pforan("S = %v\n", shape.S)
for m := 0; m < shape.Nverts; m++ {
if n == m {
errS += math.Abs(shape.S[m] - 1.0)
} else {
errS += math.Abs(shape.S[m])
}
}
}
if errS > tol {
tst.Errorf("%s failed with err = %g\n", name, errS)
return
}
// check dSdR
tol = 1e-14
h := 1.0e-1
S_temp := make([]float64, shape.Nverts)
if name == "lin5" || name == "tri15" || name == "lin4" || name == "tri10" || name == "qua12" || name == "qua16" {
tol = 1.0e-10
}
for n := 0; n < shape.Nverts; n++ {
rst := []float64{0, 0, 0}
for i := 0; i < shape.Gndim; i++ {
rst[i] = shape.NatCoords[i][n]
}
// analytical
shape.Func(shape.S, shape.dSdR, rst[0], rst[1], rst[2], true)
// numerical
for i := 0; i < shape.Gndim; i++ {
dSndRi, _ := num.DerivCentral(func(x float64, args ...interface{}) (Sn float64) {
rst_temp := []float64{rst[0], rst[1], rst[2]}
rst_temp[i] = x
shape.Func(S_temp, nil, rst_temp[0], rst_temp[1], rst_temp[2], false)
Sn = S_temp[n]
return
}, rst[i], h)
io.Pfgrey2(" dS%ddR%d @ [% 4.1f % 4.1f % 4.1f] = %v (num: %v)\n", n, i, rst[0], rst[1], rst[2], shape.dSdR[n][i], dSndRi)
tol2 := tol
if name == "tri15" && n == 11 && i == 1 {
tol2 = 1.0e-9
}
if math.Abs(shape.dSdR[n][i]-dSndRi) > tol2 {
tst.Errorf("%s dS%ddR%d failed with err = %g\n", name, n, i, math.Abs(shape.dSdR[n][i]-dSndRi))
return
}
//chk.Scalar(tst, fmt.Sprintf("dS%ddR%d", n, i), tol2, dSdR[n][i], dSndRi)
}
}
// check face vertices
tol = 1e-17
errS = 0.0
if name == "tri10" {
tol = 1e-14
}
nfaces := len(shape.FaceLocalV)
if nfaces == 0 {
continue
}
for k := 0; k < nfaces; k++ {
for n := range shape.FaceLocalV[k] {
rst := []float64{0, 0, 0}
for i := 0; i < shape.Gndim; i++ {
rst[i] = shape.NatCoords[i][n]
}
shape.Func(shape.S, shape.dSdR, rst[0], rst[1], rst[2], false)
io.Pforan("S = %v\n", shape.S)
for m := range shape.FaceLocalV[k] {
if n == m {
errS += math.Abs(shape.S[m] - 1.0)
} else {
errS += math.Abs(shape.S[m])
}
}
}
}
io.Pforan("%g\n", errS)
if errS > tol {
tst.Errorf("%s failed with err = %g\n", name, errS)
return
}
io.PfGreen("OK\n")
}
}
// testing_compare_results_u compares results with u-formulation
func TestingCompareResultsU(tst *testing.T, simfilepath, cmpfname, alias string, tolK, tolu, tols float64, skipK, verbose bool) {
// FEM structure
fem := NewFEM(simfilepath, alias, false, false, true, false, verbose, 0)
// set stage
err := fem.SetStage(0)
if err != nil {
chk.Panic("cannot set stage:\n%v", err)
}
// zero solution
err = fem.ZeroStage(0, true)
if err != nil {
chk.Panic("cannot zero stage data:\n%v", err)
}
// read file with comparison results
buf, err := io.ReadFile(cmpfname)
if err != nil {
tst.Errorf("TestingCompareResultsU: ReadFile failed\n")
return
}
// unmarshal json
var cmp_set T_results_set
err = json.Unmarshal(buf, &cmp_set)
if err != nil {
tst.Errorf("TestingCompareResultsU: Unmarshal failed\n")
return
}
// run comparisons
dom := fem.Domains[0]
dmult := 1.0
for idx, cmp := range cmp_set {
// displacements multiplier
if idx == 0 && math.Abs(cmp.DispMult) > 1e-10 {
dmult = cmp.DispMult
}
// time index
tidx := idx + 1
if verbose {
io.PfYel("\n\ntidx = %d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n", tidx)
}
// load gofem results
err = dom.Read(fem.Summary, tidx, 0, true)
if err != nil {
chk.Panic("cannot read 'gofem' results:\n%v", err)
}
if verbose {
io.Pfyel("time = %v\n", dom.Sol.T)
}
// check K matrices
if !skipK {
if verbose {
io.Pfgreen(". . . checking K matrices . . .\n")
}
for eid, Ksg := range cmp.Kmats {
if e, ok := dom.Elems[eid].(*ElemU); ok {
err = e.AddToKb(dom.Kb, dom.Sol, true)
if err != nil {
chk.Panic("TestingCompareResultsU: AddToKb failed\n")
}
chk.Matrix(tst, io.Sf("K%d", eid), tolK, e.K, Ksg)
}
if e, ok := dom.Elems[eid].(*Rod); ok {
err = e.AddToKb(dom.Kb, dom.Sol, true)
if err != nil {
chk.Panic("TestingCompareResultsU: AddToKb failed\n")
}
chk.Matrix(tst, io.Sf("K%d", eid), tolK, e.K, Ksg)
}
if e, ok := dom.Elems[eid].(*ElastRod); ok {
err = e.AddToKb(dom.Kb, dom.Sol, true)
if err != nil {
chk.Panic("TestingCompareResultsU: AddToKb failed\n")
}
chk.Matrix(tst, io.Sf("K%d", eid), tolK, e.K, Ksg)
}
}
}
// check displacements
if verbose {
io.Pfgreen(". . . checking displacements . . .\n")
}
for nid, usg := range cmp.Disp {
ix := dom.Vid2node[nid].Dofs[0].Eq
iy := dom.Vid2node[nid].Dofs[1].Eq
chk.AnaNum(tst, "ux", tolu, dom.Sol.Y[ix], usg[0]*dmult, verbose)
chk.AnaNum(tst, "uy", tolu, dom.Sol.Y[iy], usg[1]*dmult, verbose)
if len(usg) == 3 {
iz := dom.Vid2node[nid].Dofs[2].Eq
chk.AnaNum(tst, "uz", tolu, dom.Sol.Y[iz], usg[2]*dmult, verbose)
}
}
// check stresses
if true {
if verbose {
io.Pfgreen(". . . checking stresses . . .\n")
}
for eid, sig := range cmp.Sigmas {
if verbose {
io.Pforan("eid = %d\n", eid)
}
if e, ok := dom.Cid2elem[eid].(*ElemU); ok {
for ip, val := range sig {
if verbose {
io.Pfgrey2("ip = %d\n", ip)
}
σ := e.States[ip].Sig
if len(val) == 6 {
chk.AnaNum(tst, "sx ", tols, σ[0], val[0], verbose)
chk.AnaNum(tst, "sy ", tols, σ[1], val[1], verbose)
} else {
chk.AnaNum(tst, "sx ", tols, σ[0], val[0], verbose)
chk.AnaNum(tst, "sy ", tols, σ[1], val[1], verbose)
chk.AnaNum(tst, "sxy", tols, σ[3]/SQ2, val[2], verbose)
if len(val) > 3 { // sx, sy, sxy, sz
chk.AnaNum(tst, "sz ", tols, σ[2], val[3], verbose)
}
}
}
}
if e, ok := dom.Cid2elem[eid].(*Rod); ok {
for ip, val := range sig {
if verbose {
io.Pfgrey2("ip = %d\n", ip)
}
σ := e.States[ip].Sig
chk.AnaNum(tst, "sig", tols, σ, val[0], verbose)
}
}
if e, ok := dom.Cid2elem[eid].(*ElastRod); ok {
dat := e.OutIpsData()
for ip, val := range sig {
if verbose {
io.Pfgrey2("ip = %d\n", ip)
}
res := dat[ip].Calc(dom.Sol)
σ := res["sig"]
chk.AnaNum(tst, "sig", tols, σ, val[0], verbose)
}
}
}
}
}
}