2. Theoretical background

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

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

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

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

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

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

accuracy of the results necessarily implies an increment of the computational cost.

much higher computational cost.

60 Contact and Fracture Mechanics

distribution that satisfies the contact conditions.

a loss in the efficiency of the method.

problems using semi-analytical methods.

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 strategy to perform adaptive mesh refinement.
