designing optimal spatial meshes: cutting by parallel trihedra ...

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: IASSarise as many families of parallel pyramids asdifferent types of face the body has; similarly, asmany different radii of spheres as different faces. Ageometric demonstration of the compatibility of twoor more different parallel pyramids (compatibilityalong their common boundary) for making up thistype of mesh is found in [1] or [5]. Summarising,the condition of compatibility allows only a radiusof sphere to be a degree of freedom. For example,starting from the rombicosidodecahedron (3-4-5-4:1 triangle, 2 squares and 1 pentagon around eachvertex) there are three different spheres but thevalues of two of them depend on the third. Therelation is proposed in the indicated references.4. A STUDY WITH RESPECT TO THEGENERATION OF GRIDS INTO THESCALES CREATED BY PARALLELPYRAMIDSAt least, regular and semi-regular polyhedra causenew meshes shaped as intersections of spheres. Donot forget that, in all of them, the balance ofdifferent elements is optimal. Now, we can focus onthe internal triangulation:REMARK I. Whatever polyhedron is used as astarting body, every obtained scale accepts manydifferent grids of triangles so that there are always afamily of bars approaching the boundary arcs andtheir parallel.REMARK II. The optimal grid in a scale dependson the polihedron used for starting the procedure.Figure 11-a summarises the optimal pattern R1proposed by [3] (ref. [6], [7] and [8] for moredetail); this is used for making the distribution ofbars at point 2 and can be thus typified so:Feature I. The type a bars, on the arc AB, have alength R·? (R is the radius of the spherecircumscribed to the icosahedron); the type b bars,on the arc A 1 B 1 , have a length R 1·? (R 1 is the radiusof the parallel circle holding A 1 B 1 ), the type c bars,on the arc A 2 B 2 , have a length R 2·? and so on.Feature II. If there are n type a bars, there are (n-3)type b, (n-6) type c, etc.Feature III. The slab ABA 1 B 1 holds type a, b, x andy bars (with the same applying for its symmetricBCB 1 C 1 and CAC 1 A 1 ). Type x bars connect a nodeon AB with one in A 1 B 1 . There are three type ybars, connecting nodes on the main arcs AB, BCand CD. This scheme remains on the slabsA 1 B 1 A 2 B 2 , etc.Feature IV. The string of type a bars approaches thearc AB (BC and CA); that of b bars, the arc A 1 B 1(B 1 C 1 and C 1 A 1 ) and so on.A possible alternative grid R2 is shown in figure11-b, where:- Feature I remains.- Feature II has changed: if there are n type abars, there are (n-1) type b, (n-2) type c, etc.- Feature III has changed: the slab ABA 1 B 1 holdstype a, b and x bars but there are not type y bars.- Feature IV remains.Figure 11-a Figure 11-b