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bi-material.py
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Fri, Nov 1, 10:31

bi-material.py
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__copyright__ = (
"Copyright (©) 2018-2023 EPFL (Ecole Polytechnique Fédérale de Lausanne)"
"Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)"
)
__license__ = "LGPLv3"
import akantu as aka
import numpy as np
# ------------------------------------------------------------------------------
class LocalElastic(aka.Material):
def __init__(self, model, _id):
super().__init__(model, _id)
super().registerParamReal('E',
aka._pat_readable | aka._pat_parsable,
'Youngs modulus')
super().registerParamReal('nu',
aka._pat_readable | aka._pat_parsable,
'Poisson ratio')
# change it to have the initialize wrapped
self.factor = super().registerInternalReal('factor', 1)
self.quad_coords = super().registerInternalReal('quad_coordinates', 2)
def initMaterial(self):
nu = self.getReal('nu')
E = self.getReal('E')
self.mu = E / (2 * (1 + nu))
self.lame_lambda = nu * E / (
(1. + nu) * (1. - 2. * nu))
# Second Lame coefficient (shear modulus)
self.lame_mu = E / (2. * (1. + nu))
super().initMaterial()
model = self.getModel()
model.getFEEngine().computeIntegrationPointsCoordinates(
self.quad_coords, self.getElementFilter())
for elem_type in self.factor.elementTypes():
factor = self.factor(elem_type)
coords = self.quad_coords(elem_type)
factor[:] = 1.
factor[coords[:, 1] < 0.5] = .5
# declares all the parameters that are needed
def getPushWaveSpeed(self, params):
return np.sqrt((self.lame_lambda + 2 * self.lame_mu) / self.rho)
# compute small deformation tensor
@staticmethod
def computeEpsilon(grad_u):
return 0.5 * (grad_u + np.einsum('aij->aji', grad_u))
# constitutive law
def computeStress(self, el_type, ghost_type):
grad_u = self.getGradU(el_type, ghost_type)
sigma = self.getStress(el_type, ghost_type)
n_quads = grad_u.shape[0]
grad_u = grad_u.reshape((n_quads, 2, 2))
factor = self.getInternalReal('factor')(
el_type, ghost_type).reshape(n_quads)
epsilon = self.computeEpsilon(grad_u)
sigma = sigma.reshape((n_quads, 2, 2))
trace = np.einsum('aii->a', grad_u)
sigma[:, :, :] = (
np.einsum('a,ij->aij', trace,
self.lame_lambda * np.eye(2))
+ 2. * self.lame_mu * epsilon)
sigma[:, :, :] = np.einsum('aij, a->aij', sigma, factor)
# constitutive law tangent modulii
def computeTangentModuli(self, el_type, tangent_matrix, ghost_type):
n_quads = tangent_matrix.shape[0]
tangent = tangent_matrix.reshape(n_quads, 3, 3)
factor = self.getInternalReal('factor')(
el_type, ghost_type).reshape(n_quads)
Miiii = self.lame_lambda + 2 * self.lame_mu
Miijj = self.lame_lambda
Mijij = self.lame_mu
tangent[:, 0, 0] = Miiii
tangent[:, 1, 1] = Miiii
tangent[:, 0, 1] = Miijj
tangent[:, 1, 0] = Miijj
tangent[:, 2, 2] = Mijij
tangent[:, :, :] = np.einsum('aij, a->aij', tangent, factor)
# computes the energy density
def computePotentialEnergy(self, el_type):
sigma = self.getStress(el_type)
grad_u = self.getGradU(el_type)
nquads = sigma.shape[0]
stress = sigma.reshape(nquads, 2, 2)
grad_u = grad_u.reshape((nquads, 2, 2))
epsilon = self.computeEpsilon(grad_u)
energy_density = self.getPotentialEnergy(el_type)
energy_density[:, 0] = 0.5 * np.einsum('aij,aij->a', stress, epsilon)
# ------------------------------------------------------------------------------
# applies manually the boundary conditions
def applyBC(model):
nbNodes = model.getMesh().getNbNodes()
position = model.getMesh().getNodes()
displacement = model.getDisplacement()
blocked_dofs = model.getBlockedDOFs()
width = 1.
height = 1.
epsilon = 1e-8
for node in range(0, nbNodes):
if ((np.abs(position[node, 0]) < epsilon) or
(np.abs(position[node, 0] - width) < epsilon)):
blocked_dofs[node, 0] = True
displacement[node, 0] = 0 * position[node, 0] + 0.
if np.abs(position[node, 1]) < epsilon: # lower side
blocked_dofs[node, 1] = True
displacement[node, 1] = - 1.
if np.abs(position[node, 1] - height) < epsilon: # upper side
blocked_dofs[node, 1] = True
displacement[node, 1] = 1.
# register the material to the material factory
def allocator(dim, option, model, id):
return LocalElastic(model, id)
mat_factory = aka.MaterialFactory.getInstance()
mat_factory.registerAllocator("local_elastic", allocator)
# main parameters
spatial_dimension = 2
mesh_file = 'square.msh'
# read mesh
mesh = aka.Mesh(spatial_dimension)
mesh.read(mesh_file)
# parse input file
aka.parseInput('material.dat')
# init the SolidMechanicsModel
model = aka.SolidMechanicsModel(mesh)
model.initFull(_analysis_method=aka._static)
# configure the solver
solver = model.getNonLinearSolver()
solver.set("max_iterations", 2)
solver.set("threshold", 1e-3)
solver.set("convergence_type", aka.SolveConvergenceCriteria.solution)
# prepare the dumper
model.setBaseName("bimaterial")
model.addDumpFieldVector("displacement")
model.addDumpFieldVector("internal_force")
model.addDumpFieldVector("external_force")
model.addDumpField("strain")
model.addDumpField("stress")
model.addDumpField("factor")
model.addDumpField("blocked_dofs")
# Boundary conditions
applyBC(model)
# solve the problem
model.solveStep()
# dump paraview files
model.dump()
epot = model.getEnergy('potential')
print('Potential energy: ' + str(epot))

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