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用于将功能连接理论扩展到非矩形二维域的双射映射分析。

Bijective Mapping Analysis to Extend the Theory of Functional Connections to Non-Rectangular 2-Dimensional Domains.

作者信息

Mortari Daniele, Arnas David

机构信息

Aerospace Engineering, Texas A&M University, College Station, TX 77845-3141, USA.

Aeronautics and Astronautics, Massachussetts Institute of Technology, Cambridge, MA 02139, USA.

出版信息

Mathematics (Basel). 2020 Sep;8(9):1593. doi: 10.3390/math8091593. Epub 2020 Sep 16.

DOI:10.3390/math8091593
PMID:33062599
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC7553096/
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.

摘要

这项工作对使用双射映射将功能连接理论扩展到非矩形二维域进行了初步分析。具体而言，本文提出了三种不同的映射技术：(a) 复映射，(b) 投影映射，以及 (c) 多项式映射。在这方面，还为那些没有闭式逆映射的映射开发了精确的最小二乘近似逆映射。文中强调了使用这些映射的优缺点，并提供了一些示例。此外，本文展示了如何用一个单一函数来替代以分段函数序列表示的边界约束，该单一函数与已经为矩形域开发的功能连接理论兼容且为其所需。

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Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations.
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