Implicit boundary integral methods for the Helmholtz equation in exterior domains
 Chieh Chen^{1} and
 Richard Tsai^{1, 2}Email author
https://doi.org/10.1186/s406870170108y
© The Author(s) 2017
Received: 17 June 2016
Accepted: 24 April 2017
Published: 2 October 2017
Abstract
We propose a new algorithm for solving Helmholtz equations in exterior domains with implicitly represented boundaries. The algorithm not only combines the advantages of implicit surface representation and the boundary integral method, but also provides a new way to compute a class of the socalled hypersingular integrals. The keys to the proposed algorithm are the derivation of the volume integrals which are equivalent to any given integrals on smooth closed hypersurfaces, and the ability to approximate the natural limit of the singular integrals via seamless extrapolation. We present numerical results for both two and threedimensional scattering problems at near resonant frequencies as well as with nonconvex scattering surfaces.
Keywords
Helmholtz equation Hypersingular integrals Level set methods Closest point projection Boundary integrals1 Background

The implicit representation of the domain boundaries is a natural choice and is convenient for handling changes in the shapes;

Boundary integral formulation is a natural choice as typical applications assume that scattering take place in unbounded domains in which wave propagate at a constant speed;

The solution of the Helmholtz equation is needed only on a lowerdimensional set, where receivers of the scattered wave field are positioned.
1.1 Boundary integral formulations of the Helmholtz equation
While (1.1) is wellposed for a large class of \(\varOmega ,\) however, the integral equation (1.5) is not uniquely solvable if k is an eigenvalue of the interior problem that is used to derive (1.4). In such a case, the integral operator has a nontrivial subspace; see e.g. [8].
2 The implicit boundary integral methods
In this section, we first describe the implicit boundary integral formulation of [11] as it lays out the foundation of the proposed algorithm. The main contribution of this paper is presented in Sects. 2.3 and 2.4. We first introduce some essential definitions below.
Definition 1
Let \(d_{\varOmega }:{\mathbb {R}}^{m}\mapsto {\mathbb {R}}\) be the signed distance function to \(\varGamma =\partial \varOmega \) such that \(d_{\varOmega }(x)>0\) for \(x\in \bar{\varOmega }^{c}\) and \(d_{\varOmega }(x)<0\) for \(x\in \varOmega .\) Define the normal vector \(n_{x}:=\nabla d_{\varOmega }(x)\) and the normal derivative of a function f at x to be \(\partial f/\partial n=\partial f/\partial n_{x}:=\lim _{h\rightarrow 0+}\nabla f(x+hn_{x})\cdot n_{x}\).
Definition 2
2.1 Exact integral formulations using signed distance functions
The Jacobian \(J_{\varGamma }\) required by this formulation can be easily evaluated by computing the curvatures H and G directly using standard centered finite differencing applied to the distance function, as in [11], or it could be very easily computed by singular value decomposition applied to a difference approximation of \(P_{\varGamma }^{\prime }\), as in [12]. The exact integral formulation we use in this work was first proposed in [11] and extended in [12]. It is also generalized in [4] to simulate the Mullins–Sekerka dynamics of complicated interface geometry in unbounded domains.
2.2 Implicit boundary integral methods (IBIMs)
Definition 3
For convenience, we shall use the following notation for the tubular neighborhood of \(\varGamma \).
Definition 4
\(T_{(\epsilon _{1},\epsilon _{2})}=\{x\in {\mathbb {R}}^{d}:\epsilon _{1}<d_{\varOmega }(x)<\epsilon _{2}\}.\)
By construction, (2.7) has a unique solution if (2.6) does. Furthermore, the solution of (2.7) is automatically the constant extension of the solution of (2.6). The implication is that in forming the integral equation, either at the continuum level or at a discretized level, we do not need to artificially enforce the extension of u. This property is summarized below.
Proposition 5
On uniform Cartesian grids, \(K_{\varGamma }\) in (2.7) should be replaced by a suitable regularization of the singularity in K when \(xy\rightarrow 0.\) Here, we follow the approach of [11] which replaces the values of \(K_{\varGamma }(x,y)\) by a function \(\bar{K}(P_{\varGamma }(x),r_{0}),\) where \(r_{0}\) corresponds to a small regularization parameter.
2.2.1 Regularization of the doublelayer potentials
2.2.2 The Implicit Boundary Integral Method (IBIM)
2.3 Extrapolative integrals
In this section, we propose a way to approximate (1.7) using the framework described in the previous section.
Definition 6
2.3.1 A few extrapolative weight functions

for \(\theta =0.1\), \(a_{1}(r)\approx 759.2781934172483\,r+446.2604260472818\);

for \(\theta =0,\) \(a_{1}(r)\approx 261.5195892865372\,r+145.7876577089403.\)

for \(\theta =0.1\), \(a_{2}(r)\approx 14317.969703708994\,r^{2}16509.044867497203\,r+4480.224717878304\);

for \(\theta =0,\) \(a_{2}(r)\approx 3196.1015220946833\,r^{2}3457.6211113812255\,r+852.9832518883903.\)
2.4 Extrapolative boundary integral methods (EIBIMs)
Equations (2.18) and (2.19) can be discretized easily on uniform Cartesian grids as in (2.12) and in (2.13).
2.4.1 The choice of \(\xi \)
3 Numerical examples
In this section, we present numerical simulations that reveal the properties of the proposed methods. We present results in two and three dimensions, using simple shapes on which we can compare the numerical solutions with the analytical solutions, and using two standard nonconvex shapes. In the examples presented in Sects. 3.3 and 3.4, the results are obtained using the distance functions computed on the grids via the standard secondorder level set reinitialization algorithm [16].
3.1 Tests with a unit circle
 1.
When the system is well resolved by the grid, the errors are dominated by \(E_{\text {ker}}\). From \(w_{\infty ,0.1}^{(1)}\), \(E_{\text {ker}}\sim \mathcal {O}(\Delta x^{1.9})\) and \(\mathcal {O}(\Delta x^{1.5})\), respectively, for \(\epsilon =\tilde{C}_{0}\Delta x^{0.95}\) and \(\epsilon =\tilde{C}_{1}\Delta x^{0.75}.\) From \(w_{\infty ,0.1}^{(2)}\), \(E_{\text {ker}}\sim \mathcal {O}(\Delta x^{2.85})\) and \(\mathcal {O}(\Delta x^{2.25})\) for \(\epsilon =\tilde{C}_{0}\Delta x^{0.95}\) and \(\epsilon =\tilde{C}_{1}\Delta x^{0.75}.\) This regime is verified by the upper subfigure in Fig. 3.
 2.
Higherorder kernels, i.e., kernels having higher vanishing moments, do require more grid points as the constant in the quadrature error terms are larger. Thus in practice, lowerorder kernels may out perform the higherorder kernels. This case is shown in the lower subfigure in Fig. 3, where we see that \(w_{\infty ,0.1}^{(2)}\) yields worse errors, with a rate that is lower than the theoretical rate of \(E_{\text {ker}}\), while the lowerorder kernel \(w_{\infty ,1}^{(1)}\) yields comparatively smaller errors that decay at the theoretical rate of \(E_{\text {ker}}.\)
Condition numbers of the IBIM linear systems (2.12) and of the EIBIM (2.18)
\(\varvec{\Delta x}\)  \(\varvec{k=1}\)  \(\varvec{k=2.4048255577}\)  

Cond( \(\mathcal {M}_{\varvec{IBIM}}\) )  Cond( \(\mathcal {M}_{\varvec{EIBIM}}\) )  Cond( \(\mathcal {M}_{\varvec{IBIM}}\) )  Cond( \(\mathcal {M}_{\varvec{EIBIM}}\) )  
\(\frac{4}{128}\)  2.72E+00  1.16E+02  6.50E+02  1.46E+02 
\(\frac{4}{256}\)  2.70E+00  1.01E+02  9.74E+02  1.22E+02 
\(\frac{4}{512}\)  2.69E+00  1.00E+02  1.88E+03  1.21E+02 
Condition numbers of matrices formed by using different kernels
\(\varvec{N=987,\epsilon /\Delta x=5.0}\)  \(\varvec{N=3986,\epsilon /\Delta x=11.0}\)  \(\varvec{N=8972,\epsilon /\Delta x=16.0}\)  

Cond(\(w_{sc,0.1}^{(1)}\))  142.9  141.2  140.9 
Cond(\(w_{\infty ,0.1}^{(2)}\))  1666.5  1419.2  1413.01 
3.2 Tests with a unit sphere
3.3 Tests using a nonconvex “kite” shape
Farfield solution errors on 2D kite shape described in (3.1)
\(\varvec{\Delta x}\)  \({\mathbf {Re}}\,\varvec{u_{\infty }(1,0)}\)  \(\mathbf {Err}_{\mathbf {Re}}\)  \(\mathbf {Im}\,\varvec{u_{\infty }(1,0)}\)  \(\mathbf {Err}_{{\mathbf {Im}}}\)  \(\mathbf {Rate\,of} \sqrt{\mathbf {Err}_{{\mathbf {Re}}}^{\mathbf {2}}+\mathbf {Err}_{{\mathbf {Im}}}^{\mathbf {2}}}\) 

\(\frac{4}{128}\)  \(\)0.326625  0.045953  \(\)0.220888  0.077292  – 
\(\frac{4}{256}\)  \(\)0.319861  0.039189  \(\)0.239901  0.058279  0.36 
\(\frac{4}{512}\)  \(\)0.300958  0.020286  \(\)0.271519  0.026661  1.07 
\(\frac{4}{1024}\)  \(\)0.29066  0.009988  \(\)0.282961  0.015219  0.88 
\(\frac{4}{2048}\)  \(\)0.286097  0.005425  \(\)0.290469  0.007711  0.95 
\(\varvec{\Delta x}\)  \(\mathbf {Re}\,\varvec{u_{\infty }(1,0)}\)  \(\mathbf {Err}_{{\mathbf {Re}}}\)  \(\mathbf {Im}\,\varvec{u_{\infty }(1,0)}\)  \(\mathbf {Err}_{{\mathbf {Im}}}\)  \({\mathbf {Rate\, of}}\sqrt{\mathbf {Err}_{{\mathbf {Re}}}^{\mathbf {2}}+\mathbf {Err}_{{\mathbf {Im}}}^{\mathbf {2}}}\) 

\(\frac{4}{128}\)  \(\)1.928536  0.018957  \(\)1.038644  0.237263  – 
\(\frac{4}{256}\)  \(\)1.98456  0.037067  \(\)1.145499  0.130408  0.23 
\(\frac{4}{512}\)  \(\)1.960317  0.012824  \(\)1.220294  0.055613  0.95 
\(\frac{4}{1024}\)  \(\)1.958113  0.01062  \(\)1.245836  0.030071  1.02 
\(\frac{4}{2048}\)  \(\)1.955342  0.007849  \(\)1.259986  0.015921  0.88 
3.4 Scattering in three dimensions by a “Bean” shape
Differences in the solutions computed by EIBIM and IBIM using different mesh sizes
\(\varvec{\Delta x}\)  \(\mathbf {Re}\varvec{(u_{{\mathbf {EIBIM}}})}\)  \(\mathbf {Successive~differences}\)  \(\mathbf {Re}\varvec{(}\varvec{u}_{{\mathbf {IBIM}}}\varvec{)}\)  \(\varvec{}\mathbf {Re}_{{\mathbf {EIBIM}}}\mathbf {Re}_{{\mathbf {IBIM}}}\varvec{}\) 

\(\frac{4}{60}\)  0.972148  −  0.974335  0.002187 
\(\frac{4}{90}\)  0.946706  0.025442  0.918696  0.028010 
\(\frac{4}{120}\)  0.930812  0.015894  0.920763  0.010049 
\(\varvec{\Delta x}\)  \(\mathbf {Im}\varvec{(}\varvec{u}_{{\mathbf {EIBIM}}}\varvec{)}\)  \(\mathbf {Successive~differences}\)  \(\mathbf {Im}\varvec{(}\varvec{u}_{{\mathbf {IBIM}}}\varvec{)}\)  \(\varvec{}\mathbf {Im}_{{\mathbf {EIBIM}}}{\mathbf {Im}}_{{\mathbf {IBIM}}}\varvec{}\) 

\(\frac{4}{60}\)  0.262999  −  0.349435  0.0864360 
\(\frac{4}{90}\)  0.303281  0.040282  0.334712  0.031431 
\(\frac{4}{120}\)  0.318407  0.015126  0.327899  0.009492 
4 Summary
We proposed two new types of numerical methods (abbreviated above as IBIMs and EIBIMs) for solving Helmholtz equation in unbounded domains \({\mathbb {R}}^{m}{\setminus }\bar{\varOmega }\) with Neumann boundary conditions on \(\bar{\varOmega }.\) What distinguish the proposed algorithms from other related ones are: (1) the use of implicit representation of \(\varOmega \) and \(\partial \varOmega \) with out the need of explicit parametrization; (2) in the case in which hypersingular kernels are involved, EIBIMs rely on builtin extrapolation instead of solving the equivalent equations that involve partial derivatives of the unknown densities. The proposed algorithms can be easily implemented on Cartesian grids; however, a drawback is the larger linear systems resulting from the need to resolve the averaging kernel. In this regard, the algorithms are not intended to compete with the conventional highorder accurate methods for fixed and well parameterized geometries. Our algorithms are more suitable for applications in which one needs to solve the Helmholtz equation as \(\partial \varOmega \) goes through significant change in its shape and topology—applications for which implicit representation of the geometries is a natural choice.
Acknowledgements
The authors are partially supported by Simons Foundation, NSF Grants DMS1318975, DMS1217203, and ARO Grant No. W911NF1210519. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing the computing resources that have contributed to the research results reported within this paper.
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