Time‐step constraints in transient coupled finite element analysis

In transient finite element analysis, reducing the time‐step size improves the accuracy of the solution. However, a lower bound to the time‐step size exists, below which the solution may exhibit spatial oscillations at the initial stages of the analysis. This numerical ‘shock’ problem may lead to accumulated errors in coupled analyses. To satisfy the non‐oscillatory criterion, a novel analytical approach is presented in this paper to obtain the time‐step constraints using the θ‐method for the transient coupled analysis, including both heat conduction–convection and coupled consolidation analyses. The expressions of the minimum time‐step size for heat conduction–convection problems with both linear and quadratic elements reduce to those applicable to heat conduction problems if the effect of heat convection is not taken into account. For coupled consolidation analysis, time‐step constraints are obtained for three different types of elements, and the one for composite elements matches that in the literature. Finally, recommendations on how to handle the numerical ‘shock’ issues are suggested. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A combined surface integral and finite element solution for a three-dimensional contact problem

A method is described to determine contact stresses and deformation using a combination of the finite element method and a surface integral form of the Bousinesq solution. Numerical examples of contacting hypoid gears are presented.     

Three-dimensional finite element contact algorithm for metal forming

This work presents a three-dimensional rigid plastic finite element formulation. The workpiece is discretized with eight node hexahedral isoparametric elements. Friction is included in the formulation by means of a shear stress depending on the relative velocity between the workpiece and the tool. Special attention is given to the contact problems, and a three-dimensional contact algorithm based on a discretization of the tool surface with triangular elements is presented. Finally, some selected examples are solved, in order to show the capabilities of the formulation.     

An algorithm for multipoint constraints in finite element analysis

A method of introducing general constraint equations into finite element matrix equations is described. The characteristics of the method are that it requires no reordering or condensation of the equations, no large matrix operations, and no increase in the number of unknowns. The method is suitable for application in minicomputer implementations of finite element analysis unless a large number of constraints is to be applied.     

An adaptive finite element method for transient compressible flows

An adaptive finite element method for the solution of time dependent strongly compressible flows in two dimensions is described. The computational domain is represented by an unstructured assembly of linear triangular elements and the mesh adaptation is achieved by local regeneration of the grid, using an error estimation procedure coupled to an automatic triangular mesh generator. Problems involving shock propagation are solved to illustrate the numerical performance of the proposed approach.     

Linear equality constraints in finite element approximation

The typical numerical problem associated with finite element approximations is a quadratic programming problem with linear equality constraints. When nodal variables are employed, the coefficient matrix of the constraint equations, [ A ], acquires a block-diagonal structure. The transformation from polynomial coefficients to nodal variables involves finding a basis for [ A ] and computing its inverse. Simultaneous satisfaction of completeness and C¹ (or higher) continuity requirements establishes linear relationships among the nodal variables and precludes inversion of the basis by exclusively element-level operations. Linear dependencies among the constraint equations and among the nodal variables can be evaluated by the simplex method. The computational procedure is outlined.     

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.     

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

A novel finite element formulation for frictionless contact problems

This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.     

Enhanced multiple-point beam-to-beam frictionless contact finite element

In this paper a new contact finite element for beams with circular cross-sections is presented. The element is an enhancement of the previously formulated point-wise beam-to-beam contact finite elements to be used in cases when beams get in contact at very acute angles. In such situation, if beam deformations in the vicinity of the contact zone are taken into account, the contact is not point-wise but it extends to a certain area. To cover such a case in a more realistic way, two additional pairs of contact points are introduced to accompany the original single pair of contact points. The central pair is determined using the orthogonality conditions for the beam axes and the positions of two extra points are defined on one beam axis by a shift of the local co-ordinate. This shift depends on beams geometry and the current angle between tangent vectors at the central contact point. The appropriate kinematic variables for normal contact together with their finite element approximation are derived. Basing on the weak form for normal contact and its linearisation, the tangent stiffness matrix and the residual vector are derived. The new element is tested using author’s computer programs and comparisons with the point-wise contact elements are made.     

An adaptive finite element algorithm for contact problems in plasticity

Due to the fact that in contact problems the contact area is not known a priori, a sufficient discretization to obtain a convergent finite element solution cannot be supplied from the outset. Therefore it is necessary to use adaptive finite element methods to adjust automatically the mesh sizes not only in the bodies under consideration but also in the contact zone. In this paper we develop an adaptive method for geometrically linear contact problems, which also includes elastoplastic material behavior. The radial return algorithm is used to derive the error estimator for one time increment of the solution process. The error estimator is based on the Zienkiewicz-Zhu projection scheme, which is extended to account for the special situation in the contact interface.In memoriam of J. C. Simo     

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.     

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.     

Automation of finite element formulations for large deformation contact problems

The aim of this paper is to present a general method for automation of finite element formulations of large deformation contact problems. A new automatic‐differentiation‐based notation is introduced that represents a bridge between the classical mathematical notation of contact mechanics and the actual computer implementation of contact finite elements. Automation of derivation of the required formulas (e.g. element residual and tangent matrix) combined with automatic code generation makes the finite element implementation possible at a moderate effort. Accordingly, several 3D contact formulations have been implemented in this work, including penalty and augmented Lagrangian treatments of contact constraints, and several contact smoothing techniques. A typical benchmark problem could thus be executed in an objective way leading to a comprehensive study of the efficiency and the accuracy of various formulations of 3D contact finite elements. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A parallel finite element contact/impact algorithm for non-linear explicit transient analysis: Part I—The search algorithm and contact mechanics

A contact algorithm has been developed and implemented in a non-linear dynamic explicit finite element program to analyse the response of three-dimensional shell structures. The contact search algorithm accounts for initial contact, sliding, and release through the use of a parametric representation of the motion of points located on the surface of the structure combined with a contact surface representation which approximates the actual surface by means of triangular search planes. The mechanics of contact is handled by taking advantage of the fact that an explicit time integration scheme results in very small displacements during a time step. The amount of overlap of the discrete representation of the surfaces which occurs at contact is taken as a measure of the approach of the surfaces. Hence, experimental data which relates approach to normal contact pressure can be used to determine the contact pressure applied to the finite element model of the surface as contact evolves. The friction model also incorporates experimental data on the dependence of the coefficient of friction on both the relative sliding velocity and on the relative tangential displacement between surfaces in contact observed in friction tests. The parallel implementation of this contact algorithm and its performance on a 128-processor distributed-memory multiprocessor computer is discussed in Part II of this paper.