Optimization based determination of highly absorbing boundary conditions for linear finite difference schemes

Many wave propagation problems (in acoustics or in railway catenary or cable car dynamics, for example) can be solved with high efficiency if the computational domain can be truncated to a small region of interest with appropriate absorbing boundary conditions. In this paper, highly absorbing and stable boundary conditions for linear using a flexible, optimization-based formulation. The proposed optimization approach to the computation of the absorbing boundary conditions is capable of optimizing the accuracy (the absorbing quality of the boundary condition) while guaranteeing stability of the discretized partial differential equations with the absorbing boundary conditions in place. Penalty functions are proposed that explicitly quantify errors introduced by the boundary condition on the solution of the bounded domain compared to the solution of the unbounded domain problem. Together with the stability condition the described approach can be applied on various types of linear partial differential equations and is thus applicable for generic wave propagation problems. Its flexibility and efficiency is demonstrated for two engineering problems: The Euler–Bernoulli beam under axial load, which can be used to model cables as well as catenary flexural dynamics, and a twodimensional wave as commonly encountered in acoustics. The accuracy of the absorbing boundary conditions obtained by the proposed concept is compared to analytical absorbing boundary conditions. & 2015 Elsevier Ltd. All rights reserved.