# Multi-Objective Pareto Optimization of Tensile Membrane Architecture for Energy Harvesting
## 3 Methodology
### Configuration for Genes: Geometry  
This section introduces the parameters that determine the TMA geometry: shape, material stiffness, and orientation. First, the three types of shapes (hyper, conic, and barrel vault) are introduced. Because the three types differ with respect to the structural features and the major factors affecting the structure, the experiments were conducted separately [30]. A \(14\times 14\times 7\) m bounding box was assumed, and the three aforementioned types of TMAs were formed within the bounding zone (Figure 3). The orientation, compared with our previous research [12], was a newly added variable, while the rest of the bounding box and TMA shape variables were under the same conditions. Figure 2: Wind speed graphs of three different frequencies for Inner Mongolia, China (**top**) and Texas, USA (**bottom**). The hyper shape has alternating high and low support points, and the most basic hyper shape comprises two high points and two low points. More complex hypars may have larger numbers of supports, but in the present simulation, optimization was performed on a hyper with only four supports, two of which were placed diagonally and fixed to the bottom, and the other two were unfixed supports located in the range of 0-7 m along the z-axis. \"Conic\" refers to a structure consisting of a head ring and a square base support. In the case of the conic shape, the membrane fails if the minimum surface and the desired shape differ [36]. Head ring height and radius values were determined according to the feasible ranges identified by [PERSON] et al. [36]. Thus, to satisfy the feasible bounds, the radius was fixed within the range of 0-7 m, and the height was determined to be between 0 and 7 m.  
Barrel vault consisted of two elliptical curves located at the ends of two parallel lines. The lengths of the parallel straight lines were both 14 m, the distance between them was fixed at 14 m, and the y-radius of each of the two ellipses was in the range of 0-7 m.  
All alternatives of the three types of TMAs could be rotated around the center. To prevent the overlap of the geometry due to rotation, the turning radii for the hyper and barrel-vault shapes were \(<\)180 deg, and that for the conic shape was \(<\)90 deg. The rotation angle of the TMAs was defined as \"Orientation\" and determined which side of the structure was facing the wind. The main direction of the winds is different in Inner Mongolia versus Texas; orientation was the parameter which determined the direction of the TMA geometry that can have the maximum energy generation. To apply the stiffness of the membrane material to the simulation, a brief calculation method of Kangaroo was used, wherein the solver regards the membrane surface as multiple spring objects linked to each other. As shown in Figure 3, hundreds of springs comprised a curved membrane, and these springs were used to reach the desired length with a specific strength. According to [PERSON] [32], the spring objects follow [PERSON]'s law; i.e., the spring strength is calculated as EA/L, where E represents [PERSON]'s modulus, A represents the cross-sectional area of the spring, and L represents the resting length. The stiffness range in the simulation was determined according to measurement data from the research of [PERSON] [37]. The two most commonly used membrane materials from his study--polyester fiber and glass fiber--had longitudinal stiffness of 25,200 and 54,720 N/cm, respectively. In the simulation, warp and weft elastic modulus was supposed to be identical and an available material stiffness range was determined. The stiffness range was modified by the minimum and maximum of the [PERSON]'s modulus of the two materials, referred from the [PERSON]'s study [37], and the formula to get longitudinal stiffness from [PERSON]'s modulus was used. The stiffness range was established from the minimum value of polyester fiber's stiffness to the maximum value of glass fiber's stiffness and calculated as 1200 to 6840 kN/m, respectively.