# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 2 Design Variables  
This paper focuses on the optimisation of planar frame structures, using 2D beam elements with hollow circular cross-sections for structural simulations. However, the proposed method can be easily extended to 3D, applied to trusses, and adapted to encompass different types of cross-sectional profiles. The hollow circular cross-section is defined by two parameters, the outer diameter \(d\) and wall thickness \(t\), with an assumed relationship of \(t=d/20\). During the optimisation process, the outer diameters of elements are taken as design variables and are allowed to vary within the range of \([0,d_{\text{max}}]\), where \(d_{\text{max}}\) represents the predefined maximum diameter of cross-section. A threshold parameter \(d_{\text{th}}\), typically much smaller than \(d_{\text{max}}\), is introduced to differentiate the contribution of elements. Elements with \(d_{e}<d_{\text{th}}\) are termed low-stiffness elements. Moreover, analogous to the density-based topology optimisation of continuum structures, an auxiliary topology variable is defined to indicate the status of elements, which can be expressed as  
\[\rho_{e}=\left\{ \begin{aligned} 1,&\text{if }d_{e}\geq d_{\text{th}}.\\ d_{e}/d_{\text{th}},&\text{otherwise}.\end{aligned} \right. \tag{1}\]  
where elements with \(d_{e}\geq d_{\text{th}}\) are assigned \(\rho_{e}=1\), classified as solid elements, and are retained in the layout. Conversely, elements with \(d_{e}=0\) are assigned \(\rho_{e}=0\), deemed void, and removed. For elements with \(0<d_{e}<d_{\text{th}}\), \(0<\rho_{e}<1\) is assigned, classifying them as grey elements. The grey elements should be optimised towards solid or void to minimise their presence. This is achieved using the SIMP-based approach [65], with the penalisation imposed on the elements' cross-sectional diameters rather than the material elasticity modulus as in the original method. The penalised element diameter can be expressed as  
\[\hat{d}_{e}=\left\{ \begin{aligned} & d_{e},&\text{if }d_{e}\geq d_{\text{th}}.\\ &\rho_{e}{}^{\omega}d_{\text{th}},&\text{otherwise}. \end{aligned} \right. \tag{2}\]  
where \(\omega\) is the penalisation factor, usually greater than \(1\). A larger \(\omega\) results in more severe penalisation, leading to a drastic decrease in element diameters and effectively driving the removal of grey elements. The sensitivity of \(\tilde{d}_{e}\) with respect to the design variable is  
\[\frac{\partial\tilde{d}_{e}}{\partial d_{e}}=\left\{ \begin{array}{ll}1,&\text{ if }d_{e}\geq d_{\text{th}}.\\ \omega\rho_{e}\omega^{-1},&\text{ otherwise.}\end{array} \right. \tag{3}\]  
Unless otherwise specified, \(d_{\text{max}}=0.5\) m and \(d_{\text{th}}=d_{\text{max}}/10=0.05\) m are used as the default values. Given that the wall thickness of the hollow circular cross-section is set to \(t=d/20\), the cross-sectional areas and moments of inertia scale with the square and fourth powers of the diameter, respectively. Consequently, the stiffness of elements with \(d=d_{\text{th}}\) is approximately 1/10,000 to 1/100 of the stiffness of elements with \(d=d_{\text{max}}\). Therefore, elements with \(d_{e}<0.05\) m, which offer very low stiffness but compromise manufacturability, should be removed during optimisation. Other values for \(d_{\text{th}}\), such as \(d_{\text{max}}/5\) and \(d_{\text{max}}/20\), may be selected depending on stock availability and manufacturing considerations. However, a larger \(d_{\text{th}}\) might result in an increased number of grey elements (\(0<d_{\text{e}}<d_{\text{th}}\)). To prevent abrupt changes in element diameters and sudden removal, and to stabilise the optimisation process, a gradual increase in the penalisation factor \(\omega\) and a small step size for updating design variables should be employed.