Meshing and Solving Elastic Waves, Time Explicit Models

Meshing and solution time are closely linked when modeling physics based on the discontinuous Galerkin (dG) time explicit method. The computational mesh has to resolve the shortest wavelength in the model (the slowest wave), while it is the fastest wave speed and the smallest mesh element that dictate the internal time step of the solver.

In general, the maximal mesh size hmax is dictated by the smallest wavelength (the slowest wave) such that

for the default 4th (quartic) order discretization, where fmax is the maximal frequency to resolve in the model, given by the frequency content of the source, and cmin is the slowest waves speed in the model. This is typically the shear wave speed cs, but for problems with a free interface or a material discontinuities, interfacial waves also exist. For example, the classical estimate for the Rayleigh wave speed vR is

where ν is the Poisson’s ratio and cs is shear wave speed.

Internally, the time step taken by the time explicit method is given by the global minimum of the local mesh size relative to the maximal local wave speed. This is also known as the cell wave time scale, the value can be visualized by plotting the variable elte.wtc. This means that small mesh elements should be avoided, see Meshing, Discretization, and Solvers for the Convected Wave Equation documentation, for more details.

The maximal wave speed used to predict the cell wave time scale uses the pressure wave speed cp as default. The value is exact for isotropic materials but is an estimate for materials with anisotropic properties. For anisotropic materials it can be necessary to use a larger value to ensure stability of the numerical method. This can be done in the section for each material model. To display this section, click the button (

) and select in the dialog. Select (the default) or . With the option the maximum speed is set equal to the pressure wave speed, for enter another value for cmax (a larger value may be needed for stability).

When several materials are used in a model, the use of assemblies and pair features is recommended (Continuity and Pair Acoustic-Structure Boundary, Time Explicit). The advantage of using a pair feature is that the mesh does not need to be conforming on the two sides of the interface (the two parts pf the assembly). This is especially advantageous for the time explicit discontinuous Galerkin method as the time step depends on mesh size and local speed of sound.

If material properties result in large variations of the local cell wave time scale, then it can be advantageous to use the method instead of the default method.