We propose and study a nonlinear elimination preconditioned inexact Newton
method for the numerical simulation of diseased human arteries with a heterogeneous hyperelastic model. We assume the artery is made of layers of distinct tissues and also contains plaque. Traditional Newton methods often work well for smooth and homogeneous arteries but suffer from slow or no convergence due to the heterogeneousness of diseased soft tissues when the material is quasi-incompressible. The proposed nonlinear elimination method adaptively finds a small number of equations causing the nonlinear stagnation and then eliminates them from the global nonlinear system. By using the theory of affine invariance of Newton method, we provide insight into why the nonlinear elimination method can improve the convergence of Newton iterations. Our numerical results show that the combination of nonlinear elimination with an initial guess interpolated from a coarse level solution can lead to the uniform convergence of Newton method for this class of very difficult nonlinear problems.
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This content will become publicly available on April 26, 2025
A finite element method for a two-dimensional Pucci equation
A nonlinear least-squares finite element method for strong solutions of the Dirichlet boundary value problem of a two-dimensional Pucci equation on convex polygonal domains is investigated in this paper. We obtain a priori and a posteriori error estimates and present corroborating numerical results, where the discrete nonsmooth and nonlinear optimization problems are solved by an active set method and an alternating direction method with multipliers.
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- Award ID(s):
- 2208404
- PAR ID:
- 10508357
- Publisher / Repository:
- Academie des Sciences, Institut de France
- Date Published:
- Journal Name:
- Comptes Rendus. Mécanique
- Volume:
- 351
- Issue:
- S1
- ISSN:
- 1873-7234
- Page Range / eLocation ID:
- 261 to 276
- Subject(s) / Keyword(s):
- Pucci’s equation finite element method strong solution a priori and a posteriori error estimates nonlinear least-squares active set method ADMM
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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