Proceedings of 20th International Congress on Acoustics, ICA 2010 23–27 August 2010, Sydney, Australia The ultra weak variational formulation for 3D elastic wave problems Teemu Luostari (1), Tomi Huttunen (1) and Peter Monk (2) (1) Department of Physics and Mathematics, University of Eastern Finland, Kuopio, Finland (2) Department of Mathematical Sciences, University of Delaware, Newark DE, USA PACS: 43.58.Ta, 43.20.Bi, 43.40.Fz, 43.20.Gp ABSTRACT In this paper we investigate the feasibility of using the ultra weak variational formulation (UWVF) to solve the time- harmonic 3D elastic wave propagation problem. The UWVF is a non-polynomial volume based method that uses plane waves as basis functions which reduces the computational burden. More general, the UWVF is a special form of the discontinuous Galerkin method. As a model problem we consider plane wave propagation in a cubic domain. We shall show numerical results for the accuracy, conditioning and p-convergence of the UWVF. In addition, we shall investigate the effect of different ratios of the P- and S-wave basis functions. INTRODUCTION Elastic wave problems, in common with acoustic and electro- magnetic problems, are usually challenging and computation- ally demanding due to the need approximate wavelike solu- tions. An added difficulty for elastic waves is that the solutions consist of two components, S and P-waves, with different wave numbers. Therefore it is useful to develop numerical methods that reduces the computational time, and also which can ap- proximate the S and P waves separately. The approach we follow is to use non-polynomial basis func- tions that are appropriate solutions of the underlying equation (for other applications of non-polynomial basis functions see (Barnett and Betcke 2010, Cessenat and Després 1998, Gabard 2007, Farhat et al. 2001, Huttunen et al. 2002, Perrey-Debain 2006)). The analysis and implementation of methods based on non-polynomial bases is an active area of research and these methods have been compared to each other recently, see, for example, references (Gabard et al. 2010, Gamallo and Astley 2007, Huttunen et al. 2009). However, the 3D elasticity prob- lem has not been considered to date, and we shall focus on the ultra weak variational formulation (UWVF) for the 3D elastic wave problems. The UWVF was first developed for the Helmholtz and Maxwell equations by Cessenat and Després, see references (Cessenat 1996, Cessenat and Després 1998). Further development, and particularly computational aspects of the UWVF for acoustics and electromagnetism, can be found in (Huttunen et al. 2007a;b, Loeser and Witzigmann 2009). Relevant to this paper is the study of the UWVF for the 2D elastic wave problems inves- tigated in reference (Huttunen et al. 2004). The UWVF is a volume based method that uses plane wave basis functions which are efficient to compute. However, if too many plane wave basis functions are used on an element the results may become inaccurate due to the ill-conditioning. Therefore, in this paper we study the behavior of the 3D elastic UWVF with different numbers of basis functions, and different ratios between S and P-wave components. The UWVF is a spe- cial form of the discontinuous Galerkin method, shown inde- pendently in references (Huttunen et al. 2007a, Gabard 2007). Thus the UWVF shares similar properties with the DGM and similar finite element meshes are used in the UWVF. In addi- tion, convergence analysis of the non-polynomial DGM, shown in references (Hiptmair et al. 2009, Gittelson et al. 2009), are then applicable for the UWVF. The error estimates for the UWVF has been studied in reference (Buffa and Monk 2008). No estimates are specifically available for 3D elasticity. In this paper we aim to show preliminary numerical results for accu- racy, p-convergence and conditioning of the 3D elastic UWVF. This paper is organized as follows: First we give a short in- troduction of the UWVF for the Navier equation of elastic- ity with discretization. A detailed derivation of the 2D elas- tic UWVF, which is similar to 3D, can be found in reference (Huttunen et al. 2004). In the second section we show the nu- merical results for the model problem which is plane wave propagation in a cubic domain. Finally we draw conclusions from the preliminary numerical experiments. THE ULTRA WEAK VARIATIONAL FORMULATION In this section we consider the ultra weak variational formu- lation for the Navier equation, and its discretization by plane waves. The UWVF for the Navier equation Let K be a computational domain with the boundary Γ = ∂ K and let us assume that K consists of non-overlapping elements, i.e. K = ∪ N k=1 K k where N is the number of elements (a finite element grid). We consider the Navier equation μ ∆u +(λ + μ )∇(∇ · u)+ ϖ 2 ρ u = 0 in K k (1) where ϖ is the angular frequency of the field, u is the time- harmonic displacement vector, λ and μ are the Lamé constants and ρ is the density of the medium. Later we use the following notation ∆ e = μ ∆u +(λ + μ )∇(∇ · u). The Lamé constants can be expressed as μ = E 2(1 - ν ) and λ = E ν (1 + ν )(1 - 2ν ) , (2) ICA 2010 1