Explicit and implicit methods for shear band modeling at high strain rates

Berger-Vergiat, Luc; Chen, Jiayao; Waisman, Haim

doi:10.1007/s00466-018-1612-7

Cited by 3 publications

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“…In addition, one needs to compute the Jacobian of the nonlinear system in (1), which can be done numerically using a finite difference approach [33,34,35,36] or analytically, thus reducing some of the numerical burden [30]. The matrix associated with the INC scheme is much larger than that used in splitting methods (combined with explicit schemes) but typically requires fewer time steps since the stability requirements are less restrictive [37]. Nonetheless, the bottleneck of such monolithic schemes is the efficiency of the linear solver which may not be suited for these thermo-mechanical systems.…”

Section: Introduction and Problem Statementmentioning

confidence: 99%

An overlapping Domain Decomposition preconditioning method for monolithic solution of shear bands

Berger-Vergiat

Waisman

2017

Computer Methods in Applied Mechanics and Engineering

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Metals subjected to high strain rate impact often exhibit a sudden and profound drop in the material's load bearing capability, a ductile failure phenomenon known as shear banding. Shear bands, characterized as material instabilities, are driven by shear heating and described as narrow regions that have sustained intense plastic deformation and high temperature rise. This coupled thermo-mechanical localization problem can be formulated as a nonlinear system with two balance equations, momentum and energy, and two constitutive laws for elasticity and plasticity.In our formulation, mixed finite elements are used to discretize the equations in space and an implicit finite difference scheme is used to advance the system in time, where at every time step, a Newton type method is used to solve the system monolitically. To that end, a block Jacobian matrix is formed analytically using Gâteaux derivatives. The resulting Jacobian is sparse, nonsymmetric and its sparsity pattern and eigenvalue content vary with the different stages of shear bands formation, consequently posing a significant challenge to iterative solvers.To address this issue, an overlapping Domain Decomposition preconditioner that takes into account the physics of shear bands and the domain in which it forms, is proposed. The key idea is to resolve the shear band domain accurately while only solving the remaining domain approximately, with selective updates when necessary. The preconditioner is formed on the basis of an additive Schwartz method that is applied to a Schur complement partition of the system.The proposed preconditioner is implemented in parallel and tested on three dif-

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