2021/05/14 by Carl Leake, Leake, Carl
Computer Science · Energy · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #Mechanical Systems and Engineering #Optimization and Control (math.OC) #Tribology and Lubrication Engineering
paper · pdf · doi:10.48550/arxiv.2105.07070
openalex publication_date 2021/05/14 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
The Theory of Functional Connections (TFC) is a functional interpolation\nframework founded upon the so-called constrained expression: a functional that\nexpresses the family of all possible functions that satisfy some\nuser-specified, linear constraints. These constrained expressions can be\nutilized to transform constrained problems into unconstrained ones. The\nbenefits of doing so include faster solution times, more accurate solutions,\nand more robust convergence. This dissertation contains a comprehensive,\nself-contained presentation of the TFC theory beginning with simple univariate\npoint constraints and ending with general linear constraints in n-dimensions;\nrelevant mathematical theorems and clarifying examples are included throughout\nthe presentation to expand and solidify the reader's understanding.\nFurthermore, this dissertation describes how TFC can be applied to estimate\ndifferential equations' solutions, its primary application to date. In\naddition, comparisons with other state-of-the-art algorithms that estimate\ndifferential equations' solutions are included to showcase the advantages and\ndisadvantages of the TFC approach. Lastly, the aforementioned concepts are\nleveraged to estimate solutions of differential equations from the field of\nflexible body dynamics.\n