TY - JOUR

T1 - Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies

AU - Rudi, Johann

AU - Shih, Yu hsuan

AU - Stadler, Georg

N1 - Funding Information:
We thank Anton Popov and Dave May for insightful reviews and the resulting improvements of this paper. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research under Contract DE‐AC02‐06CH11357 and the Exascale Computing Project (Contract 17‐SC‐20‐SC). This research was supported by the Exascale Computing Project (17‐SC‐20‐SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration. The work was partially supported by the U.S. National Science Foundation through Grants EAR‐1646337 and DMS‐1723211. Computing time on TACC's Stampede2 supercomputer was provided through the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant ACI‐1548562.
Publisher Copyright:
©2020. American Geophysical Union. All Rights Reserved.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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.

AB - 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.

KW - Newton's method

KW - computational geodynamics

KW - incompressible Stokes

KW - solvability

KW - viscoplasticity

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U2 - 10.1029/2020GC009059

DO - 10.1029/2020GC009059

M3 - Article

AN - SCOPUS:85091661116

VL - 21

JO - Geochemistry, Geophysics, Geosystems

JF - Geochemistry, Geophysics, Geosystems

SN - 1525-2027

IS - 9

M1 - e2020GC009059

ER -