# Topology and Parametric Optimization-Based Design Processes for Lightweight Structures
## 3 The Experimental Design Process
However, it is not clear if this process is the ideal combination of TO and PO. It seems that relative research work is missing from the literature. For the scope of this research, both the CAD, the Finite Element Analysis (FEA), as well as the topology and parametric optimizations, were conducted in ANSYS software. Concerning the TO, the LSM was used in ANSYS Mechanical. Furthermore, the PO was implemented in ANSYS DesignXplorer and was divided into six steps: Step 1. Definition of factors and responses, Step 2. Selection of design exploration method, Step 3. Selection of DOE method, Step 4. Creation of a response surface, Step 5. Sensitivity analysis of the results, and Step 6. Design optimization. The implemented process is depicted in Figure 4.  
In the first step, the designer has to parameterize the design and decide all the possible factors (inputs) and responses (outputs) that will be used in the optimization process. On the one hand, as factors in a PO of a structure could be used, its length and thickness. On the other hand, some common responses are the mass, weight, and the volume of a structure, as well as its maximum stress,  
Figure 4: The experimental design process in ANSYS software. Figure 3: The ten design processes and their categorization based on the design workflow as well as the production method. deflection, and displacement. All three presented structures here share the same goals, which are the minimization of their mass and maximum stress. The second step is about the selection of the solution of the parametric optimization problem. In this paper, the Response Surface Methodology (RSM) was applied. The RSM was firstly developed by [PERSON] and [PERSON] [24] to leverage the data from DOE. It is a collection of mathematical and statistical techniques that are used to optimize simultaneously different variables of an objective function and represent their dependencies graphically [18]. The term response surface is derived from the appearance of a second-order model's plot. A common RSM is usually based on the method of steepest ascent. For example, a first-order and a second-order regression model of two factors (k = 2) can be formulated by the following polynomials [25]:  
\[\text{f}(x)=\beta_{0}+\beta_{1}\ x_{1}+\beta_{2}\ x_{2}+\varepsilon(x) \tag{10}\]  
\[\text{f}(x)=\beta_{0}+\beta_{1}\ x_{1}+\beta_{2}\ x_{2}+\beta_{11}\ x_{1}{}^{2 }+\beta_{22}\ x_{2}{}^{2}+\beta_{12}\ x_{1}\ x_{2}+\varepsilon(x) \tag{11}\]  
where:  
\(\text{f}(x)\): Response of the model  
\(x_{1}\), \(x_{2}\): First-order terms  
\(x_{1}{}^{2}\), \(x_{2}{}^{2}\): Second-order terms  
\(\beta_{\text{ij}}\): Regression coefficients  
\(\varepsilon(x)\): Model error  
The response surface plots for the two models are illustrated in Figure 5.  
There are many different methods of DOE that have been developed to fit response surfaces. Some of the most known DOE methods are the Full factorial, the Fractional factorial, the Box-Behnken, the Plackett-Burman, the Latin Hypercube Sampling (LHS), the Central composite, and the Taguchi designs [18]. The selected DOE method in this paper is the LHS. The LHS is a statistical method that is used to generate random samples based on the given factors [26]. This method was preferred due to its data accuracy, efficiency, and flexibility in the presence of a large number of parameters [26]. As was presented by [PERSON], [PERSON] [27], at the LHS, the sample values are placed in a square grid, also known under the name Latin square. Unlike random sampling, the researcher using LHS needs to decide on the number and the placement of the sample points inside the square. If an experimental design consists of p design points and k number of factors (random variables), its sampling space is a p \(\times\) k matrix. Each column of this matrix represents a variable, and each row a sample.