A semi-smooth Newton method for orthotropic plasticity and frictional contact at finite strains

Seitz, Alexander; Popp, Alexander; Wall, Wolfgang A.

doi:10.1016/j.cma.2014.11.003

Cited by 26 publications

(34 citation statements)

References 55 publications

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“…A combination of the later presented contact algorithm with small strain plasticity [34] and finite strain anisotropic plasticity [35] based on NCP functions have already been presented. In the following, we will refer to Ω (1) as the slave body (with the slave contact surface γ (1) c ) and to Ω (2) as the master body (with the master contact surface γ (2) c ).…”

Section: Problem Definition Of Finite Deformation Frictional Contactmentioning

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: Problem Definition Of Finite Deformation Frictional Contactmentioning

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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Section: Algebraic Formulation Of Linear Systemsmentioning

confidence: 99%

“…The specific properties of the used dual basis functions can be exploited to simplify the linear system (25). Due to the biorthogonality condition (17) of the dual basis functions, the mortar matrices D  and D  in (25), vanish and the mortar matrices D  and D  reduce to diagonal blocks. The diagonal form of the mortar matrix D gives rise to the condensation of the Lagrange multipliers from (25), allowing to avoid the saddle-point structure.…”

Section: Algebraic Formulation Of Linear Systemsmentioning

confidence: 99%

“…Due to the biorthogonality condition (17) of the dual basis functions, the mortar matrices D  and D  in (25), vanish and the mortar matrices D  and D  reduce to diagonal blocks. The diagonal form of the mortar matrix D gives rise to the condensation of the Lagrange multipliers from (25), allowing to avoid the saddle-point structure. First, the sixth row and column in the system matrix are removed making use of the fact that  = 0.…”

Section: Algebraic Formulation Of Linear Systemsmentioning

confidence: 99%

“…from the third row in (25). Introducing P  ∶= D −1  M  , the final condensed system has the form…”

Section: Algebraic Formulation Of Linear Systemsmentioning

confidence: 99%

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Algebraic multigrid methods for dual mortar finite element formulations in contact mechanics

Wiesner

Popp

Gee

et al. 2018

Numerical Meth Engineering

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Summary This paper proposes novel strategies to enable multigrid preconditioners within iterative solvers for linear systems arising from contact problems based on mortar finite element formulations. The so‐called dual mortar approach that is exclusively employed here allows for an easy condensation of the discrete Lagrange multipliers. Therefore, it has the advantage over standard mortar methods that linear systems with a saddle‐point structure are avoided, which generally require special preconditioning techniques. However, even with the dual mortar approach, the resulting linear systems turn out to be hard to solve using iterative linear solvers. A basic analysis of the mathematical properties of the linear operators reveals why the naive application of standard iterative solvers shows instabilities and provides new insights of how contact modeling affects the corresponding linear systems. This information is used to develop new strategies that make multigrid methods efficient preconditioners for the class of contact problems based on dual mortar methods. It is worth mentioning that these strategies primarily adapt the input of the multigrid preconditioners in a way that no contact‐specific enhancements are necessary in the multigrid algorithms. This makes the implementation comparably easy. With the proposed method, we are able to solve large contact problems, which is an important step toward the application of dual mortar–based contact formulations in the industry. Numerical results are presented illustrating the performance of the presented algebraic multigrid method.

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Algebraic multigrid methods for saddle point systems arising from mortar contact formulations

Wiesner¹,

Mayr

Popp

et al. 2021

Numerical Meth Engineering

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In this article, a fully aggregation-based algebraic multigrid strategy is developed for nonlinear contact problems of saddle point type using a mortar finite element approach. While the idea of extending multigrid methods to saddle point systems can already be found, for example, in the context of Stokes and Oseen equations in literature, the main contributions of this work are (i) the development and open-source implementation of an interface aggregation strategy specifically suited for generating Lagrange multiplier aggregates that are required for coupling structural equilibrium equations with contact constraints and (ii) a review of saddle point smoothers in the context of constrained interface problems. The new interface aggregation strategy perfectly fits into an aggregation-based multigrid framework and can easily be combined with segregated transfer operators, which allow to preserve the saddle point structure on the coarse levels. Further analysis provides insight into saddle point smoothers applied to contact problems, while numerical experiments illustrate the robustness of the new method.We have implemented the proposed algorithm within the MueLu package of the open-source Trilinos project. Numerical examples demonstrate the robustness of the proposed method in complex dynamic contact problems as well as its scalability up to 23.9 million unknowns on 480 MPI ranks.

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A semi-smooth Newton method for orthotropic plasticity and frictional contact at finite strains

Cited by 26 publications

References 55 publications

Isogeometric dual mortar methods for computational contact mechanics

Isogeometric dual mortar methods for computational contact mechanics

Algebraic multigrid methods for dual mortar finite element formulations in contact mechanics

Algebraic multigrid methods for saddle point systems arising from mortar contact formulations

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