Mesh deformation using the biharmonic operator

The use of the biharmonic operator for deforming a mesh in an arbitrary–Lagrangian–Eulerian simulation is investigated. The biharmonic operator has the advantage that two conditions can be specified on each boundary of the mesh. This allows both the position and the normal mesh spacing along a boundary to be controlled, which is important for two‐fluid interfaces and periodic boundaries. At these boundaries, we can simultaneously fix the position of the boundary and ensure that the normal mesh spacing is continuous across the boundary. In addition, results for deforming surfaces show that greater surface deformation can be tolerated when using biharmonic equations compared to approaches using second‐order partial differential equations. A final advantage is that with the biharmonic operator, the integrity of a grid in a moving boundary layer can be preserved as the boundary moves. The main disadvantage of the approach is its increased computational expense.