diff --git a/doc/_build/doctrees/demo/modal_analysis_dynamics/cantilever_modal.py.doctree b/doc/_build/doctrees/demo/modal_analysis_dynamics/cantilever_modal.py.doctree index 810391e221271b91b5ada187aa7b9756731cfbc1..760eb189abd4ac80dc1ff864ab66ec6debdcdc6a 100644 Binary files a/doc/_build/doctrees/demo/modal_analysis_dynamics/cantilever_modal.py.doctree and b/doc/_build/doctrees/demo/modal_analysis_dynamics/cantilever_modal.py.doctree differ diff --git a/doc/_build/doctrees/environment.pickle b/doc/_build/doctrees/environment.pickle index 7e00a313900a20493bb9f2c6dbca8ea519c1dc7c..622e684106f5dc36166a609758ef5b9352dc6e0d 100644 Binary files a/doc/_build/doctrees/environment.pickle and b/doc/_build/doctrees/environment.pickle differ diff --git a/doc/_build/doctrees/index.doctree b/doc/_build/doctrees/index.doctree index 2a91f760fd4323ed83d3460bee330be719b9525d..399aca990d53c9bb2b3d34faa378a2648b47f728 100644 Binary files a/doc/_build/doctrees/index.doctree and b/doc/_build/doctrees/index.doctree differ diff --git a/doc/_build/html/_downloads/cantilever_modal.py b/doc/_build/html/_downloads/cantilever_modal.py index 468800ffbaf71ee0bb755060f0fdde186090d401..ae6f54121c3b8d952d10eda6afdca143586d7ea7 100644 --- a/doc/_build/html/_downloads/cantilever_modal.py +++ b/doc/_build/html/_downloads/cantilever_modal.py @@ -10,8 +10,8 @@ # ------------- # # 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 @@ # .. 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 @@ mesh = BoxMesh(Point(0.,0.,0.),Point(L,B,H), Nx, Ny, Nz) # 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 @@ def sigma(v): 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 @@ def left(x, on_boundary): 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 @@ m_form = rho*dot(du,u_)*dx 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 @@ eigensolver.parameters['spectral_shift'] = 0. # 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) @@ -141,26 +141,26 @@ file_results.parameters["functions_share_mesh"] = True 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 @@ for i in range(N_eig): # 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 @@ for i in range(N_eig): # # 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 # ===== ============= ================= # # diff --git a/doc/_build/html/_sources/demo/modal_analysis_dynamics/cantilever_modal.py.rst.txt b/doc/_build/html/_sources/demo/modal_analysis_dynamics/cantilever_modal.py.rst.txt index 508fa5a18f5c9e48cec13907f91049d6668e2cf9..d52f9e8bb75715f1382cb7480df9f63f4cc45d2a 100644 --- a/doc/_build/html/_sources/demo/modal_analysis_dynamics/cantilever_modal.py.rst.txt +++ b/doc/_build/html/_sources/demo/modal_analysis_dynamics/cantilever_modal.py.rst.txt @@ -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 ===== ============= ================= diff --git a/doc/_build/html/_sources/index.rst.txt b/doc/_build/html/_sources/index.rst.txt index acb546200f4eab6f65ddcb17dbcb51d86df43fc9..e33a637c7b0fcd0d1114d5b94ab9fdf09c6c4e3c 100644 --- a/doc/_build/html/_sources/index.rst.txt +++ b/doc/_build/html/_sources/index.rst.txt @@ -15,7 +15,7 @@ Contents: linear_problems homogenization nonlinear_problems - demo/reissner_mindlin/reissner_mindlin.rst + beams_and_plates tips_and_tricks diff --git a/doc/_build/html/demo/2D_plasticity/vonMises_plasticity.py.html b/doc/_build/html/demo/2D_plasticity/vonMises_plasticity.py.html index c34cacba81facb26b45c34eb5550720f4905464c..332d907d467f48099d8c801f21f4912919a9fb04 100644 --- a/doc/_build/html/demo/2D_plasticity/vonMises_plasticity.py.html +++ b/doc/_build/html/demo/2D_plasticity/vonMises_plasticity.py.html @@ -24,7 +24,7 @@ - + @@ -35,7 +35,7 @@

The beam analytical solution is obtained using the eigenfrequencies of a clamped @@ -282,7 +282,8 @@ and the beam theory eigenfrequencies :

  • Elasto-plastic analysis of a 2D von Mises material
    • -
  • Reissner-Mindlin plates
    • -
  • Reissner-Mindlin plates