# A Comparative Study of the Application of Different Commercial Software for Topology Optimization
## 2 Theory  
According to Christensen and Klarbring [32], a general SO problem can be expressed mathematically by an objective function \(f(x,y)\), a design variable (\(x\)), and a state variable (\(y\)). The objective function is a function that is used to classify the designs and can usually measure compliance, weight, and displacement. On the one hand, the design variables are related to the geometry of the structure or the chosen material. On the other hand, the state variables represent the response of the structure, such as stress and strain. In addition, Christensen and Klarbring [32] categorize the design constraints into behavioral, design, and equilibrium. The behavioral constraints are related to the state variable, while the design constraints concern the design variables. Finally, the discretization of the design space \(\Omega\) creates the need for equilibrium constraints, which, in a linear problem, have the following representation:  
\[K(x)u=F(x) \tag{1}\]  
where \(K(x)\), \(u\), and \(F(x)\) are the stiffness matrix, the displacement vector, and the force vector, respectively. In the traditional compliance TO, also described by Bendsoe and Sigmund [4], the objective function measures the stiffness of the structure via compliance (\(f(x,y)=c\)), and the design variable is a pseudo-density of the elements (\(x=\rho_{e}\)), where \(\rho_{e}=1\) means material and \(\rho_{e}=0\) is the void. In addition, a volume constraint, \(g_{i}\), is used to express the maximum amount of the distributed material (\(g_{i}^{*}\)). A nested representation of the compliance TO problem is as follows:  
\[\text{min}c(\rho_{e},\ u(\rho_{e}))=\int_{\Omega}c(u(\rho_{e}),\rho)dV \tag{2}\]  
\[\text{subject to}\quad g_{0}(\rho_{e})=\int_{\Omega}\rho_{e}dV-V_{0} \tag{3}\]  
\[g_{i}(\rho_{e},u(\rho_{e}))\leq g_{i}^{*},\ i=1,\dots,\text{M} \tag{4}\]  
\[K(\rho_{e})U=F(\rho_{e}) \tag{5}\]  
\[\rho_{e}=\left\{\begin{array}{l}0\\ 1\end{array},\ e=1,\ \dots,\ \text{N}\right. \tag{6}\]  
However, the discrete formulation faces many problems, such as the checkerboard problem and computational limitations. Thus, continuous variables are used instead of binary ones. The elements' densities can now attain values between zero and one [4], as follows:  
\[0<\rho_{min}\leq\rho_{e}\leq 1,\ e=1,\ \dots\ \text{N},\ \rho_{min}\
eq 0\ (\text{ lower bound}) \tag{7}\]  
The solution of this problem can be conducted by either gradient-based optimization techniques (GTO), such as the optimality criteria (OC) and the method of moving asymptotes (MMA), and non-gradient topology optimization techniques (NGTO), such as genetic algorithms [33]. Interpolation is used for the expression of the material properties in a continuous setting. One of the most popular interpolation methodologies is the solid isotropic material with penalization (SIMP). SIMP interpolates Young's modulus of the material (\(E\)) using the following power law [4]:  
\[E(\rho_{e})=\rho_{e}^{p}E_{0},\ p\geq 1,\ E_{0}\
eq 0\ (\text{lower bound}) \tag{8}\]  
where \(p\) is a penalization parameter for the elements with intermediate densities, which usually takes a value between 1 and 3 [34]. Usually, the traditional density-based TO is called simply SIMP in the literature. Other notable TO methods are the level set [20], the evolutionary structural optimization (ESO) [35], the topological derivatives [36], and the phase field [37].