A Nonlinear Elimination Preconditioned Inexact Newton Method for Heterogeneous Hyperelasticity

Gong, Shihua; Cai, Xiao Chuan

doi:10.1137/18m1194936

Cited by 13 publications

(16 citation statements)

References 17 publications

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“…In these methods, the elimination set is static and based on the domain decomposition. In [7], a residual based, dynamic approach was introduced. A corresponding approach has been investigated for Nonlinear FETI-DP in [24].…”

Section: Nonlinear Elimination and Sqp Methodsmentioning

confidence: 99%

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

Köhler

Rheinbach

2022

Vietnam J. Math.

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The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these methods, nonlinear elimination is an important ingredient to increase the convergence radius of Newton’s method. We prove standard globalization results for SQP-based globalization of Nonlinear FETI-DP, first for the case that the elimination set is empty. We then show how to combine nonlinear elimination and SQP-based globalization. The globalization preserves the block structure of the FETI-DP operator, which is the basis of the computational parallelism.Supporting numerical experiments using homogenous and heterogeneous model problems from nonlinear structural mechanics are provided. In the numerical experiments, we consider four standard choices of different elimination sets and different problem setups including stiff or almost incompressible inclusions in every subdomain. The numerical experiments illustrate that a good elimination set is important. However, the use of the SQP-based globalization approach presented here can improve the convergence of Nonlinear FETI-DP methods further, especially, if combined with a good choice of the elimination set.

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Section: Nonlinear Elimination and Sqp Methodsmentioning

confidence: 99%

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

Köhler

Rheinbach

2022

Vietnam J. Math.

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“…Right nonlinear preconditioners are derived from nonlinear elimination (NE) [20]. Based on different partitioning strategies, they lead to algorithms such as NKS-RAS [6] in a geometry-based domain decomposition form and nonlinear elimination preconditioned inexact Newton algorithm (INB-NE) [16,31] or RS-NE [32] in a physics-based field-split form.…”

Section: A2527mentioning

confidence: 99%

Approximate Error Bounds on Solutions of Nonlinearly Preconditioned PDEs

Liu

Keyes

2021

SIAM J. Sci. Comput.

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In many multiphysics applications, an ultimate quantity of interest can be written as a linear functional of the solution to the discretized governing nonlinear partial differential equations and finding a sufficiently accurate pointwise solution may be regarded as a step toward that end. In this paper, we derive a posteriori approximate error bounds for linear functionals corresponding to quantities of interest using two kinds of nonlinear preconditioning techniques. Nonlinear preconditioning, such as the inexact Newton with backtracking and nonlinear elimination algorithm and the multiplicative Schwarz preconditioned inexact Newton algorithm, may be effective in improving global convergence for Newton's method. It may prevent stagnation of the nonlinear residual norm and reduce the number of solutions of large ill-conditioned linear systems involving a global Jacobian required at each nonlinear iteration. We illustrate the effectiveness of the new bounds using canonical nonlinear PDE models: a flame sheet model and a nonlinear coupled lid-driven cavity problem.

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“…and is called a right-preconditioned nonlinear system. By using the theory of affine invariance, Gong and Cai [18] discussed some criteria for choosing the to-be-eliminated components under certain conditions so that NE improves the convergence of Newton iterations. We refer to the details in the reference.…”

Section: )mentioning

confidence: 99%

Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

Luo¹,

Cai²,

Keyes³

2021

SIAM J. Sci. Comput.

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We consider numerical simulation of two-phase flows in porous media using implicit methods. Because of the complex features involving heterogeneous permeability and nonlinear capillary effects the nonlinear algebraic systems arising from the discretization are very difficult to solve. The traditional Newton method suffers from slow convergence in the form of a long stagnation or sometimes does not converge at all. In the paper, we develop some nonlinear preconditioning strategies for the system of two-phase flows discretized by a fully implicit discontinuous Galerkin method. The preconditioners identify and approximately eliminate the local high nonlinearities that cause the Newton method to take small updates. Specifically, we propose two elimination strategies, one is based on exploring the unbalanced nonlinearities of the pressure and the saturation fields, and the other is based on identifying certain elements of the finite element space that have much higher nonlinearities than the rest of the elements. We compare the performance and robustness of the proposed algorithms with an existing single-field elimination approach and the classical inexact Newton method with respect to some physical and numerical parameters. Experiments on three-dimensional porous media applications show that the proposed algorithms are superior to the other methods in terms of the robustness and parallel efficiency.

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A Nonlinear Elimination Preconditioned Inexact Newton Method for Heterogeneous Hyperelasticity

Cited by 13 publications

References 17 publications

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

Approximate Error Bounds on Solutions of Nonlinearly Preconditioned PDEs

Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

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