A mixed formulation for the finite element solution of contact problems

Papadopoulos, Panayiotis; Taylor, Robert L.

doi:10.1016/0045-7825(92)90061-n

Cited by 194 publications

(174 citation statements)

References 7 publications

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“…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. Moreover, a penalty Lagrangian method was proposed in Taylor and Papadopoulos [126].…”

Section: Summary Of the Main Resultsmentioning

confidence: 99%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

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Section: Summary Of the Main Resultsmentioning

confidence: 99%

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Demoures

Gay–Balmaz

Raţiu

2016

Forum of Mathematics, Sigma

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“…Originally, contact segments have been introduced by Simo et al [22] to take into account the kinematics of the contact between two discretized bodies. Similar segmentation procedures have been devised by Papadopoulos and Taylor [23], Zavarise and Wriggers [24], McDevitt and Laursen [6] and Yang et al [9].…”

Section: Elementwise Calculation Of the Mortar Contact Constraintsmentioning

confidence: 96%

A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems

Hesch

Betsch

2008

Numerical Meth Engineering

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SUMMARYIn the present work the mortar method is applied to planar large deformation contact problems without friction. In particular, the proposed form of the mortar contact constraints is invariant under translations and rotations. These invariance properties lay the foundation for the design of energy-momentum timestepping schemes for contact-impact problems. The iterative solution procedure is embedded into an active set algorithm. Lagrange multipliers are used to enforce the mortar contact constraints. The solution of generalized saddle point systems is circumvented by applying the discrete null space method. Numerical examples demonstrate the robustness and enhanced numerical stability of the newly developed energymomentum scheme.

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“…The discrete node-to-surface gap formulation does not pass the patch test, failing to represent a state of constant stress along the interface [4]. In the context of 2-D elasticity, the discrete gap function formulation passes the patch test only when the two contacting bodies are discretized with linear elements and the contact events happen at the nodes.…”

Section: Introductionmentioning

confidence: 99%

“…Such bias can be eliminated via a two-pass approach where both surfaces take roles as master and slave and the contact constraints are enforced on both sides of the interface. Such approaches, however, have been show to suffer from locking due to over constraints [4,5]. Therefore, it is recommended to use the node-to-surface method as a single-pass approach to avoid locking.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional flat shell-to-shell coupling: numerical challenges

Guo¹,

Haikal²

2017

Curved and Layered Structures

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Abstract:The node-to-surface formulation is widely used in contact simulations with finite elements because it is relatively easy to implement using different types of element discretizations. This approach, however, has a number of well-known drawbacks, including locking due to over-constraint when this formulation is used as a twopass method. Most studies on the node-to-surface contact formulation, however, have been conducted using solid elements and little has been done to investigate the effectiveness of this approach for beam or shell elements. In this paper we show that locking can also be observed with the node-to-surface contact formulation when applied to plate and flat shell elements even with a singlepass implementation with distinct master/slave designations, which is the standard solution to locking with solid elements. In our study, we use the quadrilateral four node flat shell element for thin (Kirchhoff-Love) plate and thick (Reissner-Mindlin) plate theory, both in their standard forms and with improved formulations such as the linked interpolation [1] and the Discrete Kirchhoff [2] elements for thick and thin plates, respectively. The Lagrange multiplier method is used to enforce the node-to-surface constraints for all elements. The results show clear locking when compared to those obtained using a conforming mesh configuration.

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A mixed formulation for the finite element solution of contact problems

Cited by 194 publications

References 7 publications

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

Multisymplectic Variational Integrators for Nonsmooth Lagrangian Continuum Mechanics

A mortar method for energy‐momentum conserving schemes in frictionless dynamic contact problems

Three-dimensional flat shell-to-shell coupling: numerical challenges

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