## 2 Methods
### Optimization Process Via TPO/MA  
In the section, the optimization process performed for the minimization of the total potential energy is explained. In the optimization, the objective function is the total potential energy of the system given in Equation (28). It is calculated by summing the energy values of members and subtracting the work done by all external forces. The flowchart of handling of metaheuristics to provide optimum energy as TPO/MA can be shown in Figure 2.  
As with all metaheuristic-based optimization algorithms, the method starts with the definition of nodal coordinates, material constants, nodal loads, and algorithm parameters. The nodes of the system are numbered and the nodal numbers of the members are defined. In that case, it is not needed to define an extra relation for the compatibility of displacements at the nodes. The design variables of the problem are the nodal displacements. For boundary conditions of fixed freedoms, these conditions are taken as design constants and defined as zero. All the other nodal displacements are randomly defined by considering the algorithm rules, and since the displacements are known, it is very practical to calculate directly the energy of all members and then, the whole system. Since the aim of the process is the analysis instead of design, design constraints are not used. Then, an initial solution vector that includes the candidate values of design variables is generated. These variables are randomly selected in a predefined solution range. In TPO/MA, the design variables are the possible displacements at the nodes of the system that are not correct at the beginning, and then, the exact solution is obtained for the set that has the minimum potential energy value of the system at the end of the iteration process. Once these displacements are set, the distribution of strains in the structure can be defined, followed by the determination of stresses in concordance with the stress-strain relations, whether they are linear or not. Knowing the distribution of stresses and strains, the strain energy at each element can be determined. The summation of all strain energies in the elements gives the total strain energy in the system. Algebraically adding to this value, the work done by external forces due to nodal displacements, the total potential energy of the system becomes calculated. Comparison of these energies calculated for all candidate vectors enables the determination of the best and worst vectors. According to this evaluation, the employed algorithm determines how new candidates will be determined. With this process, one approaches a better set of candidates, and repeatedly following this approach, after several iterations, the final results are obtained. Nine different metaheuristic algorithms including the genetic-based ones (GA [14], DE [15]), population or nature-based ones (PSO [16], ABC [17; 18], FPA [19], TLBO [20], JA [21], GWO [22]), and memory-based HS [23] are evaluated for the linear analysis of the first structural model. In this study, HS was applied using the modified equations. Here, two different stages exist to find the best harmonies by a musician as given in Equation (29). \[\left\{ \begin{array}{cc}if\ \text{HMCR}>\text{rand}(\ )&X_{i,\ min}+\text{rand} \ (\ )\ (X_{i,max}-X_{i,min})\\ else&X_{i,n}+\text{rand}\Big{(}\frac{-1}{2},\frac{1}{2}\Big{)}\text{FW}\ (X_{i,max}-X_{i,min}) \end{array} \right.. \tag{29}\]  
There are two different HS parameters: HMCR is harmony consideration rate, and FW is the fret width. \(X_{i,\ min}\) and \(X_{i,\ max}\) are also upper and lower bounds of \(i\)th design variable, respectively. On the other hand, \(X_{i,n}\) is \(n\)th candidate solution and is found as randomly, besides \(X_{i,\ new}\) expresses the new value for this variable. Figure 2: The flowchart of the Total Potential Optimization using Metaheuristic Algorithms (TPO/MA) including three stages used in the process. Also, an adaptive version of HS is employed in the study.