Strain localization and resulting plasticity and failure play an important role in the evolution of the lithosphere. These phenomena are commonly modeled by Stokes flows with viscoplastic rheologies. The nonlinearities of these rheologies make the numerical solution of the resulting systems challenging, and iterative methods often converge slowly or not at all. Yet accurate solutions are critical for representing the physics. Moreover, for some rheology laws, aspects of solvability are still unknown. We study a basic but representative viscoplastic rheology law. The law involves a yield stress that is independent of the dynamic pressure, referred to as von Mises yield criterion. Two commonly used variants, perfect/ideal and composite viscoplasticity, are compared. We derive both variants from energy minimization principles, and we use this perspective to argue when solutions are unique. We propose a new stress‐velocity Newton solution algorithm that treats the stress as an independent variable during the Newton linearization but requires solution only of Stokes systems that are of the usual velocity‐pressure form. To study different solution algorithms, we implement 2‐D and 3‐D finite element discretizations, and we generate Stokes problems with up to 7 orders of magnitude viscosity contrasts, in which compression or tension results in significant nonlinear localization effects. Comparing the performance of the proposed Newton method with the standard Newton method and the Picard fixed‐point method, we observe a significant reduction in the number of iterations and improved stability with respect to problem nonlinearity, mesh refinement, and the polynomial order of the discretization.

中文翻译：

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粘塑性流变斯托克斯流动力学模型的先进牛顿方法

应变局部化以及由此产生的可塑性和破坏在岩石圈的演化中起着重要作用。这些现象通常由具有粘塑性流变学的斯托克斯流模拟。这些流变学的非线性使得所得系统的数值解具有挑战性，并且迭代方法通常收敛缓慢或根本没有收敛。然而，准确的解决方案对于代表物理学至关重要。此外，对于某些流变学规律，可溶性方面仍然未知。我们研究了基本但有代表性的粘塑性流变定律。该定律涉及与动压无关的屈服应力，称为von Mises屈服准则。比较了两种常用的变体，理想/理想和复合粘塑性。我们从能量最小化原理中得出这两种变体，我们使用这种观点来争论解决方案何时是唯一的。我们提出了一种新的应力-速度牛顿求解算法，该算法在牛顿线性化过程中将应力视为一个独立变量，但仅需要通常速度-压力形式的斯托克斯系统的求解。为了研究不同的求解算法，我们实现了2D和3D有限元离散化，并生成了具有多达7个数量级的粘度对比的Stokes问题，其中压缩或拉伸会导致明显的非线性局部化效果。将建议的牛顿法的性能与标准牛顿法和Picard定点法进行比较，我们发现迭代次数显着减少，并且在问题非线性，网格细化，