Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies
Abstract 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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Wiley
2020-09-01
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| Series: | Geochemistry, Geophysics, Geosystems |
| Subjects: | |
| Online Access: | https://doi.org/10.1029/2020GC009059 |
| _version_ | 1797637950061674496 |
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| author | Johann Rudi Yu‐hsuan Shih Georg Stadler |
| author_facet | Johann Rudi Yu‐hsuan Shih Georg Stadler |
| author_sort | Johann Rudi |
| collection | DOAJ |
| description | Abstract 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. |
| first_indexed | 2024-03-11T12:55:37Z |
| format | Article |
| id | doaj.art-7a32568d59304045b967c0f8998828d2 |
| institution | Directory Open Access Journal |
| issn | 1525-2027 |
| language | English |
| last_indexed | 2024-03-11T12:55:37Z |
| publishDate | 2020-09-01 |
| publisher | Wiley |
| record_format | Article |
| series | Geochemistry, Geophysics, Geosystems |
| spelling | doaj.art-7a32568d59304045b967c0f8998828d22023-11-03T17:01:08ZengWileyGeochemistry, Geophysics, Geosystems1525-20272020-09-01219n/an/a10.1029/2020GC009059Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic RheologiesJohann Rudi0Yu‐hsuan Shih1Georg Stadler2Mathematics and Computer Science Division Argonne National Laboratory Lemont IL USACourant Institute of Mathematical Sciences New York University New York NY USACourant Institute of Mathematical Sciences New York University New York NY USAAbstract 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.https://doi.org/10.1029/2020GC009059computational geodynamicsviscoplasticityincompressible StokessolvabilityNewton's method |
| spellingShingle | Johann Rudi Yu‐hsuan Shih Georg Stadler Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies Geochemistry, Geophysics, Geosystems computational geodynamics viscoplasticity incompressible Stokes solvability Newton's method |
| title | Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies |
| title_full | Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies |
| title_fullStr | Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies |
| title_full_unstemmed | Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies |
| title_short | Advanced Newton Methods for Geodynamical Models of Stokes Flow With Viscoplastic Rheologies |
| title_sort | advanced newton methods for geodynamical models of stokes flow with viscoplastic rheologies |
| topic | computational geodynamics viscoplasticity incompressible Stokes solvability Newton's method |
| url | https://doi.org/10.1029/2020GC009059 |
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