AlternativeDPA

Important remark. This algorithm is obsolete as it is contained in the ClassicalDPA by using the option ===========================================================================================================- This algorithm is invoked if the first digit of the variable FLIQUID_ConsistentPressure_Version is put to 2. Short derivation/motivation: Let us again consider the equation of momentum \begin{align}\mathbf{\dot{v}}+\frac{1}{\rho }\nabla p=\frac{1}{\rho }\left( \nabla ^{T}\mathbf{S}_{s} \right)^{T}+\frac{1}{\rho }\left( \nabla ^{T}\mathbf{S}_{v}\left( \mathbf{v} \right) \right)^{T}+\mathbf{g}-\beta \cdot \mathbf{v}\end{align} and isolate for the target dynamic pressure gradient \begin{align}\nabla p_{dyn}^{\text{target}}=\left( \nabla ^{T}\mathbf{S}_{v}\left( \mathbf{v} \right) \right)^{T}-\rho \beta \cdot \mathbf{v}-\rho \mathbf{\dot{v}}\end{align} For simplicity, we now omit the suffix >>>dyn<<<. Between two MESHFREE points \( i\) and \( j\) we can compute the intermediate pressure value by \begin{align}\bar{p}_{ij}=p_{i}+\frac{\rho _{j}}{\rho _{i}+\rho _{j}}\cdot \left( \mathbf{x}_{j}-\mathbf{x}_{i} \right)^{T}\cdot \nabla p_{i}^{\text{target}}=p_{j}+\frac{\rho _{i}}{\rho _{i}+\rho _{j}}\cdot \left( \mathbf{x}_{i}-\mathbf{x}_{j} \right)^{T}\cdot \nabla p_{j}^{\text{target}}\end{align} So, we can write \begin{align}\frac{\rho _{i}}{\rho _{i}+\rho _{j}}\cdot \left( \mathbf{x}_{j}-\mathbf{x}_{i} \right)^{T}\cdot \nabla p_{j}^{\text{target}}+\frac{\rho _{j}}{\rho _{i}+\rho _{j}}\cdot \left( \mathbf{x}_{j}-\mathbf{x}_{i} \right)^{T}\cdot \nabla p_{i}^{\text{target}}=p_{j}-p_{i}\end{align} or even better, in order to have full symmetry, \begin{align}\frac{1}{2}\left( \mathbf{x}_{j}-\mathbf{x}_{i} \right)^{T}\cdot \frac{1}{\rho _{j}}\nabla p_{j}^{\text{target}}+\frac{1}{2}\left( \mathbf{x}_{j}-\mathbf{x}_{i} \right)^{T}\cdot \frac{1}{\rho _{i}}\nabla p_{i}^{\text{target}}=\frac{1}{2}\frac{\rho _{i}+\rho _{j}}{\rho _{i}\rho _{j}}\left( p_{j}-p_{i} \right)\end{align} For each MESHFREE point \( i\) we have as many equations of this type as there are neighbor points, which means we have an overdetermined system. However, we could require \begin{align}\sum\limits_{j=1}^{N(i)}{W_{ij}\left( \frac{1}{2}\left( \mathbf{x}_{j}-\mathbf{x}_{i} \right)^{T}\cdot \frac{1}{\rho _{j}}\nabla p_{j}^{\text{target}}+\frac{1}{2}\left( \mathbf{x}_{j}-\mathbf{x}_{i} \right)^{T}\cdot \frac{1}{\rho _{i}}\nabla p_{i}^{\text{target}} \right)}=\sum\limits_{j=1}^{N(i)}{W_{ij}\left( \frac{1}{2}\frac{\rho _{i}+\rho _{j}}{\rho _{i}\rho _{j}}\left( p_{j}-p_{i} \right) \right)}\end{align} This set forms a linear system of equations for the unknowns \( p_{i}\) . So far, only choosing \( W_{ij}=c_{ij}^{\Delta }\) provided very stable results. This special choices of the weight function, by the way, provides a nice similarity to the classical ansatz, if we also remember, that \begin{align} c_{ij}^{x}=\frac{1}{2}\left( x_{j}-x_{i} \right)\cdot c_{ij}^{\Delta },\text{ }c_{ij}^{y}=\frac{1}{2}\left( y_{j}-y_{i} \right)\cdot c_{ij}^{\Delta },\text{ }c_{ij}^{z}=\frac{1}{2}\left( z_{j}-z_{i} \right)\cdot c_{ij}^{\Delta }\end{align}