Hello all,
I am trying to solve for an undamped second-order differential equation (Mü + Ku = F
) by extracting the mass matrix (M), stiffness matrix (K), and transient load matrix (F) from COMSOL Multiphysics, and subsequently solving them in MATLAB.
Newmark's beta method is chosen to solve the system. When I use the constants corresponding for an unconditionally stable scheme (beta = 0.25 and gamma = 0.5), the method is producing the solution which is approximately 10 times amplified to that of the actual solution (obtained from COMSOL). However, when I tweak the parameter alpha to 100, it is resulting in the correct solution.
I could not explain the weird value of alpha = 100. Could someone help me out with this?
Best regards,
Nanda Kishore
https://en.wikipedia.org/wiki/Newmark-beta_method
Can you share the matrices and the load in an appropriate format (e.g. .mat)?
What are the initial conditions?
Dear Nils Wagner,
thank you for reaching out.
Kindly find the system matrices and the description of the problem (model, ICs, and BCs) attached.
Please let me know if any further information is required.
Best regards,
Nanda
Can you also share the underlying finite element model in an appropriate format?
AFAIK, Nastran (.bdf) is supported as an export format in Comsol.
I recommend testing your implementation in MATLAB using the 1 DOF spring-mass-damper system. Once you test your implementation, you can look for potential issues with the large-scale problem.
Dear Nils Wagner,
COMSOL is supporting .mph, .m (matlab), and .java as an export format. I am attaching the COMSOL model, as well as the matlab file. Kindly find them. Unfortunately, there was no option to export the model into .bdf file.
Best,
Nanda
Chennakesava Kadapa sure. Nevertheless, I did it with 2D single element model. It was the same still.
Dear all,
Nice question,
It seems to me that when the analysis method is an approximate one, it is reasonable to have the chance to set the parameters of the method (Gamma and beta for the Newmark method) such that the deviation of the result compared to an "exact" or any other result disappears (for some integration step). This is however unimportant, because in practice, there is no basis for comparison, and the computed results of time integration should be numerically stable and consistent, such that you can count on convergence of the computed result to the exact result. This explains why for approximate time integration methods the order of accuracy which implies the results' rate of convergence to the exact result is a main characteristic. Besides, the size of error implying instability is different in different problems. Accordingly, it seems to me neither what is observed using gamma=0.5 and beta=0.25 is weird, nor what observed when using gamma=0.5 and beta=100. Furthermore, if the analysis using gamma=0.5 and beta=0.25 is sequentially repeated with smaller integration steps, the computed result would likely approach the exact result with second order of accuracy.
With many thanks for your kind attention, have a nice and healthy evening and future.
Dear all,
I think it is allowable if the sum of the coefficients of time derivative satisfy mathematical constraints. For example, (1/2-beta) plus beta should always be 1/2, which is typically satisfied in the modified Taylor expansion.
I strongly believe that it is possible to use beta / 100, but accuracy may decrease. On the other hand, strong numerical dissipation may be achieved.
Thanks.
Dear Aram Soroushian and Wooram Kim thank you very much for your time and effort in the detailed answer.
Wish you a very nice day.
Best,
Nanda
An analysis that links the value of beta and other coefficients to the amount of damping that is wished for the highest frequency modes has been presented here
Article The analysis of the Generalized -Α method for non-linear dyn...
An explanation why the Newmark-based methods are not the optimal ones for structural mechanics applications has instead been presented here
Article The TR-BDF2 method for second order problems in structural mechanics
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