I usually apply the integration scheme on the global assembled matrices. However, on the local matrices, I would be able to parallelize better my code.
I think local time integration may create stability problem. Check local stability gives you global stability or not?
You can apply it locally, but you should not solve the local matrix system. There will not be any issues if you apply the time integration scheme locally but solve the system globally.
You can do this by multiplying the element matrices with the appropriate coefficients coming from the the time integration scheme. Once you form the equivalent element stiffness matrix (which consists of stiffness, damping and mass matrices in some linear combination), then you can assemble it to the global system in the regular fashion and solve the global system.
You may refer to my paper to know the details on forming the equivalent stiffness matrix.
Article On the advantages of using the first-order generalised-alpha...
R. C. Mittal Whether I apply the time integration scheme on the global matrices or on the local ones, the final system of equations to be solved is the same. I am more concerned with the efficiency (I should have made that more clear in my question).
Chennakesava Kadapa Thanks for the article. I agree there should be no problem applying the scheme locally. But what about the efficiency? Is it standard use to apply the scheme globally?
@ Michiel Fenaux , When you say " on the local matrices, I would be able to parallelize better my code", I understand that you are solving element wise separately in parallel. Stability has to be bothered about the total assembled system and not of individual ones.
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In my opinion the answer is mainly no, mass coupling makes it difficult to program a local integration scheme. As I read mr Kadapa's answer, this is what he points at when he says: 'you should not solve the local matrix system'.
However there is an option: lump the masses to the nodes, in that case (for linear models) it is possible to locally solve the matrix exuation and add up the resulting accelerations using you element assembly routine. Off course mass lumping reduces accuracy.
If you have a code that is already parallelised for the static problems, then I see no apparent reason for the parallel efficiency (for the assembly stage) to deteriorate for the dynamic problem if you apply the time integration scheme locally. (Note that whatever the additional cost incurred in applying time integration is the same for all the elements.)
However, if you apply it globally then you will have to store two global matrices, (three if you have damping), which is very inefficient for parallel codes. This why it is a standard practice to apply the time integration locally, element wise, before assembling into the global system.
Stephan D A Hannot I am afraid that your suggestion is valid only for discontinuous Galerkin methods. It is not applicable to standard FE schemes in which elements are continuous along edges/face, even when using explicit time integration schemes. You have to assemble the global lumped-mass matrix and force vector first, before solving for accelerations.
R. C. Mittal I referred to the parallelisation of the assembling process, I do not solve the system element-wise, sorry for the confusion.
Thanks to all for your answers!
Chennakesava Kadapa, I meant, if it is linear, the mass matrix is constant, thus you can determine that once, lump it. In that case with explicit schemes you can find algorithms where you acceleration vector is a linear combination that can be assembled. However, the algorithm should have acces to the complete lumped mass vector. So you have to assemble to global lumped mass matrix once as initialisation. I would not recommend this scheme though. I myself prefer an implicit newmark method on the global matrices.
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