Infinite Elements
Infinite elements are used to represent the effect of far field radiation on unbounded finite element acoustic models. They have several advantages over alternative boundary treatments for such problems. For example, they provide a direct numerical estima
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Summary. Infinite elements are used to represent the effect of far field radiation on unbounded finite element acoustic models. They have several advantages over alternative boundary treatments for such problems. For example, they provide a direct numerical estimate of the solution at all points in the outer domain and by using elements of arbitrary radial order they give an anechoic boundary condition which is accurate to an arbitrary order of approximation. In this chapter infinite elements are introduced with reference to a simple one–dimensional formulation and then extended to two and three dimensions. The application of the method to transient problems is also touched upon.
7.1 Preliminaries The solution of the acoustic wave equation on an unbounded domain presents particular problems for computation. When conventional grid–based or element–based methods are used, they must of necessity be applied to a bounded computational region which contains a discrete number of grid points or nodes. This implies the existence of a truncation surface through which acoustic energy must propagate without being reflected back into the computational domain. A generic exterior problem of this type is illustrated in Figure 7.1(a). Here the sound field in an unbounded domain Ω which lies outside a radiating or scattering surface Γ , is truncated at a computational boundary Γc . The computational domain which lies between the surfaces Γ and Γc is denoted by Ωc . The development of “anechoic” boundary conditions which can be applied on Γc and which are effective for arbitrary radiative fields poses a major challenge for domain–based numerical methods. In this chapter the Infinite Element (IE) concept is introduced as a practical technique for treating such problems. Conventional Finite Element (FE) models will be used to represent the solution within the inner domain Ωc . A variety of other methods currently exist for terminating such models. Local Non–Reflecting Boundary Conditions (NRBCs) are perhaps the most common. These are applied on the boundary itself and generally involve the normal and/or tangential derivative(s) of the solution [17,18]. Non–local NRBCs are also well developed for such problems. Galerkin
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RJ Astley
Fig. 7.1 (a) The exterior Helmholtz problem. (b) The acoustic horn.
mode–matching and the closely related FE–Dirichlet–to–Neumann (FE–DtN) approach [2] are well known examples. Such methods generally involve integration over the truncation boundary and are less appealing for large 3D models where it is important to exploit fully the sparsity of the FE matrices. Alternatively, the computational domain can be extended beyond the truncation boundary to include an “absorbing” or “perfectly matched” layer within which a non–physical operator is used to progressively remove the outgoing disturbance [19] so that it passes through the truncation boundary without significant reflection. In all of these approaches, the location of the truncation surface determines the size of the numerical problem in the inner
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