Open Access

ARTICLE

A Multi-Grid, Single-Mesh Online Learning Framework for Stress-Constrained Topology Optimization Based on Isogeometric Formulation

Kangjie Li, Wenjing Ye^*

Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong, 999077, China

* Corresponding Author: Wenjing Ye. Email:

Computer Modeling in Engineering & Sciences 2025, 145(2), 1665-1688. https://doi.org/10.32604/cmes.2025.072447

Received 27 August 2025; Accepted 10 October 2025; Issue published 26 November 2025

Abstract

Recent progress in topology optimization (TO) has seen a growing integration of machine learning to accelerate computation. Among these, online learning stands out as a promising strategy for large-scale TO tasks, as it eliminates the need for pre-collected training datasets by updating surrogate models dynamically using intermediate optimization data. Stress-constrained lightweight design is an important class of problem with broad engineering relevance. Most existing frameworks use pixel or voxel-based representations and employ the finite element method (FEM) for analysis. The limited continuity across finite elements often compromises the accuracy of stress evaluation. To overcome this limitation, isogeometric analysis is employed as it enables smooth representation of structures and thus more accurate stress computation. However, the complexity of the stress-constrained design problem together with the isogeometric representation results in a large computational cost. This work proposes a multi-grid, single-mesh online learning framework for isogeometric topology optimization (ITO), leveraging the Fourier Neural Operator (FNO) as a surrogate model. Operating entirely within the isogeometric analysis setting, the framework provides smooth geometry representation and precise stress computation, without requiring traditional mesh generation. A localized training approach is employed to enhance scalability, while a multi-grid decomposition scheme incorporates global structural context into local predictions to boost FNO accuracy. By learning the mapping from spatial features to sensitivity fields, the framework enables efficient single-resolution optimization, avoiding the computational burden of two-resolution simulations. The proposed method is validated through 2D stress-constrained design examples, and the effect of key parameters is studied.

---

Keywords

---