# Simple FEM-BEM Coupling for the Helmholtz Equation

## Background

In this tutorial, we will solve the problem of a wave travelling through a unit cube, $\displaystyle \Omega = [0,1]^3$ with different material parameters inside and outside the domain. The incident wave is given by

$\displaystyle u^\text{inc}(\mathbf{x})=\mathrm{e}^{\mathrm{i} k \mathbf{x}\cdot\mathbf{d}},$

where $\displaystyle \mathbf{x}=(x,y,z)$ and $\displaystyle \mathbf{d}$ is the direction of the incident wave. In the implementation we use, $\displaystyle \mathbf{d} = \frac{1}{\sqrt{3}}(1,1,1)$.

The PDE is

$\displaystyle \Delta u + n(\mathbf{x})^2 k^2 u = 0, \quad \text{ in } \Omega\\[2mm] \Delta u + k^2 u = 0, \quad \text{ in } \mathbb{R}^3 \backslash \Omega$

In this example, we use

$\displaystyle n(\mathbf{x}) = 0.5$

Since the interior wavenumber is constant one could have also used a BEM/BEM coupling approach. However, here we demonstrate the use of FEM for the interior problem using the FEniCS finite element package.

### FEM Part

In $\displaystyle \Omega$, the FEM part is formulated as

$\displaystyle \int_\Omega \nabla u\cdot\nabla v -k^2\int_\Omega n^2uv - \int_{\Gamma} v\frac{\partial u}{\partial \nu} = 0,$

or

$\displaystyle \langle\nabla u,\nabla v\rangle_\Omega - k^2\langle n^2u,v\rangle_\Omega - \langle \lambda,v\rangle_\Gamma=0,$

where $\displaystyle \lambda=\frac{\partial u}{\partial \nu}$.

Later, we will write this as the following operator equation

$\displaystyle \mathsf{A}u-k^2 \mathsf{M}u-\mathsf{M}_\Gamma \lambda = 0$

### BEM Part

In $\displaystyle \mathbb{R}^3 \backslash \Omega$, we let $\displaystyle u = u^\text{inc}+u^\text{s}$, where $\displaystyle u^\text{inc}$ is the incident wave and $\displaystyle u^\text{s}$ is the scattered wave. As given in Integral equation methods in scattering theory by Colton & Kress,

$\displaystyle 0 = \mathcal{K}u^\text{inc}-\mathcal{V}\frac{\partial u^{inc}}{\partial \nu},\\[2mm] u^\text{s} = \mathcal{K}u^\text{s}-\mathcal{V}\frac{\partial u^{s}}{\partial \nu},$

where $\displaystyle \mathcal{K}$ and $\displaystyle \mathcal{V}$ are the double single layer potential operators. Adding these, we get

$\displaystyle u^\text{s} = \mathcal{K}u-\mathcal{V}\lambda.$

This representation formula will be used to find $\displaystyle u^\text{s}$ for plotting later.

Taking the trace on the boundary gives

$\displaystyle u-u^\text{inc} = \left(\tfrac{1}{2}\mathsf{Id}+\mathsf{K}\right)u -\mathsf{V}\lambda.$

This rearranges to

$\displaystyle u^\text{inc} = \left(\tfrac{1}{2}\mathsf{Id}-\mathsf{K}\right)u+\mathsf{V}\lambda.$

### Full Formulation

The full blocked formulation is

$\displaystyle \begin{bmatrix} \mathsf{A}-k^2 \mathsf{M} & -\mathsf{M}_\Gamma\\ \tfrac{1}{2}\mathsf{Id}-\mathsf{K} & \mathsf{V} \end{bmatrix} \begin{bmatrix} u\\ \lambda \end{bmatrix}=\begin{bmatrix} 0\\ u^\text{inc} \end{bmatrix}.$

This formulation is not stable for all frequencies due to the possibility of interior resonances. But it is sufficient for this example and serves as a blueprint for more complex formulations.

## Implementation

We begin by importing Dolfin, the FEniCS python library, Bempp and NumPy.

import dolfin
import bempp.api
import numpy as np

Next, we set the wavenumber k and the direction d of the incoming wave.

= 6.
= np.array([1., 1., 1])
/= np.linalg.norm(d)

We create a Dolfin mesh. Later, the boundary mesh will be extracted from this.

A mesh could be created from a file changing this line to mesh = dolfin.Mesh('/path/to/file.xml').

mesh = dolfin.UnitCubeMesh(10, 10, 10)

Next, we make the Dolfin and Bempp function spaces.

The function coupling.fenics_to_bempp_trace_data will extract the trace space from the Dolfin space and create the matrix trace_matrix, which maps between the dofs (degrees of freedom) in Dolfin and Bempp.

from bempp.api import fenics_interface

fenics_space = dolfin.FunctionSpace(mesh, "CG", 1)
trace_space, trace_matrix = \
fenics_interface.coupling.fenics_to_bempp_trace_data(fenics_space)
bempp_space = bempp.api.function_space(trace_space.grid, "DP", 0)

print("FEM dofs: {0}".format(mesh.num_vertices()))
print("BEM dofs: {0}".format(bempp_space.global_dof_count))

FEM dofs: 1331
BEM dofs: 1200

We create the boundary operators that we need.

id_op = bempp.api.operators.boundary.sparse.identity(
trace_space, bempp_space, bempp_space)
mass = bempp.api.operators.boundary.sparse.identity(
bempp_space, bempp_space, trace_space)
dlp = bempp.api.operators.boundary.helmholtz.double_layer(
trace_space, bempp_space, bempp_space, k)
slp = bempp.api.operators.boundary.helmholtz.single_layer(
bempp_space, bempp_space, bempp_space, k)

We create the Dolfin function spaces and the function (or in this case constant) n.

= dolfin.TrialFunction(fenics_space)
= dolfin.TestFunction(fenics_space)
= 0.5

We make the vectors on the right hand side of the formulation.

def u_inc(x, n, domain_index, result):
result[0] = np.exp(1j * k * np.dot(x, d))
u_inc = bempp.api.GridFunction(bempp_space, fun=u_inc)

# The rhs from the FEM
rhs_fem = np.zeros(mesh.num_vertices())
# The rhs from the BEM
rhs_bem = u_inc.projections(bempp_space)
# The combined rhs
rhs = np.concatenate([rhs_fem, rhs_bem])

We are now ready to create a BlockedLinearOperator containing all four parts of the discretisation of

$\displaystyle \begin{bmatrix} \mathsf{A}-k^2 \mathsf{M} & -\mathsf{M}_\Gamma\\ \tfrac{1}{2}\mathsf{Id}-\mathsf{K} & \mathsf{V} \end{bmatrix}.$
from bempp.api.fenics_interface import FenicsOperator
from scipy.sparse.linalg.interface import LinearOperator
blocks = [[None,None],[None,None]]

trace_op = LinearOperator(trace_matrix.shape, lambda x:trace_matrix*x)

- k*** n*** u * v) * dolfin.dx)

blocks[0][0] = A.weak_form()
blocks[0][1] = -trace_matrix.T * mass.weak_form().sparse_operator
blocks[1][0] = (.5 * id_op - dlp).weak_form() * trace_op
blocks[1][1] = slp.weak_form()

blocked = bempp.api.BlockedDiscreteOperator(np.array(blocks))

Next, we solve the system, then split the solution into the parts assosiated with u and λ. For an efficient solve, preconditioning is required.

from scipy.sparse.linalg import LinearOperator

# Compute the sparse inverse of the Helmholtz operator
# Although it is not a boundary operator we can use
# the SparseInverseDiscreteBoundaryOperator function from
# BEM++ to turn its LU decomposition into a linear operator.
P1 = bempp.api.InverseSparseDiscreteBoundaryOperator(
blocked[0,0].sparse_operator.tocsc())

# For the Laplace slp we use a simple mass matrix preconditioner.
# This is sufficient for smaller low-frequency problems.
P2 = bempp.api.InverseSparseDiscreteBoundaryOperator(
bempp.api.operators.boundary.sparse.identity(
bempp_space, bempp_space, bempp_space).weak_form())

# Create a block diagonal preconditioner object using the Scipy LinearOperator class
def apply_prec(x):
"""Apply the block diagonal preconditioner"""
m1 = P1.shape[0]
m2 = P2.shape[0]
n1 = P1.shape[1]
n2 = P2.shape[1]

res1 = P1.dot(x[:n1])
res2 = P2.dot(x[n1:])
return np.concatenate([res1, res2])

p_shape = (P1.shape[0] + P2.shape[0], P1.shape[1] + P2.shape[1])
= LinearOperator(p_shape, apply_prec, dtype=np.dtype('complex128'))

# Create a callback function to count the number of iterations
it_count = 0
def count_iterations(x):
global it_count
it_count += 1

from scipy.sparse.linalg import gmres
soln, info = gmres(blocked, rhs, M=P, callback=count_iterations)

soln_fem = soln[:mesh.num_vertices()]
soln_bem = soln[mesh.num_vertices():]

print("Number of iterations: {0}".format(it_count))

Number of iterations: 340

Next, we make Dolfin and Bempp functions from the solution.

# Store the real part of the FEM solution
= dolfin.Function(fenics_space)
u.vector()[:] = np.ascontiguousarray(np.real(soln_fem))

# Solution function with dirichlet data on the boundary
dirichlet_data = trace_matrix * soln_fem
dirichlet_fun = bempp.api.GridFunction(trace_space, coefficients=dirichlet_data)

# Solution function with Neumann data on the boundary
neumann_fun = bempp.api.GridFunction(bempp_space, coefficients=soln_bem)

We now evaluate the solution on the slice $\displaystyle z=0.5$ and plot it. For the exterior domain, we use the respresentation formula

$\displaystyle u^\text{s} = \mathcal{K}u-\mathcal{V}\frac{\partial u}{\partial \nu}$

to evaluate the solution.

# Reduce the H-matrix accuracy since the evaluation of potentials for plotting
# needs not be very accurate.
bempp.api.global_parameters.hmat.eps = 1E-2

Nx=200
Ny=200
xmin, xmax, ymin, ymax=[-1,3,-1,3]
plot_grid = np.mgrid[xmin:xmax:Nx*1j,ymin:ymax:Ny*1j]
points = np.vstack((plot_grid[0].ravel(),
plot_grid[1].ravel(),
np.array([0.5]*plot_grid[0].size)))
plot_me = np.zeros(points.shape[1], dtype=np.complex128)

x,y,z = points
bem_x = np.logical_not((x>0) * (x<1) * (y>0) * (y<1) * (z>0) * (z<1))

slp_pot= bempp.api.operators.potential.helmholtz.single_layer(
bempp_space, points[:, bem_x], k)
dlp_pot= bempp.api.operators.potential.helmholtz.double_layer(
trace_space, points[:, bem_x], k)

plot_me[bem_x] += np.exp(1j * k * (points[0, bem_x] * d[0] \
+ points[1, bem_x] * d[1] \
+ points[2, bem_x] * d[2]))
plot_me[bem_x] += dlp_pot.evaluate(dirichlet_fun).flat
plot_me[bem_x] -= slp_pot.evaluate(neumann_fun).flat

fem_points = points[:, np.logical_not(bem_x)].transpose()
fem_val = np.zeros(len(fem_points))
for p,point in enumerate(fem_points):
result = np.zeros(1)
u.eval(result, point)
fem_val[p] = result[0]

plot_me[np.logical_not(bem_x)] += fem_val

plot_me = plot_me.reshape((Nx, Ny))

plot_me = plot_me.transpose()[::-1]

# Plot the image
from matplotlib import pyplot as plt
plt.imshow(np.real(plot_me), extent=[xmin, xmax, ymin, ymax])
plt.xlabel('x')
plt.ylabel('y')
plt.colorbar()
plt.title("FEM-BEM Coupling for Helmholtz")
plt.show()