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) the projection mapping, and (c) polynomial mapping. In that respect, an accurate least-squares approximated inverse mapping is also developed for those mappings with no closed-form inverse. 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 piece-wise sequence of functions with a single function, which is compatible and required by the Theory of Functional Connections already developed for rectangular domains.
Highlights
The Theory of Functional Connections (TFC) is a mathematical methodology to perform functional interpolation, i.e., the process of deriving functionals, called constrained expressions, which contain the constraints of the problem already embedded on their expression
Projection mapping is based on the selection of a set of points inside the boundaries of the domain in such a way that a straight segment can be defined between any point in the domain and at least one projection point, where this segment does not cross at any moment any of the boundaries of the domain
Connections, which was developed for rectangular domains in any dimensional space, to generic domains in 2-dimensional spaces
Summary
The Theory of Functional Connections (TFC) is a mathematical methodology to perform functional interpolation, i.e., the process of deriving functionals, called constrained expressions, which contain the constraints of the problem already embedded on their expression. The main purpose of this study consists of developing easy mappings between the unit-square domain and any other domain, where the flexibly can accommodate typical domains appearing in physics, science, and engineering problems Examples of this include computation of structures with “C”,. Conformal (angle-preserving) and area/volume-preserving mappings are important transformations since they are very interesting for their applications in physics, science, and engineering. These mappings are used to transform differential equations in order to simplify the system and make it easier to solve [16]. For several dimensions equal or higher than 3, Liouville’s theorem states that Möbius transformations are the unique transformations that are conformal
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
