Log In Sign Up

Advanced Newton Methods for Geodynamical Models of Stokes Flow with Viscoplastic Rheologies

by   Johann Rudi, et al.

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.


page 10

page 11

page 12

page 13

page 14


Successive finite element methods for Stokes equations

This paper will suggest a new finite element method to find a P^4-veloci...

Robust and Efficient Multilevel-ILU Preconditioned Newton-GMRES for Incompressible Navier-Stokes

We introduce a new preconditioned Newton-GMRES method for solving the no...

Multi-scale Deep Neural Network (MscaleDNN) Methods for Oscillatory Stokes Flows in Complex Domains

In this paper, we study a multi-scale deep neural network (MscaleDNN) as...

Conditions for Digit Stability in Iterative Methods Using the Redundant Number Representation

Iterative methods play an important role in science and engineering appl...