A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method (original) (raw)
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International Journal for Numerical Methods in Engineering, 2011
This paper focuses on the application of NURBS-based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar-based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C 0-continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface.
A dual mortar approach for 3D finite deformation contact with consistent linearization
International Journal for Numerical Methods in Engineering, 2010
In this paper, an approach for three-dimensional frictionless contact based on a dual mortar formulation and using a primal-dual active set strategy for direct constraint enforcement is presented. We focus on linear shape functions, but briefly address higher order interpolation as well. The study builds on previous work by the authors for two-dimensional problems. First and foremost, the ideas of a consistently linearized dual mortar scheme and of an interpretation of the active set search as a semi-smooth Newton method are extended to the 3D case. This allows for solving all types of nonlinearities (i.e. geometrical, material and contact) within one single Newton scheme. Owing to the dual Lagrange multiplier approach employed, this advantage is not accompanied by an undesirable increase in system size as the Lagrange multipliers can be condensed from the global system of equations. Moreover, it is pointed out that the presented method does not make use of any regularization of contact constraints. Numerical examples illustrate the efficiency of our method and the high quality of results in 3D finite deformation contact analysis.
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In this paper two different finite element formulations for frictionless large deformation contact problems with non-matching meshes are presented. Both are based on the mortar method. The first formulation introduces the contact constraints via Lagrange multipliers, the other employs the penalty method. Both formulations differ in size and the way of fulfilling the contact constraints, thus different strategies to determine the permanently changing contact area are required. Starting from the contact potential energy, the variational formulation, the linearization and finally the matrix formulation of both methods are derived. In combination with different contact detection methods the global solution algorithm is applied to different twodimensional examples.
International Journal for Numerical Methods in Engineering, 2012
A finite element formulation for three dimensional (3D) contact mechanics using a mortar algorithm combined with a mixed penalty-duality formulation from an augmented Lagrangian approach is presented. In this method, no penalty parameter is introduced for the regularisation of the contact problem. The contact approach, based on the mortar method, gives a smooth representation of the contact forces across the bodies interface, and can be used in arbitrarily curved 3D configurations. The projection surface used for integrating the equations is built using a local Cartesian basis defined in each contact element. A unit normal to the contact surface is defined locally at each element, simplifying the implementation and linearisation of the equations. The displayed examples show that the algorithm verifies the contact patch tests exactly, and is applicable to large displacements problems with large sliding motions.
Large deformation frictional contact formulation based on a velocity description
Proceedings in applied mathematics & mechanics, 2004
A special contact formulation which is compatible with the so-called 'Solid-Shell' is developed for applications involving large deformation and frictional contact. The contact conditions are considered in the covariant form from the surface geometry point of view, which is very similar to shell theory. The contact integral and the necessary kinematical values are considered on the tangent plane of the contact surface for which a special surface coordinate system is introduced. A focus is on the regularization of the frictional conditions, which leads to evolutions equations in the form of covariant derivatives. A geometrical interpretation of these equations as the parallel translation is used to overcome the problem of discontinuity of the characteristics on element boundaries. The main advantage of the developments is a more algorithmic and geometrical structure of the tangent matrix. Different integration techniques based on higher order integration formulae as well as based on the subdivision of the contact area into subdomains allow to construct elements with diminishing error for the contact patch test in the non-frictional case. The segment-to-segment and the segment-to-analytical surface approaches are developed for the frictional problems with large sliding. Within the numerical examples the focus is also on the effect where a 3D continuum approach as e.g. for the solid-shell elements appears to be beneficial in the context with frictional contact.
A contact domain method for large deformation frictional contact problems. Part 1: Theoretical basis
Computer Methods in Applied Mechanics and Engineering, 2009
In the first part of this work, the theoretical basis of a frictional contact domain method for two-dimensional large deformation problems is presented. Most of the existing contact formulations impose the contact constraints on the boundary of one of the contacting bodies, which necessitates the projection of certain quantities from one contacting surface onto the other. In this work, the contact constraints are formulated on a so-called contact domain, which has the same dimension as the contacting bodies. This contact domain can be interpreted as a fictive intermediate region connecting the potential contact surfaces of the deformable bodies. The introduced contact domain is subdivided into a non-overlapping set of patches and is endowed with a displacement field, interpolated from the displacements at the contact surfaces. This leads to a contact formulation that is based on dimensionless, strain-like measures for the normal and tangential gaps and that exactly passes the contact patch test. In addition, the contact constraints are enforced using a stabilized Lagrange multiplier formulation based on an interior penalty method (Nitsche method). This allows the condensation of the introduced Lagrange multipliers and leads to a purely displacement driven problem. An active set strategy, based on the concept of effective gaps as entities suitable for smooth extrapolation, is used for determining the active normal stick and slip patches of the contact domain.
Algorithme mortier pour des problèmes de contact en grandes déformations
Le Centre pour la Communication Scientifique Directe - HAL - memSIC, 2013
For the simulation of contact problems with the finite element method we usually use the NTS (Node-To-Segment) formulation for transmitting contact constraints from one body to another. The use of the NTS method involves a loss of accuracy in the calculation of the displacements and stresses in the contact area because of the node-wise contact constraint enforcement. We can overcome this problem by using the mortar method [1] which has been successfully applied to solve contact problems with finite deformations. The enforcement of contact constraints is applied in a weak sense throughout the contact interface. In this work, we describe several techniques for solving contact problems using the mortar approach. The penalty technique [2], the Lagrange multipliers method [2-3] and the augmented Lagrangian formulation [4-5] are used for solving various numerical problems. A simplified algorithm is described to implement these types of formulations.
Computer Methods in Applied Mechanics and Engineering, 2006
Finite element discretizations including contact usually apply the standard NTS-(node-to-segment) element. Thus a coupling with higher order solid elements leads to inconsistencies in the transmission of contact stresses. One can circumvent this using the mortar method which includes a weak projection of the contact constraints. In this paper we present a penalty formulation based on the mortar concept for two-dimensional large deformation frictional contact. The discretization of the contact surfaces contains quadratic approximations which can be used within a quadratic approximation of the solid elements. To obtain a simple and efficient matrix formulation, the moving friction cone algorithm is applied.
Improved robustness and consistency of 3D contact algorithms based on a dual mortar approach
Computer Methods in Applied Mechanics and Engineering, 2013
Mortar finite element methods have been successfully applied as discretization scheme to a wide range of contact and impact problems in recent years. The 3D finite deformation contact formulation taken up and enhanced in this paper is based on a mortar approach using so-called dual Lagrange multipliers, which substantially facilitate the treatment of interface constraints as compared with standard mortar techniques. Despite being quite well-established in the meantime, dual mortar methods may lack robustness or even consistency in certain situations, e.g., when large curvatures occur in the contact zone or when one body slides off another at a sharp edge. However, since such scenarios are regularly appearing in engineering practice, the present contribution provides several new extensions that completely resolve these issues and thus significantly improve the applicability of dual mortar formulations for challenging contact problems. The proposed extensions include a consistent biorthogonalization procedure to obtain feasible dual Lagrange multiplier shape functions close to boundaries of the contact surfaces, an improved conditioning of the global linear system of equations by nodal scaling and a novel approach to unify the advantages of standard and dual mortar methods via a Petrov-Galerkin type of Lagrange multiplier interpolation. Several numerical examples demonstrate the achievable improvements in terms of consistency and robustness for 3D contact analysis including finite deformations.