An augmented Lagrangian method for discrete large‐slip contact problems

Abstract

An augmented Lagrangian formulation is proposed for large-slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node-on-facet element is defined. A non-linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non-differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

“…). It was recognized as the proximity method applied to the dual part of the classical Lagrange multiplier method in Rockafellar [116][117][118] in convex analysis and in Alart [1], Hefgaard and Curnier [63], Pietrzak [106], Glocker [52], Leine and Glocker [78], Leine and Nijmeijer [79]'. The work of Papadopoulos and Taylor [104,126] was influenced by the augmented Lagrangian formulation.…”

Higher‐order quasicontinuum methods for elastic and dissipative lattice models: uniaxial deformation and pure bending

“…(32) and (33) can be incorporated in the governing equations via Lagrange multipliers (see e.g. [39]), but in this contribution they are incorporated by replacing the horizontal and vertical force equilibria of each (lattice or triangle) node on the right edge in Eqs. (17) and (23).…”

Algorithms for the analysis of 3D finite strain contact problems

“…Therefore, selecting either face indistinctly, as proposed in Reference [43] '...the (orthogonal) projection may fail to be unique... We circumvent these ambiguities by choosing either candidate projection' can lead to incorrect answers. In the same 2D analysis context, Reference [33], p. 154 states: 'In the unlikely event that both distances are equal, either projection is suitable for subsequent constraint calculation'.…”

“…Both these strategies contain shortcomings: the first one is inconsistent with the constraint and the second one is restricted to be adopted in a first-order strategy for the multiplier estimates. Due to the fact that each element's target face size r l depends on the converged co-ordinates alone, then g 2 = g. Finite element applications including a Rockafellar Lagrangian were presented in References [43,49], however, and according to previous developments, the following constraint potential for g 2 0 is adopted:…”

“…The friction force, f, is caused by the sticking part of the relative tangential displacement, through the use of an elastic law: (43) where is the non-dimensional relative tangential admissible displacement. The apparent area, A a (see Equation (29)) is calculated using the action-reaction principle as:…”