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

  • Masato TanakaEmail author
  • Daniel Balzani
  • Jörg Schröder
Part of the Lecture Notes in Applied and Computational Mechanics book series (LNACM, volume 81)


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.


Finite Difference Method Strain Energy Function Tangent Modulus Tangent Stiffness Matrix Roundoff Error 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Masato Tanaka
    • 1
    Email author
  • Daniel Balzani
    • 2
  • Jörg Schröder
    • 3
  1. 1.Toyota Central R&D Labs., Inc.NagakuteJapan
  2. 2.Institute of Mechanics and Shell StructuresTU DresdenDresdenGermany
  3. 3.Institute of MechanicsUniversity Duisburg-EssenEssenGermany

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