On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics

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Abstract

In this paper, the wave approach called the Variational Theory of Complex Rays (VTCR), which was developed for medium-frequency acoustics and vibrations, is revisited as a discontinuous Galerkin method. Extensions leading to a weak Trefftz constraint are introduced. This weak Trefftz discontinuous Galerkin approach enables hybrid FEM/VTCR strategies to be developed easily, and paves the way for new computational techniques for the resolution of engineering problems. This paper presents some of the fundamental properties of the approach, which is illustrated by several numerical examples.

Introduction

In recent years, the use of numerical simulation techniques in the design, analysis and optimization of systems has become an indispensable part of the industrial design process. The standard Galerkin Finite Element Method (FEM)  [1] is a well-established computer-aided engineering tool commonly used for the analysis of time-harmonic dynamic problems. However, using continuous, piecewise polynomial shape functions leads to very large numerical models and, in practice, restricts applications of this prediction technique to the low-frequency range.

Trefftz methods  [2] have been proposed as a means to overcome this limitation. They differ from the FEM in the shape functions they use for the expansion of the field variables, which are exact solutions of the governing differential equations. Compared to finite element methods, these functions often lead to a considerable reduction in model size and computational effort. Some examples of such methods are: a special version of the partition of unity method  [3], the ultra weak variational method  [4], [5], the plane wave discontinuous Galerkin method  [6], [7], the least-squares method  [8], [9], the discontinuous enrichment method  [10], [11], the element-free Galerkin method  [12], the wave boundary element method  [13], [14] and the wave-based method  [15], [16]. Some mathematical results regarding the convergence of these methods can be found in  [6], [7]. The Variational Theory of Complex Rays (VTCR), first introduced in  [17] for steady-state vibration problems and in  [18] for transient problems, also belongs to that category. The main differences in these methods lie essentially in the treatment of the transmission conditions at the boundaries of the elements or substructures.

The main characteristic of the VTCR, the method discussed in this paper, is the use of a specific weak formulation of the problem which enables the approximations within the substructures to be a priori independent of one another. Thus, any type of shape function can be used within a given substructure provided it satisfies the governing equation, thus giving the approach great flexibility.

As explained in  [17], [19], [20], the VTCR was originally developed as an extension to acoustics and vibrations of the formulation introduced in  [21]. However, since the shape functions are discontinuous, there is a link between the VTCR and the discontinuous Galerkin methods studied in this paper. Discontinuous Galerkin methods represent a vast domain (see  [22] or  [23] for an overview, unified analysis and comparisons). Our discontinuous Galerkin formulation, which is nonsymmetrical, can be viewed as the Trefftz version of Baumann–Oden’s discontinuous Galerkin formulation  [24], [25]. The main engineering applications addressed are car acoustics  [26] and pyrotechnic shock propagation in space launchers  [18].

In addition, this paper introduces extensions of the classical VTCR formulation in which the Trefftz constraint is weakened. This leads to a new numerical method which can be called the weak Trefftz discontinuous Galerkin method. These extensions allow for an easy coupling of different types of numerical models, including classical finite element models. As a consequence, they lead to new approaches to the resolution of engineering problems. Another application concerns structures which are not piecewise homogeneous. In this paper, some fundamental properties of this new method will be presented and illustrated by numerical examples. Our standard problem is the 2D or 3D acoustic problem.

First, in Section  2, the VTCR approach is reviewed. Then, in Section  3, the weak Trefftz discontinuous Galerkin formulation is introduced as a discontinuous Galerkin method and two examples of finite element approaches are examined in detail. In Section  4, hybrid FEM/VTCR approaches are introduced as special cases of the weak Trefftz discontinuous Galerkin formulation. Section  5 presents several applications which illustrate the proposed computational technique for medium-frequency vibration problems.

Section snippets

The reference problem

Our reference problem is a standard acoustic problem defined over domain Ω with boundaries Ω=1Ω2Ω (see Fig. 1 on the left): find uH1(Ω) such that {(1+iη)Δu+k2u+rd=0over  Ωu=udover  1Ω(1+iη)nu+hiku=gdover  2Ω where nu=gradun,n being the outward normal; u is the acoustic pressure; k is the wave number, which is proportional to the frequency of the problem; h is a constant related to the acoustic impedance; rd and gd are prescribed acoustic sources over Ω and 2Ω, and ud is the

The weak Trefftz discontinuous Galerkin formulation

In the VTCR, the governing equation is satisfied within each subdomain ΩE,EE (see Section  2). The objective of this section is to weaken this condition in order to enable, for example, the use of the FEM solutions. Let Uh and U0h denote respectively the working space and the associated vector space, examples of which will be given later. The weak Trefftz discontinuous Galerkin formulation consists in finding uUh such that Re(ik(E,EEΓEE(12{qun}EE{ṽ}EE12[q̃vn]EE[u]EE)dSEEΓEE

Coupling of the FEM and the VTCR

The weak Trefftz discontinuous Galerkin formulation (12) can be used to couple the FEM and the VTCR. Let us divide Ω into two parts Ω1 and Ω2 and use the VTCR in Ω1 and the FEM in Ω2 (see the rightmost part of Fig. 1). The corresponding sets of subdomains are E1 and E2. In the working space Uh=U1hU2h, for EE1, UEh={uhuhVEh;(1+iη)Δuh+k2uh+rd=0  on  ΩE} and for EE2, UEh={uhuhVEh;ΩE((1+iη)Δuh+k2uh+rd)ṽhdΩ=0vhVE,0h}.

Property 10

The coupled FEM/VTCR problem defined by the weak Trefftz discontinuous

Numerical illustrations

In this section, the weak Trefftz discontinuous Galerkin formulation described in Section  3 is tested using two benchmark problems:

  • The first problem is defined over a square domain divided into two subdomains with different wave numbers so that an incident wave gives rise to a reflected wave and a transmitted wave. This problem has an analytical solution, which is used to define the prescribed boundary conditions and to assess the quality of the approximate solution.

  • The second problem is

Conclusion

In the VTCR proposed in  [17] for the calculation of medium-frequency phenomena, the solution of a vibrational problem is sought in an approximation space which is spanned by the exact solutions of the governing equation in the form of propagative and evanescent waves. These shape functions are discontinuous. The transmission conditions at the boundaries of the substructures are satisfied, thanks to a dedicated variational formulation which can be viewed as the Trefftz version of a

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