Witold Cecot

Computational micro- and macroscopic models of contact and friction: formulation, approach and applications

Abstract This paper is dedicated to new asperity-based constitutive models of contact interfaces. These models have been obtained through a combination of finite element analysis of surface asperities and statistical homogenization techniques, to predict macroscopic, phenomenological behavior of the interface. This new approach has generalized the existing asperity-based models of contact and friction by considering realistic, complex shapes and mechanical properties of surface asperities, as opposed to previous simplified analytical solutions. This has been achieved by application, at the stage of asperity modeling, of the finite element method, which takes into account arbitrary shapes of asperities, non-linear material properties, molecular-range adhesion forces, and sliding resistance on the contact surface. The h–p adaptive mesh refinement techniques, adaptive timestepping and other adaptive methods are used to assure high accuracy of the solution. The result of this development is a new family of constitutive interface models, consistent with surface micromechanics and applicable to studies of static and dynamic friction phenomena. They are also extendible to calculation of thermal or electrical resistances, wear modeling, and other applications. This paper presents the theoretical formulation, numerical methodology and sample models of contact, adhesion and friction obtained through these homogenization techniques.

A two-dimensional infinite element for Maxwell's equations

The paper describes an implementation of the infinite element for two-dimensional steady-state Maxwells equations, proposed by L. Demkowicz and M. Pal (Comput. Meth. Appl. Mech. Engrg., in press). The element is compatible with the hp FE element discretizations for Maxwells equations in bounded domains reported in L. Demkowicz and L. Vardapetyan (Comput. Meth. Appl. Mech. Engrg. 152 (1/2) (1996) 103–124) and W. Rachowicz and L. Demkowicz (Comput. Meth. Appl. Mech. Engrg., in press).

High order FEM for multigrid homogenization

The high order FEM was applied to the multigrid based homogenization and proved to be an efficient method for upscaled solution approximation of problems with rapidly varying, possibly noncontinuous coefficients as well as non coherent (porous) domains that may or may not depict a periodic micro structure and do not have to satisfy the separation condition. Our main contribution is an improved, appropriate for bubble functions, inter-grid mapping. Consequently a fast, exponential convergence of both displacements and stresses was observed in the conducted numerical experiments with respect to the number of degrees of freedom as well as the time of computation. Particularly, the high accuracy of solution derivatives is an important advantage of this homogenization technique that is equivalent to the multiscale FEM.

Boundary element analysis of actual residual stresses in elastic plastic bodies under cyclic loading

Abstract The BEM was applied to the evaluation of an actual residual stress field in an elastic — perfectly plastic body subjected to any given cyclic loading. A new mechanical model describing this problem was used. A corresponding discrete model based on both the direct and indirect versions of the BEM was proposed. The BEM approximation of stresses applied was similar to that one used in the elasto-plastic analysis in so called modified tractions and modified body forces algorithm. Some numerical tests have been carried out where BE solutions were compared with those obtained by means of the FE and FD methods. These comparisons show that the discrete model based on the BEM is efficient in the residual stress analysis.

Estimation of computational homogenization error by explicit residual method

In this paper a new method is introduced for the estimation of modeling error resulting from homogenization of elastic heterogeneous bodies. The approach is similar to the well known explicit residual approximation error estimation. It is proved that besides the residuum of the equilibrium equation and interelement traction jump also a difference of stress divergences as well as traction jump along the material interfaces contribute to the modeling error estimate. Moreover, explicit specification and numerical evaluation of stability constants provide reasonable effectivity index of this error indicator.Selected numerical examples illustrate the promise of this approach. Therefore, the proposed methodology is a computationally inexpensive option for the other methods of modeling error assessment.

Estimation of actual residual stresses by the boundary element method

The aim of this work is the application of the boundary element method (BEM) to the evaluation of the actual residual stress field arising in an elastic-plastic body subjected to cyclic loading. As proposed in Chapter 6, this problem is solved by minimizing the complementary energy of the body in an assumed self-equilibrating stress state, and subject to the yield constraint imposed on the total stresses. The assumed state corresponding to such a minimum is the sought residual stress field.

Modeling of heterogeneous elastic materials by the multiscale hp-adaptive finite element method

We present an enhancement of the multiscale finite element method (MsFEM) by combining it with the hp-adaptive FEM. Such a discretization-based homogenization technique is a versatile tool for modeling heterogeneous materials with fast oscillating elasticity coefficients. No assumption on periodicity of the domain is required. In order to avoid direct, so-called overkill mesh computations, a coarse mesh with effective stiffness matrices is used and special shape functions are constructed to account for the local heterogeneities at the micro resolution. The automatic adaptivity (hp-type at the macro resolution and h-type at the micro resolution) increases efficiency of computation. In this paper details of the modified MsFEM are presented and a numerical test performed on a Fichera corner domain is presented in order to validate the proposed approach.

Approximation of boundary values in the boundary element method

Abstract Various techniques may be applied to the approximation of the unknown boundary functions involved in the boundary element method (BEM). Several techniques have been examined numerically to find the most efficient. Techniques considered were: Lagrangian polynomials of the zeroth, first and second orders; spline functions; and the novel weighted minimization technique used successfully in the finite difference method (FDM) for arbitrarily irregular meshes. All these approaches have been used in the BEM for the numerical analysis of plates with various boundary conditions. Both coarse and fine grids on the boundary have been assumed. Maximal errors of the deflections of each plate and the bending moments have been found and the effective computer CPU times determined. Analysis of the results showed that, for the same computer time, the greatest accuracy was obtained by the weighted FDM approach. In the case of the Lagrange approximation, higher order polynomials have proved more efficient. The spline technique yielded more accurate results, but with a higher CPU time. Two discretization approaches have been investigated: the least-squares technique and the collocation method. Despite the fact that the simultaneous algebraic equations obtained were not symmetric, the collocation approach has been confirmed as clearly superior to the least-squares technique, because of the amount of computation time used.