Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part I. Formulation in the frequency domain

Abstract

Mapped wave-envelope elements of variable radial order are presented for the computation of time-harmonic, unbounded, three-dimensional acoustical fields. Their application to transient problems is described in a companion article (Part II). Accuracy is assessed by a comparison of computed and analytic solutions for multi-pole fields generated by a vibrating sphere. Solutions are also presented for plane wave scattering. Elements of radial order m+l are shown to be capable of modeling multi-pole components of order m, although the provision of adequate transverse resolution is shown to be a stringent requirement, particularly at high frequencies. Ill-conditioning of the coefficient matrix limits the practical implementation of the method to elements of radial order eleven or less. The utility of the method for more general geometries is demonstrated by the presentation of computed solutions for the sound field generated by the vibration of a cylindrical piston in a plane baffle and of an idealised engine casing under anechoic conditions. The computed results are shown to be in close agreement with the analytic solution in the case of the cylindrical piston, and with a boundary element solution in the case of the engine casing.

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

“…At infinity the nonreflective Sommerfeld radiation condition causes the sound pressure to decay to zero. 2,4 On the surface of an obstacle, the normal fluid particle velocity v f (x) is assumed to be equal to the structural particle velocity v s (x). In other words, the coupling of the fluid and the structure is nondispersive.…”

“…These system matrices are required for further investigations in modal decomposition. The IFEM implies the Sommerfeld radiation condition, 2,4 which provides a nonreflective boundary condition and decay to zero of the sound pressure at infinity. 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.…”

“…Two approaches have been established in order to calculate acoustic exterior problems numerically: Perfectly matched layers (PML) 1 and the infinite element method (IFEM). 2,3 The method of conjugated Astley-Leis infinite elements is applied by the authors, since these elements provide the frequency-independent system matrices of stiffness, damping and mass on the basis of the corresponding FE matrices. These system matrices are required for further investigations in modal decomposition.…”

Salinas : theory manual.

“…In the early 1990's, Astley [78][79][80] derived a conjugated formulation that resulted in system matrices that were independent of frequency. This allowed the frequency domain formulation to be readily transformed to the time domain, in the same way that is typically done in linear structural dynamics.…”

“…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.…”