210 lines
9.2 KiB
TeX
210 lines
9.2 KiB
TeX
\documentclass[12pt,numbers,sort&compress]{article}
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%% Use the option review to obtain double line spacing
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%% \documentclass[authoryear,preprint,review,12pt]{elsarticle}
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%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn
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%% for a journal layout:
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%% \documentclass[final,1p,times]{elsarticle}
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%% \documentclass[final,1p,times,twocolumn]{elsarticle}
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%% \documentclass[final,3p,times]{elsarticle}
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%% \documentclass[final,3p,times,twocolumn]{elsarticle}
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%% \documentclass[final,5p,times]{elsarticle}
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%% \documentclass[final,5p,times,twocolumn]{elsarticle}
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%% if you use PostScript figures in your article
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%% use the graphics package for simple commands
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%% \usepackage{graphics}
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%% or use the graphicx package for more complicated commands
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%% \usepackage{graphicx}
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%% or use the epsfig package if you prefer to use the old commands
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%% \usepackage{epsfig}
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%% The amssymb package provides various useful mathematical symbols
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\usepackage[usenames,dvipsnames,pdftex]{color}
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\usepackage{amsmath,amssymb,amsfonts}
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\usepackage{siunitx}
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%\usepackage{subeqnarray}
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\usepackage[hang]{subfigure}
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\usepackage{verbatim}
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\usepackage{bm}
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\usepackage{tikz}
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\usetikzlibrary{arrows}
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\usepackage{booktabs}
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\usepackage[latin1]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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\usepackage{natbib}
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\newcommand{\pathToFigures}{./figures}
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\graphicspath{{\pathToFigures/}}
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\DeclareGraphicsExtensions{.pdf,.png}
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\usepackage[pdftex, % hyper-references for pdftex
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bookmarksnumbered=true,% % generate bookmarks with numbers
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%pagebackref=true,% % generate backref in biblio
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%colorlinks=true,%
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]{hyperref}%
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%% The amsthm package provides extended theorem environments
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%% \usepackage{amsthm}
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%% The lineno packages adds line numbers. Start line numbering with
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%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
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%% for the whole article with \linenumbers.
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%% \usepackage{lineno}
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\newlength{\diagramsize}
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\setlength{\diagramsize}{0.4\textwidth}
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\newcommand{\question}[1]{\textcolor{Red}{#1}}
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\newcommand{\note}[1]{\textcolor{CornflowerBlue}{#1}}
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\newcommand{\term}[1]{\textsc{#1}}
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\newcommand{\eref}[1]{Eq.~\eqref{#1}}
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\newcommand{\Eref}[1]{Eq.~\eqref{#1}}
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\newcommand{\erefs}[1]{Eqs.~\eqref{#1}}
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\newcommand{\Erefs}[1]{Eqs.~\eqref{#1}}
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\newcommand{\fref}[1]{Fig.~\ref{#1}}
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\newcommand{\Fref}[1]{Fig.~\ref{#1}}
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\newcommand{\frefs}[1]{Figs.~\ref{#1}}
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\newcommand{\Frefs}[1]{Figs.~\ref{#1}}
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\newcommand{\tref}[1]{Tab.~\ref{#1}}
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\newcommand{\Tref}[1]{Tab.~\ref{#1}}
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\newcommand{\trefs}[1]{Tabs.~\ref{#1}}
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\newcommand{\Trefs}[1]{Tabs.~\ref{#1}}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\cf}{\textit{cf.}}
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\newcommand{\Euler}{\textsc{Euler}}
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\newcommand{\Gauss}{\textsc{Gauss}}
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\newcommand{\kB}{\ensuremath{k_\text{B}}}
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\newcommand{\transpose}[1]{\ensuremath{{#1}^{\mathrm T}}}
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\newcommand{\inverse}[1]{\ensuremath{{#1}^{-1}}}
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\newcommand{\invtranspose}[1]{\ensuremath{{#1}^{\mathrm{-T}}}}
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\newcommand{\sign}[1]{\ensuremath{\operatorname{sgn}\left({#1}\right)}}
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\newcommand{\grad}[1][]{\ensuremath{\operatorname{grad}{#1}}}
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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{\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{\inc}[1]{\ensuremath{\text d{#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{\avg}[1]{\ensuremath{\bar{#1}}}
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\newcommand{\fluct}[1]{\ensuremath{\tilde{#1}}}
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\newcommand{\FT}[1]{\ensuremath{\hat{#1}}}
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\newcommand{\domain}[1]{\ensuremath{\mathcal{#1}}}
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\newcommand{\tnsrfour}[1]{\ensuremath{\mathbb{#1}}}
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\newcommand{\tnsr}[1]{\ensuremath{\bm{#1}}}
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\newcommand{\vctr}[1]{\ensuremath{\bm{#1}}}
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\newcommand{\eyetwo}{\ensuremath{\tnsr I}}
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\newcommand{\eyefour}{\ensuremath{\tnsrfour I}}
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\newcommand{\stiffness}{\ensuremath{\tnsrfour D}}
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\newcommand{\refStiffness}{\ensuremath{\avg{\tnsrfour D}}}
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\newcommand{\fPK}{\ensuremath{\tnsr P}}
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\newcommand{\sPK}{\ensuremath{\tnsr S}}
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\newcommand{\F}[1][]{\ensuremath{\tnsr F^{#1}}}
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\newcommand{\Favg}{\ensuremath{\avg{\F}}}
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\newcommand{\Ffluct}{\ensuremath{\fluct{\F}}}
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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{\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{\ld}[1]{\ensuremath{\vctr p^{(#1)}}}
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\newcommand{\velocity}[2]{\ensuremath{v^{(#1)}_\text{#2}}}
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\newcommand{\avgvelocity}[2]{\ensuremath{{\bar v}^{(#1)}_ \text{#2}}}
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\newcommand{\flux}[2]{\ensuremath{\vctr f^{(#1)}_ \text{#2}}}
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\newcommand{\averageflux}[2]{\ensuremath{\bar{\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{\dotgalpha}{\ensuremath{\dot{\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{\density}[2]{\ensuremath{\varrho^{(#1)}_ \text{#2}}}
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\newcommand{\densityfunc}[2]{\ensuremath{{\tilde\varrho}^{(#1)}_ \text{#2}}}
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\newcommand{\avgdensity}[2]{\ensuremath{{\bar\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, authors and addresses
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%% use the tnoteref command within \title for footnotes;
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%% use the tnotetext command for theassociated footnote;
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%% use the fnref command within \author or \address for footnotes;
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%% use the fntext command for theassociated footnote;
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%% use the corref command within \author for corresponding author footnotes;
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%% use the cortext command for theassociated footnote;
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%% use the ead command for the email address,
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%% and the form \ead[url] for the home page:
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%% \title{Title\tnoteref{label1}}
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%% \tnotetext[label1]{}
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%% \author{Name\corref{cor1}\fnref{label2}}
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%% \ead{email address}
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%% \ead[url]{home page}
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%% \fntext[label2]{}
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%% \cortext[cor1]{}
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%% \address{Address\fnref{label3}}
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%% \fntext[label3]{}
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\title{Implement a remeshing scheme into a spectral method based crystal plasticity code}
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%% use optional labels to link authors explicitly to addresses:
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%% \author[label1,label2]{}
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%% \address[label1]{}
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%% \address[label2]{}
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\author{M.~Diehl}
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%% \linenumbers
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% main text
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\begin{document}
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\maketitle
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% ----------------------------------------------------------------------------------------------------------------------------
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\section{General Information}
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At MPIE, the flexible crystal plasticity framework ``D<>sseldorf Advanced MAterial Simulation Kit'' (DAMASK) is developed.
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It consists of different constitutive models, homogenization schemes, and tools for post- and preprocessing \cite{Roters_etal2010}.
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It has interfaces to different solvers to the mechanical boundary value problem.
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To compute the boundary value problem, commercial FEM software like MSC.Marc or Abaqus or a solver based on a so-called spectral method \cite{Moulinec+Suquet1998,Lebensohn2001}.
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Spectral methods have advantages concerning accuracy, performance, and memory efficiency compared to the de-facto standard FEM.
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However, their use is limited to periodic boundary conditions due to the approximation of the solution by plane waves.
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The spectral method implemented at MPIE uses a finite-strain formulation proposed in \cite{Lahellec_etal2001} that is written in terms of deformation gradient \tnsr F and Piola--Kirchhoff stress \tnsr P and can therefore be used to solve the mechanical boundary value problem in the reference configuration.
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Calculations have shown that for inhomogeneous material convergence cannot be achieved any longer at strains larger than ca.~15--20 \%.
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We presently believe that this is due to the fact that the regular mesh in the reference configuration is locally heavily deformed to an extent where single points cross the path of neighboring points.
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To reach higher strains, a remeshing scheme should be implemented as follows.
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\begin{enumerate}
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\item Write out the current state
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\item Reconstruct current geometry
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\item Approximate the deformed configuration by a regular (undeformed, new) mesh with potentially $a \ne b \ne c$
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\item Translate the old state values to the new mesh (``The winner takes it all?'')
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\end{enumerate}
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\section{Remarks, Notes}
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\begin{enumerate}
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\item Which criterion? Even for converged solutions we get ``chess board patterns?''
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\end{enumerate}
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\bibliographystyle{unsrtnat}
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\bibliography{Masterthesis}
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\end{document}
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\endinput
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