An augmented Lagrangian technique combined with a mortar algorithm for modelling mechanical contact problems

Cavalieri, Federico J.; Cardona, Alberto

doi:10.1002/nme.4391

Cited by 23 publications

(26 citation statements)

References 35 publications

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“…For contact algorithms based on Lagrangian finite elements such methods can be found in e.g. [36,[41][42][43][44] and within IGA in [12,13]. Using these methods based on a standard Lagrange multiplier interpolation, a system of increased size containing both displacement and Lagrange multiplier degrees of freedom has to be solved.…”

Section: Dual Basis Functionsmentioning

confidence: 99%

Isogeometric dual mortar methods for computational contact mechanics

Seitz

Farah

Kremheller

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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In recent years, isogeometric analysis (IGA) has received great attention in many fields of computational mechanics research. Especially for computational contact mechanics, an exact and smooth surface representation is highly desirable. As a consequence, many well-known finite e lement m ethods a nd a lgorithms f or c ontact m echanics h ave b een t ransferred t o I GA. I n t he present contribution, the so-called dual mortar method is investigated for both contact mechanics and classical domain decomposition using NURBS basis functions. In contrast to standard mortar methods, the use of dual basis functions for the Lagrange multiplier based on the mathematical concept of biorthogonality enables an easy elimination of the additional Lagrange multiplier degrees of freedom from the global system. This condensed system is smaller in size, and no longer of saddle point type but positive definite. A very simple and commonly used element-wise construction of the dual basis functions is directly transferred to the IGA case. The resulting Lagrange multiplier interpolation satisfies discrete inf-sup stability and biorthogonality, however, the reproduction order is limited to one. In the domain decomposition case, this results in a limitation of the spatial convergence order to O(h 3 /2 ) in the energy norm, whereas for unilateral contact, due to the lower regularity of the solution, optimal convergence rates are still met. Numerical examples are presented that illustrate these theoretical considerations on convergence rates and compare the newly developed isogeometric dual mortar contact formulation with its standard mortar counterpart as well as classical finite elements based on first and second order Lagrange polynomials.

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Section: Dual Basis Functionsmentioning

confidence: 99%

Isogeometric dual mortar methods for computational contact mechanics

Seitz

Farah

Kremheller

et al. 2016

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

“…Numerical tests showed that this choice gives a better condition number of the iteration matrix than other choices [10]. Eventually, different sets of values of r, k can be used for the normal and for the tangential parts.…”

Section: Augmented Lagrangian Formulationmentioning

confidence: 96%

“…At each contact element, the restrictions to the element facets of the integrals needed for the computation of the weight factors n 1 AB , n 2 AB are evaluated. The weight factors are then obtained by assembling the contributions of all elements [10,53].…”

Section: Contact Element Definitionmentioning

confidence: 99%

“…The method is an extension of the frictionless mortar method presented by Cavalieri and Cardona [10]. Unlike the works mentioned above, the exact enforcement of the contact and friction constraints is obtained by using the augmented Lagrangian method proposed by Alart and Curnier [1].…”

mentioning

confidence: 99%

“…Puso [53] proposed a mortar method based on an augmented Lagrangian combined with a double loop Uzawa-type algorithm. Later, Popp et al [50] proposed a mortar algorithm where a dual Lagrange multipliers is used, and recently, Cavalieri and Cardona [10] proposed a frictionless mortar contact algorithm combined with an augmented Lagrangian method. Each one of these proposals has advantages and disadvantages as described, e.g., in [10,52].…”

mentioning

confidence: 99%

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Numerical solution of frictional contact problems based on a mortar algorithm with an augmented Lagrangian technique

Cavalieri

Cardona

2015

Multibody Syst Dyn

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This work presents a frictional contact formulation to solve three-dimensional contact problems with large finite displacements. The kinematic description of the contacting bodies is defined by using a mortar approach. The regularization of the variational frictional contact problem is solved with a mixed dual penalty approach based on an augmented Lagrangian technique. In this method, the numerical results do not depend on the definition of any user-defined penalty parameter affecting the normal or tangential component of forces. The robustness and performance of the proposed algorithm are studied and validated by solving a series of numerical examples with finite displacements and large slip.

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Dual mortar methods for computational contact mechanics – overview and recent developments

Popp

Wall

2014

GAMM-Mitteilungen

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Mortar-based finite element discretization in computational contact mechanics has seen a great thrust of research activities over the last years. In this contribution, we review the most important features of the so-called dual mortar finite element approach for finite deformation contact and friction. Advantageous properties of the devised algorithms comprise superior robustness as compared with the traditional node-to-segment (NTS) approach, the absence of any unphysical user-defined parameters (e.g. penalty parameter), the integration of all types of nonlinearities into one single iteration loop and the possibility to condense the biorthogonal discrete Lagrange multipliers from the global system of equations. Beside this general overview, some recent developments are highlighted, such as improved consistency and robustness of dual mortar methods, investigations on efficient numerical integration schemes, an extension to second-order interpolation as well as the combined treatment of contact and elastoplasticity. Several numerical examples are presented to illustrate the high quality of results obtained with the proposed methods.

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An augmented Lagrangian technique combined with a mortar algorithm for modelling mechanical contact problems

Cited by 23 publications

References 35 publications

Isogeometric dual mortar methods for computational contact mechanics

Isogeometric dual mortar methods for computational contact mechanics

Numerical solution of frictional contact problems based on a mortar algorithm with an augmented Lagrangian technique

Dual mortar methods for computational contact mechanics – overview and recent developments

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