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Author: Ehrhardt, Matthias × End date: 2009 × Start date: 2000 × Has files: Yes × Type: Text × Subject: Floquet-Bloch theory ×

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    Evaluation of exact boundary mappings for one-dimensional semiinfinite periodic arrays
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Sun, Jiguang; Zheng, Chunxiong
    Periodic arrays are structures consisting of geometrically identical subdomains, usually called periodic cells. In this paper, by taking the Helmholtz equation as a model, we consider the definition and evaluation of the exact boundary mappings for general one-dimensional semi-infinite periodic arrays for any real wavenumber. The well-posedness of the Helmholtz equation is established via the limiting absorption principle. An algorithm based on the doubling procedure and extrapolation technique is proposed to derive the exact Sommerfeld-to-Sommerfeld boundary mapping. The advantages of this algorithm are the robustness and simplicity of implementation. But it also suffers from the high computational cost and the resonance wave numbers. To overcome these shortcomings, we propose another algorithm based on a conjecture about the asymptotic behaviour of limiting absorption principle solutions. The price we have to pay is the resolution of two generalized eigenvalue problems, but still the overall computational cost is significantly reduced. Numerical evidences show that this algorithm presents theoretically the same results as the first algorithm. Moreover, some quantitative comparisons between these two algorithms are given.
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    Numerical simulation of waves in periodic structures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Han, Houde; Zheng, Chunxiong
    In this work we present a new numerical technique for solving periodic structure problems. This new approach possesses several advantages. First, it allows for a fast evaluation of the Robin-to-Robin operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. Our strategy is an improvement of the recently developed recursive doubling process by Yuan and Lu. In this paper we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and finally the Helmholtz equation with damping.