# Effect of Dynamic Loading Conditions on Maximizing Energy Dissipation of Metallic Dampers
## 2 Design Concept and Optimization Approaches
### Optimization Approach  
The purpose of the optimization process was to find the optimum shape that dissipates the greatest amount of seismic energy during an earthquake. The optimization objectives are represented by Equation (1). The amount of dissipated energy is represented by \(\mathrm{E}(x_{\mathrm{i}})\). This energy term comprises the variables that are the control points, \(x_{\mathrm{i}}\), where \(x_{\mathrm{i}}\) indicates the width of the damper at location i. However, to obtain a single valid result through the process of optimization, boundary constraints were also applied. Here, two different boundary conditions were applied. First, the coordinates of the control points were regulated by their constraints. Second, the constraint of the total volume of the damper was determined. The first boundary condition restricts the alteration with the upper and lower bounds for the control point coordinates, \(x_{i}^{ll}\) and \(x_{i}^{L}\), respectively. Moreover, the volume constraint was applied so that the volume of the changed shape might not exceed the initial volume. This constraint also helped to increase the efficiency of the device by using only a small amount of the component material.  
\[\begin{array}{ll}\mathrm{Maximize}&\mathrm{E}(x_{\mathrm{i}})\ (x_{\mathrm{i}}:\ shape\ variables)\\ \mathrm{Subjectto}&\mathrm{V}(x_{\mathrm{i}})\leq V_{0}\\ &x_{\mathrm{i}}^{L}\leq x_{\mathrm{i}}\leq x_{\mathrm{i}}^{L}\end{array} \tag{1}\]  
The amount of energy dissipation is derived by extracting the maximum ALLPD from the ABAQUS FE analysis. ALLPD represents the energy dissipation through the plastic deformation of the model during the analysis. This value is calculated by using Equation (2):  
\[\mathrm{ALLPD}=\int_{V}\int_{0}^{t}\sigma:\ \dot{e}^{pl}dtdv \tag{2}\]  
where \(\sigma\) is the total stress and \(\dot{e}^{pl}\) is the plastic strain rate that occurred on the model volume \(V\) during the analyzed time, \(t\). The \(\sigma\) and \(\dot{e}^{pl}\) are affected by the deformation of  
Figure 2: Initial geometry of the proposed damper.  
the damper under the given loading conditions. Therefore, they are eventually affected by the initial un-deformed geometry of the damper, which is defined by the control variables \(x_{\rm i}\). Through alteration of the control variables, the optimization process was conducted by repetitive FE analysis. An integrated system comprising MATLAB R2018a and FE software (ABAQUS) was established. In this system, the geometrical alteration is carried out by modifying the control point coordinates in a Python script that runs on ABAQUS. After the analysis is finished for one changed shape, the Python script extracts the dissipated energy, which becomes the objective of the total optimization process. A complete flowchart of this process is shown in Figure 3.  
The sequential quadratic programming (SQP) algorithm was used for the optimization in MATLAB processing. The SQP algorithm is one of the constrained nonlinear optimization algorithms used by the MATLAB toolbox. The algorithm consists of three main steps: updating (1) the Hessian matrix, (2) the quadratic programming solution, and (3) the line search and merit function. The SQP algorithm executes all of the iterative steps in the region limited by the constraints, and if the constraint is not satisfied, the merit function is applied by combining the objective function and the constraint function into one gain function [30]. Its proven robustness makes the algorithm suitable for this problem, where it is difficult to derive an explicit relationship between shapes and object function due to the strong geometric nonlinearity.