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# Copyright (C) 2020 Jørgen S. Dokken
#
# This file is part of DOLFINX_MPC
#
# SPDX-License-Identifier: MIT
#
# Create constraint between two bodies that are not in contact
import gmsh
import numpy as np
from dolfinx.fem import (Constant, Function, FunctionSpace,
VectorFunctionSpace, dirichletbc,
locate_dofs_topological)
from dolfinx.io import XDMFFile, gmshio
from dolfinx_mpc import LinearProblem, MultiPointConstraint
from dolfinx_mpc.utils import (create_point_to_point_constraint,
determine_closest_block,
rigid_motions_nullspace)
from mpi4py import MPI
from petsc4py import PETSc
from ufl import (Identity, Measure, SpatialCoordinate, TestFunction,
TrialFunction, as_vector, grad, inner, sym, tr)
# Mesh parameters for creating a mesh consisting of two spheres,
# Sphere(r2)\Sphere(r1) and Sphere(r_0)
r0, r0_tag = 0.4, 1
r1, r1_tag = 0.5, 2
r2, r2_tag = 0.8, 3
outer_tag = 1
inner_tag = 2
assert r0 < r1 and r1 < r2
gmsh.initialize()
if MPI.COMM_WORLD.rank == 0:
gmsh.clear()
# Create Sphere(r2)\Sphere(r1)
p0 = gmsh.model.occ.addPoint(0, 0, 0)
outer_sphere = gmsh.model.occ.addSphere(0, 0, 0, r2)
mid_sphere = gmsh.model.occ.addSphere(0, 0, 0, r1)
hollow_sphere = gmsh.model.occ.cut([(3, outer_sphere)], [(3, mid_sphere)])
# Create Sphere(r0)
inner_sphere = gmsh.model.occ.addSphere(0, 0, 0, r0)
gmsh.model.occ.synchronize()
# Add physical tags for volumes
gmsh.model.addPhysicalGroup(hollow_sphere[0][0][0], [hollow_sphere[0][0][1]], tag=outer_tag)
gmsh.model.setPhysicalName(hollow_sphere[0][0][0], 1, "Hollow sphere")
gmsh.model.addPhysicalGroup(3, [inner_sphere], tag=inner_tag)
gmsh.model.setPhysicalName(3, 2, "Inner sphere")
# Add physical tags for surfaces
r1_surface, r2_surface = [], []
hollow_boundary = gmsh.model.getBoundary(hollow_sphere[0], oriented=False)
inner_boundary = gmsh.model.getBoundary([(3, inner_sphere)], oriented=False)
for boundary in hollow_boundary:
bbox = gmsh.model.getBoundingBox(boundary[0], boundary[1])
if np.isclose(max(bbox), r1):
r1_surface.append(boundary[1])
elif np.isclose(max(bbox), r2):
r2_surface.append(boundary[1])
gmsh.model.addPhysicalGroup(inner_boundary[0][0], [inner_boundary[0][1]], r0_tag)
gmsh.model.setPhysicalName(inner_boundary[0][0], r0_tag, "Inner boundary")
gmsh.model.addPhysicalGroup(2, r1_surface, r1_tag)
gmsh.model.setPhysicalName(2, r1_tag, "Mid boundary")
gmsh.model.addPhysicalGroup(2, r2_surface, r2_tag)
gmsh.model.setPhysicalName(2, r2_tag, "Outer boundary")
# Set mesh resolution
res_inner = r0 / 5
res_outer = (r1 + r2) / 5
gmsh.model.occ.synchronize()
gmsh.model.mesh.field.add("Distance", 1)
gmsh.model.mesh.field.setNumbers(1, "NodesList", [p0])
gmsh.model.mesh.field.add("Threshold", 2)
gmsh.model.mesh.field.setNumber(2, "IField", 1)
gmsh.model.mesh.field.setNumber(2, "LcMin", res_inner)
gmsh.model.mesh.field.setNumber(2, "LcMax", res_outer)
gmsh.model.mesh.field.setNumber(2, "DistMin", r0)
gmsh.model.mesh.field.setNumber(2, "DistMax", r1)
gmsh.model.mesh.field.add("Threshold", 3)
gmsh.model.mesh.field.setNumber(3, "IField", 1)
gmsh.model.mesh.field.setNumber(3, "LcMin", res_outer)
gmsh.model.mesh.field.setNumber(3, "LcMax", res_outer)
gmsh.model.mesh.field.setNumber(3, "DistMin", r1)
gmsh.model.mesh.field.setNumber(3, "DistMax", r2)
gmsh.model.mesh.field.add("Min", 4)
gmsh.model.mesh.field.setNumbers(4, "FieldsList", [2, 3])
gmsh.model.mesh.field.setAsBackgroundMesh(4)
# Generate mesh
gmsh.model.mesh.generate(3)
gmsh.option.setNumber("General.Terminal", 1)
gmsh.model.mesh.optimize("Netgen")
gmsh.model.mesh.setOrder(2)
mesh, ct, ft = gmshio.model_to_mesh(gmsh.model, MPI.COMM_WORLD, 0, gdim=3)
gmsh.clear()
gmsh.finalize()
MPI.COMM_WORLD.barrier()
V = VectorFunctionSpace(mesh, ("Lagrange", 1))
tdim = mesh.topology.dim
fdim = tdim - 1
DG0 = FunctionSpace(mesh, ("DG", 0))
outer_dofs = locate_dofs_topological(DG0, tdim, ct.find(outer_tag))
inner_dofs = locate_dofs_topological(DG0, tdim, ct.find(inner_tag))
# Elasticity parameters
E_outer = 1e3
E_inner = 1e5
nu_outer = 0.3
nu_inner = 0.1
mu = Function(DG0)
lmbda = Function(DG0)
with mu.vector.localForm() as local:
local.array[inner_dofs] = E_inner / (2 * (1 + nu_inner))
local.array[outer_dofs] = E_outer / (2 * (1 + nu_outer))
with lmbda.vector.localForm() as local:
local.array[inner_dofs] = E_inner * nu_inner / ((1 + nu_inner) * (1 - 2 * nu_inner))
local.array[outer_dofs] = E_outer * nu_outer / ((1 + nu_outer) * (1 - 2 * nu_outer))
# Stress computation
def sigma(v):
return (2.0 * mu * sym(grad(v))
+ lmbda * tr(sym(grad(v))) * Identity(len(v)))
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
dx = Measure("dx", domain=mesh, subdomain_data=ct)
a = inner(sigma(u), grad(v)) * dx
x = SpatialCoordinate(mesh)
rhs = inner(Constant(mesh, PETSc.ScalarType((0, 0, 0))), v) * dx
rhs += inner(Constant(mesh, PETSc.ScalarType((0.01, 0.02, 0))), v) * dx(outer_tag)
rhs += inner(as_vector(PETSc.ScalarType((0, 0, -9.81e-2))), v) * dx(inner_tag)
# Create dirichletbc
owning_processor, bc_dofs = determine_closest_block(V, -np.array([-r2, 0, 0]))
bc_dofs = [] if bc_dofs is None else bc_dofs
u_fixed = np.array([0, 0, 0], dtype=PETSc.ScalarType)
bc_fixed = dirichletbc(u_fixed, np.asarray(bc_dofs, dtype=np.int32), V)
bcs = [bc_fixed]
# Create point to point constraints
mpc = MultiPointConstraint(V)
signs = [-1, 1]
axis = [0, 1]
for i in axis:
for s in signs:
r0_point = np.zeros(3)
r1_point = np.zeros(3)
r0_point[i] = s * r0
r1_point[i] = s * r1
sl, ms, co, ow, off = create_point_to_point_constraint(V, r1_point, r0_point)
mpc.add_constraint(V, sl, ms, co, ow, off)
mpc.finalize()
# Create nullspace
null_space = rigid_motions_nullspace(mpc.function_space)
petsc_options = {"ksp_rtol": 1.0e-8, "pc_type": "gamg", "pc_gamg_type": "agg",
"pc_gamg_coarse_eq_limit": 1000, "pc_gamg_sym_graph": True,
"mg_levels_ksp_type": "chebyshev", "mg_levels_pc_type": "jacobi",
"mg_levels_esteig_ksp_type": "cg", "matptap_via": "scalable",
"pc_gamg_square_graph": 2, "pc_gamg_threshold": 0.02
# ,"help": None, "ksp_view": None
}
problem = LinearProblem(a, rhs, mpc, bcs=bcs, petsc_options=petsc_options)
# Build near nullspace
null_space = rigid_motions_nullspace(mpc.function_space)
problem.A.setNearNullSpace(null_space)
u_h = problem.solve()
it = problem.solver.getIterationNumber()
unorm = u_h.vector.norm()
if MPI.COMM_WORLD.rank == 0:
print("Number of iterations: {0:d}".format(it))
# Write solution to file
u_h.name = "u"
with XDMFFile(MPI.COMM_WORLD, "results/demo_elasticity_disconnect.xdmf", "w") as xdmf:
xdmf.write_mesh(mesh)
xdmf.write_function(u_h)
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