I am extending modal analysis code for calculating X^T*M*T with fenics. Here X^T
is transpose of eigen vector/mode shape matrix, M
is the mass matrix, T
is the influence direction vector. I am facing some error.
from dolfin import *
import numpy as np
L, B, H = 20., 0.5, 1.
Nx = 200
Ny = int(B/L*Nx)+1
Nz = int(H/L*Nx)+1
mesh = BoxMesh(Point(0.,0.,0.),Point(L,B,H), Nx, Ny, Nz)
E, nu = Constant(1e5), Constant(0.)
rho = Constant(1e-3)
# Lame coefficient for constitutive relation
mu = E/2./(1+nu)
lmbda = E*nu/(1+nu)/(1-2*nu)
def eps(v):
return sym(grad(v))
def sigma(v):
dim = v.geometric_dimension()
return 2.0*mu*eps(v) + lmbda*tr(eps(v))*Identity(dim)
V = VectorFunctionSpace(mesh, 'Lagrange', degree=1)
u_ = TrialFunction(V)
du = TestFunction(V)
def left(x, on_boundary):
return near(x[0],0.)
bc = DirichletBC(V, Constant((0.,0.,0.)), left)
k_form = inner(sigma(du),eps(u_))*dx
l_form = Constant(1.)*u_[0]*dx
K = PETScMatrix()
b = PETScVector()
assemble_system(k_form, l_form, bc, A_tensor=K, b_tensor=b)
m_form = rho*dot(du,u_)*dx
M = PETScMatrix()
assemble(m_form, tensor=M)
eigensolver = SLEPcEigenSolver(K, M)
eigensolver.parameters['problem_type'] = 'gen_hermitian'
eigensolver.parameters['spectral_transform'] = 'shift-and-invert'
eigensolver.parameters['spectral_shift'] = 0.
N_eig = 6 # number of eigenvalues
print("Computing {} first eigenvalues...".format(N_eig))
eigensolver.solve(N_eig)
# Exact solution computation
from scipy.optimize import root
from math import cos, cosh
falpha = lambda x: cos(x)*cosh(x)+1
alpha = lambda n: root(falpha, (2*n+1)*pi/2.)['x'][0]
# Set up file for exporting results
file_results = XDMFFile("modal_analysis.xdmf")
file_results.parameters["flush_output"] = True
file_results.parameters["functions_share_mesh"] = True
eig_l = []
eigenmodes = []
# 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(float(E)*I_bend/(float(rho)*B*H*L**4))/2/pi
print("Solid FE: {:8.5f} [Hz] Beam theory: {:8.5f} [Hz]".format(freq_3D, freq_beam))
# Initialize function and assign eigenvector
eigenmode = Function(V,name="Eigenvector "+str(i))
eigenmode.vector()[:] = rx
eigenmodes.append(eigenmode)
eig_l.append(freq_3D)
u = Function(V)
u.interpolate(Constant((0, 1, 0)))
combined_mass = 0
q = assemble(action(action(eigenmodes.T, m_form), u))
here eigenmodes.T
should be the transpose of eigen vector/mode shape matrix.