## 3 Optimization of the Hybrid Last-Stage Launch Vehicle
This flowchart provides a structured representation of the optimization process, highlighting the parameters involved, the computational tools employed, and the performance metrics that guide the search for an optimal design and trajectory of the launch vehicle. The calculation steps involve hybrid propulsion, CATIA-based structural properties, aerodynamics characteristics, and trajectory determination. In the 6 DOF simulation, the aerodynamics, propulsion, structure, and trajectory subsystems are coupled. This coupling  
Figure 4: Workflow of the multi-objective multidisciplinary optimization process. implies that the outputs of the propulsion subsystem serve as inputs to the trajectory and structural subsystems, and vice versa. The specific couplings between subsystems are defined in Table 3. The table encompasses the input and output parameters among the subsystems of the launch vehicle. The columns indicate whether each parameter serves as an input for a given subsystem, while the rows specify whether the parameters act as outputs from the subsystems. From the propulsion subsystem, information such as thrust and propellant mass is provided to the trajectory model; stage length, propellant mass, and stage diameter are provided to the structural model; and motor length and motor diameter are provided to the aerodynamic model. The structural subsystem supplies center of gravity and moment of inertia information to the trajectory model and center of gravity information to the aerodynamic model. From the aerodynamics model, aerodynamic coefficient details are provided for the trajectory model. Conversely, the trajectory model supplies Mach number, angle of attack, and sideslip angle information to the aerodynamic model. The optimization process was executed on a workstation equipped with an Intel Xeon Silver 4216 CPU operating at 2.10 GHz, 64 cores, and 128 GB RAM. The process runs for approximately 8 h and 11 min. To expedite the process, 30 models were created and run simultaneously in parallel. The initial design parameters were selected using the design of the experiment methodology. Throughout the optimization process, the design parameters were constrained to scan a wide range. However, due to the simultaneous fulfillment of targeted trajectory parameters being a highly challenging criterion, it has been observed that as the iteration steps progress during optimization, the algorithm converges until the global optimum solution is reached, and the optimization process is terminated. A total of 30,000 iterations were performed, resulting in 29,688 unfeasible designs and 310 errors. An unfeasible solution indicates that a specific point in the design space cannot be physically realized. In other words, it denotes situations where constraints related to that design configuration are not satisfied or conflict. Errors signify a potential issue with simulation or analysis tools. This could be due to various reasons such as an error in input parameters, model configuration, or missing/corrupted data files. However, since only 1.03% of the entire solution space is affected, it does not carry critical importance in the optimization solution here. Moreover, two feasible points were obtained for the multi-objective multi-disciplinary problem. The reasons for finding a few feasible points lie in the complexity of multi-stage launch vehicles and hybrid-propellant motor optimizations. Additionally, the scarcity of feasible points indicates that solutions meeting specific constraints, such as providing required orbit parameters for placing the payload into the targeted orbit, may be limited. The constraints narrow down the solution space, reducing the number of viable points. In these optimization problems, it is crucial to ensure the physical feasibility of the found solutions, achieve practical optimal results, and performance improvement [58; 59]. Upon successful completion of the optimization process, feasible designs and the corresponding broken constraints are acquired.