Weak solvability via Lagrange multipliers for contact problems involving multi-contact zones

2014 ◽  
Vol 21 (7) ◽  
pp. 826-841 ◽  
Author(s):  
Andaluzia Matei
Keyword(s):  
Lagrange Multipliers ◽  
Contact Problems ◽  
Contact Zones ◽  
Weak Solvability

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

Weak solvability of two quasistatic viscoelastic contact problems

2012 ◽  
Vol 18 (7) ◽  
pp. 745-759 ◽  
Author(s):  
Stanisław Migórski ◽  
Anna Ochal ◽  
Mircea Sofonea
Keyword(s):  
Contact Problems ◽  
Viscoelastic Contact ◽  
Weak Solvability

Modeling Compliant Part Assembly: Mechanics of Deformation and Contact

2000 ◽  
Author(s):  
Raju Mattikalli ◽  
Saba Mahanian ◽  
Alan Jones ◽  
Greg Clark
Keyword(s):  
Finite Element ◽  
Quadratic Programming ◽  
Variational Inequalities ◽  
Lagrange Multipliers ◽  
Contact Problems ◽  
Point Of View ◽  
Contact Conditions ◽  
Compliant Part ◽  
Variational Approaches ◽  
Compliant Parts

Abstract This paper describes an approach to model the mechanics of assembly by assuming parts are compliant. The approach involves a model of contact between compliant bodies based on variational inequalities. This approach has a number of advantages over current finite element codes, which rely on traditional variational approaches such as penalty force methods and Lagrange multipliers to resolve multiple unknown contact conditions. From a mathematical point of view, contact problems among compliant parts are particularly difficult to handle due to the fact that contact constraints are not permanently active, but depend on deformations. They are inherently non-linear and irreversible in character. To obtain a more mathematically robust way of modeling contact, we present a variational inequalities based approach that produces a quadratic programming (QP) problem. The QP is solved to resolve contact situations and obtain the mechanics of parts during assembly. We apply the method to simulate and design aircraft assembly processes.

scholarly journals On the Interpretation of the Lagrange Multipliers in the Constraint Formulation of Contact Problems; or Why Are Some Multipliers Always Zero?

2014 ◽  
Author(s):  
A. L. Schwab
Keyword(s):  
Mechanical System ◽  
Lagrange Multipliers ◽  
Mechanical Systems ◽  
Contact Problems ◽  
Constraint Forces ◽  
Basic Notion ◽  
Constraint Equations ◽  
Linear Dependency ◽  
Constraint Approach ◽  
Two Bodies

One method for modeling idealized contact between two bodies in mechanical system is based on the constraint approach, where Lagrange multipiers are introduced, which serve as constraint forces. In the usage of this formulation, there exists a linear dependancy between the Lagrange multipliers. Moreover, it has been observed that some Lagrange multipliers are always identical to zero. This sort of contradicts the basic notion that Lagrange multipliers in mechanical systems act as constraint forces which, when constraints are violated, push the system back in the desired configuration. In this paper it will be shown, by theory and example, that the above-mentioned linear dependency of the Lagrange multipliers, together with specific entries in the Jacobian of the constraint equations, results in some Lagrange multipliers being identical to zero.

History-dependent quasi-variational inequalities arising in contact mechanics

2011 ◽  
Vol 22 (5) ◽  
pp. 471-491 ◽  
Author(s):  
MIRCEA SOFONEA ◽  
ANDALUZIA MATEI
Keyword(s):  
Variational Inequalities ◽  
Frictional Contact ◽  
Deformable Body ◽  
Regularity Result ◽  
Contact Problems ◽  
Uniqueness Result ◽  
Constitutive Law ◽  
Signorini Condition ◽  
Normal Damped Response ◽  
Weak Solvability

We consider a class of quasi-variational inequalities arising in a large number of mathematical models, which describe quasi-static processes of contact between a deformable body and an obstacle, the so-called foundation. The novelty lies in the special structure of these inequalities that involve a history-dependent term as well as in the fact that the inequalities are formulated on the unbounded interval of time [0, +∞). We prove an existence and uniqueness result of the solution, then we complete it with a regularity result. The proofs are based on arguments of monotonicity and convexity, combined with a fixed point result obtained in [22]. We also describe a number of quasi-static frictional contact problems in which we model the material's behaviour with an elastic or viscoelastic constitutive law. The contact is modelled with normal compliance, with normal damped response or with the Signorini condition, as well, associated to versions of Coulomb's law of dry friction or to the frictionless condition. We prove that all these models cast in the abstract setting of history-dependent quasi-variational inequalities, with a convenient choice of spaces and operators. Then, we apply the abstract results in order to prove the unique weak solvability of each contact problem.

A Partitioned Formulation for FEM/BEM Coupling in Contact Problems Using Localized Lagrange Multipliers

2014 ◽  
Vol 618 ◽  
pp. 23-48
Author(s):  
Jose A. González ◽  
K.C. Park ◽  
Ramon Abascal
Keyword(s):  
Finite Element Method ◽  
Lagrange Multipliers ◽  
Frictional Contact ◽  
State Of The Art ◽  
Contact Problems ◽  
Coupled Systems ◽  
The Finite Element Method ◽  
Numerical Techniques ◽  
Strongly Coupled ◽  
Software Modularity

This paper presents a state-of-the-art in the use of localized Lagrange multipliers (LLMs)for 3D frictional contact problems coupling the Finite Element Method (FEM) and the BoundaryElement Method (BEM). Resolution methods for the contact problem between non-matching mesheshave traditionally been based on a direct coupling of the contacting solids using classical Lagrangemultipliers. These methods tend to generate strongly coupled systems that require a deep knowledgeof the discretization characteristics on each side of the contact zone complicating the process ofmixing different numerical techniques. In this work a displacement contact frame is inserted betweenthe FE and BE interface meshes, discretized and finally connected to the contacting substructuresusing LLMs collocated at the mesh-interface nodes. This methodology will provide a partitionedformulation which preserves software modularity and facilitates the connection of non-matching FEand BE meshes.

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