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Asynchronous Variational Integrators suitable for static loading situations?
Hello everybody,
I am planning to use AVI for typical elasticity problems. I already have an implementation for 2D plane stress elements running and I am quite satisfied except for one issue. Here's the test case: a square plate is scaled (dilated) isotropically by say 1%. The boundary conditions are uniformly set to clamped and different time steps are chosen for each element (always respecting the CFL-criterion). If I now run the simulation everything first looks fine until I zoom on an internal node. This node is actually not at rest but oscillates back and forth although - correct me if I am wrong - no motion should occur.
I think the reason for this lies in the very formulation of the AVI. Although the body is in static equilibrium, impulses are exerted by the elements because the elemental gradients of the elemental elastic potentials are not zero although summing up all elemental contributions for one node (as in classical FE) would yield a zero gradient. Visually, one element first (mistakenly) assigns non-zero velocities to its nodes, then a neighbouring elements moves its nodes and corrects their velocites accordingly leading to the above mentioned oscillation.
Is it actually impossible to reach a real static equilibrium state with AVI?
Looking forward to your comments,
Bernhard
Hi Bernhard : Although I
Hi Bernhard :
Although I have not tried a 2D problem with AVI but I tried to solve simple 1D spring mass problem using AVI . And the result matched with the actual solution .
Can you post your results , I am very interested to see how AVI behaves for 2 D problem .
Eventually I am intersted to apply AVI to a multi scale multi time problem .Right now I am working on just multi space ....but after somtime will dive into multi time problem too.
Kapil Nandwana
Is Asynchronous Variational Integrator unstable?
I implemented a 1-d bar wave propagation code to test out AVI. The bar, from left to right, is discretized using 30 elements, the first 20 elements at 0.5 units each, and the last 10 at 1.0 units each. The left (free) tip is subjected to a force (exact solution is available), and the right end is fixed. All elements are set to march at CFL = 1.0. The stress time history of each element is plotted.
One problem is that there's a lot of noise in the solution, perhaps due to the discontinuous mesh. The other problem is that I am getting an unstable solution when CFL condition for each element is 1.0. If the CFL condition is less than 1.0, say 0.99, the instability grows slowly.
When I used a uniform mesh with 40 elements at 0.5 units each (this is quqivalent to the standard central difference method), there's no instability when the CFL = 1.0. The noise goes away as well. In fact, for this case the solution is very close to the exact one.
My questions are:
1) Is the AVI unstable for certain conditions?
2) Can you march at exactly CFL = 1.0 when you have mesh of different sizes and elements with different material properties?
3) Is there any way to reduce the noise of the solution?
Thanks.
Stability of AVI
I have studied the stability of AVI for 1-D harmonic oscillators and 2-D harmonic lattices and the results of this study also apply for finite element problems.
In response to your questions:
1) Yes AVI is unstable for certain conditions. For a mesh with two different element sizes, if the time step used for the large elements is an integer multiple of the half-period of the normal modes corresponding to the small elements, resonant behavior will be observed. Since the larger elements are exactly twice as large as the smaller elements in your example, the condition for resonance is satisified when CFL = 1.0 for the large elements. In addition the resonance phenomena occurs for an interval of time steps surrounding the integer multiples of the half-period. However the strength of the instability decreases as the time step for the large elements is farther away from these integer multiples. This is seen in your CFL = 0.99 simulation. It should be noted that the choice of CFL for the small elements does not matter as long it is less than or equal to 1.0.
2) Yes it is possible to march at CFL = 1.0 for meshes with different element sizes. For example you can try 20 elements at 0.5 units each and 9 elements at 10/9 units each. AVI also works for elements with different material properties. The resonant properties can also be extended as follows. If you have stiff elements and soft elements that are 25% as stiff (both are the same size) then when the soft elements are marched at CFL = 1.0 instability should be observed. However the instability will disappear if the soft elements are 20% as stiff.
3) It is possible that the noise is a manifestation of your observed instability. See if the noise is still present for a stable AVI simulation.
Let me know if you any other questions or if my suggestions don't pan out.
Stability of AVI
Hi Will,
Thank you for your reply. I have since looked at your paper with your advisor Lew, and Darve on the stability of AVIs. I will try what you suggested but in the meantime, however, these are some additional comments and questions:
1. From what you have mentioned, it appears as if one needs to design a mesh to make AVI stable. I am sure that you would agree that the mesh will generally be unstructured, coming from, say, an automatic mesh generator.
2. You mentioned in the stability paper that you are working on AVIs for MD in the context of Langevin dynamics, which provides viscosity/damping to cure the stability problem. Also, you mentioned about the mollified impulse (MOLLY) technique for stabilization. I have yet to study these in detail but I would like to know whether you think these techniques are transferable to finite element models. For example, there is this aspect of stochasticity of MOLLY which seems rather odd for a standard finite element model.
3. While there is a lot of theoretical development of AVIs by Lew, West, Ortiz, Masden, etc. the stability aspect of AVI appears to be not well developed. The numerical simulations were done on fairly complex problems with unstructured meshes. Yet, instabilities are not reported. Is it possible that some of these instabilities were damped by some dissipative mechanism that is inherent in the model, for example, artificial viscosity. Also, the courant number for many of these problems are not reported. Also, is there an explanation for the initial jerkiness for one of the helicopter blade simulations?
I am interested in applying AVI to a standard elastic wave propagation (without damping) problem on a general unstructured mesh. The asychronous nature of AVI seems great but my concern is that AVI may not be robust enough because of these instabilities. I will appreciate your thoughts and clarifications on some of these issues. I apologize in advance if there are any inaccuracies in my comments as I have just started studying AVI and MD.
Thank you in advance.
Stability of AVI for Finite Element Problems
We have a new paper on the stability of AVI that will be published in J. Comp. Phys. soon. A version of it is now available online. This paper discusses in detail why resonances are observed in MD but not in FE applications. I will summarize the results below and address your questions.
1. It turns out that AVI resonances will occur in linear elasticity problems if the element sizes do not vary smoothly across the mesh as in your discretization of the rod. For example if you have one large element surrounded by much smaller elements, instabilities will be observed for certain combinations of time steps. However if you first surround this large element with medium-sized elements (which in turn are surrounded by small elements), the strength of the resonances decreases very rapidly. Essentially the propagation of the resonance is damped by elements of similar size. With automatic mesh generation there is usually a smooth gradient of element sizes hence for most problems the resonances are still present but are too weak to manifest themselves.
2. Since the appearance of resonances in FE is extremely rare no viscocity or damping is needed.
3. The lack of observable resonances is the primary reason why stability is not a major topic of discussion in the papers you cited. A CFL of 0.1 was used in the simulations you mentioned. Unfortunately I cannot provide any insight on the initial jerkiness of the helicopter blade simulations.
Based on what I have described, it is my opinion that for linear elasticity AVI is a robust method for time integration.
I hope that your 1-D simulations are working now. Let me know if you have any additional questions or concerns.