wave propagation
Last updated
Last updated
In this example, we demonstrate an example of wave propagation in a 2D solid.
Essentially, it is a 2D dynamics problem. The element is used to model the solid. Various time integration methods will be used to compare their performance regarding numerical energy dispersion.
The model script can be downloaded .
This model contains 16641 nodes and 16384 elements. The memory usage is about 1.2 GB.
Due to small step size and need to export visualization data, the full analysis takes around 20 minutes to complete on an average PC platform (6 physical cores @ 3 GHz).
A square solid of size is used. The structured mesh can be generated by using whatever mesh generator available. It is not difficult to generate an array of squares using scripting languages such as Python or Matlab. Here, is used.
The left boundary is constrained along the horizontal direction. The bottom boundary is constrained along the vertical direction. The top and right boundaries are free. An initial velocity is assigned to the node at the centre of the top boundary.
Whether plane stress or plane strain assumption is adopted is not the focus of this example, we simply use a plane stress element with a unit thickness.
For visualisation, we define a recorder. We record von Mises stress to represent the propagation of stress field.
We use both implicit and explicit time integration methods.
An implicit integrator shall be used with an implicit step.
Most explicit methods use acceleration as the primary variable, the equations of motion are often expressed as a function of acceleration. This differs from implicit methods that often use displacement as the primary variable. In order to adopt such a difference, one needs to define an ExplicitDynamic step, similar to the setting in ABAQUS.
Please be aware that most displacement-based constraints cannot be used in explicit analysis.
Also, since the system is linear, the global stiffness/mass matrix does not change once assembled. It is possible to indicate the solver to skip iterations.
For a not-so-rigorous comparison, different spectral radii are used for different methods, mainly for the purpose of showcasing different methods.
Also, the chosen model parameters are quite arbitrary. Sufficiently accurate results often require an accurate estimation of the highest frequency of the model, which governs the time step size.
The chosen time step size is 0.01 s for all three cases. The following radii are used:
Bathe Implicit: 0.9
GSSSS Optimal: 0.8
Newmark: 1.0
The explicit methods show better numerical dispersion.
The chosen time step size is 0.001 s. The following radii are used:
Tchamwa: 0.9
Tchamwa: 0.6
Bathe Explicit: 0.9
The boundaries can be extracted by node groups.
Then BCs can be applied via .
The initial condition can be applied using command.
The implicit time integration methods are the default. If no integrator is defined, a default integrator will be used.
Since we are using a linear elastic material with the elements, the elemental stiffness is symmetric. As there are no other non-linear constraints defined in the model, the global stiffness/mass matrix is also symmetric. It is possible to then turn on symmetric banded storage to save memory space.
The two-step method appears to have the best numerical dispersion performance among implicit methods.
The optimal scheme is also fine if the spectral radius is chosen properly.
The second-order, unconditionally stable method has significant high-frequency noise. This explains why it is mainly used for structural dynamics in which the low-frequency response is of interest.
The method is first-order accurate, and does not require corrector.
The two-step explicit method, as discussed in the original reference, shows superior performance. However, it requires a corrector step, which requires an additional element-wise computation for each substep. This increases the computational cost.
