I'm struggling to understand the kernel gradient correction to restore consistency of the SPH methods on free surface or boundary. Where does the definition of M come from?
\begin{equation} \begin{aligned} \nabla W_{ij}' & = L_i \nabla W_{ij}\\ L_i & = M_i^{-1}\\ M_i & = \sum_{j\in neigh(i)}\frac{m_j}{\rho_j}\nabla W \otimes (r_i-r_j) \end{aligned} \end{equation}
Does anyone has a reference or somethings where is has been described
Okey I found the solutions. The major problem for the consistency is that on the boundary, the density is no more symmetric around a particle and so the gradient computation is biased.\ To have a good gradient computation we need : \begin{equation} \sum_{j\in neigh(i)}^N\frac{m_j}{\rho_j}(x_j-x_i)\otimes \nabla W_j(x_i) = \textbf{I} \qquad \forall i\in\text{Particles} \end{equation} To ensure that even on the boundary let's define $W_{ij}' = L_i \nabla W_{ij}$. It gives \begin{equation} \sum_{j\in neigh(i)}^N\frac{m_j}{\rho_j}(x_j-x_i)\otimes L_i \nabla W_j(x_i) = \textbf{I} \qquad \forall i\in\text{Particles}\label{kgc:cdt1} \end{equation} Now if we take $L_i = \left(\sum_{j\in neigh(i)}^N\frac{m_j}{\rho_j}\nabla W_j(x_i)\otimes(x_j-x_i)\right)^{-1}$ we have that this condition holds for every particles even them on the borders.