Stability analysis of time integration methods in nonlinear differential equations.
Solutions obtained using Newmark's linear acceleration method blow up if the time step is greater than critical time step. We know what the critical time step is. But this critical time is valid if the differential equation is linear. Right? What if the differential equation is not linear? Can the critical time step be found anyway?
You have quite a lot of bibliography on the subject. The main problem being that you will not have the same behavior as function of the non-linearity, and developments towards an exact value become tedious of impossible.
You can check for example the paper of Hughes
Hughes, Thomas JR. "Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics." Computers & Structures 6.4 (1976): 313-324.
The stability properties are indeed established for linear or slightly damped systems, so that the scheme evolution matrix is applied to a single mode system (or mass spring).
With non-linearities, the local dynamics of your system can drastically change. You will get divergence if the dynamics get to higher frequencies during your simulation, for example with contact impact, you do not have the same dynamics whether contact is active or not.
To get convergence, your time step has to be small enough so that convergence is ensured throughout your simulation. You thus have to estimate the minimum time step required by estimating the vibration level you will get, or employ an estimator that will be able to reduce your time step during the simulation if the convergence property varies.
You also have a familiy of modified Newmark schemes that allows handling non-linearities, by modifying the Jacobian formulations to ease-up convergence. This requires numeric parameters to optimize.
I hope this helps,
Mr. Roches,
It is realy helpful and if I find I will examine the paper. Thank you for your response.
The stability analysis can be performed by the von Neumann method where the iterative solution is obtained with an amplification factor : this one has to be bounded (
Group analysis of differential equations is one of the systematic approach which can be used for analysis of both ODEs and PDEs whether are linear or nonlinear moreover it is valid both for scalar and systems. For this, one can look on any standard book of this method. Further a larger number of criteria are developed for particular classes of ODEs and PDEs: both scalar and systems.
It seems like you want to extend the stability requirements for linear problems to nonlinear problems. I want to look at the problem from another perspective. Let's assume that a nonlinear restoring force is the source of nonlinearity for the given ODE. Such nonlinear restoring force can be divided into several linear segments. Now, all we have to do is to solve several linear problems. In other words, we can find the solution for each linear segment and then join the solutions for each segment to come up with the whole nonlinear solution. Therefore, the stability of the nonlinear problem will be not only a function of the critical time step for the linear segment, but also resolution of the divided segments.
I recommend you also to take a look at Anil K. Chopra's book:
"Dynamics of structures, theory and applications to earthquake engineering"
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