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# Copyright (C) 2022 Nathan Sime
#
# This file is part of DOLFINX_MPC
#
# SPDX-License-Identifier: MIT
from __future__ import annotations
from mpi4py import MPI
from petsc4py import PETSc
import basix
import dolfinx
import dolfinx.fem.petsc
import dolfinx.la as _la
import dolfinx.nls.petsc
import numpy as np
import pytest
import ufl
import dolfinx_mpc
class NonlinearMPCProblem(dolfinx.fem.petsc.NonlinearProblem):
def __init__(self, F, u, mpc, bcs=[], J=None, form_compiler_options={}, jit_options={}):
self.mpc = mpc
super().__init__(F, u, bcs=bcs, J=J, form_compiler_options=form_compiler_options, jit_options=jit_options)
def F(self, x: PETSc.Vec, F: PETSc.Vec): # type: ignore
with F.localForm() as F_local:
F_local.set(0.0)
dolfinx_mpc.assemble_vector(self._L, self.mpc, b=F)
# Apply boundary condition
dolfinx_mpc.apply_lifting(
F,
[self._a],
bcs=[self.bcs],
constraint=self.mpc,
x0=[x],
scale=dolfinx.default_scalar_type(-1.0),
)
F.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE) # type: ignore
dolfinx.fem.petsc.set_bc(F, self.bcs, x, -1.0)
def J(self, x: PETSc.Vec, A: PETSc.Mat): # type: ignore
A.zeroEntries()
dolfinx_mpc.assemble_matrix(self._a, self.mpc, bcs=self.bcs, A=A)
A.assemble()
class NewtonSolverMPC(dolfinx.cpp.nls.petsc.NewtonSolver):
def __init__(
self,
comm: MPI.Intracomm,
problem: NonlinearMPCProblem,
mpc: dolfinx_mpc.MultiPointConstraint,
):
"""A Newton solver for non-linear MPC problems."""
super().__init__(comm)
self.mpc = mpc
self.u_mpc = dolfinx.fem.Function(mpc.function_space)
# Create matrix and vector to be used for assembly of the non-linear
# MPC problem
self._A = dolfinx_mpc.cpp.mpc.create_matrix(problem.a._cpp_object, mpc._cpp_object)
self._b = _la.create_petsc_vector(mpc.function_space.dofmap.index_map, mpc.function_space.dofmap.index_map_bs)
self.setF(problem.F, self._b)
self.setJ(problem.J, self._A)
self.set_form(problem.form)
self.set_update(self.update)
def update(self, solver: dolfinx.nls.petsc.NewtonSolver, dx: PETSc.Vec, x: PETSc.Vec): # type: ignore
# We need to use a vector created on the MPC's space to update ghosts
self.u_mpc.x.petsc_vec.array = x.array_r
self.u_mpc.x.petsc_vec.axpy(-1.0, dx)
self.u_mpc.x.petsc_vec.ghostUpdate(
addv=PETSc.InsertMode.INSERT, # type: ignore
mode=PETSc.ScatterMode.FORWARD, # type: ignore
) # type: ignore
self.mpc.homogenize(self.u_mpc)
self.mpc.backsubstitution(self.u_mpc)
x.array = self.u_mpc.x.petsc_vec.array_r
x.ghostUpdate(
addv=PETSc.InsertMode.INSERT, # type: ignore
mode=PETSc.ScatterMode.FORWARD, # type: ignore
) # type: ignore
def solve(self, u: dolfinx.fem.Function):
"""Solve non-linear problem into function u. Returns the number
of iterations and if the solver converged."""
n, converged = super().solve(u.x.petsc_vec)
u.x.scatter_forward()
return n, converged
@property
def A(self) -> PETSc.Mat: # type: ignore
"""Jacobian matrix"""
return self._A
@property
def b(self) -> PETSc.Vec: # type: ignore
"""Residual vector"""
return self._b
@pytest.mark.skipif(
np.issubdtype(dolfinx.default_scalar_type, np.complexfloating),
reason="This test does not work in complex mode.",
)
@pytest.mark.parametrize("poly_order", [1, 2, 3])
def test_nonlinear_poisson(poly_order):
# Solve a standard Poisson problem with known solution which has
# rotational symmetry of pi/2 at (x, y) = (0.5, 0.5). Therefore we may
# impose MPCs on those DoFs which lie on the symmetry plane(s) and test
# our numerical approximation. We do not impose any constraints at the
# rotationally degenerate point (x, y) = (0.5, 0.5).
N_vals = np.array([4, 8, 16], dtype=np.int32)
l2_error = np.zeros_like(N_vals, dtype=np.double)
for run_no, N in enumerate(N_vals):
mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, N, N)
V = dolfinx.fem.functionspace(mesh, ("Lagrange", poly_order))
u_bc = dolfinx.fem.Function(V)
u_bc.x.array[:] = 0.0
mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)
facets = dolfinx.mesh.exterior_facet_indices(mesh.topology)
topological_dofs = dolfinx.fem.locate_dofs_topological(V, 1, facets)
zero = np.array(0, dtype=dolfinx.default_scalar_type)
bc = dolfinx.fem.dirichletbc(zero, topological_dofs, V)
bcs = [bc]
# Define variational problem
u = dolfinx.fem.Function(V)
v = ufl.TestFunction(V)
x = ufl.SpatialCoordinate(mesh)
u_soln = ufl.sin(ufl.pi * x[0]) * ufl.sin(ufl.pi * x[1])
f = -ufl.div((1 + u_soln**2) * ufl.grad(u_soln))
F = ufl.inner((1 + u**2) * ufl.grad(u), ufl.grad(v)) * ufl.dx - ufl.inner(f, v) * ufl.dx
J = ufl.derivative(F, u)
# -- Impose the pi/2 rotational symmetry of the solution as a constraint,
# -- except at the centre DoF
def periodic_boundary(x):
eps = 1000 * np.finfo(x.dtype).resolution
return np.isclose(x[0], 0.5, atol=eps) & ((x[1] < 0.5 - eps) | (x[1] > 0.5 + eps))
def periodic_relation(x):
out_x = np.zeros_like(x)
out_x[0] = x[1]
out_x[1] = x[0]
out_x[2] = x[2]
return out_x
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_periodic_constraint_geometrical(V, periodic_boundary, periodic_relation, bcs)
mpc.finalize()
# Sanity check that the MPC class has some constraints to impose
num_slaves_global = mesh.comm.allreduce(len(mpc.slaves), op=MPI.SUM)
num_masters_global = mesh.comm.allreduce(len(mpc.masters.array), op=MPI.SUM)
assert num_slaves_global > 0
assert num_masters_global == num_slaves_global
problem = NonlinearMPCProblem(F, u, mpc, bcs=bcs, J=J)
solver = NewtonSolverMPC(mesh.comm, problem, mpc)
solver.atol = 1e1 * np.finfo(u.x.array.dtype).resolution
solver.rtol = 1e1 * np.finfo(u.x.array.dtype).resolution
# Ensure the solver works with nonzero initial guess
u.interpolate(lambda x: x[0] ** 2 * x[1] ** 2)
solver.solve(u)
l2_error_local = dolfinx.fem.assemble_scalar(dolfinx.fem.form((u - u_soln) ** 2 * ufl.dx))
l2_error_global = mesh.comm.allreduce(l2_error_local, op=MPI.SUM)
l2_error[run_no] = l2_error_global**0.5
rates = np.log(l2_error[:-1] / l2_error[1:]) / np.log(2.0)
assert np.all(rates > poly_order + 0.9)
@pytest.mark.parametrize("tensor_order", [0, 1, 2])
@pytest.mark.parametrize("poly_order", [1, 2, 3])
def test_homogenize(tensor_order, poly_order):
mesh = dolfinx.mesh.create_unit_square(MPI.COMM_WORLD, 8, 8)
if tensor_order == 0:
shape = ()
elif tensor_order == 1:
shape = (mesh.geometry.dim,)
elif tensor_order == 2:
shape = (mesh.geometry.dim, mesh.geometry.dim)
else:
pytest.xfail("Unknown tensor order")
cellname = mesh.ufl_cell().cellname()
el = basix.ufl.element(basix.ElementFamily.P, cellname, poly_order, shape=shape, dtype=mesh.geometry.x.dtype)
V = dolfinx.fem.functionspace(mesh, el)
def periodic_boundary(x):
return np.isclose(x[0], 0.0)
def periodic_relation(x):
out_x = np.zeros(x.shape)
out_x[0] = 1.0 - x[0]
out_x[1] = x[1]
out_x[2] = x[2]
return out_x
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_periodic_constraint_geometrical(V, periodic_boundary, periodic_relation, [])
mpc.finalize()
# Sanity check that the MPC class has some constraints to impose
num_slaves_global = mesh.comm.allreduce(len(mpc.slaves), op=MPI.SUM)
assert num_slaves_global > 0
u = dolfinx.fem.Function(V)
u.x.petsc_vec.set(1.0)
assert np.isclose(u.x.petsc_vec.min()[1], u.x.petsc_vec.max()[1])
assert np.isclose(u.x.petsc_vec.array_r[0], 1.0)
mpc.homogenize(u)
with u.x.petsc_vec.localForm() as u_:
for i in range(V.dofmap.index_map.size_local * V.dofmap.index_map_bs):
if i in mpc.slaves:
assert np.isclose(u_.array_r[i], 0.0)
else:
assert np.isclose(u_.array_r[i], 1.0)
u.x.petsc_vec.destroy()
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