Keywords: boundary representation, orthogonal polyhedra, brinks, extreme vertices model, decomposition, parallel analyses.

A method of decomposition applied to orthogonal polyhedra is presented in this paper. The process of decomposition of a solid model is carried out for two different purposes. The first is to reconstruct elements that were merged into a single solid. The second is the division of a whole model into smaller elements as a pre-processing for subsequent parallel analyses. In the approach that is presented in the paper both purposes can be satisfied because the decomposed models are orthogonal polyhedra.

The paper shows decomposition of an orthogonal (pseudo) polyhedra based on an Extreme Vertices Model. Definition of an orthogonal polyhedra is presented followed by Aguilera [1]:

• An Orthogonal Polyhedra (OP) is a solid with all its edges and faces oriented in three orthogonal directions. Polyhedra are 2-manifold [3,4] objects in which all the edges are adjacent to exactly two faces.
• An Orthogonal Pseudo Polyhedra (OPP) is defined as a regular and orthogonal polyhedra with a non-manifold boundary. In an OPP, a non-manifold [3,4] edge is adjacent to exactly four faces and a non-manifold vertex is the apex of two cones of faces.

A model which was originally created using the boundary representation method [2,3,4] has to be modified. For this purpose new elements called brinks are introduced. The elements are taken from Aguilera [1] and they are defined as maximal uninterrupted segments, built out of a sequence of collinear and contiguous edges. It is important for further operations that the brinks are oriented along the positive x, y, z axes. Vertices ending brinks are called the extreme vertices and they are the basic elements of the Extreme Vertices Model. The decomposition consists of cutting a solid model by moving a cutting plane along three axes x, y, z independently. The process is linear, which means the cutting according to the one of the directions has to be completed before cutting in second direction can be initiated. As a result of this sequential manner six various decomposed models may be created. The models are labelled to the order in which the cutting is undertaken. For instance; the zyx-model was generated by cutting the original model by planes that were perpendicular, in order, first to the z axis, next to the y axis and finally to the x axis.

As an example of the whole procedure, let us analyse zyx decomposition. First the model is split according the z direction. The cutting plane is displaced from one to another extreme vertex along the z axis through the model. It starts at the point where the vertices have the smallest z value of the coordinates and finishes where the z value of the coordinates is the largest. At this point the model is split into two sub-models. All the brinks which create the cut model must be separated into two sub-models to complete the splitting. First the brinks in relation to the z direction are analysed by comparing their coordinates with the value of the cutting plane. Some of the brinks have to be cut into two. In the next step the brinks parallel to the y axis and the x axis are selected. The brinks which lies on the cutting plane are omitted for a while and the remaining ones are easily attached to the proper sub-model. Finally, the brinks laying on the cutting plane are separated, cut or new brinks are created if it is necessary. When the sub-models are completed in sense of a brink frame of the models the cutting plane moves to the next potential place of cutting. The same procedure is repeated until the plane reaches the maximal value in z direction. The number of sub-models are generated in that process, nevertheless, the decomposition is not finished yet. Now, each of the sub-model is treated as the basic model for decomposition according to the next direction. In the analysed example it is direction y. This analogue procedure is repeated for the y direction and subsequently for the x direction.

The original model is cut into a number of parallelepipeds formed as frame models. To finalise the decomposition the surfaces for each block must be defined. It is clearly seen that the process enables the generation of six decomposed models. It is worth noticing the models can be created independently therefore parallel processing can be used to speed up the calculation. The decomposition is a part of the pre-processing required for a parallel finite element analyses. For the generated blocks a finite element mesh will be designed and then the blocks will be split and distributed to a number of processors.

1

A. Aguilera "Orthogonal Polyhedra: Study and Application. Doctoral Thesis", Universitat Politecnica de Catalunya, Spain, April 1998.

2

H. Chiyokura "Solid Modeling with Designbase: Theory and Application", Addison-Wesley Publishing Company, 1988.

3

Ch.M. Hoffmann " Geometric & Solid Modeling an Introduction", Morgan Haufmann Publishers, Inc., San Mateo, California, 1989.

4

M. Mäntylä "An Introduction to Solid Modeling", Computer Science Press, Inc., 1988.

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