The time step is the analogous in time of the mesh size, so dependence of the results on the choice of the time step is unavoidable. Rather than a time independent test, some kind of convergence test could be performed, in which at fixed spatial resolution the time step is reduced (say halved) repeatedly until the results obtained do not change significantly. Another option is to use an adaptive choice of the time step. Standard ODE solvers such as those implemented in MATLAB do that already, in PDE solvers this is far less common, but still feasible without great effort, especially if some kind of Runge Kutta time stepping method is employed, for which economical embedded error estimators and adaptation critera are available, see e.g. the review in the classical reference

https://www.springer.com/gp/book/9783540566700

Definitely timestep is a strong factor that can affect your numerical approximations. The smaller the size of the step, the accurate the results.

A proper selection of the time step is important, especially if you are using explicit methods and/or your system is nonlinear. Time-step adaptation is an additional option. However most FEA solvers use a constant time step.

Dear Luca Bonaventura ,

Thank you so much for your valuable contribution.

Best regards.

Dear Sir/Madam,

Nice question, congratulations!

Considering the many methods proposed for analysis in time via time integration, e.g., Newmark, RungeKutta . . . , the computed results are different from each other, and become united under the umbrella of convergence, i.e. regardless of the integration method, the results obtained from successful approximate time integration computations asymptotically approach to the exact result of the mathematical model (ordinary initial value problem) at the limit of zero time steps (when disregarding the round off). This implies the significant role of time step in the accuracy of the results computed by time integration. Since in analyses using fem and a time integration method, the problem is usually discretized in space by fem and then the resulting ordinary initial value problem is solved in time by an integration method, the role of the integration step does not change, compared to analysis of problems originally as ordinary initial value problem.

It seems also important to point out that for linear problems, the dependence of the results accuracy to the integration time step is mainly according to the order of accuracy of the integration method (generally equal to two), i.e. for sufficiently small steps the log of error changes with respect to the log of time step with a slope equal to the order of accuracy. For nonlinear problems the case is practically different and the above mentioned dependence generally happens for much smaller steps depending mainly on the nonlinearity iterative method and the tolerance.

Methods and techniques for eliminating or reducing the above difference between linear and nonlinear analyses are proposed partly by myself.

With many thanks for your kind attention, have a nice day evening and future.

Dear Ifeanyi Nwankwo ,

Thank you for your reply.

I agree with you. However, in some of my cases, when I used the smaller time steps, the results not always more accurate.

Any way, thank you very much.

Regards.

Dear Nils Wagner ,

Thank you for your reply.

Yes, I agree. That's my problem, how to find the best time step.

Kind regards.

Dear Aram Soroushian ,

Thank you so much for your useful contribution.

Is there any simpler methods to find the best time step?

Best regards,

Nazar

Nazaruddin Sinaga You might be interested in

Article A priori local error estimation with adaptive time‐stepping

Dear Nazaruddin Sinaga and all,

In fact, the best time step seems to imply the largest step leading to sufficient accuracy for the results (this is slightly different for nonlinear problems). In practice, the appropriate integration step is being determined specifically for each problem, gradually through repetitive analyses, starting from larger steps. This approach has both theoretical basis in mathematics (see (Hairer and Wagner 1996)), and structural dynamics (see (Clough andd Penzien 1993)), and also has influenced the practice; see the seismic code of NewZealand (NZS1170, 2004). I have addressed these in different publications mostly available at Researchgate, e.g. Soroushian, A., 2017. Integration step size and its adequate selection in analysis of structural systems against earthquakes. In Computational methods in earthquake engineering (pp. 285-328). Springer, Cham.

Many sincere thanks for your kind attention. Have a very nice and healthy day tomorrow and all future.

PS: By the way, I forgot to mention that "best " (or even "appropriate") time step seems also depending on other algorithmic parameters of the analysis, e.g. nonlinearity iteration method, nonlinearity tolerance, and even the discretization in space; sorry. Have a nice and healthy moments ahead.

Have you tried the most basic ones? That is, The Euler and Backward Euler methods? https://en.wikipedia.org/wiki/Euler_method

Dear Nils Wagner ,

Thank you very much for sharing the reference.

Have a nice day...

Thanks dear Andreas Almqvist for your information.

Best regards.

Nazar

Dear Aram Soroushian ,

Thank you so much for your valuable contributions.

Do you agree if, for transient simulation, the result must be "time-step independence", which is similar with the concept of grid-independence?

Please give your comment.

Thank you.

Have a nice day....

Dear Nazaruddin Sinaga and all,

I agree with the concept of time-step independence and its importance for accuracy; it is even addressed in practice, e.g. see the seismic code of NewZealand. However, it seems to me that for nonlinear problems even when both independence are provided, with further changes of the sizes of grids or time steps more accurate results may be available. In other words, for nonlinear problems, the computed responses might be unchanged for some special value of time step and grid size that does not correspond to good accuracy. Separate examples exist in the literature, e.g. my papers in 2005 and 2013. These two independence seem to me necessary and not sufficient requirements for the good accuracy. Maybe they can be completed towards sufficiency, with attention to concepts such as "proper convergence".

https://www.comsol.com/blogs/automatic-time-step-and-order-selection-in-time-dependent-problems/

https://www.sciencedirect.com/topics/engineering/maximum-time-step

https://www.researchgate.net/topic/Numerical-Methods

https://www.flow3d.com/resources/cfd-101/numerical-issues/implicit-versus-explicit-numerical-methods/