A solution method for static and dynamic analysis of three-dimensional contact problems with friction

A solution method is presented for the analysis of contact between two (or more) three-dimensional bodies. The surfaces of the contacting bodies are discretized using quadrilateral surface segments. A Lagrange multiplier technique is employed to impose that, in the contact area, the surface displacements of the contacting bodies are compatible with each other. Distributed contact tractions over the surface segments are calculated from the externally applied forces, inertia forces and internal element stresses. Using the segment tractions, Coulomb's law of friction is enforced in a global sense over each surface segment. The time integration of dynamic response is performed using the Newmark method with parameters and . Using these parameters the energy and momentum balance criteria for the contacting bodies are satisfied accurately when a reasonably small time step is used.

The applicability of the algorithm is illustrated by selected sample numerical solutions to static and dynamic contact problems.     

A numerical method for the solution of static and dynamic three-dimensional elasticity problems

Kupradze's functional equation, reduced to a regular Fredholm integral equation of the first kind, was solved by applying a new numerical method, based on numerical integration, whose collocation points are chosen in self-similar surfaces. An application of the method to a particular problem of elasticity demonstrated a sufficient accuracy and stability of the method. It was shown that the proposed method is faster, simpler and more easily programmable than the existing classical methods. Finally, suggestions were made for a better use of the method and for possible improvements.     

A mathematical programming approach to three-dimensional contact problems with friction

The present study is concerned with the numerical treatment of linear elastic contact problems with friction. Making the restriction that the contact surface is constant during the loading history, we obtain a variational inequality formulation, which justifies a finite element approximation. The finitedimensional problem so obtained is shown to be equivalent to a problem of mathematical programming—namely, a parametric linear complementarity problem (LCP) involving derivatives. A solution procedure for such problems has previously been suggested. In this paper, the conditions under which this procedure defines a unique solution map, which describes the evolution of contact stresses and displacements for a prescribed load history, are given. As an application of the theory presented, the indentation of a halfspace by a flat-ended square punch is considered. Both the loading cases of a vertical displacement of the punch that increases monotonically and the gradual removal of the same displacement are considered.     

A solution method for dynamic contact problems

An efficient method is presented for analyzing the transient dynamic contact problems of elastic bodies in this paper. This approach exploits the Lagrange multiplier concept and a special time integration algorithm. Due to the introduced high-frequency dissipation in this time integration algorithm, this method can lead to the effective analysis of real response of elastic bodies with dynamic surface contact constraints. The results of numerical examples show that this method can avoid the weakness of the classical Lagrange multiplier method in dealing with dynamic contact problems with relatively high inertial forces. Stable results can be provided when the time integration step size is small. The properties of this method have also been discussed in this paper.     

A parallel spectral element method for dynamic three-dimensional nonlinear elasticity problems

We present a high-order method employing Jacobi polynomial-based shape functions, as an alternative to the typical Legendre polynomial-based shape functions in solid mechanics, for solving dynamic three-dimensional geometrically nonlinear elasticity problems. We demonstrate that the method has an exponential convergence rate spatially and a second-order accuracy temporally for the four classes of problems of linear/geometrically nonlinear elastostatics/elastodynamics. The method is parallelized through domain decomposition and message passing interface (MPI), and is scaled to over 2000 processors with high parallel performance.     

A Dirichlet–Neumann type algorithm for contact problems with friction

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We introduce a new algorithm for the numerical solution of a nonlinear contact problem with Coulomb friction between linear elastic bodies. The discretization of the nonlinear problem is based on mortar techniques. We use a dual basis Lagrange multiplier space for the coupling of the different bodies. The boundary data transfer at the contact zone is essential for the algorithm. It is realized by a scaled mass matrix which results from the mortar discretization on non-matching triangulations. We apply a nonlinear block Gauss–Seidel method as iterative solver which can be interpreted as a Dirichlet–Neumann algorithm for the nonlinear problem. In each iteration step, we have to solve a linear Neumann problem and a nonlinear Signorini problem. The solution of the Signorini problem is realized in terms of monotone multigrid methods. Numerical results illustrate the performance of our approach in 2D and 3D. Received: 20 March 2001 / Accepted: 1 February 2002 Communicated by P. Deuflhard     

A gap element for three-dimensional elasto-plastic contact problems

A simple yet efficient contact algorithm with gap elements is developed to solve three-dimensional elasto-plastic contact problems without friction. A special gap element is presented, together with a stress invariance principle, to model the contact process. The solution is achieved through an iterative procedure which adjusts the modulus of the gap elements. The elasto-plastic solid element with variable nodes and nonconforming modes is employed for the effective and economic three-dimensional finite element analysis. The validity and performance of the proposed contact algorithm with gap elements are demonstrated through various numerical tests.     

An automatic incrementation technique for contact problems with friction

A numerical method for analysis of elastostatic contact problems with friction has been developed. This class of problems are load history dependent because of the irreversible nature of frictional forces. An automatic incrementation technique of the applied load has been developed and implemented in the algorithm. The method is a direct method based on an iterative procedure applied to a set of linear equations established with the finite element method. The size of the applied load increments, automatically chosen by the algorithm, is in general influenced both by the nature of the problem and of the discretization of the bodies involved. The frictional forces occurring in the slip zone of the contact area are treated as known tangential forces calculated from the normal forces in the previous iteration. This piecewise linear treatment of the frictional contact problem requires an innermost iteration loop over the applied tangential force.The tangential force must coincide with the coefficient of friction times the normal force obtained in the last iteration, otherwise a new tangential force has to be calculated and the system of equations must be solved for a new right hand side vector. The automatic incrementation technique is based on the fact that each iteration is a linear problem. A tentative load increment is used in the solution of a certain iteration. A linear scaling of this solution is performed afterwards. A load scale factor is calculated in each contact node pair where a change of contact condition will occur. The change in contact status corresponding to the node pair with the smallest load scale factor is the only change which is accomplished in a certain iteration. The uniqueness of this kind of contact problem with friction has not been mathematically proven for a general case.The method has been applied to a case of loading and unloading of an elastic halfspace by a rigid cylindrical stamp and compared to solutions by Spence and Turner.     

A parallel multigrid method for constrained minimization problems and its application to friction,contact, and obstacle problems

The parallel solution of constrained minimization problems requires special care to be taken with respect to the information transfer between the different subproblems. Here, we present a nonlinear decomposition approach which employs an additional nonlinear correction step along the processor interfaces. Our approach is generic in the sense that it can be applied to a wide class of minimization problems with strongly local nonlinearities, including even nonsmooth minimization problems. We also describe the implementation of our nonlinear decomposition method in the object oriented library ObsLib \(++\). The flexibility of our approach and its implementation is presented along different problem classes as obstacle problems, frictional contact problems and biomechanical applications. For the same examples, number of iterations, computation time, and parallelization speedup are measured, and the results demonstrate that the implementation scales reasonably well up to 4096 processors.     

A computational method for elasto-plastic contact problems

A numerical procedure is developed for the solution of elasto-plastic contact problems. The contacting surface is assumed unbonded and frictionless. Under the assumption of small deformations, an incremental formulation is obtained as a sequence of partial differential equations with inequalities. An equivalent minimization problem is proposed and their equivalence is shown. A quadratic programming problem based on the finite element technique is used for numerical computation. Several examples tested using a modified simplex method show the practicability of the method.     

An identification method for static and coulomb friction coefficients

This paper deals with the identification of static and Coulomb friction coefficients. An approach based on limit cycles properties once the mechanical system is brought to oscillations is proposed. This leads to an explicit formulation in the regulator parameters on one part, and the oscillation amplitude and period on the other part. The method is validated by a simulation example and an experimental benchmark. Recommended by Editor Jae Weon Choi. Bastien Borsotto was graduated from Supélec, Gif sur Yvette, France in 2004, and received the Ph.D. degrees in Physics from Paris-Sud University in 2008. He is currently a Engineer in the French society Renault SA. His current interests include clutch driving for automatic transmission, and more precisely managing of creeping and takeoff stages. Emmanuel Godoy was graduated from Supélec, Gif sur Yvette, France, in 1984. He is currently a Professor in the Automatic Control Department of Supélec. His area research is about modelling and robust control of mechatronic systems and common control approaches for industrial problems. Dominique Beauvois was graduated from Supélec, Gif sur Yvette, France in 1977. He is currently a Professor in the Automatic Control Department of Supélec. His current interests include estimation, identification techniques and common control approaches for industrial problems. Emmanuel Devaud was graduated from Supélec, Gif sur Yvette, France in 1995, and received the Ph.D. degree in Physics from Paris-Sud University in 1999. He is currently a Engineer in the French society Renault SA, and his function is project manager for automatic gearbox systems.     

A multi-grid method for variational inequalities in contact problems

The convergence of the method is proved and it is shown that the objective function corresponding to the quadratic programming problem is monotonically decreasing. The results of numerical tests for an elasticity contact problem are presented.     

A simple algorithm for three-dimensional finite element analysis of contact problems

This paper addresses the numerical solution of three-dimensional frictionless contact problems by a finite element method. The two-body contact problem is considered in the context of fully non-linear kinematics. The impenetrability constraint is satisfied via a classical penalty formulation. The contacting surfaces are discretized by means of projections of the interacting element faces onto suitably chosen flat surfaces. Attention is focused on the efficiency of the overall algorithm. Numerical simulations are conducted for a series of test problems to assess the performance of the proposed methodology.     

Finite element analysis of elastodynamic sliding contact problems with friction

This paper presents a general but effective finite element technique to analyze elastodynamic sliding contact problems with friction. The deformed contact area is obtained using the constraint conditions developed by a quadratic mathematic programming technique. Lagrangian multipliers are introduced to evaluate the contact pressures due to friction and determine the adhesion or release of contact surface. Based on these, the sliding process between two contact surfaces is accurately modelled. This work also provides the correction formulae for modelling the transient response of velocities and accelerations on the contact surface when the initial impact or release of the contact surface occurred. Several numerical examples are carried out to demonstrate the validity and accuracy of this work. The complete scheme of the transient sliding response of two contact components subjected to oblique impact loadings is drawn.     

A two-level iteration method for solution of contact problems

The merits and limitations of some existing procedures for the solution of contact problems, modeled by the finite element method, are examined. Based on the Lagrangian multiplier method, a partitioning scheme can be used to obtain a small system of equation for the Lagrange multipliers which is then solved by the conjugate gradient method. A two-level contact algorithm is employed which first linearizes the nonlinear contact problem to obtain a linear contact problem that is in turn solved by the Newton method. The performance of the algorithm compared to some existing procedures is demonstrated on some test problems.     

Equivalent static displacements method for contact force optimization

Structural and Multidisciplinary Optimization - Contact force optimization problems exist extensively in engineering. However, there are few simple and universal methods to solve contact force...     

Predicting program execution times by analyzing static and dynamic program paths

This paper describes a method to predict guaranteed and tight deterministic execution time bounds of a sequential program. The basic prediction technique is a static analysis based on simple timing schema for source-level language constructs, which gives accurate predictions in many cases. Using powerful user-provided information, dynamic path analysis refines looser predictions by eliminating infeasible paths and decomposing the possible execution behaviors in a pathwise manner. Overall prediction cost is scalable with respect to desired precision, controlling the amount of information provided. We introduce a formal path model for dynamic path analysis, where user execution information is represented by a set of program paths. With a well-defined practical high-level interface language, user information can be used in an easy and efficient way. We also introduce a method to verify given user information with known program verification techniques. Initial experiments with a timing tool show that safe and tight predictions are possible for a wide range of programs. The tool can also provide predictions for interesting subsets of program executions.This research was supported in part by the Office of Naval Research under grant number N00014-89-J-1040.     

A three-dimensional torsional spring analogy method for unstructured dynamic meshes

We present a three-dimensional spring analogy method for updating the position of unstructured dynamic meshes that features torsional springs for controlling the arbitrary motion of the grid points. We show that these torsional springs can be designed to prohibit the interpenetration of neighboring tetrahedra, and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications. We illustrate the proposed spring analogy method with the solution of three-dimensional flow problems with complex moving boundaries, and where the unstructured meshes undergo significant deformations. We highlight the advantages of this method with respect to robustness and mesh quality, and address its computational performance.