Topology Optimisation for Enhanced Buckling Resistance in Continuum Structures

Topology optimisation has gained significant attention due to its ability to design efficient structures with minimal material usage. Buckling is a critical factor that should be considered in optimisation algorithms to guarantee the stability of designed structures. However, topology optimisation that incorporates buckling resistance has remained challenging in recent decades, primarily due to issues related to inaccuracies in buckling analysis and the phenomenon of mode switching. This thesis systematically addresses these challenges through a comprehensive investigation into their underlying mechanisms and the development of robust algorithms for the topology optimisation of structures designed to resist buckling.

First, a topology optimisation approach that integrates buckling constraints into the bi-directional evolutionary structural optimisation (BESO) method is proposed. Compared to density-based topology optimisation algorithms, which often encounter pseudo buckling modes in intermediate-density elements, the proposed method offers an advantage by using only two discrete states for design variables, thereby effectively alleviating the numerical issues associated with these pseudo modes. The Kreisselmeier-Steinhauser aggregation function (KS function) is introduced to aggregate multiple buckling constraints into a differentiable form. An augmented Lagrangian multiplier method is developed to integrate buckling constraints into the objective function, ensuring computational stability. Furthermore, a modified design variable update scheme enables precise control of the evolutionary rate after reaching the target volume fraction. The effectiveness of the buckling-constrained BESO method is validated by four topology optimisation design examples, demonstrating enhanced structural stability with only a modest compliance increment.

Second, while the discrete optimisation algorithm can mitigate the occurrence of pseudo buckling modes, achieving their complete elimination remains a challenge. The mechanism underlying pseudo buckling modes is comprehensively investigated. The research reveals that these modes stem from stress singularities in the finite element analysis of multi-density elements. To facilitate the implementation of stress relaxation functions that avoid singularities, a linear material interpolation scheme is introduced to the buckling optimisation problem. This linear material model offers several advantages: it eliminates the need for penalisation schemes and penalty values, and enables straightforward sensitivity analysis. The buckling optimisation algorithm is subsequently developed based on the floating projection topology optimisation (FPTO) method, effectively driving intermediate elements towards definitive 0/1 solutions. By employing the developed method, the objective of maximising the critical buckling load factor (BLF) is successfully achieved, with four design examples demonstrating significant improvements over existing algorithms.

Third, the proposed optimisation approach is further developed to address the issues of incorrect buckling analyses and high dependency on parameter variations. This study delves into the underexplored phenomenon of ‘spurious localised buckling modes’, which can occur across all densities, including solid elements. A novel local stress relief strategy is developed to eliminate these spurious modes. Additionally, an innovative problem reformulation is introduced for buckling-constrained optimisation, which decouples the critical BLF from the aggregation function. This reformulation effectively prevents the influence of aggregation parameters on the accuracy of the critical BLF approximation and enhances optimisation stability by avoiding mode switching. The effectiveness of the proposed algorithm is validated through both BLF maximisation and buckling-constrained problems. A series of 2D and 3D examples demonstrate that the proposed algorithm requires minimal parameter tuning across various challenging cases. It ensures stable convergence and attains satisfactory objective values, while accurately fulfilling the constraints.

Finally, the buckling optimisation algorithm is extended to accommodate design-dependent problems. A mixed formulation is employed to handle three different types of elements: incompressible hydrostatic fluid, elastic solid, and air. A straightforward implementation method is proposed to identify the exchangeable domains for these three element types. The buckling analysis within the mixed formulation is systematically investigated, and the corresponding sensitivities are derived. The proposed algorithm allows for a large bulk modulus value in fluid simulation without compromising numerical stability. The proposed algorithm is tested across a variety of cases, demonstrating its ability to design underwater structures with enhanced buckling resistance.

This thesis makes significant theoretical contributions to the field of topology optimisation by systematically resolving fundamental challenges, including the mechanisms behind pseudo buckling modes, spurious localised modes, mode switching, and numerical instability. These theoretical advancements have paved the way for the development of robust algorithms with strong potential for practical applications, ranging from traditional solid mechanics structures to specialised underwater designs.