\documentclass{bredelebeamer} %This looks good
%\documentclass[8pt]{beamer}
\usepackage{stmaryrd} % for \llbracket
\usepackage{multirow} % for \multirow
\usepackage{graphicx,color} % for Inkscape
\graphicspath{{images/}} % for Inkscape
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\title[Nitsche IGA]{Nitsche's method for patch coupling}

\subtitle{in isogeometric analysis}

\author[Qingyuan HU]{Qingyuan HU \inst{DUT,UL},\\ Supervisor: St\'ephane BORDAS \inst{UL}}


\institute[(University of Luxembourg)]
{
  \inst{DUT}
  Dalian University of Technology\\
  P.R. China
  
  \inst{UL}
  University of Luxembourg\\
  Luxembourg
}

%\date{15 April 2017}

%\subject{test}

\titlegraphic
{
\includegraphics[height=1cm]{images/ul.jpg}
\includegraphics[height=1cm]{images/erc.jpg}
}

%\logo{\includegraphics[scale=0.5]{images/uniLux.jpg}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\begin{frame}
  \titlepage
\end{frame}


%\begin{frame}{Outline}
%  \tableofcontents
  % possibilité d'ajouter l'option [pausesections]
%\end{frame}


\section{Motivation}

\begin{frame}{Why do we need patch coupling}

    \begin{itemize}
\item Model complex structures
    \end{itemize}

\begin{columns}
\begin{column}{0.6\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.24]{images/complex_model_nguyen.jpg}
\caption{Connecting rod \\ $\left[ V.P.Nguyen\ et.al.\ 2013 \right]$}
\end{figure}
\end{column}

\begin{column}{0.4\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.25]{images/complex_model.jpg}
\caption{Intersecting tubular shell \\ $\left[Y.Guo\ et.al.\ 2017\right]$}
\end{figure}
\end{column}
\end{columns}

\end{frame}


\begin{frame}{Why do we need patch coupling}

    \begin{itemize}
\item Model complex structures
\item Assign various materials to sub-structures
    \end{itemize}

\begin{columns}
\begin{column}{0.6\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.25]{images/iphone6.jpg}
\caption{iPHONE 6S \\ $\left[www.visualcapitalist.com\right]$}
\end{figure}
\end{column}

\begin{column}{0.4\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.26]{images/carseat.jpg}
\caption{Seat frames \\ $\left[ae\-plus.com\right]$}
\end{figure}
\end{column}
\end{columns}

\end{frame}


\begin{frame}{Why do we need patch coupling}

    \begin{itemize}
\item Model complex structures
\item Assign various materials to sub-structures
\item Calculate using suitable elements (dimensions)
    \end{itemize}

\begin{columns}
\begin{column}{0.6\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.24]{images/blendedshell_Guo.jpg}
\caption{Mixed-dimensional coupling \\ $\left[Y.Guo\ and\ M.Ruess\ 2015\right]$}
\end{figure}
\end{column}

\begin{column}{0.4\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.25]{images/blendedshell.jpg}
\caption{Simulation of metal forming \\ $\left[D.J.Benson\ et.al.\ 2012\right]$}
\end{figure}
\end{column}
\end{columns}

\end{frame}


\begin{frame}{Why do we need patch coupling}

    \begin{itemize}
\item Model complex structures
\item Assign various materials to sub-structures
\item Choose reasonable element types (dimensions)
\item Discrete model into elements of sufficient numbers
    \end{itemize}

\begin{columns}
\begin{column}{0.65\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.27]{images/cardoor.jpg}
\caption{A truck door, left: commercial software results \\ $\left[Marco\ Brino\ 2015\right]$}
\end{figure}
\end{column}

\begin{column}{0.35\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.21]{images/Saint-Venant.jpg}
\caption{Saint Venant’s Principle \\ $\left[V.P.Nguyen\ et.al.\ 2014\right]$}
\end{figure}
\end{column}
\end{columns}

\end{frame}


\begin{frame}{Why do we need patch coupling: from FEM to IGA}
\begin{itemize}
\item IGA use NURBS (non-uniform rational B-spline) instead of polynomials
\end{itemize}

\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.18]{images/lagrange.eps}
\caption{Lagrange basis functions in FEM}
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.18]{images/nurbs.eps}
\caption{NURBS basis functions in IGA}
\end{figure}
\end{column}
\end{columns}

\begin{itemize}
\item \alert{Non-interpolatory} control points
\end{itemize}

\begin{columns}
\begin{column}{0.33\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.34]{images/mesh_FEM.eps}
\caption{Meshes and nodes in FEM}
\end{figure}
\end{column}
\begin{column}{0.33\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.17]{images/patch.jpg}
\caption{Control meshes and control points in IGA}
\end{figure}
\end{column}
\begin{column}{0.33\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/curve.jpg}
\caption{NURBS curve}
\end{figure}
\end{column}
\end{columns}

\end{frame}


\section{Idea}

\begin{frame}{Stitching two fabrics together}

\begin{figure}
\centering
\includegraphics[scale=0.14]{images/stitching.jpg}
\caption{Stitching fabrics\ $\left[bigbgsd.blogspot.com\right]$}
\end{figure}

Where there are displacement gaps,  \\
there should be some kind of forces to prevent the two fabrics from separation, \\
and additional work to be done to stitching them together.

\end{frame}


\begin{frame}{Analogy}

\begin{columns}

\begin{column}{0.3\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.06]{images/stitching.jpg}
\caption{Stitching fabrics}
\end{figure}
\end{column}

\begin{column}{0.65\textwidth}
\begin{figure}
\centering
\def\svgwidth{0.9\textwidth}
\input{images/Nitsche_couple.pdf_tex}
\caption{Patch coupling}
% made by Qingyuan HU
\end{figure}
\end{column}

\end{columns}

\end{frame}


\begin{frame}{Constraints and additional work}

\begin{columns}

\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\def\svgwidth{1\textwidth}
\input{images/Nitsche_couple.pdf_tex}
\caption{Patch coupling}
% made by Qingyuan HU
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
Constraints on interface $\Gamma_C$
\begin{subequations}
\begin{align}
u_1 &= u_2 \qquad \textrm{on $\Gamma_C$} \\
F_1 &= F_2 \qquad \textrm{on $\Gamma_C$}
\end{align}
\end{subequations}
Define jump and average operators
\begin{equation}
\begin{split}
\llbracket u \rrbracket &:= u_1-u_2 \\
\langle F \rangle &:= \frac{1}{2}(F_1+F_2)
\end{split}
\end{equation}
Additional work to be done
\begin{equation}
W_{add}=\langle F \rangle \llbracket u \rrbracket
\end{equation}
\end{column}

\end{columns}

\end{frame}


\section{Nitsche}


\begin{frame}{Problem setup}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.38]{images/couple.eps}
\caption{Couple two patches}
% made by Qingyuan HU
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{subequations}
\begin{align}
\boldsymbol{u}_1 &= \boldsymbol{u}_2 \qquad\qquad\;\; \textrm{ on $\Gamma_C$} \\
\boldsymbol{\sigma}_1\cdot\boldsymbol{N}_1 &= -\boldsymbol{\sigma}_2\cdot\boldsymbol{N}_2 \qquad \textrm{on $\Gamma_C$}
\end{align}
\end{subequations}
\end{column}
\end{columns}

The jump and average operators are defined as
\begin{equation}
\begin{split}
\llbracket \boldsymbol{u} \rrbracket &:= \boldsymbol{u}_1-\boldsymbol{u}_2 \\
\langle\boldsymbol{\sigma}\boldsymbol{N} \rangle &:= \frac{1}{2}(\boldsymbol{\sigma}_1\boldsymbol{N}+\boldsymbol{\sigma}_2\boldsymbol{N})
\end{split}
\end{equation}
here $\boldsymbol{N}$ is chosen to be $\boldsymbol{N}_1$.

\end{frame}

    
\begin{frame}{Different from fabrics stretching}
\begin{itemize}
\item Instead of scalar $u$, use vector $\boldsymbol{u}$ for generalized cases, e.g. $\boldsymbol{u}=(u,v)^{\rm T}$ in 2D
\end{itemize}

\begin{itemize}
\item Instead of forces F, use traction $\boldsymbol{\sigma}(\boldsymbol{u})\boldsymbol{N}$, \\ where the stress comes from displacement field
\begin{equation}
   \boldsymbol{\sigma}(\boldsymbol{u})=\boldsymbol{D}\boldsymbol{\varepsilon}(\boldsymbol{u})=\boldsymbol{D}\nabla\boldsymbol{u}
\end{equation}
and $\boldsymbol{N}$ is the transformation matrix to collect area contribution.
\end{itemize}

\begin{columns}
\begin{column}{0.9\textwidth}
\begin{figure}
\centering
\def\svgwidth{1\textwidth}
\input{images/N.pdf_tex}
\caption{$\boldsymbol{u}$, $\boldsymbol{n}$ and $\boldsymbol{\sigma}$}
% made by Qingyuan HU
\end{figure}
\end{column}

\begin{column}{0.2\textwidth}
\begin{equation*}
\boldsymbol{N}=
\begin{bmatrix}
    n_x & 0 & n_y \\
    0 & n_y & n_x
\end{bmatrix}
\end{equation*}
\end{column}
\end{columns}
\end{frame}


\begin{frame}{Nitsche formulation}

Start from the classical weak form
\begin{equation}
a(\boldsymbol{u},\boldsymbol{w})=L(\boldsymbol{w})
\end{equation}
and introduce Nitsche contribution into the weak form
\begin{equation}
a(\boldsymbol{u},\boldsymbol{w})-\int_{\Gamma_C} \langle\boldsymbol{\sigma}(\boldsymbol{u})\boldsymbol{N} \rangle\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma-\int_{\Gamma_C}\llbracket \boldsymbol{u} \rrbracket\langle\boldsymbol{\sigma}(\boldsymbol{w})\boldsymbol{N} \rangle {\rm d}\Gamma+\alpha\int_{\Gamma_C} \llbracket \boldsymbol{u} \rrbracket\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma=L(\boldsymbol{w})
\end{equation}

\alert{Note}
\begin{itemize}
\item Two Nitsche terms are introduced to keep the stiffness matrix symmetric
\item Additional stabilisation parameter $\alpha$ to guarantee coercive (positive definite)
%\item According to Newton's third law, we use the traction force on the slave boundary
\item Boundary integrations are performed along slave boundary
\item The Nitsche contributions are made by work–conjugate pairs:
\\ for membrane element they are displacement and traction force,
\\ for thin bending plate they are rotation and bending moment
\end{itemize}

\end{frame}


\begin{frame}{Penalty and Lagrange multiplier}

\alert{Penalty method}:
\begin{equation}
a(\boldsymbol{u},\boldsymbol{w})+\frac{\alpha}{2}\int_{\Gamma_C}\llbracket \boldsymbol{u} \rrbracket\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma=L(\boldsymbol{w})
\end{equation}
where $\alpha$ is the penalty parameter.
\\~\\
\alert{Lagrange multiplier method}:
\begin{equation}
    a(\boldsymbol{u},\boldsymbol{w})+\int_{\Gamma_C}\boldsymbol{\lambda}\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma+\int_{\Gamma_C}\delta\boldsymbol{\lambda}\llbracket \boldsymbol{u} \rrbracket{\rm d}\Gamma=L(\boldsymbol{w})
\end{equation}
where $\boldsymbol{\lambda}$ is the vector of Lagrange multiplier.

\begin{table}
\begin{tabular}{l | c | c }
\hline
Methods & Pros  & Cons   \\\hline
\multirow{2}{*}{Penalty}   & No increased DOFs & Depends on penalty parameter \\   & Easy and straightforward & sometimes ill-conditioned    \\\hline
\multirow{2}{*}{Lagrange multiplier} & $\boldsymbol{\lambda}$ means traction & Increase DOFs \\ & Stable when satisfies LBB & Not positive define   \\\hline
\multirow{2}{*}{Nitsche} & No increased DOFs   & Not parameter-free \\ & Positive define, robust & Involve constitutive equation \\
\hline
\end{tabular}
\end{table}

\end{frame}


\begin{frame}{How do they work}

\alert{Penalty method}:
\begin{equation}
a(\boldsymbol{u},\boldsymbol{w})+\frac{\alpha}{2}\int_{\Gamma_C}\llbracket \boldsymbol{u} \rrbracket\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma=L(\boldsymbol{w})
\end{equation}

\alert{Lagrange multiplier method}:
\begin{equation}
    a(\boldsymbol{u},\boldsymbol{w})+\int_{\Gamma_C}\boldsymbol{\lambda}\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma+\int_{\Gamma_C}\delta\boldsymbol{\lambda}\llbracket \boldsymbol{u} \rrbracket{\rm d}\Gamma=L(\boldsymbol{w})
\end{equation}

\alert{Nitsche's method}:
\begin{equation}
a(\boldsymbol{u},\boldsymbol{w})-\int_{\Gamma_C} \langle\boldsymbol{\sigma}(\boldsymbol{u})\boldsymbol{N} \rangle\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma-\int_{\Gamma_C}\llbracket \boldsymbol{u} \rrbracket\langle\boldsymbol{\sigma}(\boldsymbol{w})\boldsymbol{N} \rangle {\rm d}\Gamma+\alpha\int_{\Gamma_C} \llbracket \boldsymbol{u} \rrbracket\llbracket \boldsymbol{w} \rrbracket{\rm d}\Gamma=L(\boldsymbol{w})
\end{equation}

\begin{columns}
\begin{column}{0.33\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.6]{images/Penalty.eps}
\caption{$\alpha$ is spring stiffness}
% made by Qingyuan HU
\end{figure}
\end{column}

\begin{column}{0.33\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.6]{images/Lagrange_multiplier.eps}
\caption{$\lambda$ is external traction}
% made by Qingyuan HU
\end{figure}
\end{column}

\begin{column}{0.33\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.6]{images/Nitsche.eps}
\caption{$\sigma$ is from slave boundary}
% made by Qingyuan HU
\end{figure}
\end{column}
\end{columns}

\end{frame}


\begin{frame}{Slave boundary to perform boundary integration}

\begin{itemize}
\item Choose slave boundary that has more elements
\end{itemize}

\begin{figure}
\centering
\includegraphics[scale=0.4]{images/gauss_new.eps}
% made by Qingyuan HU
\end{figure}

\begin{itemize}
\item Choose slave boundary that has shorter edge
\end{itemize}

\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.2]{images/slave1.jpg}
\caption{2D \\ $\left[V.P.Nguyen\ et.al.\ 2013\right]$}
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.18]{images/slave2.jpg}
\caption{3D \\ $\left[V.P.Nguyen\ et.al.\ 2013\right]$}
\end{figure}
\end{column}
\end{columns}

\end{frame}


\begin{frame}{Stiffness matrix illustration}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.45]{images/stiffness_matrix_illustration.eps}
% made by Qingyuan HU
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{enumerate}
		\item Calculate stiffness matrix \\ for slave and master \\ $\boldsymbol{K} \: = \: \sum \boldsymbol{K}_s+\sum \boldsymbol{K}_m$
		\item Calculate Nitsche contribution \\ along coupled boundary \\ $\boldsymbol{K} \: += \: \sum \boldsymbol{K}_N$
	\end{enumerate}
\end{column}
\end{columns}
\end{frame}


\begin{frame}{Stabilisation parameter $\alpha$}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.5]{images/stable_parameter_illustration.eps}
% made by Qingyuan HU
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{enumerate}
		\item Solve generalized eigenvalues $\boldsymbol{\lambda}$ \\ along coupled boundary \\ $\boldsymbol{K}_N \boldsymbol{u}_c = \lambda \boldsymbol{K} \boldsymbol{u}_c$ 
		\item $\alpha = 2 max(\lambda)$ \\
		ref A.Apostolatos et.al. IJNME. 2013
	\end{enumerate}
\end{column}
\end{columns}
\end{frame}


\section{Numerical illustrations}

\begin{frame}{Bending plate}

\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.3]{images/sp_plate.eps}
\caption{Bending plate}
\end{figure}
\end{column}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/static_plate_conforming_Hrefine_3.eps}
% This is the conforming mesh result
\end{figure}
\end{column}
\end{columns}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/static_plate_nonconforming_xita_1_Hrefine_1.eps}
% This is the non-conforming mesh result
\end{figure}
\end{column}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/static_plate_nonconforming_xita_1_Hrefine_2.eps}
% This is the non-conforming mesh result
\end{figure}
\end{column}
\end{columns}

\end{frame}


\begin{frame}{Vibration square plate}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.55]{images/sp_plate_vib.eps}
\caption{Square plate}
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.06]{images/non_dyna.jpg}
\caption{Meshes}
\end{figure}
\end{column}
\end{columns}
\end{frame}


\begin{frame}{Vibration square plate}

\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/W_11_eigenmoda.eps}
% This is the analytical solution of the 1st eigenmodal
\end{figure}
\end{column}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/W_22_eigenmoda.eps}
% This is the analytical solution of the 4th eigenmodal
\end{figure}
\end{column}
\end{columns}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/Nitsche_xita1_1.eps}
% The 1st eigenmodal
\end{figure}
\end{column}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.12]{images/Nitsche_xita1_4.eps}
% The 4th eigenmodal
\end{figure}
\end{column}
\end{columns}
\end{frame}


\begin{frame}{Clamped plate and errors [X.Du et.al. 2015]}
\begin{figure}
\centering
\includegraphics[scale=0.25]{images/clamp_plate.jpg}
\end{figure}

\begin{figure}
\centering
\includegraphics[scale=0.25]{images/clamp_circle_plate.jpg}
\end{figure}
\end{frame}


\begin{frame}{Connecting rod [V.P.Nguyen et.al. 2014]}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.3]{images/rod_result1.jpg}
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.3]{images/rod_result2.jpg}
\end{figure}
\end{column}
\end{columns}
\end{frame}


\begin{frame}{Intersecting tubular shell [Y.Guo et.al. 2017]}

\begin{columns}
\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.5]{images/tubular_displacement.jpg}
\end{figure}
\end{column}

\begin{column}{0.5\textwidth}
\begin{figure}
\centering
\includegraphics[scale=0.35]{images/tubular_stress.jpg}
\end{figure}
\end{column}
\end{columns}
\end{frame}

%\begin{frame}[allowframebreaks]
%\frametitle{References}
%\bibliographystyle{plain}
%\bibliographystyle{apalike}
%\bibliography{main.bib}
%\end{frame}

\end{document}
