An Eigenvalue-Free Implementation of the Log-Conformation Formulation

TL;DR

This paper introduces an eigenvalue-free computational method for the log-conformation formulation in viscoelastic flow simulations, applicable to 2D and 3D, avoiding eigenvalue decomposition and validated on benchmark problems.

Contribution

It presents a novel eigenvalue-free algorithm for the log-conformation formulation, enabling efficient 3D flow simulations without eigenvalue decomposition.

Findings

Successfully implemented for 2D and 3D flows

Validated on confined cylinder and sedimenting sphere benchmarks

Avoids eigenvalue decomposition in numerical simulations

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.