粘塑性Stokes流地球动力学模型的改进牛顿法
Advanced Newton Methods for Geodynamical Models of Stokes Flow with Viscoplastic Rheologies
摘要：
应变局部化及其引起的塑性和破坏在岩石圈演化中起着重要作用。这些现象通常由具有粘塑性流变性的Stokes流动来模拟。这些流变学的非线性使得所得系统的数值解具有挑战性，并且迭代法通常收敛缓慢或根本不收敛。然而，精确的解决方案是至关重要的代表物理。此外，对于某些流变学定律，可解性方面仍然是未知的。我们研究了一个基本的但具有代表性的粘塑性流变规律。该定律涉及与动压无关的屈服应力，称为冯米塞斯屈服准则。两种常用的变体，完美/理想和复合粘塑性，进行了比较。我们从能量最小化原理推导出这两种变体，并使用这种观点来论证解决方案的唯一性。我们提出了一种新的应力-速度牛顿解算法，该算法将应力作为牛顿线性化过程中的自变量，但只需要解决通常的速度-压力形式的Stokes系统。为了研究不同的解决方案的算法，我们实现了二维和三维有限元离散化，我们生成斯托克斯问题高达7个数量级的粘度对比，其中压缩或拉伸的结果显着的非线性局部化效果。与标准牛顿法和Picard不动点法的性能比较，我们观察到显着减少的迭代次数和改进的稳定性问题的非线性，网格细化，和多项式阶的离散化。
英文摘要：
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 2D and 3D 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.
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