Multi-Patch Isogeometric Convolution Hierarchical Deep-learning Neural Network

TL;DR

This paper extends the Isogeometric Convolution Hierarchical Deep-learning Neural Network (C-IGA) framework to multi-CAD-patch systems, analyzing compatibility conditions and demonstrating improved geometric and approximation capabilities.

Contribution

It generalizes C-IGA theory for multi-patch systems, providing mathematical analysis of interface compatibility and convergence, which was not previously addressed.

Findings

Validated nodal and G^0 compatibility conditions through numerical examples

Demonstrated higher order approximation without increasing degrees of freedom

Maintained exact geometrical mapping during mesh refinement

Abstract

A seamless integration of neural networks with Isogeometric Analysis (IGA) was first introduced in [1] under the name of Hierarchical Deep-learning Neural Network (HiDeNN) and has systematically evolved into Isogeometric Convolution HiDeNN (in short, C-IGA) [2]. C-IGA achieves higher order approximations without increasing the degree of freedom. Due to the Kronecker delta property of C-IGA shape functions, one can refine the mesh in the physical domain like standard finite element method (FEM) while maintaining the exact geometrical mapping of IGA. In this article, C-IGA theory is generalized for multi-CAD-patch systems with a mathematical investigation of the compatibility conditions at patch interfaces and convergence of error estimates. Two compatibility conditions (nodal compatibility and G^0 (i.e., global C^0) compatibility) are presented and validated through numerical examples.