A numerical solution for dynamic contact problems satisfying the velocity and acceleration compatibilities on the contact surface

For the numerical solution of dynamic contact problems, correct contact points and displacements are determined by iteratively reducing the displacement error vector monotonically toward zero. And spurious oscillations are prevented from the solution by enforcing the velocity and acceleration compatibilities of the contact points with the corresponding error vectors. Economic computation is possible because the accelerated iterative schemes are used and because decomposition of large matrix is not required in the iterations for the contact analysis of elastic bodies. Numerical simulations are conducted to demonstrate the accuracy of the solution and the necessity of the velocity and acceleration compatibilities on the contact surface.This research was supported in part by Korea Research Foundation, 1993.     

Stabilized numerical solution for transient dynamic contact of inelastic solids on rough surfaces

A simulation technique to deal with transient dynamic contact of tire rubber compounds on rough road surfaces is presented. The segment-to-surface approach is used for modeling the contact between tire tread rubber and road track. While the rubber components are deformable and described by a sophisticated viscoelastic damage constitutive model, the road surface is assumed to be rigid and characterized by an analytical function. A spectral approach based on an inverse computation of the 2D-Fast Fourier transform has been suggested for the reconstruction of rough surface profiles. The Newmark time-stepping method is used for the integration of transient dynamic equations. With the so-called contact-stabilized Newmark method the spurious oscillation at contact boundary has been removed. The detailed investigation on the dynamic contact of inelastic rubber block with rough road surfaces has been made. The robustness of the contact-stabilized Newmark method within a finite deformation framework is underlined by numerical studies, in which it is compared with several dissipation-based stabilization techniques selected from literature.     

Boundary variational formulations and numerical solution techniques for unilateral contact problems

In this paper the numerical solution of the elastic frictionless contact problem is obtained by means of boundary discretization techniques. Variational formulations in terms of boundary tractions are given in presence of both bilateral and unilateral constraints. The discretization of the boundary functional is examined from the point of view of the theory of approximation and it is proved that the coerciveness (but not the symmetry) of the continuum problem is preserved when standard B.E.Ms are employed. As a consequence, the contact problem can be cast as a L.C.P. having, as coefficient matrix, a generally non symmetric P matrix. A simple, but meaningful example is discussed in some detail.     

A numerical solution for integral equations

A method for the numerical approximation for the solution of Fredholm integral equations of the second kind is presented. The approximation is obtained using cubic B-splines as co-ordinate functions in the variational formulation of the problem.     

A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law

3D elastoplastic frictional contact problems with orthotropic friction law belong to the unspecified boundary problems with nonlinearities in both material and geometric forms. One of the difficulties in solving the problem lies in the determination of the tangential slip states at the contact points. A great amount of computational efforts is needed so as to obtain high accuracy numerical results. Based on a combination of the well known mathematical programming method and iterative method, a finite element model is put forward in this paper. The problems are finally reduced to linear complementarity problems. A specially designed smoothing algorithm based on NCP-function is then applied for solving the problems. Numerical results are given to demonstrate the validity of the model and the algorithm proposed.The project is jointly supported by the National Natural Science Foundation (10225212, 50178016, 10302007), the National Key Basic Research Special Foundation (G1999032805), the Special Funds for Major State Basic Research Projects and the Foundation for University Key Teacher by the Ministry of Education of China. The authors are also grateful to the referees for their careful reading and detailed remarks on an earlier version of the paper.     

Comparison of some single-step methods for the numerical solution of the structural dynamic equation

This paper compares the performance of the SS22 and SS23 (References 1 and 2) single step algorithms for the numerical solution of the second-order structural dynamic equation and a related new algorithm SS32B applied to the equivalent first-order system, with sine and step forcing functions. Various aspects of stability relevant to these equations are discussed.     

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 numerical solution for the symmetric melting of spheres

A numerical method is presented for the solution of the radially symmetric heat conduction problem in a melting sphere. The method employs the embedding technique; this permits the solution to be written in the form of an ordinary integro-differential equation which is readily solved numerically by means of a forward integration scheme. The accuracy of the method is briefly discussed and numerical results for both constant and variable heat inputs are presented.     

A combined numerical method for determining the conductance of composite bodies

We propose a new numerical method (a combination of the method of grids with Rayleigh's method) which is very promising for the calculation of potential fields, fluxes, and conductance of composite bodies, especially in the case of components with sharply differing properties.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 2, pp. 284–291, February, 1977.     

The numerical solution of radiative heat transfer for grey bodies in an absorbing medium

A new approximate analytical method for solving the integral radiation equations [1, 2] is used for the numerical calculation and investigation of the local and average characteristics of radiative heat transfer in systems of grey bodies separated by an isothermal absorbing medium.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 3, pp 520–525, September, 1969.     

A direct numerical method for the solution of field problems

A numerical method for the analysis of field problems is described. The algorithm is based upon the generalized Betti-Maxwell theorem. Using a set of known solutions to problems with similar boundary conditions produces a set of ‘integral’ equations for the required solution. Using any convenient numerical integration formula reduces the problem to the solution of a set of simultaneous algebraic equations. The accuracy of the solution depends upon the accuracy of the integration formula as applied to the problem under consideration and is independent of the known auxiliary solutions. The method is described in detail as applied to harmonic problems.     

Meshless point collocation for the numerical solution of Navier-Stokes flow equations inside an evaporating sessile droplet

The Navier-Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid-gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier-Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.     

A note on numerical computation of elastic contact problems

A simple procedure of obtaining flexibility matrices in terms of contact pressures at possible contact points of two bodies allows the frictionless contact pressures to be solved as a quasi-linear problem. Iteration is limited in general to very few cycles and, as small matrices are handled, can be very economical. Excellent accuracy of this procedure is demonstrated.     

Three‐dimensional numerical solution for wear prediction using a mortar contact algorithm

A finite element formulation to compute the wear between three‐dimensional flexible bodies that are in contact with each other is presented. The contact pressure and the bodies displacements are calculated using an augmented Lagrangian approach in combination with a mortar method, which defines the contact kinematics. The objective of this study is to characterize the wear rate coefficients for bimetallic pairs and to numerically predict the wear depths in new component designs. The proposed method is first validated with the classical pin‐on‐disc problem. Then, experimental results of wear for the metallic pairs used in internal combustion engine valves and inserts are presented and are taken as a reference solution. An example is provided that shows agreement of the numerical and experimental solution. Finally, the proposed algorithm is used to predict the wear in an application example: the wear in an internal combustion engine valve. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical Green's function for multiple cracks in anisotropic bodies

The numerical construction of a Green's function for multiple interacting planar cracks in an anisotropic elastic space is considered. The numerical Green's function can be used to obtain a special boundary-integral method for an important class of two-dimensional elastostatic problems involving planar cracks in an anisotropic body.     

A method for the evaluation of contact compliance of bodies with surface grooves

An axisymmetric model of imperfect elastic contact of an isotropic half space with a rigid base containing a system of circular grooves uniformly distributed over the surface is constructed. The model takes into account the influence of the collection of surface defects on the stress-strain state in the vicinity of a separate inhomogeneity. We propose a method for the evaluation of the integral elastic stiffness of the contact surface and perform the comparative analysis of the results obtained by using the proposed approach with the well-known solutions of three-dimensional problems. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 42, No. 6, pp. 34–38, November–December, 2006.     

A finite element model for contact analysis of multiple Cosserat bodies

The objective of this paper is to develop a finite element model for multi-body contact analysis of Cosserat materials. Based on the parametric virtual work principle, a quadratic programming method is developed for finite element analysis of contact problems. The contact problem with friction between two Cosserat bodies is treated in the same way as in plastic analysis. The penalty factors, that are normally introduced into the algorithm for contact analysis, have a direct influence on accuracy of solution. There is no available rule for choosing a reasonable value of these factors for simulation of contact problems of Cosserat materials, and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy. Compared with the conventional work on Cosserat elasticity, the newly developed model is on the contact analysis of the Cosserat materials and is seldom found in the existing literatures. Four examples are computed to illustrate the validity and importance of the model developed.     

Numerical solution of elastic bodies in contact by FEM utilising equilibrium displacement fields

In this paper a new effective formulation of the computation of the statics of elastic bodies in contact is described and demonstrated. Line contact, plane strain, and negligible friction are assumed. The formulation is an extension of the standard finite element method (FEM). With the aim to utilise the Hertz theory directly, we use the exact solution of the elastic 2-dimensional halfspace loaded by Hertzian pressure distribution and enforce the contact condition by the method of Lagrange multipliers. In numerical examples we have focussed on the demonstration and evaluation of the accuracy of the new formulation for selected applications compared with the state of the art node-to-segment contact algorithm implemented in the software system ADINA. Proposed formulation is more accurate for problems where Hertz contact dominates the strain state, especially for small number of elements, whereas we obtained a fairly good agreement with ADINA for a more general bending problem.