Generalized finite differences

Structured tiling

Structured tiling has advantages against fully unstructured meshing

structured mesh is fast to generate (also parallel mesh generation is relatively simple)
lower memory usage (mesh can be packed because of repeating blocks)
quality of mesh can be guaranteed (see below)

We have studied structured tiling alternatives and their properties in 2- and 3-dimension.

Räbinä, J., Kettunen, L., Mönkölä, S., & Rossi, T. (2018). Generalized wave propagation problems and discrete exterior calculus. ESAIM: Mathematical Modelling and Numerical Analysis.

Fig: Alternatives for 2-dimensional structured tiling

Fig: Amount of numerical dispersion in different directions. Directional dependency is decreased using grids with hexagonal base. Systematic wave speed error is eliminated using sophisticated Hodge approximation (dotted line). Value of p indicates type of wave problem.

Fig: Alternatives for 3-dimensional structured tiling

Fig: Amount of numerical dispersion in different spatial directions (A-J). Directional dependency is decreased using tetrahedral grids. The systematic error is eliminated using Hodge approximation for single frequency (dotted line).

We also employ hybrid meshes, which combine benefits of structured and unstructured meshing.