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\title[GIFT]{Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super- geometric analysis to Geometry Independent Field approximaTion (GIFT)}
\author{Elena Atroshchenko, Gang Xu, Satyendra Tomar, St\'ephane P.A. Bordas}
\institute{University of Chile, Hangzhou Dianzi University, University of Luxembourg}
\date{\today}

\begin{document}

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

% Uncomment these lines for an automatically generated outline.
%\begin{frame}{Outline}
%  \tableofcontents
%\end{frame}

\section{}
\begin{frame}{Outline of the presentation}
\begin{itemize}
  \item Motivation 
  \item Theoretical background
  \item Patch tests
  \item Numerical results
  \item Conclusions
\end{itemize}
\end{frame}

\section{Motivation}
\begin{frame}{Motivation: IsoGeometric Analysis (IGA)}
\begin{itemize}
  \item Tight link between CAD and analysis
  \item The same basis functions, which are used in CAD to represent the geometry, are used in the IGA as shape functions to approximation the unknown solution
  \item Geometry is exact at any stage of the solution refinement process
  \item Better accuracy per DOF in comparison with standard FEM
  \item Additional advantages, such as higher continuity of splines, makes IGA applicable for PDEs of higher order
%     \begin{center}
%     \includegraphics[scale=.2]{Solvation_Diagram.png}
%     \end{center}
%   \footnotetext[1]{\tiny{Silverstein, K. A. T.; Haymet, A. J. D.; Dill, K. A. Molecular Model of Hydrophobic Solvation J. Chem. Phys. \textbf{1999}, 111(17), 8000-8009}} 
\end{itemize}
\end{frame}

\begin{frame}{Disadvantages of IsoGeometric Analysis (IGA)}
\begin{itemize}
  \item Gaps can occur when different geometrical pieces are joined
  \item Additional coupling mechanisms are required for multi-patch geometries
  \item Tensor-product structure of NURBS does not allow local refinement
%     \begin{center}
%     \includegraphics[scale=.2]{Solvation_Diagram.png}
%     \end{center}
%   \footnotetext[1]{\tiny{Silverstein, K. A. T.; Haymet, A. J. D.; Dill, K. A. Molecular Model of Hydrophobic Solvation J. Chem. Phys. \textbf{1999}, 111(17), 8000-8009}} 
\end{itemize}
\end{frame}

\begin{frame}{What is ``I" in the IGA?}
\hspace*{-0.5in}
\begin{tabular}{ccc}
CAD model \\
(surface for 2d or volume for 3d) & \arrowright & Knot vectors and control points: \\
\includegraphics[scale=.3]{figures/geoCAD_1.pdf}
&  &
\hspace{-0.5cm}
\includegraphics[scale=.175]{figures/geoCAD_3.pdf} \includegraphics[scale=.2]{figures/geoCAD_4.pdf}\\
\end{tabular}
\end{frame}

\begin{frame}{What is ``I" in the IGA?}
\begin{tabular}{ccc}
on each element: & \arrowright & global assembly: \\
$\boldsymbol{x} = \sum\limits_{n = 1}^{N_{e}}\boldsymbol{C}_{i}^{e}N_{i}(\xi, \eta)$
&  & \\
\vspace{0.5cm}
$\boldsymbol{u} = \sum\limits_{n = 1}^{N_{e}}U_{i}^{e}N_{i}(\xi, \eta)$ & & $\boldsymbol{K}\boldsymbol{U} = \boldsymbol{f}$\\
$\boldsymbol{K}^{e}\boldsymbol{U}^{e} = \boldsymbol{f}^{e}$ & & \\
\end{tabular}
\end{frame}

\begin{frame}{What is ``I" in the IGA?}
\begin{tabular}{ccc}
refinement: & \arrowright & on each element:  \\
& & $\boldsymbol{x} = \sum\limits_{n = 1}^{N_{e}}\boldsymbol{C}_{i}^{e}N_{i}(\xi, \eta)$
\\
& & $\boldsymbol{u} = \sum\limits_{n = 1}^{N_{e}}U_{i}^{e}N_{i}(\xi, \eta)$\\
\hspace{-0.5cm}
\includegraphics[scale=.2]{figures/geoCAD.png} \includegraphics[scale=.15]{figures/geoCAD_6.pdf} & & $\boldsymbol{K}^{e}\boldsymbol{U}^{e} = \boldsymbol{f}^{e}$ \\
\end{tabular}
\end{frame}

\begin{frame}{What is ``I" in the IGA?}

{{\it Isogeometric} means that the same shape functions are used on each element to represent the geometrical variables $\boldsymbol{x}$ and the field $\boldsymbol{u}$: } 
$$\boldsymbol{x} = \sum\limits_{n = 1}^{N_{e}}\boldsymbol{C}_{i}^{e}N_{i}(\xi, \eta)$$

$$\boldsymbol{u} = \sum\limits_{n = 1}^{N_{e}}U_{i}^{e}N_{i}(\xi, \eta)$$

\end{frame}
\begin{frame}{What is the difference between ``iso-geometric" and ``iso-parametric"?}
In the standard FEM iso-parametric elements are used to {\bf approximate} both, the field and the computational domain, while in the IGA, refinement does not improve geometry parameterization (it remains exact and equivalent to the original model at each refinement stage), therefore geometry refinement is redundant. 

\end{frame}
\begin{frame}{Are NURBS always the best choice to approximate the solution?}
\begin{itemize}
\item Rational functions
\item No local refinement
\end{itemize} 
\end{frame}

\begin{frame}{What can we improve?}
\begin{itemize}
\item Keep the exact representation of the geometry
\item Choose more suitable approximation for the field
\end{itemize} 
\end{frame}

\section{Patch tests}

\begin{frame}{Patch test: will it pass or fail?}
\hspace*{-0.65in}
\begin{tabular}{cl}
\multirow{4}{*}{\includegraphics[scale=0.75]{figures/plot_qannu.eps}} & \\
& Laplace eqn: \\
 & 
$\Delta{u} = 0,  \,\,\,\,\text{in}\,\,\,\Omega $\\
& $u|_{\partial\Omega} (x,y) = 1 + x + y. $ \\
& \\
& Elasticity eqn: \\
& $\sigma_{ij,j}  = 0,  \,\,\,\,\text{in}\,\,\,\Omega $\\
& $t_{i} = \sigma_{0}n_{i}\,\,\,\, \text{at}\,\,\, r = 1, 2,$ \\
& $u_{2} = 0, t_{1} = 0 \,\,\,\, \text{at}\,\,\, \theta = 0,$ \\
& $u_{1} = 0, t_{2} = 0 \,\,\,\, \text{at}\,\,\, \theta = \pi/2,$ \\
\end{tabular}
\end{frame}

\begin{frame}{Patch test: will it pass or fail?}
Quarter annulus parameterizations:

\vspace*{-2mm}
\hspace*{-4mm}
\includegraphics[scale=0.45]{figures/plot_Q0.eps}
\hspace*{-8mm}
\includegraphics[scale=0.45]{figures/plot_A1.eps}
\hspace*{-8mm}
\includegraphics[scale=0.45]{figures/plot_C1.eps}

\vspace*{-2mm}
\setlength{\tabcolsep}{2.0pt}
\begin{tabular}{|c|c|c|}
\hline
$Q_{0}$ & $A_{1}$ & $C_{1}$ \\ \hline
$\Sigma = [0,0,1,1]$ & 
$\Sigma = [0,0,2/3,1,1]$ & 
$\Sigma = [0,0,2/3,1]$ \\
$\Pi = [0,0,0,1,1,1]$  & 
$\Pi = [0,0,0,1/8,1,1,1]$  & 
$\Pi = [0,0,0,1/8,1,1,1]$  \\ \hline
& $B_{1}$ & \\ \hline
& $\Sigma = [0,0,0.17,1,1]$ & \\
& $\Pi = [0,0,0,0.81,1,1,1]$ & \\ \hline
& $D_{1}$ (B-Splines) & \\ \hline
& $\Sigma = [0,0,2/3,1,1]$ & \\
& $\Pi = [0,0,0,1/8,1,1,1]$ & \\
\hline
\end{tabular}
\end{frame}

\begin{frame}{Five types of geometry parametrization}

\begin{itemize}
\item $\mathbf{Q_{0}:}$ the coarsest parameterization (single element) necessary to represent the geometry,
\item $\mathbf{A_{1}:}$ uniform parametrization (four elements), obtained by the refinement of $Q_{0}$ with knot insertion at $2/3$ and $1/8$
\item $\mathbf{B_{1}:}$ uniform parametrization (four elements), obtained by the refinement of $Q_{0}$ with knot insertion at $0.17$ and $0.81$
\item $\mathbf{C_{1}:}$ non-uniform parametrization (four elements), obtained from $A_{1}$ by moving the internal points randomly
\begin{itemize}
\item Bases $A_{2}$, $B_{2}$, $C_{2}$, are obtained by elevating degree in both directions by 1 of $A_{1}$, $B_{1}$, and $C_{1}$, respectively.
\end{itemize}
\item $\mathbf{D_{1}}$ \textbf{(B-Splines):} uniform parametrization (four elements), obtained from $A_{1}$ by setting all the weights to $1$
\begin{itemize}
\item Bases $D_{0}$ and $D_{2}$ are obtained from $D_{1}$ by reducing and elevating (respectively) degrees in both directions.
\end{itemize}
\end{itemize}

\end{frame}

%TODO_SATYEN - Would you mind explaining in a slide or next to the table what the meanings of the different labels are, so that it is easier for me to go through the table, at this point I have difficulty going through the table and explaining the results as they are written. 
\begin{frame}
\begin{table}
\tiny
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
$T^{\ell}_{G_{i}, S_{j}}$ & Laplace Eq. & Elasticity Eq.\\
\hline 
$T^{0}_{Q_{0}, A_{1}}$ & 1.3815e-15 & 3.0871e-14 \\
$T^{0}_{Q_{0}, A_{2}}$ & 5.2147e-15 & 1.7986e-14 \\
$T^{0}_{Q_{0}, C_{1}}$ & 0.0182 & 0.0050 \\
$T^{0}_{Q_{0}, C_{2}}$ & 0.0023 & 0.0012 \\
\hline \hline
$T^{1}_{A_{1}, A_{1}}$ & 1.0023e-15 & 1.1675e-14 \\
$T^{1}_{A_{1}, A_{2}}$ & 4.3958e-14 & 1.2547e-14 \\
$T^{1}_{A_{2}, A_{1}}$ & 1.4059e-15 & 1.5525e-15 \\
\hline \hline
$T^{2}_{B_{1}, A_{1}}$ & 1.4755e-15 & 2.9941e-15 \\
$T^{2}_{B_{1}, A_{2}}$ & 2.1639e-15 & 1.2118e-14 \\
$T^{2}_{B_{2}, A_{1}}$ & 1.0144e-15 & 5.4590e-15 \\
\hline \hline
$T^{3}_{C_{1}, C_{1}}$ & 1.1061e-15 & 1.6439e-14 \\
$T^{3}_{C_{1}, C_{2}}$ & 1.8263e-15 & 2.8737e-15 \\
$T^{3}_{C_{2}, C_{1}}$ & 1.2062e-15 & 5.6517e-14 \\
\hline \hline
$T^{4}_{C_{1}, A_{1}}$ & 0.0203 & 0.0085 \\
$T^{4}_{C_{1}, A_{2}}$ & 0.0016 & 0.0009 \\
$T^{4}_{C_{2}, A_{1}}$ & 0.0203 & 0.0085 \\
\hline \hline
$T^{5}_{A_{1}, D_{1}}$ & 0.0188 & 0.0214 \\
$T^{5}_{A_{1}, D_{2}}$ & 0.0121 & 0.0039 \\
$T^{5}_{A_{1}, D_{0}}$ & 0.5418 & 0.1411 \\
\hline
\end{tabular} 
\caption{Results of various patch tests, denoted by $T$. Superscript $\ell$ denotes the test case, $G_{i}$ denotes the bases for the geometry, and $S_{j}$ denotes the bases for the solution approximation}
\label{tab:res_tests}
\end{center}
\end{table}
\end{frame}

% TODO_ELENA I don't quite understand this sentence. Can you explain a bit more, please? Thanks! 
\begin{frame}{Conclusion from the patch test studies}
Any of the following combination of bases, which are equal up to operations of knot insertion or degree elevation, pass the patch test
\begin{itemize}
\item Geometry by $Q_{0}$, together with $A_{i}$ or $B_{i}$ for the solution
\item Geometry by $A_{i}$ and solution by $A_{j}$
\item Geometry by $B_{i}$ and solution by $A_{j}$
\item Geometry by $C_{i}$ and solution by $C_{j}$
\end{itemize}

\begin{block}{Patch test for}
A mixture of $C_{i}$ with either of $Q_{0}$, $A_{i}$ or $B_{i}$ fails because $C_{i}$ can not be obtained from these bases (only internal points randomly moved)
\end{block}

\end{frame}
\section{Numerical examples}
\begin{frame}{Convergence studies}
\hspace*{-0.65in}
\begin{tabular}{cl}
\multirow{4}{*}{\includegraphics[scale=0.75]{figures/plot_qannu.eps}} & \\
& Laplace eqn: \\
& 
$\Delta{u} = 0,  \,\,\,\,\text{in}\,\,\,\Omega $\\
& $u|_{\partial\Omega} (x,y) = r^{-3}\cos 3\theta. $ \\
& \\
& Elasticity eqn: \\
& $\sigma_{ij,j}  = 0,  \,\,\,\,\text{in}\,\,\,\Omega $\\
& $t_{i} = \sigma_{1}n_{i}\,\,\,\, \text{at}\,\,\, r = 1, $ \\
& $t_{i} = \sigma_{2}n_{i}\,\,\,\, \text{at}\,\,\, r = 2, $ \\
& $u_{2} = 0, t_{1} = 0 \,\,\,\, \text{at}\,\,\, \theta = 0,$ \\
& $u_{1} = 0, t_{2} = 0 \,\,\,\, \text{at}\,\,\, \theta = \pi/2,$ \\
\end{tabular}
\end{frame}
% TODO_SATYEN what can we learn from the results on this slide and the next - is it possible to please provide some explanation in words? I have a hard time deciphering the nice notations A_1, A_2, ... and am worried to fumble through the slide if I don't have a proper guide. Thanks!! 

\begin{frame}{Numerical examples: Example 1 - Laplace}

%\vspace*{-1cm}
\begin{block}{Convergence studies for various combinations of bases}
For exact representation of the geometry, all combinations of bases, including those which fail the patch test, deliver the optimal convergence
\end{block}
%\begin{tabular}{ccc}
\vspace*{-2mm}
\hspace*{-5mm}
\scalebox{0.4}{\input{figures/plot_ex_1_1}}
%\includegraphics[scale = 0.45]{figures/plot_ex_1_1_b.pdf}
%\hspace*{-2.25in}
\scalebox{0.4}{\input{figures/plot_ex_1_2}}
%\includegraphics[scale = 0.45]{figures/plot_ex_1_2_b.pdf}
%\hspace*{-2.25in}
\scalebox{0.4}{\input{figures/plot_ex_1_3}}
%\includegraphics[scale = 0.45]{figures/plot_ex_1_3_b.pdf}
%\end{tabular}
\end{frame}

\begin{frame}{Numerical examples: Example 1 - Elasticity}

\begin{block}{Convergence studies for various combinations of bases}
For exact representation of the geometry, all combinations of bases, including those which fail the patch test, deliver the optimal convergence
\end{block}
\vspace*{-2mm}
\hspace*{-5mm}
\scalebox{0.4}{\input{figures/plot_ex_3_1}}
\scalebox{0.4}{\input{figures/plot_ex_3_2}}
\scalebox{0.4}{\input{figures/plot_ex_3_3}}
%\end{tabular}
%\begin{tabular}{ccc}
%\vspace*{-5mm}
%\hspace*{-0.75in}
%\includegraphics[scale = 0.45]{figures/plot_ex_3_1_b.pdf}
%\hspace*{-2.25in}
%\includegraphics[scale = 0.45]{figures/plot_ex_3_2_b.pdf}
%\hspace*{-2.25in}
%\includegraphics[scale = 0.45]{figures/plot_ex_3_3_b.pdf}
%\end{tabular}
\end{frame}

%TODO_SATYEN - Satyen, please can you read this one more time, I'm wondering if the geo must not be properly represented to get optimal convergence. Should we say optimal convergence is attained as long as the exact geometry can be represented by the geometry basis?
\begin{frame}{Conclusion:}
\begin{block}{}
Together with the given (exact) geometry parametrization at the coarsest level, the convergence rate is entirely defined by the solution basis, and does not depend on the further refinement of the geometry parametrization:
\end{block}
\begin{itemize}
\item For a given geometry parameterization, the degree of the solution basis can be increased or decreased without changing the degree of the geometry (from iso-geometric to super-geometric and sub-geometric elements)
\item For solution approximation, using same degree B-Splines or NURBS yields almost identical results
\end{itemize}
\end{frame}
%TODO_SATYEN - same here, what is the solution that converges at rate 2. How do we compare it to the others? Are all curves superimposed (top curve?) looks like it. What is the solution converging at rate 3? Maybe we just need one slide explaining the notations...
\begin{frame}{Numerical examples: Example 2 - Laplace}

\begin{block}{Convergence studies for various combinations of bases}
Exact representation of the geometry by $\mathcal{N}_{1,4}$, various bases for the solution approximation, iso/super/sub-geometric, deliver optimal convergence (governed by the minimum degree), second order for first five choices, and third order for last two choices. $\mathcal{B}$ represents B-splines.
\end{block}
\vspace*{-10mm}
\begin{minipage}{0.45\textwidth}
\vspace*{10mm}
\begin{equation*}
u(x, y) = \ln((x + 0.1)^2 + (y + 0.1)^2)
\end{equation*}
\includegraphics[scale = 0.5]{figures/plot_wedge.eps}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\hspace*{15mm}
\scalebox{0.45}{\input{figures/plot_ex_2_1}}
%\hspace*{-0.5in}
%\includegraphics[scale = 0.5]{figures/plot_ex_2_1_b.pdf}
\end{minipage}


\end{frame}

\begin{frame}{Numerical examples: Example 3 - Elasticity}
\begin{block}{Convergence studies for various combinations of bases}
Exact representation of the geometry by $\mathcal{N}_{2,2}$, various bases for the solution approximation, iso/super-geometric, deliver optimal convergence (governed by the minimum degree), third order for first three choices, and fourth order for last three choices. $\tilde{\mathcal{N}}$ are NURBS with weights of two inner points changed.
\end{block}
\vspace*{-5mm}
\begin{minipage}{0.45\textwidth}
\vspace*{5mm}
\includegraphics[scale = 0.25]{figures/example_plate.pdf}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\hspace*{15mm}
\scalebox{0.475}{\input{figures/plot_ex_4_1}}
%\hspace*{-0.5in}
%\includegraphics[scale = 0.5]{figures/plot_ex_4_1_b.pdf}
\end{minipage}
\end{frame}

\begin{frame}{Numerical examples: Example 4 - Elasticity 3D}
\begin{block}{Convergence studies for various combinations of bases}
Same observation as in Example 2 and 3. Optimal convergence governed by the minimum degree, second order for first three choices, and third order for last two choices
\end{block}
\vspace*{-3mm}
\begin{minipage}{0.45\textwidth}
\vspace*{3mm}
\includegraphics[scale = 0.4]{figures/sphere.pdf}
\end{minipage}
\begin{minipage}{0.45\textwidth}
\hspace*{15mm}
\scalebox{0.5}{\input{figures/plot_ex_5_3D}}
%\hspace*{-0.5in}
%\includegraphics[scale = 0.5]{figures/plot_ex_5_3D_b.pdf}
\end{minipage}
\end{frame}

\begin{frame}{Numerical examples: NURBS + PHT splines}
Main properties of PHT-splines:
\begin{itemize}
\item Polynomial splines of degree $p = 3$ defined over T-meshes:
\begin{center}
\includegraphics[scale=0.25]{figures/plate_param_0.pdf}
\includegraphics[scale=0.25]{figures/plate_param_1.pdf}\\
{\tiny Global refinement (tensor-product mesh)   vs local refinement (T-mesh)}
\end{center}
\item $C^{1}$ continuity across the elements 
\end{itemize}
\end{frame}

\begin{frame}{Numerical examples: NURBS + PHT splines}
Laplace equation in the quarter annulus with the solution exhibiting a high peak:
\begin{center}
\includegraphics[scale=0.75]{figures/plot_qannu.eps}
\end{center}
\begin{equation*}
u(x,y) = (r - 1)(r - 2)\theta(\theta - \pi/2)\exp(-100(r \cos\theta - 1)^2)
\end{equation*}
\end{frame}

\begin{frame}{Numerical examples: NURBS + PHT splines}
\begin{itemize}
\item IGA with cubic NURBS (for the geometry as well as numerical solution). Note that, a quadratic NURBS is sufficient for this geometry, however, to have a fair comparison with the remaining studies, we elevate the degree while maintaining the exact geometry representation.
%
\item IGA with cubic PHT-splines (for the geometry as well as the numerical solution). Note that, in this case, the computational geometry is only approximate (not exact as in IGA with cubic NURBS).
%
\item GIFT with cubic B-splines for the numerical solution, and quadratic NURBS for exact geometry representation.
%
\item GIFT with cubic PHT-splines for the numerical solution, and quadratic NURBS for exact geometry representation.
\end{itemize}
\end{frame}
\begin{frame}{Adaptive refinement with PHT splines:}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=1.0]{figures/diskparalevel0.eps} & \includegraphics[scale=0.13]{figures/disksolutionlevel0.eps} \\
\includegraphics[scale=0.09]{figures/diskparalevel1.eps} & \includegraphics[scale=0.08]{figures/disksolutionlevel1.eps} \\
\end{tabular}
\end{center}
\end{frame}
\begin{frame}{Adaptive refinement with PHT splines:}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=1.0]{figures/diskparalevel2.eps} & \includegraphics[scale=0.15]{figures/disksolutionlevel2.eps} \\
\includegraphics[scale=1.0]{figures/diskparalevel3.eps} & \includegraphics[scale=0.15]{figures/disksolutionlevel3.eps} \\
\end{tabular}
\end{center}
\end{frame}
\begin{frame}{Numerical examples: NURBS + PHT splines}
\begin{center}
\includegraphics[scale = 0.25]{figures/errorAnnuPHT-GIFT.eps}
\end{center}
\end{frame}

\begin{frame}
Comparison between IGA with cubic PHT-spline field and GIFT with cubic PHT-spline field:
\begin{itemize}
\item The advantage of the exact geometry representation in the latter case over an approximate geometry in the former case is very minor in the given example, but in realistic industrial problems with complex domains, this advantage will become more pronounced. 
\item Use of GIFT concept eliminates the need to communicate with the original CAD model at each step of the solution refinement process, and the approximation of the boundaries.
\item Use of GIFT concept also eliminates the need to refine the original coarse geometry, as well as to store and process the refined data, which can lead to significant computational savings for big problems.
\end{itemize} 
\end{frame}

%TODO_SATYEN TODO_ELENA Please could you both go through this slide and let me know if it makes sense to you? 
\section{Conclusions}
\begin{frame}{Conclusions}
\begin{itemize}
\item It is possible to retain the advantages of IGA but decouple the geometry and the field approximation
\item Standard patch tests may not always pass, yet the convergence rates are optimal as long as the geometry is exactly represented by the geometry basis
\item With geometry exactly represented by NURBS, using same degree B-splines or NURBS for the approximation of the solution field yields almost identical results
\item With geometry exactly represented by NURBS, using PHT splines for the approximation of the solution gives additional advantage of local adaptive refinement
\end{itemize}
\end{frame}
\end{document}
