BEM in words#

Author: Andrew Gibbs

Introduction#

Physical intuition#

Suppose we have an incoming acoustic wave \(u^i\) and a scattering obstacle \(\Omega\), and we want to determine the amplitude of the scattered acoustic field \(u^s(x)\), i.e. how much sound has bounced back, at any point \(x\) in the region surrounding the obstacle?

Physically, the idea of BEM may be interpreted as covering the obstacle in lots of tiny speakers. (Each microphone is analogy for the more officially-named point source or the Green’s function.) The aim of BEM is to solve a different problem; to adjust the volume on each speaker individually so that the combined amplitude of all of the microphones, if we were listening away from the obstacle, is the same as the scattered acoustic field \(u^s\) in our original problem.

To solve the BEM problem, i.e. to fine tune the volume on each microphone, we must solve a problem on the surface of \(\Omega\), rather than in the area surrounding \(\Omega\). We will write \(\partial\Omega\) to represent the surface. Practically this can be appealing, because this problem is usually simpler: in a lower spatial dimensions and on a bounded domain. For example, modelling scattering by a cube, most of the computational work will be done on the circumference on the square faces of the cube.

Representation in terms of point sources#

We will write \(\Phi(x,y)\) to mean the point-source / Green’s function / metaphorical speaker at point \(y\) on \(\partial \Omega\), observed at point \(x\) on \(\Omega\). Naturally we would expect \(\Phi(x,y)\) to be a wave, and to get larger as \(y\) moves towards \(x\).

It follow’s from Green’s third identity that

\[ u^s(x) = -\int_{\partial\Omega}\Phi(x,y)\frac{\partial u}{\partial n}(y)\ \mathrm{d}s(y) + \int_{\partial\Omega}\frac{\partial \Phi(x,y)}{\partial n(y)}u(y)\ \mathrm{d}s(y), \]

\(\Phi\) denotes the fundamental solution / point source / Green’s function / figurative speaker:

\[\begin{split} \Phi(x,y) := \left\{ \begin{array}{ll} \frac{\mathrm{i}}{4}H^{(1)}_0(k|x-y|),&\quad \text{two dimensions},\\ {\exp({\mathrm{i}k|x-y|}})/({4\pi|x-y|}),&\quad \text{three dimensions}, \end{array} \right. \end{split}\]

where \(H^{(1)}_0\) is the Hankel function of the first kind order zero.

Here \(x\) is some point away from the obstacle \(\Omega\). If we consider two types of speakers, \(\Phi(x,y)\) and \(\frac{\partial \Phi(x,y)}{\partial n(y)}\), then the above should be interpreted as a weighted sum of a large number of these point-sources/speakers, over the surface of the obstacle. Conveniently, the unknown density which weights these sources (i.e. the other term in the integrals), has been expressed in terms of our total field \(u\).

Depending on boundary conditions, it may be possible to remove one of the two integrals above. For example, a sound-hard obstacle has boundary condition \(u=0\) on \(\partial\Omega\).

The steps to the Boundary Element Method#

  1. Modify (1) to obtain an integral equation on the boundary \(\partial \Omega\), where \(u\) and/or \(\frac{\partial u}{\partial n}\) on \(\partial \Omega\) are the unknown quantities. This is called a Boundary Integral Equation (BIE). At this point, no approximation has taken place.

  2. Approximately solve the BIE, using a finite element method on the boundary. Hence the name Boundary Element Method.

  3. Plug our approximation to \(u\) and/or \(\frac{\partial u}{\partial n}\) into the representation formula (1), to obtain an approximation to \(u^s\) away from the boundary.

For the remainder of this document, we will describe each of these steps in more detail.

Constructing a Boundary Integral Equation#

We want to solve (1) for the unknown densities.

The next step is to move the problem onto the boundary \(\partial \Omega\), so that \(x\) and \(y\) live on \(\partial \Omega\), and then our original problem is reduced to an (arguably simpler) problem on \(\partial \Omega\).

This process of moving the problem onto the boundary is commonly referred to as a trace (no relation to the matrix operation of the same name). There are three or four commonly used traces, which warrants a separate discussion in its own right. This can be skipped for now, and taken on trust.

Click here to show optional details on trace operators

Trace operators#

The simplest of the trace operators is the Dirichlet trace \(\gamma_D\), which can be interpreted physically as moving \(x\) to the boundary \(\partial\Omega\). This trace gives us

\[ \gamma_Du^s(x) = -\int_{\partial\Omega}\Phi(x,y)\frac{\partial u}{\partial n}(y)\mathrm{d}s(y) + \int_{\partial\Omega}\frac{\partial\Phi(x,y)}{\partial n(y)}u(y)\mathrm{d}s(y) + \frac{1}{2}u(x), \]

for \(x\) on \(\partial \Omega\). This is often compactly written as

\[ \gamma_Du^s = S\frac{\partial u}{\partial n} - \left(D+\frac{1}{2}\mathcal{I}\right)u,\quad\text{on }\partial\Omega. \]

The second most commonly used trace is the Neumann trace, which can be interpreted as moving \(\nabla u\) onto \(\partial\Omega\) (non-tangentially) and then taking the dot product with the normal derivative at this limit point. This gives us:

\[\begin{split} \begin{align*} \gamma_Nu^s(x) =& -\int_{\partial\Omega}\frac{\Phi(x,y)}{\partial n(x)}\frac{\partial u}{\partial n}(y)\mathrm{d}s(y) +\frac{1}{2}\frac{\partial u}{\partial n}(x)\\ &+ \frac{\partial}{\partial n(x)}\int_{\partial\Omega}\frac{\partial\Phi(x,y)}{\partial n(y)}u(y)\mathrm{d}s(y) \end{align*},\quad x\text{ on }\partial\Omega, \end{split}\]

which has the compact form

\[ \gamma_N u^s = \left(-D'+\frac{1}{2}\mathcal{I}\right)\frac{\partial u}{\partial n} + Hu. \]

Both of these trace equations can be used to construct our boundary integral equation which follows.

After taking a trace, we obtain an integral equation of the form

\[ \begin{equation} (\chi\mathcal{I}+\mathcal{K})v(x) = f(x),\quad x\text{ on }\partial \Omega. \end{equation} \]

where our unknown \(v\) depends on the boundary conditions, i.e. the material of \(\Omega\) as follows:

\[\begin{split} v = \left\{\begin{array}{ll} {\partial u}/{\partial n}& \text{sound-soft / Dirichlet}\\ u&\text{sound-hard / Neumann}\\ ({\partial u}/{\partial n}, u ) &\text{impedance / Robin} \end{array}\right. \end{split}\]

In the third case, our unknown quantity \(v\) is a vector of two unknown functions. For problems of scattering by thin screens/plates, the quantities above are replaced by their jump in value from either side of the screen. For example, \(u\) would be replaced by \(u^+-u^-\), where \(u+\) and \(u^-\) are respectively the limiting values of \(u\) above and below the screen.

In (1) \(\mathcal{I}\) denotes the identity operator, which maps a function to its self. Some BIEs will contain the identity operator in which case \(\chi=\pm\frac{1}{2}\), more details follow below. If there are no identity terms, clearly \(\chi=0\).

In (1) \(\mathcal{K}\) is a boundary integral operator (BIO), meaning it maps functions on the boundary \(\partial \Omega\) to functions on \(\partial \Omega\)

\[ \mathcal{K}\psi(x) := \int_{\partial \Omega}K(x,y)\psi(y)\mathrm{d} s(y), \]

where \(K(x,y)\) is a known function called the kernel (no relation to the computer component or algebraic objects of the same name) and will depend on the choice of trace taken, but in the simplest case, with Dirichlet BCs / sound-soft obstacle, \(K=\Phi\).

Similarly, \(f\) is known explicitly, and will depend on the choice of trace used and the incoming wave \(u^i\). In the simplest case, we have \(f = u^i\).

BIE directory#

This section contains a list of the formulations required for common acoustic scattering problems. Before we define these formulations, it is necessary to introduce the five main integral operators, their names are listed to the right:

\[\begin{split} \begin{align*} \mathcal{I}\psi(x) &= \psi(x),\quad&&\text{Identity}\\ S\psi(x) &= \int_{\partial\Omega}\Phi(x,y)\psi(y),\mathrm{d}s(y)\quad&&\text{Single Layer}\\ D\psi(x) &=\int_{\partial\Omega}\frac{\partial\Phi(x,y)}{\partial n(y)}\psi(y)\mathrm{d}s(y)\quad&&\text{Double Layer}\\ D'\psi(x) &=\int_{\partial\Omega}\frac{\partial\Phi(x,y)}{\partial n(x)}\psi(y)\mathrm{d}s (y)\quad&&\text{Adjoint double Layer}\\ H\psi(x) &=\int_{\partial\Omega}\frac{\partial^2\Phi(x,y)}{\partial n(x)\partial n(y)}\psi(y)\mathrm{d}s (y)\quad&&\text{Hypersingular} \end{align*} \end{split}\]

For each type of problem solvable by BEM, the starting point is a relevant boundary integral equation, which makes use of some combination of the above operators. These can be derived using the trace identities in the above (optional) section.

Depending on the shape of \(\Omega\) and the boundary conditions, there may be one or many possible BIE formulations of the form (2). The appropriate formulations are given below, sometimes there are multiple formulations which could be used. The formulation can often be simplified when considering problems on thin screens/plates, by volume we refer to more typical scattering obstacles such as polygons in two dimensions and tetrahedra in three dimensions. By plates/screens, we refer to obstacles which are thin in one direction, for example a square plate in three dimensions. And as is explained in the table, certain formulations are not well-posed at certain wavenumbers \(k\), meaning that they may have multiple solutions. If our BIE is not well posed, then the approximate problem we solve later has no chance!

Material / BCs

Screen / volume

BIO \(\mathcal{K}\)

RHS \(f\)

Solvable?

Sound-soft / Dirichlet

Screen

\(S\)

\(u^i\)

Always

Volume

\(S\)

\(u^i\)

Not at certain \(k\)

Volume

\(\frac{1}{2}\mathcal{I}+D'\)

\(\frac{\partial u^i}{\partial n}\)

Not at certain \(k\)

Volume

\(\frac{1}{2}\mathcal{I}+D'-\mathrm{i}\eta S\)

\(\frac{\partial u^i}{\partial n}-\mathrm{i}\eta u^i\)

If \(\eta\neq0\)

Sound-Hard / Neumann

Screen

\(H\)

\(\frac{\partial u^i}{\partial n}\)

Always

Volume

\(\frac{1}{2}\mathcal{I}-D\)

\(u^i\)

Not at certain \(k\)

Volume

\(H\)

\(\frac{\partial u^i}{\partial n}\)

Not at certain \(k\)

Volume

\(\frac{1}{2}\mathcal{I}-\mathcal{D}-\mathrm{i}\eta H\)

\(u^i+\mathrm{i}\eta\frac{\partial u^i}{\partial n}\)

If \(\eta\neq0\)

Regarding the BIEs which are not valid at certain \(k\), you could consider yourself unlucky if you happened to encounter such a \(k\) in practice, as they are rare; corresponding to eigenvalues of interior Laplace problems. The simple BIEs (with fewer operators in \(\mathcal{K}\)) are sometimes more popular in practice, because these bad wavenumbers \(k\) are rare, and implementation is much easier.

The term \(\eta\) is referred to as the coupling parameter, and can be chosen to be any non-zero real number to ensure the BIE is well posed. For example, in the Dirichlet case, the two equations which are ill-posed at certain \(k\) values are never ill-posed at the same \(k\) value. A classical idea is to couple the equations via \(\eta\), so that the coupled equation is well-posed for all \(k\). In practice, a good choice is \(\eta=k\).

Impedance/Robin problems are considerably more complicated, click here for details.

Impedance on the screen#

Now consider more general Impedance / Robin problems on the screen, with boundary conditions

\[ \gamma^\pm_N u\pm\lambda^\pm\gamma_D^\pm u = -(\gamma_N^\pm u^i\pm\lambda^\pm\gamma_D^\pm u^i), \]

where \(\gamma^\pm_D\) and \(\gamma^\pm_N\) denote the Dirichlet and Neumann traces taken from above/below the screen, and \(\lambda^pm\) are the impedance parameters describing the material. Here the boundary integral equation takes a matrix form, with

\[\begin{split}\mathcal{K} = \left[\begin{array}{ll} -\frac{1}{2}(\lambda^++\lambda^-)\mathcal{I}-2H&(\lambda^+-\lambda^-)S \\ \frac{1}{2}(\lambda^+-\lambda^-)\mathcal{I} & \mathcal{I}-(\lambda^++\lambda^-)S\end{array}\right]\end{split}\]

and

\[\begin{split} f = \left[\begin{array}{c}(\lambda^+-\lambda^-)\frac{\partial u^i}{\partial n}\\ (\lambda^--\lambda^+)u^i\end{array} \right]. \end{split}\]

This is uniquely solvable for \(\lambda^++\lambda^-\neq0\).

Constructing a Boundary Element Method (BEM)#

The main aim of the BEM is to approximate \(v_N\) by approximately solving (2), then plug this approximation into (1), to obtain an approximation for \(u^s\). This is done by writing

\[ v(x)\approx v_N(x)=\sum_{n=1}^Nc_n\phi_n(x), \]

where the \(\phi_n\) are basis functions, for example piecewise linear, piecewise constants. Hence the name Boundary Element Method; we are implementing a finite element method on the boundary \(\partial \Omega\). We then solve either the collocation or Galerkin problem.

Collocation BEM#

The idea behind collocation is to force (2) to hold at \(N\) collocation points \(x_1,\ldots,x_N\), on the surface \(\partial \Omega\). This can be expressed as

\[ \sum_{n=1}^Nc_n(\chi\mathcal{I}+\mathcal{K})\phi_n(x_m) = f(x_m),\quad\text{for }m=1,\ldots,N, \]

where the unknowns are the coefficients \(c_n\). This is equivalent to solving the linear system:

\[ \left[\chi\phi_n(x_m)+\int_{\mathrm{supp}\ \phi_n}K(x_m,y)\phi_n(y)\mathrm{d} s(y)\right]_{n,m=1}^N \left[c_n \right]_{n=1}^N = \left[f(x_m)\right]_{m=1}^N. \]

Collocation has the practical advantage over Galerkin (which will be summarised next) because there are only single integrals. However, there are few theoretical guarantees about the above linear system being well-conditioned, or even solvable, and little is known about the best way to choose \(x_m\). Here’s a summary of what is known:

  • Taking more collocation points than basis functions is known as oversampling. By doing this, and reformulating as a least-squares problem, one can often overcome the instabilities associated with collocation.

  • Another technique is to supplement the linear system with some collocation points inside of \(\Omega\) satisfying a different equation, which follows from (1), noting that \(u=0\) in \(\Omega\). There are known as CHIEF points.

  • When \(\phi_n\) are piecewise linear functions, e.g. hat functions, choosing collocation points as the midpoints of \(\mathrm{supp}\phi_n\) is actually a bad idea, and can lead to the linear system being unsolvable.

Galerkin BEM#

The idea behind Galerkin BEM is similar to Galerkin FEM, we force (2) to hold when integrate against each of our basis functions

\[ \sum_{n=1}^Nc_n\left<(\chi\mathcal{I}+\mathcal{K})\phi_n,\phi_m\right> = \left<f,\phi_m\right>,\quad\text{for }m=1,\ldots,N, \]

where the unknowns are the coefficients \(c_n\), and the triangular brackets denote the inner product

\[\left<\psi,\varphi\right>=\int_{\partial\Omega}\psi(x)~\overline{\varphi}(x)~\mathrm{d}s(x).\]

This is equivalent to solving the linear system:

\[\begin{split} \left[\chi\int_{\mathrm{supp}\phi_m}\phi_n(x)\overline{\phi_m}(x)\mathrm{d}s(x) + \int_{\mathrm{supp}\ \phi_m}\int_{\mathrm{supp}\ \phi_n}K(x,y)\phi_n(y)\overline{\phi_m}(x)\mathrm{d} s(y)\mathrm{d} s(x)\right]_{n,m=1}^N \left[c_n \right]_{n=1}^N \\= \left[\int_{\partial\Omega}f(x)\overline{\phi_m}(x)\mathrm{d}s(x)\right]_{m=1}^N. \end{split}\]

The only disadvantage of Galerkin (when compared against collocation) is the extra integral, and the double integral can be tricky to implement, especially on a two-dimensional surface, this will likely be an integral over four spatial variables.

Sometimes this can be worth it, because the system to solve is often much better behaved in practice. There are some theoretical guarantees about solvability and accuracy, which follow when the operator \(\mathcal{K}\) satisfies the coercivity property. For this reason, mathematicians often prefer Galerkin BEM, and engineers prefer collocation.

Obtaining an approximate representation#

Finally, we can plug our approximation \(v_N\) in place of \(u\) or \(\frac{\partial u}{\partial n}\) in (1) to obtain our approximation to \(u^s(x)\).

Sound-soft/Dirichlet representation#

Here we have \(\phi_h\approx \frac{\partial u}{\partial n}\), so

\[ \begin{equation} u^s(x)\approx u^s_h(x) = -\int_{\partial\Omega}\Phi(x,y)v_N(y)\ \mathrm{d}s(y) = -\sum_{n=1}^Nc_n\int_{\partial\Omega}\Phi(x,y)\phi_n(y)\ \mathrm{d}s(y), \end{equation} \]

Sound-hard/Neumann representation#

Here we have \(\phi_h\approx u\), so

\[ u^s(x)\approx u^s_h(x) = \int_{\partial\Omega}\frac{\partial \Phi(x,y)}{\partial n(y)}v_N\ \mathrm{d}s(y) =\sum_{n=1}^Nc_n\int_{\partial\Omega}\frac{\partial \Phi(x,y)}{\partial n(y)}\phi_h(y)\ \mathrm{d}s(y), \]

Some final comments#

Comparison with FEM#

The following table summarises the pros and cons of BEM, when compared against FEM for solving the same problem.

Property

FEM

BEM

Spatial dimension of unknown

Same as original problem (with the exception of Trefftz DG FEM)

One less than original problem

Matrix

Sparse and large

Dense and not so large (due to lower spatial dimension)

Matrix entries

Smooth integrals

Singular integrals

Size of unknown domain

Unbounded, typically addressed using an artificial boundary, e.g. Perfectly Matched Layers

Bounded, on the surface of \(\Omega\)

Choice of quadrature#

In the coded example in the next tutorial, we will use a one-point quadrature rule for our integrals, which is the most basic approximation conceivable. For smooth integrands \((m\neq n)\), Gauss-Legendre quadrature is very popular in practice, as this converges much faster. In higher dimensional integrals, a popular approach is to use Gauss quadrature in each direction. This is sub-optimal, cubature rules are the most efficient way to do this, but are rarely used in practice.

For singular integrals \((m=n)\), grading can be used as a one-size-fits all approach. However, we often know the precise singular behaviour, so grading can be overkill. A more informed approach is that of singularity subtraction, where the singular part of the integrand is evaluated analytically, and the remaining part is evaluated using standard quadrature. A second informed approach is to use generalised Gaussian quadrature, which is designed to be accurate for integrals containing a certain type of singularity.

For singular double integrals, when the singularity is along the diagonal of the integration domain, the Duffy transform can be used to convert to two integrals over a square/cube/hypercube with singularities at the edges, making it more amenable to techniques for 1D singular integrals.

Quadrature is the main difficulty when implementing a BEM. If possible, use BEM software such as bempp, where quadrature has been implemented carefully and efficiently. If you are hellbent on implementing your own BEM, get your quadrature routines from a colleague who has tried and tested them for similar problems, otherwise prepare yourself for several days/weeks/months of painful debugging.

Summary#

  • Certain acoustic scattering problems can be reformulated as a problem on the boundary, where the unknown density determines the amplitude of lots of tiny sources/speakers

  • BEMs are FEMs on the boundary/surface of the obstacle

  • Certain BIEs and/or certain choices of collocation points can lead to numerical instabilities

  • Implementing and understanding BEMs can be harder than FEMs, but there are computational advantages