ICAMS / Interdisciplinary Centre for Advanced Materials Simulation

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Numerical calculation of thermo-mechanical problems at large strains based on complex step derivative approximation of tangent stiffness matrices

D. Balzani, A. Gandhi, M. Tanaka, J. Schröder.

Computational Mechanics, 55, 861-871, (2015)

Abstract
In this paper a robust approximation scheme for the numerical calculation of tangent stiffness matrices is presented in the context of nonlinear thermo-mechanical finite element problems and its performance is analyzed. The scheme extends the approach proposed in Kim et al. (Comput Methods Appl Mech Eng 200:403–413, 2011) and Tanaka et al. (Comput Methods Appl Mech Eng 269:454–470, 2014 and bases on applying the complex-step-derivative approximation to the linearizations of the weak forms of the balance of linear momentum and the balance of energy. By incorporating consistent perturbations along the imaginary axis to the displacement as well as thermal degrees of freedom, we demonstrate that numerical tangent stiffness matrices can be obtained with accuracy up to computer precision leading to quadratically converging schemes. The main advantage of this approach is that contrary to the classical forward difference scheme no round-off errors due to floating-point arithmetics exist within the calculation of the tangent stiffness. This enables arbitrarily small perturbation values and therefore leads to robust schemes even when choosing small values. An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of thermo-elastic and thermo-elastoplastic boundary value problems at finite strains the performance of the proposed approach is analyzed. © 2015, Springer-Verlag Berlin Heidelberg.


Keyword(s): finite element method, nonlinear thermo-mechanics, numerical differentiation, quadratic convergence, complex numbers, CSDA approach
Cite as: https://www.scopus.com/inward/record.uri?eid=2-s2.0-84937817954&doi=10.1007%2fs00466-015-1139-0&partnerID=40&md5=92e6947c3f04a861daff1fb680cc0410
DOI: 10.1007/s00466-015-1139-0
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