A stable 3D energetic Galerkin BEM approach for wave propagation interior problems A. Aimi a , M. Diligenti a,n , A. Frangi b , C. Guardasoni a a Department of Mathematics, Universita ´ di Parma, V.le delle Scienze 53/A, 43124 Parma, Italy b Department of Structural Engineering, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 23133 Milano, Italy article info Article history: Received 19 March 2012 Accepted 17 June 2012 Keywords: Wave propagation Boundary integral equation Energetic Galerkin boundary element method abstract We consider 3D interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potentials. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the energetic Galerkin boundary element method. Numerical results are presented and discussed in order to show the stability and accuracy of the proposed technique. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Time-dependent problems that are frequently modelled by hyperbolic partial differential equations can be dealt with the boundary integral equation (BIE) method in which the transfor- mation of the problem to a BIE follows the same well-known method as for elliptic boundary value problems. Although the basic integral equation formulations for time-dependent pro- blems were known for many years their systematic use for obtaining numerical solutions is a rather recent event. The boundary value problem is formulated directly in terms of boundary values and the solution at interior points does not need to be considered, although it may be evaluated at these points directly from the integral representation, once the BIE has been solved. Boundary element methods (BEMs) have been success- fully applied at the discretization level both in the frequency- domain and in the time-domain (see, e.g., the reviews by Beskos [8] and Costabel [9]). Focusing on the latter case, the representation formula in terms of single layer and double layer potentials employs the fundamental solution of the hyperbolic partial differential equa- tion and jump relations, giving rise to the so-called ‘‘retarded’’ BIEs. Usual numerical discretization procedures include the col- location technique with explicit evaluation of the time-convolu- tion, which is very appealing thanks to its simplicity and limited demand in terms of computing effort. However, this approach has been always plagued with limited robustness. Indeed, the time marching scheme of classical space–time collocation methods often turns unstable in an erratic way upon variation of the time step adopted in the analysis (see, e.g., [15,16]). Even if this issue can be somehow cured in BEM only approaches by means of a trial and error choice of the time step, improving the basic formulation seems imperative, e.g., when a coupling with other numerical methods like the FEM is envisaged. It has been shown (see, e.g., [26]), indeed, that in staggered iterative coupling procedure between BEM and FEM the stability and accuracy of the solution in the two subdomains impose requirements which may be contradictory. It is worth mentioning that an alternative to the standard approach for the direct explicit evaluation of the time-convolu- tion is represented by the convolution quadrature method which has been developed in [19] and then extended to different applications in [13,22–24]. It provides a straightforward way to obtain an efficient time stepping scheme in terms of the Laplace transform of the kernel function, which is crucial for many time- dependent problems where only the transformed fundamental solutions are available. Although stability and convergence are assured only under strong regularity assumptions on problem data, it is empirically found that the stability of the algorithm is improved compared to the standard approach. Also the time-stepping method, based on the finite difference approximation of time derivatives, is not yet well developed. This is partly because accuracy and stability of the solution are affected by the time step size. Properties of the numerical solution computed by this method have been already studied in [5,11]. It is however only with variational techniques that the most promising results concerning stability have been obtained, also from the theoretical point of view. The application of Galerkin Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enganabound.2012.06.003 n Corresponding author. E-mail address: mauro.diligenti@unipr.it (M. Diligenti). Engineering Analysis with Boundary Elements 36 (2012) 1756–1765