From 50651df99b3b0a6a4685f157b7652d55f49955c3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Marek=20Ne=C4=8Dada?= Date: Mon, 29 Jul 2019 15:24:16 +0300 Subject: [PATCH] Finite systems theory almost done. Former-commit-id: 179d3ac047b53e1f670619409036c8136c6d0f26 --- lepaper/arrayscat.lyx | 24 ++++++ lepaper/finite.lyx | 186 +++++++++++++++++++++++++++++++++++++++++- 2 files changed, 209 insertions(+), 1 deletion(-) diff --git a/lepaper/arrayscat.lyx b/lepaper/arrayscat.lyx index f00e958..9262776 100644 --- a/lepaper/arrayscat.lyx +++ b/lepaper/arrayscat.lyx @@ -274,6 +274,16 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\rcoeff}{a} +\end_inset + + +\begin_inset FormulaMacro +\newcommand{\rcoeffinc}{a^{\mathrm{inc.}}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\rcoeffptlm}[4]{\rcoeffp{#1,#2#3#4}} \end_inset @@ -294,6 +304,11 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\outcoeff}{f} +\end_inset + + \begin_inset FormulaMacro \newcommand{\outcoeffp}[1]{f_{#1}} \end_inset @@ -354,6 +369,11 @@ \end_inset +\begin_inset FormulaMacro +\newcommand{\truncate}[2]{\left[#1\right]_{#2}} +\end_inset + + \begin_inset FormulaMacro \newcommand{\dlmfFer}[2]{\mathsf{P}_{#1}^{#2}} \end_inset @@ -673,6 +693,10 @@ Consistent notation of balls. Abstract. \end_layout +\begin_layout Itemize +Truncation notation. +\end_layout + \begin_layout Itemize Example results. \end_layout diff --git a/lepaper/finite.lyx b/lepaper/finite.lyx index 9e8f2f6..00cc2eb 100644 --- a/lepaper/finite.lyx +++ b/lepaper/finite.lyx @@ -1014,7 +1014,7 @@ jaksetometuje? Then the EM field inside each such volume can be expanded in a way similar to \begin_inset CommandInset ref -LatexCommand ref +LatexCommand eqref reference "eq:E field expansion" plural "false" caps "false" @@ -1083,6 +1083,190 @@ noprefix "false" \end_inset . + For each scatterer, we also have its +\begin_inset Formula $T$ +\end_inset + +-matrix relation as in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:T-matrix definition" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, +\begin_inset Formula +\[ +\outcoeffp q=T_{q}\rcoeffp q. +\] + +\end_inset + +Together with +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:particle total incident field coefficient a" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, this gives rise to a set of linear equations +\begin_inset Formula +\begin{equation} +\outcoeffp p-T_{p}\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pq\outcoeffp q=T_{p}\rcoeffincp p,\quad p\in\mathcal{P}\label{eq:Multiple-scattering problem} +\end{equation} + +\end_inset + +which defines the multiple-scattering problem. + If all the +\begin_inset Formula $p,q$ +\end_inset + +-indexed vectors and matrices (note that without truncation, they are infinite-d +imensional) are arranged into blocks of even larger vectors and matrices, + this can be written in a short-hand form +\begin_inset Formula +\begin{equation} +\left(I-T\trops\right)\outcoeff=T\rcoeffinc\label{eq:Multiple-scattering problem block form} +\end{equation} + +\end_inset + +where +\begin_inset Formula $I$ +\end_inset + + is the identity matrix and +\begin_inset Formula $T$ +\end_inset + +is a block-diagonal matrix containing all the individual +\begin_inset Formula $T$ +\end_inset + +-matrices. +\end_layout + +\begin_layout Standard +In practice, the multiple-scattering problem is solved in its truncated + form, in which all the +\begin_inset Formula $l$ +\end_inset + +-indices related to a given scatterer +\begin_inset Formula $p$ +\end_inset + + are truncated as +\begin_inset Formula $l\le L_{p}$ +\end_inset + +, laeving only +\begin_inset Formula $N_{p}=2L_{p}\left(L_{p}+2\right)$ +\end_inset + + different +\begin_inset Formula $\tau lm$ +\end_inset + +-multiindices left. + The truncation degree can vary for different scatterers (e.g. + due to different physical sizes), so the truncated block +\begin_inset Formula $\tropsp pq$ +\end_inset + + has shape +\begin_inset Formula $N_{p}\times N_{q}$ +\end_inset + +, not necessarily square. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Such truncation of the translation operator +\begin_inset Formula $\tropsp pq$ +\end_inset + + is justified by the fact on the left, TODO +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +If no other type of truncation is done, there remain +\begin_inset Formula $2L_{p}\left(L_{p}+2\right)$ +\end_inset + + different +\begin_inset Formula $\tau lm$ +\end_inset + +-multiindices for +\begin_inset Formula $p$ +\end_inset + +-th scatterer, so that the truncated version of the matrix +\begin_inset Formula $\left(I-T\trops\right)$ +\end_inset + + is a square matrix with +\begin_inset Formula $\left(\sum_{p\in\mathcal{P}}N_{p}\right)^{2}$ +\end_inset + + elements in total. + The truncated problem +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:Multiple-scattering problem block form" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + can then be solved using standard numerical linear algebra methods. +\end_layout + +\begin_layout Standard +Alternatively, the multiple scattering problem can be formulated in terms + of the regular field expansion coefficients, +\begin_inset Formula +\begin{align*} +\rcoeffp p-\sum_{q\in\mathcal{P}\backslash\left\{ p\right\} }\tropsp pqT_{q}\rcoeffp q & =\rcoeffincp p,\quad p\in\mathcal{P},\\ +\left(I-\trops T\right)\rcoeff & =\rcoeffinc, +\end{align*} + +\end_inset + +but this form is less suitable for numerical calculations due to the fact + that the regular VSWF expansion coefficients on both sides of the equation + are typically non-negligible even for large multipole degree +\begin_inset Formula $l$ +\end_inset + +, and the truncation is not justified in this case. + +\begin_inset Note Note +status open + +\begin_layout Plain Layout +TODO less bulshit. +\end_layout + +\end_inset + + \end_layout \begin_layout Subsubsection