A Fast Algorithm for the Inversion of the Biharmonic in Plate Dynamics Applications

EasyChair Preprint 10298

Abstract

In this paper, we present a numerical method for solving the biharmonic equation using finite difference methods, which can be used for fast acoustic simulation with nonlinear plate dynamics. With the simply supported boundary condition, the linear system could be regarded as a composition of two Poisson's equations, and these Poisson's equations are solved by the Thomas algorithm for a series of tridiagonal systems after transpositions and linear transformations for vectors in the systems and all non-empty blocks of the Laplacian matrix. We also point out that the eigendecomposition used for these linear transformations has a closed-form formula, which is easy to be pre-computed and also space-saving. Furthermore, since this solver is computed block by block and does not need sparse matrix operations, this method is good for single instruction multiple data (SIMD) parallelization using advanced vector extensions (AVX) intrinsics on central processing units (CPUs), which makes it possible to execute at fast speeds for real-time music applications. We also show that this solver for the simply supported boundary condition can also be easily adapted for other boundary conditions using Woodbury matrix identity with a little extra complexity. Numerical experiments show that the C++ implementation of this method is faster than decomposition-based solvers (like LU or Cholesky decomposition) of some well-known C++ libraries at the scale of applications in the field of musical acoustics.

Keyphrases: Biharmonic operator, SIMD intrinsics, musical acoustics, nonlinear plate dynamics, numerical methods

@booklet{EasyChair:10298,
  author    = {Zehao Wang and Miller Puckette},
  title     = {A Fast Algorithm for the Inversion of the Biharmonic in Plate Dynamics Applications},
  howpublished = {EasyChair Preprint 10298},
  year      = {EasyChair, 2023}}