Multilevel fast multipole algorithm for elastic wave scattering by large three-dimensional objects

Mei Song Tong, Weng Cho Chew

Research output: Contribution to journalArticle


Multilevel fast multipole algorithm (MLFMA) is developed for solving elastic wave scattering by large three-dimensional (3D) objects. Since the governing set of boundary integral equations (BIE) for the problem includes both compressional and shear waves with different wave numbers in one medium, the double-tree structure for each medium is used in the MLFMA implementation. When both the object and surrounding media are elastic, four wave numbers in total and thus four FMA trees are involved. We employ Nyström method to discretize the BIE and generate the corresponding matrix equation. The MLFMA is used to accelerate the solution process by reducing the complexity of matrix-vector product from O (N2) to O (N log N) in iterative solvers. The multiple-tree structure differs from the single-tree frame in electromagnetics (EM) and acoustics, and greatly complicates the MLFMA implementation due to the different definitions for well-separated groups in different FMA trees. Our Nyström method has made use of the cancellation of leading terms in the series expansion of integral kernels to handle hyper singularities in near terms. This feature is kept in the MLFMA by seeking the common near patches in different FMA trees and treating the involved near terms synergistically. Due to the high cost of the multiple-tree structure, our numerical examples show that we can only solve the elastic wave scattering problems with 0.3-0.4 millions of unknowns on our Dell Precision 690 workstation using one core.

Original languageEnglish (US)
Pages (from-to)921-932
Number of pages12
JournalJournal of Computational Physics
Issue number3
StatePublished - Feb 20 2009


  • Boundary integral equation
  • Elastic wave scattering
  • Multilevel fast multipole algorithm

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Physics and Astronomy (miscellaneous)
  • Physics and Astronomy(all)
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

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