\NeedsTeXFormat{LaTeX2e} \ProvidesPackage{macros}[2014/02/04 reza macros] %%%%%%%%%%%%% four subdivision subsubsubsection = paragraph: %\setcounter{secnumdepth}{4} %\titleformat{\paragraph} %{\normalfont\normalsize\bfseries}{\theparagraph}{1em}{} %\titlespacing*{\paragraph}{0pt}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex} %\usepackage{graphicx} \usepackage{amssymb,amsfonts,stmaryrd} \usepackage{amsmath} \usepackage{dcolumn}% Align table columns on decimal point \usepackage{bm} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{isomath} \usepackage{mathrsfs} % %\usepackage[pdftex]{graphicx,color} %\usepackage[pdftex]{hyperref} \usepackage{subfigure} %\usepackage{cite} \usepackage{xspace} \usepackage{verbatim} \usepackage{undertilde} \usepackage{afterpage} \usepackage{accents} %\usepackage{psfrag} %\usepackage{afterpage} %\usepackage{undertilde} %\newcommand{\eqncntr}{section} %\DeclareBoolOption{OMLmathbf} \newcommand{\stfm}{Conductivity Matrix } %\newcommand{\stfm}{Stiffness Matrix } %\setcounter{secnumdepth}{5} %\setcounter{tocdepth}{5} \DeclareMathOperator{\sym}{sym} %\newtheorem{lemma}{Lemma}[\eqncntr] %\newtheorem{definition}{Definition}[\eqncntr] %\newtheorem{proposition}{Proposition}[\eqncntr] %\newtheorem{corollary}{Corollary}[\eqncntr] %\newtheorem{problem}{Problem}[\eqncntr] %\newtheorem{remark}{Remark}[\eqncntr] %\newtheorem{example}{Example}[\eqncntr] %\newtheorem{theorem}{Theorem}[\eqncntr] %\renewcommand{\thetheorem}{\arabic{theorem}} %\renewcommand{\theremark}{\arabic{remark}} %\renewcommand{\theproblem}{\arabic{problem}} %\renewcommand{\thedefinition}{\arabic{definition}} %\renewcommand{\theproposition}{\arabic{proposition}} %\renewcommand{\thecorollary}{\arabic{corollary}} %\renewcommand{\theexample}{\arabic{example}} %\renewcommand{\thelemma}{\arabic{lemma}} %%\renewcommand{\the}{\arabic{}} %\newcommand{\bt}{\begin{theorem}} %\newcommand{\et}{\end{theorem}} %\newcommand{\bd}{\begin{definition}} %\newcommand{\ed}{\end{definition}} %\newcommand{\bl}{\begin{lemma}} %\newcommand{{\end{lemma}} %\newcommand{\bpr}{\begin{proposition}} %\newcommand{\epr}{\end{proposition}} %\newcommand{\bc}{\begin{corollary}} %\newcommand{\ec}{\end{corollary}} %\newcommand{\br}{\begin{remark}} %\newcommand{\er}{\end{remark}} %\newtheorem{problem}{Problem} % \newcommand{\vc}[1]{\ensuremath{\mathbf{#1}}} % vector field % \newcommand{\tns}[1]{\ensuremath{\mathbf{#1}}} % tensor field % \newcommand{\fm}[1]{\ensuremath{\bm{#1}}} % form \newcommand{\fms}[1]{\ensuremath{\bm{#1}}} % form with scalar coefficients \newcommand{\fmv}[1]{\ensuremath{\bm{#1}}} % form with vector coefficients \newcommand{\fmt}[1]{\ensuremath{\bm{#1}}} % form with tensor coefficients % \newcommand{\wt}[1]{\hat{#1}} % \newcommand{\eps}{\bm{\varepsilon}} % %\newcommand{\tnssigma}{\bm \upsigma} % \newcommand{\stress}{s} \newcommand{\tnssigma}{\tns \stress} % \newcommand{\insr}{\mathbf{i}} % % \newcommand{\sigR}[2]{(\sigma^{#1#2})^R} % \newcommand{\siga}[2]{(\sigma^{#1#2})^\alpha} % \newcommand{\sigb}[2]{(\sigma^{#1#2})^\beta} % % \newcommand{\bd}{\ensuremath{\mathbf{d}}} % \newcommand{\bmsf}[1]{\ensuremath{\bm{\mathsf{#1}}}} % \newcommand{\bdx}{\ensuremath{\mathbf{dx}}} % \newcommand{\doper}{\ensuremath{\mathrm{d}}} \newcommand{\dx}{\ensuremath{\mathrm{d}x}} \newcommand{\dxhat}{\ensuremath{\mathrm{d}\hat{\mathrm{x}}}} % \newcommand{\bdxhat}{\ensuremath{\mathbf{d}\hat{\mathbf{x}}}} % \newcommand{\dt}{\ensuremath{\mathrm{d}t}} %\newcommand{\bdt}{\ensuremath{\mathbf{dt}}} % \newcommand{\sdt}{{\ensuremath{\star \dt}}} % \newcommand{\sdx}{{\ensuremath{\star \dx}}} % %\newcommand{\bsdt}{\ensuremath{{\star \bdt}}} % \newcommand{\bsdx}{{\ensuremath{\star \bdx}}} % %\newcommand{\e}{\ensuremath{\mathbf{e}}} % \newcommand{\be}{\ensuremath{\vc{e}}} % \newcommand{\he}{\ensuremath{\hat{\varepsilon}}} % \newcommand{\hbe}{\ensuremath{\hat{\mathbf{\varepsilon}}}} % \newcommand{\pq}{\ensuremath{\partial \mathcal{Q}}} % \newcommand{\calV}{\ensuremath{\mathcal{V}}} % \newcommand{\VQh}{\ensuremath{\mathcal{V}^{\calQ}_h}} % \newcommand{\calQ}{\ensuremath{\mathcal{Q}}} % \newcommand{\pd}{\ensuremath{\partial \mathcal{D}}} % \newcommand{\pdb}{{\partial \calD^{\rm b}}} % \newcommand{\pdi}{{\partial \calD^{\rm i}}} % \newcommand{\pdf}{{\partial \calD^{\rm f}}} % \newcommand{\calD}{\ensuremath{\mathcal{D}}} % \newcommand{\calU}{\ensuremath{\mathcal{U}}} % \newcommand{\calP}{\ensuremath{\mathcal{P}}} % \newcommand{\calS}{\ensuremath{\mathcal{S}}} % \newcommand{\tf}{\ensuremath{t_{\rm f}}} % %\newcommand{\Df}{\ensuremath{\mathcal{D}_{\rm f}}} % %\newcommand{\gc}{\ensuremath{\Gamma^{\rm C}}} % \newcommand{\gc}{\ensuremath{\tilde{\Gamma}}} % RBH \newcommand{\gvert}{\ensuremath{\Gamma}} % RBH %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\newcommand{\cal}[1]{\ensuremath{\mathcal{#1}}} % % common textual idioms that are easy to get wrong (must include package xspace) \def\etal{\textit{et al.\@}\xspace} \def\cf{\textit{cf.\@}\xspace} \def\ie{\textit{i.e.,}\xspace} \def\eg{\textit{e.g.,}\xspace} \def\etc{\textit{etc.}\@\xspace} %\newcommand{\ETAL}{{et al.}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Local coordinate: %\newcommand{\lcOp}[1]{\underline{#1}} %RBH \newcommand{\lcOp}[1]{{#1}} %RBH %\newcommand{\lcOp}[1]{\underset{\sim}{#1}} %\newcommand{\localCrd}{\lcOp{x}} \newcommand{\localCrd}{\xi} \newcommand{\localCrdV}{\bm \localCrd} \newcommand{\unitvl}{\vc \lcOp{e}} \newcommand{\unitvld}[1]{\lcOp{\vc e}_{#1}} \newcommand{\unitvlu}[1]{\lcOp{\bm e}^{#1}} %RBH should be bold italic for basis covector %\newcommand{\unitvl}{\epsilon} %\newcommand{\bdxl}{\ensuremath{\mathbf{d} \upxi}} \newcommand{\bdxl}{\ensuremath{\mathbf{d\lcOp{x}}}} % \newcommand{\dxl}{\ensuremath{\mathrm{d}\localCrd}} %\newcommand{\dxil}[1]{\dxl^{#1}} %\newcommand{\dtl}{\dt} %\newcommand{\sdtl}{\sdt} \newcommand{\bsdxl}{{\ensuremath{\star \bdxl}}} % \newcommand{\sdxl}{{\star \dxl}} % \newcommand{\sdxOnel}{\sdxl^{1}} % \newcommand{\symgrad}{\ensuremath{\accentset{\diamond}{\nabla}}} %RBH \newcommand{\starred}[1]{\accentset{\star}{#1}} % RBH creates an accent based on \star (requires accents package) \newcommand{\dxtime}[1]{\ensuremath{#1\text{d}\times\text{time}}} \newcommand{\charval}{w} \newcommand{\charv}[1]{\charval^{#1}} %RBH \newcommand{\icbc}[1]{\bar{#1}} \newcommand{\barcharval}{\bar{\charval}} %RA \newcommand{\barcharv}[1]{\bar{\charval}^{#1}} \newcommand{\delayt}{\tau^\prime} %RBH \newcommand{\normalizeda}{\tau c_R} \newcommand{\tonset}{t_0} \newcommand{\vonset}{v_0} \newcommand{\sigmaij}[2]{\upsigma^{#1#2}} \newcommand{\sigmabarij}[2]{\underline{\upsigma}^{#1#2}} %RBH \newcommand{\sigmabarijzero}[2]{\underline{\upsigma}_0^{#1#2}} %RBH \newcommand{\tractionbar}{\breve{\sigma}} \newcommand{\tractionbaro}{\bar{\sigma}} \newcommand{\tractionbarLateral}{\bar{\pi}} \newcommand{\velbars}{\breve{v}} \newcommand{\sradiuss}{r} %RBH %\newcommand{\sradius}[1]{\sradiuss_{#1}} \newcommand{\velradiuss}{{\sradiuss_v}} %\newcommand{\velradius}[1]{\velradiuss_{#1}} %\newcommand{\velradiuss}{{r_v}} \newcommand{\vmaxSign}{\infty} \newcommand{\vmax}{v^\infty} \newcommand{\tracinf}{\bar{\trac}_\infty} \newcommand{\traccrk}{\bar{\trac}} %\newcommand{\lval}{\tractionbar} %\newcommand{\lrat}{\lval / \sigmac} \newcommand{\lval}{\bar{\sigma}} \newcommand{\lrat}{\sigma^\prime} \newcommand{\labct}{\mathbf{p}_\mathrm c} \newcommand{\posct}{\bm{x}_\mathrm c} %RBH \newcommand{\labpzt}{\mathbf{p}_\mathrm T} \newcommand{\pospzt}{\bm{x}_\mathrm T} %RBH \newcommand{\delpzt}{\delta_\mathrm T} \newcommand{\labpzl}{\mathbf{p}_\mathrm L} \newcommand{\pospzl}{\bm{x}_\mathrm L} %RBH \newcommand{\delpzl}{\delta_\mathrm L} \newcommand{\labctpzl}{\posct, \labpzlc} %\newcommand{\sigmaij}[2]{\upsigma_{#1#2}} %\newcommand{\alps}{\alpha_s} %\newcommand{\alpd}{\alpha_d} %\newcommand{\thetas}{\theta_s} %\newcommand{\thetad}{\theta_d} %\newcommand{\gammas}{\gamma_s} %\newcommand{\gammad}{\gamma_d} %\newcommand{\cs}{c_s} %\newcommand{\cd}{c_d} \newcommand{\alps}{\alpha_2} \newcommand{\alpd}{\alpha_1} \newcommand{\thetas}{\theta_2} \newcommand{\thetad}{\theta_1} \newcommand{\gammas}{\gamma_2} \newcommand{\gammad}{\gamma_1} \newcommand{\cs}{c_2} \newcommand{\cd}{c_1} \newcommand{\CS}{c_{\text s}} \newcommand{\CD}{c_{\text d}} %RBH \newcommand{\CR}{c_{\text R}} \newcommand{\mI}{I} \newcommand{\mII}{{I\hspace{-0.4ex}I}} \newcommand{\mIII}{{I\hspace{-0.4ex}I\hspace{-0.4ex}I}} \newcommand{\bareps}{\ensuremath{\bar\eps}} % Dissipation and Fracture energy \newcommand{\errD}{\epsilon_\mathrm{D}^{\mathrm Q}} %{e_\mathrm{D}^{\mathrm Q}} \newcommand{\ErrD}{\epsilon_\mathrm{D}} %{\varphi^{\mathrm Q}} \newcommand{\tolD}{\tilde{\epsilon}_\mathrm{D} } %{\varphi^*} \newcommand{\tolDU}{\underline{\epsilon}_\mathrm{D}} %{\underline{\varphi}} \newcommand{\tolDL}{\overline{\epsilon}_\mathrm{D}} %{\overline{\varphi}} \newcommand{\errC}{\epsilon_\mathrm{C}^{\mathrm Q}} %{e_\mathrm{C}^{\mathrm Q}} \newcommand{\ErrC}{\epsilon_\mathrm{C}} %{\varphi^{\mathrm Q}} %\newcommand{\ErrNC}{\overline{\epsilon}_\mathrm{C}} % total cohesive error \newcommand{\tolC}{\tilde{\epsilon}_\mathrm{C} } %{\varphi^*} \newcommand{\tolCU}{\overline{\epsilon}_\mathrm{C}} %{\overline{\varphi}} \newcommand{\tolCL}{\underline{\epsilon}_\mathrm{C}} %{\underline{\varphi}} \newcommand{\powMNBase}[1]{-{#1}} %{1 \me-{#1}~\mathrm J} \newcommand{\powM}[1]{10^{-{#1}}} %{1 \me-{#1}~\mathrm J} \newcommand{\powMJ}[1]{10^{-{#1}} ~\mathrm J} %\newcommand{\implicitRelation}{31} \newcommand{\sigmaNC}{\ \sigmac} %RBH \newcommand{\sigmaTC}{\tau_\mathrm{C}} \newcommand{\sigmac}{\ensuremath{\tilde{\sigma}}} %RBH %\newcommand{\sigmac}{\sigma_\mathrm{C}} \newcommand{\momentumc}{\ensuremath{\tilde{m}}} %\newcommand{\sigmaci}[1]{{\sigma_\mathrm{C}}_{#1}} %\newcommand{\deltac}{\delta_\mathrm{C}} \newcommand{\deltac}{\ensuremath{\tilde{\delta}}} %RBH \newcommand{\phic}{\ensuremath{\tilde{\phi}}} %RBH \newcommand{\tauc}{\ensuremath{\tilde{\tau}}} \newcommand{\velc}{\ensuremath{\tilde{v}}} %RBH \newcommand{\Ec}{\ensuremath{\tilde{E}}} \newcommand{\pzsize}{\Lambda} \newcommand{\pzsizez}{\Lambda^0} %RBH \newcommand{\pzratio}{\beta} \newcommand{\pzRadiusf}{\gamma} \newcommand{\pzsizefactor}{\zeta}%{\kappa} \newcommand{\fractoughness}{\ensuremath{\Gamma_0}} \newcommand{\fractoughnessname}{specific fracture energy} \newcommand{\energyReleaseRate}{G} \newcommand{\modeIndC}{\ensuremath{{(k)}}} % RBH \newcommand{\modeInds}{k} %{k(\kappa)} \newcommand{\modeIndFun}{\tilde{\kappa}} %\newcommand{\modeeq}{,\quad \modeIndC \in \{\mI, \mII \}} \newcommand{\modeeq}{} \newcommand{\indeq}{,\quad \modeInds = 1,2} \newcommand{\sradiusm}{\sradiuss_{\modeIndC}} \newcommand{\sradiusi}[1]{\sradiuss_{#1}} \newcommand{\velradiusm}{\velradiuss_{\modeIndC}} %\newcommand{\pzsizevi}[1]{{\pzsize_v}_{#1}} %\newcommand{\pzsizezi}[1]{{\pzsizez}_{#1}} \newcommand{\pzsizem}{\pzsize_{\modeIndC}} %RBH \newcommand{\pzsizei}[1]{\pzsize_{#1}} \newcommand{\pzsizevim}{{f\vmaxim\pzsize_v}_{\modeIndC}} %RBH \newcommand{\pzsizezim}{\pzsizez_{\modeIndC}} %RBH %\newcommand{\pzsizezIplot}{{\pzsizez}_{\mI}} \newcommand{\pzsizezIplot}{{\pzsizez}} \newcommand{\vmaximIplot}[1]{v_{#1}} \newcommand{\absvmaximIplot}{v} \newcommand{\velcIplot}{\velc} \newcommand{\pzsizefactorm}{\pzsizefactor_{\modeIndC}} %RBH \newcommand{\pzratiom}{\pzratio_{\modeIndC}} \newcommand{\phici}[1]{\phic_{#1}} \newcommand{\phicim}{\phic_\modeIndC} %RA %RBH \newcommand{\sigmacim}{{{\sigmac}_{\modeIndC}}} %RA %RBH \newcommand{\deltaik}{\delta_{\modeInds}} \newcommand{\cm}{c_{\modeInds}} %\newcommand{\velci}[1]{{v_\mathrm{C}}_{#1}} \newcommand{\velcim}{{v_\mathrm{C}}_{\modeInds}} \newcommand{\tractionbari}[1]{{\breve{\sigma}}_{#1}} %RBH \newcommand{\tractionbarim}{{\breve{\sigma}}_{k}} %RBH \newcommand{\tractionbario}[1]{\underline{\sigma}_{#1}} %RBH \newcommand{\tractionbarimo}{{\underline{\sigma}}_{k}} %RBH \newcommand{\vcrack}{\hat{v}} \newcommand{\velbari}[1]{{\velbars}_{#1}} \newcommand{\velbarim}{{\velbars}_{\modeIndC}} \newcommand{\vmaxim}{|v_{\modeIndC}|_\vmaxSign} \newcommand{\vmaxi}[1]{|v_{#1}|_\vmaxSign} %\newcommand{\modeeq}{X} \newcommand{\sigmaf}{\bar{\sigma}} \newcommand{\velf}{\bar{v}} \newcommand{\inituf}{\bar{u}} \newcommand{\initEf}{\bar{E}} \newcommand{\initpf}{\bar{p}} \newcommand{\bforcef}{\bar{b}} \newcommand{\lengthc}{\ensuremath{\tilde{L}}} %RBH \newcommand{\linmomc}{\ensuremath{\tilde{p}}} %RBH \newcommand{\factorc}{\gamma} \newcommand{\trac}{s} \newcommand{\traction}{{\vc \trac}} \newcommand{\tractioni}[1]{\trac^{#1}} %\newcommand{\tractionc}{{\vc \trac}_\mathrm{C}} \newcommand{\tractionc}{\ensuremath{\tilde{\vc \trac}}} \newcommand{\tractionci}[1]{\trac_\mathrm{C}^{#1}} %\newcommand{\sigmafmc}{\fmt{\sigma}_\mathrm{C}} \newcommand{\tractionfmc}{\ensuremath{\tilde{\fmv{\trac}}}} %RBH \newcommand{\tractionfm}{\fmv{\trac}} %\newcommand{\deltauc}{\Delta {\vc u}_\text{c}} \newcommand{\deltauc}{\ensuremath{\jump{\vc u}_{\gc}}} %\newcommand{\Deltac}{\ensuremath{\mathbf{\Delta}}} \newcommand{\deltauci}[1]{\jump u_{#1}} \newcommand{\deltaucn}{\jump{u_1}} \newcommand{\deltauct}{\jump{u_2}} \newcommand{\mbf}[1]{\mathbf{#1}} \renewcommand{\bm}{\boldsymbol} \newcommand{\ER}[1]{\mathbb E^#1\times\mathbb R} \newcommand{\jump}[1]{\llbracket{#1}\rrbracket} \newcommand{\om}{\fmt{\omega}} \newcommand{\Om}{\fms{\Omega}} %\newcommand{\scaled}[1]{{{#1}^\prime} } \newcommand{\scale}[1]{\breve{#1}} %\newcommand{\sfactor}[1]{{\tilde{#1}}} \newcommand{\sfactor}[1]{{\lambda_{#1}}} \newcommand{\sfactorx}{\lambda_{\vc x}} \newcommand{\upi}{u^{\prime i}} \newcommand{\uddoti}{{\ddot{u}}^{\prime i}} \newcommand{\scldx}{\ensuremath{\mathrm{d{\scaled{x}}}}} % \newcommand{\sclsdx}{{\star \scldx}} % %\newcommand{\crdxt}{ (\bm x, t)} %\newcommand{\scaledcrdxt}{(\scaledxs, \scaledts)} \newcommand{\crdxt}{} \newcommand{\scaledcrdxt}{} %\newcommand{\sfactordeltaC}{{\sfactor{\delta}}_\mathrm{C}} %\newcommand{\sfactorsigmaC}{{\sfactor{\sigma}}_\mathrm{C}} %RA \newcommand{\sfactordeltaC}{\lambda_{\tilde{\delta}}} %RBH \newcommand{\sfactorsigmaC}{\ensuremath{\lambda_{\tractionfmc}}} %RBH % \newcommand{\sfactorphiC}{{\sfactor{\phi}}_\mathrm{C}} \newcommand{\scaledsigmaC}{\ensuremath{\scaled{\sigmac}}} \newcommand{\scaleddeltaC}{\ensuremath{\scaled{\deltac}}} \newcommand{\scaledphiC}{{\scaled{\phi}}_\mathrm{C}} \newcommand{\sfactort}{{\sfactor{t}}} \newcommand{\sfactorc}{\sfactor{c}} \newcommand{\sfactoru}{\sfactor{\vc u}} \newcommand{\sfactorubc}{\sfactor{\bar{\vc{u}}}} \newcommand{\sfactorubcz}{\sfactorubc_0} \newcommand{\scaledu}{\scaled{\vc u}} \newcommand{\scaledubc}{\scaled{\bar{\vc{u}}}} \newcommand{\scaledubcz}{\scaledubc_0} \newcommand{\scaledx}{\scaled{\bm x}} \newcommand{\scaledt}{\scaled{t}} \newcommand{\scaledxv}{\frac{\bm x}{\tauc \CD}} \newcommand{\scaledtv}{\frac{t}{\tauc}} \newcommand{\scaledxs}{\scaledx} \newcommand{\scaledts}{\scaledt} \newcommand{\sfactorE}{\sfactor{\tns E}} \newcommand{\sfactorEbc}{\sfactor{\bar{\tns{E}}}} \newcommand{\sfactorEForm}{\sfactor{\fmt E}} \newcommand{\sfactorEFormbc}{\sfactor{\bar{\fmt E}}} \newcommand{\scaledE}{\scaled{\fmt E}} \newcommand{\scaledEbc}{\scaled{\bar{\fmt{E}}}} \newcommand{\sfactorv}{\sfactor{\vc v}} \newcommand{\sfactorvbc}{\sfactor{\bar{\vc{v}}}} \newcommand{\sfactorvForm}{\sfactor{\fmv v}} \newcommand{\sfactorvFormbc}{\sfactor{\bar{\fmv v}}} \newcommand{\scaledv}{\scaled{\vc v}} \newcommand{\scaledvbc}{\scaled{\bar{\vc{v}}}} \newcommand{\sfactorCten}{\sfactor{\tns C}} \newcommand{\measureC}{\| \tns C \|} \newcommand{\sfactorStrs}{\sfactor{\tns s}} \newcommand{\sfactorStrsbc}{\sfactor{\bar{\tns s}}} \newcommand{\sfactorTraction}{\sfactor{\traction}} %RA following modified \newcommand{\sfactorTractionc}{\sfactor{\traction}_\mathrm{C}} \newcommand{\sfactorStrsForm}{\sfactor{\fmt{\sigma}}} \newcommand{\sfactorStrsFormbc}{\sfactor{\bar{\fmt{\sigma}}}} \newcommand{\scaledStrs}{\scaled{\tns s}} \newcommand{\scaledStrsbc}{\scaled{\bar{\tns s}}} \newcommand{\sfactorp}{\sfactor{\vc{p}}} \newcommand{\sfactorpbc}{\sfactor{\bar{\vc{p}}}} \newcommand{\sfactorpForm}{\sfactor{\fmv{p}}} \newcommand{\sfactorpFormbc}{\sfactor{\bar{\fmv{p}}}} \newcommand{\scaledp}{\scaled{\vc p}} \newcommand{\scaledpbc}{\scaled{\bar{\vc{p}}}} \newcommand{\sfactorMForm}{\sfactor{\fmt{M}}} \newcommand{\sfactorb}{\sfactor{\vc{b}}} \newcommand{\sfactorbbar}{\sfactor{\bar{\vc{b}}}} \newcommand{\sfactorfmbbar}{\sfactor{\bar{\fmv{b}}}} \newcommand{\sfactorbForm}{\sfactor{\fmv{b}}} \newcommand{\scaledb}{\scaled{\vc b}} \newcommand{\sfactoreps}{\sfactor{\eps}} \newcommand{\dydx}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{\dydxc}[2]{{\partial{#1}}/{\partial{#2}}} \newcommand{\DyDx}[2]{\frac{\mathrm{d}{#1}}{\mathrm{d}{#2}}} \newcommand{\dydxn}[3]{\frac{\partial^{#3}{#1}}{\partial{#2}^{#3}}} \newcommand{\DyDxn}[3]{\frac{\mathrm{d}^{#3}{#1}}{\mathrm{d}{#2}^{#3}}} \newcommand{\DyDxc}[2]{{\mathrm{d}{#1}}/{\mathrm{d}{#2}}} %\newcommand{\pP}{\partial P} %\newcommand{\gbf}[1]{\mbox{\boldmath#1}} %\newcommand{\RiemannSym}{\mathrm{R}} %\newcommand{\RiemannSymI}{{\RiemannSym_\mI}} %\newcommand{\tractionSym}{s} %\newcommand{\tractionVecSym}{\vc \tractionSym} %\newcommand{\tractionLoad}{\bar{\tractionSym}} %\newcommand{\tractionR}[1]{{\tractionSym^{#1}}^\RiemannSym} %\newcommand{\tractionS}[1]{{\tractionSym^{#1}}^*} %\newcommand{\tractionI}[1]{\tractionSym^{#1}} %\newcommand{\tractionRs}{{\tractionSym}^\RiemannSym} %\newcommand{\tractionSs}{{\tractionSym}^*} %\newcommand{\tractionIs}{\tractionSym} %\newcommand{\tractionRleg}{\tractionRs} %\newcommand{\tractionSleg}{\tractionSs} %\newcommand{\tractionIleg}{\tractionIs} %%%%%% NEW \STARRED NOTATION \newcommand{\Mstar}{\starred{\fmt M}} \newcommand{\ustar}{\starred{\vc u}} \newcommand{\epsstar}{\starred{\eps}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% REZA FEM presentations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage[nomessages]{fp} % to do math calculations \usepackage{empheq} % for boxing equations %sample %\begin{empheq}[box=\widefbox]{equation} %eqn %\end{empheq} \newcommand{\dVolU}{\mathrm{d V}} \newcommand{\dVolL}{\mathrm{d v}} \newcommand{\dAreaU}{\mathrm{d S}} \newcommand{\dAreaL}{\mathrm{d s}} \newcommand{\dAreaTnsU}{\mathrm{d \tns{S}}} \newcommand{\dAreaTnsL}{\mathrm{d \tns{s}}} \newcommand{\orderD}{{M_u}} \newcommand{\orderN}{{M_f}} \newcommand{\rproof}{{\bfseries Proof}} \newcommand{\realn}{\mathbb{R}} \newcommand{\tfs}[1]{{#1}^h} \newcommand{\tfun}{\phi} \newcommand{\tfuni}[1]{\tfun_{#1}} \newcommand{\tfunp}{\tfun_{ {p}}} \newcommand{\wfun}{w} \newcommand{\wfuni}[1]{\wfun_{#1}} \newcommand{\ttfun}{{\boldsymbol \phi}} \newcommand{\ttfuni}[1]{\ttfun_{#1}} \newcommand{\ttfunp}{\ttfun_{ {p}}} \newcommand{\twfun}{\tns w} \newcommand{\twfuni}[1]{\twfun_{#1}} \newcommand{\desa}{k} \newcommand{\desA}{K} \newcommand{\desAfp}{\desA_{fp}} \newcommand{\tdesAfp}{{\tns \desA}_{fp}} \newcommand{\tdesAefp}{{\tns \desA}_{fp}^e} \newcommand{\tdesAeefp}[1]{{\tns \desA}_{fp}^{\elni{#1}}} \newcommand{\tdesaefp}{{\tns \desa}_{fp}^e} \newcommand{\tdesaeefp}[1]{{\tns \desa}_{fp}^{\elni{#1}}} \newcommand{\tKff}{{\tns \desA}_{ff}} \newcommand{\tKfp}{{\tns \desA}_{fp}} \newcommand{\tKpf}{{\tns \desA}_{pf}} \newcommand{\tKpp}{{\tns \desA}_{pp}} \newcommand{\tFf}{{\tns F}_{f}} \newcommand{\tFp}{{\tns F}_{p}} \newcommand{\tUf}{{\tns U}_{f}} \newcommand{\tUp}{{\tns U}_{p}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\sfun}{N} \newcommand{\tsfuni}[1]{{\tns \sfun}_{#1}} \newcommand{\sfuni}[1]{\sfun_{#1}} \newcommand{\tsfun}{\tns N} %%%% prescribed \newcommand{\sfunp}{\npop{\sfun}} \newcommand{\tsfunp}{\npop{\tsfun}} \newcommand{\sfunpi}[1]{\sfunp_{\npop{#1}}} \newcommand{\tsfunpi}[1]{{\tns \sfunp}_{\npop{#1}}} \newcommand{\sfunpib}[1]{\sfunp_{{#1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\bfuni}[1]{B_{#1}} \newcommand{\tbfun}{\tns B} \newcommand{\tbfuni}[1]{{\tns B}_{#1}} \newcommand{\Dpara}{\tns D} %%%% prescribed \newcommand{\tbfunp}{\npop{\tns B}} \newcommand{\bfunpi}[1]{\npop{B}_{\npop{#1}}} \newcommand{\tbfunpi}[1]{\npop{\tns B}_{\npop{#1}}} \newcommand{\tCee}[1]{\eli{\tns C}{#1}} \newcommand{\tCe}{\el{\tns C}} \newcommand{\Ceij}[2]{\el{C}_{#1#2}} \newcommand{\Ceeij}[3]{\eli{C}{#1}_{#2#3}} \newcommand{\tCpee}[1]{\eli{\npop{\tns C}}{#1}} \newcommand{\tCpe}{\el{\npop{\tns C}}} \newcommand{\Cpeij}[2]{\el{\npop{C}}_{#1#2}} \newcommand{\Cpeeij}[3]{\eli{\npop{C}}{#1}_{#2#3}} \newcommand{\tMee}[1]{\eli{\tns M}{#1}} \newcommand{\tMe}{\el{\tns M}} \newcommand{\tMte}{\el{\tns M}_t} \newcommand{\tMtee}[1]{{\eli{\tns M}{#1}}_t} \newcommand{\Mei}[1]{\el{M}({#1})} \newcommand{\Meei}[2]{\eli{M}{#1}_{#2}} \newcommand{\tMpee}[1]{\eli{\npop{\tns M}}{#1}} \newcommand{\tMpe}{\el{\npop{\tns M}}} \newcommand{\Mpei}[1]{\el{\npop{M}}({#1})} \newcommand{\Mpeei}[2]{\eli{\npop{M}}{#1}_{#2}} %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\sfunei}[1]{\el{\sfun}_{#1}} \newcommand{\sfunlei}[1]{\el{{\sfun_L}}_{#1}} \newcommand{\sfuneei}[2]{\eli{\sfun}{#1}_{#2}} \newcommand{\sfuneij}[2]{\sfun^{e_{#1}}_{#2}} \newcommand{\sbfuneij}[2]{B^{e_{#1}}_{#2}} \newcommand{\tceij}[2]{\tc^{e_#1}_{#2}} %%%% prescribed \newcommand{\tsfune}{{\el{\tns N}}} \newcommand{\tsfunle}{{\el{ {\tns N}_L}}} \newcommand{\sfunpe}{\el{\npop{\sfun}}} \newcommand{\tsfunpe}{\el{\npop{\tsfun}}} \newcommand{\sfunpei}[1]{\el{\sfunp}_{\npop{#1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\bfunei}[1]{\el{B}_{#1}} \newcommand{\tbfune}{{\el{\tns B}}} %%%% prescribed \newcommand{\tbfunpe}{\el{\npop{\tns B}}} \newcommand{\bfunpei}[1]{\el{\npop{B}}_{\npop{#1}}} \newcommand{\FreeSym}{\mathrm{F}} \newcommand{\PresSym}{\mathrm{P}} \newcommand{\freeSym}{{\mathrm{f}}} \newcommand{\presSym}{{\mathrm{p}}} \newcommand{\desb}{f} \newcommand{\desB}{F} \newcommand{\el}[1]{#1^e} \newcommand{\elni}[1]{{e_#1}} \newcommand{\eli}[2]{#1^\elni{#2}} \newcommand{\nni}[2]{#1^{\nI{#2}}} %\newcommand{\BCDe}{{\partial \el{\mathcal{D}}_u}} %\newcommand{\BCNe}{{\partial \el{\mathcal{D}}_f}} %\newcommand{\BCNee}[1]{{\partial \mathcal{D}_f^\elni{#1}}} \newcommand{\BCDe}{{\partial e_u}} \newcommand{\BCNe}{{\partial e_f}} \newcommand{\BCNee}[1]{{\partial e_f^{#1}}} \newcommand{\nn}{n_{\mathrm{n}}} \newcommand{\nen}{\el{\nn}} \newcommand{\nel}{n_{\mathrm{e}}} \newcommand{\neqnpn}{{n_{\mathrm{dpn}}}} \newcommand{\ndim}{n_{\mathrm{dim}}} \newcommand{\nI}[1]{n_{#1}} \newcommand{\nei}[1]{\el{n}_{#1}} \newcommand{\neei}[2]{\eli{n}{#1}_{#2}} \newcommand{\neqn}{{n_\freeSym}} \newcommand{\neqnSet}{S_\freeSym} \newcommand{\neqnSeti}[1]{{\neqnSet}_#1} \newcommand{\neqnaSetee}[1]{\eli{S}{#1}} \newcommand{\neqnaSeteei}[2]{\eli{S}{#1}_{#2}} \newcommand{\neqnSetee}[1]{\eli{S}{#1}_\freeSym} \newcommand{\neqnSeteei}[2]{{\eli{S}{#1}_\freeSym}_{#2}} \newcommand{\neeqn}{{\el{n_\freeSym}}} \newcommand{\neqnp}{{n_\presSym}} \newcommand{\neqnpSet}{S_\presSym} \newcommand{\neqnpSeti}[1]{{\neqnpSet}_#1} \newcommand{\neqnpSetee}[1]{\eli{S}{#1}_\presSym} \newcommand{\neqnpSeteei}[2]{{\eli{S}{#1}_\presSym}_{#2}} \newcommand{\neeqnp}{{\el{n_\presSym}}} \newcommand{\ndof}{{n_{\mathrm{dof}}}} \newcommand{\nedof}{{\el{n}_{\mathrm{dof}}}} \newcommand{\npop}[1]{\bar{#1}} \newcommand{\npopneqnp}{\npop{\neqnp}} \newcommand{\tc}{a} \newcommand{\ttc}{\tns \tc} \newcommand{\tci}[1]{\tc_{#1}} \newcommand{\tcp}{\npop{\tc}} \newcommand{\ttcp}{\npop{\tns \tc}} \newcommand{\tcpi}[1]{\npop{\tc}_{\npop{#1}}} \newcommand{\pind}[2]{\npop{#1}_{\npop{#2}}} \newcommand{\ttce}{\el{\tns \tc}} \newcommand{\tcei}[1]{\el{\tc}_{#1}} \newcommand{\tceei}[2]{\eli{\tc}{#1}_{#2}} %\newcommand{\tnni}[3]{\nni{#1}{#2}_{#3}} \newcommand{\tnni}[3]{{#1}_{#2#3}} \newcommand{\tcnni}[2]{\nni{\tc}{#1}_{#2}} \newcommand{\ttcei}[1]{\el{\tns \tc}_{#1}} \newcommand{\tcpe}{\el{\npop{\tc}}} \newcommand{\ttcpe}{\el{\npop{\tns \tc}}} \newcommand{\tcpei}[1]{\el{\npop{\tc}}_{\npop{#1}}} \newcommand{\KT}{\desA} \newcommand{\tKT}{{\tns \desA}} \newcommand{\tkT}{{\tns \desa}} \newcommand{\tKTff}{{\tns \desA}_{\freeSym\freeSym}} \newcommand{\tKTfp}{{\tns \desA}_{\freeSym\presSym}} \newcommand{\tKTpf}{{\tns \desA}_{\presSym\freeSym}} \newcommand{\tKTpp}{{\tns \desA}_{\presSym\presSym}} \newcommand{\KTij}[2]{\KT_{#1#2}} \newcommand{\eKT}{\el{\desA}} \newcommand{\teKT}{\el{\tns \desA}} \newcommand{\teeKT}[1]{\eli{\tns \desA}{#1}} \newcommand{\eKTi}[2]{\el{\KT}_{#1#2}} \newcommand{\ekT}{\el{\desa}} \newcommand{\tekT}{\el{\tns\desa}} \newcommand{\teekT}[1]{\eli{\tns\desa}{#1}} \newcommand{\ekTij}[2]{\el{\desa}_{#1#2}} \newcommand{\eekTij}[3]{\eli{\desa}{#1}_{#2#3}} \newcommand{\Frs}{r} \newcommand{\FDs}{D} \newcommand{\FNs}{N} \newcommand{\Fns}{n} \newcommand{\Fes}{e} \newcommand{\Fos}{o} %\newcommand{\FT}{\desB} \newcommand{\Fr}{\desB_\Frs} \newcommand{\FD}{\desB_\FDs} \newcommand{\FN}{\desB_\FNs} \newcommand{\Fn}{\desB_\Fns} \newcommand{\fr}{\desb_\Frs} \newcommand{\fD}{\desb_\FDs} \newcommand{\fN}{\desb_\FNs} \newcommand{\tFT}{{\tns \desB}} \newcommand{\tfT}{{\tns \desb}} \newcommand{\tUT}{{\tns U}} \newcommand{\tuT}{{\tns u}} \newcommand{\tFr}{\tns \desB_\Frs} \newcommand{\tFD}{{\tns \desB}_\FDs} \newcommand{\tFN}{{\tns \desB}_\FNs} \newcommand{\tFn}{{\tns \desB}_\Fns} \newcommand{\tFe}{{\tns \desB}_\Fes} \newcommand{\FTi}[1]{{\FT}_{#1}} \newcommand{\Fri}[1]{{\Fr}_{#1}} \newcommand{\FDi}[1]{{\FD}_{#1}} \newcommand{\FNi}[1]{{\FN}_{#1}} \newcommand{\Fni}[1]{{\Fn}_{#1}} \newcommand{\Fnpi}[1]{{\npop{\desB}_\Fns}_{\npop{#1}}} \newcommand{\FTe}[1]{{\FT}^{e}} \newcommand{\Fre}[1]{{\Fr}^{e}} \newcommand{\FDe}[1]{{\FD}^{e}} \newcommand{\FNe}[1]{{\FN}^{e}} \newcommand{\FTee}[1]{{\tns \desB}^{\elni{#1}}} \newcommand{\Free}[1]{{\tns \desB}_\Frs^{\elni{#1}}} \newcommand{\FDee}[1]{{\tns \desB}_\FDs^{\elni{#1}}} \newcommand{\FNee}[1]{{\tns \desB}_\FNs^{\elni{#1}}} \newcommand{\fTe}{{\tns \desb}^{e}} \newcommand{\fre}{{\tns \desb}_\Frs^{e}} \newcommand{\fDe}{{\tns \desb}_\FDs^{e}} \newcommand{\fNe}{{\tns \desb}_\FNs^{e}} \newcommand{\fee}{{\tns \desb}_\Fes^{e}} \newcommand{\fnDe}{{\tns \desb}_\Fes^{e}} \newcommand{\foe}{{\tns \desb}_\Fos^{e}} \newcommand{\fTee}[1]{{\tns \desb}^{\elni{#1}}} \newcommand{\free}[1]{{\tns \desb}_\Frs^{\elni{#1}}} \newcommand{\fDee}[1]{{\tns \desb}_\FDs^{\elni{#1}}} \newcommand{\fNee}[1]{{\tns \desb}_\FNs^{\elni{#1}}} \newcommand{\feee}[1]{{\tns \desb}_\Fes^{\elni{#1}}} \newcommand{\tFTe}{{\tns F}^{e}} \newcommand{\tFre}{{\tns \desB}_\Frs^{e}} \newcommand{\tFDe}{{\tns \desB}_\FDs^{e}} \newcommand{\tFNe}{{\tns \desB}_\FNs^{e}} \newcommand{\tFee}{{\tns \desB}_\Fes^{e}} \newcommand{\tFoe}{{\tns \desB}_\Fos^{e}} \newcommand{\Frei}[1]{{{\desB}_\Frs^{e}}_{#1}} \newcommand{\frei}[1]{{{\desb}_\Frs^{e}}_{#1}} \newcommand{\fDei}[1]{{{\desb}_\FDs^{e}}_{#1}} \newcommand{\fNei}[1]{{{\desb}_\FNs^{e}}_{#1}} \newcommand{\tFTee}[1]{{\tns \FT}^{\elni{#1}}} \newcommand{\tFree}[1]{{\tns \desB}_\Frs^{\elni{#1}}} \newcommand{\tFDee}[1]{{\tns \desB}_\FDs^{\elni{#1}}} \newcommand{\tFNee}[1]{{\tns \desB}_\FNs^{\elni{#1}}} \newcommand{\tfTe}{{\tns \fT}^{e}} \newcommand{\tfre}{{\tns \desb}_\Frs^{e}} \newcommand{\tfDe}{{\tns \desb}_\FDs^{e}} \newcommand{\tfNe}{{\tns \desb}_\FNs^{e}} \newcommand{\tfee}{{\tns \desb}_\Fes^{e}} \newcommand{\tfoe}{{\tns \desb}_\Fos^{e}} \newcommand{\tfTee}[1]{{\tns \fT}^{\elni{#1}}} \newcommand{\tfree}[1]{{\tns \desb}_\Frs^{\elni{#1}}} \newcommand{\tfDee}[1]{{\tns \desb}_\FDs^{\elni{#1}}} \newcommand{\tfNee}[1]{{\tns \desb}_\FNs^{\elni{#1}}} \newcommand{\tfeee}[1]{{\tns \desb}_\Fes^{\elni{#1}}} %\newcommand{\tfeeei}[2]{{\desb}_{\Fes_{#2}}^{\elni{#1}}} \newcommand{\tfeeei}[2]{{\desb}_{{#2}}^{\elni{#1}}} %\newcommand{\eFT}{\el{\desB}} %\newcommand{\eFr}{\el{\desB}_\Frs} %\newcommand{\eFD}{\el{\desB}_\FDs} %\newcommand{\eFN}{\el{\desB}_\FNs} %\newcommand{\eFn}{\el{\desB}_\Fns} \newcommand{\teFT}{{\el{\tns \desB}}} \newcommand{\teFr}{ \el{\tns \desB}_\Frs} \newcommand{\teFD}{ \el{\tns \desB}_\FDs} \newcommand{\teFN}{ \el{\tns \desB}_\FNs} \newcommand{\teFn}{ \el{\tns \desB}_\Fns} %\newcommand{\eFTi}[1]{{\eFT}_{#1}} %\newcommand{\eFri}[1]{{\eFr}_{#1}} %\newcommand{\eFDi}[1]{{\eFD}_{#1}} %\newcommand{\eFNi}[1]{{\eFN}_{#1}} %\newcommand{\eFni}[1]{{\eFn}_{#1}} %\newcommand{\efT}{\el{\desb}} %\newcommand{\efr}{\el{\desb}_\Frs} %\newcommand{\efD}{\el{\desb}_\FDs} %\newcommand{\efN}{\el{\desb}_\FNs} %\newcommand{\efn}{\el{\desb}_\Fns} \newcommand{\tefT}{{\tns \el{\desb}}} \newcommand{\tefr}{ \el{\tns \desb}_\Frs} \newcommand{\tefD}{ \el{\tns \desb}_\FDs} \newcommand{\tefN}{ \el{\tns \desb}_\FNs} \newcommand{\tefn}{ \el{\tns \desb}_\Fns} %\newcommand{\efTi}[1]{{\efT}_{#1}} %\newcommand{\efri}[1]{{\efr}_{#1}} %\newcommand{\efDi}[1]{{\efD}_{#1}} %\newcommand{\efNi}[1]{{\efN}_{#1}} %\newcommand{\efni}[1]{{\efn}_{#1}} \newcommand{\bls}{\mathcal{A}} \newcommand{\blop}[2]{\bls\left(#1, #2\right)} %\newcommand{\lro}[2]{\left(#1, #2\right)} \newcommand{\lrop}[2]{\left(#1, #2\right)_{r}} \newcommand{\lnop}[2]{\left(#1, #2\right)_{N}} \newcommand{\leop}[2]{\blop{#1, #2\right}_{E}} \newcommand{\deseA}{\el{\desA}} \newcommand{\deseB}{\el{\desB}} \newcommand{\BCD}{{\partial \mathcal{D}_u}} \newcommand{\BCN}{{\partial \mathcal{D}_f}} \newcommand{\BC}[1]{\bar{#1}} \newcommand{\wspace}{\mathcal{W}} \newcommand{\wspacez}{\mathcal{W}_0} \newcommand{\vspc}{\mathcal{V}} \newcommand{\vspcz}{\mathcal{V}_0} \newcommand{\vWR}{\mathcal{V}^h} \newcommand{\wWR}{\mathcal{W}^h} \newcommand{\vzWR}{\mathcal{V}_0^h} \newcommand{\vWK}{\mathcal{V}^h} \newcommand{\wWK}{\mathcal{W}^h} \newcommand{\vzWK}{\mathcal{W}^h} \newcommand{\dopx}{\:\doper x} \newcommand{\dopy}{\:\doper y} \newcommand{\dopxi}{\:\doper \xi} \newcommand{\dopv}{\:\doper \mathrm{v}} \newcommand{\dopA}{\:\doper \mathrm{A}} \newcommand{\dops}{\:\doper \mathrm{s}} \newcommand{\dopeta}{\:\doper \eta} \newcommand{\dopsv}{\:\doper {\tns s}} \newcommand{\Lmw}{L_m^w} \newcommand{\wf}{\wfun^f} \newcommand{\wfi}[1]{\wfun^f_{#1}} \newcommand{\twf}{\twfun^f} \newcommand{\twfi}[1]{\twfun^f_{#1}} \newcommand{\wfc}{w^f} \newcommand{\twfc}{{\tns w}^f} \newcommand{\wfb}{\tns \wf} \newcommand{\trans}{\mathrm{T}} \newcommand{\charfun}[1]{\chi_{#1}} \newcommand{\barf}[2]{\tfs{u}_{{#1}{#2}}} \newcommand{\lcrd}{\xi} \newcommand{\pts}[1]{({\bfseries {#1} Points})} \newcommand{\ptsr}[1]{\fbox{{\bfseries {#1} Points}}} \newcommand{\ptsrt}[1]{\fbox{{\bfseries {#1} Total Points}}} \newcommand{\gI}{I} \newcommand{\gJ}{J} \newcommand{\txtcr}[1]{\textcolor{red}{#1}} \newcommand{\txtcb}[1]{\textcolor{blue}{#1}} \newcommand{\txtcg}[1]{\textcolor{green}{#1}} \newcommand{\txtcm}[1]{\textcolor{magenta}{#1}} \newcommand{\txtcc}[1]{\textcolor{cyan}{#1}} \newcommand{\txtcy}[1]{\textcolor{yellow}{#1}} \newcommand{\txtcgy}[1]{\textcolor{gray}{#1}} \newcommand{\tnatflux}{\bar{\tns f}} \newcommand{\remat}{r} \newcommand{\remate}{\el{\tns \remat}} \newcommand{\rematee}[1]{\eli{\tns \remat}{#1}} \newcommand{\qee}[1]{\eli{\tns q}{#1}} \newcommand{\Lel}{\el{L}} \newcommand{\Leli}[1]{\eli{L}{#1}} \newcommand{\xmel}{\el{x_m}} \newcommand{\TuU}{{\tns T_{\tns u\tns U}}} \newcommand{\TFf}{{\tns T_{\tns F\tns f}}} \newcommand{\TT}{{\tns T}} \newcommand{\kb}{{\tns k}_b} \newcommand{\yei}[1]{\el{y}_{#1}} \newcommand{\ypei}[1]{\el{\theta}_{#1}} \newcommand{\mcly}[1]{\left<{#1}\right>_+} \newcommand{\sgn}[1]{\mathrm{sign}\left({#1}\right)} \newcommand{\vset}[1]{\{#1\}} \newcommand{\LEM}{\mathrm{LEM}} \newcommand{\LEMee}[1]{\eli{\mathrm{LEM}}{#1}} \newcommand{\cmnt}[1]{\txtcgy{#1}} \newcommand{\Mcv}[1]{\txtcg{#1}} \newcommand{\Mcc}[1]{\txtcr{#1}} \newcommand{\Mvec}[1]{\Mcv{(}#1\Mcv{)}} \newcommand{\Mcel}[1]{\Mcc{\{}#1\Mcc{\}}} \endinput