Commit 1aae6b06 by Michael Turner

Merge remote-tracking branch 'upstream/master' into feature/projection-meshing

parents 92868911 e2bd965e
 ... ... @@ -66,7 +66,7 @@ zero. If we now integrate the 1st, 2nd and last term in equation (\ref{eqn.weakp}) by parts we can obtain the weak pressure equation \begin{align} \int_{\Omega} \nabla q \cdot \nabla p^{n+1} &= \int_{\Omega} q\, \nabla \cdot \left ( \frac{\partial \mathbf{u}}{\partial t}^{n+1} + \mathbf{N}(\mathbf{u})^{n+1} \right ) \nonumber \\ &- \int_{\partial \Omega} q \left ( \frac{\partial \mathbf{u}}{\partial t}^{n+1} + \mathbf{N}(\mathbf{u})^{n+1} - \nu \nabla \times \nabla \times \mathbf{u}^{n+1} \right ) \cdot \mathbf{n} &- \int_{\partial \Omega} q \left ( \frac{\partial \mathbf{u}}{\partial t}^{n+1} + \mathbf{N}(\mathbf{u})^{n+1} + \nu \nabla \times \nabla \times \mathbf{u}^{n+1} \right ) \cdot \mathbf{n} \label{eqn.weakp1} \end{align} where $\partial \Omega$ is the boundary of the domain and we have used ... ... @@ -98,7 +98,7 @@ which to decouple the system we impose that \nabla \cdot \int_{\Omega} \nabla q \cdot \nabla p^{n+1} &= \int_{\Omega} q \, \nabla \cdot \left (-\frac{\hat{\mathbf{u}}}{\Delta t} + \mathbf{N}(\mathbf{u})^{*,n+1} \right ) \nonumber \\ &- \int_{\partial \Omega} q \left ( \frac{\partial \mathbf{u}}{\partial t}^{n+1} + \mathbf{N}(\mathbf{u})^{*.n+1} - \nu (\nabla \times \nabla \times \mathbf{u})^{*,n+1} \right ) \cdot \mathbf{n} &- \int_{\partial \Omega} q \left ( \frac{\partial \mathbf{u}}{\partial t}^{n+1} + \mathbf{N}(\mathbf{u})^{*.n+1} + \nu (\nabla \times \nabla \times \mathbf{u})^{*,n+1} \right ) \cdot \mathbf{n} \label{eqn.weakpfinal} \end{align} We note this can be recast into an equivalent strong form of the ... ... @@ -109,7 +109,7 @@ which to decouple the system we impose that\nabla \cdot with consistent Neumann boundary conditions prescribed as \frac{\partial p}{\partial n}^{n+1}= - \Bigl[ \frac{\partial \mathbf{u}}{\partial t}^{n+1} - \nu (\nabla \times \nabla \times \mathbf{u})^{*,n+1} + \mathbf{N}^{*,n+1}\Bigr]\cdot \mathbf{n} \frac{\partial p}{\partial n}^{n+1}= - \Bigl[ \frac{\partial \mathbf{u}}{\partial t}^{n+1} + \nu (\nabla \times \nabla \times \mathbf{u})^{*,n+1} + \mathbf{N}^{*,n+1}\Bigr]\cdot \mathbf{n} \label{eqn.pressurebcs} ... ...