Coupling of mapped wave infinite elements and plane wave basis finite elements for the Helmholtz equation in exterior domains

Abstract

SUMMARYThe theory for coupling of mapped wave inÿnite elements and special wave ÿnite elements for the solution of the Helmholtz equation in unbounded domains is presented. Mapped wave inÿnite elements can be applied to boundaries of arbitrary shape for exterior wave problems without truncation of the domain. Special wave ÿnite elements allow an element to contain many wavelengths rather than having many ÿnite elements per wavelength like conventional ÿnite elements. Both types of elements include trigonometric functions to describe wave behaviour in their shape functions. However the wave directions between nodes on the ÿnite element=inÿnite element interface can be incompatible. This is because the directions are normally globally constant within a special ÿnite element but are usually radial from the 'pole' within a mapped wave inÿnite element. Therefore forcing the waves associated with nodes on the interface to be strictly radial is necessary to eliminate this internode incompatibility. The coupling of these elements was tested for a Hankel source problem and plane wave scattering by a cylinder and good accuracy was achieved. This paper deals with unconjugated inÿnite elements and is restricted to two-dimensional problems.

Short Waves

“…Another approach to the integral of Eq. The method can be extended to three dimensions [29,30] and can be linked to conventional infinite elements [31]. In Bettess et al's method the constant Jacobian for simpler finite elements is exploited to transform the integrations into the local coordinates.…”

Exact integration of polynomial–exponential products with application to wave‐based numerical methods

“…An overview of wave-based methods is given by Bettess [1,Chapter 12]. Examples of wave-based numerical methods include the partition of unity method originally proposed by Melenk and Babuška [2,3] and which has since received particular attention [4][5][6], the discontinuous enrichment method [7], the ultra-weak 239 expensive quadrature schemes as exact solutions can be obtained in closed form. The remainder of this paper is as follows.…”