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.

翻译：对数构型公式虽然取得了巨大成功，但自提出之初便被构建为一个包含未知场特征值分解的偏微分方程——这种分解在偏微分方程领域并不常见。时至今日，绝大多数数值实现仍基于此类或其类似的特征值分解，而Knechtges等人（2014）的研究是二维流动领域唯一显著的例外。本文提出一种适用于二维与三维流动的无特征值算法，用于计算对数构型公式的本构方程。首先，我们证明本构方程中具有挑战性的项可表示为对数构型场微调矩阵的矩阵函数。随后给出该术语与更常见对数构型公式等价性的证明。基于此公式，我们开发了一种无特征值算法来评估该矩阵函数。最终，该完整公式首先通过有限体积法进行离散化，并在受限圆柱与沉降球基准测试中得到验证。