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Energy conservation in nonlinear finite element explicit dynamics

Robin Selinger's picture

I am writing to ask about the state of the art in finite element simulation using nonlinear elasticity and explicit dynamics.

Consider, for instance, a 3-d simulation of a hyperelastic beam that's fixed on one end, then  twisted about its long axis by 360 degrees and released. If we apply no friction or viscosity, the sum of kinetic plus potential energy should remain constant as the material springs back and oscillates.

Which FEM codes do the best at conserving KE+PE for a simulation of this type?  If you drop the time step, can you get energy conservation to many significant figures, as one can with, for instance, molecular dynamics simulation? 

I'm curious because I've recently written my own 3-d nonlinear explicit dynamics code that provides very high precision energy conservation, and I'm wondering if it's any better or worse than the nonlinear explicit dynamics codes already available.

Thanks for any information you can offer. If you'd like to discuss privately, please drop me a line at

Many thanks!

 -Robin Selinger,  Professor, Chemical Physics Interdisciplinary Program, Liquid Crystal Institute, Kent State University



You will find many papers on this topic over the past decade or so in the finite element literature. My colleague Tod Laursen at Duke University has written a few papers on the topic of "energy consistency". It turns out that most of the standard time integrators do not conserve energy very well, and this can actually trigger numerical instabilities at some point.

You may also wish to look at the recent literature on asynchronous variational integrators by Lew, Marsden, and Ortiz. These are designed, in part, with discrete energy conservation in mind. Here are some references to start:

Laursen TA, Meng XN, A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elastodynamics, COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, v. 190 (46-47): pp. 6309-6322, 2001

Lew A, Marsden J, Ortiz M, & West M., "Variational Time Integrators," INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, v. 60: pp. 153-212, 2004


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