Geometric Bases of Parallel Computing in Computer Modeling and Computer-Aided Design Systems

Abstract

The concept of developing a geometric CAD kernel based on the invariants of parallel projection of geometric objects on the axes of the global coordinate system, which combines the potential of constructive geometric modeling methods that can provide paralleling of geometric constructions by tasks (message passing), and the mathematical apparatus "Point calculus" capable of implementing data paralleling by means of subordinate calculations (data parallel) is proposed. Use of subordinate calculation of point equations allows not only to parallelize calculations along coordinate axes, but also to provide coherence of computational operations by threads, which significantly reduces downtime and optimizes the performance of CPU to achieve the maximum effect of parallel computations. The greater the dimensionality of the modeled geometric object, the more it lends itself to paralleling computational flows. This leads to the fact that the computation time of a multidimensional problem becomes a value independent of the number of measurements. All calculations will run in parallel and finish simultaneously. The example of parallel computational algorithm for topographic surface modeling demonstrates the possibilities of realization of the offered concept for definition of continuous and discrete geometrical objects, the analytical description of which is carried out in point-calculus. As a result, to build a single 16-point patches, the distribution of parallel computing on 12 threads for the 4 direction lines and 3 threads for the formative line is obtained. Further, the number of simultaneously involved computational threads is a value proportional to the number of 16-point patches and can be further optimized by calculating several forming lines in parallel. In the above example, all computational threads are fully balanced in the number of calculations, which greatly minimizes the downtime of calculations and optimizes the performance of the processor. Also the proposed approach to the organization of parallel computations can be effectively used for the numerical solution of differential equations using geometric interpolants, which together with the development of models of geometric objects in the point calculus creates a closed loop digital production, which by analogy with the isogeometric method eliminates the need to coordinate geometric information in the interaction between CAD and FEA systems.