# Topology Optimization of an Aerospace Bracket: Numerical and Experimental Investigation
## 1 Introduction
Topology optimization systematically modifies an object's geometry by changing its topology and shape with the aim of producing designs with improved performance and mechanical strength while minimizing material usage and weight. Topology optimization software packages coupled with computational tools for structural analysis can calculate stresses and reduce or eliminate regions with redundant material usage. Therefore, designers can optimize material distribution as dictated by the objective functions of choice, including maximization or minimization of relevant quantities, such as load, stiffness, deformation, and others. This permits the identification of the best possible geometry of a given component subject to a given set of performance criteria and engineering constraints [12]. Therefore, incorporating topology optimization into additive manufacturing enhances the arsenal of methods which enable the realization of designs that minimize material usage while increasing the stiffness-to-weight ratio, promoting eco-friendliness, achieving cost-effectiveness by placing material where necessary, reducing production time and cost, and finally, facilitating fast iterative processes [13]. According to [14; 15], a typical topology optimization problem can be expressed by the following general constrained minimization problem:  
\[\min_{\rho}F=F(\mathbf{u}(\rho),\rho)=\int_{\Omega}f(\mathbf{u}(\rho),\rho)dV, \tag{1}\]  
subject to:  
\[ \begin{split} G_{0}(\rho)=\int_{\Omega}\rho dV-V_{0}\leq 0,\\ G_{j}(\mathbf{u}(\rho),\rho)\leq 0,\text{ and }H_{i}(\mathbf{u}(\rho), \rho)=0,\quad\text{with }j=1,\dots,m,\text{ and }i=0,\dots,n\end{split}  \tag{2}\]  
where the objective function, \(F(\mathbf{u}(\rho),\rho)\), commonly corresponds to compliance which, when minimized, leads equivalently to maximization of the structure's stiffness. As for the remaining symbols, \(\rho(\mathbf{x})\) denotes the unknown variable, indicating whether material is present at location \(\mathbf{x}\), with 1 corresponding to material presence and 0 to absence; note that this variable can be also treated as a pseudo density, assuming any value in [0, 1] and thus simplifying the solution of the optimization problem. The symbol \(\mathbf{u}(\rho)\) denotes a relevant state field depending on the unknown variable (see also [16]), whereas \(\Omega\) corresponds to the design space, which may be defined by additional constraints determining the allowable volume, fixed model regions, and other factors. Finally, functions \(H_{i},~{}G_{j},~{}i=0,\ldots,n,~{}\text{and}~{}j=0,\ldots m\), correspond to functional constraints (equalities and inequalities), with the first inequality constraint (\(j=0\)) corresponding to a given volume constraint (\(V_{0}\)). The finite element method (FEM) is commonly used to estimate the value of \(\mathbf{u}(\rho)\) as there are typically no analytical solutions for the differential equations in general domains that correspond to \(\mathbf{u}\); see also [14]. Following the introduction of TO in [17, 18], several methods--such as density, level set, topological derivative, phase field, evolutionary, and others--have emerged as techniques for performing topology optimization [16]. Nevertheless, the authors in [13] noted that the density-based method, evolutionary structural optimization (ESO), and the level-set method (LSM) are the most dominant. The discretization of the design domain into small finite elements is a typical prerequisite in a broad range of methods dealing with TO, although LSM uses the level set of a hypersurface to determine and modify the boundary of the object at hand. For methods requiring voxelization--which commonly needs to be sufficiently fine to allow for any shape intricacies--the material density, which can be considered as either a binary or continuous variable, needs to be defined for each element/cell. However, targeting complex topologies and shapes through an increased number of elements leads to expensive computational simulations.