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 numpy as np
import scipy.sparse
import scipy.sparse.linalg

import ufl
from dolfinx import fem, la, mesh, plot

from mpi4py import MPI

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)
        grid.point_data["u"] = filter(np.insert(u.x.array.reshape(x.shape[0], V.dofmap.index_map_bs), 2, 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.logical_or(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)
    fem.set_bc(b.array, [bc])

    # Create a SciPy CSR  matrix that shares data with A and solve
    As = scipy.sparse.csr_matrix((A.data, A.indices, A.indptr))
    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.logical_or(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], block_mode=la.BlockMode.expanded)
    A.scatter_reverse()
    assert A.block_size == [1, 1]

    b = fem.assemble_vector(L0)
    fem.apply_lifting(b.array, [a0], bcs=[[bc]])
    b.scatter_reverse(la.InsertMode.add)
    fem.set_bc(b.array, [bc])

    # Create a SciPy CSR  matrix that shares data with A and solve
    As = scipy.sparse.csr_matrix((A.data, A.indices, A.indptr))
    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)
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)
uh = elasticity(dtype=np.complex64)
uh = elasticity(dtype=np.complex128)
display_vector(uh, "elasticity", np.real)