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

Hesch, Christian; Betsch, Peter

doi:10.1002/nme.2466

Cited by 77 publications

(70 citation statements)

References 32 publications

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“…The term W 0 (F , J) is usually defined via a standard hyperelastic constitutive model (see equations (8) or (12)) and thus, the complete closure of the system requires the definition of the entropy function η 0 (F , J). In its simplest form, η 0 can be assumed to depend only on the Jacobian of the motion J, that is η 0 (F , J) = η 0 (J).…”

Section: Mie-grüneisen Equation Of Statementioning

confidence: 99%

“…In the computational mechanics community, the ability to perform calculations on tetrahedral meshes has become increasingly important. For these reasons, the automated tetrahedral mesh generators by means of Delaunay and advancing front techniques [6] have recently received increasing attention in a number of important application areas, such as cardiovascular tissue modelling [7], crash impact simulation [8], blast and fracture mechanics and complex multi-physics problems [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

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An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Gil

Bonet

et al. 2015

Journal of Computational Physics

152

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A vertex centred Jameson-Schmidt-Turkel (JST) finite volume algorithm was recently introduced by the authors (Aguirre et al., 2014 [1]) in the context of fast solid isothermal dynamics. The spatial discretisation scheme was constructed upon a Lagrangian two-field mixed (linear momentum and the deformation gradient) formulation presented as a system of conservation laws [2][3][4]. In this paper, the formulation is further enhanced by introducing a novel upwind vertex centred finite volume algorithm with three key novelties. First, a conservation law for the volume map is incorporated into the existing two-field system to extend the range of applications towards the incompressibility limit (Gil et al., 2014 [5]). Second, the use of a linearised Riemann solver and reconstruction limiters is derived for the stabilisation of the scheme together with an efficient edge-based implementation. Third, the treatment of thermo-mechanical processes through a Mie-Grüneisen equation of state is incorporated in the proposed formulation. For completeness, the study of the eigenvalue structure of the resulting system of conservation laws is carried out to demonstrate hyperbolicity and obtain the correct time step bounds for non-isothermal processes. A series of numerical examples are presented in order to assess the robustness of the proposed methodology. The overall scheme shows excellent behaviour in shock and bending dominated nearly incompressible scenarios without spurious pressure oscillations, yielding second order of convergence for both velocities and stresses.

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Section: Mie-grüneisen Equation Of Statementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Aguirre

Gil

Bonet

et al. 2015

Journal of Computational Physics

152

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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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“…In this case, the use of the mortar method proves very important to solve a contact problem because it can smooth different constraints (contact, friction thermal …) in the interface under an integral form [7,9,10]. If we add to the mortar method for the contact constraints enforcement a direct smoothing technique applied in the interface, as in [17,24,25], we can guarantee a superior robustness in the resolution algorithm.…”

Section: Mechanical Modelmentioning

confidence: 99%

“…Mortar approach is a method of contact interaction treatment through an exact evaluation of surface integrals contributing to the weak formulation. It will be combined with discreet satisfaction of contact constraints [6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

2d Frictional B-Spline Smoothed Mortar Contact Problems Part I: Matching Phase

Kallel¹,

Bouabdallah²

2017

J Appl Mech Eng

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Simple averaged normal vector, Hermit polynomial, Cubic B-Spline curve and other technique are used to smooth surface for frictional contact problem between deformable bodies in context of large deformation. Mortar approach is combined with augmented Lagrange formulation to treat the contact constraints. A spline interpolation is also employed for the linearization of the kinematic contact constraints. Cubic B-Spline is applied at the boundary of contact element while maintaining a classical Lagrange interpolation at the interior. The quality of the contact pressure obtained with B-Spline is better than that achieved with a hall Lagrange discretization. The performance of the proposed framework is illustrated with representative two dimensional numerical examples.

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

Cited by 77 publications

References 32 publications

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

An upwind vertex centred Finite Volume solver for Lagrangian solid dynamics

Isogeometric dual mortar methods for computational contact mechanics

2d Frictional B-Spline Smoothed Mortar Contact Problems Part I: Matching Phase

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