## 2 Methods
### Mathematical Formulation and Computation Set-Up  
Density-based topology optimization is formulated through the finite element method (FEM). Element density acts as a design variable, so that an optimal distribution of elements layout may be provided within the chosen design space. Design space is approximated by a finite element analysis (FEA) mesh. FEA determines the displacement and stress gradients that help TO in the identification of elements that do not affect the optimization problem (mass reduction, compliance minimization, frequency response, etc.). During each iteration, sensitivity analysis is performed by the solver to evaluate the effect of the element density variable on the objective function and, subsequently, the element densities are updated in the design space for the next iteration and the same process is continued until a final convergence is obtained. Mathematically, for convergence, the derivative of the objective function with respect to the design variable, the element density, is studied. Basically, in density-based TO, the elements are assigned a density value from 0 to 1, where 0 represents a void, 1 represents a solid and values from 0 to 1 represent intermediate densities. In order to enforce the solution to the solids and voids discrete combination, density-based TO exploits the SIMP (solid isotropic material penalization) approach to eliminate the elements with intermediate densities with the help of power law (as shown below):  
\[k_{i}=\left(\rho_{i}\right)^{P}k_{o} \tag{1}\]  
where \"\(i\)\" is the number of elements in the design space, \(i=1,2,3,\ldots,L\); \"\(\rho\),\" is the density of the \(i\)th element, \"\(P\)\" is the penalization factor in the range \(1<P\leq 3\); \"\(k_{0}\)\" and \"\(k_{i}\)\" are the stiffnesses of \(i\)th element before and after penalization, respectively. TO with SIMP approach has been declared the most implemented and developed approach with respect to math simplicity and computational efficiency. As per numerical experimentations, a penalty factor of value 3 has been found the most suitable one for penalization purpose. Distinction between design and non-design space allows an easy set-up of the conceptual geometric domain, assuming as non-design space regions constrained by functional interfaces (e.g., displacement constraints, contact surfaces, other assigned boundary conditions). The optimization problem solved by SIMP is a minimization problem related to design space volume fraction or compliance (\(C\)), or in another words, maximization of the stiffness of the structure that can be defined for the overall domain or only for the design space. According to compliance theory, compliance is inversely proportional to the stiffness of the structure. The global compliance is the sum of the element's elastic characteristics or strain energies (as shown below). \[C=\frac{1}{2}FU^{T}=\frac{1}{2}\sum\limits_{i=1}^{L}f_{i}\;{u_{i}}^{T} \tag{2}\]  
where \"\(F\)\" is the external load vector, \"\(U\)\" is the global displacement vector and \"\(L\)\" is the overall number of finite elements. According to Hook's Law \(F=KU\), therefore \(C=\frac{1}{2}F^{2}K^{-1}\) hence \(\frac{1}{2}\) and \(F\) are constants, therefore \(C\)\(\alpha\)\(K^{-1}\) where \(K\) is the global stiffness matrix. Using the SIMP approach \(K\) is computed as:  
\[K=\sum\limits_{i=1}^{L}k_{i}=\sum\limits_{i=1}^{L}(\rho_{i})^{P}k_{o} \tag{3}\]  
where \"\(\rho_{i}\)\" is in the range \(\eta\leq\rho_{i}\leq 1\) and \(i=1,\;2,\;3\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\), \(L\); and \(\eta\) is a very small number but greater than zero to prevent the global matrix from being singular and so ensure the FEA stability. Different constraints may be set (e.g., mass, load, displacement in specific ranges) and also weighted customized objective functions may be defined. Similarly, TO may be subjected to manufacturing constraints to achieve feasible manufacturing shapes (e.g., direction of protrusion, forging shapes, etc.) [1].