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Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems pp 1–23Cite as

Robust Numerical Schemes for an Efficient Implementation of Tangent Matrices: Application to Hyperelasticity, Inelastic Standard Dissipative Materials and Thermo-Mechanics at Finite Strains

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Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 81))

Abstract

In this contribution robust numerical schemes for an efficient implementation of tangent matrices in finite strain problems are presented and their performance is investigated through the analysis of hyperelastic materials, inelastic standard dissipative materials in the context of incremental variational formulations, and thermo-mechanics. The schemes are based on highly accurate and robust numerical differentiation approaches which use non-real numbers, i.e., complex variables and hyper-dual numbers. The main advantage of these approaches are that, contrary to the classical finite difference scheme, no round-off errors in the perturbations due to floating-point arithmetics exist within the calculation of the tangent matrices. This results in a method which is independent of perturbation values (in case of complex step derivative approximations if sufficiently small perturbations are chosen). An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of hyperelastic, finite strain elastoplastic, and thermo-elastoplastic boundary value problems, the performance of the proposed approaches is analyzed.

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Acknowledgments

Financial funding by the DFG Priority Program 1648 (SPPEXA “Software for Exascale Computing”), projects BA 2823/8-1 and SCHR 570/19-1 is greatly acknowledged by D. Balzani and J. Schröder. Furthermore, assistance for the thermo-mechanical calculations by Ashutosh Gandhi is appreciated.

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Authors and Affiliations

  1. Toyota Central R&D Labs., Inc., Nagakute, Japan

    Masato Tanaka

  2. Institute of Mechanics and Shell Structures, TU Dresden, Dresden, Germany

    Daniel Balzani

  3. Institute of Mechanics, University Duisburg-Essen, Essen, Germany

    Jörg Schröder

Authors
  1. Masato Tanaka
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  2. Daniel Balzani
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Correspondence to Masato Tanaka .

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Editors and Affiliations

  1. FB 11 - Maschinenbau, Universität Siegen, Siegen, Germany

    Kerstin Weinberg

  2. Politecnico di Milano, Milano, Italy

    Anna Pandolfi

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Tanaka, M., Balzani, D., Schröder, J. (2016). Robust Numerical Schemes for an Efficient Implementation of Tangent Matrices: Application to Hyperelasticity, Inelastic Standard Dissipative Materials and Thermo-Mechanics at Finite Strains. In: Weinberg, K., Pandolfi, A. (eds) Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. Lecture Notes in Applied and Computational Mechanics, vol 81. Springer, Cham. https://doi.org/10.1007/978-3-319-39022-2_1

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  • DOI: https://doi.org/10.1007/978-3-319-39022-2_1

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