Nurbs Based Isogeometric Analysis to Perform Topology Optimisation of Continuum Structures Using Evolutionary Algorithms

Abstract

This research investigates the application of Isogeometric Analysis (IGA) for topology optimisation of continuum structures under various loading and boundary conditions. IGA, which utilizes Non-Uniform Rational B-Splines (NURBS) for both geometric representation and solution approximation, offers a seamless integration of CAD and analysis phases. This significantly enhances the accuracy and efficiency of simulations compared to traditional Finite Element Analysis (FEA), where mesh generation remains a time-consuming and error-prone step. The study focuses on minimizing structural weight while satisfying constraints on stress and displacement using the Firefly Algorithm for optimisation. Optimisation was carried out for three benchmark problems: a 2D cantilever plate, a 2D simply supported beam, and a 3D simply supported cube. Results were evaluated in terms of volume reduction and design efficiency, and validated through comparison with prior studies and commercial FEA software (MIDAS NFX®). The IGA-based approach demonstrated higher material savings and smoother convergence trends, confirming its potential in structural optimisation. Notably, the 3D cube retained only 11.5% of the initial volume, outperforming previous FEA results. This work reinforces the capability of IGA for advanced structural design and highlights its suitability for future integration into commercial analysis tools.

Introduction

The main goal of structural design is to safely carry loads while managing internal stresses through proper design, ensuring safety, durability, and functionality. Modern demands for lighter, cost-effective, and high-performance structures have led to structural optimisation, which improves size, shape, or topology within constraints to maximize or minimize objectives like weight or stress.

Topology optimisation is a key approach that determines efficient material layout. Two main methods exist: element-based (with constant variables but limited geometric accuracy) and nodal-based (more accurate and adaptable, suited for meshless methods like Isogeometric Analysis, IGA). IGA, introduced in 2005, integrates CAD geometry and analysis by using NURBS basis functions, eliminating mesh generation and reducing modeling time while improving accuracy.

The study applies IGA to topology optimise 2D plate structures and a 3D cube under specific loads and boundary conditions. Using C++ and the Firefly Algorithm for optimisation, the project aims to minimize weight while meeting stress and displacement constraints.

Results for 2D structures (cantilever plate and simply supported beam) show that IGA produces optimal material layouts comparable to commercial FEA software, demonstrating IGA’s efficiency and accuracy in structural topology optimisation.

Conclusion

This study applied the Firefly Algorithm, inspired by firefly behavior and introduced by X.S. Yang (2009), for topology optimisation of continuum structures under in-plane loading. Two 2D and one 3D problems were optimized. The 2D cases included a cantilever beam and a simply supported (Michell) beam. Both were optimized using 396 control points, significantly fewer than the 1617 used by Hassani et al., yet yielding similar material distributions. The cantilever results were also verified using MIDAS NFX®. The Michell beam, optimized for symmetry, achieved a final volume of 27.84% compared to Hassani’s 20%, likely due to mesh differences. The 3D problem was solved using linear B-splines in all directions and achieved a volume of 11.5%, outperforming the 17.6% volume from Abolbashari’s 2006 study using traditional FEA. These results confirm that Isogeometric Analysis (IGA) is a highly efficient and accurate approach for structural mechanics, as it uses the same NURBS basis for geometry and displacement fields. IGA\'s ability to precisely represent geometry enhances solution accuracy over standard FEA. Given its advantages, IGA is expected to be integrated into commercial FEA tools, with LS-DYNA® already exploring its implementation.

References

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Copyright

Copyright © 2025 Hitesh Bharti, Pradeep Ilamkar, Prof. Kshitij Thate. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.