“Cut-Glue” Approximation Method for Strongly Nonlinear and Multidimensional Object Dependencies Modeling

Abstract

The main difficulties in modeling a variety of technical systems are experienced when creating appropriate mathematical objects to simulate their behavior. It is well known that such inter-objects dependences are defined with their variable with strong nonlinear and multidimensional characteristics. The mathematical models (MM) dependences are approximated with advance numerical methods, such as polynomial decomposition, spline functions, etc., which are today still very time-consuming and laborious to be correctly created and applied, also considering their precision. In this paper, the authors have created and investigated the high-precision analytical approximation method to model the nonlinear MM dependences, which are defined only by appropriate analytical functions. These approaches have been already studied in details, where the Cut-Glue approximation method defines 1-dimensional dependences, and to 2-dimensional dependences were approximated with analytical functions of 2 arguments. The important advantage of the Cut-Glue method is that it well approximates the differentiability of the proposed MM dependencies, as its enables to investigate analytically the related modeling functions and thus, use them efficiently in applying MM in dynamical systems simulations. In this work, the Cut-Glue method has been further developed: (1) to prove its applicability by creating nonlinear models of any dimension, (2) to analyze its performance at all the stages, in which the “Cut-Glue” approximation is applied, and (3) to implement this formal algorithm, which allows numerical verification and validation of its applicability. The considered optimization criteria for both respective issues, accuracy and complexity, have been applied to the investigated MM-s. The proposed method is formalized by the optimal splitting of its experimental dependence into separate parts, which are then numerically defined and implemented within the proposed software, developed in this work. In this paper, the different possibilities of applying the optimal multidimensional “Cut-Glue” approximation method are illustrated by examples. The achieved results represent a strong base to significantly expand the proposed method applicability, and further on, they indicate potential opportunities to improve the existing solutions. Especially, when solving a variety of problems, which requires mathematical modeling of any type of technical objects, to simulate overall systems dynamics.