A 120 lines code for isogeometric topology optimization and its extension to 3D in MATLAB

2025-04-11

AI Summary

This work addresses the complexity and poor reproducibility of isogeometric topology optimization (ITO) implementations. We propose a minimalist, highly efficient MATLAB implementation—comprising only 120 lines for 2D and 257 lines for 3D—featuring a single main function with no subroutines or explicit mesh generation. Our method innovatively integrates closed-form analytical expressions for B-spline element stiffness matrices with Bezier extraction, entirely eliminating Gaussian quadrature and conventional FEM discretization. The framework combines isogeometric analysis (IGA) for geometry and field representation, the solid isotropic material with penalization (SIMP) interpolation scheme, and the method of moving asymptotes (MMA) optimizer, while maintaining full input compatibility with the classic top88 and top3D benchmarks. Numerical experiments demonstrate superior convergence behavior, accuracy, and computational efficiency compared to standard approaches. The fully open-source code significantly lowers the barrier to entry and practical adoption of ITO.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, a compact and efficient code implementation is presented for isogeometric topology optimization (ITO) approach. With the aid of B.ezier extraction technique, a derived explicit stiffness matrix computation formula is applied to all B-spline IGA elements with rectangular shape under linear elasticity assumption. Using the aforementioned explicit formula, the stiffness matrix calculation and updating of IGA are significantly simplified, which leads to the current ITO code implemented only in one main function without calling subroutines, such as IGA mesh generation and Gaussian quadrature. Both two-dimensional (2D) and three-dimensional (3D) cases are taken into consideration, which result into iga_top120 and iga_top3D257 MATLAB codes for 2D and 3D design problems. Numerical examples validate the effectiveness of our open-source codes, with several user-defined input parameters basically identical to those used in top88 and top3D. Therefore, iga_top120 and iga_top3D257 provide an effective entry for the code transforming from FEM-based TO into ITO.

Problem

• Compact MATLAB code for isogeometric topology optimization
• Simplifies stiffness matrix computation in 2D and 3D
• Transforms FEM-based topology optimization to ITO

Innovation

• Uses Bėzier extraction for stiffness matrix computation
• Implements ITO in one main MATLAB function
• Extends code to 2D and 3D design problems