Ejemplo n.º 1
0
// 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]
}
Ejemplo n.º 2
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
}
Ejemplo n.º 3
0
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")
}