1. Introduction

The contact stress analysis plays an important role during the design process of several mechanical elements like bearings, gears, etc. In order to accomplish a contact analysis, the so-called contact problem must be solved to obtain the following relevant information:


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Different approaches have been used to solve contact problems, which can be classified into three groups: numerical, analytical, and semi-analytical methods. Compared to the numerical methods, it can be said that the analytical methods are more efficient in terms of computational cost, but they have severe applicability limitations imposed by the hypotheses of the underlying theory. On the other hand, the numerical methods can overcome these limitations, but at a much higher computational cost.

is based on the discrete rate of change of any magnitude that is related with the solution of the contact problem. This strategy allows for a reduction of the computational cost of the method, while its accuracy remains unaffected. The theoretical background and the computational implementation of the method are described, and its performance is illustrated with numerical

Adaptive Mesh Refinement Strategy for the Semi-Analytical Solution of Frictionless Elastic Contact Problems

This section describes the theoretical background under which the proposed approach to solve frictionless elastic contact problems is developed. The concept of pressure element is described, as well as those considerations required to solve contact problems between bodies of finite dimensions. Finally, the quadtree decomposition of the domain is introduced, which is a useful

Consider a body that, because of its main features, can be approached to an elastic half-space, as the one shown in Figure 1a. A Cartesian coordinate system is defined over the surface of this body, which X and Y axes define a plane that is coincident with its surface, and the Z axis points inward him. A normal pressure distribution ð Þp is applied over the surface of the body,

Now consider a generic point C within the area S, whose position is defined by the vector

1 � ν 2πG

ð Þ x; y; z , being z ¼ 0. Consider another point H in the surface of the body, whose position is defined by the vector rð Þ x; y; z , being z ¼ 0. The normal elastic deflection produced at a point H due to a normal pressure distribution applied over the area S can be determined by the

> ð S

where ν is the Poisson coefficient and G is the shear modulus of the material of the considered body.

p r<sup>0</sup> ð Þ

<sup>r</sup> � <sup>r</sup><sup>0</sup> j j dS (1)

http://dx.doi.org/10.5772/intechopen.72422

61

2.1. Pressure elements and surface normal deflection in an elastic half-space

ωð Þ¼ r

examples.

r0

2. Theoretical background

strategy to perform adaptive mesh refinement.

acting over an area that is denoted by S.

superposition of the Boussinesq relation [3]:

Figure 1. Pressure distributions applied over an elastic half-space.

The semi-analytical methods (SAMs) can be considered as an intermediate approach: they are potentially faster than the numerical methods, while they allow overcoming some of the limitations of the analytical methods. SAMs are usually based on the discretization of the potential contact area into a mesh of n pressure elements, with a uniform pressure distribution assumed to be acting over each one of them. Influence coefficients are used to relate the pressure applied over each pressure element with the displacements that this pressure produces at the centroid of the other elements of the mesh. Using these influence coefficients, the solution to the contact problem can be numerically found in terms of the contact pressure distribution that satisfies the contact conditions.

As usual, in numerical methods based on the discretization of the domain, the election of the number of pressure elements in which the domain is divided involves a commitment between accuracy and computational cost. Kalker [1] stated that the computational cost of these semianalytical methods can be defined by the number of influence coefficients that need to be calculated to solve the contact problem (that, in general, is proportional to n2). He also argued that the accuracy of the solution to the contact problem, in terms of contact area and contact pressure distribution, depends on the refinement of the pressure element mesh, especially in those regions close to the border of the contact area. Consequently, an improvement of the accuracy of the results necessarily implies an increment of the computational cost.

When both shape and location of the true contact area are known in advance, the efficiency of the method can be maximized by discretizing an area similar to the true contact area. But when the true contact area is unknown, it is difficult to optimize the efficiency of the method, since the whole potential contact area must be discretized to consider any possible shape and location of the true contact area. In those cases, it is common to use a uniform pressure element mesh for the whole domain, being more or less dense depending on the desired accuracy and on the capabilities of the computer used to solve the contact problem. In consequence, there could be many pressure elements in the discretization out of the true contact area, what causes a loss in the efficiency of the method.

These difficulties could be partially overcome using adaptive mesh refinement strategies. These techniques have been previously used to improve the efficiency of numerical methods based in the discretization of the domain, especially in FEM procedures [2]. However, no previous use of adaptive refinement has been found in the literature for the solution of contact problems using semi-analytical methods.

In this work, an approach to solve frictionless elastic contact problems is presented, whose main purpose is to improve the efficiency of the semi-analytical methods that are used to solve contact problems. To do so, an adaptive refinement of the pressure element mesh is implemented, which is based on the discrete rate of change of any magnitude that is related with the solution of the contact problem. This strategy allows for a reduction of the computational cost of the method, while its accuracy remains unaffected. The theoretical background and the computational implementation of the method are described, and its performance is illustrated with numerical examples.
