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- Committer: Harish Narayanan
- Date: 2010-01-25 17:42:28 UTC
- Revision ID: hnarayanan@gmail.com-20100125174228-zcx2ctvhiu7qc5g3
Massive simplification of the bibtex file
- files added:
- auxiliary/sample-appendix.tex
- files removed:
- auxiliary/additional-proofs.tex
- files modified:
- Makefile
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auxiliary/supplementary-considerations.tex
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\chapter{Supplementary topics}
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\label{supplementary-considerations}
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\section{Frame invariance and flux contribution from acceleration}
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\label{acceleration-objectivity}
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In our earlier treatment \citep{growthpaper}, the constitutive
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relation for the fluid flux had a driving force contribution arising
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from the acceleration of the solid phase,
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$-\rho_0^\mathrm{f}\bF^{\mathrm{T}}\frac{\partial \bV}{\partial t}$.
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This term, being motivated by the reduced dissipation inequality, does
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not violate the Second Law and supports an intuitive understanding
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that the acceleration of the solid skeleton in one direction must result in
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an inertial driving force on the fluid in the opposite
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direction. However, as defined, this acceleration is obtained by the
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time differentiation of kinematic quantities,\footnote{And not in terms
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of acceleration {\em relative to fixed stars} for e.g., as discussed
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in \cite[][Page 43]{TruesdellNoll:65}.} and does not transform in a
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frame-indifferent manner. Unlike the superficially similar term
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arising from the gravity vector,\footnote{Where every observer has an
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implicit knowledge of the directionality of the field relative to a
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fixed frame, allowing it to transform objectively. Specifically, under
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a time-dependent rigid body motion imposed on the current
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configuration carrying $\bx$ to $\bx^+ = \bc(t) + \bQ(t)\bx$, where
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$\bc(t) \in \mathbb{R}^3$ and $\bQ(t) \in \mbox{SO}(3)$, it is
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understood that the acceleration due to gravity in the transformed
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frame is $\bg^+ = \bQ^\mathrm{T}\bg$ and is therefore
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frame-invariant. However, $\ba^+ = \ddot{\bc} + 2\dot{\bQ}\bv +
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\ddot{\bQ}\bx + \bQ\ba$ , and is therefore not frame-invariant.} the
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acceleration
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term presents an improper dependence on the frame of the
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observer. Thus, its use in constitutive relations is inappropriate,
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and the term has been dropped in \mbox{Equation (\ref{fluidflux})}.
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%% \todo{Cite Einstein's general relativity here.}
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\section{Stabilisation of the simplified solute transport equation}
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\label{stabilisation-solute-transport}
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In weak form, the SUPG-stabilised method \citep{Paper6} for
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Equation~(\ref{morestdform}) is,
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\begin{equation}
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\begin{split}
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&\int_{\Omega} w^{\mathrm{h}} \left(
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\frac{\mathrm{d}\rho^{\mathrm{s}^{h}}}{\mathrm{d}t} +
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\bm^f\cdot\mathrm{grad}\left[\frac{
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\rho^{\mathrm{s}^{h}}}{\rho^f}\right] \right)
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d\Omega\\ &+\int_{\Omega} \left( \mathrm{grad}
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\left[w^{\mathrm{h}}\right] \cdot \bar{\bD^\mathrm{s}} \mathrm{grad}
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\left[ \rho^{\mathrm{s}^{h}}\right] \right)\ d\Omega\\ +&
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\sum_{\mathrm{e}=1}^{\mathrm{n_{el}}} \int_{\Omega_{\mathrm{e}}}
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\tau \frac{\bm^{f}}{\rho^f} \cdot \mathrm{grad}
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\left[w^{\mathrm{h}}\right] \left(
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\frac{\mathrm{d}\rho^{\mathrm{s}^{h}}}{\mathrm{d}t} +
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\bm^f\cdot\mathrm{grad}\left[\frac{
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\rho^{\mathrm{s}^{h}}}{\rho^f}\right] \right) \ d\Omega\\ -&
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\sum_{\mathrm{e}=1}^{\mathrm{n_{el}}} \int_{\Omega_{\mathrm{e}}}
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\tau \frac{\bm^{f}}{\rho^f} \cdot \mathrm{grad}
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\left[w^{\mathrm{h}}\right]
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\left(\mathrm{div}\left[\bar{\bD^\mathrm{s}}\ \mathrm{grad} \left[
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\rho^{\mathrm{s}^{h}}\right]\right]\right) \ d\Omega\\ = &
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\int_{\Omega} w^{\mathrm{h}} \pi^\mathrm{s} \ d\Omega +
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\int_{\Gamma_{\mathrm{h}}} w^{\mathrm{h}} h \ d\Gamma\\ +&
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\sum_{\mathrm{e}=1}^{\mathrm{n_{el}}} \int_{\Omega_{\mathrm{e}}}
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\tau \frac{\bm^{f}}{\rho^f} \cdot \mathrm{grad}
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\left[w^{\mathrm{h}}\right] \pi^\mathrm{s} \ d\Omega,
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\label{stabilizedmassbal}
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\end{split}
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\end{equation}
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\noindent where quantities with the superscript $\mathrm{h}$ represent
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finite-di\-men\-sion\-al approximations of infinite-dimensional field
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variables, $\Gamma_{\mathrm{h}}$ is the Neumann boundary, and this
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equation introduces a numerical stabilisation parameter, $\tau$, which
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we calculate from the $\mathrm{L}_{2}$~norms of element level
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matrices, as described in \cite{tezduyarsupg}.
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