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where Li is the central normal vector of the target surface.Consequently, the undercut volume shown in Figs. 8 and 9 can be calculated based on the target surface and its positive undercut feature edges. The equation is as follows.
where AT is the area of the target surface, AUE is the area bounded by the vertices of the positive undercut edges (not in the target surface, if the number of positive undercut edges is equal to 2, AUE=0) and UE is the average length of the positive undercut edges projected onto the central normal vector of the target surface.
3.2. V-map and draw-range of undercut feature
The V-map of undercut feature V-map(U) is the intersection of all V-maps of the surfaces of which the undercut feature is composed. V-map(U) can be defined aswhere n is the number of surfaces belonging to the undercut feature. V-map(Fi) is the V-map of the undercut surface Fi.For the undercut feature shown in Fig. 10(a), the corresponding V-map(U) is illustrated in Fig. 10(b). In the figure,Fj1 and Fj2 are one group of the first adjacent surfaces, and Hj1 and Hj2 are another group of the first adjacent surfaces.The draw-range of undercut features refers to the range in
which the local tools can be withdrawn from the moulding.In V-map(U), the boundary of V-map constitutes the drawrange of the undercut features. In practice, not all withdrawal directions in the draw-range are preferred withdrawal directions, only the directions in which the local tools can be effectively withdrawn are considered. These preferred withdrawal directions are called undercut feature directions in this paper.
3.3. Undercut feature directions
The direction of undercut feature can be defined as the directions in which the local tool can be withdrawn most effectively from the undercut features to avoid the warpage of the moulding. Based on the V-maps of undercut features,it is found that any point in the V-map(U) can be a feasible withdrawal direction of the local tool. For any given undercut feature, there exist the most preferred withdrawal directions, which are different from case to case. For inside external or inside internal undercut features, the undercut direction can be determined uniquely. For the outside external or outside internal undercut features, there is more than one preferred withdrawal direction and thus more than one undercut direction. For the case shown in Fig. 10, the preferred direction is in the intersectional line of two bisectional planes between the first adjacent surfaces Fj1 and Fj2 ,Fh1 and Fh2 .
4. Outside external undercut features
4.1. Three-edge-OEU (3-OEU)
In this case, the edges of the target surface are three and at least one of the edges is concave, the target surface and its first adjacent surfaces will form an undercut feature.Supposing that the target surface Fi has three edges. Let Li be the normal vector of Fi at the surface centre and Lji (j=1; 2; 3; the first subscript refers to the adjacent surface and the second subscript refers to the coordinate axis) be the normal vector of the first adjacent surfaces.Ecave(Fi) and Evex(Fi) are the concave and convex edges number of the target surface respectively. The undercut features, V-map and its undercut directions are summarized in Table 1.
4.2. Four-edge-OEU (4-OEU)
In this case, the target surface has four edges and four adjacent surfaces. Let the first adjacent surfaces be pided into two groups (i.e. groups j and h) in which the two opposite surfaces are in the same group. The normal vectors at the surface centre of the two groups are Lji and Lhi (j=j1; j2; h=h1; h2) .The undercut criteria, V-maps andundercut directions can be determined based on the geometrical information and topological relationships of Fi with its adjacent surfaces and are summarized in Table 2.
4.3. More than four-edge-OEU ( 4-OEU)
If the edges of the target surface are more than four, there are more than four first adjacent surfaces. The target surface and its adjacent surfaces can constitute the different convex and concave portions in the moulded part, which can become undercut features. All the possible cases are listed as follows.