// 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
}
}
}
}
func main() {
// define function and derivative function
y_fcn := func(x float64) float64 { return math.Sin(x) }
dydx_fcn := func(x float64) float64 { return math.Cos(x) }
d2ydx2_fcn := func(x float64) float64 { return -math.Sin(x) }
// run test for 11 points
X := utl.LinSpace(0, 2*math.Pi, 11)
io.Pf(" %8s %23s %23s %23s\n", "x", "analytical", "numerical", "error")
for _, x := range X {
// analytical derivatives
dydx_ana := dydx_fcn(x)
d2ydx2_ana := d2ydx2_fcn(x)
// numerical derivative: dydx
dydx_num, _ := num.DerivCentral(func(t float64, args ...interface{}) float64 {
return y_fcn(t)
}, x, 1e-3)
// numerical derivative d2ydx2
d2ydx2_num, _ := num.DerivCentral(func(t float64, args ...interface{}) float64 {
return dydx_fcn(t)
}, x, 1e-3)
// check
chk.PrintAnaNum(io.Sf("dy/dx @ %.6f", x), 1e-10, dydx_ana, dydx_num, true)
chk.PrintAnaNum(io.Sf("d²y/dx² @ %.6f", x), 1e-10, d2ydx2_ana, d2ydx2_num, true)
}
// generate 101 points for plotting
X = utl.LinSpace(0, 2*math.Pi, 101)
Y := make([]float64, len(X))
for i, x := range X {
Y[i] = y_fcn(x)
}
// plot
plt.SetForPng(0.75, 300, 150)
plt.Plot(X, Y, "'b.-', clip_on=0, markevery=10, label='y(x)=sin(x)'")
plt.Gll("x", "y", "")
plt.SaveD("/tmp/gosl", "num_deriv01.png")
}
func main() {
// define function and derivative function
y_fcn := func(x float64) float64 { return math.Sin(x) }
dydx_fcn := func(x float64) float64 { return math.Cos(x) }
d2ydx2_fcn := func(x float64) float64 { return -math.Sin(x) }
// run test for 11 points
X := utl.LinSpace(0, 2*math.Pi, 11)
for _, x := range X {
// analytical derivatives
dydx_ana := dydx_fcn(x)
d2ydx2_ana := d2ydx2_fcn(x)
// numerical derivative: dydx
dydx_num, _ := num.DerivCentral(func(t float64, args ...interface{}) float64 {
return y_fcn(t)
}, x, 1e-3)
// numerical derivative d2ydx2
d2ydx2_num, _ := num.DerivCentral(func(t float64, args ...interface{}) float64 {
return dydx_fcn(t)
}, x, 1e-3)
// check
chk.PrintAnaNum(io.Sf("dy/dx @ %.6f", x), 1e-10, dydx_ana, dydx_num, true)
chk.PrintAnaNum(io.Sf("d²y/dx² @ %.6f", x), 1e-10, d2ydx2_ana, d2ydx2_num, true)
}
// generate 101 points
X = utl.LinSpace(0, 2*math.Pi, 101)
Y := make([]float64, len(X))
for i, x := range X {
Y[i] = y_fcn(x)
}
// plot
plt.Plot(X, Y, "'b.-'")
plt.Gll("x", "y", "")
plt.Show()
}
// CheckEigenprojsDerivs checks the derivatives of eigen projectors w.r.t defining tensor
func CheckEigenprojsDerivs(a []float64, tol float64, ver bool, zero float64) {
// compute eigenvalues and eigenprojectors
ncp := len(a)
λ := make([]float64, 3)
P := la.MatAlloc(3, ncp)
docalc := func() {
err := M_EigenValsProjsNum(P, λ, a)
if err != nil {
chk.Panic("eigenprojs.go: CheckEigenprojsDerivs failed:\n %v", err.Error())
}
}
// compute derivatives of eigenprojectors
docalc()
dPda := utl.Deep3alloc(3, ncp, ncp)
err := M_EigenProjsDerivAuto(dPda, a, λ, P)
if err != nil {
chk.Panic("%v", err)
}
// check
var tmp float64
has_error := false
for k := 0; k < 3; k++ {
for i := 0; i < ncp; i++ {
for j := 0; j < ncp; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
docalc()
a[j] = tmp
return P[k][i]
}, a[j], 1e-6)
err := chk.PrintAnaNum(io.Sf("dP%d[%d]/da[%d]", k, i, j), tol, dPda[k][i][j], dnum, ver)
if err != nil {
has_error = true
}
}
}
if ver {
io.Pf("\n")
}
}
if has_error {
chk.Panic(_eigenprojs_err8)
}
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
}
}
}
}
func Test_ops03(tst *testing.T) {
//verbose()
chk.PrintTitle("ops03")
nonsymTol := 1e-15
dtol := 1e-9
dver := chk.Verbose
nd := test_nd
for idxA := 0; idxA < len(test_nd)-3; idxA++ {
//for idxA := 0; idxA < 1; idxA++ {
// tensor and eigenvalues
A := test_AA[idxA]
a := M_Alloc2(nd[idxA])
Ten2Man(a, A)
io.PfYel("\n\ntst # %d ###################################################################################\n", idxA)
io.Pfblue2("a = %v\n", a)
// inverse
Ai := Alloc2()
ai := M_Alloc2(nd[idxA])
detA, err := Inv(Ai, A)
if err != nil {
chk.Panic("%v", err)
}
deta_ := M_Det(a)
deta, err := M_Inv(ai, a, MINDET)
if err != nil {
chk.Panic("%v", err)
}
Ai_ := Alloc2()
Man2Ten(Ai_, ai)
aia := M_Alloc2(nd[idxA])
err = M_Dot(aia, ai, a, nonsymTol)
if err != nil {
chk.Panic("%v", err)
}
chk.Scalar(tst, "detA", 1e-14, detA, deta)
chk.Scalar(tst, "deta", 1e-14, deta, deta_)
chk.Matrix(tst, "Ai", 1e-14, Ai, Ai_)
chk.Vector(tst, "ai*a", 1e-15, aia, Im[:2*nd[idxA]])
io.Pforan("ai*a = %v\n", aia)
// derivative of inverse
dtol_tmp := dtol
if idxA == 5 {
dtol = 1e-8
}
var tmp float64
ai_tmp := M_Alloc2(nd[idxA])
daida := M_Alloc4(nd[idxA])
M_InvDeriv(daida, ai)
io.Pforan("ai = %v\n", ai)
for i := 0; i < len(a); i++ {
for j := 0; j < len(a); j++ {
//dnum, _ := num.DerivForward(func(x float64, args ...interface{}) (res float64) {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
_, err := M_Inv(ai_tmp, a, MINDET)
a[j] = tmp
if err != nil {
chk.Panic("daida failed:\n%v", err)
}
return ai_tmp[i]
}, a[j], 1e-6)
chk.AnaNum(tst, io.Sf("dai/da[%d][%d]", i, j), dtol, daida[i][j], dnum, dver)
}
}
dtol = dtol_tmp
}
}
func Test_ops01(tst *testing.T) {
//verbose()
chk.PrintTitle("ops01")
// basic derivatives
dver := chk.Verbose
dtol := 1e-5
// invariants derivatives
dveri := false
dtoli1 := []float64{1e-6, 1e-6, 1e-6, 1e-6, 1e-6, 1e-6, 1e-6, 1e-6, 1e-6, 1e-6}
dtoli2 := []float64{1e-6, 1e-5, 1e-6, 1e-4, 1e-5, 1e-6, 1e-6, 1e-6, 1e-6, 1e-6}
dtoli3 := []float64{1e-6, 1e-3, 1e-6, 1e-3, 1e-3, 1e-6, 1e-6, 1e-6, 1e-5, 1e-6}
// lode derivatives
dverw := chk.Verbose
dtolw := 1e-8
nd := test_nd
for m := 0; m < len(test_nd)-3; m++ {
//for m := 0; m < 3; m++ {
A := test_AA[m]
a := M_Alloc2(nd[m])
Ten2Man(a, A)
trA := Tr(A)
tra := M_Tr(a)
detA := Det(A)
deta := M_Det(a)
devA := Dev(A)
deva := M_Dev(a)
devA_ := Alloc2()
a2 := M_Alloc2(nd[m])
A2 := Alloc2()
A2_ := Alloc2()
trDevA := Tr(devA)
deva__ := M_Alloc2(nd[m])
devA__ := Alloc2()
s2 := M_Alloc2(nd[m])
M_Sq(a2, a)
M_Sq(s2, deva)
Man2Ten(A2, a2)
Man2Ten(devA_, deva)
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
for k := 0; k < 3; k++ {
A2_[i][j] += A[i][k] * A[k][j]
}
}
}
for i := 0; i < len(a); i++ {
for j := 0; j < len(a); j++ {
deva__[i] += Psd[i][j] * a[j]
}
}
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
for k := 0; k < 3; k++ {
for l := 0; l < 3; l++ {
devA__[i][j] += M2TT(Psd, i, j, k, l) * A[k][l]
}
}
}
}
// check basic
if math.Abs(trA-tra) > 1e-17 {
chk.Panic("tra failed. diff = %g", trA-tra)
}
if math.Abs(detA-deta) > 1e-14 {
chk.Panic("detA failed. diff = %g", detA-deta)
}
if math.Abs(trDevA) > 1e-13 {
chk.Panic("trDevA failed. error = %g", trDevA)
}
chk.Matrix(tst, "devA", 1e-13, devA, devA_)
chk.Matrix(tst, "devA", 1e-13, devA, devA__)
chk.Vector(tst, "devA", 1e-13, deva, deva__)
chk.Matrix(tst, "A²", 1e-11, A2, A2_)
// check tr(s2)
io.Pfblue2("tr(s²) = %v\n", M_Tr(s2))
if M_Tr(s2) < 1 {
chk.Panic("Tr(s2) failed")
}
// check derivatives
da2da := M_Alloc4(nd[m])
a2tmp := M_Alloc2(nd[m]) // a2tmp == a²
M_SqDeriv(da2da, a)
var tmp float64
for i := 0; i < len(a); i++ {
for j := 0; j < len(a); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
M_Sq(a2tmp, a)
a[j] = tmp
return a2tmp[i]
}, a[j], 1e-6)
chk.AnaNum(tst, io.Sf("da²/da[%d][%d]", i, j), dtol, da2da[i][j], dnum, dver)
}
}
// characteristic invariants
I1, I2, I3 := M_CharInvs(a)
I2a := 0.5 * (tra*tra - M_Tr(a2))
I1_, I2_, I3_, dI1da, dI2da, dI3da := M_CharInvsAndDerivs(a)
if math.Abs(I1-tra) > 1e-17 {
chk.Panic("I1 failed (a). error = %v", I1-tra)
}
if math.Abs(I2-I2a) > 1e-12 {
chk.Panic("I2 failed (a). error = %v (I2=%v, I2_=%v)", I2-I2a, I2, I2a)
}
if math.Abs(I3-deta) > 1e-17 {
chk.Panic("I3 failed (a). error = %v", I3-deta)
}
if math.Abs(I1-I1_) > 1e-17 {
chk.Panic("I1 failed (b). error = %v", I1-I1_)
}
if math.Abs(I2-I2_) > 1e-17 {
chk.Panic("I2 failed (b). error = %v", I2-I2_)
}
if math.Abs(I3-I3_) > 1e-17 {
chk.Panic("I3 failed (b). error = %v", I3-I3_)
}
// dI1da
for j := 0; j < len(a); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
i1, _, _ := M_CharInvs(a)
a[j] = tmp
return i1
}, a[j], 1e-6)
chk.AnaNum(tst, io.Sf("dI1/da[%d]", j), dtoli1[m], dI1da[j], dnum, dveri)
}
// dI2da
for j := 0; j < len(a); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
_, i2, _ := M_CharInvs(a)
a[j] = tmp
return i2
}, a[j], 1e-6)
chk.AnaNum(tst, io.Sf("dI2/da[%d]", j), dtoli2[m], dI2da[j], dnum, dveri)
}
// dI3da
for j := 0; j < len(a); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
_, _, i3 := M_CharInvs(a)
a[j] = tmp
return i3
}, a[j], 1e-6)
chk.AnaNum(tst, io.Sf("dI3/da[%d]", j), dtoli3[m], dI3da[j], dnum, dveri)
}
// dDet(a)/da
DdetaDa := make([]float64, len(a))
M_DetDeriv(DdetaDa, a)
for j := 0; j < len(a); j++ {
chk.AnaNum(tst, io.Sf("dDet(a)/da[%d]", j), dtoli3[m], dI3da[j], DdetaDa[j], dveri)
}
// lode angle
if true {
s := M_Alloc2(nd[m])
dwda := M_Alloc2(nd[m])
dwda_ := M_Alloc2(nd[m])
d2wdada := M_Alloc4(nd[m])
p, q, w := M_pqws(s, a)
M_LodeDeriv1(dwda, a, s, p, q, w)
M_LodeDeriv2(d2wdada, dwda_, a, s, p, q, w)
chk.Vector(tst, "s", 1e-13, deva, s)
chk.Vector(tst, "dwda", 1e-13, dwda, dwda_)
// dwda
for j := 0; j < len(a); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
res = M_w(a)
a[j] = tmp
return res
}, a[j], 1e-6)
chk.AnaNum(tst, io.Sf("dw/da[%d]", j), dtolw, dwda[j], dnum, dverw)
}
// d2wdada
s_tmp := M_Alloc2(nd[m])
dwda_tmp := M_Alloc2(nd[m])
for i := 0; i < len(a); i++ {
for j := 0; j < len(a); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, a[j] = a[j], x
p_tmp, q_tmp, w_tmp := M_pqws(s_tmp, a)
M_LodeDeriv1(dwda_tmp, a, s_tmp, p_tmp, q_tmp, w_tmp)
a[j] = tmp
return dwda_tmp[i]
}, a[j], 1e-6)
chk.AnaNum(tst, io.Sf("d2w/dada[%d][%d]", i, j), dtolw, d2wdada[i][j], dnum, dverw)
}
}
}
}
}
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")
}
}
// CheckDerivs check derivatives computed by isotropic function
func (o *IsoFun) CheckDerivs(A []float64, tol, tol2, tolq, tol3 float64, ver bool, args ...interface{}) (err error) {
// L and invariants
chk.IntAssert(len(A), o.ncp)
L := make([]float64, 3)
err = M_EigenValsNum(L, A)
if err != nil {
return
}
p, q, err := GenInvs(L, o.n, o.a)
if err != nil {
return
}
io.Pforan("L = %v\n", L)
io.Pforan("p, q = %v, %v\n", p, q)
// constants
h := 1e-6
var has_error error
// derivatives of callback functions ///////////////
// df/dp, df/dq, d²f/dp², d²f/dq², d²f/dpdq
dfdp, dfdq := o.gfcn(p, q, args...)
d2fdp2, d2fdq2, d2fdpdq := o.hfcn(p, q, args...)
if ver {
io.Pfpink("\nd w.r.t invariants . . . \n")
}
// check df/dp
dfdp_num, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
return o.ffcn(x, q, args...)
}, p, h)
err = chk.PrintAnaNum("df/dp ", tol, dfdp, dfdp_num, ver)
if err != nil {
has_error = err
}
// check df/dq
dfdq_num, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
return o.ffcn(p, x, args...)
}, q, h)
err = chk.PrintAnaNum("df/dq ", tol, dfdq, dfdq_num, ver)
if err != nil {
has_error = err
}
// check d²f/dp²
d2fdp2_num, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
dfdp_tmp, _ := o.gfcn(x, q, args...)
return dfdp_tmp
}, p, h)
err = chk.PrintAnaNum("d²f/dp² ", tol, d2fdp2, d2fdp2_num, ver)
if err != nil {
has_error = err
}
// check d²f/dq²
d2fdq2_num, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
_, dfdq_tmp := o.gfcn(p, x, args...)
return dfdq_tmp
}, q, h)
err = chk.PrintAnaNum("d²f/dq² ", tol, d2fdq2, d2fdq2_num, ver)
if err != nil {
has_error = err
}
// check d²f/dpdq
d2fdpdq_num, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
dfdp_tmp, _ := o.gfcn(p, x, args...)
return dfdp_tmp
}, q, h)
err = chk.PrintAnaNum("d²f/dpdq", tol, d2fdpdq, d2fdpdq_num, ver)
if err != nil {
has_error = err
}
// derivatives w.r.t eigenvalues ///////////////////
// df/dL and d²f/dLdL
_, err = o.Gp(L, args...)
if err != nil {
chk.Panic("Gp failed:\n%v", err)
}
err = o.HafterGp(args...)
if err != nil {
chk.Panic("HafterGp failed:\n%v", err)
}
dfdL := make([]float64, 3)
d2pdLdL := la.MatAlloc(3, 3)
d2qdLdL := la.MatAlloc(3, 3)
d2fdLdL := la.MatAlloc(3, 3)
copy(dfdL, o.DfdL)
la.MatCopy(d2pdLdL, 1, o.d2pdLdL)
la.MatCopy(d2qdLdL, 1, o.d2qdLdL)
la.MatCopy(d2fdLdL, 1, o.DgdL)
// check df/dL
if ver {
io.Pfpink("\ndf/dL . . . . . . . . \n")
}
var fval, tmp float64
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
tmp, L[j] = L[j], x
defer func() { L[j] = tmp }()
fval, err = o.Fp(L, args...)
if err != nil {
chk.Panic("Fp failed:\n%v", err)
}
return fval
}, L[j], h)
err := chk.PrintAnaNum(io.Sf("df/dL[%d]", j), tol, dfdL[j], dnum, ver)
if err != nil {
has_error = err
}
}
// check d²p/dLdL
if ver {
io.Pfpink("\nd²p/dLdL . . . . . . . . \n")
}
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
tmp, L[j] = L[j], x
defer func() { L[j] = tmp }()
_, err = o.Gp(L, args...)
if err != nil {
chk.Panic("Gp failed\n%v", err)
}
return o.dpdL[i]
}, L[j], h)
err := chk.PrintAnaNum(io.Sf("d²p/dL[%d]dL[%d]", i, j), tol2, d2pdLdL[i][j], dnum, ver)
if err != nil {
has_error = err
}
}
}
// check d²q/dLdL
if ver {
io.Pfpink("\nd²q/dLdL . . . . . . . . \n")
}
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
tmp, L[j] = L[j], x
defer func() { L[j] = tmp }()
_, err = o.Gp(L, args...)
if err != nil {
chk.Panic("Gp failed\n%v", err)
}
return o.dqdL[i]
}, L[j], h)
err := chk.PrintAnaNum(io.Sf("d²q/dL[%d]dL[%d]", i, j), tolq, d2qdLdL[i][j], dnum, ver)
if err != nil {
has_error = err
}
}
}
// check d²f/dLdL
if ver {
io.Pfpink("\nd²f/dLdL . . . . . . . . \n")
}
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
tmp, L[j] = L[j], x
defer func() { L[j] = tmp }()
_, err = o.Gp(L, args...)
if err != nil {
chk.Panic("Gp failed\n%v", err)
}
return o.DfdL[i]
}, L[j], h)
err := chk.PrintAnaNum(io.Sf("d²f/dL[%d]dL[%d]", i, j), tol2, d2fdLdL[i][j], dnum, ver)
if err != nil {
has_error = err
}
}
}
// derivatives w.r.t full tensor ///////////////////
// dfdA and d²f/dAdA
ncp := len(A)
dfdA := make([]float64, ncp)
d2fdAdA := la.MatAlloc(ncp, ncp)
_, err = o.Ga(dfdA, A, args...) // also computes P, L and Acpy
if err != nil {
chk.Panic("Ga failed:\n%v", err)
}
err = o.HafterGa(d2fdAdA, args...) // also computes dPdA
if err != nil {
chk.Panic("HafterGa failed:\n%v", err)
}
// check df/dA
if ver {
io.Pfpink("\ndf/dA . . . . . . . . \n")
}
for j := 0; j < ncp; j++ {
dnum, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
tmp, A[j] = A[j], x
defer func() { A[j] = tmp }()
fval, err = o.Fa(A, args...)
if err != nil {
chk.Panic("Fa failed:\n%v", err)
}
return fval
}, A[j], h)
err := chk.PrintAnaNum(io.Sf("df/dA[%d]", j), tol, dfdA[j], dnum, ver)
if err != nil {
has_error = err
}
}
// check dP/dA
if false {
for k := 0; k < 3; k++ {
if ver {
io.Pfpink("\ndP%d/dA . . . . . . . . \n", k)
}
for i := 0; i < ncp; i++ {
for j := 0; j < ncp; j++ {
dnum, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
tmp, A[j] = A[j], x
defer func() { A[j] = tmp }()
err = M_EigenValsProjsNum(o.P, o.L, o.Acpy)
if err != nil {
chk.Panic("M_EigenValsProjsNum failed:\n%v", err)
}
return o.P[k][i]
}, A[j], h)
err := chk.PrintAnaNum(io.Sf("dP%d/dA[%d]dA[%d]", k, i, j), tol, o.dPdA[k][i][j], dnum, ver)
if err != nil {
has_error = err
}
}
}
}
}
// check d²f/dAdA
if ver {
io.Pfpink("\nd²f/dAdA . . . . . . . . \n")
}
dfdA_tmp := make([]float64, ncp)
for i := 0; i < ncp; i++ {
for j := 0; j < ncp; j++ {
dnum, _ := num.DerivCentral(func(x float64, notused ...interface{}) (res float64) {
tmp, A[j] = A[j], x
defer func() { A[j] = tmp }()
_, err = o.Ga(dfdA_tmp, A, args...)
if err != nil {
chk.Panic("Ga failed:\n%v", err)
}
return dfdA_tmp[i]
}, A[j], h)
dtol := tol2
if i == j && (i == 3 || i == 4 || i == 5) {
dtol = tol3
}
err := chk.PrintAnaNum(io.Sf("d²f/dA[%d]dA[%d]", i, j), dtol, d2fdAdA[i][j], dnum, ver)
if err != nil {
has_error = err
}
}
}
// any errors?
if has_error != nil {
err = has_error
}
return
}
func Test_geninvs01(tst *testing.T) {
//verbose()
chk.PrintTitle("geninvs01")
// coefficients for smp invariants
smp_a := -1.0
smp_b := 0.5
smp_β := 1e-1 // derivative values become too high with
smp_ϵ := 1e-1 // small β and ϵ @ zero
// constants for checking derivatives
dver := chk.Verbose
dtol := 1e-6
dtol2 := 1e-6
// run tests
nd := test_nd
for idxA := 0; idxA < len(test_nd); idxA++ {
//for idxA := 10; idxA < 11; idxA++ {
// tensor and eigenvalues
A := test_AA[idxA]
a := M_Alloc2(nd[idxA])
Ten2Man(a, A)
L := make([]float64, 3)
M_EigenValsNum(L, a)
// SMP derivs and SMP director
dndL := la.MatAlloc(3, 3)
dNdL := make([]float64, 3)
d2ndLdL := utl.Deep3alloc(3, 3, 3)
N := make([]float64, 3)
F := make([]float64, 3)
G := make([]float64, 3)
m := SmpDerivs1(dndL, dNdL, N, F, G, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpDerivs2(d2ndLdL, L, smp_a, smp_b, smp_β, smp_ϵ, m, N, F, G, dNdL, dndL)
n := make([]float64, 3)
SmpUnitDirector(n, m, N)
// SMP invariants
p, q, err := GenInvs(L, n, smp_a)
if err != nil {
chk.Panic("SmpInvs failed:\n%v", err)
}
// output
io.PfYel("\n\ntst # %d ###################################################################################\n", idxA)
io.Pfblue2("L = %v\n", L)
io.Pforan("n = %v\n", n)
io.Pforan("p = %v\n", p)
io.Pforan("q = %v\n", q)
// check invariants
tvec := make([]float64, 3)
GenTvec(tvec, L, n)
proj := make([]float64, 3) // projection of tvec along n
tdn := la.VecDot(tvec, n) // tvec dot n
for i := 0; i < 3; i++ {
proj[i] = tdn * n[i]
}
norm_proj := la.VecNorm(proj)
norm_tvec := la.VecNorm(tvec)
q_ := GENINVSQEPS + math.Sqrt(norm_tvec*norm_tvec-norm_proj*norm_proj)
io.Pforan("proj = %v\n", proj)
io.Pforan("norm(proj) = %v == p\n", norm_proj)
chk.Scalar(tst, "p", 1e-14, math.Abs(p), norm_proj)
chk.Scalar(tst, "q", 1e-13, q, q_)
// dt/dL
var tmp float64
N_tmp := make([]float64, 3)
n_tmp := make([]float64, 3)
tvec_tmp := make([]float64, 3)
dtdL := la.MatAlloc(3, 3)
GenTvecDeriv1(dtdL, L, n, dndL)
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n_tmp, m_tmp, N_tmp)
GenTvec(tvec_tmp, L, n_tmp)
L[j] = tmp
return tvec_tmp[i]
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("dt/dL[%d][%d]", i, j), dtol, dtdL[i][j], dnum, dver)
}
}
// d²t/dLdL
io.Pfpink("\nd²t/dLdL\n")
dNdL_tmp := make([]float64, 3)
dndL_tmp := la.MatAlloc(3, 3)
dtdL_tmp := la.MatAlloc(3, 3)
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
for k := 0; k < 3; k++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[k] = L[k], x
m_tmp := SmpDerivs1(dndL_tmp, dNdL_tmp, N_tmp, F, G, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n_tmp, m_tmp, N_tmp)
GenTvecDeriv1(dtdL_tmp, L, n_tmp, dndL_tmp)
L[k] = tmp
return dtdL_tmp[i][j]
}, L[k], 1e-6)
dana := GenTvecDeriv2(i, j, k, L, dndL, d2ndLdL[i][j][k])
chk.AnaNum(tst, io.Sf("d²t[%d]/dL[%d]dL[%d]", i, j, k), dtol2, dana, dnum, dver)
}
}
}
// change tolerance
dtol_tmp := dtol
switch idxA {
case 5, 11:
dtol = 1e-5
case 12:
dtol = 0.0013
}
// first order derivatives
dpdL := make([]float64, 3)
dqdL := make([]float64, 3)
p_, q_, err := GenInvsDeriv1(dpdL, dqdL, L, n, dndL, smp_a)
if err != nil {
chk.Panic("%v", err)
}
chk.Scalar(tst, "p", 1e-17, p, p_)
chk.Scalar(tst, "q", 1e-17, q, q_)
var ptmp, qtmp float64
io.Pfpink("\ndp/dL\n")
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n_tmp, m_tmp, N_tmp)
ptmp, _, err = GenInvs(L, n_tmp, smp_a)
if err != nil {
chk.Panic("DerivCentral: SmpInvs failed:\n%v", err)
}
L[j] = tmp
return ptmp
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("dp/dL[%d]", j), dtol, dpdL[j], dnum, dver)
}
io.Pfpink("\ndq/dL\n")
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n_tmp, m_tmp, N_tmp)
_, qtmp, err = GenInvs(L, n_tmp, smp_a)
if err != nil {
chk.Panic("DerivCentral: SmpInvs failed:\n%v", err)
}
L[j] = tmp
return qtmp
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("dq/dL[%d]", j), dtol, dqdL[j], dnum, dver)
}
// recover tolerance
dtol = dtol_tmp
// change tolerance
io.Pforan("dtol2 = %v\n", dtol2)
dtol2_tmp := dtol2
switch idxA {
case 5:
dtol2 = 1e-5
case 10:
dtol2 = 0.72
case 11:
dtol2 = 1e-5
case 12:
dtol2 = 544
}
// second order derivatives
dpdL_tmp := make([]float64, 3)
dqdL_tmp := make([]float64, 3)
d2pdLdL := la.MatAlloc(3, 3)
d2qdLdL := la.MatAlloc(3, 3)
GenInvsDeriv2(d2pdLdL, d2qdLdL, L, n, dpdL, dqdL, p, q, dndL, d2ndLdL, smp_a)
io.Pfpink("\nd²p/dLdL\n")
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDerivs1(dndL_tmp, dNdL_tmp, N_tmp, F, G, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n_tmp, m_tmp, N_tmp)
GenInvsDeriv1(dpdL_tmp, dqdL_tmp, L, n_tmp, dndL_tmp, smp_a)
L[j] = tmp
return dpdL_tmp[i]
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("d²p/dL[%d][%d]", i, j), dtol2, d2pdLdL[i][j], dnum, dver)
}
}
io.Pfpink("\nd²q/dLdL\n")
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDerivs1(dndL_tmp, dNdL_tmp, N_tmp, F, G, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n_tmp, m_tmp, N_tmp)
GenInvsDeriv1(dpdL_tmp, dqdL_tmp, L, n_tmp, dndL_tmp, smp_a)
L[j] = tmp
return dqdL_tmp[i]
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("d²q/dL[%d][%d]", i, j), dtol2, d2qdLdL[i][j], dnum, dver)
}
}
// recover tolerance
dtol2 = dtol2_tmp
}
}
func Test_noncteM01(tst *testing.T) {
//verbose()
chk.PrintTitle("noncteM01")
prms := []string{"φ", "Mfix"}
vals := []float64{32, 0}
var o NcteM
o.Init(prms, vals)
// check
if math.Abs(o.M(1)-o.Mcs) > 1e-17 {
chk.Panic("M(+1) failed. err = %v", o.M(1)-o.Mcs)
}
if o.Mfix {
if math.Abs(o.M(-1)-o.Mcs) > 1e-17 {
chk.Panic("M(-1) failed. err = %v", o.M(-1)-o.Mcs)
}
} else {
Mext := 6.0 * math.Sin(32*math.Pi/180) / (3 + math.Sin(32*math.Pi/180))
if math.Abs(o.M(-1)-Mext) > 1e-15 {
chk.Panic("M(-1) failed. err = %v", o.M(-1)-Mext)
}
}
ver, tol := chk.Verbose, 1e-9
var tmp float64
for _, w := range utl.LinSpace(-1, 1, 11) {
dMdw := o.DMdw(w)
d2Mdw2 := o.D2Mdw2(w)
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, w = w, x
res, w = o.M(w), tmp
return
}, w, 1e-6)
chk.AnaNum(tst, "dM/dw ", tol, dMdw, dnum, ver)
dnum, _ = num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, w = w, x
res, w = o.DMdw(w), tmp
return
}, w, 1e-6)
chk.AnaNum(tst, "d²M/dw²", tol, d2Mdw2, dnum, ver)
}
ver, tol = chk.Verbose, 1e-9
nd := test_nd
for m := 0; m < len(test_nd)-3; m++ {
//for m := 0; m < 3; m++ {
A := test_AA[m]
σ := M_Alloc2(nd[m])
Ten2Man(σ, A)
s := M_Alloc2(nd[m])
dMdσ := M_Alloc2(nd[m])
d2Mdσdσ := M_Alloc4(nd[m])
p, q, w := M_pqws(s, σ)
o.Deriv2(d2Mdσdσ, dMdσ, σ, s, p, q, w)
io.Pforan("σ = %v\n", σ)
io.Pforan("tr(dMdσ) = %v\n", M_Tr(dMdσ))
if math.Abs(M_Tr(dMdσ)) > 1e-16 {
chk.Panic("tr(dMdσ)=%v failed", M_Tr(dMdσ))
}
I_dc_d2Mdσdσ := M_Alloc2(nd[m]) // I:d²M/dσdσ
for j := 0; j < len(σ); j++ {
for k := 0; k < len(σ); k++ {
I_dc_d2Mdσdσ[j] += Im[k] * d2Mdσdσ[k][j]
}
}
//io.Pfblue2("I_dc_d2Mdσdσ = %v\n", I_dc_d2Mdσdσ)
chk.Vector(tst, "I_dc_d2Mdσdσ", 1e-15, I_dc_d2Mdσdσ, nil)
// dMdσ
for j := 0; j < len(σ); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, σ[j] = σ[j], x
w := M_w(σ)
σ[j] = tmp
return o.M(w)
}, σ[j], 1e-6)
chk.AnaNum(tst, io.Sf("dM/dσ[%d]", j), tol, dMdσ[j], dnum, ver)
}
// d²Mdσdσ
s_tmp := M_Alloc2(nd[m])
dMdσ_tmp := M_Alloc2(nd[m])
for i := 0; i < len(σ); i++ {
for j := 0; j < len(σ); j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, σ[j] = σ[j], x
p_tmp, q_tmp, w_tmp := M_pqws(s_tmp, σ)
o.Deriv1(dMdσ_tmp, σ, s_tmp, p_tmp, q_tmp, w_tmp)
σ[j] = tmp
return dMdσ_tmp[i]
}, σ[j], 1e-6)
chk.AnaNum(tst, io.Sf("d²M/dσdσ[%d][%d]", i, j), tol, d2Mdσdσ[i][j], dnum, ver)
}
}
}
}
// Check checks derivatives
func Check(tst *testing.T, mdl Model, pc0, sl0, pcf float64, npts int, tolCc, tolD1a, tolD1b, tolD2a, tolD2b float64, verbose bool, pcSkip []float64, tolSkip float64, doplot bool) {
// nonrate model
nr_mdl, is_nonrate := mdl.(Nonrate)
io.Pforan("is_nonrate = %v\n", is_nonrate)
// for all pc stations
Pc := utl.LinSpace(pc0, pcf, npts)
Sl := make([]float64, npts)
Sl[0] = sl0
var err error
for i := 1; i < npts; i++ {
// update and plot
Sl[i], err = Update(mdl, Pc[i-1], Sl[i-1], Pc[i]-Pc[i-1])
if err != nil {
tst.Errorf("Update failed: %v\n", err)
return
}
if doplot {
plt.PlotOne(Pc[i], Sl[i], "'ko', clip_on=0")
}
// skip point on checking of derivatives
if doskip(Pc[i], pcSkip, tolSkip) {
continue
}
// wetting flag
wet := Pc[i]-Pc[i-1] < 0
// check Cc = dsl/dpc
io.Pforan("\npc=%g, sl=%g, wetting=%v\n", Pc[i], Sl[i], wet)
if is_nonrate {
// analytical Cc
Cc_ana, err := mdl.Cc(Pc[i], Sl[i], wet)
if err != nil {
tst.Errorf("Cc failed: %v\n", err)
return
}
// numerical Cc
Cc_num, _ := num.DerivCentral(func(x float64, args ...interface{}) float64 {
return nr_mdl.Sl(x)
}, Pc[i], 1e-3)
chk.AnaNum(tst, "Cc = ∂sl/∂pc ", tolCc, Cc_ana, Cc_num, verbose)
}
// compute all derivatives
L, Lx, J, Jx, Jy, err := mdl.Derivs(Pc[i], Sl[i], wet)
if err != nil {
tst.Errorf("Derivs failed: %v\n", err)
return
}
L_ana_A := L
L_ana_B, err := mdl.L(Pc[i], Sl[i], wet)
if err != nil {
tst.Errorf("L failed: %v\n", err)
return
}
Lx_ana := Lx
Jx_ana := Jx
Jy_ana := Jy
J_ana_A := J
J_ana_B, err := mdl.J(Pc[i], Sl[i], wet)
if err != nil {
tst.Errorf("J failed: %v\n", err)
return
}
// numerical L = ∂Cc/∂pc
L_num, _ := num.DerivCentral(func(x float64, args ...interface{}) float64 {
Cctmp, _ := mdl.Cc(x, Sl[i], wet)
return Cctmp
}, Pc[i], 1e-3)
chk.AnaNum(tst, "L = ∂Cc/∂pc ", tolD1a, L_ana_A, L_num, verbose)
// numerical Lx := ∂²Cc/∂pc²
Lx_num, _ := num.DerivCentral(func(x float64, args ...interface{}) float64 {
Ltmp, _, _, _, _, _ := mdl.Derivs(x, Sl[i], wet)
return Ltmp
}, Pc[i], 1e-3)
chk.AnaNum(tst, "Lx = ∂²Cc/∂pc² ", tolD2a, Lx_ana, Lx_num, verbose)
// numerical J := ∂Cc/∂sl (version A)
J_num, _ := num.DerivCentral(func(x float64, args ...interface{}) float64 {
Ccval, _ := mdl.Cc(Pc[i], x, wet)
return Ccval
}, Sl[i], 1e-3)
chk.AnaNum(tst, "J = ∂Cc/∂sl ", tolD1b, J_ana_A, J_num, verbose)
// numerical Jx := ∂²Cc/(∂pc ∂sl)
Jx_num, _ := num.DerivCentral(func(x float64, args ...interface{}) float64 {
Ltmp, _, _, _, _, _ := mdl.Derivs(Pc[i], x, wet)
return Ltmp
}, Sl[i], 1e-3)
chk.AnaNum(tst, "Jx = ∂²Cc/∂pc∂sl", tolD2b, Jx_ana, Jx_num, verbose)
// numerical Jy := ∂²Cc/∂sl²
Jy_num, _ := num.DerivCentral(func(x float64, args ...interface{}) float64 {
Jtmp, _ := mdl.J(Pc[i], x, wet)
return Jtmp
}, Sl[i], 1e-3)
chk.AnaNum(tst, "Jy = ∂²Cc/∂sl² ", tolD2b, Jy_ana, Jy_num, verbose)
// check A and B derivatives
chk.Scalar(tst, "L_A == L_B", 1e-17, L_ana_A, L_ana_B)
chk.Scalar(tst, "J_A == J_B", 1e-17, J_ana_A, J_ana_B)
}
}
func Test_smpinvs02(tst *testing.T) {
//verbose()
chk.PrintTitle("smpinvs02")
// coefficients for smp invariants
smp_a := -1.0
smp_b := 0.5
smp_β := 1e-1 // derivative values become too high with
smp_ϵ := 1e-1 // small β and ϵ @ zero
// constants for checking derivatives
dver := chk.Verbose
dtol := 1e-9
dtol2 := 1e-8
// run tests
nd := test_nd
for idxA := 0; idxA < len(test_nd); idxA++ {
//for idxA := 0; idxA < 1; idxA++ {
//for idxA := 10; idxA < 11; idxA++ {
// tensor and eigenvalues
A := test_AA[idxA]
a := M_Alloc2(nd[idxA])
Ten2Man(a, A)
L := make([]float64, 3)
M_EigenValsNum(L, a)
// SMP director
N := make([]float64, 3)
n := make([]float64, 3)
m := SmpDirector(N, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n, m, N)
// output
io.PfYel("\n\ntst # %d ###################################################################################\n", idxA)
io.Pforan("L = %v\n", L)
io.Pforan("N = %v\n", N)
io.Pforan("m = %v\n", m)
io.Pfpink("n = %v\n", n)
chk.Vector(tst, "L", 1e-12, L, test_λ[idxA])
chk.Scalar(tst, "norm(n)==1", 1e-15, la.VecNorm(n), 1)
chk.Scalar(tst, "m=norm(N)", 1e-14, m, la.VecNorm(N))
// dN/dL
var tmp float64
N_tmp := make([]float64, 3)
dNdL := make([]float64, 3)
SmpDirectorDeriv1(dNdL, L, smp_a, smp_b, smp_β, smp_ϵ)
io.Pfpink("\ndNdL = %v\n", dNdL)
for i := 0; i < 3; i++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[i] = L[i], x
SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
L[i] = tmp
return N_tmp[i]
}, L[i], 1e-6)
chk.AnaNum(tst, io.Sf("dN/dL[%d][%d]", i, i), dtol, dNdL[i], dnum, dver)
}
// dm/dL
n_tmp := make([]float64, 3)
dmdL := make([]float64, 3)
SmpNormDirectorDeriv1(dmdL, m, N, dNdL)
io.Pfpink("\ndmdL = %v\n", dmdL)
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
L[j] = tmp
return m_tmp
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("dm/dL[%d]", j), dtol, dmdL[j], dnum, dver)
}
// dn/dL
dndL := la.MatAlloc(3, 3)
SmpUnitDirectorDeriv1(dndL, m, N, dNdL, dmdL)
io.Pfpink("\ndndL = %v\n", dndL)
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpUnitDirector(n_tmp, m_tmp, N_tmp)
L[j] = tmp
return n_tmp[i]
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("dn/dL[%d][%d]", i, j), dtol, dndL[i][j], dnum, dver)
}
}
// change tolerance
dtol2_tmp := dtol2
if idxA == 10 || idxA == 11 {
dtol2 = 1e-6
}
// d²m/dLdL
dNdL_tmp := make([]float64, 3)
dmdL_tmp := make([]float64, 3)
d2NdL2 := make([]float64, 3)
d2mdLdL := la.MatAlloc(3, 3)
SmpDirectorDeriv2(d2NdL2, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpNormDirectorDeriv2(d2mdLdL, L, smp_a, smp_b, smp_β, smp_ϵ, m, N, dNdL, d2NdL2, dmdL)
io.Pfpink("\nd2mdLdL = %v\n", d2mdLdL)
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[j] = L[j], x
m_tmp := SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpDirectorDeriv1(dNdL_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpNormDirectorDeriv1(dmdL_tmp, m_tmp, N_tmp, dNdL_tmp)
L[j] = tmp
return dmdL_tmp[i]
}, L[j], 1e-6)
chk.AnaNum(tst, io.Sf("d2m/dL[%d]dL[%d]", i, j), dtol2, d2mdLdL[i][j], dnum, dver)
}
}
// d²N/dLdL
io.Pfpink("\nd²N/dLdL\n")
for i := 0; i < 3; i++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[i] = L[i], x
SmpDirectorDeriv1(dNdL_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
L[i] = tmp
return dNdL_tmp[i]
}, L[i], 1e-6)
chk.AnaNum(tst, io.Sf("d²N[%d]/dL[%d]dL[%d]", i, i, i), dtol2, d2NdL2[i], dnum, dver)
}
// d²n/dLdL
io.Pfpink("\nd²n/dLdL\n")
dndL_tmp := la.MatAlloc(3, 3)
d2ndLdL := utl.Deep3alloc(3, 3, 3)
SmpUnitDirectorDeriv2(d2ndLdL, m, N, dNdL, d2NdL2, dmdL, n, d2mdLdL, dndL)
for i := 0; i < 3; i++ {
for j := 0; j < 3; j++ {
for k := 0; k < 3; k++ {
dnum, _ := num.DerivCentral(func(x float64, args ...interface{}) (res float64) {
tmp, L[k] = L[k], x
m_tmp := SmpDirector(N_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpDirectorDeriv1(dNdL_tmp, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpNormDirectorDeriv1(dmdL_tmp, m_tmp, N_tmp, dNdL_tmp)
SmpUnitDirectorDeriv1(dndL_tmp, m_tmp, N_tmp, dNdL_tmp, dmdL_tmp)
L[k] = tmp
return dndL_tmp[i][j]
}, L[k], 1e-6)
chk.AnaNum(tst, io.Sf("d²n[%d]/dL[%d]dL[%d]", i, j, k), dtol2, d2ndLdL[i][j][k], dnum, dver)
}
}
}
// recover tolerance
dtol2 = dtol2_tmp
// SMP derivs
//if false {
if true {
io.Pfpink("\nSMP derivs\n")
dndL_ := la.MatAlloc(3, 3)
dNdL_ := make([]float64, 3)
d2ndLdL_ := utl.Deep3alloc(3, 3, 3)
N_ := make([]float64, 3)
F_ := make([]float64, 3)
G_ := make([]float64, 3)
m_ := SmpDerivs1(dndL_, dNdL_, N_, F_, G_, L, smp_a, smp_b, smp_β, smp_ϵ)
SmpDerivs2(d2ndLdL_, L, smp_a, smp_b, smp_β, smp_ϵ, m_, N_, F_, G_, dNdL_, dndL_)
chk.Scalar(tst, "m_", 1e-14, m_, m)
chk.Vector(tst, "N_", 1e-15, N_, N)
chk.Vector(tst, "dNdL_", 1e-15, dNdL_, dNdL)
chk.Matrix(tst, "dndL_", 1e-13, dndL_, dndL)
chk.Deep3(tst, "d2ndLdL_", 1e-11, d2ndLdL_, d2ndLdL)
}
}
}
// M_EigenProjsDerivNum returns the derivatives of the eigenprojectors w.r.t its defining tensor
// using the finite differences method.
// Input:
// a -- tensor in Mandel basis
// h -- step size for finite differences
// Output:
// dPda -- derivatives [3][ncp][ncp]
func M_EigenProjsDerivNum(dPda [][][]float64, a []float64, h float64) (err error) {
ncp := len(a)
λ := make([]float64, 3)
P := la.MatAlloc(3, ncp)
Q := Alloc2()
A := Alloc2()
q2p := func(k int) {
switch k {
case 0:
P[0][0] = Q[0][0] * Q[0][0]
P[0][1] = Q[1][0] * Q[1][0]
P[0][2] = Q[2][0] * Q[2][0]
P[0][3] = Q[0][0] * Q[1][0] * SQ2
if ncp == 6 {
P[0][4] = Q[1][0] * Q[2][0] * SQ2
P[0][5] = Q[2][0] * Q[0][0] * SQ2
}
case 1:
P[1][0] = Q[0][1] * Q[0][1]
P[1][1] = Q[1][1] * Q[1][1]
P[1][2] = Q[2][1] * Q[2][1]
P[1][3] = Q[0][1] * Q[1][1] * SQ2
if ncp == 6 {
P[1][4] = Q[1][1] * Q[2][1] * SQ2
P[1][5] = Q[2][1] * Q[0][1] * SQ2
}
case 2:
P[2][0] = Q[0][2] * Q[0][2]
P[2][1] = Q[1][2] * Q[1][2]
P[2][2] = Q[2][2] * Q[2][2]
P[2][3] = Q[0][2] * Q[1][2] * SQ2
if ncp == 6 {
P[2][4] = Q[1][2] * Q[2][2] * SQ2
P[2][5] = Q[2][2] * Q[0][2] * SQ2
}
}
}
var tmp float64
failed := false
for k := 0; k < 3; k++ {
for i := 0; i < ncp; i++ {
for j := 0; j < ncp; j++ {
dPda[k][i][j], _ = num.DerivCentral(func(x float64, args ...interface{}) float64 {
tmp, a[j] = a[j], x
defer func() { a[j] = tmp }()
Man2Ten(A, a)
_, err = la.Jacobi(Q, λ, A)
if err != nil {
failed = true
return 0
}
q2p(k)
return P[k][i]
}, a[j], h)
if failed {
return
}
}
}
}
return
}