Commit b7bef5d1 by Jeremy BLEYER

 ... ... @@ -10,8 +10,8 @@ Introduction ------------- This program performs a dynamic modal analysis of an elastic cantilever beam represented by a 3D solid continuum. The eigenmodes are computed using the **SLEPcEigensolver** and compared against an analytical solution of beam theory. represented by a 3D solid continuum. The eigenmodes are computed using the **SLEPcEigensolver** and compared against an analytical solution of beam theory. The corresponding file can be obtained from :download:cantilever_modal.py. ... ... @@ -20,14 +20,14 @@ The first four eigenmodes of this demo will look as follows: .. image:: vibration_modes.gif :scale: 80 % The first two fundamental modes are on top with bending along the weak axis (left) and along the strong axis (right), the next two modes are at the bottom. The first two fundamental modes are on top with bending along the weak axis (left) and along the strong axis (right), the next two modes are at the bottom. --------------- Implementation --------------- After importing the relevant modules, the geometry of a beam of length :math:L=20 After importing the relevant modules, the geometry of a beam of length :math:L=20 and rectangular section of size :math:B\times H with :math:B=0.5, H=1 is first defined:: from fenics import * ... ... @@ -43,7 +43,7 @@ and rectangular section of size :math:B\times H with :math:B=0.5, H=1 is fir Material parameters and elastic constitutive relations are classical (here we take :math:\nu=0) and we also introduce the material density :math:\rho for take :math:\nu=0) and we also introduce the material density :math:\rho for later definition of the mass matrix:: E, nu = 1e5, 0. ... ... @@ -59,7 +59,7 @@ later definition of the mass matrix:: dim = v.geometric_dimension() return 2.0*mu*eps(v) + lmbda*tr(eps(v))*Identity(dim) Standard FunctionSpace is defined and boundary conditions correspond to a Standard FunctionSpace is defined and boundary conditions correspond to a fully clamped support at :math:x=0:: V = VectorFunctionSpace(mesh, 'Lagrange', degree=1) ... ... @@ -73,7 +73,7 @@ fully clamped support at :math:x=0:: bc = DirichletBC(V, Constant((0.,0.,0.)), left) The system stiffness matrix :math:[K] and mass matrix :math:[M] are The system stiffness matrix :math:[K] and mass matrix :math:[M] are respectively obtained from assembling the corresponding variational forms:: k_form = inner(sigma(du),eps(u_))*dx ... ... @@ -86,14 +86,14 @@ respectively obtained from assembling the corresponding variational forms:: M = PETScMatrix() assemble(m_form, tensor=M) Matrices :math:[K] and :math:[M] are first defined as PETSc Matrix and Matrices :math:[K] and :math:[M] are first defined as PETSc Matrix and forms are assembled into it to ensure that they have the right type. Note that boundary conditions have been applied to the stiffness matrix using assemble_system so as to preserve symmetry (a dummy l_form and right-hand side vector have been introduced to call this function). vector have been introduced to call this function). Modal dynamic analysis consists in solving the following generalized Modal dynamic analysis consists in solving the following generalized eigenvalue problem :math:[K]\{U\}=\lambda[M]\{U\} where the eigenvalue is related to the eigenfrequency :math:\lambda=\omega^2. This problem can be solved using the SLEPcEigenSolver. :: ... ... @@ -121,7 +121,7 @@ We now ask SLEPc to extract the first 6 eigenvalues by calling its solve functio and extract the corresponding eigenpair (first two arguments of get_eigenpair correspond to the real and complex part of the eigenvalue, the last two to the real and complex part of the eigenvector):: N_eig = 6 # number of eigenvalues print "Computing %i first eigenvalues..." % N_eig eigensolver.solve(N_eig) ... ... @@ -136,31 +136,31 @@ real and complex part of the eigenvector):: file_results = XDMFFile("modal_analysis.xdmf") file_results.parameters["flush_output"] = True file_results.parameters["functions_share_mesh"] = True # Extraction for i in range(N_eig): # Extract eigenpair r, c, rx, cx = eigensolver.get_eigenpair(i) # 3D eigenfrequency freq_3D = sqrt(r)/2/pi # Beam eigenfrequency if i % 2 == 0: # exact solution should correspond to weak axis bending I_bend = H*B**3/12. else: #exact solution should correspond to strong axis bending I_bend = B*H**3/12. freq_beam = alpha(i/2)**2*sqrt(E*I_bend/(rho*B*H*L**4))/2/pi print("Solid FE: {0:8.5f} [Hz] Beam theory: {1:8.5f} [Hz]".format(freq_3D, freq_beam)) # Initialize function and assign eigenvector (renormalize by stiffness matrix) eigenmode = Function(V,name="Eigenvector "+str(i)) eigenmode.vector()[:] = rx/omega eigenmode.vector()[:] = rx The beam analytical solution is obtained using the eigenfrequencies of a clamped beam in bending given by :math:\omega_n = \alpha_n^2\sqrt{\dfrac{EI}{\rho S L^4}} where :math:S=BH is the beam section, :math:I the bending inertia and where :math:S=BH is the beam section, :math:I the bending inertia and :math:\alpha_n is the solution of the following nonlinear equation: .. math:: ... ... @@ -173,9 +173,9 @@ and the other along the strong axis (:math:I=I_{\text{strong}} = BH^3/12). Sin for the considered numerical values, the strong axis bending frequency will be twice that corresponsing to bending along the weak axis. The solution :math:\alpha_n are computed using the scipy.optimize.root function with initial guess given by :math:(2n+1)\pi/2. With Nx=400, we obtain the following comparison between the FE eigenfrequencies and the beam theory eigenfrequencies : and the beam theory eigenfrequencies : ===== ============= ================= ... ... @@ -184,11 +184,11 @@ Mode Eigenfrequencies # Solid FE [Hz] Beam theory [Hz] ===== ============= ================= 1 2.04991 2.01925 2 4.04854 4.03850 2 4.04854 4.03850 3 12.81504 12.65443 4 25.12717 25.30886 4 25.12717 25.30886 5 35.74168 35.43277 6 66.94816 70.86554 6 66.94816 70.86554 ===== ============= =================
 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Eulerian buckling of a beam\n", "\n", "In this numerical tour, we will compute the critical buckling load of a straight beam under normal compression, the classical Euler buckling problem. Usually, buckling is an important mode of failure for slender beams so that a standard Euler-Bernoulli beam model is sufficient. However, since FEniCS does not support Hermite elements ensuring $C^1$-formulation for the transverse deflection, implementing such models is not straightforward and requires using advanced DG formulations for instance, see the fenics-shell [implemntation of the Love-Kirchhoff plate model](http://fenics-shells.readthedocs.io/en/latest/demo/kirchhoff-love-clamped/demo_kirchhoff-love-clamped.py.html) or the [FEniCS documented demo on the biharmonic equation](http://fenics.readthedocs.io/projects/dolfin/en/2017.2.0/demos/biharmonic/python/demo_biharmonic.py.html).\n", "\n", "As a result, we will simply formulate the buckling problem using a Timoshenko beam model.\n", "\n", "## Timoshenko beam model formulation\n", "\n", "We first formulate the stiffness bilinear form of the Timoshenko model given by:\n", "\\n", "k((w,\\theta),(\\widehat{w},\\widehat{\\theta}))= \\int_0^L EI \\dfrac{d\\theta}{dx}\\dfrac{d\\widehat{\\theta}}{dx} dx + \\int_0^L \\kappa \\mu S \\left(\\dfrac{dw}{dx}-\\theta\\right)\\left(\\dfrac{d\\widehat{w}}{dx}-\\widehat{\\theta}\\right) dx\n", "\\n", "where $I=bh^3/12$ is the bending inertia for a rectangular beam of width $b$ and height $h$, $S=bh$ the cross-section area, $E$ the material Young modulus and $\\mu$ the shear modulus and $\\kappa=5/6$ the shear correction factor. We will use a $P^2/P^1$ interpolation for the mixed field $(w,\\theta)$. " ] }, { "cell_type": "raw", "metadata": {}, "source": [ "For issues related to shear-locking and reduced integration formulation, we refer to the :ref:ReissnerMindlinQuads tour." ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [], "source": [ "from dolfin import *\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib notebook\n", "\n", "L = 10.\n", "thick = Constant(0.03)\n", "width = Constant(0.01)\n", "E = Constant(70e3)\n", "nu = Constant(0.)\n", "\n", "EI = E*width*thick**3/12\n", "GS = E/2/(1+nu)*thick*width\n", "kappa = Constant(5./6.)\n", "\n", "\n", "N = 100\n", "mesh = IntervalMesh(N, 0, L) \n", "\n", "U = FiniteElement(\"CG\", mesh.ufl_cell(), 2)\n", "T = FiniteElement(\"CG\", mesh.ufl_cell(), 1)\n", "V = FunctionSpace(mesh, U*T)\n", "\n", "u_ = TestFunction(V)\n", "du = TrialFunction(V)\n", "(w_, theta_) = split(u_)\n", "(dw, dtheta) = split(du)\n", "\n", "\n", "k_form = EI*inner(grad(theta_), grad(dtheta))*dx + kappa*GS*dot(grad(w_)[0]-theta_, grad(dw)[0]-dtheta)*dx\n", "l_form = Constant(1.)*u_[0]*dx" ] }, { "cell_type": "raw", "metadata": {}, "source": [ "As in the :ref:ModalAnalysis tour, a dummy linear form l_form is used to call the assemble_system function which retains the symmetric structure of the associated matrix when imposing boundary conditions. Here, we will consider clamped conditions on the left side :math:x=0 and simple supports on the right side :math:x=L." ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "( *' at 0x7f9df3cf8780> >,\n", " *' at 0x7f9df3cf8ae0> >)" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def both_ends(x, on_boundary):\n", " return on_boundary\n", "def left_end(x, on_boundary):\n", " return near(x[0], 0) and on_boundary\n", "\n", "bc = [DirichletBC(V.sub(0), Constant(0.), both_ends),\n", " DirichletBC(V.sub(1), Constant(0.), left_end)]\n", "\n", "K = PETScMatrix()\n", "assemble_system(k_form, l_form, bc, A_tensor=K)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Construction of the geometric stiffness matrix\n", "\n", "The buckling analysis amounts to solving an eigenvalue problem of the form:\n", "\n", "\\n", "(\\mathbf{K}+\\lambda\\mathbf{K_G})\\mathbf{U} = 0\n", "\\n", "\n", "in which the geometric stiffness matrix $\\mathbf{K_G}$ depends (linearly) on a prestressed state, the amplitude of which is represented by $\\lambda$. The eigenvalue/eigenvector $(\\lambda,\\mathbf{U})$ solving the previous generalized eigenproblem respectively correspond to the critical buckling load and its associated buckling mode. For a beam in which the prestressed state correspond to a purely compression state of intensity $N_0>0$, the geometric stiffness bilinear form is given by:\n", "\n", "\\n", "k_G((w,\\theta),(\\widehat{w},\\widehat{\\theta}))= -\\int_0^L N_0 \\dfrac{dw}{dx}\\dfrac{d\\widehat{w}}{dx} dx\n", "\\n", "\n", "which is assembled below into the KG PETScMatrix (up to the negative sign)." ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [], "source": [ "N0 = Constant(1e-3)\n", "kg_form = N0*dot(grad(w_), grad(dw))*dx\n", "KG = PETScMatrix()\n", "assemble(kg_form, tensor=KG)\n", "for bci in bc:\n", " bci.zero(KG)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that we made use of the zero method of DirichletBC making the rows of the matrix associated with the boundary condition zero. If we used instead the apply method, the rows would have been replaced with a row of zeros with a 1 on the diagonal (as for the stiffness matrix K). As a result, we would have obtained an eigenvalue equal to 1 for each row with a boundary condition which can make more troublesome the computation of eigenvalues if they happen to be close to 1. Replacing with a full row of zeros in KG results in infinite eigenvalues for each boundary condition which is more suitable when looking for the lowest eigenvalues of the buckling problem.\n", "\n", "## Setting and solving the eigenvalue problem\n", "\n", "Up to the negative sign cancelling from the previous definition of KG, we now formulate the generalized eigenvalue problem $\\mathbf{KU}=-\\lambda\\mathbf{K_G U}$ using the SLEPcEigenSolver. The only difference from what has already been discussed in the dynamic modal analysis numerical tour is that buckling eigenvalue problem may be more difficult to solve than modal analysis in certain cases, it is therefore beneficial to prescribe a value of the spectral shift close to the critical buckling load." ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Computing 3 first eigenvalues...\n" ] }, { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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