Generalized finite differences - Space-time meshing

Space-time meshing

The software package is applied for space-time discretization

We need orthogonal duality using a Minkowski metric. Then, a time integration scheme is obtained automatically. For more details see our space-time presentation.

Fig: 2-dimensional meshes with orthogonal duality using Euclidean metric on the left and Minkowski metric on the right.

Fig: Moving cavity: 1+1-dimensional space-time mesh and a solution of discrete wave equation after initialization of impulse at t=0.

SpiralBoomerang1080p.mp4

Fig: Wave propagation in rotating cavity. The simulation was produced similarly to previous figure, but in 2+1-dimensions.

Fig: To refine spatial grid one also has to refine time step size. With space-time meshing, this could be done on the fly. This is a 1+1-dimensional example.

Fig: 2+1-dimensional stability test.

Further studies consider benefits of space-time discretization

We could apply transforming cavities to take moving, spinning or transforming objects into account. Probably we could also decrease computational cost by following the wave path or by refining the grid on the fly.

Fig: Illustration of 2+1-dimensional space-time mesh. This is a BCC grid rotated such that diagonal axis is chosen as time axis. We have already performed preliminary simulations and analysis with this grid.

Fig: Illustration of 3+1-dimensional space-time mesh. This grid is achieved by modifying a 4d Cartesian grid by inserting vertex at the center of each hypercube.

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