An Eigenvalue-Free Implementation of the Log-Conformation Formulation

Abstract

The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows. In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks.

arXiv comment23 pages, 6 figures, 6 tables

References in corpus (4)

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• Fully-implicit log-conformation formulation of constitutive laws
• The fully-implicit log-conformation formulation and its application to three-dimensional flows
• Automatic implementation of material laws: Jacobian calculation in a finite element code with TAPENADE