# Resolving time-dependent waves

 Solution Number: 1118 Title: Resolving time-dependent waves Platform: All Platforms Applies to: COMSOL Multiphysics, RF Module, MEMS Module, Structural Mechanics Module, Acoustics Module, Wave Optics Module Versions: All versions Categories: Mesh, Mesh Keywords:

## Problem Description

I am solving a time-dependent wave-type problem, what mesh and solver settings should I adjust for better accuracy and runtimes?

## Solution

The accuracy of a time-dependent model solving for electromagnetic, acoustic, structural, or any other kind of wave-type model is limited by how well the finite element mesh resolves the waves in space and by the how well the time steps resolve the temporal variations. This article describes how to control the mesh and the solver settings for such problems.

Before starting your modeling, you should know two things. First, the maximum frequency content of your excitation. Define this as a Global Parameter, for example: f_max. Second, the material properties in all domains. This information governs what size mesh you will need in all domains, since the material properties govern the shortest possible wavelength. If there are any material or boundary condition nonlinearities, consider that these may lead to frequency doubling or other phenomena that may lead to much higher frequency content. If you anticipate any step changes in the loads or boundary conditions, first read: Knowledge Base 1244: Solving Wave-Type Problems with Step Changes in the Loads

Once you know the maximum frequencies that you expect to model, and the material properties, you can build and mesh your model. By default, second order, (quadratic) elements are used in the electromagnetic, acoustic, and structural transient interfaces. At a bare minimum, the mesh must fulfill the Nyquist criterion, so at least two elements per local wavelength in each domain. However, this will usually be of unacceptably low accuracy. Practically speaking, at least five second order elements per local wavelength should be used, but very rarely more than ten elements per wavelength. Set up a User-controlled mesh with different Size features applied to the different material domains. The Free Triangular (in 2D models) and Free Tetrahedral (for 3D models) meshes are preferred, but the other element types (Hexahedral, Triangular Prisms, and Pyramids) can be used if there are high aspect ratio structures, thin layers, material anisotropy, or other reasons to expect strong variations in the solution in only one or two directions. Also define a Global Parameter (for example: N) to control the number of elements per local wavelength and a set of Variables to define the local wavelengths in different domains, as demonstrated in the screenshot below.

Manually setting the element sizes

By default, the Time-Dependent Solver will continuously adjust the time step in order to fulfill your specified tolerances. The Relative tolerance field in the Time Dependent settings (shown below) controls how the solver will adjust the timestep. Smaller numbers will lead to smaller timesteps and higher accuracy. Note that changing the number of output Times in the Time Dependent node controls only the output times, but has little effect on the time steps actually taken by the solver.

The output timesteps and relative tolerance settings

However, since you already know your maximum local mesh size and maximum frequency, it is more computationally efficient to manually set the timestep. The time step should resolve the wave equally well in time as the mesh does in space. Longer time steps will not make optimal use of the mesh, and any shorter time steps will lead to longer solution times with no considerable improvements to the results. The relationship between wavespeed, c, maximum mesh size, h, and time step length, Δt, is known as the CFL number:

CFL = cΔt/h

We have already manually defined the maximum mesh size to be 1/N of the local wavelength (as described above), so this can re-written in terms of the frequency f, and N the number of elements per local wavelength:

CFL = fNΔt

or, rearranging:

Δt = CFL/Nf

With the default second order, quadratic, mesh elements, the CFL number should be less that 0.2, and a value of 0.1 proves to be near optimal, it is helpful to define this CFL number as a Global Parameter as well.

Implement the manual timestep size in the Time-Dependent Solver settings, usually found at Study > Solver Configurations > Solution > Time-Dependent Solver, Time Stepping settings. (If these settings are not already in the Study branch, then right-click on Study and select Show Default Solver.) Set the Time Stepping Method: to Generalized alpha, set the Steps taken by solver: to Manual and set the Time Step to: `CFL/(N*f_max)`

The Generalized alpha solver is preferred for the reasons outlined in: Knowledge Base 1062: BDF, Generalized Alpha, and Runge-Kutta Methods

Manually setting the time step size

For an example of these mesh and solver settings, see the Gaussian Explosion in the Application Library of the Acoustics Module.

For an example of modeling a nonlinear material within the RF Module, see the Second Harmonic Generation of a Gaussian Beam. The same example is also available within the Wave Optics Module: Second Harmonic Generation of a Gaussian Beam

Note: If you are solving model using any Time Explicit formulation, such as the Pressure Acoustics, Time Explicit, Convected Wave Equation, Time Explicit or the Electromagnetic Waves, Time Explicit physics interfaces, this knowledgebase does not apply.