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

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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.

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Acknowledgments

Financial funding by the DFG German Priority Program SPP 1648 (SPPEXA “Software for Exascale Computing”), projects BA 2823/8-1 and SCHR 570/19-1 is greatly acknowledged by D. Balzani, A. Gandhi and J. Schröder.

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Authors and Affiliations

  1. Faculty of Civil Engineering, Institute of Mechanics and Shell Structures, Technical University of Dresden, Nürnbergerstr. 31a, 01187, Dresden, Germany

    Daniel Balzani & Ashutosh Gandhi

  2. Toyota Central R&D Labs. Inc., 41-1 Yokomichi, Nagakute, Aichi, 480-1192, Japan

    Masato Tanaka

  3. Department of Civil Engineering, Faculty of Engineering, Institute of Mechanics, University of Duisburg-Essen, Universitätsstr. 15, 45117, Essen, Germany

    Jörg Schröder

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  1. Daniel Balzani
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  2. Ashutosh Gandhi
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  3. Masato Tanaka
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  4. Jörg Schröder
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Correspondence to Daniel Balzani.

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Balzani, D., Gandhi, A., Tanaka, M. et al. Numerical calculation of thermo-mechanical problems at large strains based on complex step derivative approximation of tangent stiffness matrices. Comput Mech 55, 861–871 (2015). https://doi.org/10.1007/s00466-015-1139-0

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  • DOI: https://doi.org/10.1007/s00466-015-1139-0

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