Bijective Mapping Analysis to Extend the Theory of Functional Connections to Non-rectangular 2-dimensional Domains

TL;DR

This paper explores bijective mappings to adapt the Theory of Functional Connections for non-rectangular 2D domains, proposing three mapping techniques and inverse approximation methods to handle complex domain shapes.

Contribution

It introduces three novel mapping techniques and inverse approximation methods to extend the Theory of Functional Connections to non-rectangular domains.

Findings

Complex, projection, and polynomial mappings each have unique advantages and disadvantages.

A least-squares inverse mapping approach effectively handles non-invertible mappings.

The method simplifies boundary constraint representation for complex domains.

Abstract

This work presents an initial analysis of using bijective mappings to extend the Theory of Functional Connections to non-rectangular two-dimensional domains. Specifically, this manuscript proposes three different mappings techniques: a) complex mapping, b) projection mapping, and c) polynomial mapping. In that respect, an accurate least-squares approximated inverse mapping is also developed for those mappings having no closed-form inverse. The advantages and disadvantages of using these mappings are highlighted and a few examples are provided. Additionally, the paper shows how to replace boundary constraints expressed in terms of a piecewise sequence of functions with a single function, that is compatible and required by the Theory of Functional Connections already developed by rectangular domains.