127 lines
6.4 KiB
TeX
127 lines
6.4 KiB
TeX
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\documentclass[a4paper,12pt]{article}
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\usepackage{mathbbol} %poor substitute for mathbb. Changed to mathbbg to have an option for greek bb
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\newcommand{\ie}{i.e.}
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\newcommand{\eg}{e.g.}
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\newcommand{\field}[1]{\ensuremath{\mathcal{#1}}}
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\newcommand{\tnsr}[1]{\ensuremath{\bm{{#1}}}}
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\newcommand{\tnsrfour}[1]{\ensuremath{\mathbb{#1}}}
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\newcommand{\abs}[1]{\ensuremath{\left|{#1}\right|}}
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\newcommand{\norm}[1]{\ensuremath{\left|\left|{#1}\right|\right|}}
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\newcommand{\vctr}[1]{\ensuremath{\bm{#1}}}
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\newcommand{\inc}[1]{\ensuremath{{\rm d}{#1}}}
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\newcommand{\sign}[1]{\ensuremath{\operatorname{sign}\left({#1}\right)}}
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\newcommand{\grad}[1][]{\ensuremath{\operatorname{grad}{#1}}}
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\newcommand{\divergence}[1][]{\ensuremath{\operatorname{div}{#1}}}
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\newcommand{\totalder}[2]{\ensuremath{\frac{\inc{#1}}{\inc{#2}}}}
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\newcommand{\partialder}[2]{\ensuremath{\frac{\partial{#1}}{\partial{#2}}}}
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\newcommand{\identity}{\ensuremath{\tnsr I}}
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\newcommand{\F}[1][]{\ensuremath{\tnsr F^{{\rm #1}}}}
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\newcommand{\Favg}{{\ensuremath{\overline{\F}^{(n)}}}}
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\newcommand{\Fp}[1][]{\ensuremath{\tnsr F_\text{p}^{#1}}}
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\newcommand{\Fe}[1][]{\ensuremath{\tnsr F_\text{e}^{#1}}}
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\newcommand{\Lp}{\ensuremath{\tnsr L_\text{p}}}
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%\newcommand{\Q}[1]{\ensuremath{\tnsr Q^{(#1)}}}
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%\newcommand{\x}[2][]{\ensuremath{\vctr x^{(#2)}_\text{#1}}}
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%\newcommand{\y}[2][]{\ensuremath{\vctr x^{(#2)}_\text{#1}}}
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\newcommand{\dg}[2][]{\ensuremath{\Delta\vctr g^{(#2)}_\text{#1}}}
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\newcommand{\g}[1][]{\ensuremath{\vctr g_\text{#1}}}
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%\newcommand{\A}[2][]{\ensuremath{A^{(#2)}_\text{#1}}}
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\newcommand{\N}[2]{\ensuremath{\varrho^{(#1)}_\text{#2}}}
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\newcommand{\Burgers}[1]{\ensuremath{\vctr s^{(#1)}}}
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\newcommand{\n}[1]{\ensuremath{\vctr n^{(#1)}}}
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\newcommand{\m}[2]{\ensuremath{\vctr m^{(#1)}_{#2}}}
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\newcommand{\flux}[2]{\ensuremath{\vctr f^{(#1)}_ \text{#2}}}
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\newcommand{\averageflux}[2]{\ensuremath{\overline{\vctr f}^{(#1)}_ \text{#2}}}
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\newcommand{\interfaceflux}[2]{\ensuremath{\tilde{\vctr f}^{(#1)}_ \text{#2}}}
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\newcommand{\transmissivity}[1]{\ensuremath{\chi^{(#1)}}}
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\newcommand{\galpha}{\ensuremath{\gamma^{(\alpha)}}}
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\newcommand{\taualpha}{\ensuremath{\tau^{(\alpha)}}}
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\newcommand{\taualphamax}{\ensuremath{\hat\tau^{(\alpha)}}}
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\newcommand{\densityfunc}[2]{\ensuremath{{\tilde\varrho}^{(#1)}_ \text{#2}}}
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\newcommand{\avgdensity}[2]{\ensuremath{{\overline\varrho}^{(#1)}_ \text{#2}}}
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\newcommand{\dotdensity}[2]{\ensuremath{\dot{\varrho}^{(#1)}_ \text{#2}}}
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\newcommand{\densityexcess}[2]{\ensuremath{\Delta\varrho^{(#1)}_ \text{#2}}}
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\newcommand{\cs}[2][]{\ensuremath{\sigma^{(#1)}_ \text{#2}}}
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\title{Geometry reconstruction using Fast Fourier transform}
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\date{\today}
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\author{Martin Diehl}
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\begin{document}
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\maketitle
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The presented method allows the shape reconstruction of a volume element with periodic boundary conditions.
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The deformation gradient on each point of a regular, three-dimensional lattice in undeformed configuration must be known.
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The deformation gradient maps each point (or a line in the infinitesimal neighborhood of the point) into the current configuration.
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It is defined as:
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\begin{equation}
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\F(\vctr x) = \left(
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\begin{array}{ccc}
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\frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \frac{\partial y_1}{\partial x_3} \\
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\frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \frac{\partial y_2}{\partial x_3} \\
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\frac{\partial y_3}{\partial x_1} & \frac{\partial y_3}{\partial x_2} & \frac{\partial y_3}{\partial x_3} \\
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\end{array}
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\right)
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\end{equation}
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with \vctr y\ are the coordinates in current configuration and \vctr x\ are the coordinates in reference configuration.
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The three-dimensional field of second order tensors is transformed to the Fourier space, giving three-dimensional field of second order tensors that depend on the three dimensional wave vector and not on the vector of spatial coordinates:
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\begin{equation}
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\mathcal F \left( \F(\vctr x) \right)= \hat{\F}(\vctr k)
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\end{equation}
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Integration in Fourier space works is defined for the one dimensional case as:
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\begin{equation}
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\mathcal{F} \left( \int_{-\infty}^{x} g (\tau) d \tau \right) = \frac{\mathcal{F}{g(x)}}{i2 \pi k} + c \delta(k)
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\end{equation}
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where the last term is the integration constant.
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Constant (or linear after integration) terms cannot properly handled when using the integration property of the Fourier domain, as a division by the ``constant wave'' ($k=0$) is not possible.
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Thus, to carry out the integration the function is separated into an average and a locally fluctuating part.
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The locally fluctuating part is integrated in Fourier space, while the integration of the constant part is easier in the real domain.
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The fluctuation field of the position vector in deformed configuration in Fourier space reads as:
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\begin{equation}
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\hat{\tilde{y}}_{\rm j}(\vctr k) = \hat{F_{\rm ji}}(\vctr k) \left(k_{\rm i} i 2 \pi \right)^{-1} \forall k_i \neq 0
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\end{equation}
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The average part is set to zero in Fourier space.
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The inverse Fourier transform gives the locally fluctuating part of each position in current configuration:
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\begin{equation}
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\mathcal{F}^{-1}\left(\hat{\tilde{\vctr y}}(\vctr k) \right) = \tilde{\vctr y}(\vctr x)
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\end{equation}
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and the position vector in undeformed configuration is given as:
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\begin{equation}
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\vctr y(\vctr x) = \overline{\F}: \vctr x + \tilde{\vctr y}(\vctr x)
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\end{equation}
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\end{document}
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