From f8d657022231576e46fb8fa472c1787235b86233 Mon Sep 17 00:00:00 2001 From: regislebrun Date: Mon, 1 Apr 2024 23:18:35 +0200 Subject: [PATCH] Added a stiffened panel buckling model --- .../_static/stiffened_panel_description.png | Bin 0 -> 334324 bytes .../_static/stiffened_panel_illustration.jpg | Bin 0 -> 169343 bytes .../_static/stiffened_panel_simulation.png | Bin 0 -> 1453132 bytes python/doc/bibliography.rst | 2 + .../reliability/plot_stiffened_panel.py | 143 ++++++++++++ .../doc/usecases/use_case_stiffened_panel.rst | 207 ++++++++++++++++++ python/doc/usecases/usecases.rst | 1 + python/doc/user_manual/usecases.rst | 1 + python/src/usecases/stiffened_panel.py | 130 +++++++++++ python/test/CMakeLists.txt | 1 + python/test/t_stiffened_panel.py | 34 +++ 11 files changed, 519 insertions(+) create mode 100644 python/doc/_static/stiffened_panel_description.png create mode 100644 python/doc/_static/stiffened_panel_illustration.jpg create mode 100644 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z{sgqW`q1!ld0C878QHFO?3C}ArP%WTC9mvZlIW0jjZo@${9=ykqQoFZVIq%cG~4J= z+Yb1Y7z6AEp6WMHVT|Dw^D}85PthUv)n;^ys5m(glU6zdX;jB%Rqs8m~&Q0pcc zUKG0^&r7}N@6K_5j!B;xxRB5whwm1tIY-;aLjA|ix&KZ5zxng`AM{&nhalj`=!a{) l^abe8kN@xI|JM^Z0v9QBI8{%w*ZcYF=gwNADvhtk{U1Ep%TE9R literal 0 HcmV?d00001 diff --git a/python/doc/bibliography.rst b/python/doc/bibliography.rst index 831259b78fc..f22d1fef0ab 100644 --- a/python/doc/bibliography.rst +++ b/python/doc/bibliography.rst @@ -186,6 +186,8 @@ Bibliography .. [knio2010] Le MaƮtre, O., & Knio, O. M. (2010). *Spectral methods for uncertainty quantification: with applications to computational fluid dynamics.* Springer Science & Business Media. +.. [ko1994] William L. Ko, Raymond H. Jackson, + *Share Buckling Analysis of a Hat-Stiffend Panel*, NASA Technical Memorandum 4644 (November 1994). .. [koay2006] Koay C.G., Basser P.J., *Analytically exact correction scheme for signal extraction from noisy magnitude MR signals*, Journal of magnetics Resonance 179, 317-322, 2006. diff --git a/python/doc/examples/reliability_sensitivity/reliability/plot_stiffened_panel.py b/python/doc/examples/reliability_sensitivity/reliability/plot_stiffened_panel.py new file mode 100644 index 00000000000..41ce1bfc013 --- /dev/null +++ b/python/doc/examples/reliability_sensitivity/reliability/plot_stiffened_panel.py @@ -0,0 +1,143 @@ +""" +Estimate a buckling probability +=============================== +""" +# %% +# +# In this example, we estimate the probability that the output of a function +# exceeds a given threshold with the FORM method, the SORM method and an advanced +# sampling method. + +# We consider the :ref:`stiffened panel model `. + +# %% +# Define the model +# ---------------- + +# %% +from openturns.usecases import stiffened_panel +import openturns as ot +import openturns.viewer as viewer + +ot.Log.Show(ot.Log.NONE) + +# %% +# We load the stiffened panel model from the usecases module : +panel = stiffened_panel.StiffenedPanel() +distribution = panel.distribution +model = panel.model + +# %% +# See the input distribution +distribution + +# %% +# See the model +model.getOutputDescription() + +# %% +# Draw the distribution of a sample of the output. +sampleSize = 1000 +inputSample = distribution.getSample(sampleSize) +outputSample = model(inputSample) +graph = ot.HistogramFactory().build(outputSample).drawPDF() +_ = viewer.View(graph) + +# %% +# Define the event +# ---------------- + +# %% +# Then we create the event whose probability we want to estimate. + +# %% +vect = ot.RandomVector(distribution) +G = ot.CompositeRandomVector(model, vect) +N0 = 165 +event = ot.ThresholdEvent(G, ot.Less(), N0) +event.setName("buckling") + +# %% +# Estimate the probability with FORM +# ---------------------------------- + +# %% +# Define a solver. + +# %% +optimAlgo = ot.Cobyla() +optimAlgo.setMaximumEvaluationNumber(1000) +optimAlgo.setMaximumAbsoluteError(1.0e-10) +optimAlgo.setMaximumRelativeError(1.0e-10) +optimAlgo.setMaximumResidualError(1.0e-10) +optimAlgo.setMaximumConstraintError(1.0e-10) + +# %% +# Run FORM. + +# %% +startingPoint = distribution.getMean() +algo = ot.FORM(optimAlgo, event, startingPoint) +n0 = model.getCallsNumber() +algo.run() +n1 = model.getCallsNumber() +result = algo.getResult() +standardSpaceDesignPoint = result.getStandardSpaceDesignPoint() + +# %% +# Retrieve results. + +# %% +result = algo.getResult() +probability = result.getEventProbability() +print("Pf (FORM)=%.3e" % probability, "nb evals=", n1 - n0) + +# %% +# Importance factors. + +# %% +graph = result.drawImportanceFactors() +view = viewer.View(graph) + +# %% +# Estimate the probability with SORM +# ---------------------------------- + +# %% +# Run SORM. + +# %% +algo = ot.SORM(optimAlgo, event, startingPoint) +n0 = model.getCallsNumber() +algo.run() +n1 = model.getCallsNumber() + +# %% +# Retrieve results. + +# %% +result = algo.getResult() +probability = result.getEventProbabilityBreitung() +print("Pf (SORM)=%.3e" % probability, "nb evals=", n1 - n0) + +# %% +# We see that the FORM and SORM approximations give significantly different +# results. Use a simulation algorithm to get a confidence interval. + +# %% +# Estimate the probability with PostAnalyticalControlledImportanceSampling +# ------------------------------------------------------------------------ + +# %% +algo = ot.PostAnalyticalControlledImportanceSampling(result) +algo.setBlockSize(100) +algo.setMaximumOuterSampling(100) +algo.setMaximumCoefficientOfVariation(0.1) +n0 = model.getCallsNumber() +algo.run() +n1 = model.getCallsNumber() +result = algo.getResult() +Pf = result.getProbabilityEstimate() +print("Pf (sim) = %.3e" % Pf, "nb evals=", n1 - n0) +width = result.getConfidenceLength(0.95) +print("C.I (95%)=[" + "%.3e" % (Pf - 0.5 * width), ",%.3e" % (Pf + 0.5 * width), "]") diff --git a/python/doc/usecases/use_case_stiffened_panel.rst b/python/doc/usecases/use_case_stiffened_panel.rst new file mode 100644 index 00000000000..ac6bfb9a8ca --- /dev/null +++ b/python/doc/usecases/use_case_stiffened_panel.rst @@ -0,0 +1,207 @@ +.. _use-case-stiffened-panel: + +Stiffened panel buckling +======================== + + +Introduction +------------ + +The following figure presents a stiffed panel subject to buckling on a military aircraft. + +This use-case implements a simplified model of buckling for a stiffened panel, detailed in [ko1994]_. + +.. figure:: ../_static/stiffened_panel_illustration.jpg + :align: center + :alt: buckling illustration + :width: 100% + + **Figure 1.** Buckling of a stiffened panel. + + +.. figure:: ../_static/stiffened_panel_simulation.png + :align: center + :alt: buckling simulation + :width: 100% + + **Figure 2.** 3D simulation of buckling. + + +.. figure:: ../_static/stiffened_panel_description.png + :align: center + :alt: stiffened panel geometry + :width: 100% + + **Figure 3.** Parameterization of the stiffened panel. + + +This test case is composed of nine random variables: + +- :math:`E\sim\mathcal{TN}(110.0e9, 55.0e9, 99.0e9, 121.0e9)` : Young modulus (Pa) + +- :math:`nu\sim\mathcal{U}(0.3675, 0.3825)` : Poisson coefficient (-) + +- :math:`h_c\sim\mathcal{U}(0.0285, 0.0315)` : Distance between the mean surface of the hat and the foot of the Stiffener (m) + +- :math:`\ell\sim\mathcal{U}(0.04655, 0.05145)` : Length of the stiffener flank (m) + +- :math:`f_1\sim\mathcal{U}(0.0266, 0.0294)` : Width of the stiffener foot (m) + +- :math:`f_2\sim\mathcal{U}(0.00627, 0.00693)` : Width of the stiffener hat (m) + +- :math:`t\sim\mathcal{U}(8.02e-5, 8.181e-5)` : Thickness of the panel and the stiffener (m) + +- :math:`a\sim\mathcal{U}(0.6039, 0.6161)` : Width of the panel (m) + +- :math:`b_0\sim\mathcal{U}(0.04455, 0.04545)` : Distance between two stiffeners (m) + +- :math:`p\sim\mathcal{U}(0.03762, 0.03838)` : Half-width of the stiffener (m) + +The output of interest is: + +- :math:`(N_{xy})_{cr}`: the critical shear force (N) + +We assume that the input variables are independent except the :math:`f_1` and +:math:`f_2` for which we measure a Spearman correlation of :math:`\rho^S_{12}=-0.8`, +modelled using a :class:`~openturns.NormalCopula`. + +The critical load :math:`(\tau_{xy})_{cr}` of a stiffened panel subject to shear load is given by + +.. math:: + (\tau_{xy})_{cr}=k_{xy}\frac{\pi^2D}{b_0^2t_s} + +where + +- :math:`a` is the width of the panel; + +- :math:`b_0` is the width between too consecutive stiffener feet; + +- :math:`t_s` is the thickness of the panel main surface; + +- :math:`E_s` is the Young modulus of the panel main surface; + +- :math:`\nu_s` is the Poisson coefficient of the panel main surface; + +- :math:`D=\frac{E_st_s^3}{12(1-\nu_s^2)}` is the bending coefficient of the + panel main surface; + +- :math:`k_{xy}` is the load factor associated to shear buckling. It is given as + a function of :math:`\frac{b_0}{a}` through the empirical relation + :math:`k_{xy}=5.35 + 4\left(\frac{b_0}{a}\right)^2` + +It is more convenient to use the shear force :math:`N_{xy}` instead of the shear +stress component :math:`\tau_{xy}`. It leads to the relation: + +.. math:: + N_{xy}=q_1+q_c + +where :math:`q_1` abd :math:`q_c` are the shear fluxes in the panel main surface +and its stiffener. They are given by: + +.. math:: + q_1=\tau_{xy}t_s=2G_sh_0t_s\frac{\partial^2w}{\partial x\partial y} + +and + +.. math:: + q_c=\frac{G_ct_cp}{\ell}\left[h-2h_0+\frac{h_c}{2p}(f_1-f_2)\right]\frac{\partial^2w}{\partial x\partial y} + +where + +- :math:`G_s=\frac{E_s}{2(1+\nu_s)}` is the shear modulus of the panel main + surface; + +- :math:`\frac{\partial^2w}{\partial x\partial y}` is the torsion strain of the + panel main surface; + +- :math:`G_c=\frac{E_c}{2(1+\nu_c)}` is the shear coefficient of the stiffener; + +- :math:`t_c` is the thickness of the stiffener; + +- :math:`h_c` is the distance between the mean surfaces of the stiffener hat and + foot; + +- :math:`h=h_c+\frac{t_c+t_s}{2}` is the distance between the mean surfaces of + the stiffener hat and the panel main surface; + +- :math:`f_1` is the width of the foot of the stiffener; + +- :math:`f_2` is the width of the hat of the stiffener; + +- :math:`p` is the half-widht of the stiffener; + +- :math:`R` is the radius of the circular part of the stiffener; + +- :math:`\theta` is the angle of the circular part of the stiffener; + +- :math:`\ell` is the length of the stiffener flank; + +- :math:`d=\frac{\ell-f_2}{2}-R\theta- :math:` is the half-lenght of the straight + part of the flank of the stiffener; + +- :math:`A=\ell t_c` is the area of the section of an half-ondulation; + +- :math:`\bar{A}=A+pt_s+\frac{1}{2}(f_1-f_2)t_c` is the area of the section of + the full panel (main surface and stiffener) bounded by :math:`p`; + +- :math:`h_0=\frac{1}{2\bar{A}}\left[A(h_c+t_c+t_s)+\frac{1}{2}t_c(f_1-f_2)(t_c+t_s)\right]` + is the distance between the mean surface of the panel main surface and the + global geometric center of the pa,nel; + +It leads to: + +.. math:: + N_{xy}=q_1(1+q_c/q1)=\tau_{xy}t_s\left[1+\frac{1}{4}\frac{G_ct_c}{G_st_s}\frac{\left(2p(h-2h_0)-h_c(f_1-f_2)\right)}{h_0\ell}\right] + +and finally, :math:`(N_{xy})_{cr}` is given by: + +.. math:: + (N_{xy})_{cr}=\bigg[5.35 + 4\left(\frac{b_0}{a}\right)^2\bigg]\bigg[\frac{\pi^2}{b_0^2}\frac{E_st_s^3}{12(1-\nu_s^2)}\bigg]\bigg[1+\frac{1}{4}\frac{G_ct_c}{G_st_s}\frac{(2p(h-2h_0)-h_c(f_1-f_2))}{h_0\ell}\bigg] + +For industrial constraints, the stiffener and the main surface are cut in the +same metal sheet, so :math:`E_c=E_s=E`, :math:`\nu_c=\nu_s=\nu`, :math:`t_c=t_s=t`. +The final expression of the critical shear force is then: + +.. math:: + (N_{xy})_{cr}=\bigg[5.35 + 4\left(\frac{b_0}{a}\right)^2\bigg]\bigg[\frac{\pi^2}{b_0^2}\frac{Et^3}{12(1-\nu^2)}\bigg]\bigg[1+\frac{1}{4}\frac{(2p(h-2h_0)-h_c(f_1-f_2))}{h_0\ell}\bigg] + +with: + +- :math:`A=\ell t` + +- :math:`\bar{A}=A+t\left(p+\frac{f_1-f_2}{2}\right)` + +- :math:`h_0=\frac{1}{2\bar{A}}\left[A(h_c+2t)+t^2(f_1-f_2)\right]` + +- :math:`h=h_c+t` + + +References +---------- + +* [ko1994]_ + +Load the use case +----------------- + +We can load this model from the use cases module as follows : + +.. code-block:: python + + >>> from openturns.usecases import stiffened_panel + >>> sp = stiffened_panel.StiffenedPanel() + >>> # Load the stiffened panel use case + >>> model = sp.model() + +API documentation +----------------- + +.. currentmodule:: openturns.usecases.stiffened_panel + +.. autoclass:: StiffenedPanel + :noindex: + +Examples based on this use case +------------------------------- + +.. minigallery:: openturns.usecases.stiffened_panel.StiffenedPanel diff --git a/python/doc/usecases/usecases.rst b/python/doc/usecases/usecases.rst index 592311a51d0..285299a19a8 100644 --- a/python/doc/usecases/usecases.rst +++ b/python/doc/usecases/usecases.rst @@ -27,3 +27,4 @@ Contents use_case_oscillator coles use_case_linthurst + use_case_stiffened_panel diff --git a/python/doc/user_manual/usecases.rst b/python/doc/user_manual/usecases.rst index fd01dcac602..8c3ebfbc6a7 100644 --- a/python/doc/user_manual/usecases.rst +++ b/python/doc/user_manual/usecases.rst @@ -24,3 +24,4 @@ Use cases from the usecases module usecases.fireSatellite_function.FireSatelliteModel usecases.wingweight_function.WingWeightModel usecases.oscillator.Oscillator + usecases.stiffened_panel.StiffenedPanel diff --git a/python/src/usecases/stiffened_panel.py b/python/src/usecases/stiffened_panel.py new file mode 100644 index 00000000000..13358a4d6c3 --- /dev/null +++ b/python/src/usecases/stiffened_panel.py @@ -0,0 +1,130 @@ +""" +Use case : stiffened panel +========================== +""" +import openturns as ot + + +class StiffenedPanel: + r""" + Data class for the stiffened panel model. + + + Attributes + ---------- + + dim : Constant, the dimension of the problem. + dim=10 + + model : `SymbolicFunction` + Model of the critical shearing load. + The model has input dimension 10 and output dimension 1. + More precisely, we have :math:`\vect{X} = (E, \nu, h_c, \ell, f_1, f_2, t, a, b_0, p)` + and :math:`Y = ((N_{xy})_{cr})`. + + E : Young modulus (Pa), :class:`~openturns.TruncatedNormal` distribution + `ot.TruncatedNormal(110.0e9, 55.0e9, 99.0e9, 121.0e9)` + + nu : Poisson coefficient (-), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.3675, 0.3825)` + + h_c : Distance between the mean surface of the hat and the foot of the Stiffener (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.0285, 0.0315)` + + ell : Length of the stiffener flank (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.04655, 0.05145)` + + f_1 : Width of the stiffener foot (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.0266, 0.0294)` + + f_2 : Width of the stiffener hat (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.00627, 0.00693)` + + t : Thickness of the panel and the stiffener (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(8.02e-5, 8.181e-5)` + + a : Width of the panel (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.6039, 0.6161)` + + b_0 : Distance between two stiffeners (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.04455, 0.04545)` + + p : Half-width of the stiffener (m), :class:`~openturns.Uniform` distribution + `ot.Uniform(0.03762, 0.03838)` + + distribution : :class:`~openturns.ComposedDistribution` + The joint distribution of the input parameters. + + Examples + -------- + >>> from openturns.usecases import stiffened_panel + >>> # Load the stiffened panel model + >>> panel = stiffened_panel.StiffenedPanel() + >>> print("Inputs:", panel.model.getInputDescription()) + Inputs: [F,L,a,De,di,E] + >>> print("Outputs:", panel.model.getOutputDescription()) + [Deflection,Left angle,Right angle] + """ + + def __init__(self): + self.dim = 10 + + # Here we use substitutions in order to get a unique formula for the model. + # This way, we get a symbolic derivation of the gradient and the hessian, + # which is not possible if we use the exprtk embedded language + formula = '(5.35+4*(b_0/a)^2) * (pi_^2/b_0^2*E*t^3/(12*(1-nu^2))) * (1+(2*p*(h-2*h_0)-h_c*(f_1-f_2))/(4*h_0*l))' + # Substitutions + A = '(l*t)' + A_bar = '(A+t*(p+(f_1-f_2)/2))' + h_0 = '((A*(h_c+2*t)+t^2*(f_1-f_2))/(2*A_bar))' + h = '(h_c + t)' + formula = formula.replace('h_0', h_0) + formula = formula.replace('A_bar', A_bar) + formula = formula.replace('A', A) + # Attention! If the substitution of h by its formula is done without + # caution h_c will be replaced by (h_c + t)_c! + # First, we protect h_c + formula = formula.replace('h_c', 'H_C') + # Then we replace h by its formula + formula = formula.replace('h', h) + # On restaure les occurrences de h_c + formula = formula.replace('H_C', 'h_c') + self.model = ot.SymbolicFunction( + ['E', 'nu', 'h_c', 'l', 'f_1', 'f_2', 't', 'a', 'b_0', 'p'], + [formula] + ) + self.model.setOutputDescription([r'(N_{xy})_{cr}']) + + self.E = ot.TruncatedNormal(110.0e9, 55.0e9, 99.0e9, 121.0e9) + self.E.setDescription(['E (Pa)']) + + self.nu = ot.Uniform(0.3675, 0.3825) + self.nu.setDescription(['nu (-)']) + + self.h_c = ot.Uniform(0.0285, 0.0315) + self.h_c.setDescription(['h_c (m)']) + + self.ell = ot.Uniform(0.04655, 0.05145) + self.ell.setDescription(['ell (m)']) + + self.f_1 = ot.Uniform(0.0266, 0.0294) + self.f_1.setDescription(['f_1 (m)']) + + self.f_2 = ot.Uniform(0.00627, 0.00693) + self.f_2.setDescription(['f_2 (m)']) + + self.t = ot.Uniform(8.02e-5, 8.181e-5) + self.t.setDescription(['t (m)']) + + self.a = ot.Uniform(0.6039, 0.6161) + self.a.setDescription(['a (m)']) + + self.b_0 = ot.Uniform(0.04455, 0.04545) + self.b_0.setDescription(['b_0 (m)']) + + self.p = ot.Uniform(0.03762, 0.03838) + self.p.setDescription(['p (m)']) + + self.distribution = ot.ComposedDistribution( + [self.E, self.nu, self.h_c, self.ell, self.f_1, self.f_2, self.t, self.a, self.b_0, self.p] + ) diff --git a/python/test/CMakeLists.txt b/python/test/CMakeLists.txt index 2c8d3541f21..b0b9c1da11f 100644 --- a/python/test/CMakeLists.txt +++ b/python/test/CMakeLists.txt @@ -46,6 +46,7 @@ ot_pyinstallcheck_test (stressed_beam IGNOREOUT) ot_pyinstallcheck_test (fireSatellite_function IGNOREOUT) ot_pyinstallcheck_test (wingweight_function IGNOREOUT) ot_pyinstallcheck_test (oscillator IGNOREOUT) +ot_pyinstallcheck_test (stiffened_panel IGNOREOUT) if (NUMPY_FOUND) ot_pyinstallcheck_test (viscous_free_fall IGNOREOUT) endif() diff --git a/python/test/t_stiffened_panel.py b/python/test/t_stiffened_panel.py new file mode 100755 index 00000000000..92d0084379f --- /dev/null +++ b/python/test/t_stiffened_panel.py @@ -0,0 +1,34 @@ +#! /usr/bin/env python + +import openturns as ot +from openturns.testing import assert_almost_equal +from openturns.usecases import stiffened_panel as stiffened_panel + + +ot.TESTPREAMBLE() +ot.PlatformInfo.SetNumericalPrecision(20) + +print("Test the StiffenedPanel class.") +sp = stiffened_panel.StiffenedPanel() + +print(sp.distribution.getMean()) +# test parameter +assert sp.dim == 10 + +# test marginals means +assert_almost_equal(sp.E.getMean()[0], 110000000000, 1e-12) +assert_almost_equal(sp.nu.getMean()[0], 0.375, 1e-12) +assert_almost_equal(sp.h_c.getMean()[0], 0.03, 1e-12) +assert_almost_equal(sp.ell.getMean()[0], 0.049, 1e-12) +assert_almost_equal(sp.f_1.getMean()[0], 0.028, 1e-12) +assert_almost_equal(sp.f_2.getMean()[0], 0.0066, 1e-12) +assert_almost_equal(sp.t.getMean()[0], 8.1005e-5, 1e-12) +assert_almost_equal(sp.a.getMean()[0], 0.61, 1e-12) +assert_almost_equal(sp.b_0.getMean()[0], 0.045, 1e-12) +assert_almost_equal(sp.p.getMean()[0], 0.038, 1e-12) + +meanInput = sp.distribution.getMean() +print("meanInput = ", meanInput) +meanOutput = sp.model(meanInput) +print("meanOutput = ", meanOutput) +assert_almost_equal(meanOutput, [197.76], 1.0e-4)

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