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

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VOL. 41 (2000) n. 1332. CUTTING BY PARALLEL TRIHEDRAConsideration 1. We will refer to the sphere{O, R} as the one circumscribed to the regularicosahedron; starting from the spherical triangleABC (see fig. 1), which has its centre in O and isrelated to a face of the regular icosahedron, thetrihedron OABC is defined. Point M, on the spherecircumscribed to the polyhedron, defines with O anorder-3 rotation symmetry axis on the generatedshape ABC.Consideration 2. Let it be OM and let us considernew vertices of trihedra O 1 , O 2 , etc. (see fig. 2)parallel to OABC on it. They make new sphericalpatches A 1 B 1 C 1 , A 2 B 2 C 2 , etc. on the sphericaltriangle ABC. Every one of them is bounded by anarc A 1 B 1 , A 2 B 2 , etc. of minor circle on the formersphere {O, R}.Figure 2PROPERTY I. A distribution of edges as the oneshown in fig. 3 is optimal (the number of elements,nodes, panels and edges is minimum [3]) if thelength of edges along the arc AB, A 1 B 1 , A 2 B 2 , etc.(and their symmetric BC, B 1 C 1 , B 2 C 2 , etc.CA, C 1 A 1 , C 2 A 2 , etc.) have the general expressionl i = R i ·? , where ? is the only degree of freedom, l iis the length of the edge along the arc A i B i and R i isthe radius of the circle holding the arc A i B i .Obviously, all the bars on an A i B i string have thesame length.PROPERTY II. Having assumed these conditions,the distances OO 1 , O 1 O 2 , O 2 O 3 , etc. needed forplacing the vertexes O 1 , O 2 , O 3 of the paralleltrihedra are defined [1].Figure 3PROPERTY III. The value of the parameter ? isrelated to the number n of bars proposed forapproaching the geodesic arc AB. Indeed, it resultsin: ? =arc(AOB)/n. From here, the rest of thegeometric parameters needed for defining the patchis obtained, more or less, immediately (the readercould find it in [1] or [5]). Once the grid is solved, itturns out that the sequence of distances OO 1 , O 1 O 2 ,O 2 O 3 , etc. becomes convergent and the number ofbars on the arcs AB, A 1 B 1 , A 2 B 2 , etc. produce anarithmetic succession of ratio –3. The last string, onthe arc A n B n is closed with 1, 2 or 3 bars, depending Figure 4
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: IASSon the number n proposed at the beginning. Therange of length of edges obtained by this type ofdiscreetization result to be especially favorablewhen the starting polyhedron is the regularicosahedron; however, the procedure is useful forany other regular one, with the same properties butlosing the excellent range of length in the bars.PROPERTY IV. Following with an associatedicosahedron, its own symmetry properties make thejoin of trihedra as the OABC around the vertex Opossible: 20 of them make up the dome. We shouldremember that any position for the vertexes O 1 , O 2 ,O 3 , etc. does not generate an optimal mesh butsolves the problem (see fig. 4); when the situationof them is chosen according to PROP I and III, thespatial mesh also has that feature: its balance ofelements is optimal [1].PROPERTY V. Let the trihedron O i A i B i C i ,obtained during the latter procedure, be considered(see fig. 5). Since it is parallel to OABC, it has in O ithe same solid angle ? /5; again, the union around O iof 20 similar trihedra O i A i B i C i creates a closedsurface made by 20 spherical triangular patches theboundaries of which are now minor circle arcs. Inother words, a shape composed by the intersectionof 20 spheres the intersection lines of which are allarcs identical to A i B i . Notice that every one of these20 equal patches is a SCALE of knowndiscreetisation (prop. I to IV). This idea ofconstructing meshes based on scales has beenproposed in [2] in this same bulletin. In figure 6 itcan be seen that, once again, the placement ofvertexes can always be discretionary but onlyaccording to prop III, the balance of elementsremains optimal.Figure 5PROPERTY VI. Let us, at last, consider again therhythm of generation of the parallel trihedraproposed in fig. 2. The sequence of vertexes O igradually approaches to M from O; however, wecan propose some other O-1 , O-2, O-3 , etc withthem moving away. In this latter group, anypreviously described property is lost; the newness isthat the resulting mesh is more vault-shaped thanthe first one.Figure 6
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