Improved conditioning of infinite elements for exterior acoustics

Abstract

SUMMARYAn optimized version of the so-called mapped wave envelope elements, also known as Astley-Leis elements, is introduced. These elements extend to inÿnity in one dimension and therefore provide an approach to the simulation of exterior acoustical problems in both frequency and time domains. Their formulation is improved signiÿcantly through the proper choice of polynomial bases in the direction of radiation. In particular, certain Jacobi polynomials are identiÿed which behave well with respect to conditioning of the system matrices. As a consequence, the size of the ÿnite element discretization may reduce considerably without any loss of accuracy. In addition, the new polynomial bases lead to superior performance of the inÿnite elements in conjunction with iterative solvers.

Infinite Elements and Their Influence on Normal and Radiation Modes in Exterior Acoustics

“…The condition number κ(A) is given by the ratio of the largest to the lowest eigenvalue of the frequency-dependent matrix A and is calculated by power iteration and inverse iteration, respectively. As shown by Babuška et al 41 Cremers et al 42 and Dreyer et al 6 the matrix condition increases with the polynomial order, which leads to ill-conditioned matrices for polynomials of degree > 8 in the case of Lagrange polynomials, whereas Legendre and Jacobi polynomials provide distinctly better-conditioned matrices. Tests performed by the authors of this paper have confirmed this observation, although the choice between the four polynomials had no considerable effect on the modes in exterior acoustics, as will be shown subsequently.…”

“…Investigations by von Estorff and Dreyer et al 6,34 prove poor suitability of Lagrangian polynomials for higher orders and achieve better performance with Legendre and Jacobi polynomials. This was shown by calculating the condition number of the dynamic stiffness matrix A in the system of linear equations in Eq.…”

“…The sound pressure field in the radial direction in the domain with the infinite elements is interpolated by polynomials such as Lagrange polynomials, Legendre polynomials or Jacobi polynomials, which lead to differences in the matrix condition number of the discrete, global system matrices. 5,6 The subject of this paper is the investigation of the influence of the choice of finite elements and the polynomial type for infinite element interpolation in the radial direction as well as the choice of the order of these polynomials on modes in exterior acoustics. The authors consider acoustic radiation modes (ARMs) [7][8][9] and normal modes (NMs), [10][11][12] which are based on eigenvalue problems of the acoustic impedance matrix Z R and of a statespace formulation consisting of the discrete system matrices of stiffness, damping and mass, respectively.…”

Salinas : theory manual.

“…Various choices of Q j (x) have been investigated, including Lagrangian, 78,79 Legendre, 82 Jacobi, 83 and rational (integrated Jacobi). 84 Lagrangian shape functions result in very poorly conditioned infinite element matrices.…”

“…The other three choices all appear to give acceptable levels of conditioning. Dreyer 83 showed that the Jacobi polynomials in general give a better condition than the Legendre polynomials. Regardless of the choice for Q(x), equations D.57 and D.64 imply that P(x) will be a function of the master element coordinates r, s,t, and thus can be integrated over the master element.…”