\form#0:$V_{ijk}$ \form#1:$i$ \form#2:$j$ \form#3:$k$ \form#4:$ijk$ \form#5:$jik$ \form#6:$V_{111} V_{112} V_{113} V_{114} V_{121} V_{122} V_{123} V_{124} V_{211} V_{212} ... $ \form#7:$ V_{111} V_{112} V_{113} V_{114} V_{211} V_{212} V_{213} V_{214} V_{311} V_{312} ... V_{121} V_{122} V_{123} $ \form#8:$ P_i $ \form#9:$ [-\pi/2,\pi/2 ] $ \form#10:$ (BP_i) $ \form#11:$ N_{edges} = V.size()/2 $ \form#12:$ Edge_i i \in [0,N_{edges}-1] $ \form#13:$ V[i+N_{edge}] $ \form#14:$ Edge_i $ \form#15:$ N_edges = V.size() $ \form#16:$ T_i\cap S_j$ \form#17:$ S_j $ \form#18:$ T_i $ \form#19:$ W_{ij} $ \form#20:$ [0,n-1] $ \form#21:$ [1,n] $ \form#22:$ \phi_s $ \form#23:$ \phi_t $ \form#24:\[ \phi_t=W.\phi_s. \] \form#25:$W$ \form#26:$ \phi $ \form#27:\[ \int_{T_i} \phi_t = \sum_{S_j\cap T_i \neq \emptyset} \int_{T_i\cap S_j} \phi_s. \] \form#28:\[ \sum_{S_j} Vol(T_i\cap S_j) = Vol(T_i),\hspace{1cm} and \hspace{1cm} \sum_{T_i} Vol(S_j\cap T_i) = Vol(S_j) \] \form#29:$Vol(T_i)$ \form#30:$\sum_{S_j} Vol(T_i\cap S_j)$ \form#31:\[ \int_{T_i} \phi = Vol(T_i).\phi_{T_i}. \] \form#32:\[ \sum_{S_j\cap T_i \neq \emptyset} \int_{T_i\cap S_j} \phi = \sum_{S_j\cap T_i \neq \emptyset} {Vol(T_i\cap S_j)}.\phi_{S_j}. \] \form#33:$ W $ \form#34:\[ W_{ij}=\frac{Vol(T_i\cap S_j)}{ Vol(T_i) }. \] \form#35:\[ \int_{T_i} \phi = P_{T_i}, \] \form#36:\[ \sum_{S_j\cap T_i \neq \emptyset} \int_{T_i\cap S_j} \phi = \sum_{S_j\cap T_i \neq \emptyset} \frac{Vol(T_i\cap S_j)}{ Vol(S_j)}.P_{S_j}. \] \form#37:\[ W_{ij}=\frac{Vol(T_i\cap S_j)}{ Vol(S_j) }. \] \form#38:\[\frac{Vol(T_i\cap S_j)}{ Vol(T_i)}\] \form#39:\[ \frac{Vol(T_i\cap S_j)}{ \sum_{T_i} Vol(S_j\cap T_i) }\] \form#40:\[\frac{Vol(T_i\cap S_j)}{ \sum_{S_j} Vol(T_i\cap S_j)}\] \form#41:\[\frac{Vol(T_i\cap S_j)}{ Vol(S_j) }\] \form#42:\[ M1=\left[\begin{tabular}{cc} 0.125 & 0.75 \\ \end{tabular}\right] \] \form#43:$ M_{Conservative Volumic} $ \form#44:\[ M_{Conservative Volumic}=\left[\begin{tabular}{cc} $\displaystyle{\frac{0.125}{0.125+0.75}}$ & $\displaystyle{\frac{0.75}{0.125+0.75}}$ \\ \end{tabular}\right]=\left[\begin{tabular}{cc} 0.14286 & 0.85714 \\ \end{tabular}\right] \] \form#45:\[ FT=\left[\begin{tabular}{cc} $\displaystyle\frac{0.125}{0.875}$ & $\displaystyle\frac{0.75}{0.875}$ \\ \end{tabular}\right].\left[\begin{tabular}{c} 4 \\ 100 \\ \end{tabular}\right] =\left[\begin{tabular}{c} 86.28571\\ \end{tabular}\right] \] \form#46:$ M_{Integral} $ \form#47:\[ M_{Integral}=\left[\begin{tabular}{cc} $\displaystyle{\frac{0.125}{9}}$ & $\displaystyle{\frac{0.75}{3}}$ \\ \end{tabular}\right]=\left[\begin{tabular}{cc} 0.013888 & 0.25 \\ \end{tabular}\right] \] \form#48:\[ FT=\left[\begin{tabular}{cc} $\displaystyle{\frac{0.125}{9}}$ & $\displaystyle{\frac{0.75}{3}}$ \\ \end{tabular}\right].\left[\begin{tabular}{c} 4 \\ 100 \\ \end{tabular}\right] =\left[\begin{tabular}{c} 25.055\\ \end{tabular}\right] \] \form#49:$ FS_{vol} $ \form#50:$ M1 $ \form#51:\[ FS_{vol}=\left[\begin{tabular}{c} $\displaystyle{\frac{4}{9}}$ \\ $\displaystyle{\frac{100}{3}}$ \\ \end{tabular}\right] \] \form#52:$ FS $ \form#53:$ M_{IntegralGlobConstraint} $ \form#54:\[ M_{IntegralGlobConstraint}=\left[\begin{tabular}{cc} $\displaystyle{\frac{0.125}{0.125}}$ & ${\displaystyle\frac{0.75}{0.75}}$ \\ \end{tabular}\right]=\left[\begin{tabular}{cc} 1 & 1 \\ \end{tabular}\right] \] \form#55:\[ FT=\left[\begin{tabular}{cc} 1 & 1 \\ \end{tabular}\right].\left[\begin{tabular}{c} 4 \\ 100 \\ \end{tabular}\right] =\left[\begin{tabular}{c} 104\\ \end{tabular}\right] \] \form#56:$ M_{RevIntegral} $ \form#57:\[ M_{RevIntegral}=\left[\begin{tabular}{cc} $\displaystyle{\frac{0.125}{1.5}}$ & $\displaystyle{\frac{0.75}{1.5}}$ \\ \end{tabular}\right]=\left[\begin{tabular}{cc} 0.083333 & 0.5 \\ \end{tabular}\right] \] \form#58:\[ FT=\left[\begin{tabular}{cc} $\displaystyle{\frac{0.125}{1.5}}$ & $\displaystyle{\frac{0.75}{1.5}}$ \\ \end{tabular}\right].\left[\begin{tabular}{c} 4 \\ 100 \\ \end{tabular}\right] =\left[\begin{tabular}{c} 50.333\\ \end{tabular}\right] \] \form#59:$ W/m^2 $ \form#60:$M_s$ \form#61:$F_s$ \form#62:$M_t$ \form#63:$F_t$ \form#64:$ x_0,y_0,z_0,x_1,y_1,z_1,x_2,y_2,z_2,x_3,y_3,z_3,x_4,y_4,z_4 $ \form#65:$kg.m^{-3}$ \form#66:$W.m^{-3}$ \form#67:$K$ \form#68:$Pa$ \form#69:$kg$ \form#70:$m^3$ \form#71:$kg.m.s^{-1}$ \form#72:$(W$ \form#73:$ T $ \form#74:$ v_x $ \form#75:$ \sigma_{xy} $ \form#76:$ i^{th} $ \form#77:$ j^{th} $ \form#78:$ x_1,y_1,z_1,x_2,y_2,z_2,\ldots,x_n,y_n,z_n $ \form#79:$ x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n,z_1,z_2,\ldots,z_n $ \form#80:$ n_{types}+1 $ \form#81:$ type[i] $ \form#82:$ index[i] $ \form#83:$ index[i+1] $ \form#84:$ index[i] \leq j < index[i+1] $ \form#85:$n_{cell}+1$ \form#86:$index[i]$ \form#87:$index[i+1]$ \form#88:$index[i] \leq j < index[i+1]$ \form#89:\[ Area=\int_{Polygon} dS \] \form#90:\[ \int_{Polygon} x \cdot dS=\sum_{0 \leq i < nb of edges} -\int_{Edge_{i}}ydx=\sum_{0 \leq i < nb of edges} AreaOfZone_{Edge_{i}} \] \form#91:$ AreaOfZone_{i} $ \form#92:\[ AreaOfZone_{i}=\int_{Edge_{i}} -ydx \] \form#93:\[ Area \cdot x_{G}=\int_{Polygon} x \cdot dS \] \form#94:\[ Area \cdot y_{G}=\int_{Polygon} y \cdot dS \] \form#95:\[ \int_{Polygon} x \cdot dS=\sum_{0 \leq i < nb of edges} -\int_{Edge_{i}}yxdx \] \form#96:\[ \int_{Polygon} y \cdot dS=\sum_{0 \leq i < nb of edges} -\int_{Edge_{i}}\frac{y^{2}}{2}dx \] \form#97:$ -\int_{Edge_{i}}yxdx $ \form#98:$ -\int_{Edge_{i}}\frac{y^{2}}{2}dx $ \form#99:\[ \int_{Current Edge} -ydx \] \form#100:\[ bary[0]=\int_{Current Edge} -yxdx \] \form#101:\[ bary[1]=\int_{Current Edge} -\frac{y^{2}}{2}dx \] \form#102:\[ x=x_{0}+Radius \cdot cos(\theta) \] \form#103:\[ y=y_{0}+Radius \cdot sin(\theta) \] \form#104:\[ dx=-Radius \cdot sin(\theta) \cdot d\theta \] \form#105:\[ y=y_{1}+\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1}) \] \form#106:\[ \sqrt{\sum_{0 \leq i < nbOfEntity}val[i]*val[i]} \] \form#107:\[ \max_{0 \leq i < nbOfEntity}{abs(val[i])} \] \form#108:\[ \frac{\sum_{0 \leq i < nbOfEntity}|val[i]*Vol[i]|}{\sum_{0 \leq i < nbOfEntity}|Vol[i]|} \] \form#109:\[ \sqrt{\frac{\sum_{0 \leq i < nbOfEntity}|val[i]^{2}*Vol[i]|}{\sum_{0 \leq i < nbOfEntity}|Vol[i]|}} \] \form#110:$ j^{th}$ \form#111:$ k^{th}$ \form#112:$ i^{th}$ \form#113:$ (i,j,k)$ \form#114:$ (i, j, k )$ \form#115:$ (i,j,k) $ \form#116:$ (i+1)^th $ \form#117:$ A $ \form#118:$A \cap B$ \form#119:\[ I_{source}=\sum_{i=1}^{n_{source}}V_{i}.|\Phi^{source}_{i}|^2, \] \form#120:\[ I_{target}=\sum_{i=1}^{n_{target}}V_{i}.|\Phi^{target}_{i}|^2, \] \form#121:\[ \Phi^{target}:=\Phi^{target}.\sqrt{I_{source}/I_{target}}. \] \form#122:$ \phi_t=W.\phi_s$ \form#123:\[ \begin{tabular}{|cccc|} 0.72 & 0 & 0.2 & 0 \\ 0.46 & 0 & 0.51 & 0.03\\ 0.42 & 0.53 & 0 & 0.05\\ 0 & 0 & 0.92 & 0.05 \\ \end{tabular} \] \form#124:\[ \begin{tabular}{|cccc|} 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{tabular} \] \form#125:$ M_k $