THE INVARIANT GEOMETRIC STRUCTURE OF ENGINEERING MODELING

Abstract

The paper considers engineering modeling as a process of constructing models of technical objects, in which an invariant geometric structure is identified as its fundamental basis. Model construction is not limited to the formulation of mathematical or numerical relationships. Typically, model construction is preceded by the determination of the object’s shape, the structure of interactions between its elements, boundary contours (topology), and boundary conditions. The combination of these spatial and structural characteristics forms the geometric structure of the object, which remains unchanged under variations in the mathematical description, the type of equations, or the numerical method, thereby being invariant.

It is substantiated that the set of possible models of any object is determined by its invariant geometric structure, which defines the framework for further formalization. It is emphasized that the geometric component constitutes an essential basis of engineering modeling, since any model relies on a previously established spatial configuration of the system that determines the admissible forms of its mathematical representation.

The modeling process is viewed as a sequence of transitions from a real object to its mathematical model through the identification of the geometric component and the extraction of the invariant geometric structure. This approach makes it possible to generalize the logic of model construction regardless of the nature of the problem and the mathematical framework.

The proposed approach is illustrated by examples of modeling a thin-walled shell, a parametric surface, and a kinematic scheme of a mechanism. The considered problems belong to different domains; however, they demonstrate a common logic of transition from a real object to its mathematical or numerical model, which confirms the general character of the invariant geometric structure.

The obtained results make it possible to consider the geometric stage not only as an independent component of the engineering modeling process, but also as a methodological basis for constructing models of complex technical systems.

Keywords: engineering modeling; modeling methodology; geometric modeling; geometric structure; spatial organization of systems; topology; geometric component of a model; invariant geometric structure