%BND_none%

ATTENTION: this type of boundary condition is, theoretically, not valid in general if used with FLIQUID_ConsistentPressure_Version =?1?? , i.e. a 1 at the second digit. Here, the accelerations are considered to be quasistationary, hence %BND_none% will make a mistake if used in a non-quasistationary setting. With the same ansatz as in AlternativeDPA , we solve for the boundary point \( i\) the equation \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} Remember that \( \frac{1}{\rho_{j}}\nabla p_{j}^{\text{target}}\) might contain only the stationary part of the substantial derivative \( \frac{d \mathbf{v}}{dt} \approx \left( \mathbf{v}^T \cdot \nabla \right) \mathbf{v}\), so we provide the option: AllowBoundaryAcceleration: has to be bigger than zero. If the optional parameter is given, we enhance the equation to \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 \left( \frac{1}{\rho_{i}}\nabla p_{i}^{\text{target}} - \frac{ d \mathbf{v}_i^{BND} }{ dt } \right) \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} where we restrict the magnitude of the acceleration of the boundary to the optional value given, i.e. \( \left\| \frac{ d \mathbf{v}_i^{BND} }{ dt } \right\| \leq \text{AllowBoundaryAcceleration}\) In order to allow the true acceleration, set the value hugh enough. DEFAULT: 0 WeightKernel: the %BND_none% conditions can be put into practice ONLY treating the boundary point acording to AlternativeDPA . This requires a weight kernel, which is defined by this optional parameter. If set to 0, the classical Neumann ctencil is used as a weight. Otherwise, \( W_{ij} = exp\left( -\alpha \cdot r_{ij}^2 \right)\). The value of WeightKernel: then defines the parameter \( \alpha\). DEFAULT: 0