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：对数构型公式虽极为成功，但从提出之初便被构造成一个包含特征值分解（这在偏微分方程中并不常见）的偏微分方程。迄今为止，大多数数值实现仍基于此类或类似的特征值分解，仅在二维流动中Knechtges等人（2014）的研究是显著的例外。本文提出了一种适用于二维和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。为此，我们首先证明本构方程中具有挑战性的项可表示为对数构型场经轻微修正后的矩阵的函数。我们给出了该项与常用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。最终完整的公式首先采用有限体积法进行离散化，然后通过受限圆柱和沉降球基准测试进行验证。