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

An Eigenvalue-Free Implementation of the Log-Conformation Formulation