Solving PDEs with different scalar (float) types
This demo (demo_types.py) shows:
How to solve problems using different scalar types, .e.g. single or double precision, or complex numbers
Interfacing with SciPy sparse linear algebra functionality
import sys
from mpi4py import MPI
import numpy as np
import scipy.sparse
import scipy.sparse.linalg
import ufl
from dolfinx import fem, la, mesh, plot
SciPy solvers do not support MPI, so all computations will be performed on a single MPI rank
comm = MPI.COMM_SELF
Create a function that solves the Poisson equation using different precision float and complex scalar types for the finite element solution.
def display_scalar(u, name, filter=np.real):
"""Plot the solution using pyvista"""
try:
import pyvista
cells, types, x = plot.vtk_mesh(u.function_space)
grid = pyvista.UnstructuredGrid(cells, types, x)
grid.point_data["u"] = filter(u.x.array)
grid.set_active_scalars("u")
plotter = pyvista.Plotter()
plotter.add_mesh(grid, show_edges=True)
plotter.add_mesh(grid.warp_by_scalar())
plotter.add_title(f"{name}: real" if filter is np.real else f"{name}: imag")
if pyvista.OFF_SCREEN:
pyvista.start_xvfb(wait=0.1)
plotter.screenshot(f"u_{'real' if filter is np.real else 'imag'}.png")
else:
plotter.show()
except ModuleNotFoundError:
print("'pyvista' is required to visualise the solution")
def display_vector(u, name, filter=np.real):
"""Plot the solution using pyvista"""
try:
import pyvista
V = u.function_space
cells, types, x = plot.vtk_mesh(V)
grid = pyvista.UnstructuredGrid(cells, types, x)
bs = V.dofmap.index_map_bs
grid.point_data["u"] = filter(np.insert(u.x.array.reshape(x.shape[0], bs), bs, 0, axis=1))
plotter = pyvista.Plotter()
plotter.add_mesh(grid.warp_by_scalar(), show_edges=True)
plotter.add_title(f"{name}: real" if filter is np.real else f"{name}: imag")
if pyvista.OFF_SCREEN:
pyvista.start_xvfb(wait=0.1)
plotter.screenshot(f"u_{'real' if filter is np.real else 'imag'}.png")
else:
plotter.show()
except ModuleNotFoundError:
print("'pyvista' is required to visualise the solution")
def poisson(dtype):
"""Poisson problem solver
Args:
dtype: Scalar type to use.
"""
# Create a mesh and locate facets by a geometric condition
msh = mesh.create_rectangle(
comm=comm,
points=((0.0, 0.0), (2.0, 1.0)),
n=(32, 16),
cell_type=mesh.CellType.triangle,
dtype=np.real(dtype(0)).dtype,
)
facets = mesh.locate_entities_boundary(
msh, dim=1, marker=lambda x: np.isclose(x[0], 0.0) | np.isclose(x[0], 2.0)
)
# Define a variational problem.
V = fem.functionspace(msh, ("Lagrange", 1))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
x = ufl.SpatialCoordinate(msh)
fr = 10 * ufl.exp(-((x[0] - 0.5) ** 2 + (x[1] - 0.5) ** 2) / 0.02)
fc = ufl.sin(2 * np.pi * x[0]) + 10 * ufl.sin(4 * np.pi * x[1]) * 1j
gr = ufl.sin(5 * x[0])
gc = ufl.sin(5 * x[0]) * 1j
a = ufl.inner(ufl.grad(u), ufl.grad(v)) * ufl.dx
L = ufl.inner(fr + fc, v) * ufl.dx + ufl.inner(gr + gc, v) * ufl.ds
# In preparation for constructing Dirichlet boundary conditions, locate
# facets on the constrained boundary and the corresponding
# degrees-of-freedom.
dofs = fem.locate_dofs_topological(V=V, entity_dim=1, entities=facets)
# Process forms. This will compile the forms for the requested type.
a0 = fem.form(a, dtype=dtype)
if np.issubdtype(dtype, np.complexfloating):
L0 = fem.form(L, dtype=dtype)
else:
L0 = fem.form(ufl.replace(L, {fc: 0, gc: 0}), dtype=dtype)
# Create a Dirichlet boundary condition
bc = fem.dirichletbc(value=dtype(0), dofs=dofs, V=V)
# Assemble forms
A = fem.assemble_matrix(a0, [bc])
A.scatter_reverse()
b = fem.assemble_vector(L0)
fem.apply_lifting(b.array, [a0], bcs=[[bc]])
b.scatter_reverse(la.InsertMode.add)
bc.set(b.array)
# Create a SciPy CSR matrix that shares data with A and solve
As = A.to_scipy()
uh = fem.Function(V, dtype=dtype)
uh.x.array[:] = scipy.sparse.linalg.spsolve(As, b.array)
return uh
Create a function that solves the linearised elasticity equation using different precision float and complex scalar types for the finite element solution.
def elasticity(dtype) -> fem.Function:
"""Linearised elasticity problem solver."""
# Create a mesh and locate facets by a geometric condition
msh = mesh.create_rectangle(
comm=comm,
points=((0.0, 0.0), (2.0, 1.0)),
n=(32, 16),
cell_type=mesh.CellType.triangle,
dtype=np.real(dtype(0)).dtype,
)
facets = mesh.locate_entities_boundary(
msh, dim=1, marker=lambda x: np.isclose(x[0], 0.0) | np.isclose(x[0], 2.0)
)
# Define the variational problem.
gdim = msh.geometry.dim
V = fem.functionspace(msh, ("Lagrange", 1, (gdim,)))
ω, ρ = 300.0, 10.0
x = ufl.SpatialCoordinate(msh)
f = ufl.as_vector((ρ * ω**2 * x[0], ρ * ω**2 * x[1]))
E, ν = 1.0e6, 0.3
μ, λ = E / (2.0 * (1.0 + ν)), E * ν / ((1.0 + ν) * (1.0 - 2.0 * ν))
def σ(v):
"""Return an expression for the stress σ given a displacement field"""
return 2.0 * μ * ufl.sym(ufl.grad(v)) + λ * ufl.tr(ufl.sym(ufl.grad(v))) * ufl.Identity(
len(v)
)
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = ufl.inner(σ(u), ufl.grad(v)) * ufl.dx
L = ufl.inner(f, v) * ufl.dx
dofs = fem.locate_dofs_topological(V=V, entity_dim=1, entities=facets)
# Process forms. This will compile the forms for the requested type.
a0, L0 = fem.form(a, dtype=dtype), fem.form(L, dtype=dtype)
# Create a Dirichlet boundary condition
bc = fem.dirichletbc(np.zeros(2, dtype=dtype), dofs, V=V)
# Assemble forms (CSR matrix)
A = fem.assemble_matrix(a0, [bc])
A.scatter_reverse()
b = fem.assemble_vector(L0)
fem.apply_lifting(b.array, [a0], bcs=[[bc]])
b.scatter_reverse(la.InsertMode.add)
bc.set(b.array)
# Create a SciPy CSR matrix that shares data with A and solve
As = A.to_scipy()
uh = fem.Function(V, dtype=dtype)
uh.x.array[:] = scipy.sparse.linalg.spsolve(As, b.array)
return uh
Solve problems for different types
uh = poisson(dtype=np.float32)
uh = poisson(dtype=np.float64)
if not sys.platform.startswith("win32"):
uh = poisson(dtype=np.complex64)
uh = poisson(dtype=np.complex128)
display_scalar(uh, "poisson", np.real)
display_scalar(uh, "poisson", np.imag)
uh = elasticity(dtype=np.float32)
uh = elasticity(dtype=np.float64)
if not sys.platform.startswith("win32"):
uh = elasticity(dtype=np.complex64)
uh = elasticity(dtype=np.complex128)
display_vector(uh, "elasticity", np.real)