I am interested to know whether there exists any general rule to explain the effect of physical damping on the order of accuracy of arbitrary time integration method. In further detail, if one plans to check the consistency and order of accuracy of an arbitrary time integration method numerically (without discussing the truncation error) can he or she reduce the models to be studied by considering a special value of damping? Thank you very much for your kind attention, and sorry if this is not a good question.
time integration methods are developed to solve ode/pde. therefore the accuracy depends upon the choice of method and integration increment size. with damping, the structural frequency increases (i.e. natural period decreases). as a rule of thumb, your increment size should be at least 1/20th of smallest natural period of interest in the response.
Dear Abhimanyu Kumar
Many thanks for your kind attention and response. However order of accuracy is different from accuracy in that order of accuracy is the slope of the line when expressing errors with respect to the integration step size for sufficiently small steps and in a log-log diagram; for consistent time integration, order of accuracy is to be a positive integer equal or larger than one; recommended to be at least two. An appropriate reference for the order of accuracy (in very close relation to rate of convergence) is
Belytschko, T., and Hughes, T. J. R., 1983, Computational Methods for Transient Analysis, Elsevier, The Netherlands.
Therefore, your response is somehow irrelevant to my question, though a very interesting issue especially while the smallest natural period is sometimes very small. Much better comments compared to Tmin/20 exist about which we would rather discuss elsewhere. Again thank you very much.
As you mentined, the order of accuracy is usually measured numerically by the slope of the line plotted in a log-log figure representing the numerical error with respect to the time step size. However, it is very important to notice that the critical situation in which the order of accuracy could be measured is when the physical damping is zero. In other words, presence of physical damping usually helps the order of accuracy.
As for the numerical damping, this is completely different. For the sake of inducing numerical damping in most of the time integration methods it is common to change the original value of the taylor series constant parameter; this change sometimes ends up in lowering the order of accuracy by one.
Dear Saeed,
Many thanks for your kind interest in the question above.
As stated in the explanation in the text of the question (and in few minutes also in the title of the question) the discussion is on physical damping so concentrating on the first paragraph of your explanation, I am looking for a general rule valid for the influence of physical damping on the consistency of arbitrary time integration method. Therefore, if you have seen some reference in this regard or have any mathematical proof for your claim, as a general rule for arbitrary integration method, I would be very thankful to hear about them from you. Have a nice day.
I believe that it depends on time integration methods. For example, stability and order of accuracy of some fully explicit methods are decreased in the presence of viscous damping. These situations are often observed in the central difference method with the explicit treatment of velocity, the Noh and Bathe method, the Kim and Lee method, and the explicit methods used in the LS-DYNA and Abaqus. Please see Kim(2019, IJNME) for related discussions. Thank you.
Dear Wooram,
Hello and many thanks for your kind and detailed guidance, and moreover for introducing some papers to me (about one of them I will write to you later).
In view of a similar research on numerical stability reported in
"A General Rule for the Influence of Physical Damping on the Numerical Stability of Time Integration Analysis." Journal of Applied and Computational Mechanics 4 (2018): 467-481,
and the Lax-Richtmyer equivalence theorem I conceptually agree with you. However, still the general rule to explain the influence of viscous damping on numerical consistency should be found.
Again many sincere thanks for your kind attention interest guidance and papers.
With Kind Regards,
Aram
Dear Aram,
I recommend that you refer to Chapter 22 of the following textbook for further information (this book has good mathematical proofs):
Paultre, P. (2013). Dynamics of structures. John Wiley & Sons.
Dear Mohammad,
Many thanks for your kind attention and guidance. I had checked this book before, and had other brief look to it these days. I could not find anything new regarding the question there (the presented material on consistency seems almost the same as that stated in the book of Belytschko and Hughes (1983)). Kindly please write me if you have noticed something special to moves us closer to a general rule for the effect of damping on consistency. Again many thanks.
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