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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.
#
# Original implementation in DOLFIN by Kristian B. Oelgaard and Anders Logg
# This implementation can be found at:
# https://bitbucket.org/fenics-project/dolfin/src/master/python/demo/documented/periodic/demo_periodic.py
#
# 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 Dict, Union
from mpi4py import MPI
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 VTXWriter
from dolfinx.mesh import CellType, create_unit_cube, locate_entities_boundary, meshtags
from numpy.typing import NDArray
from ufl import (
SpatialCoordinate,
TestFunction,
TrialFunction,
as_vector,
dx,
exp,
grad,
inner,
pi,
sin,
)
import dolfinx_mpc.utils
from dolfinx_mpc import LinearProblem
# Get PETSc int and scalar types
complex_mode = True if np.dtype(default_scalar_type).kind == "c" else False
def demo_periodic3D(celltype: CellType):
# Create mesh and finite element
if celltype == CellType.tetrahedron:
# Tet setup
N = 10
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N)
V = fem.functionspace(mesh, ("Lagrange", 1, (mesh.geometry.dim,)))
else:
# Hex setup
N = 10
mesh = create_unit_cube(MPI.COMM_WORLD, N, N, N, CellType.hexahedron)
V = fem.functionspace(mesh, ("Lagrange", 2, (mesh.geometry.dim,)))
tol = float(5e2 * np.finfo(default_scalar_type).resolution)
def dirichletboundary(x: NDArray[Union[np.float32, np.float64]]) -> NDArray[np.bool_]:
return np.logical_or(
np.logical_or(np.isclose(x[1], 0, atol=tol), np.isclose(x[1], 1, atol=tol)),
np.logical_or(np.isclose(x[2], 0, atol=tol), np.isclose(x[2], 1, atol=tol)),
)
# Create Dirichlet boundary condition
zero = default_scalar_type([0, 0, 0])
geometrical_dofs = fem.locate_dofs_geometrical(V, dirichletboundary)
bc = fem.dirichletbc(zero, geometrical_dofs, V)
bcs = [bc]
def PeriodicBoundary(x):
"""
Full surface minus dofs constrained by BCs
"""
return np.isclose(x[0], 1, atol=tol)
facets = locate_entities_boundary(mesh, mesh.topology.dim - 1, PeriodicBoundary)
arg_sort = np.argsort(facets)
mt = meshtags(mesh, mesh.topology.dim - 1, facets[arg_sort], np.full(len(facets), 2, dtype=np.int32))
def periodic_relation(x):
out_x = np.zeros(x.shape)
out_x[0] = 1 - x[0]
out_x[1] = x[1]
out_x[2] = x[2]
return out_x
with Timer("~~Periodic: Compute mpc condition"):
mpc = dolfinx_mpc.MultiPointConstraint(V)
mpc.create_periodic_constraint_topological(V.sub(0), mt, 2, periodic_relation, bcs, default_scalar_type(1))
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
dz_ = x[2] - 0.1
f = as_vector(
(
x[0] * sin(5.0 * pi * x[1]) + 1.0 * exp(-(dx_ * dx_ + dy_ * dy_ + dz_ * dz_) / 0.02),
0.1 * dx_ * dz_,
0.1 * dx_ * dy_,
)
)
rhs = inner(f, v) * dx
petsc_options: Dict[str, Union[str, float, int]]
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": str(tol),
"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,
}
problem = LinearProblem(a, rhs, mpc, bcs, petsc_options=petsc_options)
u_h = problem.solve()
# --------------------VERIFICATION-------------------------
print("----Verification----")
u_ = fem.Function(V)
u_.x.array[:] = 0
org_problem = fem.petsc.LinearProblem(a, rhs, u=u_, bcs=bcs, petsc_options=petsc_options)
with Timer("~Periodic: Unconstrained solve"):
org_problem.solve()
it = org_problem.solver.getIterationNumber()
print(f"Unconstrained solver iterations: {it}")
# Write solutions to file
ext = "tet" if celltype == CellType.tetrahedron else "hex"
u_.name = "u_" + ext + "_unconstrained"
# NOTE: Workaround as tabulate dof coordinates does not like extra ghosts
u_out = fem.Function(V)
old_local = u_out.x.index_map.size_local * u_out.x.block_size
old_ghosts = u_out.x.index_map.num_ghosts * u_out.x.block_size
mpc_local = u_h.x.index_map.size_local * u_h.x.block_size
assert old_local == mpc_local
u_out.x.array[: old_local + old_ghosts] = u_h.x.array[: mpc_local + old_ghosts]
u_out.name = "u_" + ext
outdir = Path("results")
outdir.mkdir(exist_ok=True, parents=True)
fname = outdir / f"demo_periodic3d_{ext}.bp"
out_periodic = VTXWriter(mesh.comm, fname, u_out, engine="BP4")
out_periodic.write(0)
out_periodic.close()
root = 0
with Timer("~Demo: Verification"):
dolfinx_mpc.utils.compare_mpc_lhs(org_problem.A, problem.A, mpc, root=root)
dolfinx_mpc.utils.compare_mpc_rhs(org_problem.b, 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(org_problem.A, root=root)
K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root)
L_np = dolfinx_mpc.utils.gather_PETScVector(org_problem.b, root=root)
u_mpc = dolfinx_mpc.utils.gather_PETScVector(u_h.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, rtol=tol, atol=tol)
if __name__ == "__main__":
for celltype in [CellType.hexahedron, CellType.tetrahedron]:
demo_periodic3D(celltype)
list_timings(MPI.COMM_WORLD, [TimingType.wall])
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