Implementation of a relaxed linear micromorphic continuum

The static case of this model is discussed in

  Neff, Ghiba, Lazar, Madeo

  "The relaxed linear micromorphic continuum:
   Well-posedness of the static problem and relations to the
   gauge theory of dislocations"

   DOI: 10.1093/qjmam/hbu027

doc/.gitignore

+*.aux
+*.bbl
+*.bcf
+*.blg
+*.log
+*.out
+*.run.xml
+*.synctex.gz
+*.toc
+
+relaxed-micromorphic-continuum.pdf

doc/relaxed-micromorphic-continuum.bib

+@Article{neff_ghiba_lazar_madeo:2015,
+ author = {Patrizio Neff and I.D. Ghiba and M. Lazar and Angela Madeo},
+ title = {The relaxed linear micromorphic continuum: Well-posedness of the static problem and relations to the gauge theory of dislocations},
+ journal = {Quarterly Journal of Mechanics and Applied Mathematics},
+ year = {2015},
+ volume = {68},
+ number = {1},
+ pages = {53--84},
+ doi = {10.1093/qjmam/hbu027}
+}

doc/relaxed-micromorphic-continuum.tex

+\documentclass[a4paper,10pt]{scrartcl}
+\usepackage[utf8]{inputenc}
+
+\usepackage[defernumbers=true,
+           maxbibnames=99,  % Never write 'et al.' in a bibliography
+           giveninits=true, % Abbreviate first names
+           sortcites=true,  % Sort citations when there are more than one
+           natbib=true,     % Allow commands like \citet and \citep
+           backend=biber]{biblatex}
+\addbibresource{relaxed-micromorphic-continuum.bib}
+
+%%% PACKAGES
+
+\usepackage{amssymb, amsmath, amsthm}
+\usepackage{graphicx}
+
+\usepackage{colonequals}
+\usepackage{lmodern}
+
+\usepackage{microtype}
+\usepackage{todonotes}
+\newcommand{\todosander}[1]{\todo[inline,color=green,author=OS]{#1}}
+
+\usepackage{tikz}
+\usepackage{hyperref}
+
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+
+\renewcommand{\skew}{\operatorname{skew}}
+\newcommand{\sym}{\operatorname{sym}}
+\newcommand{\tr}{\operatorname{tr}}
+\newcommand{\dev}{\operatorname{dev}}
+\newcommand{\Curl}{\operatorname{Curl}}
+
+\newcommand{\bu}{\mathbf{u}}
+\newcommand{\bP}{\mathbf{P}}
+
+\newcommand{\abs}[1]{\lvert #1 \rvert}
+\newcommand{\norm}[1]{\lVert #1 \rVert}
+\newcommand{\tnorm}[1]{||| #1 |||}
+
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator*{\argmin}{arg\,min}
+
+\newtheorem{definition}{Definition}
+\newtheorem{theorem}[definition]{Theorem}
+\newtheorem{cor}[definition]{Corollary}
+\newtheorem{lemma}[definition]{Lemma}
+\newtheorem{remark}[definition]{Remark}
+\newtheorem{problem}[definition]{Problem}
+\theoremstyle{remark}
+
+
+%%  All graphics files may be in this subdirectory
+\graphicspath{{gfx/}}
+
+
+% Title Page
+\title{Simulating the relaxed linear micromorphic continuum}
+\author{Oliver Sander}
+
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+We investigate finite element discretizations of the relaxed linear micromorphic continuum.
+\end{abstract}
+
+\tableofcontents
+
+\section{The static relaxed linear micromorphic continuum}
+
+Let $\Omega$ be a domain in $\R^d$ with $d=2$ or $d=3$.  On $\Omega$ we consider two fields:
+First, a displacement field
+\begin{equation*}
+ \bu \in H^1(\Omega,\R^d),
+\end{equation*}
+and secondly a micro-distortion density tensor
+\begin{equation*}
+ \bP \in H(\Curl; \Omega, \R^{d \times d}).
+\end{equation*}
+Note that the $\bP$ are not necessarily symmetric.  The matrix curl is defined row-wise,
+i.e., for a matrix field $\bP : \R^3 \subset \Omega \to \R^{3 \times 3}$ we define the
+matrix curl as
+\begin{equation*}
+ (\Curl P)_i
+ \colonequals
+ \Big( \frac{\partial P_{i3}}{\partial x_2} - \frac{\partial P_{i2}}{\partial x_3}, \quad
+       \frac{\partial P_{i3}}{\partial x_2} - \frac{\partial P_{i3}}{\partial x_2}, \quad
+       \frac{\partial P_{i3}}{\partial x_2} - \frac{\partial P_{i3}}{\partial x_2} \Big).
+\end{equation*}
+\todosander{Wie würde das Modell in 2d aussehen?}
+
+We define the quantities
+\begin{alignat*}{2}
+ e & \colonequals \nabla \bu - \bP   & \qquad & \text{(elastic distortion)} \\
+ \varepsilon_p & \colonequals \sym \bP &      & \text{(micro-strain)} \\
+ \alpha        & \colonequals \Curl e = - \Curl \bP & & \text{(micro-dislocation)}
+\end{alignat*}
+
+We consider the minimization of the energy functional
+\begin{align*}
+ \mathcal{E}(e,\varepsilon_p,\alpha)
+ & =
+ \mu_e \norm{\sym(\nabla u - P)}^2 + \mu_c \norm{\skew(\nabla u - P)}^2 + \frac{\lambda_e}{2} \abs{\tr(\nabla u - P)}^2 \\
+ & \quad
+ + \mu_h \norm{\sym P}^2 + \frac{\lambda_h}{2}\abs{\tr P}^2 \\
+ & \quad
+ + \frac{\alpha_1}{2} \norm{\dev \sym \Curl P}^2
+ + \frac{\alpha_2}{2} \norm{\skew \Curl P}^2
+ + \frac{\alpha_3}{2} \abs{\tr \Curl P}^2
+\end{align*}
+where $\mu_e$, $\mu_c$, $\lambda_e$, $\mu_h$, $\lambda_h$, $\alpha_1$, $\alpha_2$, $\alpha_3$
+are material parameters subject to
+\begin{align*}
+ \mu_e > 0 \qquad 2\mu_e + 3\lambda_e > 0, \qquad \mu_h > 0, \qquad 2\mu_h + 3\lambda_h > 0,\\
+ \alpha_1 > 0, \qquad \alpha_2 > 0, \qquad \alpha_3 > 0.
+\end{align*}
+
+% The weak formulation of the problem is: Find $(u,P) \in H^1 \times H_0(\Curl)$ such that
+% \begin{equation}
+%  \label{eq:weak_formulation}
+%  a((u,P),(v,Q)) = \ell(v,Q)
+%  \qquad
+%  \text{for all $(v,Q) \in H^1 \times H_0(\Curl)$}.
+% \end{equation}
+% Here, the bilinear form $a(\cdot,\cdot)$ is
+% \begin{multline*}
+% \label{eq:bilinear_form}
+%  a((u_1,P_1),(u_2,P_2))
+%  \colonequals
+%  \int_\Omega \bigg( \langle \mathbb{C} : \sym(\nabla u_1 - P_1),\sym(\nabla u_2 - P_2) \rangle
+%             + \langle \mathbb{H} \sym P_1, \sym P_2 \rangle \\
+%             + \langle \mathbb{L}_c \Curl P_1, \Curl P_2 \rangle \bigg)\,dx.
+% \end{multline*}
+When the material is isotropic, this reduces to
+\todosander{Prüfen!}
+\begin{multline*}
+\label{eq:bilinear_form_isotropic}
+ a((u_1,P_1),(u_2,P_2))
+ \colonequals
+ \int_\Omega \bigg( \mu_e \langle \sym(\nabla u_1 - P_1), \sym(\nabla u_2 - P_2) \rangle \\
+                  + \mu_c \langle \skew(\nabla u_1 - P_1), \skew(\nabla u_2 - P_2) \rangle
+                  + \frac{\lambda_e}{2} (\tr (\nabla u_1 - P_1) \cdot \tr(\nabla u_2 - P_2) \\
+                  + \mu_h \langle \sym P_1, \sym P_2 \rangle + \frac{\lambda_h}{2} \tr P_1 \tr P_2 \\
+                  + \frac{\alpha_1}{2} \langle \dev \sym \Curl P_1, \dev \sym \Curl P_2 \rangle \\
+                  + \frac{\alpha_2}{2} \langle \skew \Curl P_1, \skew \Curl P_2 \rangle
+                  + \frac{\alpha_3}{2} \langle \tr \Curl P_1, \tr \Curl P_2 \rangle
+            \bigg)\,dx.
+\end{multline*}
+
+
+
+\subsection{Well-posedness}
+
+To investigate well-posedness of the problem, \cite{von_Neff_zitierte_Quelle}
+introduced
+\begin{equation*}
+ \tnorm{P}^2 \colonequals \norm{\sym P}^2_{L^2(\Omega)} + \norm{\Curl P}^2_{L^2(\Omega)}.
+\end{equation*}
+This is only a semi-norm on $H(\Curl)$---all skew-symmetric curl-free matrix fields
+map to zero.  It is, however, a norm on
+\begin{equation*}
+ H_0(\Curl)
+ \colonequals
+ \Big\{ P \in H(\Curl) \; : \; \text{$P(x)\tau = 0$ for almost all $x$ on $\partial \Omega$
+                                      and $\tau$ tangent to $\partial \Omega$} \Big\}.
+\end{equation*}
+Even more, it is equivalent there to the canonical $H(\Curl)$-norm.
+\todosander{Vermutung: Wir brauchen diese ungewöhnliche Norm, um den Fall $\mu_c = 0$
+ abdecken zu können.  Für $\mu_c > 0$ könnte man vielleicht Elliptizität in der
+ $H(\Curl)$-norm zeigen.}
+
+Using this particular norm, \citet{neff_ghiba_lazar_madeo:2015} showed ellipticity
+of the bilinear form~\eqref{eq:bilinear_form} on $H_0(\Curl)$.  For this, define
+the norm
+\begin{equation*}
+ \tnorm{(u,P)}^2_{\mathcal{X}} \colonequals \norm{u}^2_{H^1(\Omega)} + \tnorm{P}^2.
+\end{equation*}
+
+\begin{theorem}[Thm.\,3.2 in~\cite{neff_ghiba_lazar_madeo:2015}]
+ \label{thm:ellipticity}
+ Assume that (i) the constitutive coefficients satisfy the symmetry relations~\eqref{}
+ and the inequalities~\eqref{}; and (ii) the loads satisfy the regularity conditions~\eqref{}.
+ Then the bilinear form $a(\cdot,\cdot)$ defined in~\eqref{eq:bilinear_form}
+ is continuous and coercive on $H^1 \times H_0(\Curl)$ with respect to the $\tnorm{\cdot}_{\mathcal{X}}$-norm.
+\end{theorem}
+
+\todosander{Bei Neff ist immer $\mu_c = 0$, aber in der Darstellung hier
+soll auch der Fall $\mu_c > 0$ behandelt werden.}
+
+Cauchy--Schwarz- and Poincaré inequalities imply that the linear operator $\ell$
+is bounded. Hence existence of a unique solution of \eqref{eq:weak_formulation}
+follows by the Lax--Milgram lemma.
+
+
+
+
+\section{Discretization by finite elements}
+
+We now consider a finite element discretization of the relaxed linear micromorphic model.
+As the natural spaces for the weak formulation~\eqref{eq:weak_formulation} are
+$H^1$ for the displacements and $H(\Curl)$ for the micro-distortion, we select
+first-order Lagrange finite elements $V_h^\text{La}$ for the displacements, and
+\begin{multline*}
+ V_h^\text{Né}
+ \colonequals
+ \Big\{ P \in H(\Curl) \; : \; \text{$P_i$ is a first-order Nédélec function of the first kind} \\
+     \text{for $i=1,\dots,3$} \Big\}.
+\end{multline*}
+As this is a conforming discretization for the pair of spaces $H^1 \times H(\Curl)$,
+from Theorem~\eqref{thm:ellipticity} we immediately get coercivity and continuity
+of $a(\cdot,\cdot)$ on $V_h^\text{La} \times V_h^\text{Né}$ with respect to the
+norm $\tnorm{\cdot}_\mathcal{X}$.
+\todosander{Diskutiere den Fall $\mu_c > 0$!}
+
+\begin{theorem}
+ {[ASSUMPTIONS]} the finite element discretization of the relaxed micromorphic
+ model has a unique solution.
+\end{theorem}
+
+
+\section{The dynamic case}
+
+\printbibliography
+\end{document}

src/CMakeLists.txt

 if(ADOLC_FOUND AND IPOPT_FOUND AND PYTHONLIBS_FOUND AND dune-uggrid_FOUND)
   set(programs finite-strain-elasticity
-               linear-elasticity)
+               linear-elasticity
+               relaxed-micromorphic-continuum)
 
   foreach(_program ${programs})
     add_executable(${_program} ${_program}.cc)