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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Perfectly matched layer ： ウィキペディア英語版 Perfectly matched layer A perfectly matched layer (PML) is an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods to simulate problems with open boundaries, especially in the FDTD and FE methods.〔S. G. Johnson, (Notes on Perfectly Matched Layers ), online MIT course notes (Aug. 2007).〕 The key property of a PML that distinguishes it from an ordinary absorbing material is that it is designed so that waves incident upon the PML from a non-PML medium do not reflect at the interface—this property allows the PML to strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior. PML was originally formulated by Berenger in 1994 for use with Maxwell's equations, and since that time there have been several related reformulations of PML for both Maxwell's equations and for other wave-type equations, such as elastodynamics, the linearized Euler equations, Helmholtz equations, and poroelasticity. Berenger's original formulation is called a split-field PML, because it splits the electromagnetic fields into two unphysical fields in the PML region. A later formulation that has become more popular because of its simplicity and efficiency is called uniaxial PML or UPML, in which the PML is described as an artificial anisotropic absorbing material. Although both Berenger's formulation and UPML were initially derived by manually constructing the conditions under which incident plane waves do not reflect from the PML interface from a homogeneous medium, ''both'' formulations were later shown to be equivalent to a much more elegant and general approach: stretched-coordinate PML. In particular, PMLs were shown to correspond to a coordinate transformation in which one (or more) coordinates are mapped to complex numbers; more technically, this is actually an analytic continuation of the wave equation into complex coordinates, replacing propagating (oscillating) waves by exponentially decaying waves. This viewpoint allows PMLs to be derived for inhomogeneous media such as waveguides, as well as for other coordinate systems and wave equations. ==Technical description== Specifically, for a PML designed to absorb waves propagating in the ''x'' direction, the following transformation is included in the wave equation. Wherever an ''x'' derivative $\backslash partial/\backslash partial\; x$ appears in the wave equation, it is replaced by: :$\backslash frac\; \backslash to\; \backslash frac\}\; \backslash frac$ where $\backslash omega$ is the angular frequency and $\backslash sigma$ is some function of ''x''. Wherever $\backslash sigma$ is positive, propagating waves are attenuated because: :$e^\; \backslash to\; e^\; \backslash int^x\; \backslash sigma(x\text{'})\; dx\text{'}\}\; ,$ where we have taken a planewave propagating in the +''x'' direction (for $k\; >\; 0$) and applied the transformation (analytic continuation) to complex coordinates: $x\; \backslash to\; x\; +\; \backslash frac\; \backslash int^x\; \backslash sigma(x\text{'})\; dx\text{'}$, or equivalently $dx\; \backslash to\; dx\; (1\; +\; i\backslash sigma/\backslash omega)$. The same coordinate transformation causes waves to attenuate whenever their ''x'' dependence is in the form $e^$ for some propagation constant ''k'': this includes planewaves propagating at some angle with the ''x'' axis and also transverse modes of a waveguide. The above coordinate transformation can be left as-is in the transformed wave equations, or can be combined with the material description (e.g. the permittivity and permeability in Maxwell's equations) to form a UPML description. Note also that the coefficient σ/ω depends upon frequency—this is so the attenuation rate is proportional to ''k''/ω, which is dependent of frequency in a homogeneous material (not including material dispersion, e.g. for vacuum) because of the dispersion relation between ω and ''k''. However, this frequency-dependence means that a time domain implementation of PML, e.g. in the FDTD method, is more complicated than for a frequency-independent absorber, and involves the auxiliary differential equation (ADE) approach (equivalently, ''i''/ω appears as an integral or convolution in time domain). Perfectly matched layers, in their original form, only attenuate propagating waves; purely evanescent waves (exponentially decaying fields) oscillate in the PML but do not decay more quickly. However, the attenuation of evanescent waves can also be accelerated by including a real coordinate stretching in the PML: this corresponds to making σ in the above expression a complex number, where the imaginary part yields a real coordinate stretching that causes evanescent waves to decay more quickly. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Perfectly matched layer」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.