On the contact domain method: A comparison of penalty and Lagrange multiplier implementations

doi:10.1016/j.cma.2011.01.011

Computer Methods in Applied Mechanics and Engineering

Volumes 205–208, 15 January 2012, Pages 68-82

https://doi.org/10.1016/j.cma.2011.01.011 Get rights and content

Abstract

This work focuses on the assessment of the relative performance of the so-called contact domain method, using either the Lagrange multiplier or the penalty strategies. The mathematical formulation of the contact domain method and the imposition of the contact constraints using a stabilized Lagrange multiplier method are taken from the seminal work (as cited later), whereas the penalty based implementation is firstly described here. Although both methods result into equivalent formulations, except for the difference in the constraint imposition strategy, in the Lagrange multiplier method the constraints are enforced using a stabilized formulation based on an interior penalty method, which results into a different estimation of the contact forces compared to the penalty method. Several numerical examples are solved to assess certain numerical intricacies of the two implementations. The results show that both methods perform similarly as one increases the value of the penalty parameter or decreases the value of the stabilization factor (in case of the Lagrange multiplier method). However there seems to exist a clear advantage in using the Lagrange multiplier based strategy in a few critical situations, where the penalty method fails to produce convincing results due to excessive penetration.

Section snippets

Motivation

The computational modeling and analysis of structural contact problems have been important subjects of interest over the past several decades. Despite the significant progress achieved in the subject, it still poses a challenge in non-linear problems, especially when the aim is to develop accurate and efficient algorithms based on implicit methods.

While developing a contact formulation two main ingredients may be basically chosen:

•
A scheme to discretize the contact surfaces or the interface

Review of the contact domain method: stabilized Lagrange multiplier implementation

The contact domain method was originally proposed for 2D contacting bodies in [14] and validated with a number of numerical examples in [9]. An extension to the frictionless 3D case was given in [11]. The method poses some specific features for modeling contact between two largely deformable bodies, namely:

•
The discretization scheme allowing the contact restrictions to be applied in a manifold (the contact domain) of the same dimension as the contacting bodies, which is in contrast with other

Variational form using a (regularized) penalty method

For the penalty strategy, the contribution to the virtual work due to contact can be written as follows: $δ Π_{cont} (u^{(α)}; δ u^{(α)}) = \underset{normal contact}{\underset{︸}{\int_{D_{n}^{N}} {\tilde{t}}_{N} ({\bar{g}}_{N}) δ {\bar{g}}_{N}}} dD + \underset{stick}{\underset{︸}{\int_{D_{n}^{T}} {\tilde{t}}_{T} ({\bar{g}}_{T}) δ {\bar{g}}_{T}}} dD + \underset{slip}{\underset{︸}{\int_{D_{n}^{N} / D_{n}^{T}} \tilde{T} δ {\bar{g}}_{T}}} dD \forall δ u^{(α)} \in V_{0} .$ Comparing Eq. (39) with (34) it can be noticed that, in the case of the penalty method the contact term is a function of the penalized tractions ${\tilde{t}}_{N}$ and ${\tilde{t}}_{T}$ , which is different from the Lagrange multiplier method, the latter method is a function of the Lagrange multipliers, which in

Equivalence between Lagrange multiplier and penalty based approaches

To point out the equivalence between the two methods derived in the previous sections, we start from the variational constraint equations for the Lagrange multiplier method (SLMM) as given in (37), (38), and in (39) for the penalty method. For conciseness in the derivation, let us focus on the normal constraint part, as the others constraint conditions can be treated in a similar way. For a linear triangular contact patch, the integration of Eq. (37) results into a simplified form such as: $\frac{1}{2} H {\bar{g}}_{N}$

Contact patch test

To assess the implementations of the proposed contact algorithms, i.e. whether they are able to exactly transmit constant stress field from one body to another along an arbitrary non-conforming contact surface, a patch test has to be considered. Different patch test setups have been proposed in the literature and the one proposed by Asghar and Lyons [1] recently has been chosen here. According to it, linear displacement fields are assumed for the horizontal and vertical components u and v in a

Concluding remarks

The present work summarizes the performance of the contact domain method (CDM) in terms of two different implementations namely, the Lagrange multiplier based implementation and the penalty strategy. In particular the treatment of the constraints using the penalty method has been derived, while the Lagrange multiplier based formulation is summarized from [14].

Emphasis has been given on pointing out the equivalence between the two strategies, and mathematically it has been shown that indeed a

Acknowledgments

The Spanish Ministry of Science and Innovation and the Catalan Government Research Department are gratefully acknowledged for their financial support under Grants BIA2008-00411 and 2009 SGR 1510, respectively.

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On the contact domain method: A comparison of penalty and Lagrange multiplier implementations

Abstract

Section snippets

Motivation

Review of the contact domain method: stabilized Lagrange multiplier implementation

Variational form using a (regularized) penalty method

Equivalence between Lagrange multiplier and penalty based approaches

Contact patch test

Concluding remarks

Acknowledgments

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Finite Elem. Anal. Des.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Finite Elem. Anal. Des.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.