A contact formulation based on localized Lagrange multipliers: formulation and application to two-dimensional problems (original) (raw)
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A Lagrange multiplier method for the finite element solution of frictionless contact problems
Mathematical and Computer Modelling, 1998
This article proposes a novel Lagrange multiplier-based formulation for the finite element solution of the quasi-static two-body contact problem in the presence of finite motions and deformations. The main idea rests in the interpretation of the two-body contact ss a composition of two simultaneous Signorini-like problems, which naturally yield geometrically unbiased approximations of the kinematics and kinetics of frictionless contact. A two-dimensional finite element is introduced that exactly satisfies the impenetrability constraint and allows for the direct computation of consistent pressure distributions on the interacting surfaces.
Partitioned formulation of frictional contact problems using localized Lagrange multipliers
Communications in Numerical Methods in Engineering, 2005
Interface treatment methods for the contact problem between non-matching meshes have traditionally been based on a direct coupling of the contacting solids employing a master-slave strategy or classical Lagrange multipliers. These methods tend to generate strongly coupled systems that is dependent on the discretization characteristics on each side of the contact zone. In this work a displacement frame is intercalated between the interface meshes. The frame is then discretized so that the discrete frame nodes are connected to the contacting substructures using the localized Lagrange multipliers collocated at the interface nodes. The resulting methodology alleviates the need for master-slave book-keeing and provides a partitioned formulation which preserves software modularity, facilitates non-matching mesh treatment and passes the contact patch test. Frictional contact problems are used to demonstrate the salient features of the proposed method.
Numerical models based on the penalty and Lagrange multiplier method for contact problems with friction are compared in this paper. The presented approaches, with use of Coulomb's frictional law, elasto-plastic tangential slip decomposition, and consistent linearization, result in quadratic rates of convergence with the Newton-Raphson iteration. A standard contact search algorithm independent of the formulation is used for the detection of contact between previously separate meshes and for the application of displacement constraints where contact was identified. The models have been implemented into a version of the computational finite element program PAK [3]. Numerical examples that illustrate performance of the described procedures are given.
A Finite Element Formulation for Nonlinear 3D Contact Problems
Mecánica …, 2007
A finite element formulation to deal with friction contact between an elastic body and a rigid obstacle is presented. Contact between flexible solids or between a flexible and a rigid solid is defined using a non-penetration condition which is based on a representation of the interacting deforming surfaces. A large number of contact algorithms based on the imposition of inequality constraints were developed in the past to represent the non penetration condition. We can mention penalty methods, Lagrange multiplier methods, augmented Lagrangian methods and many others. In this work, we developed an augmented Lagrangian method using a slack variable, which incorporates a modified Rockafellar Lagrangian to solve non linear contact mechanics problems. The use of this method avoids the utilization of the well known Hertz-Signorini-Moreau conditions in contact mechanics problems (coincident with Kuhn-Tucker complementary conditions in optimization theory). The contact detection strategy makes use of a node-surface algorithm. Examples are provided to demonstrate the robustness and accuracy of the proposed algorithm. The contact element we present can be used with typical linear 3-D elements. The program was written in C++ under the OOFELIE environment. Finally, we present several applications of validation.
On the contact domain method: A comparison of penalty and Lagrange multiplier implementations
Computer Methods in Applied Mechanics and Engineering, 2012
Contact domain method Lagrange multiplier method Penalty method Regularized penalty method Interior penalty method a b s t r a c t 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.
Normal contact with high order finite elements and a fictitious contact material
Computers & Mathematics with Applications, 2015
Contact problems in solid mechanics are traditionally solved using the h-version of the finite element method. The constraints are enforced along the surfaces of e.g. elastic bodies under consideration. Standard constraint algorithms include penalty methods, Lagrange multiplier methods and combinations thereof. For complex scenarios, a major part of the solution time is taken up by operations to identify points that come into contact. This paper presents a novel approach to model frictionless contact using high order finite elements. Here, we employ an especially designed material model that is inserted into two-respectively threedimensional regions surrounding contacting bodies. Contact constraints are thus enforced on the same manifold as the accompanying structural problem. The application of the current material formulation leads to a regularization of the Karush-Kuhn-Tucker conditions. Our formulation can be classified as a barrier-type method. Results are obtained for two-and three-dimensional problems, including a Hertzian contact problem. Comparisons to a commercial FEA package are provided. The proposed formulation works well for non-matching discretizations on adjacent contact interfaces and handles self-contact naturally. Since the non-penetrating conditions are solved in a physically consistent manner, there is no need for an explicit contact search.
A special focus on 2D formulations for contact problems using a covariant description
International Journal for Numerical Methods in Engineering, 2006
A fully covariant description, based on the consideration of contact conditions especially for the 2D case is proposed. The description is based on a reconsideration of contact kinematics and all necessary operations such as derivatives in a specially chosen curvilinear coordinate system based on a curved geometry in plane. In addition, details of the finite element implementation are presented for the simple linear contact element. Special cases, requiring the update of history variables as well as their careful transfer over the element boundaries are illustrated by numerical examples. With these procedures artificial jumps in the contact forces can be avoided.
A novel finite element formulation for frictionless contact problems
International Journal for Numerical Methods in Engineering, 1995
This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.
A contact domain method for large deformation frictional contact problems. Part 1: Theoretical basis
Computer Methods in Applied Mechanics and Engineering, 2009
In the first part of this work, the theoretical basis of a frictional contact domain method for two-dimensional large deformation problems is presented. Most of the existing contact formulations impose the contact constraints on the boundary of one of the contacting bodies, which necessitates the projection of certain quantities from one contacting surface onto the other. In this work, the contact constraints are formulated on a so-called contact domain, which has the same dimension as the contacting bodies. This contact domain can be interpreted as a fictive intermediate region connecting the potential contact surfaces of the deformable bodies. The introduced contact domain is subdivided into a non-overlapping set of patches and is endowed with a displacement field, interpolated from the displacements at the contact surfaces. This leads to a contact formulation that is based on dimensionless, strain-like measures for the normal and tangential gaps and that exactly passes the contact patch test. In addition, the contact constraints are enforced using a stabilized Lagrange multiplier formulation based on an interior penalty method (Nitsche method). This allows the condensation of the introduced Lagrange multipliers and leads to a purely displacement driven problem. An active set strategy, based on the concept of effective gaps as entities suitable for smooth extrapolation, is used for determining the active normal stick and slip patches of the contact domain.
This paper presents a unified formulation for the combination of the finite element method (FEM) and the boundary element method (BEM) in 3D frictional contact problems that is based on the use of localized Lagrange multipliers (LLMs). Resolution methods for the contact problem between non-matching meshes have traditionally been based on a direct coupling of the contacting solids using classical Lagrange multipliers. These methods tend to generate strongly coupled systems that require a deep knowledge of the discretization characteristics on each side of the contact zone complicating the process of mixing different numerical techniques. In this work a displacement contact frame is inserted between the FE and BE interface meshes, discretized and finally connected to the contacting substructures using LLMs collocated at the mesh-interface nodes. This methodology will provide a partitioned formulation which preserves software modularity and facilitates the connection of non-matching FE and BE meshes.