MODELLING OF COMPUTATIONAL 2D-TEMPLATES AND CUBATURES AS THE PROBLEMS OF SYSTEM ANALYSIS

Abstract

Numerical integration is becoming an increasingly important procedure in the modern method of finite elements (MFE). It is clear that the overwhelming majority of known cubatures is associated with triangles and squares. Unfortunately, not all cubatures are suitable for practical use. For example, there are cubatures with negative weight number. According to modern American mathematicians G. Strang and J. Fix, the problem of constructing the cubatures even on triangular and square patterns remains relevant. To obtain new cubatures pseudo-random numbers and the Monte-Carlo quasi-method are used. By the example of well-known systems of triangular and square Pythagorean numbers in the 50s of the twentieth century the systems of triangular and square computational templates and corresponding cubatures appeared in the MFE. The peculiarity of system analysis is that on one template there may be several alternative cubatures within the law of conservation of weight balance. In these cases the problem of segment testing of new bases (for compatibility) arises. Efforts spent to stratify the selection result in improved quality of evaluation. The paper considers systems of computational 2D-templates which are formed following the example of arithmetic systems and geometry of triangular and square Pythagorean numbers. The purpose of the study is to illustrate the possibilities and advantages of the procedure of stratification of selective applicates, to emphasize the important role of centered models (with integration node in the barycenter of a triangle, square) on the examples of computational 2D-templates and random cubatures. The study found the following: if the number of integration nodes and their location was recorded, it is necessary to find out which criterion should be used to determine the coefficients of the linear combination of applicates. In the system of alternative cubatures none of the stratification criteria has a significant advantage over the others. For each criterion one can choose an example in which it will be the best. To find the most effective cubature for a particular task a special analysis is required.