Example of normal projection in a spherical surface

The purpose of this section is to highlight the importance of the surface description method used in the numerical simulation of frictional contact problems involving large sliding. The discretization of the master surfaces using low order finite elements leads to sudden changes in the surface normal field, which can cause convergence problems in the solution procedure. Moreover, some blind spots arise in the normal projection of the slave nodes on the faceted master segments (see Figure 3.19), causing severe difficulties in the local search detection procedure. Therefore, a simple example is selected to evaluate the impact of using a non-smooth master surface on the efficiency of the normal projection method, and consequently in the local search procedure.

Two surfaces are involved in the proposed example, a flat surface representing the slave surface and a spherical surface describing the master surface. Besides, two different configurations are analysed, i.e. the master surface can be either convex or concave in relation to the slave surface. In both configurations the master surface is discretized by 16

previous time step

current time step

m 1 2 ( , ) n n n ξ ξ x m n x s n x 1 n n 1 m n x 1 s n _x 1 n g s n_x m 1 2 ( , ) n n n ξ ξ x

previous time step

current time step

1 s n _x 1 n g 1 n n 1 m n x n 0 g  n m x

104

bilinear quadrilateral finite elements, as shown in Figure 3.21. On the other hand, a fine grid of points (300 divisions in each direction) is created over the square flat surface, producing a total of 90,601 points. Then, the aforementioned normal projection algorithm is applied to each of these points in order to calculate its closest point on the master surface. The relative position of the two surfaces is presented in Figure 3.21, where two views of each configuration are shown.

(a) (b)

Figure 3.21. Configuration of the problem composed by a flat surface and a spherical surface (lateral and top views): (a) convex surface; (b) concave surface.

The normal projection of a slave point on the curved master surface described by bilinear quadrilateral finite elements (Figure 3.21) may have multiple solutions (concave surface) or no solution (convex surface) near the common edges of the master finite elements. Figure 3.22 shows the colour map denoting the finite elements on which the slave points are projected with smallest normal gap. Some deadzones (white colour) arise in the case of convex surface, which are larger for points located more distant to the surface, due to the pyramidal shape of the blind spots, as illustrated previously in Figure 3.19 (a). Therefore, the points of the slave surface located within the white regions shown in Figure 3.22 do not have any normal projection with any master finite element. On the other hand, the normal projection of the slave points on the concave surface does not comprise any external blind spot, as shown in Figure 3.19 (b), providing a continuous projection field (Figure 3.22 (b)).

General Finite Element Framework

105

(a) (b)

Figure 3.22. Areas of flat surface with normal projection on the non-smoothed spherical surface: (a) convex surface; (b) concave surface. Each colour denotes a different finite

element.

The blind spots in the normal projection, created by the finite element discretization of the master surfaces with linear elements (Figure 3.22), can be avoided through the surface smoothing, ensuring a continuous projection on the master surface. This strategy is adopted in the present study to improve the accuracy of the local search detection and avoid convergence problems in large sliding contact problems. The next chapter is entirely dedicated to surface smoothing with Nagata patches. Thus, in order to highlight the advantages of adopting this approach, the finite element discretization of the spherical surface presented in Figure 3.21 is smoothed with Nagata patches. After that, the same contact search procedure performed for the faceted surface description is carried out for the smoothed spherical surface, in order to compare the results of the normal projection. The colour map denoting the patches on which the slave points are projected with smallest normal gap is presented in Figure 3.23. The blind spots observed in Figure 3.22 (a) for the convex surface modelled by faceted elements are strongly reduced using a smoothed master surface. In fact, the areas of the slave surface where no normal projection exist (white colour) are now located in a very narrow range near the edges of the patches. Since the configuration with the concave surface (Figure 3.21 (b)) does not contain any external blind spot, the normal projection field is continuous. Although not shown here, the zones with multiple projections are also drastically reduced. Also, the pattern is slightly different from the one obtained with the non-smoothed spherical surface, as observed by comparing

Figure 3.22 (b) with Figure 3.23 (b). Note that both configurations of the selected example originate a square pattern of the normal projection when the spherical surface is smooth, as shown in Figure 3.23. This is also a consequence of the smoothing procedure applied to the surface.

106

(a) (b)

Figure 3.23. Areas of flat surface with normal projection on the smoothed spherical surface: (a) convex surface; (b) concave surface. Each colour denotes a different patch.

Surface Smoothing with Nagata Patches

107