Welcome to this Ansys video.
Let’s begin by discussing a problem that looks simple enough, but seems to be giving strange results.
Here we have a long plate that is fixed at one end and is subjected to an impact load on the other end.
Since we have an impact load, we solve the problem using an explicit method.
Intuitively, we expect the plate to vibrate with a decreasing amplitude.
If we consider a case where there’s no damping due to air friction, we expect it to vibrate with a constant amplitude.
Why does this plate exhibit unphysical behavior then?
Did we make a mistake in the boundary conditions?
Or did we define material properties incorrectly?
When using an explicit method, you may have come across cases where you set up the model correctly, but somehow, the results don’t seem to agree with what you would expect intuitively.
The reason in such cases may be the conditional stability of explicit methods.
Unlike implicit methods, explicit time integration methods are conditionally stable, meaning that they are stable only when the time step size is less than a certain limit.
For this example, we probably used a time step that was too large, causing the solution to be unstable.
In this video, we will learn how to calculate the upper limit on the time step size in an explicit analysis, and ways to increase this upper limit to reduce the solve time.
Let’s get started!
Let us first talk about the limit on the time step size.
To enforce stability in explicit solutions, it is necessary that the time step is less than the critical time step.
This limit on allowable time step size is the primary reason explicit methods are not used to simulate phenomena occurring over longer timescales of minutes, hours or days.
The critical time step size is proportional to the time taken by a stress wave to propagate through the length of the smallest element in the model.
If delta ‘t’ represents the time step size and delta ‘t critical’ represents the critical time step size, then the equation here represents the condition for a stable explicit solution.
Hence the critical time step size is expressed in terms of f, L, and c,
where ‘L’ is the characteristic length of the smallest element in the mesh,
‘c’ is the speed of sound in the material being analyzed,
and ‘f’ is a scale factor usually equal to or less than 1.
This condition for stability is known as the Courant-Friedrichs-Lewy or the CFL condition.
So how does the mesh affect the critical time step?
Element type and size determine the characteristic length.
For 1D beam elements, the length of the beam element along its axis defines the characteristic length.
For 2D shell elements, the characteristic length may be calculated in different ways depending upon the software in use.
It may be calculated as the ratio of the area of the element to the largest element edge length or the largest diagonal length,
or it may be calculated as the square root of the area.
For 3D solid elements, the characteristic length is the ratio of the volume of the element to the largest element face area.
Essentially, the smaller the element size, the smaller the timestep needed to prevent a stress wave from completely passing through the element.
It’s important to note that characteristic length changes when an element deforms under applied loads.
For example, under tensile or compressive loading the element size will increase or decrease respectively.
Hence, the characteristic length and, in turn, the critical time step size may change in each loading cycle.
Next, let’s discuss how the critical time step is dependent on the material properties.
If we look at the expression for the critical time step, we see that it is inversely proportional to the speed at which sound waves travel in a material.
The speed of sound in beam, shell and solid elements can be expressed in terms of Young’s modulus, Poisson’s ratio and density as indicated here.
In general, the critical time step will be smaller for materials that are very stiff and lightweight and larger for compliant, heavy materials.
Now, let’s look at a simple simulation example of a flat steel plate impacting the ground.
Here, we model the ground plane as a rigid body and hence, it does not affect the critical time step.
The steel plate is meshed non-uniformly with rectangular shell elements of varying sizes.
The area and the longest side of the smallest element are used to calculate the characteristic length for this mesh.
Using the material properties of Steel, we calculate the speed of sound c in Steel and finally, we calculate the critical time step.
Here is a plot showing the CFL time step at different locations of the plate.
An important point to remember is that the minimum value determines the critical time step for the entire model.
For a given simulation with an end time t, the solver needs to solve the FEM equations for a certain number of cycles, which is given by t divided by the time step size.
Now what if we wanted to reduce the time required to solve this problem?
We would need to decrease the number of cycles for the problem, which can be achieved by increasing the critical time step.
One way to do that is to increase the characteristic length which is determined by the smallest element in the entire mesh.
Hence, in explicit dynamics, it is important to use as uniform a mesh as possible.
As we see from the CFL time step plot, the overall critical time step for the entire model is driven by a few small elements.
Using a uniform mesh will ensure that the CFL time step is almost the same throughout.
In our example, remeshing the plate uniformly tends to increase the CFL time step for the overall model.
If need be, we may have to use a coarser mesh, to increase the critical time step.
But we may not always be able to control the characteristic length in a model, especially when the geometry is complex.
Consider this example of crushing of a can.
In this case, we see that using a coarser mesh does not lead to any significant increase in the critical time step.
The reason for this is that critical time step for this geometry is controlled by the elements at the rim of the can
and using a coarser mesh does not correspondingly increase the characteristic length of these rim elements.
So what do we do in such cases?
Are we stuck with an inefficient, slow simulation that may take days to run?
No!
In such cases, we turn our attention to the other term controlling the critical time step.
We can increase the critical time step by increasing the mass of the part or model.
More specifically, if we selectively increase the mass of only those elements with the smallest CFL timestep, we can effectively increase the overall timestep of the solution.
This is called mass scaling.
Note that not all elements are affected by mass-scaling.
Since only a small portion of the elements is subjected to mass scaling, the global accuracy of the analysis should not be affected.
This is an important point to keep in mind while using mass-scaling.
If the mass of too many elements is scaled, the global inertia changes and this in turn changes the physics of the problem.
Thus, abusing mass-scaling can lead to erroneous results.
So how do we ensure that mass-scaling does not change the physics of the problem?
We know that explicit solutions do not perform equilibrium iterations to enforce balance in the system.
Instead, they rely on energy balance in the system.
So the first check would be to look at the energy summary and ensure that the initial kinetic energy in the simulation is what is expected theoretically.
If too much artificial mass has been added to the system, the initial kinetic energy in the simulation will be significantly higher than the expected value.
The second check would be to ensure that the mass added per part is a small percentage of the actual mass of the part.
A good rule of thumb is to limit the added mass to less than 5% of the actual mass of the part.
A third way to ensure the accuracy of our results would be to solve the problem again using a reduced time step.
If the results don’t change significantly, we can safely conclude that mass-scaling is not changing the physics of the problem.
Let us use mass-scaling to optimize the runtime of the can crushing problem.
Let’s try setting the target time step to 3e-7 seconds, or twice the time required without mass-scaling.
Hence the number of cycles required to reach the end time is halved.
When we solve this problem, first we check the initial kinetic energy in the Energy summary.
In this case however, only the rigid indenter has an initial non-zero velocity and since rigid bodies do not affect the critical time step, it does not undergo any mass scaling.
Hence the initial kinetic energy is not a good indicator of the effects of mass-scaling in this case.
Next, we check the mass added for the can.
We see that it is less than 5% of the actual mass of the can.
Make sure that you compare the added mass to the mass of the can and not to the mass of the entire system.
We can also look at the number of elements affected by mass-scaling.
This plot shows the added mass for all the elements.
We see that very few elements have a non-zero value.
So we can conclude that this is a reasonable target time step to use.
Indeed, the results obtained in this simulation are quite close to those obtained without mass-scaling.
Thus, we have achieved a reasonable solve time without compromising on the accuracy of the problem.
Now let’s summarize the important takeaways.
When performing simulations using explicit time integration techniques, we are interested in balancing numerical accuracy with computational efficiency.
While a fine mesh in the region of interest may give us a high resolution of the stress field, it may require very long solution times.
This is because the overall solution time is directly related to the critical time step size, which is influenced by element size and material properties.
Thus, we want to manage the critical time step size, either by ensuring that there are no small elements that drive the critical timestep size down,
or we can use mass scaling that selectively modifies the density to raise the critical timestep size.
I hope you have found this video informative.
Thank you for watching and do check out our other courses to discover more useful learning resources.