Elastoviscoplastic Finite Element analysis in 100 lines of Matlab

J. Numer. Math., Vol. 10, No. 3, pp. 157–192 (2002) c VSP 2002 Elastoviscoplastic Finite Element analysis in 100 lines of Matlab C. Carstensen and R. Klose † Received 3 July, 2002 Abstract — This paper provides a short Matlab implementation with documentation of the P 1 finite element method for the numerical solution of viscoplastic and elastoplastic evolution problems in 2D and 3D for von-Mises yield functions and Prandtl-Reuß flow rules. The material behaviour includes perfect plasticity as well as isotropic and kinematic hardening with or without a viscoplastic penali- sation in a dual model, i.e. with displacements and the stresses as the main variables. The numerical realisation, however, eliminates the internal variables and becomes displacement-oriented in the end. Any adaption from the given three time-depending examples to more complex applications can easily be performed because of the shortness of the program and the given documentation. In the numerical 2D and 3D examples an efficient error estimator is realized to monitor the stress error. Keywords: finite element method, viscoplasticity, elastoplasticity, Matlab 1. INTRODUCTION Elastoplastic time-evolution problems usually require universal, complex, commercial computer codes running on workstations or even super computers [11,15]. The argument of keeping commercial secrets hidden inside a black-box, leaves the typ- ical user without any idea what exactly is going on behind the program’s user- friendly surface. The difference between a mathematical and a numerical model is fuzzy. Often, users do not care about nasty details such as quadrature rules or the exact material laws realised. But sometimes it does matter whether a regu- larised or penalised discrete model is solved, how the termination of an iterative process is steered, and what post-processing led to both, equilibrium and admissi- bility of the approximate stress field. In particular, if the solution process is part of guaranteed error control, it is quite important to know if discrete equilibrium is fulfilled exactly or not. Finally, the use of more than one black-box in a chain is likely to be less efficient. In summary, to aim efficiency (e.g. by well-adapting au- tomatic mesh-refinements) and reliability we have to prevent, at least confess, all Institute for Applied Mathematics and Numerical Analysis, Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria † Mathematisches Seminar, Christian-Albrechts-Universit¨ at zu Kiel, Ludewig-Meyn-Str. 4, D- 24098 Kiel, Germany

Elastoviscoplastic Finite Element analysis in 100 lines of Matlab

Documents

finite element method

viscoplasticity

elastoplasticity

matlab