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# This demo program solves Poisson's equation
#
# - div grad u(x, y) = f(x, y)
#
# on the unit square with homogeneous Dirichlet boundary conditions
# at y = 0, 1 and periodic boundary conditions at x = 0, 1.
#
# Copyright (C) Jørgen S. Dokken 2020-2022.
#
# This file is part of DOLFINX_MPCX.
#
# SPDX-License-Identifier: MIT
from __future__ import annotations
from pathlib import Path
from typing import Union
from mpi4py import MPI
from petsc4py import PETSc
import dolfinx.fem as fem
import numpy as np
import scipy.sparse.linalg
from dolfinx import default_scalar_type
from dolfinx.common import Timer, TimingType, list_timings
from dolfinx.io import XDMFFile
from dolfinx.mesh import create_unit_square, locate_entities_boundary
from ufl import (
SpatialCoordinate,
TestFunction,
TrialFunction,
as_vector,
dx,
exp,
grad,
inner,
pi,
sin,
)
import dolfinx_mpc.utils
from dolfinx_mpc import LinearProblem, MultiPointConstraint
# Get PETSc int and scalar types
complex_mode = True if np.dtype(default_scalar_type).kind == "c" else False
# Create mesh and finite element
NX = 50
NY = 100
mesh = create_unit_square(MPI.COMM_WORLD, NX, NY)
V = fem.functionspace(mesh, ("Lagrange", 1, (mesh.geometry.dim,)))
tol = 250 * np.finfo(default_scalar_type).resolution
def dirichletboundary(x):
return np.logical_or(np.isclose(x[1], 0, atol=tol), np.isclose(x[1], 1, atol=tol))
# Create Dirichlet boundary condition
facets = locate_entities_boundary(mesh, 1, dirichletboundary)
topological_dofs = fem.locate_dofs_topological(V, 1, facets)
zero = np.array([0, 0], dtype=default_scalar_type)
bc = fem.dirichletbc(zero, topological_dofs, V)
bcs = [bc]
def periodic_boundary(x):
return np.isclose(x[0], 1, atol=tol)
def periodic_relation(x):
out_x = np.zeros_like(x)
out_x[0] = 1 - x[0]
out_x[1] = x[1]
out_x[2] = x[2]
return out_x
with Timer("~PERIODIC: Initialize MPC"):
mpc = MultiPointConstraint(V)
mpc.create_periodic_constraint_geometrical(V, periodic_boundary, periodic_relation, bcs)
mpc.finalize()
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
a = inner(grad(u), grad(v)) * dx
x = SpatialCoordinate(mesh)
dx_ = x[0] - 0.9
dy_ = x[1] - 0.5
f = as_vector((x[0] * sin(5.0 * pi * x[1]) + 1.0 * exp(-(dx_ * dx_ + dy_ * dy_) / 0.02), 0.3 * x[1]))
rhs = inner(f, v) * dx
# Setup MPC system
with Timer("~PERIODIC: Initialize varitional problem"):
problem = LinearProblem(a, rhs, mpc, bcs=bcs)
solver = problem.solver
# Give PETSc solver options a unique prefix
solver_prefix = "dolfinx_mpc_solve_{}".format(id(solver))
solver.setOptionsPrefix(solver_prefix)
petsc_options: dict[str, Union[str, int, float]]
if complex_mode or default_scalar_type == np.float32:
petsc_options = {"ksp_type": "preonly", "pc_type": "lu"}
else:
petsc_options = {
"ksp_type": "cg",
"ksp_rtol": 1e-6,
"pc_type": "hypre",
"pc_hypre_type": "boomeramg",
"pc_hypre_boomeramg_max_iter": 1,
"pc_hypre_boomeramg_cycle_type": "v", # ,
# "pc_hypre_boomeramg_print_statistics": 1
}
# Set PETSc options
opts = PETSc.Options() # type: ignore
opts.prefixPush(solver_prefix)
if petsc_options is not None:
for k, v in petsc_options.items():
opts[k] = v
opts.prefixPop()
solver.setFromOptions()
with Timer("~PERIODIC: Assemble and solve MPC problem"):
uh = problem.solve()
# solver.view()
it = solver.getIterationNumber()
print("Constrained solver iterations {0:d}".format(it))
# Write solution to file
outdir = Path("results")
outdir.mkdir(exist_ok=True, parents=True)
uh.name = "u_mpc"
outfile = XDMFFile(mesh.comm, outdir / "demo_periodic_geometrical.xdmf", "w")
outfile.write_mesh(mesh)
outfile.write_function(uh)
print("----Verification----")
# --------------------VERIFICATION-------------------------
bilinear_form = fem.form(a)
A_org = fem.petsc.assemble_matrix(bilinear_form, bcs)
A_org.assemble()
linear_form = fem.form(rhs)
L_org = fem.petsc.assemble_vector(linear_form)
fem.petsc.apply_lifting(L_org, [bilinear_form], [bcs])
L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE) # type: ignore
fem.petsc.set_bc(L_org, bcs)
solver.setOperators(A_org)
u_ = fem.Function(V)
solver.solve(L_org, u_.x.petsc_vec)
it = solver.getIterationNumber()
print("Unconstrained solver iterations {0:d}".format(it))
u_.x.scatter_forward()
u_.name = "u_unconstrained"
outfile.write_function(u_)
root = 0
comm = mesh.comm
with Timer("~Demo: Verification"):
dolfinx_mpc.utils.compare_mpc_lhs(A_org, problem._A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(L_org, problem._b, mpc, root=root)
is_complex = np.issubdtype(default_scalar_type, np.complexfloating) # type: ignore
scipy_dtype = np.complex128 if is_complex else np.float64
# Gather LHS, RHS and solution on one process
A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh.x.petsc_vec, root=root)
if MPI.COMM_WORLD.rank == root:
KTAK = K.T.astype(scipy_dtype) * A_csr.astype(scipy_dtype) * K.astype(scipy_dtype)
reduced_L = K.T.astype(scipy_dtype) @ L_np.astype(scipy_dtype)
# Solve linear system
d = scipy.sparse.linalg.spsolve(KTAK, reduced_L)
# Back substitution to full solution vector
uh_numpy = K.astype(scipy_dtype) @ d.astype(scipy_dtype)
assert np.allclose(uh_numpy.astype(u_mpc.dtype), u_mpc, atol=float(tol))
list_timings(MPI.COMM_WORLD, [TimingType.wall])
L_org.destroy()
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