A primal–dual active set strategy for non-linear multibody contact problems

Hüeber, S.; Wohlmuth, Barbara

doi:10.1016/j.cma.2004.08.006

Cited by 227 publications

(251 citation statements)

References 20 publications

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“…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. In this work, we follow a different approach using dual shape functions for the Lagrange multiplier, which were initially introduced in domain decomposition problems [19,45] and extended to contact problems in [20][21][22][23][24][25][26][27] and recently reviewed in [29]. While dual mortar methods are meanwhile well-established in finite elements, the present work, to the authors knowledge, is the first application of dual basis functions in the context of IGA for both domain decomposition and finite deformation frictional contact.…”

Section: Dual Basis Functionsmentioning

confidence: 99%

“…With these function spaces, the strong form of the contact problem can be transformed into an equivalent mixed variational form by applying the method of weighted residuals and subsequent integration by parts, see e.g. [20] for a detailed derivation. As a consequence of the balance of linear momentum at the contact interface, the contact terms can be formulated as slave side integrals only and one obtains:…”

Section: Problem Definition Of Finite Deformation Frictional Contactmentioning

confidence: 99%

“…Proposing an alternative approximation of the Lagrange multiplier, so-called dual mortar methods were originally introduced in the context of domain decomposition [19] and later extended to small deformation frictionless contact in 2D [20] and 3D [21], small deformation frictional contact [22], finite deformation frictionless contact [23,24] and finite deformation frictional contact [25]. Since NURBS are naturally associated with higher order approximations, the extension of the dual mortar method to second order Lagrange finite elements in [26,27] including optimal a priori error estimates is also worth mentioning.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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%

Section: Problem Definition Of Finite Deformation Frictional Contactmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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“…To account for the contact conditions we make use of an active set strategy proposed by Hüeber and Wohlmuth [8], see also Hartmann et al [12] and the references cited in these works. Substituting for v n+1 from (79) 1 into (79) 2 , yields the non-linear system of equations…”

Section: Methodsmentioning

confidence: 99%

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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“…In recent years segment-to-segment formulations like the mortar method [8] have been successfully applied to solving a wide variety of contact problems in 2D [35,27,55] and 3D [39,38], with linear and quadratic elements [28,53], in large and small deformations including Coulomb friction [40,41,17,18,42,50,20] and dynamic problems [24]. The theoretical basis of the mortar method is well known [15,28,32,30,31]. The compatibility of the displacement field and the contact stresses allows the Brezzi-Babuska-InfSup condition to be fulfilled, so the optimal convergence rate of the finite element solution can be achieved.…”

Section: Introductionmentioning

confidence: 99%

A modified perturbed Lagrangian formulation for contact problems

et al. 2015

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The aim of this work is to propose a formulation to solve both small and large deformation contact problems using the finite element method. We consider both standard finite elements and the so-called immersed boundary elements. The method is derived from a stabilized Nitsche formulation. After introduction of a suitable Lagrange multiplier discretization the method can be simplified to obtain a modified perturbed Lagrangian formulation. The stabilizing term is iteratively computed using a smooth stress field. The method is simple to implement and the numerical results show that it is robust. The optimal convergence rate of the finite element solution can be achieved for linear elements.

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A primal–dual active set strategy for non-linear multibody contact problems

Cited by 227 publications

References 20 publications

Isogeometric dual mortar methods for computational contact mechanics

Isogeometric dual mortar methods for computational contact mechanics

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

A modified perturbed Lagrangian formulation for contact problems

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