From 27f7bf373007dcc7eb5797d495e07041052bcb30 Mon Sep 17 00:00:00 2001
From: Spencer Smith <smiths@mcmaster.ca>
Date: Thu, 21 Sep 2017 09:56:21 -0400
Subject: [PATCH] Updates to lectures, especially L05 and L06.

---
 .../L05_ProgramFamilies/ProgramFamilies.pdf   |  Bin 1105371 -> 400672 bytes
 .../L05_ProgramFamilies/ProgramFamilies.tex   | 1148 ----------------
 .../L06_ProgFamContinued/ProgFamContinued.pdf |  Bin 0 -> 1026543 bytes
 .../L06_ProgFamContinued/ProgFamContinued.tex | 1218 ++++++++++++++++-
 .../VerificAndValid.pdf                       |  Bin 0 -> 266160 bytes
 .../VerificAndValid.tex                       |  132 ++
 6 files changed, 1334 insertions(+), 1164 deletions(-)
 create mode 100644 Lectures/L06_ProgFamContinued/ProgFamContinued.pdf
 create mode 100644 Lectures/L09_VerificationAndValidation/VerificAndValid.pdf
 create mode 100755 Lectures/L09_VerificationAndValidation/VerificAndValid.tex

diff --git a/Lectures/L05_ProgramFamilies/ProgramFamilies.pdf b/Lectures/L05_ProgramFamilies/ProgramFamilies.pdf
index c5ece29f23d6627d2f180f0614239fcfdf5aa97b..6fc7c913ae2ba61968fb4cba7c42a2b2fa27fd39 100644
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diff --git a/Lectures/L05_ProgramFamilies/ProgramFamilies.tex b/Lectures/L05_ProgramFamilies/ProgramFamilies.tex
index 5395b60..700d305 100755
--- a/Lectures/L05_ProgramFamilies/ProgramFamilies.tex
+++ b/Lectures/L05_ProgramFamilies/ProgramFamilies.tex
@@ -672,1154 +672,6 @@ leads to weak error bounds
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\begin{frame}
-\frametitle{Overview of Process}
-
-\begin{center}
- \includegraphics[width=1.0\textwidth]{../Figures/Waterfall.pdf}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{CA to SRS to Design}
-
-\begin{center}
-\includegraphics[width=1.0\textwidth]{../Figures/OverviewOfMethodology.pdf}
-\end{center}
-
-% mention an overview of what goes in the boxes
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Proposed Methodology}
-
-\begin{enumerate}
-
-\item Identify family of interest
-\begin{itemize}
-\item Specific physical model?
-\item Multipurpose tool?
-\end{itemize}
-\item Commonality Analysis (CA)
-\begin{itemize}
-\item Terminology
-\item Commonalities
-\item Variabilities
-\item Parameters of variation
-\item Binding time
-\end{itemize}
-\item Domain Specific Language (DSL)
-\item Generation of family members
-
-\end{enumerate}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{CA Template From \cite{Smith2006}}
-
-\scalebox{1.0}{
-\begin{minipage}{0.94\columnwidth}
-\begin{enumerate}
-
-\item{Reference Material:}
-{a) Table of Contents}
-{b) Table of Symbols}
-{c) Abbreviations and Acronyms}
-
-\item{Introduction:} 
-{a) Purpose of the Document}
-{b) Organization of the Document}
-
-\item{General System Description:}
-{a) \structure{Potential} System Contexts}
-{b) \structure{Potential} User Characteristics}
-{c) \structure{Potential} System Constraints}
-
-\item Commonalities:
-{a) Background Overview}
-{b) Terminology Definition}
-\structure{c) Goal Statements}
-\structure{d) Theoretical Models}
-
-\item \structure{Variabilities:
-{a) Input Assumptions}
-{b) Calculation}
-{c) Output}}
-
-\item{Traceability Matrix}
-
-\item{References}
-
-\end{enumerate}
-\end{minipage}
-}
-% highlight parts to talk about
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Abstract Requirements}
-
-% explain how CAnal addresses this challenge
-\begin{itemize}
-\item Appropriate level of abstraction by refining from goal to theory to input assumptions
-\item A goal is a functional objective the software should achieve: \newline
-\structure{\textbf{G1:} Find the roots of an equation}
-\item Goals are refined into theoretical models: \newline
-% presented as they would be presented in a mathematics textbook 
-\structure{\textbf{T1:} Given a function $f(x)$ and an interval $ \{ x | x_{lower} \leq x_{upper} \} $,
-return the points where $f(x) = 0$}
-\item Introduce simplifying assumptions to allow theoretical model to be solved:\newline
-\structure{\textbf{VA1,2:} $f(x)$ is continuous on the interval and/or $f(x)$ has at least one sign change on the
-interval}
-%traceability matrix?
-\end{itemize}
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Abstract Requirements (Continued)}
-
-\begin{itemize}
-\item Each variability has an associated parameter of variation and a binding time
-\begin{itemize}
-\item Specification time
-\item Compile time
-\item Run time
-\end{itemize}
-\end{itemize}
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Capture Existing Knowledge}
-
-\begin{itemize}
-\item Systematic consideration from general to specific
-\item Communication between experts
-\item Standard template allows comparison
-\item Convenient framework for summarizing existing literature
-\item Eventually a library of requirements documentation
-% \item Same goal may have different theoretical models
-\item CA refined by a family of SRSs
-\end{itemize}
-\end{frame}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{System Requirements Specification (SRS)}
-
-\begin{itemize}
-
-\item Based on IEEE Standard 830 and Volere requirements specification template
-\item Sections from CA are refined in SRS
-\item ``Potential'' descriptions are made specific
-\item Variabilities are set
-\item Binding times are set
-
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{SRS Template}
-
-\scalebox{1.0}{
-\begin{minipage}{0.94\columnwidth}
-\begin{enumerate}
-
-\item{Reference Material}%:
-%{a) Table of Contents}
-%{b) Table of Symbols}
-%{c) Abbreviations and Acronyms}
-
-\item{Introduction}%:
-%{a) Purpose of the Document}
-%{b) Scope of the Software Product}
-%{c) Organization of the Document}
-
-\item{General System Description}%:
-%{a) System Context}
-%{b) User Characteristics}
-%{c) System Constraints}
-
-\item {Specific System Description:}
-{a) Background Overview,}
-{b) Terminology Definition,}
-{c) Goal Statements}
-{d) Theoretical Models,}
-{e) Assumptions,}
-{f) Data Constraints,}
-{g) System Behaviour}
-
-\structure{\item Non-functional Requirements:
-{a) Accuracy of Input Data,}
-{b) Sensitivity of the Model,}
-{c) Tolerance of Solution,}
-{d) Performance,}
-{...}
-{i) Portability,}}
-
-\structure {\item {Solution Validation Strategies,}}
-
-\item{Other System Issues:}
-%{ Open Issues}
-%{ Off-the-Shelf Solutions}
-%{ New Problems}
-%{ Waiting Room}
-
-\item{Traceability Matrix}
-
-%\item List of Possible Changes in the Requirements
-
-%\item{Values of Auxiliary Constants}
-
-%\item{References}
-
-\end{enumerate}
-\end{minipage}
-}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{NFRs}
-
-\begin{itemize}
-
-\item Rather than absolute quantification of NFRs, use relative comparison between other program family members
-\item Specify requirements in big O notation
-% makes the requirement validatable
-%\item Identify benchmark test problems %system test plan
-%\item A posteriori description, rather than a priori specification
-\item Relative importance between NFRs using Analytic Hierarchy Process (AHP) \cite{Saaty1980}
-\begin{itemize}
-\item Addresses challenge of comparing attributes that are measured in different (or hard to quantify) units
-\item Series of pair-wise comparisons between attributes
-\item 1 for equal importance, 3 for moderately strong importance, ..., 9 for extreme importance
-\end{itemize}
-
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Validatable Requirements}
-
-\begin{itemize}
-
-%\item Rather than absolute quantification of NFRs, use relative comparison between other program family members
-% makes the requirement validatable
-\item Relative comparison between programs is a validatable requirement
-\item Focus on a posteriori description, rather than a priori specification
-% experiment with the software, as experiment with nature
-\item Solution validation strategies
-\begin{itemize}
-\item Solve using different techniques
-\item Identify benchmark test problems %system test plan
-\item Test cases built starting from assumed solutions (Method of Manufactured Solutions)
-\item Partially validate for a simpler subset where the solution is known
-\end{itemize}
-
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-\frametitle{Mesh Generating Software}
-\begin{figure}
-\begin{center}
-\rotatebox{0}
-{
- \includegraphics[width=0.9\textwidth]{../Figures/ExampleMesh.pdf}
-}
-%\caption{Example finite element mesh}
-\end{center}
-\end{figure}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-\frametitle{Commonality Analysis for a Mesh Generator}
-
-From Chen's work \cite{Chen2003, SmithAndChen2004, SmithAndChen2004b}.
-Alternate approach in \cite{BastarricaAndHitschfeld2006, RosselEtAl2014,
-  BastarricaAndHitschfeld-Kahler2004,
-  Bastarrica2002, BastarricaAndHitschfeld2004}
-
-\begin{itemize}
-
-\item Terminology
-\begin{itemize}
-\item requirement
-\item structured mesh, ...
-\end{itemize}
-
-\item Commonalities
-\begin{itemize}
-\item discretization
-\item input from user is required, ...
-\end{itemize}
-
-\item Variabilities
-\begin{itemize}
-\item shape of elements
-\item coordinate system used, ...
-\end{itemize}
-
-\item Parameters of variation
-\begin{itemize}
-\item line, triangle, quadrilateral, tetrahedral, hexahedral
-\item Cartesian, polar, spherical, ...
-\end{itemize}
-
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-\frametitle{Definition of a Mesh}
-Let $\Omega$ be a closed bounded domain in $\mathbb{R}$ or $\mathbb{R}^2$ or $\mathbb{R}^3$
-and let $K$
-be a simple shape, such as a line segment in 1D, a triangle or a quadrilateral in 2D, or a tetrahedron or hexahedron
-in 3D.  A mesh of $\Omega$, denoted by $\tau$, has the following properties:
-\begin{enumerate}
-\item $\Omega \approx \cup(K | K \epsilon \tau : K)$, where $\cup$ is first closed and then opened
-\item the length of every element $K$, of dimension 1, in $\tau$ is greater than zero
-\item the interior of every element $K$, of dimension 2 or greater, in $\tau$ is nonempty
-\item the intersection of the interior of two elements is empty
-\end{enumerate}
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-\frametitle{Example Commonality}
-
-\newcommand{\colACwidth}{0.25\textwidth}
-\newcommand{\colBCwidth}{0.7\textwidth}
-
-\begin {tabular}{| p{\colACwidth} | p{\colBCwidth}|}
-\hline
-{\bf Item Number} & C1\\
-\hline
-{\bf Description} & A mesh generator discretizes a given computational domain (closed boundary $\Omega$) into a covering
-up of a finite number of simpler shapes.\\
-\hline
-{\bf Related Variability} & V6, V8, V12, V14, V15, V16, V17, V18\\
-\hline
-{\bf History} & Created - May 7, 2004\\
-\hline
-\end {tabular}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{frame}
-
-\frametitle{Mesh Generator (MG) Goals}
-
-\begin{enumerate}[G1]
-
-\item Input spatial domain $\Omega$ output a mesh $M$ that covers this domain.
-
-\item Transform information on the materials, material properties and the
-locations of the different materials
-
-\item Transform information on the boundary condition types, values and locations
-
-\item Transform system information, such as numerical algorithm parameters
-%values needed by the subsequent
-%processing step (for instance initial time, time step size, degree of implicitness, etc.)
-
-\end{enumerate}
-
-\end{frame}
-
-% TO DO - add a slide for the theoretical model? - no room
-% TO DO - mention output file format variability via XSLT? - mention during talk, but no room on slides
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Element Variability}
-
-Location of nodes: sequence of LocationT \\
-Number of dof at nodes: sequence of $\mathbb{N}$ \\
-LocationT = tuple of ($L_1: \mbox{natT}$, $L_2: \mbox{natT}$, $L_3: \mbox{natT}$)\\
-natT = \{ $s: \mathbb{R} | 0 \leq s \leq 1 : s$ \}\\
-
-\begin{center}
-%\rotatebox{-90}
-{
- \includegraphics[width=0.5\textwidth]{../Figures/TriangleElement.pdf}
-}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Local Topology Variability}
-
-\begin{center}
-{
-\includegraphics[width=0.65\textwidth]{../Figures/TopologyPatterns.pdf}
-}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}[fragile]
-
-\frametitle{DSL Using XML}
-
-\lstset{language=XML,linewidth=.9\textwidth,xleftmargin=0.5cm}
-\begin{lstlisting}%[frame=single]
-
-<elementSet>
-    <geometrySpec>
-        <shape>triangle1</shape>
-        <nodeGeo count="3">
-            <node id="1">
-                <location>1,0,0</location>
-            </node>
-            <node id="2">
-                <location>0,1,0</location>
-            </node>
-            ...
-        </nodeGeo>
-    </geometrySpec>
-</elementSet>
-\end{lstlisting}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Proof of Concept Implementation}
-
-From Cao's work \cite{Cao2006, SmithMcCutchanAndCao2007}
-\begin{itemize}
-\item XML document that customizes a Java object
-\item The Java object customizes the general purpose MG as it is loaded
-\item General purpose MG
-\begin{itemize}
-\item All variabilities bound at run-time
-\item Corresponds to an empty XML specification
-\end{itemize}
-\end{itemize}
-
-\begin{center}
-{
-\includegraphics[width=0.4\textwidth]{../Figures/elementdialog2.png}
-}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-\frametitle{Linear Systems of Equations}
-
-$$A x = b$$
-
-Commonality analysis presented in \cite{Smith2006}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%\begin{frame}
-
-%\frametitle{Terminology}
-
-%\begin{tabular}{ l p{5.5cm}}
-%$n: \mathbb{N}$ & number of linear equations and the number of unknowns\\
-%$A: \mathbb{R}^{n \times n}$ & $n \times n$ real matrix\\
-%$x: \mathbb{R}^{n \times 1}$ & $n \times 1$ real column vector\\
-%$b: \mathbb{R}^{n \times 1}$ & $n \times 1$ real column vector\\
-%$I: \mathbb{R}^{n \times n}$ & an $n \times n$ matrix where all entries are $0$, except for the diagonal entries, which
-%are $1$\\
-%$A^{-1}: \mathbb{R}^{n \times n}$ & the inverse matrix, with the property that $A^{-1} A = I$\\
-%$|| v || $ & the norm (estimate of magnitude) of vector $v$\\
-%residual & $|| b - A x ||$\\
-%singular & matrix $A$ is singular if $A^{-1}$ does not exist\\
-%\end{tabular}
-
-%\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Goal and Theoretical Model}
-
-\textbf{G1}: Given a system of $n$
-linear equations represented by matrix $A$ and column vector $b$, return $x$ such that $A x = b$, if possible
-
-\textbf{T1}: Given square matrix $A$ and column vector $b$, the possible
-solutions for $x$ are as follows:
-
-\begin{enumerate}
-\item A unique solution $x = A^{-1} b$, if $A$ is nonsingular
-\item An infinite number of solutions if $A$ is singular and $b \in span(A)$
-\item No solution if $A$ is singular and $b \notin span(A)$
-\end{enumerate}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Variabilities for Input Assumptions}
-
-\begin{tabular}{| p{\colA} | p{\colB} | }
-\hline
-\textbf{Variability} & \textbf{Parameter of Variation} \\
-\hline
-Allowed structure  $A$ & Set of \{ full, sparse, banded, tridiagonal, block triangular, ...,
-%block structured, diagonal, upper triangular, lower triangular, 
-Hessenberg \} \\
-\hline
-Allowed definiteness $A$ & Set of \{ not definite, positive definite, ...,
-%positive semi-definite, negative definite, 
-negative semi-definite \} \\
-\hline
-Allowed class of $A$ & Set of \{ diagonally dominant, Toeplitz, Vandermonde \} \\
-\hline
-Symmetry\newline assumed? & boolean \\
-\hline
-Possible values for $n$ & set of $\mathbb{N}$ \\
-\hline
-Possible entries in $A$ & set of $\mathbb{R}$ \\
-\hline
-... & ... \\
-%Possible entries in $b$ & set of $\mathbb{R}$ \\
-%\hline
-%Source\newline of input & Set of \{ from a file, through the user interface, passed in memory \} \\
-%\hline
-%Encoding of input & Set of \{binary, text \} \\
-%\hline
-%Format\newline of input $A$ & Set of \{arbitrary, by row, by column, by diagonal \} \\
-%\hline
-%Format\newline of input $b$ & Set of \{arbitrary, ordered \} \\\hline
-\end{tabular}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Variabilities for Calculation}
-
-\begin{tabular}{| p{\colA} | p{\colB} | }
-\hline
-\textbf{Variability} & \textbf{Parameter of Variation} \\
-\hline
-Check input? & boolean (false if the input is assumed to satisfy the input assumptions)\\ 
-\hline
-Exceptions generated? & boolean (false if the goal is non-stop arithmetic)\\ 
-\hline
-Norm used for residual & Set of \{1-norm, 2-norm, $\infty$-norm \} \\
-\hline
-\end{tabular}
-
-% mention turning off input checking helps with problem of theoretical model refined by assumpts
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Variabilities for Output}
-
-\begin{tabular}{| p{\colA} | p{\colB} | }
-\hline
-\textbf{Variability} & \textbf{Parameter of Variation} \\
-\hline
-Destination for output $x$ & Set of \{ to file, to screen, to memory \} \\
-\hline
-Encoding of output $x$ & Set of \{binary, text \} \\
-\hline
-Format of output $x$ & Set of \{arbitrary, ordered \} \\
-\hline
-Output residual & boolean (true if the program returns the residual) \\
-\hline
-Possible entries in $x$ & set of $\mathbb{R} \cup \{- \infty, \infty, undef \}$ \\
-\hline
-\end{tabular}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Analytic Hierarchy Process}
-
-\begin{itemize}
-\item Example 1
-\begin{itemize}
-\item Embedded real-time system for digital signal processing
-\item $n = 10$
-\item $A$ is assumed to be Toeplitz
-\end{itemize}
-\end{itemize}
-
-\begin{tabular}{| l | l | l | l | l |}
-\hline
-~ & \textbf{Speed} & \textbf{Accuracy} & \textbf{Portability} & \textbf{Priority}\\
-\hline
-\textbf{Speed} & 1 & 3 & 5 & 0.64\\
-\textbf{Accuracy} & 1/3 & 1 & 3 & 0.26\\
-\textbf{Portability} & 1/5 & 1/3 & 1 & 0.11\\
-\hline
-\end{tabular}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-%\begin{frame}
-
-%\frametitle{AHP Continued}
-
-%\begin{itemize}
-%\item Example 2
-%\begin{itemize}
-%\item Structural mechanics software for education
-%\item $10 \leq n \leq 1000$
-%\item $A$ is assumed to be positive definite
-%\item Would have different relative priorities
-%\end{itemize}
-%\end{itemize}
-
-%\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Solution Validation Strategies}
-
-\begin{itemize}
-\item Create test cases with known solutions
-\begin{itemize}
-\item Assume $A$ and $x$, calculate $b$
-\item Given $A$ and $b$ calculate $x^*$ and compare to the assumed $x$
-\end{itemize}
-\item Comparison with Matlab
-\item Comparison with NAG library
-\item Where possible compare solution to interval arithmetic solution
-\item Experiments to describe how accuracy changes with increasing condition number
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Connection to Design}
-
-\begin{itemize}
-
-\item Abstract requirements to concrete design decisions
-%\item Software packages likely already exist within the program family
-\item Reuse existing packages within the program family
-\item Summarize existing software by the parameters of variation and binding time
-%\item Run time binding can often be changed to specification time binding
-\item If functional requirements match, then use NFRs
-\begin{itemize}
-\item AHP to compare each design against each of the NFRs %as done in SRS 
-%cols and rows are des alternatives, compared against a given NFR
-%repeat for each NFR
-\item Contribution of each NFR for each design alternative is found by multiplying the contribution of
-each alternative to the given NFR with the corresponding priority of that NFR
-\item Sum the contributions
-\item The highest overall score is the ``winning'' alternative
-\end{itemize}
-
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{A Family of Material Models}
-
-From McCutchan's work \cite{CaretteEtAl2008, SmithMcCutchanAndCao2007,
-  SmithEtAl2008, CaretteEtAl2007_TR, SmithMcCutchanAndCarette2017, McCutchan2007}
-
-\begin{center}
-{
-\includegraphics[width=0.5\textwidth]{../Figures/TestSpecimenUniaxial.pdf}
-}
-\end{center}
-
-\begin{center}
-\begin{tabular}{cc}
-\includegraphics[width=0.45\textwidth]{../Figures/Plastic.pdf} &
-\includegraphics[width=0.45\textwidth]{../Figures/Viscoplastic.pdf}\\
-\end{tabular}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Terminology Definitions}
-
-\begin{minipage}{\textwidth}
-\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
-\hline
-\rowcolor[gray]{0.9}
-Label: & D\_YieldFunction\\ \hline
-Symbol: & $F = F(\bm{\sigma}, \kappa)$\\ \hline
-Type: & $(\mbox{tensor2DT} \times \mathbb{R}) \rightarrow \mathbb{R}$\\ \hline
-%Units: & --\\ \hline
-Related: & {D\_Stress}, {D\_HardeningParameter}\\ \hline %originally related items, shorted for space
-Sources: & ...\\
-\hline Descrip: & The yield function defines a surface $F = 0$ in the six dimensional stress space ...\\
-%\hline History: & Created -- June 15, 2007\\
-\hline
-\end{tabular}
-\end{minipage}
-
-\begin{center}
-{
-\includegraphics[width=0.35\textwidth]{../Figures/YieldHardeningAndPlastPotential.pdf}
-}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Goal Statement}
-
-\noindent
-\begin{minipage}{\textwidth}
-\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
-\hline
-\rowcolor[gray]{0.9}
-Label: & G\_StressDetermination\\ \hline
-Descrip: & Given the initial stress and the deformation history of a material particle, determine the stress
-within the material particle.\\
-\hline
-Refine: & {T\_ConstitEquation}\\ \hline
-%History: & Created -- June 8, 2007\\ \hline
-\end{tabular}
-\end{minipage}
-
-\begin{center}
-{
-\includegraphics[width=0.5\textwidth]{../Figures/StressTensor.pdf}
-}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Assumptions}
-
-\begin{minipage}{\textwidth}
-\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
-\hline
-\rowcolor[gray]{0.9}
-Label: & A\_AdditivityPostulate\\ \hline
-Related: & {D\_StrainRate}\\ \hline
-Equation: & $\dot{\bm{\epsilon}} = \dot{\bm{\epsilon}}^e + \dot{\bm{\epsilon}}^{\mathit{vp}}$\newline
-with the following types and units\newline
-$\dot{\bm{\epsilon}}: \mbox{tensor2DT}$ (1/t) (1/s)\newline
-$\dot{\bm{\epsilon}}^e: \mbox{tensor2DT}$ (1/t) (1/s)\newline
-$\dot{\bm{\epsilon}}^{\mathit{vp}}: \mbox{tensor2DT}$ (1/t) (1/s)\\ \hline
-Descrip: & The total strain rate ($\dot{\bm{\epsilon}}$) is assumed to decompose into elastic
-($\dot{\bm{\epsilon}}^e$) and viscoplastic ($\dot{\bm{\epsilon}}^{\mathit{vp}}$)
-strain rates.\\ \hline 
-Rationale & This is a standard assumption for elastoplastic and elastoviscoplastic materials.  The 
-appropriateness of this assumption is born out by the success of theories built upon it.\\ \hline
-Source: & [6, page 339]; [7, page 181]\\ \hline %references hard-coded
-%History: & Created -- June 11, 2007\\ \hline
-\end{tabular}
-\end{minipage}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Theoretical Model}
-
-\begin{minipage}{\textwidth}
-\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
-\hline
-\rowcolor[gray]{0.9}
-Label: & T\_ConstitEquation\\
-\hline
-Related: & {A\_CauchyStress}, {A\_DeformationHistory}, {A\_PerzynaConstit}, {A\_AdditivityPostulate},
-{A\_ElasticConstit}, {A\_DescriptionOfMotion}, {V\_MaterialProperties}\\
-\hline
-Input: & $\bm{\sigma}_0: \mbox{tensor2DT}$ (StressU) (Pa)\newline
-$t_{\mathit{begin}}: \mathbb{R}$ (t) (s) \newline
-$t_{\mathit{end}}: \mathbb{R} $ (t) (s) \newline
-$\dot{\bm{\epsilon}}(t) : \{ t: \mathbb{R} | t_{\mathit{begin}} \leq t \leq t_{\mathit{end}} : t \} \rightarrow
-\mbox{   } \mbox{   } \mbox{  tensor2DT}$ (1/t) (1/s)\newline
-$\mathit{mat\_prop\_val}: \mbox{string} \rightarrow \mathbb{R}$\newline
-$E: \mathbb{R}^{+}$ (StressU) (Pa)\newline
-$\nu: \mbox{poissonT}$ (dimensionless)
-\\
-\hline
-\end{tabular}
-\end{minipage}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Theoretical Model Continued}
-
-\begin{minipage}{\textwidth}
-\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
-\hline
-\rowcolor[gray]{0.9}
-Label: & T\_ConstitEquation\\
-\hline
-Output: & $\bm{\sigma}(t): \{ t: \mathbb{R} | t_{\mathit{begin}} \leq t \leq t_{\mathit{end}} : t \} \rightarrow
-\mbox{tensor2DT}$ such that 
-$$\dot{\bm{\sigma}} = \mathbf{D} \left ( \dot{\bm{\epsilon}} - \gamma < \phi (F(\bm{\sigma},\kappa) ) >
-\frac {
-\partial Q (\bm{\sigma})}{
-\partial \bm{\sigma}} \right )
-$$ 
-and $\bm{\sigma}(t_{\mathit{begin}}) = \bm{\sigma}_0$,
-the components of $\bm{\sigma}$ have the units of StressU (Pa)\\
-\hline
-Derive: & The governing differential equation is found by first solving for $\dot{\bm{\epsilon}}^e$
-in {A\_AdditivityPostulate} and then ...\\
-\hline
-Descrip: & The theoretical model is only completely defined once the associated
-variabilities~({V\_MaterialProperties}) that define the
-material have been set.  ...\\
-\hline History: & Created -- June 14, 2007\\
-\hline
-\end{tabular}
-\end{minipage}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Variabilities}
-
-\begin{itemize}
-\item $F = F(\bm{\sigma}, \kappa): \mathbb{R}^6 \times \mathbb{R} \rightarrow \mathbb{R}$
-\item $Q = Q(\bm{\sigma}): \mathbb{R}^6 \rightarrow \mathbb{R}$
-\item $\kappa = \kappa(\bm{\epsilon}^{\mathit{vp}}): \mathbb{R}^6 \rightarrow \mathbb{R}$
-\item $\phi = \phi(F): \mathbb{R} \rightarrow \mathbb{R}$
-\item $\gamma: \mathbb{R}$
-\item $\mathit{mat\_prop\_names}: \mbox{set of string}$
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Dependency Graph} %between commonalities
-
-\begin{center}
-{
-\includegraphics[width=1\textwidth]{../Figures/DependenceGraph.pdf}
-}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Dependency Graph Between Commonalities and Variabilities}
-
-\begin{center}
-{
-\includegraphics[width=1\textwidth]{../Figures/DependenceOfVariabilities.pdf}
-}
-\end{center}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Example}
-
-\begin{minipage}{\textwidth}
-\begin{tabular}{| p{0.24\textwidth} | p{0.65\textwidth}|}
-\hline
-\rowcolor[gray]{0.9}
-Label: & E\_StrainHardening\\ \hline
-{V\_MatName} & $\mathit{name = }$``Strain-Hardening Viscoelastic''\\
-\hline
-{V\_YieldFunct} & $F =  q \kappa^{\frac{n-1}{m}}$ (StressU) (Pa)\\ \hline
-{V\_PlasticPot} & $Q = q$ (StressU) (Pa)\\ \hline 
-{V\_HardParam} & $\kappa = \epsilon_q^{\mathit{vp}}$ (L/L) (m/m)\\
-\hline
-{V\_Phi} & $\phi = F^{\frac{m}{n}}$ ($\mbox{StressU}^{\frac{m}{n}}$) ($\mbox{Pa}^{\frac{m}{n}}$)\\
-\hline
-{V\_FluParam} & $\gamma = n A^{\frac{1}{n}}$ ($\mbox{StressU}^{-m} \mbox{t}^{-1}$) ($\mbox{Pa}^{-m} \mbox{s}^{-1}$)\\ \hline
-{V\_MatProps} & $\mathit{mat\_prop\_names} = $ \{``$A$'', ``$m$'', ``$n$'' \}, where the type of the
-material properties are ...\\
-%There is likely an upper limit on the values for $m$ and $n$, but at this time the value of these limits is unclear. 
-%With respect to units, $m$ and $n$ do not have units and $A$ has units of $\mbox{StressU}^{-m} \mbox{t}^{-1}$ ($\mbox{Pa}^{-m}%%
-%\mbox{s}^{-1}$)
-\hline
-{V\_Description} & $\mathit{descript =}$ ``This constitutive equation combines a power-law viscoelastic
-material with a strain hardening (softening) material.  ...''
-\\ \hline
-%Source & \cite{Smith2001}
-%\\ \hline
-%History & Created -- Sept 21, 2007\\ \hline
-\end{tabular}
-\end{minipage}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Code Generation}
-
-\begin{itemize}
-\item Specify variabilities
-\item Symbolically calculate terms needed by numerical algorithm, including $\frac{ \partial Q}{ \partial
-\bm{\sigma}}$, $\frac{\partial F}{\partial \bm{\sigma}}$, etc.
-\item Symbolic processing avoids tedious and error-prone hand calculations
-\begin{itemize}
-\item Reduces workload
-\item Allows non-experts to deal with new problems
-\item Increases reliability
-\end{itemize}
-\item Use Maple Computer Algebra System for model manipulation
-\item Convert math expressions into C expressions using ``CodeGeneration''
-\item Inline into a C++ class defining the material model
-\item A finite element program can this interface to realize the numerical algorithm
-\end{itemize}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}[fragile]
-
-\frametitle{BNF of DSL for $F$}
-
-\begin{grammar}
-[(colon){$\rightarrow$}]
-[(semicolon){$|$}]
-[(comma){}]
-[(period){\\}]
-[(quote){\begin{bf}}{\end{bf}}]
-[(nonterminal){$\langle$}{$\rangle$}]
-<expression>:<number>;\\
-(<expression>);\\
-<expression>,\^{ },<expression>;\\
-<expression>,$*$,<expression>;\\
-\end{grammar}
-~~~~~~~...
-\begin{grammar}
-[(colon){$\rightarrow$}]
-[(semicolon){$|$}]
-[(comma){}]
-[(period){\\}]
-[(quote){\begin{bf}}{\end{bf}}]
-[(nonterminal){$\langle$}{$\rangle$}]
-<simulation-variable-F>;<user-defined-constants>.
-<simulation-variable-F>:"Kappa";<simulation-variable-stress>;<simulation-variable-stress-macros>.
-<simulation-variable-stress>:"SigmaXX";"SigmaYY";"SigmaZZ";"SigmaXY";\\
-"SigmaYZ";"SigmaXZ".
-<simulation-variable-stress-macros>:"Sxx";"Syy";"Szz";"Sxy";"Syz";"Sxz";"Sm";"J2";"J3";"q".
-<user-defined-constants>:<string>.
-\end{grammar}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-\frametitle{Concluding Remarks}
-\begin{itemize}
-\item Case studies of applying software engineering methodologies to mesh generating systems and linear solvers
-\item Appropriate and advantageous to apply program family strategy
-\item Challenges for software engineers
-\item General purpose scientific software is best studied as a program family
-\begin{itemize}
-\item Variabilities are assumptions about problems that can be handled
-\item Derive requirements from commonality analysis
-\end{itemize}
-\item Eventually hope for automatic code generation
-\end{itemize}
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Concluding Remarks (Continued)}
-
-A new methodology for documenting requirements for general purpose scientific computing software
-
-\setcounter{temp}{0}
-
-\begin{enumerate}
-\setcounter{enumi}{\value{temp}}
-
-\item Validatable requirements
-\begin{itemize}
-\item Relative comparison between program family members
-\item Focus on description rather than specification
-\item Solution validation strategy
-\end{itemize}
-\item Abstract
-\begin{itemize}
-\item Refine goal statement to theoretical model to input assumptions
-\item In some cases one may want to turn off input checking
-\item Connection to design
-\end{itemize}
-
-\setcounter{temp}{\value{enumi}}
-\end{enumerate}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle{Concluding Remarks (Continued)}
-
-\begin{enumerate}
-\setcounter{enumi}{\value{temp}}
-
-\item NFRs
-\begin{itemize}
-\item Relative comparison
-\item AHP
-\end{itemize}
-\item Capture and reuse
-\begin{itemize}
-\item Systematic consideration from general to specific
-\item CA refined by a family of SRSs
-\item CA and SRS summarize existing knowledge and currently available software
-\item Standard template allows comparison
-\item Convenient framework for summarizing existing literature
-\end{itemize}
-
-\setcounter{temp}{\value{enumi}}
-\end{enumerate}
-
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle<presentation>{Concluding Remarks}
-
-\begin{itemize}
-\item A new template for a family of models of physical phenomena
-\item Refinement of \structure{Goals} to \structure{Theoretical Models} using \structure{Data Definitions} and
-\structure{Assumptions}
-\item \structure{Variabilities} are identified in the Theoretical Model
-\item A constitutive equation can be written using a (declarative) DSL and the code can be generated
-\item A DSL has been built, using Maple, for a virtual material testing laboratory
-\end{itemize}
-  
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{frame}
-
-\frametitle<presentation>{Concluding Remarks}
-
-\begin{itemize}
-\item SC software is a great candidate for development as a program family
-\item Produce programs that are as special or general purpose as needed % run time bindings
-\item Improve reusability, usability and reliability
-\item Potential to improve performance
-\item A commonality analysis facilitates the design of a DSL
-\item Symbolic processing and code generation are very useful techniques
-\item \structure{We will return to code generation later}
-\end{itemize}
-  
-\end{frame}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
 \begin{frame}[allowframebreaks]
 \frametitle{References}
 
diff --git a/Lectures/L06_ProgFamContinued/ProgFamContinued.pdf b/Lectures/L06_ProgFamContinued/ProgFamContinued.pdf
new file mode 100644
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diff --git a/Lectures/L06_ProgFamContinued/ProgFamContinued.tex b/Lectures/L06_ProgFamContinued/ProgFamContinued.tex
index 13d8735..6730495 100755
--- a/Lectures/L06_ProgFamContinued/ProgFamContinued.tex
+++ b/Lectures/L06_ProgFamContinued/ProgFamContinued.tex
@@ -62,8 +62,6 @@
 \bi
 \item Administrative details
 \item Questions?
-\item License and copyright
-\item Motivation
 \item Proposed Family Methods
 \item Family of Mesh Generators
 \item Family of Linear Solvers
@@ -77,19 +75,8 @@
 \frametitle{Administrative Details}
 
 \bi
-\item Add me to your GitHub repos, my GitHub id is \texttt{smiths}
-\item Assign me an issue to review your problem statements
-\bi
-\item Clearly state that you would like me to review your problem statement
-\item Include a link to your problem statement
-\ei
-\item Updates to SRS template
-\item Commonality analysis should start from SRS template
-\item Presentations
-\bi
-\item VGA by default, ask if need adapter
-\item Can you my laptop
-\ei
+\item Marking scheme for SRS
+\item Grade columns in Avenue
 \ei
 
 \end{frame}
@@ -117,11 +104,1210 @@ Final Documentation & Week 13 & Dec 6\\
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+\begin{frame}
+\frametitle{Administrative Details: Presentation Schedule}
+
+\bi
+\item{SRS Present}
+\bi
+\item Tuesday: Paul, Isobel, Keshav
+\item Friday: Devi, Shushen, Xiaoye
+\ei
+\item V\&V Present
+\bi
+\item Tuesday: Steven, Alexandre P., Alexander S.
+\item Friday: Geneva, Jason, Yuzhi
+\ei
+\item MG Present
+\bi
+\item Tuesday: Xiaoye, Shushen, Devi, Keshav, Alex P, Paul
+\item Friday: Yuzhi, Jason, Geneva, Alex S, Isobel, Steven
+\ei
+\item MIS Present
+\bi
+\item Tuesday: Isobel, Keshav, Paul
+\item Friday: Shushen, Xiaoye, Devi
+\ei
+\item Impl.\ Present
+\bi
+\item Tuesday: Alexander S., Steven, Alexandre P.
+\item Friday: Jason, Geneva, Yuzhi
+\ei
+
+\ei
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 \begin{frame}
 \frametitle{Questions?}
 \begin{itemize}
-\item Questions about ...
+\item Questions about
+\bi
+\item SRS presentations?
+\item SRS documents?
+\item LaTeX?
+\ei
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Overview of Process}
+
+\begin{center}
+ \includegraphics[width=1.0\textwidth]{../Figures/Waterfall.pdf}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{CA to SRS to Design}
+
+\begin{center}
+\includegraphics[width=1.0\textwidth]{../Figures/OverviewOfMethodology.pdf}
+\end{center}
+
+% mention an overview of what goes in the boxes
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Proposed Methodology}
+
+\begin{enumerate}
+
+\item Identify family of interest
+\begin{itemize}
+\item Specific physical model?
+\item Multipurpose tool?
+\end{itemize}
+\item Commonality Analysis (CA)
+\begin{itemize}
+\item Terminology
+\item Commonalities
+\item Variabilities
+\item Parameters of variation
+\item Binding time
+\end{itemize}
+\item Domain Specific Language (DSL)
+\item Generation of family members
+
+\end{enumerate}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{CA Template From \cite{Smith2006}}
+
+\scalebox{1.0}{
+\begin{minipage}{0.94\columnwidth}
+\begin{enumerate}
+
+\item{Reference Material:}
+{a) Table of Contents}
+{b) Table of Symbols}
+{c) Abbreviations and Acronyms}
+
+\item{Introduction:} 
+{a) Purpose of the Document}
+{b) Organization of the Document}
+
+\item{General System Description:}
+{a) \structure{Potential} System Contexts}
+{b) \structure{Potential} User Characteristics}
+{c) \structure{Potential} System Constraints}
+
+\item Commonalities:
+{a) Background Overview}
+{b) Terminology Definition}
+\structure{c) Goal Statements}
+\structure{d) Theoretical Models}
+
+\item \structure{Variabilities:
+{a) Input Assumptions}
+{b) Calculation}
+{c) Output}}
+
+\item{Traceability Matrix}
+
+\item{References}
+
+\end{enumerate}
+\end{minipage}
+}
+% highlight parts to talk about
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Abstract Requirements}
+
+% explain how CAnal addresses this challenge
+\begin{itemize}
+\item Appropriate level of abstraction by refining from goal to theory to input assumptions
+\item A goal is a functional objective the software should achieve: \newline
+\structure{\textbf{G1:} Find the roots of an equation}
+\item Goals are refined into theoretical models: \newline
+% presented as they would be presented in a mathematics textbook 
+\structure{\textbf{T1:} Given a function $f(x)$ and an interval $ \{ x | x_{lower} \leq x_{upper} \} $,
+return the points where $f(x) = 0$}
+\item Introduce simplifying assumptions to allow theoretical model to be solved:\newline
+\structure{\textbf{VA1,2:} $f(x)$ is continuous on the interval and/or $f(x)$ has at least one sign change on the
+interval}
+%traceability matrix?
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Abstract Requirements (Continued)}
+
+\begin{itemize}
+\item Each variability has an associated parameter of variation and a binding time
+\begin{itemize}
+\item Specification time
+\item Compile time
+\item Run time
+\end{itemize}
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Capture Existing Knowledge}
+
+\begin{itemize}
+\item Systematic consideration from general to specific
+\item Communication between experts
+\item Standard template allows comparison
+\item Convenient framework for summarizing existing literature
+\item Eventually a library of requirements documentation
+% \item Same goal may have different theoretical models
+\item CA refined by a family of SRSs
+\end{itemize}
+\end{frame}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{System Requirements Specification (SRS)}
+
+\begin{itemize}
+
+\item Based on IEEE Standard 830 and Volere requirements specification template
+\item Sections from CA are refined in SRS
+\item ``Potential'' descriptions are made specific
+\item Variabilities are set
+\item Binding times are set
+
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{SRS Template}
+
+\scalebox{1.0}{
+\begin{minipage}{0.94\columnwidth}
+\begin{enumerate}
+
+\item{Reference Material}%:
+%{a) Table of Contents}
+%{b) Table of Symbols}
+%{c) Abbreviations and Acronyms}
+
+\item{Introduction}%:
+%{a) Purpose of the Document}
+%{b) Scope of the Software Product}
+%{c) Organization of the Document}
+
+\item{General System Description}%:
+%{a) System Context}
+%{b) User Characteristics}
+%{c) System Constraints}
+
+\item {Specific System Description:}
+{a) Background Overview,}
+{b) Terminology Definition,}
+{c) Goal Statements}
+{d) Theoretical Models,}
+{e) Assumptions,}
+{f) Data Constraints,}
+{g) System Behaviour}
+
+\structure{\item Non-functional Requirements:
+{a) Accuracy of Input Data,}
+{b) Sensitivity of the Model,}
+{c) Tolerance of Solution,}
+{d) Performance,}
+{...}
+{i) Portability,}}
+
+\structure {\item {Solution Validation Strategies,}}
+
+\item{Other System Issues:}
+%{ Open Issues}
+%{ Off-the-Shelf Solutions}
+%{ New Problems}
+%{ Waiting Room}
+
+\item{Traceability Matrix}
+
+%\item List of Possible Changes in the Requirements
+
+%\item{Values of Auxiliary Constants}
+
+%\item{References}
+
+\end{enumerate}
+\end{minipage}
+}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{NFRs}
+
+\begin{itemize}
+
+\item Rather than absolute quantification of NFRs, use relative comparison between other program family members
+\item Specify requirements in big O notation
+% makes the requirement validatable
+%\item Identify benchmark test problems %system test plan
+%\item A posteriori description, rather than a priori specification
+\item Relative importance between NFRs using Analytic Hierarchy Process (AHP) \cite{Saaty1980}
+\begin{itemize}
+\item Addresses challenge of comparing attributes that are measured in different (or hard to quantify) units
+\item Series of pair-wise comparisons between attributes
+\item 1 for equal importance, 3 for moderately strong importance, ..., 9 for extreme importance
+\end{itemize}
+
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Validatable Requirements}
+
+\begin{itemize}
+
+%\item Rather than absolute quantification of NFRs, use relative comparison between other program family members
+% makes the requirement validatable
+\item Relative comparison between programs is a validatable requirement
+\item Focus on a posteriori description, rather than a priori specification
+% experiment with the software, as experiment with nature
+\item Solution validation strategies
+\begin{itemize}
+\item Solve using different techniques
+\item Identify benchmark test problems %system test plan
+\item Test cases built starting from assumed solutions (Method of Manufactured Solutions)
+\item Partially validate for a simpler subset where the solution is known
+\end{itemize}
+
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Mesh Generating Software}
+\begin{figure}
+\begin{center}
+\rotatebox{0}
+{
+ \includegraphics[width=0.9\textwidth]{../Figures/ExampleMesh.pdf}
+}
+%\caption{Example finite element mesh}
+\end{center}
+\end{figure}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Commonality Analysis for a Mesh Generator}
+
+From Chen's work \cite{Chen2003, SmithAndChen2004, SmithAndChen2004b}.
+Alternate approach in \cite{BastarricaAndHitschfeld2006, RosselEtAl2014,
+  BastarricaAndHitschfeld-Kahler2004,
+  Bastarrica2002, BastarricaAndHitschfeld2004}
+
+\begin{itemize}
+
+\item Terminology
+\begin{itemize}
+\item requirement
+\item structured mesh, ...
+\end{itemize}
+
+\item Commonalities
+\begin{itemize}
+\item discretization
+\item input from user is required, ...
+\end{itemize}
+
+\item Variabilities
+\begin{itemize}
+\item shape of elements
+\item coordinate system used, ...
+\end{itemize}
+
+\item Parameters of variation
+\begin{itemize}
+\item line, triangle, quadrilateral, tetrahedral, hexahedral
+\item Cartesian, polar, spherical, ...
+\end{itemize}
+
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Definition of a Mesh}
+Let $\Omega$ be a closed bounded domain in $\mathbb{R}$ or $\mathbb{R}^2$ or $\mathbb{R}^3$
+and let $K$
+be a simple shape, such as a line segment in 1D, a triangle or a quadrilateral in 2D, or a tetrahedron or hexahedron
+in 3D.  A mesh of $\Omega$, denoted by $\tau$, has the following properties:
+\begin{enumerate}
+\item $\Omega \approx \cup(K | K \epsilon \tau : K)$, where $\cup$ is first closed and then opened
+\item the length of every element $K$, of dimension 1, in $\tau$ is greater than zero
+\item the interior of every element $K$, of dimension 2 or greater, in $\tau$ is nonempty
+\item the intersection of the interior of two elements is empty
+\end{enumerate}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Example Commonality}
+
+\newcommand{\colACwidth}{0.25\textwidth}
+\newcommand{\colBCwidth}{0.7\textwidth}
+
+\begin {tabular}{| p{\colACwidth} | p{\colBCwidth}|}
+\hline
+{\bf Item Number} & C1\\
+\hline
+{\bf Description} & A mesh generator discretizes a given computational domain (closed boundary $\Omega$) into a covering
+up of a finite number of simpler shapes.\\
+\hline
+{\bf Related Variability} & V6, V8, V12, V14, V15, V16, V17, V18\\
+\hline
+{\bf History} & Created - May 7, 2004\\
+\hline
+\end {tabular}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{frame}
+
+\frametitle{Mesh Generator (MG) Goals}
+
+\begin{enumerate}[G1]
+
+\item Input spatial domain $\Omega$ output a mesh $M$ that covers this domain.
+
+\item Transform information on the materials, material properties and the
+locations of the different materials
+
+\item Transform information on the boundary condition types, values and locations
+
+\item Transform system information, such as numerical algorithm parameters
+%values needed by the subsequent
+%processing step (for instance initial time, time step size, degree of implicitness, etc.)
+
+\end{enumerate}
+
+\end{frame}
+
+% TO DO - add a slide for the theoretical model? - no room
+% TO DO - mention output file format variability via XSLT? - mention during talk, but no room on slides
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Element Variability}
+
+Location of nodes: sequence of LocationT \\
+Number of dof at nodes: sequence of $\mathbb{N}$ \\
+LocationT = tuple of ($L_1: \mbox{natT}$, $L_2: \mbox{natT}$, $L_3: \mbox{natT}$)\\
+natT = \{ $s: \mathbb{R} | 0 \leq s \leq 1 : s$ \}\\
+
+\begin{center}
+%\rotatebox{-90}
+{
+ \includegraphics[width=0.5\textwidth]{../Figures/TriangleElement.pdf}
+}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Local Topology Variability}
+
+\begin{center}
+{
+\includegraphics[width=0.65\textwidth]{../Figures/TopologyPatterns.pdf}
+}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}[fragile]
+
+\frametitle{DSL Using XML}
+
+\lstset{language=XML,linewidth=.9\textwidth,xleftmargin=0.5cm}
+\begin{lstlisting}%[frame=single]
+
+<elementSet>
+    <geometrySpec>
+        <shape>triangle1</shape>
+        <nodeGeo count="3">
+            <node id="1">
+                <location>1,0,0</location>
+            </node>
+            <node id="2">
+                <location>0,1,0</location>
+            </node>
+            ...
+        </nodeGeo>
+    </geometrySpec>
+</elementSet>
+\end{lstlisting}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Proof of Concept Implementation}
+
+From Cao's work \cite{Cao2006, SmithMcCutchanAndCao2007}
+\begin{itemize}
+\item XML document that customizes a Java object
+\item The Java object customizes the general purpose MG as it is loaded
+\item General purpose MG
+\begin{itemize}
+\item All variabilities bound at run-time
+\item Corresponds to an empty XML specification
+\end{itemize}
 \end{itemize}
+
+\begin{center}
+{
+\includegraphics[width=0.4\textwidth]{../Figures/elementdialog2.png}
+}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Linear Systems of Equations}
+
+$$A x = b$$
+
+Commonality analysis presented in \cite{Smith2006}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\begin{frame}
+
+%\frametitle{Terminology}
+
+%\begin{tabular}{ l p{5.5cm}}
+%$n: \mathbb{N}$ & number of linear equations and the number of unknowns\\
+%$A: \mathbb{R}^{n \times n}$ & $n \times n$ real matrix\\
+%$x: \mathbb{R}^{n \times 1}$ & $n \times 1$ real column vector\\
+%$b: \mathbb{R}^{n \times 1}$ & $n \times 1$ real column vector\\
+%$I: \mathbb{R}^{n \times n}$ & an $n \times n$ matrix where all entries are $0$, except for the diagonal entries, which
+%are $1$\\
+%$A^{-1}: \mathbb{R}^{n \times n}$ & the inverse matrix, with the property that $A^{-1} A = I$\\
+%$|| v || $ & the norm (estimate of magnitude) of vector $v$\\
+%residual & $|| b - A x ||$\\
+%singular & matrix $A$ is singular if $A^{-1}$ does not exist\\
+%\end{tabular}
+
+%\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Goal and Theoretical Model}
+
+\textbf{G1}: Given a system of $n$
+linear equations represented by matrix $A$ and column vector $b$, return $x$ such that $A x = b$, if possible
+
+\textbf{T1}: Given square matrix $A$ and column vector $b$, the possible
+solutions for $x$ are as follows:
+
+\begin{enumerate}
+\item A unique solution $x = A^{-1} b$, if $A$ is nonsingular
+\item An infinite number of solutions if $A$ is singular and $b \in span(A)$
+\item No solution if $A$ is singular and $b \notin span(A)$
+\end{enumerate}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Variabilities for Input Assumptions}
+
+\begin{tabular}{| p{\colA} | p{\colB} | }
+\hline
+\textbf{Variability} & \textbf{Parameter of Variation} \\
+\hline
+Allowed structure  $A$ & Set of \{ full, sparse, banded, tridiagonal, block triangular, ...,
+%block structured, diagonal, upper triangular, lower triangular, 
+Hessenberg \} \\
+\hline
+Allowed definiteness $A$ & Set of \{ not definite, positive definite, ...,
+%positive semi-definite, negative definite, 
+negative semi-definite \} \\
+\hline
+Allowed class of $A$ & Set of \{ diagonally dominant, Toeplitz, Vandermonde \} \\
+\hline
+Symmetry\newline assumed? & boolean \\
+\hline
+Possible values for $n$ & set of $\mathbb{N}$ \\
+\hline
+Possible entries in $A$ & set of $\mathbb{R}$ \\
+\hline
+... & ... \\
+%Possible entries in $b$ & set of $\mathbb{R}$ \\
+%\hline
+%Source\newline of input & Set of \{ from a file, through the user interface, passed in memory \} \\
+%\hline
+%Encoding of input & Set of \{binary, text \} \\
+%\hline
+%Format\newline of input $A$ & Set of \{arbitrary, by row, by column, by diagonal \} \\
+%\hline
+%Format\newline of input $b$ & Set of \{arbitrary, ordered \} \\\hline
+\end{tabular}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Variabilities for Calculation}
+
+\begin{tabular}{| p{\colA} | p{\colB} | }
+\hline
+\textbf{Variability} & \textbf{Parameter of Variation} \\
+\hline
+Check input? & boolean (false if the input is assumed to satisfy the input assumptions)\\ 
+\hline
+Exceptions generated? & boolean (false if the goal is non-stop arithmetic)\\ 
+\hline
+Norm used for residual & Set of \{1-norm, 2-norm, $\infty$-norm \} \\
+\hline
+\end{tabular}
+
+% mention turning off input checking helps with problem of theoretical model refined by assumpts
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Variabilities for Output}
+
+\begin{tabular}{| p{\colA} | p{\colB} | }
+\hline
+\textbf{Variability} & \textbf{Parameter of Variation} \\
+\hline
+Destination for output $x$ & Set of \{ to file, to screen, to memory \} \\
+\hline
+Encoding of output $x$ & Set of \{binary, text \} \\
+\hline
+Format of output $x$ & Set of \{arbitrary, ordered \} \\
+\hline
+Output residual & boolean (true if the program returns the residual) \\
+\hline
+Possible entries in $x$ & set of $\mathbb{R} \cup \{- \infty, \infty, undef \}$ \\
+\hline
+\end{tabular}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Analytic Hierarchy Process}
+
+\begin{itemize}
+\item Example 1
+\begin{itemize}
+\item Embedded real-time system for digital signal processing
+\item $n = 10$
+\item $A$ is assumed to be Toeplitz
+\end{itemize}
+\end{itemize}
+
+\begin{tabular}{| l | l | l | l | l |}
+\hline
+~ & \textbf{Speed} & \textbf{Accuracy} & \textbf{Portability} & \textbf{Priority}\\
+\hline
+\textbf{Speed} & 1 & 3 & 5 & 0.64\\
+\textbf{Accuracy} & 1/3 & 1 & 3 & 0.26\\
+\textbf{Portability} & 1/5 & 1/3 & 1 & 0.11\\
+\hline
+\end{tabular}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\begin{frame}
+
+%\frametitle{AHP Continued}
+
+%\begin{itemize}
+%\item Example 2
+%\begin{itemize}
+%\item Structural mechanics software for education
+%\item $10 \leq n \leq 1000$
+%\item $A$ is assumed to be positive definite
+%\item Would have different relative priorities
+%\end{itemize}
+%\end{itemize}
+
+%\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Solution Validation Strategies}
+
+\begin{itemize}
+\item Create test cases with known solutions
+\begin{itemize}
+\item Assume $A$ and $x$, calculate $b$
+\item Given $A$ and $b$ calculate $x^*$ and compare to the assumed $x$
+\end{itemize}
+\item Comparison with Matlab
+\item Comparison with NAG library
+\item Where possible compare solution to interval arithmetic solution
+\item Experiments to describe how accuracy changes with increasing condition number
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Connection to Design}
+
+\begin{itemize}
+
+\item Abstract requirements to concrete design decisions
+%\item Software packages likely already exist within the program family
+\item Reuse existing packages within the program family
+\item Summarize existing software by the parameters of variation and binding time
+%\item Run time binding can often be changed to specification time binding
+\item If functional requirements match, then use NFRs
+\begin{itemize}
+\item AHP to compare each design against each of the NFRs %as done in SRS 
+%cols and rows are des alternatives, compared against a given NFR
+%repeat for each NFR
+\item Contribution of each NFR for each design alternative is found by multiplying the contribution of
+each alternative to the given NFR with the corresponding priority of that NFR
+\item Sum the contributions
+\item The highest overall score is the ``winning'' alternative
+\end{itemize}
+
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{A Family of Material Models}
+
+From McCutchan's work \cite{CaretteEtAl2008, SmithMcCutchanAndCao2007,
+  SmithEtAl2008, CaretteEtAl2007_TR, SmithMcCutchanAndCarette2017, McCutchan2007}
+
+\begin{center}
+{
+\includegraphics[width=0.5\textwidth]{../Figures/TestSpecimenUniaxial.pdf}
+}
+\end{center}
+
+\begin{center}
+\begin{tabular}{cc}
+\includegraphics[width=0.45\textwidth]{../Figures/Plastic.pdf} &
+\includegraphics[width=0.45\textwidth]{../Figures/Viscoplastic.pdf}\\
+\end{tabular}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Terminology Definitions}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
+\hline
+\rowcolor[gray]{0.9}
+Label: & D\_YieldFunction\\ \hline
+Symbol: & $F = F(\bm{\sigma}, \kappa)$\\ \hline
+Type: & $(\mbox{tensor2DT} \times \mathbb{R}) \rightarrow \mathbb{R}$\\ \hline
+%Units: & --\\ \hline
+Related: & {D\_Stress}, {D\_HardeningParameter}\\ \hline %originally related items, shorted for space
+Sources: & ...\\
+\hline Descrip: & The yield function defines a surface $F = 0$ in the six dimensional stress space ...\\
+%\hline History: & Created -- June 15, 2007\\
+\hline
+\end{tabular}
+\end{minipage}
+
+\begin{center}
+{
+\includegraphics[width=0.35\textwidth]{../Figures/YieldHardeningAndPlastPotential.pdf}
+}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Goal Statement}
+
+\noindent
+\begin{minipage}{\textwidth}
+\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
+\hline
+\rowcolor[gray]{0.9}
+Label: & G\_StressDetermination\\ \hline
+Descrip: & Given the initial stress and the deformation history of a material particle, determine the stress
+within the material particle.\\
+\hline
+Refine: & {T\_ConstitEquation}\\ \hline
+%History: & Created -- June 8, 2007\\ \hline
+\end{tabular}
+\end{minipage}
+
+\begin{center}
+{
+\includegraphics[width=0.5\textwidth]{../Figures/StressTensor.pdf}
+}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Assumptions}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
+\hline
+\rowcolor[gray]{0.9}
+Label: & A\_AdditivityPostulate\\ \hline
+Related: & {D\_StrainRate}\\ \hline
+Equation: & $\dot{\bm{\epsilon}} = \dot{\bm{\epsilon}}^e + \dot{\bm{\epsilon}}^{\mathit{vp}}$\newline
+with the following types and units\newline
+$\dot{\bm{\epsilon}}: \mbox{tensor2DT}$ (1/t) (1/s)\newline
+$\dot{\bm{\epsilon}}^e: \mbox{tensor2DT}$ (1/t) (1/s)\newline
+$\dot{\bm{\epsilon}}^{\mathit{vp}}: \mbox{tensor2DT}$ (1/t) (1/s)\\ \hline
+Descrip: & The total strain rate ($\dot{\bm{\epsilon}}$) is assumed to decompose into elastic
+($\dot{\bm{\epsilon}}^e$) and viscoplastic ($\dot{\bm{\epsilon}}^{\mathit{vp}}$)
+strain rates.\\ \hline 
+Rationale & This is a standard assumption for elastoplastic and elastoviscoplastic materials.  The 
+appropriateness of this assumption is born out by the success of theories built upon it.\\ \hline
+Source: & [6, page 339]; [7, page 181]\\ \hline %references hard-coded
+%History: & Created -- June 11, 2007\\ \hline
+\end{tabular}
+\end{minipage}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Theoretical Model}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
+\hline
+\rowcolor[gray]{0.9}
+Label: & T\_ConstitEquation\\
+\hline
+Related: & {A\_CauchyStress}, {A\_DeformationHistory}, {A\_PerzynaConstit}, {A\_AdditivityPostulate},
+{A\_ElasticConstit}, {A\_DescriptionOfMotion}, {V\_MaterialProperties}\\
+\hline
+Input: & $\bm{\sigma}_0: \mbox{tensor2DT}$ (StressU) (Pa)\newline
+$t_{\mathit{begin}}: \mathbb{R}$ (t) (s) \newline
+$t_{\mathit{end}}: \mathbb{R} $ (t) (s) \newline
+$\dot{\bm{\epsilon}}(t) : \{ t: \mathbb{R} | t_{\mathit{begin}} \leq t \leq t_{\mathit{end}} : t \} \rightarrow
+\mbox{   } \mbox{   } \mbox{  tensor2DT}$ (1/t) (1/s)\newline
+$\mathit{mat\_prop\_val}: \mbox{string} \rightarrow \mathbb{R}$\newline
+$E: \mathbb{R}^{+}$ (StressU) (Pa)\newline
+$\nu: \mbox{poissonT}$ (dimensionless)
+\\
+\hline
+\end{tabular}
+\end{minipage}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Theoretical Model Continued}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{| p{\colAwidth} | p{\colBwidth}|}
+\hline
+\rowcolor[gray]{0.9}
+Label: & T\_ConstitEquation\\
+\hline
+Output: & $\bm{\sigma}(t): \{ t: \mathbb{R} | t_{\mathit{begin}} \leq t \leq t_{\mathit{end}} : t \} \rightarrow
+\mbox{tensor2DT}$ such that 
+$$\dot{\bm{\sigma}} = \mathbf{D} \left ( \dot{\bm{\epsilon}} - \gamma < \phi (F(\bm{\sigma},\kappa) ) >
+\frac {
+\partial Q (\bm{\sigma})}{
+\partial \bm{\sigma}} \right )
+$$ 
+and $\bm{\sigma}(t_{\mathit{begin}}) = \bm{\sigma}_0$,
+the components of $\bm{\sigma}$ have the units of StressU (Pa)\\
+\hline
+Derive: & The governing differential equation is found by first solving for $\dot{\bm{\epsilon}}^e$
+in {A\_AdditivityPostulate} and then ...\\
+\hline
+Descrip: & The theoretical model is only completely defined once the associated
+variabilities~({V\_MaterialProperties}) that define the
+material have been set.  ...\\
+\hline History: & Created -- June 14, 2007\\
+\hline
+\end{tabular}
+\end{minipage}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Variabilities}
+
+\begin{itemize}
+\item $F = F(\bm{\sigma}, \kappa): \mathbb{R}^6 \times \mathbb{R} \rightarrow \mathbb{R}$
+\item $Q = Q(\bm{\sigma}): \mathbb{R}^6 \rightarrow \mathbb{R}$
+\item $\kappa = \kappa(\bm{\epsilon}^{\mathit{vp}}): \mathbb{R}^6 \rightarrow \mathbb{R}$
+\item $\phi = \phi(F): \mathbb{R} \rightarrow \mathbb{R}$
+\item $\gamma: \mathbb{R}$
+\item $\mathit{mat\_prop\_names}: \mbox{set of string}$
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Dependency Graph} %between commonalities
+
+\begin{center}
+{
+\includegraphics[width=1\textwidth]{../Figures/DependenceGraph.pdf}
+}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Dependency Graph Between Commonalities and Variabilities}
+
+\begin{center}
+{
+\includegraphics[width=1\textwidth]{../Figures/DependenceOfVariabilities.pdf}
+}
+\end{center}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Example}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{| p{0.24\textwidth} | p{0.65\textwidth}|}
+\hline
+\rowcolor[gray]{0.9}
+Label: & E\_StrainHardening\\ \hline
+{V\_MatName} & $\mathit{name = }$``Strain-Hardening Viscoelastic''\\
+\hline
+{V\_YieldFunct} & $F =  q \kappa^{\frac{n-1}{m}}$ (StressU) (Pa)\\ \hline
+{V\_PlasticPot} & $Q = q$ (StressU) (Pa)\\ \hline 
+{V\_HardParam} & $\kappa = \epsilon_q^{\mathit{vp}}$ (L/L) (m/m)\\
+\hline
+{V\_Phi} & $\phi = F^{\frac{m}{n}}$ ($\mbox{StressU}^{\frac{m}{n}}$) ($\mbox{Pa}^{\frac{m}{n}}$)\\
+\hline
+{V\_FluParam} & $\gamma = n A^{\frac{1}{n}}$ ($\mbox{StressU}^{-m} \mbox{t}^{-1}$) ($\mbox{Pa}^{-m} \mbox{s}^{-1}$)\\ \hline
+{V\_MatProps} & $\mathit{mat\_prop\_names} = $ \{``$A$'', ``$m$'', ``$n$'' \}, where the type of the
+material properties are ...\\
+%There is likely an upper limit on the values for $m$ and $n$, but at this time the value of these limits is unclear. 
+%With respect to units, $m$ and $n$ do not have units and $A$ has units of $\mbox{StressU}^{-m} \mbox{t}^{-1}$ ($\mbox{Pa}^{-m}%%
+%\mbox{s}^{-1}$)
+\hline
+{V\_Description} & $\mathit{descript =}$ ``This constitutive equation combines a power-law viscoelastic
+material with a strain hardening (softening) material.  ...''
+\\ \hline
+%Source & \cite{Smith2001}
+%\\ \hline
+%History & Created -- Sept 21, 2007\\ \hline
+\end{tabular}
+\end{minipage}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Code Generation}
+
+\begin{itemize}
+\item Specify variabilities
+\item Symbolically calculate terms needed by numerical algorithm, including $\frac{ \partial Q}{ \partial
+\bm{\sigma}}$, $\frac{\partial F}{\partial \bm{\sigma}}$, etc.
+\item Symbolic processing avoids tedious and error-prone hand calculations
+\begin{itemize}
+\item Reduces workload
+\item Allows non-experts to deal with new problems
+\item Increases reliability
+\end{itemize}
+\item Use Maple Computer Algebra System for model manipulation
+\item Convert math expressions into C expressions using ``CodeGeneration''
+\item Inline into a C++ class defining the material model
+\item A finite element program can this interface to realize the numerical algorithm
+\end{itemize}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}[fragile]
+
+\frametitle{BNF of DSL for $F$}
+
+\begin{grammar}
+[(colon){$\rightarrow$}]
+[(semicolon){$|$}]
+[(comma){}]
+[(period){\\}]
+[(quote){\begin{bf}}{\end{bf}}]
+[(nonterminal){$\langle$}{$\rangle$}]
+<expression>:<number>;\\
+(<expression>);\\
+<expression>,\^{ },<expression>;\\
+<expression>,$*$,<expression>;\\
+\end{grammar}
+~~~~~~~...
+\begin{grammar}
+[(colon){$\rightarrow$}]
+[(semicolon){$|$}]
+[(comma){}]
+[(period){\\}]
+[(quote){\begin{bf}}{\end{bf}}]
+[(nonterminal){$\langle$}{$\rangle$}]
+<simulation-variable-F>;<user-defined-constants>.
+<simulation-variable-F>:"Kappa";<simulation-variable-stress>;<simulation-variable-stress-macros>.
+<simulation-variable-stress>:"SigmaXX";"SigmaYY";"SigmaZZ";"SigmaXY";\\
+"SigmaYZ";"SigmaXZ".
+<simulation-variable-stress-macros>:"Sxx";"Syy";"Szz";"Sxy";"Syz";"Sxz";"Sm";"J2";"J3";"q".
+<user-defined-constants>:<string>.
+\end{grammar}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Concluding Remarks}
+\begin{itemize}
+\item Case studies of applying software engineering methodologies to mesh generating systems and linear solvers
+\item Appropriate and advantageous to apply program family strategy
+\item Challenges for software engineers
+\item General purpose scientific software is best studied as a program family
+\begin{itemize}
+\item Variabilities are assumptions about problems that can be handled
+\item Derive requirements from commonality analysis
+\end{itemize}
+\item Eventually hope for automatic code generation
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Concluding Remarks (Continued)}
+
+A new methodology for documenting requirements for general purpose scientific computing software
+
+\setcounter{temp}{0}
+
+\begin{enumerate}
+\setcounter{enumi}{\value{temp}}
+
+\item Validatable requirements
+\begin{itemize}
+\item Relative comparison between program family members
+\item Focus on description rather than specification
+\item Solution validation strategy
+\end{itemize}
+\item Abstract
+\begin{itemize}
+\item Refine goal statement to theoretical model to input assumptions
+\item In some cases one may want to turn off input checking
+\item Connection to design
+\end{itemize}
+
+\setcounter{temp}{\value{enumi}}
+\end{enumerate}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle{Concluding Remarks (Continued)}
+
+\begin{enumerate}
+\setcounter{enumi}{\value{temp}}
+
+\item NFRs
+\begin{itemize}
+\item Relative comparison
+\item AHP
+\end{itemize}
+\item Capture and reuse
+\begin{itemize}
+\item Systematic consideration from general to specific
+\item CA refined by a family of SRSs
+\item CA and SRS summarize existing knowledge and currently available software
+\item Standard template allows comparison
+\item Convenient framework for summarizing existing literature
+\end{itemize}
+
+\setcounter{temp}{\value{enumi}}
+\end{enumerate}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle<presentation>{Concluding Remarks}
+
+\begin{itemize}
+\item A new template for a family of models of physical phenomena
+\item Refinement of \structure{Goals} to \structure{Theoretical Models} using \structure{Data Definitions} and
+\structure{Assumptions}
+\item \structure{Variabilities} are identified in the Theoretical Model
+\item A constitutive equation can be written using a (declarative) DSL and the code can be generated
+\item A DSL has been built, using Maple, for a virtual material testing laboratory
+\end{itemize}
+  
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+
+\frametitle<presentation>{Concluding Remarks}
+
+\begin{itemize}
+\item SC software is a great candidate for development as a program family
+\item Produce programs that are as special or general purpose as needed % run time bindings
+\item Improve reusability, usability and reliability
+\item Potential to improve performance
+\item A commonality analysis facilitates the design of a DSL
+\item Symbolic processing and code generation are very useful techniques
+\item \structure{We will return to code generation later}
+\end{itemize}
+  
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+\begin{frame}[allowframebreaks]
+\frametitle{References}
+
+\bibliography{../../ReferenceMaterial/References}
+
 \end{frame}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/Lectures/L09_VerificationAndValidation/VerificAndValid.pdf b/Lectures/L09_VerificationAndValidation/VerificAndValid.pdf
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diff --git a/Lectures/L09_VerificationAndValidation/VerificAndValid.tex b/Lectures/L09_VerificationAndValidation/VerificAndValid.tex
new file mode 100755
index 0000000..f0fac85
--- /dev/null
+++ b/Lectures/L09_VerificationAndValidation/VerificAndValid.tex
@@ -0,0 +1,132 @@
+%\documentclass[handout]{beamer} 
+\documentclass[t,12pt,numbers,fleqn]{beamer}
+%\documentclass[ignorenonframetext]{beamer}
+
+\newif\ifquestions
+%\questionstrue
+\questionsfalse
+
+\usepackage{pgfpages} 
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,
+    linkcolor=blue,
+    citecolor=blue,
+    filecolor=blue,
+    urlcolor=blue,
+    unicode=false}
+\urlstyle{same}
+
+\bibliographystyle{plain}
+
+%\usetheme{Iimenau}
+
+\useoutertheme{split} %so the footline can be seen, without needing pgfpages
+
+%\pgfpagesuselayout{resize to}[letterpaper,border shrink=5mm,landscape]  %if this is uncommented, the hyperref links do not work
+
+\mode<presentation>{}
+
+\input{../def-beamer}
+
+\newcommand{\topic}{09 Verification and Validation}
+
+\input{../titlepage}
+
+\begin{document}
+
+\input{../footline}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Verification and Validation}
+
+\bi
+\item Administrative details
+\item Questions?
+\item License and copyright
+\ei
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Administrative Details}
+
+\bi
+\item GitHub issues for colleagues
+\ei
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Administrative Details: Deadlines}
+~\newline
+\begin{tabular}{l l l}
+\textbf{SRS} & Week 05 & Oct 4\\
+\textbf{V\&V Present} & Week 06 & Week of Oct 16\\
+\textbf{V\&V Plan} & Week 07 & Oct 25\\
+MG Present & Week 08 & Week of Oct 30\\
+MG & Week 09 & Nov 8\\
+MIS Present & Week 10 & Week of Nov 13\\
+MIS & Week 11 & Nov 22\\
+Impl.\ Present & Week 12 & Week of Nov 27\\
+Final Documentation & Week 13 & Dec 6\\
+\end {tabular}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Administrative Details: Presentation Schedule}
+
+\bi
+\item V\&V Present
+\bi
+\item Tuesday: Steven, Alexandre P., Alexander S.
+\item Friday: Geneva, Jason, Yuzhi
+\ei
+\item MG Present
+\bi
+\item Tuesday: Xiaoye, Shushen, Devi, Keshav, Alex P, Paul
+\item Friday: Yuzhi, Jason, Geneva, Alex S, Isobel, Steven
+\ei
+\item MIS Present
+\bi
+\item Tuesday: Isobel, Keshav, Paul
+\item Friday: Shushen, Xiaoye, Devi
+\ei
+\item Impl.\ Present
+\bi
+\item Tuesday: Alexander S., Steven, Alexandre P.
+\item Friday: Jason, Geneva, Yuzhi
+\ei
+
+\ei
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}
+\frametitle{Questions?}
+\begin{itemize}
+\item Questions about ...
+\end{itemize}
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\begin{frame}[allowframebreaks]
+\frametitle{References}
+
+\bibliography{../../ReferenceMaterial/References}
+
+\end{frame}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\end{document}
\ No newline at end of file
-- 
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