A Trefftz based method for solving Helmholtz problems in semi-infinite domains Bart Bergen à , Bert Pluymers, Bert Van Genechten, Dirk Vandepitte, Wim Desmet K.U.Leuven, Department of Mechanical Engineering, division PMA, Celestijnenlaan 300B - box 2420, B-3001 Leuven, Belgium article info Article history: Received 8 December 2010 Accepted 14 April 2011 Available online 9 September 2011 Keywords: Trefftz method Wave based method Helmholtz problems Semi-unbounded problems abstract The wave based method (WBM), which is based on an indirect Trefftz approach, is a deterministic prediction method posed as an alternative to the element-based methods. It uses wave functions, which are exact solutions of the underlying differential equation, to describe the dynamic field variables. In this way, it can avoid the pollution errors associated with the polynomial element-based approximations. As a consequence, a dense element discretization is no longer required, yielding a smaller numerical system. The resulting enhanced computational efficiency of the WBM as compared to the element-based methods has been proven for the analysis of both bounded and unbounded acoustic problems. This paper extends the applicability of the WBM to semi-infinite domains. An appropriate function set is proposed, together with a calculation procedure for both semi-infinite radiation and scattering problems, and transmission or diffraction problems containing a rigid baffle. The resulting technique is validated on two numerical examples. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction Ever increasing restrictive legal regulations regarding noise and vibration exposure as well as the customer’s growing demand for comfort, force industrial designers to take into account the acoustic behaviour of their products in the design optimization. As time is money and less is more, Computer Aided Engineering (CAE) tools have become an essential part in the product devel- opment process. The possibility of analyzing and optimizing virtual prototypes has relaxed the need for very expensive and time-consuming physical prototype testing. Both the finite element method (FEM) and the boundary element method (BEM) are well established deterministic CAE tools which are commonly used for the analysis of real-life acoustic problems. The FEM [1] discretizes the entire problem domain into a large but finite number of small elements. Within these elements, the dynamic response variables are described in terms of simple, polynomial shape functions. Because the FEM is based on a discretization of the problem domain into small elements, it cannot handle unbounded problems inherently. An artificial boundary is needed to truncate the unbounded problem into a bounded problem. Special techniques are then required to reduce spurious reflection of waves at the truncation boundary. Three strategies are applied to this end [2,3]: absorbing boundary conditions [4], infinite elements [5] or absorbing layers [6]. The BEM [7] is based on a boundary integral formulation of the problem. As a result, only the boundary of the considered domain has to be discretized. Within the applied boundary elements, some acoustic boundary variables are expressed in terms of simple, polynomial shape functions, similar to the FEM. Since the boundary integral formulation inherently satisfies the Sommerfeld radiation condition, the BEM is particularly suited for the treatment of problems in unbounded domains. However, since the simple shape functions used in both the FEM and BEM are no exact solutions of the governing differential equations, a very fine discretization is required to suppress the associated pollution errors [8] and to obtain a reasonable predic- tion accuracy. The resulting large numerical models limit the practical applicability of these methods to low-frequency problems [9], due to the prohibitively large computational cost. To overcome those limitations, several enhanced methods are proposed [2]. Often, those methods apply a wave-like basis for the approximation to better match the nature of the dynamic field. To this end, the spectral methods use a global basis based on oscillating functions, such as harmonics (sine and cosine) or Chebyshev polynomials [10,11]. The Trefftz methods [12] take this approach a step further, selecting the basis functions to satisfy a priori the governing dynamic equations. Several methods have been developed following this philosophy. The Equivalent Source Method (ESM) [13] is a prominent example, which employs a weighted distribution of particular solutions to Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2011.04.007 à Corresponding author. E-mail address: bart.bergen@mech.kuleuven.be (B. Bergen). Engineering Analysis with Boundary Elements 36 (2012) 30–38