Isoparametric,finite element,variational solution of integral equations for three-dimensional fields

An improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh–Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.     

A finite element solution method for contact problems with friction

A new finite element solution method for the analysis of frictional contact problems is presented. The contact problem is solved by imposing geometric constraints on the pseudo equilibrium configuration, defined as a configuration at which the compatibility conditions are violated. The algorithm does not require any a priori knowledge of the pairs of contactor nodes or segments. The contact condition of sticking, slipping, rolling or tension release is determined from the relative magnitudes of the normal and tangential global nodal forces. Contact iterations are in general found to converge within one or two iterations. The analysis method is applied to selected problems to illustrate the applicability of the solution procedure.     

A mixed finite element model for the elastic contact problem

The static elastic contact problem is approached using Lagrange multipliers, leading to a mixed finite element problem. A non-linear friction law is introduced explicitly and the non-local character of the friction phenomena is implicitly assumed. In order to avoid stress oscillations near singular points, a perturbed Lagrangian functional is considered. The algorithms herein proposed do not impose nodal dependencies over the contact surfaces, allowing for the independent discretization of both bodies. The method is able to model simultaneous contact over different regions of any geometrical shape. Computer code, examples and results presented here are restricted to axisymmetrical and bidimensional cases.     

Numerical solution of the point contact problem using the finite element method

The elastohydrodynamic lubrication problem, in which the lubricant pressure and film thickness are sensitive to surface deformation, is solved by using a finite element procedure and the Newton method. The numerical procedure is applied to the point contact problem, in which a thin lubricant film is maintained between two balls loaded together by a high load under conditions of pure rolling. The present analysis shows that pressure spikes are formed near the outlet region, a result which has been found in the line contact problem and which has been conjectured in the present problem.     

A computational method for frictional contact problem using finite element method

Using the finite element method a numerical procedure is developed for the solution of the two-dimensional frictional contact problems with Coulomb's law of friction. The formulation for this procedure is reduced to a complementarity problem. The contact region is separated into stick and slip regions and the contact stress can be solved systematically by applying the solution technique of the complementarity problem. Several examples are given to demonstrate the validity of the present formulation.     

Lagrange constraints for transient finite element surface contact

A new approach to enforce surface contact conditions in transient non-linear finite element problems is developed in this paper. The method is based on the Lagrange multiplier concept and is compatible with explicit time integration operators. Compatibility with explicit operators is established by referencing Lagrange multipliers one time increment ahead of associated surface contact displacement constraints. However, the method is not purely explicit because a coupled system of equations must be solved to obtain the Lagrange multipliers. An important development herein is the formulation of a highly efficient method to solve the Lagrange multiplier equations. The equation solving strategy is a modified Gauss-Seidel method in which non-linear surface contact force conditions are enforced during iteration. The new surface contact method presented has two significant advantages over the widely accepted penalty function method: surface contact conditions are satisfied more precisely, and the method does not adversely affect the numerical stability of explicit integration. Transient finite element analysis results are presented for problems involving impact and sliding with friction. A brief review of the classical Lagrange multiplier method with implicit integration is also included.     

Comparison between a finite element and a composite method for a three-dimensional potential problem

This paper is concerned with the solution of the three-dimensional potential problem for electromagnetic river gauging. It extends previous ideas of joining finite elements in an interior region to one infinite external element treated by the boundary integral method^1,2 to this case where there are two external infinite elements representing the river and the ground. The boundary continuity conditions on the infinite river–ground interface, as well as the internal–external interfaces, are dealt with by introducing a variational principle with relaxed continuity requirement³.     

A finite element solution for the two-dimensional elastic contact problems with friction

A numerical procedure developed for solving the two-dimensional elastic contact problems with friction is presented. This is a generalization of a procedure developed by Francavilla and Zienkiewicz to include frictional effects under proportionate loading. The method uses the flexibility matrix obtained by inversion of condensed stiffness matrix formed by eliminating all the nodes except those where contact is likely to take place and those with external forces. Compatibility of displacements for both normal and tangential directions is applied to those nodes which do not slip. However, for the nodes which slip, compatibility of displacements is applied for normal direction only and slip condition is applied in the tangential direction. The technique has been applied to several problems and very good results have been obtained. The number of iterations needed are very small.     

A finite element formulation of three-dimensional contact problems with slip and friction

The paper describes a special finite element for three-dimensional, large displacement analysis of contact problems with slip and friction. This element may be used to model contact between several finite element bodies or contact between a finite element body and a flexible or rigid geometrical surface fixed in space or moving with time. The contact formulation is based on the concept of a spring-supported, moving disk that transfers normal contact forces and Coulomb friction forces. The contact surface has a finite, prescribed boundary.The contact element has been incorporated into the general-purpose, nonlinear, finite element program FENRIS. Three examples of its application are described in the paper.J. W. Simons was previously NTNF Fellow, Division of Technology, Trondheim, Norway     

A mortar finite element approach for point,line, and surface contact

An approach for investigating finite deformation contact problems with frictional effects with a special emphasis on nonsmooth geometries such as sharp corners and edges is proposed in this contribution. The contact conditions are separately enforced for point contact, line contact, and surface contact by employing 3 different sets of Lagrange multipliers and, as far as possible, a variationally consistent discretization approach based on mortar finite element methods. The discrete unknowns due to the Lagrange multiplier approach are eliminated from the system of equations by employing so‐called dual or biorthogonal shape functions. For the combined algorithm, no transition parameters are required, and the decision between point contact, line contact, and surface contact is implicitly made by the variationally consistent framework. The algorithm is supported by a penalty regularization for the special scenario of nonparallel edge‐to‐edge contact. The robustness and applicability of the proposed algorithms are demonstrated with several numerical examples.     

Finite element solution of a boundary integral equation for mode I embedded three-dimensional fractures

The singular integral equation governing the opening of a mode I embedded three-dimensional fracture in an infinite solid was solved by applying the finite element method. The strategy is to formulate the equation into weak form, and to transfer the differentiation from the singular term, 1/r, in the equation to the test function. A numerical algorithm was thus developed. The numerical solutions for circular and elliptical fractures under the action of polynomial pressure distributions were compared with the analytical solutions by Green and Sneddon,¹² Irwin,¹³ Shah and Kobayashi¹⁴ and Nishioka and Atluri.¹⁶ The results have demonstrated that the numerical method reported is accurate and efficient.     

Surface integral and finite element hybrid method for two- and three-dimensional fracture mechanics analysis

This paper summarizes the development of the surface integral and finite element hybrid method for two and three dimensional fracture mechanics analysis. The fracture, which is a discontinuity in the displacement field, is modeled explicitly and efficiently by use of dislocations for two dimensional analysis and by dipoles of point forces for three dimensional applications. The boundary value problem of a fracture in a finite domain is solved by (incremental) superposition of a finite element model of the finite body without the crack and a surface integral model of an infinite body with the crack, ensuring proper traction and displacement matching at the boundaries. Finite elements are also used to model nonhomogeneity and plasticity, though isotropic kernels are used for the integral equation. A variety of two and three dimensional problems have been modeled and excellent agreement with analytical solutions has been obtained. Propagation problems in two dimensions have also been modeled and the predicted results agree very well with experimental observations.

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Résumé On résume le développement de la méthode de l'intégrale de surface et des éléments finis hybrides pour l'analyse de la mécanique de rupture à deux et à trois dimensions. La rupture, considérée comme une discontinuité dans un champ de déplacement, est représentée de manière explicite et avec efficience en recourant aux dislocations dans le cas de l'analyse à deux dimensions, et aux dipoles de forces ponctuelles dans le cas des applications à trois dimensions. Le problème de la valeur aux limites d'une rupture dans un domaine fini est solutionné par la superposition par incréments d'un modèle à éléments finis d'un corps fini dépourvu de fissures et d'un modèle en intégrale de surface d'un corps infini pourvu d'une fissure, en s'assurant que les conditions appropriées de traction et de déplacement s'accordent aux limites. On utilise également les éléments finis pour représenter une non homogénité ou de la plasticité, bien que des kernels isotropes soient employés pour l'équation intégrale. On a représenté divers problèmes à deux et à trois dimensions et on a obtenu un excellent accord avec les solutions analytiques. Les problèmes de propagation en deux dimensions ont également été modélisés, et ces résultats prévus sont en excellent accord avec les observations expérimentales.

     

A finite element method for contact using a third medium

The numerical simulation of contact problems is nowadays a standard procedure in many engineering applications. The contact constraints are usually formulated using either the Lagrange multiplier, the penalty approach or variants of both methodologies. The aim of this paper is to introduce a new scheme that is based on a space filling mesh in which the contacting bodies can move and interact. To be able to account for the contact constraints, the property of the medium, that imbeds the bodies coming into contact, has to change with respect to the movements of the bodies. Within this approach the medium will be formulated as an isotropic/anisotropic material with changing characteristics and directions. In this paper we will derive a new finite element formulation that is based on the above mentioned ideas. The formulation is presented for large deformation analysis and frictionless contact.     

Combination of finite element and integral transform techniques in a heat conduction quasi-static problem

A hybrid method, which consists in using the finite element method for fixed bodies and analytical techniques for moving ones, was developed for the thermal analysis of tribological situations. Unlike the well-known finite element method, the hybrid method gives accurate results even at high sliding speed. From a practical point of view, both computer time and storage requirements are greatly reduced. The method appears as a valuable technique to predict surface temperature in mechanical assemblies.     

A finite element for the three-dimensional deformation of a circular ring

A finite element is developed for the three-dimensional deformation of a circular ring. The derivation is based on ring theory. The unique feature of the element is that it provides for coupling between the in-plane and out-of-plane loads and deflections. The element is derived in terms of polar co-ordinates so that no co-ordinate transformation is needed to assemble the elements. Three ring deflection problems are solved using an assembly of the elements. Two of the solutions are compared with analytical results and one is compared with experimental results. The comparisons verify the theory.     

An adaptive finite element approach for the free surface seepage problem

A posteriori error estimates and adaptive mesh refinements are now on a rigorous mathematical foundation for linear, elliptic boundary-value problems of second order. Yet, for non-linear problems only a few results have been obtained till now. In this paper we consider as a non-linear model problem the two-dimensional fluid flow with free surface and show how results from linear a posteriori theory can be used to control the non-linear iteration and to refine the mesh adaptively. A numerical example shows that, similar to linear problems, considerable improvement of the accuracy is obtained by an adaptive mesh refinement and that the influence of singularities on the order of convergence disappears.     

A frontal solution program for finite element analysis

The program given here assembles and solves symmetric positive–definite equations as met in finite element applications. The technique is more involved than the standard band–matrix algorithms, but it is more efficient in the important case when two-dimensional or three-dimensional elements have other than corner nodes. Artifices are included to improve efficiency when there are many right hand sides, as in automated design. The organization of the program is described with reference to diagrams, full notation, specimen input data and supplementary comments on the ASA FORTRAN print-out.     

A three-dimensional least-squares finite element technique for deformation analysis

The development of a three-dimensional least-squares finite element technique suitable for deformation analysis was presented. By adopting a spatial viewpoint, a consistent rate formulation that treats deformation as a process was established. The technique utilized the least-squares variational principle that minimizes the squares of errors encountered in any attempt to meet the field equations exactly. Both velocity and Cauchy stress rate fields were discretized by the same linear interpolation function. The discretization always yields a sparse, symmetric, and positive-definite coefficient matrix. A conjugate gradient iterative solver with incomplete-Choleski preconditioner was used to solve the resulting linear system of equations. Issues such as finite element formulation, mesh design, code efficiency, and time integration were addressed. A set of linear elastic problems was used for patch-test; both homogeneous and non-homogeneous deformations were considered. Additionally, two finite elastic deformation problems were analysed to gauge the overall performance of the technique. The results demonstrated the computational feasibility of a three-dimensional least-squares finite element technique for deformation analysis. © 1997 John Wiley & Sons, Ltd.     

A general finite element approach for contact stress analysis

This paper presents a general finite element approach for the treatment of contact stress problems. Stanctard shape function routines are used for the detection of contact between previously separate meshes and for the application of displacement constraints where contact has been identified. The mesh contact routines are installed in an incremental approach whereby the contact constraints are imposed by using either penalty functions or Lagrange multipliers.