# Layout Optimisation of Frame Structures with Multiple Constraints and Geometric Complexity Control
## 4 Problem Formulation  
Conventionally, material volume serves as a metric reflecting the economy of a structure. To facilitate control over geometric complexity in structural layouts during the optimisation process, an additional component representing the fabrication cost of the structure is heuristically integrated with material volume to form the objective function. This modified objective function can be expressed as  
\[E(\mathbf{d})=\frac{V(\mathbf{d})}{V_{0}}+\alpha\frac{C(\mathbf{d})}{C_{0}} \tag{31}\]where \(V(\mathbf{d})\) is the material volume and \(V_{0}\) is its initial value calculated at the first iteration. \(C(\mathbf{d})\) represents the fabrication cost and \(C_{0}\) is its initial value. \(\alpha\geq 0\) is a weighting factor.  
The structural fabrication cost is obtained by summing up the elemental fabrication cost:  
\[C(\mathbf{d})=\sum_{e=1}^{N}\zeta(d_{e}) \tag{32}\]  
where \(\zeta(d_{e})\) represents the elemental fabrication cost and is approximated by a smooth Heaviside function:  
\[\zeta(d_{e})=1-e^{-\beta\,d_{e}/d_{\text{max}}}+\frac{d_{e}}{d_{\text{max}}}\,e ^{-\beta} \tag{33}\]  
where \(\beta\) is a parameter controlling the sharpness of the Heaviside projection. The larger the \(\beta\), the more aggressive the projection. For \(d_{e}\in[0,d_{\text{max}}]\), \(\zeta(d_{e})\) adopts the value of \([0,1]\), with \(0\) indicating the removal and \(1\), the presence of elements.  
The differential of \(\zeta(d_{e})\) with respect to the design variable is  
\[\frac{\partial\zeta(d_{e})}{\partial d_{e}}=\frac{\beta}{d_{\text{max}}}e^{- \beta\,d_{e}/d_{\text{max}}}+\frac{1}{d_{\text{max}}}\,e^{-\beta} \tag{34}\]  
The plots of \(\zeta(d_{e})\) and its differential \(\partial\zeta(d_{e})/\partial d_{e}\), corresponding to \(d_{e}\in[0,0.5]\) and different values of \(\beta\) as depicted in Figure 3, reveal that with an increase in \(\beta\), \(\zeta(d_{e})\) converges more rapidly towards \(1\). Moreover, \(\partial\zeta(d_{e})/\partial d_{e}\) exhibits higher values for smaller \(d_{e}\), indicating a more pronounced penalisation of elements with smaller diameters, thereby rendering them more susceptible to removal. In this study, a moderate value of the sharpness parameter is used (\(\beta=8\)), in conjunction with different values of the weighting factor \(\alpha\). Through the modulation of \(\alpha\) in Equation (31), varying degrees of penalisation, as imposed by the fabrication cost, can be achieved. This adjustment consequently influences the rate at which the values of design variables decrease, implicitly governing the elimination of elements and exerting control over the overall number of elements, i.e., the geometric complexity present in the structural layout.  
Building upon the previously introduced objective and constraint functions, the optimisation problem for the layout of frame structures, incorporating constraints related to displacement, stress, and local and global stability, alongside the modified objective function comprising material volume and fabrication cost of structures, can thus be formulated as  
Figure 3: Element fabrication cost and its sensitivity.  
local and global stability constraints are included, the minimum allowable buckling load factor is \(\underline{\lambda}=5\).