### Added beam buckling example

parent 5682d82b
 ... ... @@ -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 ===== ============= =================
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