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 Estimate of Maximum Wave Speed section for each material model. To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog. Select Automatic (the default) or User defined. With the Automatic option the maximum speed is set equal to the pressure wave speed, for User defined 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 Adam-Bashforth 3 (local) method instead of the default Runge-Kutta method.
Physics-Controlled Mesh
The Elastic Waves, Time Explicit interface supports physics-controlled mesh. To get an appropriate mesh, remember to set the value of the Maximum frequency to resolve in the Transient Mesh Settings section. The automatically generated mesh will use this value and an estimate of the slowest wave speed (the default being the shear wave speed), defined in the model, to set up an appropriate mesh that follows the best practices.
On the main Mesh node select the Sequence Type to be Physics-controlled mesh and then select the physics interface in the Contributor list. The Maximum mesh element size control parameter is either From physics (it will use the value from the Transient Mesh Settings section) or Frequency. For Frequency enter a value for the Maximum frequency. Select the Number of mesh elements per wavelength; for Automatic this is results 1.5 elements per wavelength, or enter a User defined value. Finally, select the Minimum wave speed used to define the shortest wavelength; for Automatic the shear wave speed is used, for User defined enter another value, for example, a bending wave speed or a Rayleigh wave speed, if necessary.