A wideband fast multipole accelerated boundary integral equation method for time‐harmonic elastodynamics in two dimensions

Abstract

SUMMARYThis article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two-dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low-frequency FMM and the high-frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura's group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low-frequency FMM and the quadrature order for the high-frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton-Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, N log N , where N is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite-size phononic crystal was demonstrated.

Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics

“…Improved versions of this approach were more recently proposed in [19,82], with numerical examples involving BE models of size up to N = O(10 6 ). Moreover, a 2D elastodynamic version of the wideband approach has been recently proposed in [77], while periodic 3D elastodynamic problems are addressed in [46] using a low-frequency separable decomposition of periodic Green's tensors.…”

On the efficiency of nested GMRES preconditioners for 3D acoustic and elastodynamic H-matrix accelerated Boundary Element Methods

“…Recently, BEM solvers have been speed up with acceleration techniques yielding to fast BEMs. One well-known fast BEM is the Fast Multipole accelerated BEM (FM-BEM) [4,5,6]. The Fast Multipole Method (FMM) [7,8] allows to compute efficiently the application of the integral operator to a given field.…”

An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation

“…The computational complexity of solving such a dense system using a direct method like a Gaussian elimination is O(N 3 ), whereas resolution via an iterative method like GMRES is O(N iter N 2 ) if N iter is the number of iterations. Many approaches have been proposed to speedup the iterative resolution of these dense systems [31,30,13,12], among which Fast Multipole Methods (FMMs) have enjoyed considerable success in mechanical engineering problems [18,62,14,48] by enabling a fast evaluation of the matrix-vector product required by the iterative solver. Initially developed for N-body simulations, the FMM has since been extended to oscillatory kernels [27,37,54] and thus expanded its efficacy to many applications.…”