## 1 Introduction
In this context, a multi-objective formulation of the aircraft climb path optimization problem was deemed advantageous over a single-objective one because the generated Pareto fronts may better represent aircraft climb potential and allow for comparisons between different configurations to be made on a wider basis. Under this scope and given that the computational cost per simulation is rather small, a multi-objective, population-based optimization scheme was selected, also capable of being directly interfaced with the University's engine performance software, to further simplify case setup; a user will only need to specify an engine geometry and a generic aircraft model to obtain results for the climb potential of their combination, allowing for the method to also be used for educational purposes. The authors' intention is therefore not to present a climb path methodology that will compete with present gradient-based methods, but to introduce an easy-to-implement, non-mathematical, multi-objective formulation to the traditional climb path optimization problem to be used as a tool for aircraft-engine integration studies. The well-tried and tested Multi-Objective Particle Swarm Optimization (MOPSO) method [21] is selected to conduct the intended multi-objective search for optimal climb paths because it combines simplicity with fine global search characteristics. Energy-Maneuverability (E-M) theory is exploited to increase the effectiveness of the search. This is achieved in two ways: Firstly, in place of producing control histories and contrary to similar methods, the optimizer uses Bezier splines to construct candidate flight paths in the form of curves in the Altitude (h)-Mach number (M) plane. Since the general form of these paths can be easily predicted by E-M, this facilitates the selection of the design parameters. To avoid limiting the optimizer's degrees of freedom by inserting equality constraints to satisfy the aircraft state equations, a path-tracking technique for guidance in the h-M plane based on the Carrot Chasing guidance scheme [22] is introduced and used to fly an aircraft model into the designed trajectories. Secondly, a two-level optimization scheme is employed to boost convergence: An initial low-level optimization run is performed using E-M as a low-cost, low-fidelity approximate of the actual objective functions; its solutions are used to initialize a second, high-level optimization run, which employs a simulation to accurately assess the outcome of candidate flight paths. Better initialization has a positive effect on the algorithm's convergence speed and leads to improved results for a given number of fitness function evaluations. To demonstrate a practical application of the proposed method on a realistic aircraft-engine integration scenario, a model of the F-4 Phantom II is selected as the reference airframe for this application. This represents an aircraft type still in operational service, a fact that is combined with a wide database of aerodynamic and performance data [34; 23; 24] that have become available during the aircraft's long operational career: On the basis of the latter, a reasonably accurate representation of the aircraft may be constructed. Throttle-dependent forces are also included to account for installation effects and allow for a more accurate integrated engine representation. Cranfield University's in-house gas turbine performance code, Turbomatch, is used to construct engine models, outputting thrust, air mass flow, nozzle pressure ratio and specific fuel consumption. Turbomatch comprises several pre-programmed modules, which correspond to models of individual gas turbine components. They can be called up to simulate the action of the different components of the engine, resulting finally in the output of engine thrust or power, specific fuel consumption, etc. Its modularity, which is supported by the implementation of generic component maps, enables the detailed design of any gas turbine configuration.