Golang chk.AnaNum函数代码示例

// CheckDerivT checks derivatives w.r.t to t for fixed coordinates x
func CheckDerivT(tst *testing.T, o Func, t0, tf float64, xcte []float64, np int, tskip []float64, sktol, dtol, dtol2 float64, ver bool) {
	t := utl.LinSpace(t0, tf, np)
	for i := 0; i < np; i++ {
		g := o.G(t[i], xcte)
		h := o.H(t[i], xcte)
		skip := false
		for _, val := range tskip {
			if math.Abs(val-t[i]) < sktol {
				skip = true
				break
			}
		}
		if skip {
			continue
		}
		dnum := num.DerivCen(func(t float64, args ...interface{}) (res float64) {
			return o.F(t, xcte)
		}, t[i])
		chk.AnaNum(tst, io.Sf("G(%10f)", t[i]), dtol, g, dnum, ver)
		dnum2 := num.DerivCen(func(t float64, args ...interface{}) (res float64) {
			return o.G(t, xcte)
		}, t[i])
		chk.AnaNum(tst, io.Sf("H(%10f)", t[i]), dtol2, h, dnum2, ver)
	}
}

// CheckDerivs compares analytical with numerical derivatives
func (o *Nurbs) CheckDerivs(tst *testing.T, nn int, tol float64, ver bool) {
	dana := make([]float64, 2)
	dnum := make([]float64, 2)
	for _, u := range utl.LinSpace(o.b[0].tmin, o.b[0].tmax, nn) {
		for _, v := range utl.LinSpace(o.b[1].tmin, o.b[1].tmax, nn) {
			uu := []float64{u, v}
			o.CalcBasisAndDerivs(uu)
			for i := 0; i < o.n[0]; i++ {
				for j := 0; j < o.n[1]; j++ {
					l := i + j*o.n[0]
					o.GetDerivL(dana, l)
					o.NumericalDeriv(dnum, uu, l)
					chk.AnaNum(tst, io.Sf("dR[%d][%d][0](%g,%g)", i, j, uu[0], uu[1]), tol, dana[0], dnum[0], ver)
					chk.AnaNum(tst, io.Sf("dR[%d][%d][1](%g,%g)", i, j, uu[0], uu[1]), tol, dana[1], dnum[1], ver)
				}
			}
		}
	}
}

func Test_hyperelast02(tst *testing.T) {

	//verbose()
	chk.PrintTitle("hyperelast02 (linear)")

	E, ν := 1500.0, 0.25
	K := Calc_K_from_Enu(E, ν)
	G := Calc_G_from_Enu(E, ν)
	io.Pforan("K = %v\n", K)
	io.Pforan("G = %v\n", G)

	var m HyperElast1
	m.Init(2, false, []*fun.Prm{
		&fun.Prm{N: "K0", V: K},
		&fun.Prm{N: "G0", V: G},
		&fun.Prm{N: "le", V: 1},
	})
	io.Pforan("m = %+v\n", m)

	ε := []float64{-0.001, -0.002, -0.003}
	σ := make([]float64, 3)
	m.L_update(σ, ε)
	io.Pfblue2("ε = %v\n", ε)
	io.Pfcyan("σ = %v\n", σ)

	D := la.MatAlloc(3, 3)
	m.L_CalcD(D, ε)
	la.PrintMat("D", D, "%14.6f", false)

	tol := 1e-11
	verb := io.Verbose
	var tmp float64
	for i := 0; i < 3; i++ {
		for j := 0; j < 3; j++ {
			dnum := num.DerivCen(func(x float64, args ...interface{}) (res float64) {
				tmp, ε[j] = ε[j], x
				m.L_update(σ, ε)
				res = σ[i]
				ε[j] = tmp
				return
			}, ε[j])
			chk.AnaNum(tst, io.Sf("D%d%d", i, j), tol, D[i][j], dnum, verb)
		}
	}
}

// CheckDerivX checks derivatives w.r.t to x for fixed t
func CheckDerivX(tst *testing.T, o Func, tcte float64, xmin, xmax []float64, np int, xskip [][]float64, sktol, dtol float64, ver bool) {
	ndim := len(xmin)
	dx := make([]float64, ndim)
	for i := 0; i < ndim; i++ {
		dx[i] = (xmax[i] - xmin[i]) / float64(np-1)
	}
	x := make([]float64, ndim)
	g := make([]float64, ndim)
	nz := 1
	if ndim == 3 {
		nz = np
	}
	xtmp := make([]float64, ndim)
	for k := 0; k < nz; k++ {
		if ndim == 3 {
			x[2] = xmin[2] + float64(k)*dx[2]
		}
		for j := 0; j < np; j++ {
			x[1] = xmin[1] + float64(j)*dx[1]
			for i := 0; i < np; i++ {
				x[0] = xmin[0] + float64(i)*dx[0]
				o.Grad(g, tcte, x)
				for l := 0; l < ndim; l++ {
					skip := false
					for _, val := range xskip {
						if math.Abs(val[l]-x[l]) < sktol {
							skip = true
							break
						}
					}
					if skip {
						continue
					}
					dnum := num.DerivCen(func(s float64, args ...interface{}) (res float64) {
						copy(xtmp, x)
						xtmp[l] = s
						return o.F(tcte, xtmp)
					}, x[l])
					chk.AnaNum(tst, io.Sf("dFdX(t,%10v)[%d]", x, l), dtol, g[l], dnum, ver)
				}
			}
		}
	}
}

func Test_hyperelast03(tst *testing.T) {

	//verbose()
	chk.PrintTitle("hyperelast03 (nonlinear)")

	var m HyperElast1
	m.Init(2, false, []*fun.Prm{
		&fun.Prm{N: "kap", V: 0.05},
		&fun.Prm{N: "kapb", V: 20.0},
		&fun.Prm{N: "G0", V: 1500},
		&fun.Prm{N: "pr", V: 2.2},
		&fun.Prm{N: "pt", V: 11.0},
	})
	io.Pforan("m = %+v\n", m)

	ε := []float64{-0.001, -0.002, -0.003}
	σ := make([]float64, 3)
	m.L_update(σ, ε)
	io.Pfblue2("ε = %v\n", ε)
	io.Pfcyan("σ = %v\n", σ)

	D := la.MatAlloc(3, 3)
	m.L_CalcD(D, ε)
	la.PrintMat("D", D, "%14.6f", false)

	tol := 1e-7
	verb := io.Verbose
	var tmp float64
	for i := 0; i < 3; i++ {
		for j := 0; j < 3; j++ {
			dnum := num.DerivCen(func(x float64, args ...interface{}) (res float64) {
				tmp, ε[j] = ε[j], x
				m.L_update(σ, ε)
				res = σ[i]
				ε[j] = tmp
				return
			}, ε[j])
			chk.AnaNum(tst, io.Sf("D%d%d", i, j), tol, D[i][j], dnum, verb)
		}
	}
}

// check performs the checking of Kb using numerical derivatives
func (o *testKb) check(label string, d *Domain, e Elem, Imap, Jmap []int, Kana [][]float64, restore func()) {
	var imap, jmap []int
	if o.ni < 0 {
		imap = Imap
	} else {
		if o.ni <= len(Imap) {
			imap = Imap[:o.ni]
		}
	}
	if o.nj < 0 {
		jmap = Jmap
	} else {
		if o.nj <= len(Jmap) {
			jmap = Jmap[:o.nj]
		}
	}
	step := 1e-6
	//derivfcn := num.DerivForward
	//derivfcn := num.DerivBackward
	derivfcn := num.DerivCentral
	var tmp float64
	for i, I := range imap {
		for j, J := range jmap {
			dnum, _ := derivfcn(func(x float64, args ...interface{}) (res float64) {
				tmp, d.Sol.Y[J] = d.Sol.Y[J], x
				for k := 0; k < d.Ny; k++ {
					o.Fbtmp[k] = 0
					d.Sol.ΔY[k] = d.Sol.Y[k] - o.Yold[k]
				}
				restore()
				e.Update(d.Sol)
				e.AddToRhs(o.Fbtmp, d.Sol)
				res = -o.Fbtmp[I]
				d.Sol.Y[J] = tmp
				return res
			}, d.Sol.Y[J], step)
			chk.AnaNum(o.tst, io.Sf(label+"%3d%3d", i, j), o.tol, Kana[i][j], dnum, o.verb)
		}
	}
}

// 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 {
//.........这里部分代码省略.........

//.........这里部分代码省略.........
	//verbose()
	chk.PrintTitle("spo751a")

	// start simulation
	analysis := NewFEM("data/spo751.sim", "", true, false, false, false, chk.Verbose, 0)

	// set stage
	err := analysis.SetStage(0)
	if err != nil {
		tst.Errorf("SetStage failed:\n%v", err)
		return
	}

	// initialise solution vectros
	err = analysis.ZeroStage(0, true)
	if err != nil {
		tst.Errorf("ZeroStage failed:\n%v", err)
		return
	}

	// domain
	dom := analysis.Domains[0]

	// nodes and elements
	chk.IntAssert(len(dom.Nodes), 23)
	chk.IntAssert(len(dom.Elems), 4)

	// check dofs
	for _, nod := range dom.Nodes {
		chk.IntAssert(len(nod.Dofs), 2)
	}

	// check equations
	nids, eqs := get_nids_eqs(dom)
	chk.Ints(tst, "nids", nids, []int{0, 5, 7, 2, 3, 6, 4, 1, 10, 12, 8, 11, 9, 15, 17, 13, 16, 14, 20, 22, 18, 21, 19})
	chk.Ints(tst, "eqs", eqs, utl.IntRange(23*2))

	// check solution arrays
	ny := 23 * 2
	nλ := 9 + 9
	nyb := ny + nλ
	chk.IntAssert(len(dom.Sol.Y), ny)
	chk.IntAssert(len(dom.Sol.Dydt), 0)
	chk.IntAssert(len(dom.Sol.D2ydt2), 0)
	chk.IntAssert(len(dom.Sol.Psi), 0)
	chk.IntAssert(len(dom.Sol.Zet), 0)
	chk.IntAssert(len(dom.Sol.Chi), 0)
	chk.IntAssert(len(dom.Sol.L), nλ)
	chk.IntAssert(len(dom.Sol.ΔY), ny)

	// check linear solver arrays
	chk.IntAssert(len(dom.Fb), nyb)
	chk.IntAssert(len(dom.Wb), nyb)

	// check umap
	umaps := [][]int{
		{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15},
		{2, 3, 16, 17, 18, 19, 4, 5, 20, 21, 22, 23, 24, 25, 10, 11},
		{16, 17, 26, 27, 28, 29, 18, 19, 30, 31, 32, 33, 34, 35, 22, 23},
		{26, 27, 36, 37, 38, 39, 28, 29, 40, 41, 42, 43, 44, 45, 32, 33},
	}
	for i, ele := range dom.Elems {
		e := ele.(*ElemU)
		io.Pforan("e%d.umap = %v\n", e.Id(), e.Umap)
		chk.Ints(tst, "umap", e.Umap, umaps[i])
	}

	// constraints
	chk.IntAssert(len(dom.EssenBcs.Bcs), nλ)
	var ct_uy_eqs []int // constrained uy equations [sorted]
	var ct_incsup_xeqs []int
	var ct_incsup_yeqs []int
	αrad := 120.0 * math.Pi / 180.0
	cα, sα := math.Cos(αrad), math.Sin(αrad)
	for _, c := range dom.EssenBcs.Bcs {
		chk.IntAssertLessThanOrEqualTo(1, len(c.Eqs)) // 1 ≤ neqs
		io.Pforan("c.Key=%s c.Eqs=%v\n", c.Key, c.Eqs)
		if len(c.Eqs) == 1 {
			if c.Key == "uy" {
				ct_uy_eqs = append(ct_uy_eqs, c.Eqs[0])
				continue
			}
		} else {
			if c.Key == "incsup" {
				ct_incsup_xeqs = append(ct_incsup_xeqs, c.Eqs[0])
				ct_incsup_yeqs = append(ct_incsup_yeqs, c.Eqs[1])
				chk.AnaNum(tst, "cos(α)", 1e-15, c.ValsA[0], cα, false)
				chk.AnaNum(tst, "sin(α)", 1e-15, c.ValsA[1], sα, false)
				continue
			}
		}
		tst.Errorf("key %s is incorrect", c.Key)
	}
	sort.Ints(ct_uy_eqs)
	sort.Ints(ct_incsup_xeqs)
	sort.Ints(ct_incsup_yeqs)
	chk.Ints(tst, "constrained uy equations", ct_uy_eqs, []int{1, 3, 9, 17, 21, 27, 31, 37, 41})
	chk.Ints(tst, "incsup x equations", ct_incsup_xeqs, []int{4, 6, 12, 18, 24, 28, 34, 38, 44})
	chk.Ints(tst, "incsup y equations", ct_incsup_yeqs, []int{5, 7, 13, 19, 25, 29, 35, 39, 45})
}

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
//.........这里部分代码省略.........

//.........这里部分代码省略.........
		sort.Ints(ct_uy_eqs)
		sort.Ints(ct_pl_eqs)
		chk.Ints(tst, "equations with ux prescribed", ct_ux_eqs, []int{0, 3, 6, 9, 14, 18, 22, 25, 28, 32, 36, 39, 42, 46, 50, 53, 56, 60})
		chk.Ints(tst, "equations with uy prescribed", ct_uy_eqs, []int{1, 4, 13})
		chk.Ints(tst, "equations with pl prescribed", ct_pl_eqs, []int{2, 5})

	}

	// initial values @ nodes
	io.Pforan("initial values @ nodes\n")
	for _, nod := range dom.Nodes {
		z := nod.Vert.C[1]
		for _, dof := range nod.Dofs {
			u := dom.Sol.Y[dof.Eq]
			switch dof.Key {
			case "ux":
				chk.Scalar(tst, io.Sf("nod %3d : ux(@ %4g)= %6g", nod.Vert.Id, z, u), 1e-17, u, 0)
			case "uy":
				chk.Scalar(tst, io.Sf("nod %3d : uy(@ %4g)= %6g", nod.Vert.Id, z, u), 1e-17, u, 0)
			case "pl":
				plC, _, _ := Global.HydroSt.Calc(z)
				chk.Scalar(tst, io.Sf("nod %3d : pl(@ %4g)= %6g", nod.Vert.Id, z, u), 1e-13, u, plC)
			}
		}
	}

	// intial values @ integration points
	io.Pforan("initial values @ integration points\n")
	for _, ele := range dom.Elems {
		e := ele.(*ElemUP)
		for idx, ip := range e.P.IpsElem {
			s := e.P.States[idx]
			z := e.P.Shp.IpRealCoords(e.P.X, ip)[1]
			chk.AnaNum(tst, io.Sf("sl(z=%11.8f)", z), 1e-17, s.A_sl, 1, chk.Verbose)
		}
	}

	// parameters
	ν := 0.2            // Poisson's coefficient
	K0 := ν / (1.0 - ν) // earth pressure at rest
	nf := 0.3           // porosity
	sl := 1.0           // saturation
	ρL := 1.0           // intrinsic (real) density of liquid
	ρS_top := 2.0       // intrinsic (real) density of solids in top layer
	ρS_bot := 3.0       // intrinsic (real) density of solids in bottom layer
	h := 5.0            // height of each layer
	g := 10.0           // gravity

	// densities
	nl := nf * sl         // volume fraction of luqid
	ns := 1.0 - nf        // volume fraction of solid
	ρl := nl * ρL         // partial density of liquid
	ρs_top := ns * ρS_top // partial density of solids in top layer
	ρs_bot := ns * ρS_bot // partial density of solids in bottom layer
	ρ_top := ρl + ρs_top  // density of mixture in top layer
	ρ_bot := ρl + ρs_bot  // density of mixture in bottom layer

	// absolute values of stresses
	σV_z5 := ρ_top * g * h     // total vertical stress @ elevation z = 5 m (absolute value)
	σV_z0 := σV_z5 + ρ_bot*g*h // total vertical stress @ elevation z = 0 m (absolute value)
	io.Pfyel("ρ_top       = %g\n", ρ_top)
	io.Pfyel("ρ_bot       = %g\n", ρ_bot)
	io.Pfyel("|ΔσV_top|   = %g\n", ρ_top*g*h)
	io.Pfyel("|ΔσV_bot|   = %g\n", ρ_bot*g*h)
	io.PfYel("|σV|(@ z=0) = %g\n", σV_z0)
	io.PfYel("|σV|(@ z=5) = %g\n", σV_z5)

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
//.........这里部分代码省略.........

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)

//.........这里部分代码省略.........

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
//.........这里部分代码省略.........