A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

Abstract

Abstract.We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decompositioncoordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented.Mathematics Subject Classification. 65N30, 65K10, 74S05, 74M15, 74R99.

Meso-scale fracture simulation using an augmented Lagrangian approach

“…Doyen et al (2010) ) probed that the constant is α = θ B − kc J c X , and it can be related with physical properties of the numerical model, being θ B proportional to the norm of the constitutive tensor in the bulk phase Ω. The Lipschitz continuity constant k grows with the decrease of cohesive forces.…”

“…Doyen et al. (2010) proved that the constant is α=θB-kboldcJboldcX, and it can be related to physical properties of the numerical model, with θB being proportional to the norm of the constitutive tensor in the bulk phase Ω .…”

Multi-field Modeling of Nonsmooth Problems of Continuum Mechanics, Differential Mixed Variational Inequalities and Their Stability

“…Then in view of the boundary condition (10) 3 we impose 2 H 0 .div ;˝/, multiply (11) by with a test function 2 H 0 .div ;˝/, integrate by part on˝, use the boundary conditions (10) 1 , (10) 2 to get…”

“…More recent work in this direction is [10] on error estimates of mixed FEM applied to Signorini elastic contact and [47] on mixed formulations and mixed FEM for a class of variational inequalities in a more general framework; see also the recent exhaustive exposition [48] on variationally consistent discretization schemes for the frictional contact problem. More related to, but different from the present paper is [11] that investigates nonconvex unilateral contact problems and analyzes a three-field augmented Lagrangian formulation in the triple of the bulk displacement, normal boundary displacement, and normal boundary stress fields.…”

Quasi-explicit time-integration schemes for dynamic fracture with set-valued cohesive zone models

“…-In the quasi-static case with a reversible cohesive zone model (with perfect adhesion or not), the existence is proven. The solution is in general not unique [10]. -In the quasi-static case with an irreversible cohesive zone model, the existence of a solution is proven in [8,4].…”