Abstract
SummaryThe variational theory of complex rays (VTCR) is an indirect Trefftz method designed to study systems governed by Helmholtz‐like equations. It uses wave functions to represent the solution inside elements, which reduces the dispersion error compared with classical polynomial approaches, but the resulting system is prone to be ill‐conditioned. This paper gives a simple and original presentation of the VTCR using the discontinuous Galerkin framework, and it traces back the ill‐conditioning to the accumulation of eigenvalues near zero for the formulation written in terms of wave amplitude. The core of this paper presents an efficient solving strategy that overcomes this issue. The key element is the construction of a search subspace where the condition number is controlled at the cost of a limited decrease of attainable precision. An augmented LSQR solver is then proposed to solve efficiently and accurately the complete system. The approach is successfully applied to different examples. Copyright © 2015 John Wiley & Sons, Ltd.
Highlights
In the last decades, the use of numerical simulation techniques in design, analysis and optimization of systems has become an indispensable part of the industrial design process
Many weak formulations of this system can be proposed, see [3] for a review, in this paper we investigate the Variational Theory of Complex Rays (VTCR) which can be viewed as an indirect Trefftz method applied within a discontinuous-Galerkin framework
We evaluate the classical VTCR equipped with various solvers: Matlab’s direct solver “\”, Moore-Penrose pseudo inverse (Matlab’s “pinv”); and we compare them to the new reduced approach used either directly with Matlab’s solver “\” (O-VTCR, which in that case corresponds to a Cholesky solver) or used as the augmented LSQR approach (A-VTCR), both for β of equation (33) equal to β = 0.25, 0.1, 10−4, 10−6
Summary
The use of numerical simulation techniques in design, analysis and optimization of systems has become an indispensable part of the industrial design process. The most used computer aided engineering tool is the standard Galerkin Finite Element Method (FEM [1]) It is based on the use of continuous, piecewisepolynomial shape functions supported by a mesh. The subspace that we build is quasi-optimal in the sense that, starting from an initial search space, it is a controlled approximation of the largest subspace where a chosen level of coercivity (or condition number) is preserved Note that this strategy is different from previous work as [14], it does not set up an upper limit for the number of waves, but selects all combinations of waves which result in pressure fields containing at least a certain amount of energy. The first one (Section 5) possesses an analytical solution which enables us to fully understand the properties of the selected subspace and of the augmented solver; the second one (Section 6) is more realistic and enables us to illustrate the potential of the method in terms of computational performance
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