Stability of boundary element methods for the two dimensional wave equation in time domain revisited
Introduction
Among various numerical methods for partial differential equations such as finite difference methods or finite element methods, etc., boundary element methods (BEMs) are often said to be advantageous in wave problems because they can be applied to scattering problems easily. It is certainly true that BEMs in frequency domain are easy to use, but the same does not necessarily apply to time domain methods. As a matter of fact, BEMs for the wave equation in time domain have a long standing stability problem and there have been many efforts to stabilise BEMs for wave equations. For example, Ha Duong and his colleagues (e.g., [1]) showed the stability of some time domain BEMs in 3D based on space-time variational (Galerkin) formulations. Their argument depends on the energy conservation which is why their variational formulation includes time derivatives (e.g., the time derivative of single layer potential). Aimi et al. [2] presented some numerical results in 2D using time or space differentiated integral equations and a space-time variational approach. Abboud et al. [3] considered a coupling of space-time variational BEMs with discontinuous Galerkin methods. Unfortunately, however, implementing computational codes for the full space-time variational formulation is not very easy. Coding becomes easier if one uses variational approaches only spatially and use collocation in time. Van’t Wout et al. [4] have shown a way to find a stable time-collocated variational approach based on space-time variational methods. In spite of these efforts, the standard collocation approaches remain the preferred choice in engineering, although known mathematical stability results in collocation are rather limited (see Davies and Dancan [5] for example. We remark that this reference [5] is the only paper cited as an example of known stability results for time domain collocation BEM for hyperbolic problems in the related chapter in Encyclopedia of Computational Mechanics [6] indicating how little we know about this issue). Various numerical stabilisation techniques for collocation have been proposed, from which we cite just a few relatively new ones. Parot et al. discussed the removal of a non-oscillatory instability in a hypersingular integral equation [7] as well as the stabilisation of oscillatory ones by scaling [8]. Jang and Ih proposed to use the time domain version of CHIEF method and a filtering technique to stabilise BEMs for exterior problems [9]. They also consider stabilisation for interior problems using a filtering technique [10]. Pak and Bai [11] proposed a variable-weight multi-step collocation scheme with time projection in their regularised BEM for elastodynamics, which is combined with an eigenvalue analysis [12]. These numerical stabilisation techniques have been shown to be effective via numerical examples, although they require numerical solutions of large eigenvalue problems. For further literatures on numerical stabilisation techniques, we refer the reader to the lists of references of above mentioned papers. Some other investigations take viewpoints similar to ours in that they seek stabilisation based on the choices of integral equations. For example, the use of time differentiated integral equations has been advocated by several authors [13], [14]. Ergin et al. [15] proposed to use the Burton-Miller (BM) integral equation to achieve stability guided by an observation that the instability of BEMs for scattering problems is related to fictitious eigenfrequencies (internal resonance). Chappell et al. [16], [17] gave further insight as well as the implementation details of the BM formulation. This formulation has been utilised recently in practical applications [18]. These approaches based on the choices of integral equations are of interest because they are directly related to the cause of instability thus providing intuition for the stabilisation strategies, albeit qualitatively. Finally we mention recent developments of CQM by Lubich [19], [20], [21] which is a stable method of computing convolutions. CQM has been applied successfully to engineering applications (e.g., Schanz [22]). However, implementing CQM is still not as simple as the standard collocation methods, which is the reason we consider the conventional approach in this paper.
The above brief review of the works on the stability of time domain BEMs for the wave equation covers just a small part of what have been done so far. Indeed, the cause of the instability is now fairly well understood in connection with the spectra of the integral operators and the error introduced by discretisation (e.g., [15], [16], [23]), particularly in exterior problems. In spite of these efforts by predecessors, however, there seem to exist no definite and simple criteria of stability for the collocation methods. One still needs to carry out a quantitative assessment numerically in order to see if a particular scheme is stable or not. A standard method to check the stability of collocation BEMs in time domain is to compute characteristic roots by solving a polynomial eigenvalue problem (see (11)) after reducing it to an equivalent linear eigenvalue problem for the companion matrix (See, e.g., Walker et al. [24]). This method is effective in 3D where the fundamental solution has a finite “tail” (i.e., it vanishes after a finite time). However, this approach needs linear eigensolvers for sparse, but large, matrices. One may possibly solve polynomial eigenvalue problems directly to reduce the size of the matrix, but this will lead to a non-linear eigenvalue problem. Fortunately, recent developments of eigensolvers based on contour integrals such as the Sakurai-Sugiura method (SSM) [25] made the solution of non-linear eigenvalue problems feasible. In 2D problems, however, the same approach is not very practical because the fundamental solution in 2D is very slow to decay in time. In this paper we propose to resolve this difficulty by carrying out the required stability analysis in frequency domain. Namely, we convert the stability analysis for BEMs in two dimensional wave equation to a non-linear eigenvalue problem similar to those for the Helmholtz equation and solve it with SSM using techniques proposed in Misawa et al. [26], [27]. The benefit of this approach is twofold. Firstly, this method provides an alternative to the standard stability analysis in time domain. The proposed approach is applicable to 2D problems without ambiguity and remains valid in 3D problems as well. In contrast to this the standard time domain approach is not applicable to 2D problems in an unequivocal manner. Secondly, the proposed approach has an additional benefit of providing new intuition into the subject. Specifically, it is useful in selecting stable and highly accurate integral equations in various problems. Indeed, we shall use the proposed method to investigate the stability of various time domain integral equations for transmission problems, which have not been investigated very much so far.
As a basic study in this subject, however, this paper considers only very simple problems as examples of the use of the proposed stability analysis. Namely, we restrict our attention mainly to exterior Dirichlet problems and transmission problems for domains bounded by a circle. We first present a stability analysis for the exterior Dirichlet problem using frequency domain tools. The question of stability is then reduced to the computation of the characteristic roots which are eigenvalues of a certain non-linear equation. After solving this eigenvalue problem with SSM ignoring the effect of the spatial discretisation, we identify potentials which yield stable numerical schemes with piecewise linear time basis functions and a particular choice of discretisation parameters. We then proceed to transmission problems in which we show that even the time domain counterparts of “resonance free” BEMs may lead to instability. To stabilise these integral equations we modify them using potentials which have been concluded to yield stable numerical schemes. We show that numerical schemes derived from these modified formulations do lead to stability via numerical experiments in time domain as well as our stability analysis. After examining the influence of the spatial discretisation on the characteristic roots, we present numerical examples of transmission problems for non-circular domains solved with the modified integral equations, which appear to be stable.
Section snippets
Formulation
Let be a bounded domain whose boundary is smooth and let D1 be the exterior of D2, i.e., . Also, let n be the unit normal vector on Γ directed towards D1. We are interested in the following initial- boundary value problem (Dirichlet problem):
Find u which satisfies the two dimensional wave equation in D1:the homogeneous Dirichlet boundary condition on Γ for u:the homogeneous initial conditions in D1:and the
“Stable potentials”
Motivated by the results in the previous section, we examine the stability of integral equations on the unit circle derived from potentials which may appear in BIEs. These potentials include the single layer traces of the normal derivatives of S denoted by DT ± , the time derivative of S denoted by and the traces of the double layer D ± . Although we are also interested in the normal derivative of D denoted by N, it turned out that the simplified approach presented in 2.4
Transmission problems
We are now interested in finding u which satisfies (1),the transmission boundary conditions on Γ:and the homogeneous initial conditionsin addition to the homogeneous initial and radiation conditions for usca in (2), where c2 is the wave speed in D2 given by and (s2, ρ2) are the shear modulus and density in D2, respectively. The superscript stands for the trace to Γ from D1 (D2), respectively.
Effects of space discretisation
So far, we have neglected the effect of spatial discretisation in the discussion of stability. This section discusses how the distribution of the eigenvalues is influenced by the space discretisation. We restrict our attention to the circular scatterer case using the Fourier series expansion in order to keep the discussion as analytical as possible so that we can obtain insights.
Non-circular boundary
Finally, we test if the modified formulations remain stable for boundaries other than circle. We consider transmission problems for a “star” (Fig. 13(a)) given byand a “kite” (Fig. 13(b)) given byThe incident wave is the quadratic one in (21) and parameters such as material constants, number of boundary subdivision, Δt etc. are the same as those in the transmission problems considered in Section 4. The
Concluding remarks
This paper revisited stability issues for BEMs for the two dimensional wave equation in time domain. We presented a stability analysis based on integral equations in frequency domain and showed its validity and usefulness in simple exterior or transmission problems for circular domains. The resulting non-linear eigenvalue problems for the characteristic roots have been solved numerically with SSM. We identified layer potentials which lead to stable integral equations with linear time
Acknowledgement
This work has been supported by JSPS KAKENHI Grant Number 18H03251.
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