Connection and comparison between frequency shift time integration and a spectral transformation preconditioner

Abstract

The numerical study of exterior acoustics problems is usually carried out in the frequency domain. Finite element analyses often require the solution of large-scale algebraic linear systems. For very large problems, sometimes the time domain is used. Implicit time integration requires linear system solves, but these are often far easier than those from the frequency domain. This paper shows a connection between a spectral transformation preconditioner and a frequency shift time integration. This preconditioner is close to the shifted Laplace preconditioner. The preconditioned iterative method appears to be faster than time integration.

3 Case studies of model order reduction for acoustics and vibrations

“…For undamped acoustics (C = 0), the incomplete factorization was applied to K + α 2 M [33] with an optimal choice of α, which is currently known as the shifted Laplace preconditioner. For damped acoustics with nonzero C, one could apply the preconditioner to K + ωC + ωC + 2 ω 2 M (e. g., see [36]). We now present numerical examples to illustrate properties of direct methods, i. e., methods based on a sparse lower-upper (LU) factorization, and preconditioned Krylov solvers.…”

“…This shows that the factorization cost is highly dominant. Consider another example from [36], which is a finite element problem with spherical infinite elements. The mesh is given in Figure 3.2b.…”

“…Timings on a Dell Latitude 6400 in 2009 showed that the direct solve with MUMPS required 119.13 seconds (LU factorization and forward and backward solve). BiCGStab with the ILU preconditioner from [36] required 52.20 seconds at a frequency of 200 Hz and 103.71 seconds at 500 Hz. It is well known that iterative methods perform typically badly for higher frequencies, which is confirmed by this experiment.…”

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

“…The first preconditioners of this kind were the Laplacian and the positively shifted Laplacian introduced in [Bayliss et al 1983], later generalised to complex-valued shifts [Erlangga et al 2004;Erlangga et al 2006]. Alternative preconditioners and solution methods are derived from frequency shift time integration [Meerbergen and Coyette 2009], moving perfectly matched layers [Engquist and Ying 2011], a transformation of the Helmholtz equation to a reaction-advection-diffusion problem [Haber and MacLachlan 2011], separation of variables [Plessix and Mulder 2003], the wave-ray approach [Brandt and Livshits 1997], Krylov subspace methods as smoother substitute [Elman et al 2001], or algebraic multilevel methods [Bollhöfer et al 2009;Tsuji and Tuminaro 2015]. This list is not comprehensive.…”

A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation

“…An alternative preconditioner that, in addition to shift, also scales the Laplacian was derived from frequency shift time integration by Meerbergen and Coyette [17]. By appropriately choosing the shift and the scale it is possible to restrict the spectrum of precondi-tioning matrix into one quadrant of the complex plane.…”