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On the stability of explicit methods for the numerical integration of the equations of motion in finite element methods
Author(s) -
Fu C. C.
Publication year - 1972
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620040111
Subject(s) - finite element method , mathematics , stability (learning theory) , upper and lower bounds , simple (philosophy) , numerical integration , equations of motion , numerical analysis , mathematical analysis , moment (physics) , computer science , classical mechanics , physics , philosophy , epistemology , machine learning , thermodynamics
This paper deals with the stability of the numerical solutions of a dynamic finite element analysis. The solutions are obtained through a stepwise integration of the equations of motion. Upper bounds on the steplength of the integration are obtained from a stability analysis of using a simple finite difference approximation for the equations of motion, and are shown to depend strongly on the particular element is use and on how the mass of the element is distributed at its nodes. As an example, the two‐dimensional wave propagation in a semi‐infinite plate subjected to a suddenly applied moment along its edge is studies. Through the example, we show that the bound on the steplength, obtained from the simple analysis, can provide a useful guide on choosing the steplength in other higher order integration methods. In particular, we show that, for stability considerations, the upper bound on the steplength should also hold for a fourth order explicit method. In order to achieve an acceptable accuracy of the solution, we show that the steplength should be approximately one half of the bound for the higher order explicit method as well as a higher order implicit method. Solution of the example has been compared with that of the Timoshenko theory.