A Cell Method Formulation of 3-D Electrothermomechanical Contact Problems With Mortar Discretization (original) (raw)
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A Mortar Cell Method for Electro-Thermal Contact Problems
IEEE Transactions on Magnetics, 2000
A 3-D mortar domain decomposition approach based on the cell method (CM) for analyzing electro-thermal contact problems is presented. The computational domain is subdivided into non-overlapping regions discretized according to the CM, where variables and field equations are expressed directly in integral form suitable for coupling the contact problem to the electro-thermal one in the bulk regions. Voltage and temperature discontinuities at contact interfaces are modeled by diagonal conductance matrices. The electrical and thermal continuity between contacting regions is enforced by means of dual Lagrange multipliers. Nonlinear equations are cast into a saddle-point form ensuring existence and uniqueness of the solution. Problem size is reduced by the Schur complement approach. A 3-D finite-element software for multiphysics problems is used to validate the mortar cell method.
Dual Quadratic Mortar Finite Element Methods for 3D Finite Deformation Contact
SIAM Journal on Scientific Computing, 2012
Mortar finite element methods allow for a flexible and efficient coupling of arbitrary nonconforming interface meshes and are by now quite well established in nonlinear contact analysis. In this paper, a mortar method for three-dimensional (3D) finite deformation contact is presented. Our formulation is based on so-called dual Lagrange multipliers, which in contrast to the standard mortar approach generate coupling conditions that are much easier to realize, without impinging upon the optimality of the method. Special focus is set on second-order interpolation and on the construction of novel discrete dual Lagrange multiplier spaces for the resulting quadratic interface elements (8-node and 9-node quadrilaterals, 6-node triangles). Feasible dual shape functions are obtained by combining the classical biorthogonality condition with a simple basis transformation procedure. The finite element discretization is embedded into a primal-dual active set algorithm, which efficiently handles all types of nonlinearities in one single iteration scheme and can be interpreted as a semismooth Newton method. The validity of the proposed method and its efficiency for 3D contact analysis including Coulomb friction are demonstrated with several numerical examples.
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.
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.
Domain Decomposition Algorithm for Solving Contact of Elastic Bodies
Lecture Notes in Computer Science, 2002
A non-overlapping domain decomposition is applied to a multibody unilateral contact problem with given friction (Tresca's model). Approximations are proposed on the basis of the primary variational formulation (in terms of displacements) and linear finite elements. For the discretized problem we employ the concept of local Schur complements, grouping every two subdomains which share a contact area. The proposed algorithm of successive approximations can be recommended for "short" contacts only, since the contact areas are not divided by interfaces.
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.
Use of Matlab for Domain Decomposition Method for Contact Problem in Elasticity
2005
In the present paper we will deal with numerical solution of a generalized semicoercive contact problem in linear elasticity, for the case that several bodies of arbitrary shapes are in mutual contacts and are loaded by external forces, by using the non-overlapping domain decomposition and finite elements method. The numerical example will be presented.
Mortar Methods for Single-and Multi-Field Applications in Computational Mechanics
2013
Mortar finite element methods are of great relevance as a non-conforming discretization technique in various single-field and multi-field applications. In computational contact analysis, the mortar approach allows for a variationally consistent treatment of non-penetration and frictional sliding constraints despite the inevitably non-matching interface meshes. Other single-field and multi-field problems, such as fluid-structure interaction (FSI), also benefit from the increased modeling flexibility provided by mortar methods. This contribution gives a review of the most important aspects of mortar finite element discretization and dual Lagrange multiplier interpolation for the aforementioned applications. The focus is on parallel efficiency, which is addressed by a new dynamic load balancing strategy and tailored parallel search algorithms for computational contact mechanics. For validation purposes, simulation examples from solid dynamics, contact dynamics and FSI will be discussed.
FETI domain decomposition method to solution of contact problems with large displacements
2005
The solution to contact problems between solid bodies poses difficulties to solvers because in general neither the distributions of the contact tractions throughout the areas currently in contact nor the configurations of these areas are known a priori. This implies that the contact problems are inherently strongly nonlinear. Probably the most popular solution method is based on direct iterations with the non-penetration conditions imposed by the penalty method ([Z93] or [W02]). The method enables easily enhance other non-linearity such as in the case of large displacements. In this paper we are concerned with application of a variant of the FETI domain decomposition method that enforces feasibility of Lagrange multipliers by the penalty [DH04b]. The dual penalty method, which has been shown to be optimal for small displacements is used in inner loop of the algorithm that treats large displacements. We give results of numerical experiments that demonstrate high efficiency of the FET...
Modeling of contact problems by boundary element and finite element methods
2007
This paper is concerned with the numerical modeling of contact problems in elastostatics with Coulomb friction. The boundary element method (BEM) and the finite element method (FEM) are presented and compared from an algorithmic point of view. Analysis of a contact problem between two elastic bodies is preformed so as to demonstrate the developed algorithms and to highlight their performance.