Numerical calculation of thermo-mechanical problems at large strains based on a robust approximation scheme for tangent stiffness matrices

Research output: Contribution to journalResearch articleContributed

Contributors

  • Daniel Balzani - (Author)
  • Ashutosh Gandhi - (Author)
  • Masato Tanaka - (Author)
  • Jörg Schröder - (Author)

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.

Details

Original languageEnglish
Pages (from-to)861-871
Number of pages11
JournalComputational Mechanics
Volume55
Issue number5
Publication statusPublished - 2015
Peer-reviewedNo

External IDs

Scopus 84937817954

Keywords

Keywords

  • Finite element method, Numerical differentiation, Quadratic convergence, Nonlinear thermo-mechanics, Complex numbers, CSDA approach