Displacement control in time-adaptive non-linear finite-element analysis

Autor: Stefan Hartmann, Ahmad-Wahadj Hamkar, Karsten J. Quint
Rok vydání: 2008
Předmět:
Zdroj: ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik. 88:342-364
ISSN: 1521-4001
0044-2267
DOI: 10.1002/zamm.200800002
Popis: On the one hand, displacement controlled processes are frequently applied in Computational Mechanics using the finite-element method, on the other hand, there are some shortcomings regarding the study and the presentation of this particular case. In this article, we focus our theoretical considerations on quasi-static problems with constitutive equations of evolutionary type. In this case, it is currently known that the solution procedure of global and local iterations within the “Newton-Raphson method” is related to the method of lines, where one arrives at a system of differential-algebraic equations after the spatial discretization by means of the finite element method. This could be solved by means of the Backward-Euler method or more appropriately using time-adaptive, stiffly accurate, diagonally implicit Runge-Kutta methods in combination with the Multilevel-Newton method. In this respect the theoretical basis of currently applied implicit non-linear finite element analyses is extended to displacement controlled processes. In this article, the calculation of the reaction forces using the method of Lagrange multipliers and the penalty method are of special interest in view of the new DAE-approach. Accordingly, an important objective lies in a consistent notation so that the dependence of known and unknown variables as well as the transition from local to global level becomes obvious. These investigations are of utmost importance for micro-macro homogenization techniques in FE$^2$ approaches and the development of new global time and space-adaptive integration methods.
Databáze: OpenAIRE
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