\appendixchapter{\texttt{IndexMap} Element}{n}
\label{appendixC}

Full support for an arbitrary \texttt{IndexMap} is now considered an extension to the CLF format as it complicated some implementation decisions. A reader should still recognize the token \texttt{IndexMap} structure and if only two elements are present should treat them as described. All other dimensions of an \texttt{IndexMap} are considered extensions and a user should be notified if the current reader does not support this extension.

\subsection*{\texttt{IndexMap} definition}
The mapping of input code values to indexes of the table is modifiable allowing for remapping the range of applicable input values, changing the spacing of the input sampling function, and reshaping of the index function for lookups into the LUT. For example, with a 16-bit floating-point code value, the table may be defined to only operate upon input values in the range from 0.0 to 1.0. Values outside of this range are not able to be looked up within the LUT range and extrapolation is not required. A reshaping function is useful to modify the density of sampled values in different regions of the LUT. Reshaping changes the relationship between input code values and the lookup position into the LUT.

An \texttt{IndexMap} is specified with a list containing pairs of values,  $inValue@i$, which replaces the original $inValue$ with the new $inValue$ at each entry $i$ in the table.

In this example of reshaping, the precision of the higher input values is increased with more samples in that region of the function with the following set of index values:

IndexMap:   $0@0 \quad 512@1 \quad 756@2 \quad 900@3 \quad 1023@4$

This changes a table from the behavior of the left side to the new behavior on the right side. Note that the pixel output results of each of these transforms would be different.

\begin{center}
\begin{tabularx}{2in}{|X|X|X|}
\hline
\multicolumn{3}{|c|}{Without \texttt{IndexMap}} \\\hline
index & inValue & outValue \\\hline
0 & 0 & 0 \\\hline
1 & 255	& 10 \\\hline
2 & 511	& 100 \\\hline
3 & 767	& 1000 \\\hline
4 & 1023 &1023 \\\hline
\end{tabularx}
\quad
\begin{tabularx}{2in}{|X|X|X|}
\hline
\multicolumn{3}{|c|}{With \texttt{IndexMap}} \\\hline
index & inValue & outValue \\\hline
0 & 0 & 0 \\\hline
1 & 512 & 10 \\\hline
2 & 756 & 100 \\\hline
3 & 900 & 1000 \\\hline
4 & 1023 & 1023 \\\hline
\end{tabularx}
\end{center}

\subsection*{XML Example}
As an example of an \texttt{IndexMap}, in a 5 position LUT1D with 10-bit input values, the IndexMap could be the following:

\lstinline!<IndexMap dim=5> 60@0  301@1  542@2  782@3  1023@4 </IndexMap>!

An input pixel value of 301 would be calculated using the output value in lut position `1.'

The first value in the \texttt{IndexMap} represents the minimum expected input value for the LUT and must contain the new input value for $lut[0]$. The last value in the \texttt{IndexMap} must always define the maximum expected input value, which is associated with the last entry in the LUT, $lut[n-1]$.  Values outside of the range of the \texttt{IndexMap} are looked up at either the minimum or the maximum input values.

In a default \texttt{IndexMap}, the range of available inValues is spread equally (i.e. linear interpolation) across all of the available array indexes.  Each index position in the array has a new inValue corresponding to that index position.

In the simplest case, an \texttt{IndexMap} listing only two values, $inMin$ and $inMax$ defines the range of inValues that the LUT operates on where the $inValue$  for each index in the LUT ($n=4$) is linearly calculated as:

\begin{center}
\begin{tabularx}{3.5in}{|l|X|l|}
\hline
$i$ & $inValue$ & $outValue$ \\\hline
0 & $inMin$	& 5 \\\hline
1 & $\left(\frac{i}{(n-1)}\times(inMax-inMin)\right)+inMin$	& 100 \\\hline
2 & $\left(\frac{i}{(n-1)}\times(inMax-inMin)\right)+inMin$	& 1000 \\\hline
3 & $inMax$ & 10000 \\\hline
\end{tabularx}
\end{center}

Values outside of the range of the \texttt{IndexMap} are looked up at either the minimum or the maximum input values. For example, with 16-bit floating-point codes (\texttt{inBitDepth="16"}), a LUT may have been constructed to use only the input values of $[0 \ldots 1.0]$, in which case, the minimum and maximum input values will be $0.0$ and $1.0$, respectively. If there was an input value of $-1.0$, the lookup for the minimum input value would be used.

In the absence of an \texttt{IndexMap}, the default case, the minimum and maximum input values are determined using the full range of available code values based on the input bit depth and type.

In a complete \texttt{IndexMap}, there is a value provided for each index into the table. This is desired particularly in the case of a LUT3D so that the exact input value of each entry is available.

An \texttt{IndexMap} will never contain more entries than the lookup table itself.

If only one \texttt{IndexMap} is present in a LUT node, it is applied for all the color components. If there is more than one \texttt{IndexMap}, there must be one \texttt{IndexMap} for each component (in RGB order) for a total of three.

\subsection*{Example}

\lstset{frame=single}
\begin{lstlisting}[caption=Example of a partially enumerated ``Shaper LUT'']
<LUT1D id="lut-25" name="shaper LUT" inBitDepth="10i" outBitDepth="16f">
	<Description> 1D LUT with shaper </Description>
	<IndexMap dim=4>0@0 10@100 20@250 30@360 40@440 445@445
	    700@600 800@700 900@850 950@1023</IndexMap>
	<Array dim="1024 1">
        0.00
        0.32
        0.50
        <1020 entries omitted>
        1.0
	</Array>
</LUT1D>
\end{lstlisting}

An \texttt{IndexMap} (optional extension) is used to reshape the sampling function in the LUT.  While it is sometimes possible to combine LUT calculations so that a single LUT would suffice, there are also cases where it is convenient to separate the array data from the sampling function.

In a partial \texttt{IndexMap}, there are greater than 2 items in the list but fewer items than there are index positions into the table. This is the case, for example, where you want to use a 10-point function to change the input shape of a 10-bit LUT that has 1023 entries.

\lstset{frame=none}
\begin{lstlisting}
<IndexMap dim=10>0@0 10@100 20@250 30@360 40@440 445@445 700@600 800@700 900@850
    950@1023</IndexMap>
\end{lstlisting}
\lstset{frame=single}

Each index position in the table will have a new $inValue$ calculated using the function in the \texttt{IndexMap} list.

In this example, the input sample function is changed to stretch the regions of the LUT that process the shadows and highlights. An input value of 10 is instead placed at position 100 in the LUT allowing greater definition in the LUT for all values between 0 and 10 (positions $0 \ldots 100$ in the LUT). In the highlights, an input value of 800 is instead placed at 700, and 900 is instead at position 850. This gives the highlights 150 positions in the LUT as compared to the original 100 positions. Needless to say the accuracy of the middle region of the LUT is compressed.

Another possible use is to apply an overall delta to the data in the array, perhaps with an offset or gain function. If the input data is `density' (log with a gamma), and the LUT represents a film print emulation, then applying an offset to the lookup with a shaper LUT will simulate the effect of a printer light change.

\subsection*{Indexing Calculations with \texttt{IndexMap}}
A \texttt{ProcessNode} using LUT tables must perform an index calculation to take the range of input values and ratio them to the input `index' range of the table (i.e. the minimum and maximum index positions into the table). This allows the LUT location calculation to be easily achieved as the normalized index function can be multiplied by the number of entries in the LUT to get a direct hash function to the appropriate LUT locations. For integer inputs, this is straightforward as the inBitDepth attribute may be used to apply the whole range of input across the whole range of index positions.

For floating-point inputs, the \texttt{IndexMap} element can be used to select the beginning and end values of the input floats, and apply them across the range of index positions. The LUT designer  may specify range selection with an \texttt{IndexMap}, and/or convert floats in a prior \texttt{Range} node or 1DLUT. In the absence of an \texttt{IndexMap}, the [0 to 1.0] range is applied across the input index range.

Two methods are possible for increasing the accuracy of the LUT calculation by changing the density of samples in a particular region of the LUT. One is to use a 1DLUT node as a `shaper LUT' ahead of a main LUT. The other method which has a more compact representation is to use the \texttt{IndexMap} as a re-indexing function. The array in the main LUT node and the \texttt{IndexMap} are specified at the same time to provide better sampling using the selected interpolation method. The \texttt{IndexMap} defines values at specific array coordinates.  For this second method, the \texttt{IndexMap} values are first used to calculate a hash function (the \texttt{IndexMap} table) that should have as many entries as the number of entries in the target LUT. This table contains output entries that are fractional coordinates into the array of the main LUT. The first lookup into the \texttt{IndexMap} uses the standard LUT access method ($range fraction \times numEntries$) to find the second LUT entry point. This second LUT output value is used for direct access to the main LUT whether a 1DLUT or a 3DLUT where the fractional amount provides the interpolation ratio to the next coordinate of the LUT. A different \texttt{IndexMap} will require revision to the main LUT node to place entries in new positions.


\chapter{Web Search}
\begin{multicols*}{2}

\noindent Types of user needs:
\begin{itemize}
    \item Informational: want to learn about something
    \item Navigational: want to go to a page
    \item Transactional: want to do something
    \item Gray areas
\end{itemize}

\noindent Search Engine Optimisation: manipulate page contents to appear high up in search results for selected keywords.

\section{Size of the Web}
Relative size of search engines, example:
$$A\cup B = \frac{1}{2} \text{SizeA}  = \frac{1}{6}\text{SizeB}$$
$$\frac{\text{SizeA}}{\text{SizeB}} = \frac{1}{3}$$

\section{Duplicate Detection}
\noindent Exact match can be detected with fingerprints. If fingerprints are equal, the pages are identical. \\

\noindent Near-duplication can be detected using shingles (n-grams) and jaccard coefficient.

\section{Basic Crawler Operation}
\noindent Algorithm:
\begin{itemize}
    \item Begin with known seed URL
    \item Fetch and parse them and place the extracted URL on a queue
    \item Fetch each URL on the queue
    \item Repeat
\end{itemize}

\noindent Crawler must be polite (only crawl allowed pages and respect robots.txt) and robust

\section{PageRank}

\noindent PageRank of a page $u$ is:
$$P(u) = \sum_{v \in B_u} \frac{PR(v)}{L_v}$$

\noindent where $B_u$ is the set of pages that point to $u$, and $L_v$ is the number of outgoing links from page $v$.\\

\noindent A Markov chain consists of $N$ states and a transition probability matrix. Transition probability matrix contains the transition from $N$ states to all other $N$ states, forming a $N\times N$ matrix. To build the transition probability matrix:

\begin{itemize}
    \item If there is a hyperlink from page $i$ to page $j$, then $A_{ij} = 1$
    \item Divide each 1 in $A$ by the number of 1 in its row
    \item Multiply the resulting matrix by $1 - \alpha$
    \item Add $\alpha / N$ to every entry
\end{itemize}
\noindent We use teleportation rate $\alpha$ to prevent sticking in a dead end. By doing so, we get an ergodic Markov chain. For any ergodic Markov chain, there is a unique long-term visit rate for each state.

\section{Hyperlink-induced Topic Search}
\noindent Relevance type 1: A hub page has a good list of links to pages answering the information need\\

\noindent Relevance type 2: An authority page is a direct answer to the information need\\

\noindent A good hub page links to many authority pages. A good authority page is linked to by many hub pages\\

\noindent To compute the hubs and authorities scores, first initialise $h(d) = a(d) = 1$, then do the following iteratively until converge:
$$\forall d, h(d) = \sum_{d\rightarrow y} a(y)$$
$$\forall d, a(d) = \sum_{d\leftarrow y} d(y)$$

\section{Missing List}

\begin{itemize}
    \item Lecture 2: Example for 3rd approach in context-sensitive correction
    \item Lecture 2: General issues in spell correction
    \item Lecture 2: Soundex spell correction
    \item Lecture 5: Problem of Relevant Feedback
    \item Lecture 6: Introduction to classification
    \item Lecture 6: Supervised classification
    \item Lecture 6: Classification Methods
    \item Lecture 6: Probability and conditional probability
    \item Lecture 6: Naive Bayer Learning and Classifying steps
    \item Lecture 6: Naive Bayes complexity
    \item Lecture 6: Naive Bayes Pros and Cons
    \item Lecture 6: Classification using Vector Spaces
    \item Lecture 6: kNN learning algorithm and discussion
    \item Lecture 6: Support Vector Machine Formula
    \item Lecture 6: k-fold cross-validation
    \item Lecture 7: Partition Clustering, Hierarchical Clustering, Hard and soft clustering
    \item Lecture 7: Time complexity of k-mean clustering
    \item Lecture 7: Seed choice for k-mean clustering
    \item Lecture 7: Total Cost and Total Benefit of k-mean clustering
    \item Lecture 7: Discriminative labeling
    \item Lecture 8: History of Search Engine, Anchor text
\end{itemize}

\end{multicols*}


%The extension of the material point method to the DG approximation has been motivated in the previous section and is carried out hereinafter.
After a brief historical review of DG methods, the Discontinuous Galerkin Material Point Method is derived within the large strain framework with a total Lagrangian formulation.
It will be seen that this new numerical approach makes use of the approximate-state Riemann solver developed in section \ref{sec:riemann_solvers} to compute intercell terms whose purpose is to connect elements together.
At last, the DGMPM solution scheme will be provided for hyperbolic problems.

\subsection{The discontinuous Galerkin approximation}
The DG approximation was first introduced in the context of the finite element method for the solution of the neutron transport equation \cite{NeutronDG}.
This hyperbolic equation describes the advection of the angular flux which describes the probable number of neutrons at a given location.
Since neutrons can lie in a cell of a finite element mesh while its neighbors are empty, the need for a discontinuous approximation of the primal field across element interfaces within a FEM context arises. % The need for description of discontinuities
Hence, an approximate solution was sought by the Galerkin method, in a domain discretized with triangular elements by means of Lagrange polynomials that can be discontinuous across the cells. This approach amounts to duplicate the nodes of the mesh so that the support of each shape function reduces to one finite element. This early work have launched a series of developments of the \textbf{Discontinuous Galerkin Finite Element Method} (DGFEM) for parabolic \cite{Arnold_IPM}, elliptic \cite{Hansbo_DGsolid,Noel_HEDG}, and hyperbolic problems \cite{Cockburn,FVilar_DG}. Indeed, the DGFEM gained more and more popularity since the 80's, even for problems that do not involve discontinuities, on account of its ability to locally handle high-order approximation and its highly parallelizable nature.
Research conducted in the context of hyperbolic problems, of particular interest here, enabled the introduction of numerical tools developed for Finite Volume Methods (FVM) within finite element schemes.
Namely, the use of suitable \textit{slope limiters} \cite{vanLeer_Limiters} based on the \textit{total variation} \cite{Harten} enables the formulation of flexible numerical methods in which good resolution of discontinuities is possible without destroying the accuracy in smooth regions. Furthermore, these approaches can easily handle mesh-adaption strategies thanks to the relaxation of the fields continuity. Nevertheless mesh tangling problems do not vanish.
Thus, the introduction of DG approximation in the MPM should lead to a numerical method that benefits from both FEM and FVM features and enables local high-order approximation while avoiding mesh entanglement instabilities.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % Hyperbolic
% The \textit{discontinuous Galerkin (DG)} approximation enables to build numerical schemes that benefit from both finite element and finite volume methods.
% Parler des limiteurs \cite{vanLeer_Limiters} pour atteindre la notion de schéma TVB et TVDM (TVD \cite{Harten_TVD}). Extension to RK so that the scheme can reach locally high order accuracy. In addition, the same order of accuracy is reached for velocity and gradients within a finite element framework when the weak form is based on a conservation laws system.

% % In parallel, for parabolic or elliptic
% \cite[parabolic+penalties]{Arnold_IPM},\cite{Hansbo_DGsolid},\cite[elliptic]{Noel_HEDG}: Three field Hu-Washizu variational formulation ; assumed form of deformation gradient ; Total Lagrangian ; penalization of displacement jumps; Advantages--Drawbacks in note books

% % Introduction of HDG
% \cite{Cockburn_HDG0},\cite{Cockburn_HDG1},\cite{Cockburn_HDG2}: HDG for elliptic problems (differrence between LDG-H and HDG ?)

% % Recently extended to hyperbolic problems in solid dynamics
% \cite{NGuyen_HDG} for application to solid mechanics and extension to hyperbolic problems. Solved for displacement with enforced continuity across elements interfaces. Does not use the characteristic structure as what is done in finite volumes or original DGFEM

% However, all those approaches based on the primal field do not use the characterisitc structure. Dans l'état, ça ne peut pas être appliqué à la méca des solides puisque l'on est obligé d'imposer la continuité du champ de déplacement et donc de formuler le problème en u.
% \begin{itemize}
% \item \cite{Chavent_Salzano,Chavent_Cockburn,Cockburn_Shu,DGFEM_CFL,Cockburn}:
% \item \cite{Chavent_Salzano}--\cite{Cockburn} developments of DGFEM (mainly for fluid mechanics ?)
% \item \cite[parabolic+penalties]{Arnold_IPM},\cite{Hansbo_DGsolid},\cite[elliptic]{Noel_HEDG}: Three field Hu-Washizu variational formulation ; assumed form of deformation gradient ; Total Lagrangian ; penalization of displacement jumps; Advantages--Drawbacks in note books
% \item \cite{Cockburn_HDG0},\cite{Cockburn_HDG1},\cite{Cockburn_HDG2}: HDG for elliptic problems (differrence between LDG-H and HDG ?)
% \item \cite{NGuyen_HDG} for application to solid mechanics and extension to hyperbolic problems. Solved for displacement with enforced continuity across elements interfaces. Does not use the characteristic structure as what is done in finite volumes or original DGFEM
% \item \cite{DGPIC,DGPIC_maxwell}: application to PIC
% \end{itemize}

% RKDG - limiters - TVD - CFL - HDG (steady convection-diffusion problem (elliptic problem)) - HE DG
% Applied to steady solid mechanics problems for the ability of local high order approximation ?
% Continuity of displacement enforced through Lagrange multipliers. Benefits from superconvergence properties. A priori on utilise plus du HDG en méca du solide \cite[ALE]{NGuyen_HDG} + postprocessing pour le champ de déplacement + implicit en dynamique. Originally developed in \cite{Cockburn_HDG0}: "we may define more generally as a hybrid method any finite element method based on a formulation where one unknown is a function, or some of its derivatives, on the set $\Omega$, and the other unknown is the trace of some of its derivatives of the same function, or the trace of the function itself, along the boundaries of the set K". Numerical traces \cite{Cockburn_HDG1}

\subsection{Derivation of the DGMPM}
Consider again a continuum solid body with volume $\Omega_t$ within the time interval $\tau$. The DGMPM is expected to provide a material description of a deformation so that an approximate solution of a Lagrangian system of conservation laws written in conservative form is sought. Recall that such a conservative form for some vector of conserved quantities $\Ucb$ reads, in Cartesian coordinates:
\begin{equation}
  \label{eq:conservative_form}
  \drond{\Ucb}{t} + \sum_{\alpha=1}^D \drond{\Fcb\cdot \vect{E}_\alpha}{X_\alpha} = \Scb \quad \forall \vect{X},t \in \Omega_0 \times \tau
\end{equation}
%where $\Omega_0$ denotes the reference configuration of the continuum.
\subsubsection{The DGMPM discretization}
As for MPM, a continuum body $\Omega_t$ is discretized within the time interval $\tau$ into a set of $N_p$ material points in an arbitrary Cartesian grid made of  $N_n$ nodes and $E$ non-overlapping cells of volume $\Omega^e$. The boundary of the domain is again defined by the set of edges separating empty cells from those containing particles (see figure \ref{fig:domain} for a two-dimensional example).
In addition, the reference mass density is described in the computational grid by means of the Dirac delta function and particle masses:
\begin{equation}
  \label{eq:mass_density_DGMPM}
  \rho_0\(\vect{X}\) =  \sum_{p=1}^{N_p} m_p \delta\(\vect{X}^p - \vect{X}\)
\end{equation}
In a similar manner to FEM and MPM, the vector of conserved quantities is approximated on the mesh by:
\begin{equation}
  \label{eq:DGMPM_node2points}
  \Ucb(\vect{X},t) = \sum^{N_n}_{i=1} S_{i}(\vect{X})\Ucb^i(t)
\end{equation}
with $\Ucb^i$ the vector of conserved quantities at node $i$, and $S_{i}(\vect{X})$ the shape function attached to it. Note that the convention of denoting particle and nodal fields by $p$ and $(i,j)$ still holds in this section.

\subsubsection{Weak formulation of the continuum problem}
Multiplying equation \eqref{eq:conservative_form} by a test function $\Vcb$ yields the weak formulation of the problem:
\begin{equation}
  \label{eq:weak_form}
  \begin{aligned}
    &\text{Find $\Ucb \in \Vscr_h^1$ such that} \\
    &\int_{\Omega_t} \drond{\Ucb}{t} \vect{\Vc} \: d\Omega + \int_{\Omega_t}   \drond{\Fcb_\alpha}{X_\alpha}\vect{\Vc} \: d\Omega    = \int_{\Omega_t} \Scb \vect{\Vc} \: d\Omega \quad \forall \: \vect{\Vc},t \in  \Vscr_h^1\times \tau
  \end{aligned}
\end{equation}
The key idea of DG methods is to allow the jump of fields across mesh elements faces by using broken polynomial spaces for the approximate solution \cite[Ch.1]{DiPietro}:
\begin{equation}
\Vscr^k = \{ \Vcb \in H^k(\Omega^e) \} \quad ;\quad \Vscr_h^k = \{\Vcb \in \Pscr^k(\Omega^e) \} \subset \Vscr^k
\end{equation}
with $H^k(\Omega^e)$, the Sobolev space and $\Pscr^k(\Omega^e)$, the space of polynomials of degree $k$ in $\Omega^e$. We restrict our attention here to linear polynomials ($k=1$). These broken polynomial spaces allow the rewriting of the weak form element-wise. After integration by parts, one gets:
\begin{equation}
  \label{eq:DGMPM_weak_form}
  \begin{aligned}
    &\text{Find $\Ucb \in \Vscr_h^1$ such that} \\
    &\int_{\Omega^e} \drond{\Ucb}{t} \vect{\Vc} \: d\Omega - \int_{\Omega^e} \Fcb_\alpha  \drond{\vect{\Vc}}{X_\alpha} \: d\Omega   + \int_{\partial \Omega^e} \(\Fcb\cdot \vect{N}\)  \vect{\Vc} \: d\Gamma = \int_{\Omega^e} \Scb \vect{\Vc} \: d\Omega \quad \forall \: \vect{\Vc},e,t \in  \Vscr_h^1\times \[1,E\]\times \tau
  \end{aligned}
\end{equation}
where $\partial \Omega^e$ is the boundary of the $e$th element with outward normal vector $\vect{N}$. The dot operator $\Fcb\cdot \vect{N}$ denotes the inner product between the outward normal vector and every component of the flux, thus yielding the intercell flux, written $\Fcb_N$ for simplicity. Next, the introduction of specific fields:
\begin{equation}
  \label{eq:specific_quantities}
  \Ucb = \rho_0 \bar{\Ucb} \quad ; \quad \Fcb_\alpha = \rho_0 \bar{\Fcb}_\alpha \quad ; \quad \Scb = \rho_0 \bar{\Scb}
\end{equation}
combined with the definition of mass density \eqref{eq:mass_density_DGMPM}, leads to the following Lagrangian formulation:
%Such a discretization of the reference mass density combined with the writing of Lagrangian conservation laws \eqref{eq:conservative_form} yields a total Lagrangian formulation, and equation \eqref{eq:DGMPM_weak_form} thus reads:
\begin{equation}
  \label{eq:DGMPM_discrete_weak}
  \sum_{p=1}^{N_p} m_p\[\drond{\bar{\Ucb}}{t}  \vect{\Vc} - \bar{\Fcb}_{\alpha} \drond{\vect{\Vc}}{X_\alpha} -\bar{\Scb}  \vect{\Vc} \]_{|\vect{X}=\vect{X}^p} + \int_{\partial \Omega^e} \Fcb_N  \vect{\Vc} \: d\Gamma = 0 \quad \forall \: \vect{\Vc},e,t \in  \Vscr_h^1\times \[1,E\]\times \tau
\end{equation}
As a first development step of the method, only the total Lagrangian formulation is considered in order to eliminate the grid crossing instability.

Introduction of the DGMPM approximation \eqref{eq:DGMPM_node2points} and arbitrariness of the test field in the weak form \eqref{eq:DGMPM_discrete_weak} finally provide the semi-discrete system that must be solved on the grid:
\begin{equation}
  \label{eq:DGMPM_semi_discrete}
  \sum_{p=1}^{N_p}\[ S_{ip} m_p S_{jp} \drond{\bar{\Ucb}^j}{t}  - \drond{S_{ip}}{X_\alpha} m_p S_{jp} \bar{\Fcb}^j_{\alpha} - S_{ip} m_p \bar{\Scb}^p\] + \int_{\partial \Omega^e} S_i(\vect{X}) \Fcb_N  \: d\Gamma =  0  \quad \forall \: e,t \in  \[1,E\]\times \tau
\end{equation}
or, in matrix form:
\begin{equation}
  \label{eq:DGMPM_semi_discrete_matrix}
  M_{ij} \drond{\bar{\Ucb}^j}{t} - K^\alpha_{ij} \bar{\Fcb}^j_{\alpha} - \Scb^i + \vect{\hat{\Fc}}^i = \vect{0}
\end{equation}
Here again, particles play the role of integration points in volume integrals owing to the Dirac delta function. Hence, the consistent mass matrix $M_{ij}$ may also be singular due to reduced integration so that the diagonally lumped mass matrix $M^L_i$ is used.
\begin{remark}
  \label{rq:DGPIC}
  An extension of PIC to DG approximation for the solution of Maxwell's equations is proposed in \cite{DGPIC_maxwell} and \cite{Stindl_DGPIC} in which different projections of fields between the grid and particles are used. Although those methods allow local high-order approximation, particles do not carry every field so that the DGPIC, as the original PIC, cannot be considered as a fully Lagrangian approach. In addition, the use of the Gauss quadrature rule for volume integrals of the weak form makes the approach different from that developed in the following.
\end{remark}
The discrete system is derived by discretizing the time interval $\tau$ into $N_t$ subintervals and using the explicit forward Euler method:
%Finally, the explicit forward Euler time discretization of $\tau$ in $N_t$ subinterval is perfomed, leading to the discrete system:
\begin{equation}
  \label{eq:DGMPM_discrete}
  M^L_i \frac{\bar{\Ucb}^{i,n+1} - \bar{\Ucb}^{i,n}}{\Delta t^{n} } = K^\alpha_{ij} \bar{\Fcb}_{\alpha}^{j,n} + \Scb^{i,n}- \vect{\hat{\Fc}}^{i,n}
\end{equation}
where again, the superscripts $(\bullet)^{k,l}$ denote a field evaluated at node $k$ and time step $l$. Note that in general the source term $\Scb$ may depend on the vector of conserved quantities, hence the superscript $n$ in equation \eqref{eq:DGMPM_discrete}.
Alternatively, a \textit{second-order Runge-Kutta (RK2)} explicit time discretization may be employed, leading to the following two-stage discrete form:
\begin{equation}
  \label{eq:DGMPM_discrete_RK2}
  \begin{aligned}
    & M^L_i \frac{\bar{\Ucb}^{i,n+1/2} - \bar{\Ucb}^{i,n}}{\Delta t^{n} } = \frac{1}{2}\(K^\alpha_{ij} \bar{\Fcb}_{\alpha}^{j,n} + \Scb^{i,n}- \vect{\hat{\Fc}}^{i,n}\)  \\
    & M^L_i \frac{\bar{\Ucb}^{i,n+1} - \bar{\Ucb}^{i,n}}{\Delta t^{n} } = K^\alpha_{ij} \bar{\Fcb}_{\alpha}^{j,n+1/2} + \Scb^{i,n+1/2}- \vect{\hat{\Fc}}^{i,n+1/2}
  \end{aligned}
\end{equation}
\begin{remark}
  We chose here one existing two-stage second order Runge-Kutta method among others. See for instance \cite[Sec.~10.4.2]{Leveque} for a Total Variation Diminishing version of the RK2 time discretization.
\end{remark}

\subsection{Non-homogeneous hyperbolic system}
\label{sec:splitting}
Solid mechanics equations may lead to source terms in the conservative form even for neglected body forces. For instance, the writing of equations in cylindrical and spherical coordinates systems, in which the gradient operators involve terms that are not derivatives, yields a right-hand side in system \eqref{eq:conservative_form} that depends on $\Ucb$ and called a geometric source term \cite[Ch.17]{Leveque}.
%%
This is also the case for relaxation systems as that identified in section \ref{sec:general-formulation} for elasto-viscoplasticity. In the latter case, a small relaxation time $\tau$ compared to the time scale governed by the convective part (\textit{i.e. the characteristic speeds of the problem}) leads to a \textit{stiff system} \cite{Stiff_problems}.

The solution of this class of problems on underresolved grids, that is $\tau \ll \Delta t$, requires the use of asymptotically convergent, accurate and stable numerical methods in order to avoid nonphysical results \cite{Leveque_stiff}.
Thus, the numerical scheme must enable the use of underresolved grids whose time step is dictated by the Courant-Friedrichs-Lewy (CFL) stability condition. The method also has to be \textit{asymptotic preserving} in the limit $\tau \rightarrow 0$ for fixed $\Delta t$ and $\Delta X$ \cite{Stiff_problems}, that is the limiting scheme is a good discretization of the equilibrium system even if the source term is underresolved. At last, the order of accuracy in the stiff limit and stability at the discrete level must be preserved \cite{Pareschi_stiff}, and the numerical scheme must be able to preserve the steady state numerically \cite{Toro_stiff}.

A commonly used approach to solve non-homogeneous systems consists in solving alternatively a homogeneous PDE system and a system of ODEs, namely:
\begin{subequations}
  \begin{alignat}{1}
    \label{eq:Splitting_advec}
    & \drond{\Ucb}{t} + \sum_{\alpha=1}^D \drond{\Fcb\cdot \vect{E}_\alpha}{X_\alpha} = \vect{0}\\
    \label{eq:Splitting_ODE}
    & \ddroit{\Ucb}{t} = \Scb
  \end{alignat}
\end{subequations}
Equation \eqref{eq:Splitting_advec} is solved by applying the DGMPM discretizations \eqref{eq:DGMPM_discrete} or \eqref{eq:DGMPM_discrete_RK2}, while the solution of equation \eqref{eq:Splitting_ODE} is determined by some ODE solver.
If the discrete solution operators associated with equations \eqref{eq:Splitting_advec} and \eqref{eq:Splitting_ODE} for one time step are denoted by $H^{(\Delta t)}$ and $F^{(\Delta t)}$ respectively, the discrete solution reads \cite{Toro}:
\begin{equation}
  \label{eq:godunov_splitting}
  \Ucb^{n+1} = F^{(\Delta t)} H^{(\Delta t)} (\Ucb^n)
\end{equation}
The \textit{fractional-step method} \eqref{eq:godunov_splitting}, known as Godunov's splitting, is only first–order accurate in time when H and F are at least first–order accurate solution operators. On the other hand, Strang splitting:
  \begin{equation}
    \label{eq:Strang_splitting}
    \Ucb^{n+1} = F^{(\Delta t/2)} H^{(\Delta t)} F^{(\Delta t/2)} (\Ucb^n)
  \end{equation}
  is second-order accurate if each solution operator is at least second-order accurate \cite{Leveque}.

Two sub-problems are thus solved separately at each time step, the solution of the first one being used as initial conditions in the second. Thus, fractional-step methods \eqref{eq:godunov_splitting} and \eqref{eq:Strang_splitting} enable to take advantage of efficient tools already developed for homogeneous systems of conservation laws and of ODE solvers. However, it is worth noticing that even though the use of an implicit solver for equation \eqref{eq:Splitting_ODE} eliminates the influence of the relaxation time on the CFL condition, an L-stable solver as the Euler implicit method \cite{Book_ODE} is required to compute the correct solution.

\subsection{Intercell fluxes}
\label{subsec:interface_fluxes}
The DGMPM solution scheme reduces to the discrete solution operator $H^{(\Delta t)}$, or $H$ for simplicity, which in turn, aims at finding approximate similarity solutions since it only solves homogeneous systems (recall remark \ref{rq:similarity_solution} in section \ref{sec:characteristic_analysis}).
For a DGMPM space-time discretization made of $N_p$ material points and $N_T$ time increments, those solutions can be written:
% It has been established so far that the DGMPM scheme provides an approximate solution $\Qc(\vect{X}^i,t^{n,+1})= \Qc^{i,n+1}$,  with $i=1,...,N_n$ and $n=1,...,N_T$, that depends on the value of $\Qc$ at other points and previous time step, that is:
\begin{equation}
  \label{eq:general_scheme}
  \Qc^{p,n+1}= H \(\Qc^{j,n}\) \qquad p=1...,N_p \: ; \:j=1...,N_n \: ; \: n=0,...,N_T-1
\end{equation}
where the set of nodes $j$ having an influence on $\Qc^p$ defines the \textit{stencil} of the method.
\begin{definition}
  \label{def:monotonicity}
  A numerical scheme is said \textbf{monotone} if it satisfies:
  \begin{equation}
    \drond{H}{\Qc^j} \geq 0 \quad \forall j
  \end{equation}
\end{definition}
The following statement then holds:
\begin{theorem}[Godunov \cite{Godunov_method}]
  \label{th:Godunov}
  Monotone linear numerical schemes can be at most first-order accurate.
\end{theorem}

Intercell fluxes of the weak form propagate information across cells by taking into account the different values that fields can take on each side of the interface. DG methods for hyperbolic problems are based on the requirement of ensuring monotonicity of the scheme for piecewise constant approximations \cite{Cockburn}. %, namely, when the discrete system identifies with a first-order finite volume scheme.
Such a numerical method is monotone for flux functions $\Fcb_N$ that are Lipschitz continuous, consistent and monotone, namely, they must be \textit{E-fluxes} \cite{Osher}. One possibility, which is widely used and adopted here, is the \textit{Godunov flux function}.
%The computation of interface fluxes $\hat{\Fcb}^{i,n}$ in the DGMPM system of discrete equations \eqref{eq:DGMPM_discrete} is now developed. We can here benefit from the finite volume technologies which are essentially based on the way those fluxes are computed.
\subsubsection*{The Godunov flux}
The Godunov method \cite{Godunov_method} has been proposed in the context of finite difference schemes in which the piecewise constant approximation of the solution naturally allows the definition of a local Riemann problem at cell interfaces.
That is, two cells $i$ and $i+1$ with interface having normal vector $\vect{N}$, define a Riemann problem in the direction $X_N=\vect{X}\cdot{N}$ whose \textit{stationary solution} is used to compute the intercell numerical flux $\Fcb \cdot \vect{N}$. The stationary solution $\Ucb^*$ is the similarity solution along the vertical characteristic in the $(X_N,t)$ plane of the Riemann problem:
% Hence, the intercell numerical flux $\Fcb \cdot \vect{N}()$ is that of the stationary solution, that is, the similarity solution along the vertical straight line in the $(x,t)$ plane $\Ucb(x/t=0)$, of the Riemann problem:
\begin{equation}
  \label{eq:RP_mesh}
  \begin{aligned}
    &\drond{\Ucb}{t} + \drond{\Fcb_N}{X_N} = \vect{0}  \\
    & \Ucb(X_N,0)= \left\lbrace
      \begin{aligned}
        & \Ucb_{X_N^-} \text{ if } X_N <0 \\
        & \Ucb_{X_N^{+}} \text{ if } X_N > 0
      \end{aligned}
        \right.
  \end{aligned}
\end{equation}
where $\Ucb_{X_N^-}$ and $\Ucb_{X_N^{+}}$ are the states lying infinitely close to the interface in cells $i$ and $i+1$ respectively.
Godunov's method hence allows accounting for the complete wave structure of the solution within the numerical scheme.
Since this method is known to be based on E-fluxes, intercell fluxes involved in boundary integrals of the weak form of DG-methods can be computed as Godunov's ones.
To this end, Riemann problems are defined at cells faces by considering that initial data are piecewise constant even for high-order approximations. Note that this strategy is also followed in FVM when high-order reconstruction techniques of the fields are used.

Let us consider as an illustration of the computation of interface fluxes within the DGMPM, the two-dimensional case depicted in figure \ref{fig:2D_edge}.
One Riemann problem per interface is considered in order to avoid a dramatic increase in computational times.
% Since intercell fluxes must be calculated in addition to the solution discrete equations \eqref{eq:DGMPM_discrete}, only one Riemann problem per interface is considered rather than nodal ones, in order to avoid a dramatical increase in computational time.
Thus, by averaging nodal fields on each side of the interface, one obtains mean \textit{downwind} and \textit{upwind} states $\Ucb_{X_N^-}$ and $\Ucb_{X_N^+}$ that correspond to the initial conditions of the Riemann problem.
\begin{figure}[h!]
  \centering
  \input{chapter3/pgfFigures/2DEdge}
  \caption{Duplication of nodes at an interface and building of initial conditions of the Riemann problem (2D).}
  \label{fig:2D_edge}
\end{figure}
Furthermore, as mentioned in section \ref{sec:riemann_solvers}, the exact (time consuming) solution of the Riemann problem is not necessarily desired since only the stationary solution is kept. As a consequence, the stationary states are approximated here by means of an approximate-state Riemann solver, and the corresponding Godunov fluxes are determined.
\begin{remark}
  Hyperbolic systems having zero eigenvalues lead to stationary waves, propagating at zero celerity, across which the solution of the Riemann problem may have discontinuities. In that case, the associated Godunov flux is also discontinuous so that one stationary state must be considered on both sides of the characteristic. This approach leads to the computation of two fluxes, each contributing to one cell only.
\end{remark}

Recall that hyperelasticity and elastoplasticity conservative forms (equations \eqref{eq:vectors_hyperelasticity} and \eqref{eq:vectors_plasticity}) involve strains in the vector of conserved quantities, and stresses in the flux vector.
% At last, solid mechanics conservative forms involve for hyperelasticity and elastoplasticity (equations \eqref{eq:vectors_hyperelasticity} and \eqref{eq:vectors_plasticity}), strains in the vector of conserved quantities, and stresses in the flux vector.
Hence, the calculation of Godunov fluxes requires the integration of constitutive laws that can be time-consuming for non-linear problems. Nevertheless, the introduction of an auxiliary vector of conserved quantities $\Qcb$ and the Riemann problem it provides:
\begin{equation}
  \label{eq:RP_quasilinear}
  \begin{aligned}
    &\drond{\Qcb}{t} + \Jbsf \drond{\Qcb}{X_N} = \vect{0}  \\
    & \Qcb(X_N,0)= \left\lbrace
      \begin{aligned}
        & \Qcb_{X_N^-} \text{ if } X_N <0 \\
        & \Qcb_{X_N^{+}} \text{ if } X_N \geq 0
      \end{aligned}
        \right.
  \end{aligned}
\end{equation}
avoids the computation of constitutive equations. Indeed, the auxiliary vectors introduced in section \ref{sec:general-formulation} are rearrangements of the flux components, that is $\Fcb_N(\Ucb)=\Lbsf \Qcb$ with $\Lbsf$ some localization matrix. Then, the intercell flux is $\Fcb_N(\Ucb^*)=\Lbsf \Qcb^*$ where $\Qcb^*$  is the stationary solution of Riemann problem \eqref{eq:RP_quasilinear}. Note that this approach requires that stress as well as velocity and strain are projected back and forth between material points and nodes within the DGMPM.
\begin{remark}
  \label{rq:RK2_constitutive_update_nodes}
  Moreover, solving Riemann problems for the auxiliary vector $\Qcb^*$ does not avoid the integration of constitutive equations on the grid when a multi-stage time integrator as the RK2 is used. Indeed, every subsequent step $0 \leq k \leq 1$ of the time integrator requires an initial auxiliary vector $\Qcb^{n+k}$, which is computed from the vector of conserved quantities $\Ucb^{n+k}$.
\end{remark}
%Indeed, the stationary solution $\Qcb^*$ of \eqref{eq:RP_quasilinear} is a rearrangement of the flux components, that is $\Qcb^*=\matrice{\vect{v}^* \\ \tens{\sigma}^*}$ for the infinitesimal theory and $\Qcb^* = \matrice{\vect{v}^* \\ \tens{\Pi}^*}$ for finite deformations. Thus, stress as well as velocity and strain must be projected back and forth between material points and nodes within the DGMPM. $\Fcb_N(\Ucb^*)=\Lbsf \Qcb^*$

\subsubsection*{Transverse corrections}

The method derived above for the computation of normal fluxes can be viewed as the \textit{Donor-Cell Upwind (DCU)} method \cite{Leveque} in which only contributions from upwind cells sharing an edge (in two dimensions) with the current one are considered.
For multidimensional problems waves can travel in several directions such that contributions coming from corner cells must be taken into account in order to improve accuracy and stability of the numerical scheme.
The \textit{Corner Transport Upwind (CTU)} method \cite{Colella_CTU} consists in considering contributions propagating in bias and coming from upwind cells sharing only a node (in two dimensions) with another.
This approach allows improving the Courant condition especially for solid mechanics problems for which strain components are coupled through Poisson's effect.
At each cell interface, one defines left-going and right-going fluctuations as:
\begin{equation}
  \Acb^-(\Delta \Ucb) = \Fcb_N(\Ucb^*) - \Fcb_N(\Ucb_{X_N^-}) \qquad ;  \qquad \Acb^+(\Delta \Ucb) = \Fcb_N(\Ucb_{X_N^+})-\Fcb_N(\Ucb^*)
\end{equation}
\begin{figure}[h!]
  \centering
  \input{chapter3/pgfFigures/CTU}
  \caption{Normal and transverse fluctuations defined from edge $i$.}
  \label{fig:CTU}
\end{figure}
Let's consider the patch of grid cells shown in figure \ref{fig:CTU}, and focus on the edge denoted $(i)$ whose local normal vector is $\vect{N}^i$. The Riemann problem defined at this edge gives rise to normal fluctuations $\Acb^{i,-}(\Delta \Ucb)$ and $\Acb^{i,+}(\Delta \Ucb)$ contributing to cells L and R respectively.
These terms lead to the computation of transverse fluctuations giving contribution to neighboring cells across edges $(j)$ and $(k)$ for cell L, and across edges $(m)$ and $(l)$ for cell R.
Transverse fluctuations are computed by projecting normal fluctuations onto the characteristic basis associated with the Riemann problem \eqref{eq:RP_mesh} defined on the adjacent edge, hence the name transverse Riemann solver.
The spectral analysis of the corresponding Jacobian matrix, carried out in \cite{Kluth}, leads to right eigenvectors that are be written $\Rcb_{\Wc}^i$ hereinafter.
The negative normal fluctuation is, for instance, decomposed onto the characteristic basis associated with edge $(j)$ as:
\begin{equation}
\Acb^{i,-}(\Delta \Ucb) = \sum_{m=1}^{M} \beta_m \vect{\Rc}^{j,m}_\Wc
\end{equation}
where $\vect{\Rc}_\Wc^{j,m}$ is based on the normal vector $\vect{N}^j$ but also on different tangent moduli between grid cell L and its neighbor T.
Since only waves with positive characteristic speeds with respect to the orientation defined by the outward normal vector to the considered edge will contribute to the transverse fluctuation, only the positive operator $\Bcb^+$ is used:
\begin{equation}
\Bcb^{j,+} \Acb^{i,-}(\Delta \Ucb) = \sum_{\underset{c_m >0}{m=1}}^{M} c_m \beta_m \vect{\Rc}_\Wc^{j,m} \label{eq:transverse_fluctuations}
\end{equation}
with $c_m$, the celerity of wave $m$.
An additional numerical flux defined at edges is hence built from these transverse fluctuations:
\begin{equation}
\Fcb^{j,\text{tran}} = \frac{\Delta t}{2 \Delta X^j} \Bcb^{j,+} \Acb^{i,-}(\Delta \Ucb) \label{eq:transverse_corrections}
\end{equation}
which contributes to the flux between cell $L$ and $T$ ($\Delta X^j$ being the length of edge $(j)$).
In-going transverse corrections must be counted positively while outgoing ones must be subtracted from intercell fluxes. At last, numerical fluxes computed at interfaces can be then integrated over cells faces in order to complete the discrete system.%With duplicated nodes introduced by the DG approximation this contribution must be counted negatively for nodes belonging to cell L (outgoing fluctuation) and positively for nodes belonging to cell T (incoming fluctuation).

%With duplicated nodes introduced by the DG approximation this contribution must be counted negatively for nodes belonging to cell L (outgoing fluctuation) and positively for nodes belonging to cell T (incoming fluctuation).

\begin{remark}
  For linear elasticity, there is no need for an auxiliary vector of conserved quantities so that transverse contributions are computed with the same eigenbasis as that of the approximate-state Riemann solver.
\end{remark}

\subsubsection*{Boundary conditions enforcement}
As in MPM \cite{Love,BC_MPM}, boundary conditions (BC) are treated at nodes that compose the domain boundary.
By introducing \textit{ghost nodes} on boundary interfaces, one can widen the use of the approximate-state Riemann solver in order to enforce both Dirichlet and Neumann BC.
Similarly to finite volumes \cite{Leveque}, a state vector $\Qcb_G$ is ascribed to those nodes so that the stationary solutions of Riemann problems are consistent with boundary conditions.
Dirichlet BC enforcement for instance consists of (i) setting stress and free velocity components at ghost nodes equal to that of the associated interior node and (ii) solving the approximate Riemann problem for the velocity that must be enforced on ghost nodes knowing the stationary one that is equal to that of the BC.
The same procedure holds for Neumann boundary conditions by interchanging velocity and stress.
%This vector is determined for one ghost node by (i) setting the \textit{free quantities} (i.e: velocity (\textit{resp. stress}) components for Neumann (\textit{resp. Dirichlet}) BC) equal to that of the associated interior node and (ii) solving the inverse approximate Riemann problem for \textit{enforced quantities} on ghost nodes knowing the stationary one.

Notice that the use of the auxiliary vector of conserved quantities so that state vectors contain stress %regardless of the consistutive model considered
allows the easy enforcement of Neumann boundary conditions. %Hence, the use of the quasi-linear form \eqref{eq:RP_quasilinear} also enables the easy treatment of boundary conditions. However, this requires that the auxiliary vector is projected onto the grid with the vector of conserved quantities.
\begin{figure}[ht]
  \centering
  \input{chapter3/pgfFigures/cornerNodes}
  \caption{Corner ghost nodes in a two-dimensional DGMPM mesh.}
  \label{fig:corner_ghost}
\end{figure}
Transverse corrections further require the introduction of corner ghost nodes equivalent to finite volume corner cells \cite{Leveque}.
Consider the two-dimensional case depicted in figure \ref{fig:corner_ghost} in which boundary edges are represented by red lines.
%First, boundary interfaces are extended to the corner cell (vertical and horizontal dashed lines).
First, an inverse Riemann problem is solved between the corner ghost node $G_c$ and one regular ghost node, say $G_B$, so that the stationary solution of the Riemann problem between those ghost nodes corresponds to the boundary condition holding on the vertical edge.
Second, this procedure is repeated between the corner ghost $G_c$ node and $G_L$ in order to enforce the boundary condition holding on the bottom boundary between them.

\begin{remark}
  \label{rq:BC_ghostnode}
  The solution of the Riemann problem involving ghost nodes is made possible by extending material properties of the adjacent cell so that the characteristic structure of the solution can be computed.
  This implies that the deformation of interior nodes is duplicated at ghost nodes for problems such as hyperelasticity, whose eigenstructure depends on the deformation gradient.
  Hence, for such problems ghost nodes may carry stress and strain that are not related by constitutive equations.
  However, the deformation gradient at ghost nodes has no physical sense and is only used to compute the correct wave speeds.
\end{remark}

\subsection{DGMPM solution scheme}
Let us assume that the vector of specific conserved quantities $\bar{\Ucb}^n$ as well as the auxiliary vector $\Qcb^n$ are known at every material point that discretizes a continuum body $\Omega_0$, in a grid made of $N_{n}$ nodes at a given time $t^n$.
The computational procedure followed within the DGMPM between two time steps $n$ and $n+1$ can now be summarized.
We consider cases based on the use of an auxiliary vector, the others being only particular cases.
% , which requires to (i) solve the hyperbolic system \eqref{} in $\Omega$ with associated boundary conditions and (ii) update the vector $\bar{\Qcb}^n$ on material points.
%The scheme has been established with a total Lagrangian formulation, therefore, the lumped mass and \textit{pseudo-stiffness} matrices $M^L_{ij}$ and $K^\alpha_{ij}$ of the semi-discrete form \eqref{eq:DGMPM_semi_discrete} are computed once and for all at the beginning of the calculation.
The procedure then reads:
\begin{itemize}
  % \item[] $\bar{\Ucb}, \:\Qcb$ known at material points
\item[(a)] The lumped mass and \textit{pseudo-stiffness} matrices $M^L_{ij}$ and $K^\alpha_{ij}$ of the semi-discrete form \eqref{eq:DGMPM_semi_discrete} are computed.
\item[(b)] The discrete equation \eqref{eq:DGMPM_discrete} and Riemann problems at cell interfaces \eqref{eq:RP_quasilinear} require a projection of fields onto the grid:
  \begin{equation}
    \label{eq:DGMPM_points2nodes}
    M^L_i \bar{\Ucb}^i = \sum_{p=1}^{N_p} S_{ip} m_p \bar{\Ucb}^p \qquad \text{and} \qquad M^L_i \vect{\Qc}^i = \sum_{p=1}^{N_p} S_{ip} m_p \vect{\Qc}^p
  \end{equation}
  to be solved for each $\bar{\Ucb}^i$ and $\vect{\Qc}^i$ respectively. The projection of fields from particles to nodes hence follows the weighted least squares interpolation used in MPM.
%\item[(b)] Compute time step: in the general case the tangent modulus depends on the deformation gradient as well as the waves speeds. It is therefore needed to compute the time step so that the fastest wave can at most cross the smallest cell of the mesh according to Courant condition.
\item[(c)] The specific flux vectors $\bar{\Fcb}^i_{\alpha}$ involved in the equations are computed from $\vect{\Qc}^i$ knowing $\rho_0$, thus avoiding the computation of constitutive equations.
\item[(d)] Enforce the boundary conditions on ghost nodes.
\item[(e)] Computation of interface fluxes:
  \begin{itemize}
  \item[1-] Build the state vectors $\vect{\Qc}_{X_N^{\pm}}$ based on $\vect{\Qc}^{i,n}$ where $i$ denotes the nodes belonging to the face on both sides of an interface.
  \item[2-] Compute the stationary solution $\Qcb^*$ by means of the approximate-state Riemann solver.
  \item[3-] Calculate the corresponding Godunov flux $\Fcb_N \(\Qcb^*\)$ by either using the DCU or the CTU approach.
  \end{itemize}
\item[(f)] Advance the solution in time by solving the discrete equation \eqref{eq:DGMPM_discrete} at each node.
\item[(g)] Back-mapping: as motivated at the end of section \ref{sec:MPM}, the nodal updated solution is projected to material points with the classical interpolation as in PIC:
  \begin{equation}
    \bar{\Ucb}^{p,n+1} = \sum_{i=1}^{N} S_{ip}\bar{\Ucb}^{i,n+1}
  \end{equation}
\item[(h)] Material point kinematics and constitutive model: The new solution $\bar{\Ucb}^{p,n+1}$ allows incrementing the deformation $\vect{\varphi}(\vect{X},t)$ and updating stress components, which will be used in the auxiliary vector for the next time step, through hyperelastic constitutive equations:
  \begin{align}
    & \vect{\varphi}^{p,n+1} = \vect{X}^p + \Delta t \vect{v}^{p,n+1} \\
    & \tens{\Pi}^{p,n+1} =  \drond{\Psi}{\tens{F}}(\tens{F}^{p,n+1})
  \end{align}
  The grid may then be discarded and reconstructed and in particular by means of adaptive algorithms applied in the reference configuration, in order to improve wave front tracking in the current one.
  The reconstruction of the grid is responsible for the presence of step (a).
\end{itemize}

Let's now recall or highlight significant differences between the DGMPM and the original MPM schemes.
% Ouai ?
%First, while the DGMPM uses a classical interpolation to mapping back the updated solution to material points (step f), FLIP method and MPM require an additional time integration on material points.
% ok
First, the use of conservation laws \eqref{eq:conservative_form} instead of the momentum equation in the weak form implies that both velocity and gradients are solved at nodes making this new approach close to finite volume methods, which provides the same order of accuracy for both fields.
% ok
Next, since the deformation gradient is no longer calculated with shape function gradients, the task of choosing between USF and USL algorithms vanishes. In that sense the DGMPM scheme is simpler.
At last, the solution of Riemann problems at every edge of the mesh increases computational time. Fortunately, the use of discontinuous Galerkin approximation makes this numerical method highly parallelizable \cite{Cockburn}.

The numerical scheme derived above is analyzed in terms of stability and convergence in the next section.

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\label{ch:theory}
\chapter{The Standard Model of Particle Physics}
\label{SM chapter}
\section{Introduction}
By determining the dynamics of the most elementary degrees of freedom, particle physics hopes to uncover the fundamental laws of the universe. The definition of elementary has evolved with time and currently refers to quantum fields, which are operators on spacetime used to compute probability distributions. The quantized excitations of these fields gives rise to particles such as the electron. The Standard Model of Particle Physics (SM) describes the interactions between particles due to three of the four fundamental forces: weak, strong, and electromagnetic. Gravity is not included in the SM and still under investigation.
\section{Quantum Field Theory}
In the SM, forces (and particles) are represented as fields. In this context, fields are mathematical objects that define a tensor (e.g. scalar, vector, etc)  at every point on a manifold, here the manifold is space-time. These fields obey laws dictated by Quantum Field Theory (QFT). Particles arise naturally in QFT as quantizied field excitations localized in spacetime.

According to Noether's theorem, symmetries of a field give rise to conserved quantities (e.g. time-translation invariance leads to energy conservation).  Often in the history of physics, a conserved quantity of a field is found and then the underlying symmetry of the field is inferred. Gauge symmetries are symmetries among the internal degrees of freedom of the field (components of the tensor), which give rise to conserved quantities associated with fields. By specifying the symmetries of a system the dynamics and conserved quantities of the system may be succinctly defined.
\section{$U(1)_{EM}$ Local Gauge Invariance}
The Lagrangian of Quantum Electrodynamics (QED) describes the electromagnetic force. QED may be derived by requiring local $U(1)_{EM}$ gauge invariance of the free Dirac fermion Lagrangian, $\psi$:
\begin{equation}
\mathcal{L} = i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi - m\bar{\psi}{\psi}
\end{equation}
This symmetry may be represented as a complex number with unit modulus, $e^{i\theta}$. U(1) gauge invariance requries this gauge transformation of $\psi$ will leave the Lagrangian unchanged:

\begin{equation}
\psi(x) \rightarrow \psi'(x) = e^{i\theta(x)}\psi(x)
\end{equation}

NB: This transformation is a local gauge transformation as $\theta$ depends on the spacetime coordinate.

This transformation does not impact the mass term, but the kinetic term is modified due to $\theta(x)$.
\begin{equation}
\mathcal{L} \rightarrow \mathcal{L}' = i\bar{\psi} e^{-i\theta(x)}\gamma^{\mu}\partial_{\mu}\psi e^{i\theta(x)} - m \bar{\psi}e^{-i\theta(x)}\psi e^{i\theta(x)}
\end{equation}
\begin{equation}
= i\psi \gamma^{\mu}(\partial_{ \mu }\psi + i \psi \partial_{ \mu } \theta)- m \bar{ \psi } \psi
\end{equation}

The $\partial_{\mu}\theta$ terms breaks the gauge invariance of the Lagrangian. By introducing a new field, $A_{\mu}$, the gauge invariance of the derivative can be recovered. Now redefining the derivative as the covariant derivative:

\begin{equation}
D_{\mu}\psi \equiv (\partial_{\mu} - iqA_{\mu})\psi
\end{equation}

And letting $A_{\mu}$ transform under U(1) as:
\begin{equation}
A_{\mu} \rightarrow A_{\mu} +  \delta A_{\mu}
\end{equation}

The transformed covariant derivative becomes:

\begin{equation}
D_{\mu}\psi \rightarrow D_{\mu}\psi' =  (\partial_{\mu} - iq A_{\mu})\psi'
\end{equation}
\begin{equation}
=  (\partial_{\mu} - iq(A_{\mu} +  \delta A_{\mu} ))\psi e^{i\theta}
\end{equation}
\begin{equation}
=  e^{i\theta}D_{\mu}+ie^{i\theta}\psi(\partial_{\mu}\theta - q\delta A_{\mu}))
 \end{equation}

The covariant derivative can be made gauge invariant by setting the last term to zero.

\begin{equation}
\delta A_{\mu} = \frac{1}{q}\partial_{\mu}\theta
\end{equation}

So now $A_{\mu}$ transforms as:

\begin{equation}
A_{\mu} \rightarrow A_{\mu} + \frac{1}{q}\partial_{\mu}\theta
\end{equation}

Finally, replacing the derivative with the covariant derivative the Dirac Lagrangian we have:

\begin{equation}
\mathcal{L} = i \bar{\psi}\gamma^{\mu}D_{\mu}\psi - m\bar{\psi}\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}
\end{equation}
\begin{equation}
=\mathcal{L}_{QED}
\end{equation}

Here $F_{\mu\nu}\equiv \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$. This last term in the Lagrangian is the kinetic energy of the gauge boson field.

So we have derived the QED Lagrangian! By requiring the free Dirac Lagrangian to be invariant under local U(1) transformations we have generated a new gauge boson field, $A_{\mu}$, which describes the photon. As expected, the photon interacts with fermions.

Stepping back, a global U(1) gauge symmetry of the free Dirac Lagrangian implies we cannot measure the absolute phase of a charged particle. A local U(1) gauge symmetry changes the phase of fields differently across space time. For this type of transformation to leave the Lagrangian invariant, we had to introduce an additional field, $A_{\mu}$, which "communicates" these phase changes across space-time. In less formal language this effectively means: if the field at one location changes, this change is conferred to other particles via $A_{\mu}$.

\section{Yang-Mills Gauge Theories}
Requiring $U(1)_{EM}$ gauge invariance of the free Dirac Lagrangian gave us QED. Requiring different gauge symmetries we can derive the structure of other interactions. Any gauge symmetry may be written as:

\begin{equation}
\psi_{i} \rightarrow \exp(i\theta^{a}T^{a}_{ij})\psi_{j}
\end{equation}

Here $\theta$ is a dimensionless real parameter and T is the generator of the gauge symmetry group. With this, the covariant derivative can be written as:

\begin{equation}
D_{\mu}\psi_{i} \equiv \partial_{\mu}\psi_{i} + igA^{a}_{\mu}T^{a}_{ij}\psi_{j}
\end{equation}

Then the gauge field must transform as:

\begin{equation}
A^{a}_{\mu} \rightarrow A^{a}_{\mu} - \frac{1}{g}\partial_{\mu}\theta^{a} - f^{abc} \theta^{b} A_{\mu}^{c}
\end{equation}

Here $f$ is the structure constant of the gauge group. The field strength tensor is given by:

\begin{equation}
F^{a}_{\mu\nu} \equiv \partial_{\mu}A^{a}_{\nu} - \partial_{\nu}A^{a}_{\mu} - gf^{abc}A^{b}_{\mu}A^{c}_{\nu}
\end{equation}
\begin{equation}
F^{a}_{\mu\nu} \rightarrow F^{a}_{\mu\nu} - f^{abc} \theta^{b}F^{c}_{\mu\nu}
\end{equation}

This gives the Yang-Mills Lagrangian:
\begin{equation}
\mathcal{L}_{YM}= -\frac{1}{4}F^{a\mu\nu}F^{a}_{\mu\nu}+i\bar{\psi_{i}}\gamma^{\mu}D_{\mu}\psi_{i}+m\bar{\psi_{i}\psi_{i}}
\end{equation}
\section{Particles in the Standard Model}
The SM is a $SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}$ Yang-Mills theory, that explains the quantum behavior of electromagnetic, weak, and strong force. It consists of fermions (half-interger spin matter constituents) and bosons (integer spin force meditators). Fermions are spinor representations of the Poincare group and can be further separated into leptons and quarks. Bosons are the result of requiring a particular symmetry among the spinor fields:

\begin{equation}
SU(3)_{C} \times SU(2)_{L} \times U(1)_{Y}
\end{equation}

$SU(3)_{C}$ is the symmetry group of the strong force and generates eight gluon fields, $G_{\mu}$. $SU(2)_{L}$ is the symmetry group of the electroweak force and generates three electroweak boson fields. The mixing of this $SU(2)_{L}$ and $U(1)_{Y}$ gives rise to the photon field, where Y is the weak-hypercharge:

\begin{equation}
Y= 2(Q-T_{3})
\end{equation}

Q is the electromagnetic charge, and $ T_{3}$ is the z-component of the weak isospin. Weak isospin is the charge associated with the $SU(2)_{L}$ symmetry. The corresponding covariant derivative is then:

\begin{equation}
D_{\mu}\phi \equiv (\partial_{\mu} + ig_{1}B_{\mu}Y_{L/R} + [ig_{2}W^{\alpha}_{\mu}T^{\alpha}]_{L} + [ig_{3}G^{\alpha}_{\mu}\tau^{\alpha}]_{C})\psi
\end{equation}

For a given fermion to interact with a given gauge field it must have a non-zero corresponding Noether charge for that gauge symmetry. If the corresponding Noether charge is zero, that fermion transforms as a singlet and does not participate in that gauge interaction.

Fermions are divided into quarks and leptons based on their transformations under $SU(3)_{C}$. Quarks transform as color triplets. Leptons transform as color singlets and consequently do not interact with gluons.
Fermions may be further classified by their $SU(2)_{L}$ interactions. Only the left-chiral part of fermions (denoted by L here) transform as $SU(2)_{L}$ doublets, the right-chiral part forms singlets under this gauge. Lastly, all these groups of particles come in three generations, each a heavier copy of the previous, but with differing flavor quantum numbers. This is summarized in Table ~\ref{tbl:SMparticles} and shown in Figures ~\ref{fig:sm_particles} and \ref{fig:sm_particle_interactions}.

\begin{table}[!hbt]
  \resizebox{\textwidth}{!}{\begin{tabular}{|l|c|c|c|c|}
    \hline
    SM Fermion Gauge Group & First Generation & Second Generation & Third Generation & $(SU(3)_{C}, SU(2)_{L},U(1)_{Y})$ Representations \\
    \hline
    \hline
    Left-handed quarks & $\begin{pmatrix} u_{L}^{r} & u_{L}^{g} & u_{L}^{b} \\d_{L}^{r} & d_{L}^{g} & d_{L}^{b} \end{pmatrix}$ & $\begin{pmatrix} c_{L}^{r} & c_{L}^{g} & c_{L}^{b} \\s_{L}^{r} & s_{L}^{g} & s_{L}^{b} \end{pmatrix}$ & $\begin{pmatrix} t_{L}^{r} & t_{L}^{g} & t_{L}^{b} \\b_{L}^{r} & b_{L}^{g} & b_{L}^{b} \end{pmatrix}$ & $(3,2,\frac{1}{6})$ \\
    \hline

    Right-handed quarks & $( u_{R}^{r}, u_{R}^{g}, u_{R}^{b})$ & $(c_{R}^{r}, c_{R}^{g}, c_{R}^{b})$ & $(t_{R}^{r}, t_{R}^{g}, t_{R}^{b})$ & $(3,1,\frac{2}{3})$ \\
    & $( d_{R}^{r}, d_{R}^{g}, d_{R}^{b})$ & $(s_{R}^{r}, s_{R}^{g}, s_{R}^{b})$ & $(b_{R}^{r}, b_{R}^{g}, b_{R}^{b})$ & $(3,1,-\frac{1}{3})$ \\
    \hline
    Left-handed leptons & $\begin{pmatrix} \nu_{e}^{L}  \\ e_{L} \end{pmatrix}$ &$\begin{pmatrix} \mu_{e}^{L}  \\ \mu_{L} \end{pmatrix}$ & $\begin{pmatrix} \tau_{e}^{L}  \\ \tau_{L} \end{pmatrix}$ & $(1,2,-\frac{1}{2})$ \\
    \hline
    Right-handed leptons & $e_{R}$ & $\mu_{R}$ & $\tau_{R}$ & $(1,1,-1)$ \\
    \hline
\end{tabular}}
  \caption{Representations of the SM fermions under $SU(3)_{C}\times SU(2)_{L} \times U(1)_{Y}$ gauge symmetry group. Rows are correspond to different weak isospin states and columns to different QCD color states.}
  \label{tbl:SMparticles}
\end{table}

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/theory/particles.png}
  \caption{The particles of the Standard Model.}
  \label{fig:sm_particles}
\end{figure}
\FloatBarrier

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/theory/particle_interactions.pdf}
  \caption{Summary of SM particle interactions.}
  \label{fig:sm_particle_interactions}
\end{figure}
\FloatBarrier

Now we can understand the SM Lagrangian density below as a $SU(3)_{C}\times SU(2)_{L} \times U(1)_{Y}$ Yang-Mills theory with an additional $SU(2)$ complex scalar Higgs field doublet that will be discussed later.

\begin{flalign}\nonumber
\mathcal{L} _{SM} &= \underbrace{-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}W^{a}_{\mu\nu}W^{a\mu\nu}-\frac{1}{4}G^{\alpha}_{\mu\nu}G^{\alpha\mu\nu}}_\text{Kinetic Energies and Self-Interactions of Gauge Bosons}&&\\\nonumber
&+\underbrace{\bar{L_{i}}\gamma^{\mu}(i\partial_{\mu}-\frac{1}{2}g_{1}Y_{iL}B_{\mu}-\frac{1}{2}g_{2}\sigma^{a}W_{\mu}^{a})L_{i}}_\text{Kinetic Energies and EW Interactions of Left-handed Fermions}&&\\\nonumber
&+\underbrace{\bar{R_{i}}\gamma^{\mu}(i\partial_{\mu}-\frac{1}{2}g_{1}Y_{iR}B_{\mu})R_{i}}_\text{Kinetic Energies and EW Interactions of Right-Handed Fermions}&&\\\nonumber
&+\underbrace{\frac{ig_{3}}{2}\bar{Q_{j}}\gamma^{\mu}\lambda^{\alpha}G^{\alpha}_{\mu}Q_{j}}_\text{Strong Interactions between Quarks and Gluons}&&\\\nonumber
&+\underbrace{\frac{1}{2}|(i\partial_{\mu}-\frac{1}{2}g_{1}B_{\mu}-\frac{1}{2}g_{2}\sigma^{a}W_{\mu}^{a})\Phi|^{2} - V(\Phi)}_\text{Electroweak Boson Masses and Higgs Couplings}&&\\\nonumber
&-(\underbrace{y^{d}_{kl}\bar{L_{k}}\Phi R_{l}+y^{u}_{kl}\bar{R_{k}}\widetilde{\Phi}L_{l}+h.c.)}_\text{Fermion Mass terms and Higgs Couplings}&&\\\nonumber
\end{flalign}

Here several abstract spaces are being spanned:
\begin{itemize}
\item[--]$a$ spans the three $SU(2)_{L}$ gauge fields with generators expanded in Pauli matrics: $T^{\alpha}=\frac{1}{2}\sigma^{a}$
\item[--]$\alpha$ spans the eight $SU(3)_{C}$ gauge fields, with generators expanded in Gell-Mann matrices: $\tau^{\alpha}=\frac{1}{2}\lambda^{\alpha}$
\item[--]L/R represent left and right projections of Dirac fermion fields. The Strong interaction is not chiral, so Q = L+R
\item[--]$\mu$ and $\nu$ are four-vector indices
\item[--]$i$,$j$,$k$ are summed over the three generations of SM particles
\end{itemize}

\section{Higgs Mechanism}
The SM Lagrangian without the addition of a Higgs field does not allow for gauge boson and fermion mass terms, $\frac{1}{2}m_{A}^{2}A_{\mu}A_{\mu}$ and $m(\bar{\psi}\psi)$,  as these terms are not gauge invariant. By introducing the Higgs field, mass terms for these particles may be included in a gauge invariant way. This field is a complex doublet with a potential $V(\Phi)$:

\begin{equation}
\Psi = \begin{pmatrix} \Phi^{\dagger} \\ \Phi^{0} \end{pmatrix}
\end{equation}
\begin{equation}
V(\Phi)=\mu^{2}\Phi^{\dagger}\Phi + \lambda |\Phi^{\dagger}\Phi|^{2}
\end{equation}

The minima of this field occurs for $|\Phi|= \sqrt{\frac{\mu^{2}}{2\lambda}} \equiv \frac{v}{2}$. This yields degenerate minima, this symmetry is broken by choosing a specific minima (known as spontaneous symmetry breaking). By convention  $\Phi_{min} = \frac{1}{\sqrt{2}}\begin{pmatrix}0 \\ v \end{pmatrix}$ is chosen. This means the ground state of the Higgs field (Higgs vacuum) is non-zero, $\sqrt{\frac{-\mu^{2}}{\lambda}}$. The Higgs Field may now be expanded around this new ground state as:

\begin{equation}
\Phi(x)=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v+h(x)\end{pmatrix}
\end{equation}

This non-zero Higgs vacuum generates mass terms for the gauge bosons:

\begin{equation}
|(-\frac{1}{2}g_{1}B_{\mu}-\frac{1}{2}g_{2}\sigma^{a}W_{\mu}^{a})\Phi|^{2}=\frac{1}{2}m_{W}^{2}W_{\mu}^{+}W^{-\mu}+\frac{1}{2}m_{Z}^{2}Z_{\mu}Z^{\mu}
\end{equation}

where:
\begin{equation}
W^{\pm}_{\mu} \equiv \frac{1}{\sqrt{2}}(W^{1}_{\mu} \mp iW^{2}_{\mu})
\end{equation}
\begin{equation}
Z_{\mu} \equiv \frac{1}{\sqrt{g_{1}^{2}+g_{2}^{2}}}(g_{2}W^{2}_{\mu}-g_{1}B_{\mu})
\end{equation}
\begin{equation}
m_{W} = \frac{vg_{2}}{\sqrt{2}}
\end{equation}
\begin{equation}
m_{Z} = \frac{v}{\sqrt{2}}\sqrt{g_{1}^{2} + g_{2}^{2}}
\end{equation}

The Higgs field also generates a mass terms for fermions and the Higgs boson.
\section{Electroweak Theory}
$SU(2)_{L}$ generates $W^{\pm}, W^{0}$ gauge bosons, which would be massless if $SU(2)_{L}$ was a perfect symmetry. These bosons are massive as this symmetry is broken.

Above the electroweak scale (246~GeV) the electromagnetic and weak forces are unified. This can be seen from the mass eigenstates, $Z$ and $A_{\mu}$ given by:
\begin{equation}
\begin{pmatrix} Z_{\mu} \\ A_{\mu} \end{pmatrix} = \begin{pmatrix} \cos \theta_{W} & -\sin \theta_{W} \\ \sin \theta_{W} & \cos \theta_{W} \end{pmatrix} \begin{pmatrix} W^{3}_{\mu} \\ B_{\mu} \end{pmatrix}
\end{equation}

Here $\theta_{W}$ is the Weinberg angle given by:
\begin{equation}
\cos\theta_{W}=\frac{g_{2}}{\sqrt{g_{1}^{2}+g_{2}^{2}}} = \frac{m_{W}}{m_{Z}}
\end{equation}
\section{Quantum Chromodynamics}
As mentioned earlier the strong force, which binds the proton together, is mediated by gluons. Quantum Chromodynamics is the QFT which describes the interactions of quarks and gluons via $SU(3)_C$ gauge interactions. QCD contains features not present in electroweak Interactions due to $SU(3)_C$ generators not commuting (i.e. $SU(3)_C$ is a non-abelian group) and the number of quark flavors ($n_{f}$). For example, in QCD there is color confinement and asymptotic freedom due to the structure constants being non-zero (to be discussed more later). Requiring $SU(3)_C$ local gauge invariance implies:

\begin{equation}
\psi(x) \rightarrow \psi(x)^{'} = \exp[ig_{S}\alpha(x)\cdot\hat{T}]\psi(x)
\end{equation}

where $\alpha(x)$ is the local phase function, $g_{S}$ is the strong coupling constant, and $\hat{T}$ are the eight generators of $SU(3)$ (note $\hat{T^{a}}=\frac{1}{2}\lambda^{a} a$, where $\lambda^{a}$ are the Gell-Mann matrices). As the Gell-Mann matrices are 3x3, this means $\psi$ has three degrees of freedom under these $SU(3)$ rotations. So, we represent $\psi$ as:

\begin{equation}
\psi = \begin{pmatrix} \psi_{red} \\ \psi_{green} \\ \psi_{blue}\end{pmatrix}
\end{equation}

Physicists describe these components of $\psi$, as color components (red, green, and blue). A particle's corresponding antiparticle has the corresponding anticolor. This color is the QCD charge which conserved under $SU(3)$ rotations. Combining colors, color neutral states (e.g. red and antired, or red, green and blue) may be created.
For the free Dirac Lagrangian to remain invariant under $SU(3)$ transformations, we must again postulate a boson field that modifies the derivative. The gluon field tensor is given by ($\alpha=1,...,8)$:

\begin{equation}
G_{\mu\nu}^{k}  = \partial^{\mu}G^{\nu}_{\alpha}-\partial^{\nu}G^{\mu}_{\alpha}-g_{S}f^{\alpha\beta\gamma}G^{\mu}_{\beta}G^{\nu}_{\gamma}
\end{equation}

Here $f^{\alpha\beta\gamma}$ are the structure constants of $SU(3)$. Combining all this gives the QCD Lagrangian:

\begin{equation}
\mathcal{L}_{QCD} = \bar{\psi_{qi}}i\gamma^{\mu} (D_{\mu})_{ij}\psi^{qj} - m\bar{\psi^{qi}}\psi_{qi} - \frac{1}{4}G^{\alpha}_{\mu\nu}G^{\alpha\mu\nu}
\end{equation}

Here $i$ are the color indicies, and $q$ are the quark flavors. It is important to note that quarks transform under the fundamental representation of $SU(3)$, while gluons transform under the adjoint representation. This means quarks carry a single color charge (red, green, blue, antired, antigreen, antiblue) and gluons carry a color and anticolor charge.

Figure \ref{fig:QCDinteractions} shows the three dominant QCD interactions. This figure shows that gluons interact with each other since they have color charge. This does not occur in QED, as photons do not have electric charge and therefore do not interact with each other. In QED, a bare electron's effective charge is largest closest to the electron and decreases as a function of distance. This is because the QED vacuum fills with particle antiparticle pairs spontaneously, which screen the charge of the bare electron. The larger the distance from the electron, the smaller the effective charge and therefore the weaker the force.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Theory/QCD_vertices.png}
  \caption{This figure shows the three dominant QCD interactions \cite{pich}}f.
  \label{fig:QCDinteractions}
\end{figure}
\FloatBarrier

As the distance from a quark increases it's effective color charge increases due to the vacuum polarization in QCD. Color charge grows as the distance from the source increases (i.e. color is anti-screened in QCD).  In this way, strong interactions become stronger at large distances (low momenta interactions).  At small distances (large momenta interactions) strong interactions are significantly weaker and considered nearly free. This effect of referred to as asymptotic freedom. At large distances, a quark's effective charge is large and the strong force is more significant. This force becomes so strong that quarks form colorless bound states instead of remaining free particles. This effect is known as color confinement. This scaling of QCD field strength (and the other SM fields) is shown in Figure ~\ref{fig:sm_couplings}.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Theory/runningcouplings.png}
  \caption{Strength of the $U(1)$, $SU(2)$, and $SU(3)$ gauge couplings as a function of the energy scale of the interaction (Q). From Ref. \cite{runningcouplings}}.
  \label{fig:sm_couplings}
\end{figure}
\FloatBarrier

Commonly the change in a particle's effective charge under a given force is quantified with  $\beta(r) \equiv \frac{-de(r)}{d\ln r}$, where e(r) is the effective charge of a given particle under a force. In QED this function is positive but in QCD this function is negative leading to confinement and asymptotic freedom. Moreover, one can calculate how the coupling ($\alpha$) of a force varies with energies. More deeply this amounts to incorporating renormalization and vacuum polarization in the boson propagators. For QCD this is:

\begin{equation}
\alpha_{S}(Q^{2}) = \frac{ \alpha_{s}(\mu^{2}) } { 1+ \frac{ \alpha_{s}( \mu^{2} ) } { 12\pi } ( 33-2n_{f} ) ln( Q^{2}/\mu^{2})}
\end{equation}

where $Q$ is the momentum of the the force is probed at, $\mu^{2}$ is the renormalization scale, $n_{f}$ is the number of quark flavors. There are six quark flavors in SM QCD, making $33-2n_{f} > 0$. Since this factor is positive and the $ln(Q^{2}/\mu^{2})$ is in the denominator, as $Q^{2}$ increases $\alpha_{s}$ decreases. Meaning, for large $Q^{2}$, $\alpha_{s}$ is small and SM QCD is asymptotically free, while for small $Q^{2}$, $\alpha_{s}$ is large and SM QCD is confined, as mentioned earlier.

Particle experiments to date have not observed isolated quarks or gluons due to color confinement. Instead the bound colorless states have been observed. Hadronization is the process by which quarks and gluons form these colorless bound states, called hadrons. The process of hadronization is still an active area of research. One qualitative description is shown in Figure \ref{fig:colorstring}. In this figure, as two quarks separate the color field between them is restricted to a tube with energy density of ~1 GeV/fm. As they separate further, the energy in the color field increases, until there is enough energy to produce $q\bar{q}$ pairs, which breaks the color field. This process repeats until quarks and antiquarks have low enough energy to form colorless hadrons. The resulting spray of hadrons is called a jet.

Since quarks and gluons carry different color charges, their respective jets have different properties. As quarks carry only a single color charge (vs. gluons which have color and anticolor charge), their jets have less constituent particles. More precisely, the Altarelli-Parisi splitting functions ~\cite{altarelli} contain a factor $C_{A}$ for gluon radiation off a gluon, and $C_{F}$ for gluon radiation off a quark ($C_{A}/C_{F} = 9/4$). These color factors are the prefactors in the Feynman diagrams for these processes ~\cite{colorfactor}, which leads to gluon jets having more constituents (and more charged particle tracks in detectors) than quark jets. Gluon jets also tend to have a larger radius with lower momentum constituents than quarks. There are many novel techniques to distinguish quarks from gluons. For this study the number of charged particles will be the focus.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Theory/color_string.pdf}
  \caption{A cartoon of string breaking: the QCD string spanned between quark Q and antiquark $\bar{Q}$ breaks due to $q\bar{q}$ creation \cite{colorstring}.}.
  \label{fig:colorstring}
\end{figure}
\FloatBarrier

\chapter{Standard Model Successes and Limitations}
The Standard Model has accurately described numerous underlying principles of nature. It has correctly predicted cross sections for strong and electroweak processes spanning over ten orders of magnitude, as shown in Figure \ref{fig:SM cross sections}, and contains no known logical inconsistencies. Despite the strength and reality of the Standard Model, it fails to describe some important aspects of reality and suffers from aesthetic issues.
To date, dark matter and dark energy comprise $\sim 95\%$ of the universe, but the SM offers no explanation of their nature. Additionally, neutrinos are known to have mass, but the SM contains no mass generation mechanism for left-handed neutrinos without right-handed neutrinos (which do not exist). There are other mechanisms for introducing massive neutrinos in the SM, but these mechanisms create hierarchy problems.

Possibly the most significant aesthetic issue is the hierarchy between the electroweak and Planck scales. The electroweak scale is the scale of electroweak symmetry breaking. The Planck scale is the scale where the gravitational force is comparable in strength to the other forces. The SM breaks down at the Planck scale, as there is not an experimentally verified theory of quantum gravity, and at this scale gravity cannot be ignored (like it can at the electroweak scale). These scales differ by $\sim30$ orders of magnitude. Understanding the difference in these energy scales may help explain the weakness of gravity at electroweak scales, and possibly a QFT for gravity. (NB: This hierarchy can also be framed in terms of the corrections to the Higgs mass, which depend on the UV cutoff scale - where the SM breaks (i.e. the Planck scale). This leads the quantum corrections to the Higgs mass that would force the Higgs mass to $\sim 10^{18}$TeV.)

These stark contrasts in scales may indicate that a more fundamental theory exists. It is hoped that such a theory would explain and motivate some of the ad-hoc features of the SM. In particular, the SM does not offer an explanation for the values of the 19 SM parameters (6 quark masses, 3 charged lepton masses, 3 gauge couplings, Higgs parameters ($\mu^{2}$, $\lambda$)) and the structure of the fermion representations.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Theory/SM_xs.pdf}
  \caption{A comparison of cross section measurements at $\sqrt{s}=7,8,13$ TeV from ATLAS compared to theoretical measurements. From Ref. \cite{SM_XS_Comparison}}.
  \label{fig:SM cross sections}
\end{figure}
\FloatBarrier

\chapter{New Physics Models with Diboson Resonances}
\label{BSM chapter}

\section{Randall Sundrum Bulk Model}
%Top down approach (solve gravity problem)
The  Planck electroweak hierarchy can be explained in extra dimensional models, such as the 5D Randall Sundrum (RS) Bulk Model (\cite{rs2},\cite{rs1}). The RS bulk model posits one extra warped spatial dimension, $y$, extending a distance $L$. The metric in this model is:
\begin{equation}
ds^{2}=e^{-2k|y|}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}
\end{equation}
where $e^{-k|y|}$ is the warp factor of the extra dimension, which is compactified on a $S^{1}/Z_{2}$ orbifold (i.e. a circle where $y \rightarrow -y$). This can be visualized as every point in space time having a line extending from it a distance L, representing this fifth dimension. At the end of this line is the Planck brane. This fourth spatial dimension separates two 4-D branes: Planck brane and TeV brane. We live on the TeV brane, as shown in Figure ~\ref{fig:RS_cartoon}. The Higgs field (and to a lesser degree the top quark and graviton fields) is localized near the TeV Brane, while the light fermion fields are localized closer to the Planck brane.

Fundamental parameters are set on the Planck brane. The warp factor may be scaled away from all dimensionless SM terms by field redefinitions. However, the only dimensionful parameter, $m_{H}^{2}=v^{2}$ is rescaled by $\tilde{v}\sim e^{-kL}M_{Pl}\sim 1$TeV for $kL\sim 35$. So, the electroweak and Planck scales are consistent on the Planck brane, and only the electroweak scale is significantly warped down on the TeV brane, explaining the large difference between the two scales. Also, by the localization of the light fermion fields near the Planck brane and top and graviton fields near the TeV brane provides an explanation for the fermion mass hierarchy.

The two free parameters of this theory are $M_{PL}$ and $k$. Based on this RS Bulk model, all SM particles should have Kaluza-Klein (KK) excitations. In particular, the graviton would have KK excitations ($G_{KK}$) that prefer to decay to $WW$ or $ZZ$. This search is sensitive to the $WW$ decay mode.

\begin{figure}[h!]
  \centering
  \includegraphics[width=\hsize]{figures/Theory/RS_cartoon.pdf}
  \caption{Cartoon of RS Bulk Model}.
  \label{fig:RS_cartoon}
\end{figure}
\FloatBarrier

\begin{comment}
\section{Extended Scalar Sector}
%tries to solve physics problem but also somewhat ad-hoc (just a parameterization)
A further striking asymmetry of the SM is the simplicity of the scalar sector in comparison to the boson and fermion sectors. To date, the scalar sector has only one member, the Higgs boson. Therefore, it is natural to posit an extension to the scalar sector. From a theoretical standpoint this could also help generate baryon asymmetry through additional sources of CP violation. This analysis searches for a simple extension to the scalar sector as proposed in Ref. \cite{scalarsinglet}. The extended scalar sector includes a real Higgs singlet (S) and complex $SU(2)_{L}$ doublet ($\Phi$) (the SM Higgs), where mass eigenstates are mixtures of the fields. S has a vev of $v$ and $\Phi$ has a vev of $x$. This then gives a Lagrangian of:

\begin{equation}
\mathcal{L}\supset (D^{\mu}\Phi)^{\dag}D_{\mu}\Phi + \partial^{\mu}S\partial_{\mu}S - m^{2}\Phi^{\dag}\Phi-\mu^{2}S^{2} + \lambda_{1}(\Phi^{\dag}\Phi)^{2} + \lambda_{2}S^{4}+\lambda_{3}\Phi^{\dag}\Phi S^{2}
\end{equation}

The mass eigenstates of the scalar sector are then mixtures of $S$ and $\Phi$ and the free parameters of the theory are $m_{H}$, $\sin\alpha$, and $tan\beta=v/x$. The fields are then given by:

\begin{equation}
\Phi \equiv \begin{pmatrix} 0 \\ \frac{\tilde{h}+v}{\sqrt{2}} \end{pmatrix}
\end{equation}
\begin{equation}
S \equiv \frac{h'+x}{\sqrt{2}}
\end{equation}

Diagonalizing the mass matrix leads to the mass eigenstates h (discovered Higgs boson) and H (the physical particles):

\begin{equation}
\begin{pmatrix} h \\ H \end{pmatrix} = \begin{pmatrix} \cos\alpha &-\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}
\end{equation}
This suppressed h and H production and SM H couplings:

\begin{equation}
BR_{H \rightarrow SM}=\sin^{2}\alpha \times \frac{\Gamma_{SM, H \rightarrow SM}}{\Gamma_{tot}}
\end{equation}
Moreover, in the case that $m_{H} > m_{h}$, $H\rightarrow hh$ is possible. This further suppresses $H \rightarrow VV/ff$. This search is most sensitive to $H \rightarrow WW$.
\end{comment}

\section{Simple Standard Model Extensions}
\label{HVT chapter}
The RS Bulk model is motivated by resolving SM hierarchies, but it does not address other SM issues. There are many other interesting and well motivated new physics frameworks that address these issues, but there is a lack of completely predictive models, due to model flexibility (free parameters). It is difficult for experimentalists to know which theories to search for in data. Therefore, developing a model-independent resonance search that can be reinterperted in the context of a given BSM theory is ideal.

This search is sensitive to the resonance mass and its couplings, but not all of a given BSM model's parameters. Therefore, the BSM Lagrangian may be reduced to only retain this information (mass parameters and couplings) following the procedure in \cite{hvt}.
In this simplified approach, the new resonance searched for is represented as an additional heavy vector triplet (HVT), which is a real vector field in the adjoint representation of $SU(2)_{L}$ with vanishing hypercharge. This results in one neutral and two charged bosons, defined as:

\begin{equation}
V^{\pm}=\frac{V^{1}_{\mu}\mp iV^{2}_{\mu}}{\sqrt{2}}
\end{equation}
\begin{equation}
V^{0}_{\mu}=V^{3}_{\mu}
\end{equation}

The SM Lagrangian is then augmented with the additional terms:
\begin{equation}
\label{hvt_lagrangian}
\mathcal{L}\supset -\frac{1}{4}D_{[\mu}V^{a}_{\nu]}D^{[\mu}V^{\mu]a} + \frac{m_{V}^{2}}{2}V^{a}_{\mu}V^{a\mu} + ig_{V}c_{H}V^{a}_{\mu}H^{\dag}\tau^{a}\overset\leftrightarrow{D^{\mu}}H+\frac{g^{2}}{g_{V}}c_{F}V^{a}_{\mu}J_{F}^{\mu a}
\end{equation}

In order the terms represent: the kinetic, $V$ mass, Higgs-$V$ interaction, and $V$-left-handed fermion interaction terms. The $g_{V}$ coupling factor determines the coupling of the new resonance to left-handed fermions and the Higgs boson.

As benchmark models, this search considers resonances from Extended Gauge Models (EGM) and composite Higgs models as discussed in \cite{hvt} . The EGM model predicts weakly coupled resonances, where $g_{V} = 1$, referred to later as Model A. The composite Higgs Model is a strongly coupled model, where $g_{V}=3$, and later referred to as Model B.
As shown in Eq. \ref{hvt_lagrangian}, the coupling of these resonances to fermions scales as $g_{f} = g^{2}c_{F}/g_{V}$, where $g$ is the SM $SU(2)_{L}$ gauge coupling and $c_{F}$ is a free model parameter. Therefore in Model B, the coupling to fermions is suppressed relative to Model A, leading to a smaller DY production rate and branching ratio (BR) to fermionic final states. The coupling of $V$ to SM bosons scales as $g_{H}=g_{V}c_{H}$, where $c_{H}$ is a free model parameter on the order of one for Model A and B. Consequently Model A resonances have a smaller the BR to gauge bosons than Model B. For the $pp$ collision data used, Model A predicts larger production cross sections decaying to leptons and fermions than Model B which decays primarily to gauge bosons.

Model A and B vectors are produced via quark-anti-quark annihilation and vector-boson-fusion. Both production modes are probed in this search.


% !TEX root = ./Basilisk-sunSafePoint-20180427.tex

\section{Model Description}

\begin{figure}[t]
	\centerline{
		\includegraphics{Figures/sunHeading}
	}
	\caption{Body Vector Illustrations.}
	\label{fig:Fig1}
\end{figure}

\subsection{Module Goal}
This attitude guidance module has the goal of aligning a commanded body-fixed spacecraft vector $\hat{\bm s}_{c}$ with another input vector $\bm s$.  If $\hat{\bm s}_{c}$ is for example the solar panel normal vector, and $\bm s$ is the current sun heading vector, this module will compute the attitude tracking errors to align the solar panels towards the sun, i.e. achieve sun pointing.  Sun pointing is a mode for general recharging the spacecraft, but is also a common guidance scenario with Safe Mode.

Besides $\bm s$, the second input vector is the inertial body angular velocity vector $\bm\omega_{B/N}$.  The sun pointing frame is assumed to be at rest, thus the attitude rate tracking error is set either equal to the body rates to bring the body to rest, or difference with a specified rotation about the sun heading vector to achieve a final roll about this heading vector.

As the desired sun pointing orientation is inertial, the inertial reference frame  acceleration $\dot{\bm\omega}_{R/N}$ is set to zero, while the reference rate is either zero or the desired sun-heading roll vector.

Note that this module does not establish a unique sun-pointing reference frame.  Rather, the pointing condition, align $\hat{\bm s}_{c}$ with $\bm s$ is an under-determined 2 degree of freedom condition.  Thus, the rotation angle about $\bm s$ is left to be arbitrary in this sun pointing module.  For the sun pointing applications this is a very practical result as the power generation does not depend on the orientation about $\bm s$.

\subsection{Equations}
\subsubsection{Good Sun Direction Vector Case}\label{sec:withSun}
In the following mathematical developments all vectors are assumed to be taken with respect to a body-fixed frame $\cal B$.  The attitude of the body $\cal B$ relative to the reference frame $\cal R$ is written as a principal rotation from $\cal R$ to $\cal B$.  Thus, the associated principal rotation vector $\hat{\bm e}$ is
\begin{equation}
	\label{eq:ssp:1}
	\hat{\bm e} = \frac{\bm s \times \hat{\bm s}_{c}}{|\bm s \times \hat{\bm s}_{c}|}
\end{equation}
Note that the sun direction vector $\bm s$ does not have to be a normalized input vector.

The principal rotation angle between the two vectors is given through
\begin{equation}
	\label{eq:ssp:2}
	\Phi = \arccos \left( \frac{\bm s  \cdot \hat{\bm s}_{c} }{|\bm s|} \right)
\end{equation}

Next, this rotation from $\cal R$ to $\cal B$ is written as a set of MRPs through
\begin{equation}
	\label{eq:ssp:3}
	\bm\sigma_{B/R} = \tan\left(\frac{\Phi}{4}\right) \hat{\bm e}
\end{equation}
The set $\bm\sigma_{B/R}$ is the attitude error of the output attitude guidance message.

The module allows for a nominal spin rate about the sun heading axis by specifying the module parameter {\tt sunAxisSpinRate} called $\dot \theta$ in this description.  The nominal spin rate is thus given by
\begin{equation}
	\leftexp{B}{\bm\omega}_{R/N} = \leftexp{B}{\hat{\bm s}} \dot\theta
\end{equation}
Note that this constant nominal spin is only for the case where the sun is visible and the sun-heading vector measurement is available.

If the spacecraft is to be brought to rest $\bm\omega_{R/N} = \bm 0$, then $\dot\theta$ should be set to zero.  The tracking error angular velocity vector is computed using.
\begin{equation}
	\label{eq:ssp:4}
	\bm\omega_{B/R} = \bm\omega_{B/N} - \bm\omega_{R/N}
\end{equation}

Finally, the attitude guidance message must specify the inertial reference frame  acceleration vector.  This is set to zero as the roll about the sun heading is assumed to have a constant magnitude and inertial heading.
\begin{equation}
	\dot{\bm \omega}_{R/N} = \bm 0
\end{equation}

\subsubsection{No Sun Direction Vector Case}
 If $\Phi$ is less then then module parameter {\tt minUnitMag}, then it is assumed that no good sun heading vector is available and the attitude tracking error $\bm\sigma_{B/R}$ is set to zero.

 Further, if the sun is not visible, the module allows for a non-zero body rate to be prescribed.  This allows the spacecraft to engage in a constant rate tumble specified through the module configuration vector {\tt omega\_RN\_B}.  In this case the tracking error rate is evaluate through
 \begin{equation}
 	\label{eq:ssp:6}
	\bm\omega_{B/R} = \bm\omega_{B/N} - \bm\omega_{R/N}
 \end{equation}
 and the output message reference rate is set equal to the prescribed $\bm\omega_{R/N}$ while the reference acceleration vector $\dot{\bm \omega}_{R/N}$ is set to zero.

 \subsubsection{Collinear Commanded and Sun Heading Vectors}
First consider the case where $\bm s \approx \hat{\bm s}_{c}$.  In this case the cross product in Eq.~\eqref{eq:ssp:1} is not well defined.  Let $\epsilon$ be a pre-determined small angle.  Then, if $\Phi < \epsilon$ the attitude tracking error is set to
$$
	\bm\sigma_{B/R} = \bm 0
$$

However, if $\bm s \approx -\hat{\bm s}_{c}$, then an eigen-axis $\hat{\bm e}$ that is orthogonal to $\hat{\bm s}_{c}$ must be determined.  Let the body frame be defined through \frameDefinition{B}.  The eigen-axis is determined first by taking a cross product with $\hat{\bm b}_{1}$:
 \begin{equation}
	\label{eq:ssp:7}
	\hat{\bm e}_{180} = \frac{ \hat{\bm s}_{c} \times \hat{\bm b}_{1}}{| \hat{\bm s}_{c} \times \hat{\bm b}_{1}|}
\end{equation}
If $\hat{\bm s}_{c} \approx \hat{\bm b}_{1}$, then this $\hat{\bm e}$ vector will have a small norm.  In this ill-determined case, the $\hat{\bm e}$ vector is determined using
 \begin{equation}
	\label{eq:ssp:8}
	\hat{\bm e}_{180} = \frac{ \hat{\bm s}_{c} \times \hat{\bm b}_{2}}{| \hat{\bm s}_{c} \times \hat{\bm b}_{2}|}
\end{equation}
As $ \hat{\bm s}_{c}$ cannot both be aligned with $\hat{\bm b}_{1}$ and $\hat{\bm b}_{2}$, this algorithm determines a unique eigen axis $\hat{\bm e}_{180}$ for the case that the principal rotation angle is close to 180 degrees, or $\pi - \Phi < \epsilon$.  This special case eigen axis is only computed once in the module reset routine.

In this scenario the angular velocity tracking error is evaluated using the same method as outlined in section~\ref{sec:withSun}.