Model based aircraft design and optimization: a case study with cessna 172N aircraft

This study aims to create an interdisciplinary, multi-level design approach for fixed wing aircrafts. A case study is performed using Cessna 172N aircraft characteristics that is based on a mathematical model having six degrees of freedom (DOF). A Model Based Aircraft Design Software (MAD) is developed using a mathematical dynamic model in Python environment for use in the optimization phase of the Cessna 172N aircraft. The MAD environment is composed of analysis tools that communicate with each other by an automated process chain system. The model includes aerodynamics, engine, mass, control, atmosphere, and landing gear submodules. In addition, the model has trim and 6 DOF simulation analysis capabilities which provide a diversity of analyses that can be considered in early design phases. In order to get trimming and simulation capability, MAD software does not need any precalculated aerodynamic database. The force and moment calculations are performed instantly via DATCOM and AVL solvers for each iteration of simulation or trim. Thanks to this method, alternative aircraft geometries can be compared at the first phase of the design and optimization processes without creating an overall database. All methodologies of model submodules, performance, stability and control calculations, design, and optimization applications are implemented in the MAD environment.


I xx
Body x-axis inertia are the following phases of the aircraft design procedure to refine the selected design in the conceptual design phase (Raymer 2018).Table 1 shows the comparison of the capabilities of five different aircraft design software available in the literature (Kaenel et al 2008, Liersch and Hepperle 2011, Lukaczyk et al 2015, Hasan et al 2018, Prakasha et al 2018, Roskam 2018, Lietzau 2023).
The MAD software has an advantage when compared to other tools in the literature, which is defined as instant trim and simulation capability.Thanks to this methodology, performance and stability analyses are based on 6-DOF simulation and trim study.The solver algorithm does not need an aerodynamic database to calculate trim points and perform simulations.Therefore, even at the first iteration of the optimization phase, all performance and stability outputs of the alternative geometries can be obtained.At the further steps of the design process, high fidelity database can be utilized to get more accurate force and moment values by minimizing the number of design alternatives (Böhnke et al 2013).
This capability of the software allows users to analyze any unconventional design in the early design phase of the aircraft.Model-Based Aircraft Design software (MAD) is created in an object-oriented Python environment to provide different aspects of view to the aircraft design and optimization processes.The main motivation of this study is to perform rapid aircraft design with the help of the instant simulation method by performing analyses based on trim and simulations, in the conceptual design process.With the current approach, it is possible to observe the advantages and disadvantages of all possible design alternatives in the early design phase.It is also possible to generate replaceable plug-in & out analysis tools which create flexible design environment having multi-fidelity analysis capability.

MAD software and model methodology
In order to get desired performance and stability outputs from the MAD software, a 6-DOF model structure is developed.In the MAD environment (Ozdemir and Kurtulus 2023), the generic trim and simulation analysis capability allows the base model structure to gain flexibility.
To validate analysis results, a short-range version of the Cessna 172N (40-gallon fuel capacity) aircraft is modeled (figure 1).From the various versions of the Cessna 172 model aircraft, the N version, is selected for validation study, which has 160 horsepower engine and 2300 lb.gross weight.Specifications of the validation aircraft are shown in table 2.Moreover, positive sign conventions are presented in table 3. The performance properties of the aircraft are also presented in detail in appendix.

Model blocks
In order to create a multidisciplinary workflow for the design and optimization process, each discipline should have connected.In this scope, the base structure of the MAD tool consists of submodules that can work collaboratively.Disciplines that are represented by submodules are shown in figure 2.
The basic modules inside of the MAD software developed (Ozdemir and Kurtulus 2023) are: v: Included/o: partially included/x: excluded.
• Weight, inertia, and center of gravity estimations.
• Primary and secondary control surface effect analysis.
• Static and dynamic coefficient effect analysis.
• System linearization and mode analysis.
• Maneuver controller via PID and inverse simulation controller.
• Visualization of simulation results and animations.
According to input parameters, force and moment calculations are performed via engine, aerodynamic, landing gear, weight, and atmosphere blocks.The Python-based 6-DOF dynamic model performs trim and simulation analyses by using related input parameters.Force and moment values are used to find the trim points by following the equations of the motion block.In the design procedure, simulations are initiated from the trim points and performed by integrating differential equations of motion.The MAD environment provides three choices for numerical integration which are Euler, second-order and fourth-order Runge-Kutta methods.The user can select one of these methods according to the required accuracy level.These integration methodologies that are utilized to conduct simulations are very crucial for design process but not sufficient to get desired maneuver performance results.To obtain related maneuver performance, PID and Inverse Simulation Controllers are utilized.All the above procedures enable to preparation of the required environment to get desired performance and stability outputs of the concept aircraft (table 4).
In the analysis part, each combination of input parameters gives related performance outputs with the help of sub-blocks.To obtain a relation between design inputs and outputs, numerous analyses are performed via the MAD environment.As a result of this process, each output parameter can be represented as a set of polynomial equations which is defined in terms of design variables.By utilizing these analytical equations set, the global optimum point can be achieved in the constrained design space.

Aerodynamic block
To get aerodynamic force and moments, a new methodology is developed which is based on calculating the aerodynamic force and moments simultaneously for a specific state combination about each iteration of Newton Raphson and Runge-Kutta methods (Özdemir and Kurtulus 2021).This concept allows to use 6-DOF model at each phase of the design process of the aircraft.
Table 5 shows general comparisons of the different aerodynamic coefficient prediction software widely used in conceptual design phases of fixed-wing aircraft.
According to table 5, DATCOM considers most of the effects in the analysis process however, force and moment coefficients are not provided for all axis in DATCOM (Fink 1960).On the other hand, OpenVSP has a wide range of analysis capabilities but, analysis time is an important disadvantage of the tool when compared to DATCOM and AVL (McDonald and Gloudemans 2022).The last alternative, AVL has both a wide range of analysis capabilities and less calculation time however, stall and parasite drag prediction capability is not included in the tool (Drela and Youngren 1988).In order to adopt aerodynamic analysis tools to the MAD environment, the best choice was combining all these analysis tools according to their strongest skills.
Besides the comparison of the analysis capability of the tools, the results are also compared with NASA flight test results of Cessna 172 aircraft (Suit and Cannaday 1979).From table 6 it can be observed that the error norms of DATCOM and AVL are close to each other.Also, OpenVSP has larger error norms than the other tools.Moreover, from table 6, it is seen that the configurations with fuselage provide more accurate results than the configurations without fuselage models.When considering the accuracy level and scope of capabilities of the software, it is reasonable to use AVL as a base aerodynamic calculation tool by supporting it with DATCOM.In this scope, DATCOM is used to support AVL in the analysis of parasite drag and stall prediction that AVL is not capable to perform.

Equation of motion block
The equation of motion block can be regarded as the main block of the dynamic model.The block contains required differential and analytical formulations in order to perform trim and simulation analyses.All  (Chudoba and Cook 2003).In addition, the transformation matrices and constraint equations are included in the equations set.
To simulate all types of maneuvers and detect trim conditions of the aircraft, required unknowns, and equations should be defined.Some of the trim conditions that are included in the MAD environment are given in table 8.
In the conventional trim search methodology, a pre-calculated aerodynamic database of full-body aircraft is required for analyses.In the conventional method, the model reads the related aerodynamic coefficients from the database for each Newton-Raphson iteration by performing interpolation between the nearest two breakpoints.In the conventional trim search methodology, all aerodynamic database of each geometric alternative should be had in order to obtain the trim point, which is a very time-consuming issue.
On the other hand, the instant trim and simulation algorithm aims to perform trim and simulation analysis without a pre-calculated aerodynamic database need.To find trim points in the instant trim and simulation method, aerodynamic analysis is performed quickly for only required state conditions at each Newton-Raphson , , , TAS w w w iteration.By utilizing empirical and vortex lattice-based solvers, a variety of geometric alternatives can be trimmed and simulated instantly to use in the optimization process.
Figure 3 shows the difference between the methodologies from the point of database needs.According to figures, in the conventional trim method, aerodynamic equations should be solved at all state combinations before the trim study, however for the instant trim and simulation method, it is enough to solve aerodynamic equations for only needed state combinations at the process of trim and simulation iterations.

Stability and control block
The stability and control module is created in order to analyze flight characteristics and control the aircraft while performing desired maneuvers.To observe flight characteristics, MAD software benefits from linearization and mode analysis methods.Moreover, in the control section inverse simulation method is utilized.
In order to assess the flight characteristics of the aircraft, equilibrium points should be initially obtained.Since stability is a property of the trim point.If the states are not in equilibrium, then stability would be a meaningless term for that point.Thus, in a Linear Time-Invariant (LTI) system, states of the system should not change in time.Derivatives of the following the states ( ) a b q f V p q r , , , , , , , TAS should be zero to represent the system as LTI.After determining the trim values of the states, stability and control matrices can be attained by linearizing the system at the equilibrium point.
Equation (20) expresses equations of the linearized system, which is followed up for flight characteristics analysis.By using equation (20), the state derivatives can be expressed in terms of the states and input parameters of the system.According to this concept, system and control matrices can be written as equations ( 21) and (22).

TAS y
Quasi trim parameters elevator aileron rudder

Constrained Equations
Freeze g Control Matrix: In the MAD environment, while calculating derivatives of the equations, the central difference derivation method is used.To perform mode analysis of the aircraft, eigenvalues of the system matrix are obtained by solving equation (23) for λ.

Conceptual design and optimization methodology
A methodology for the design and optimization process of Cessna 172N aircraft is explained.In order to get the optimum design of the aircraft, physics-based analyses based on instant trim and simulation algorithms are utilized.
2.2.1.Design process 2.2.1.1.Customer requirements For this study, the Cessna 172N aircraft is optimized as a sample case.The main goal is to improve the flight performance and the stability properties of the Cessna 172N aircraft, by optimizing the geometrical parameters of the aircraft according to customer requirements.In table 9 customer requirements and importance levels are tabulated.

Quality function deployment (QFD) analysis
In this study, the QFD matrix is utilized in order to analyze customer requirements.In the early phase of aircraft design, the verbal demands of the customers are quite challenging to turn into engineering requirements.The QFD table makes a connection between customers and engineers by determining how the requirements can be satisfied and which type of analyses are needed for it.In this context, table 10 is created to see the effects of customer needs on engineering parameters.

Overall evaluation criteria (OEC)
The technical importance scores of the performance outputs are used to determine the coefficients of the OEC.OEC equation can be regarded as the objective function that is aimed to be maximized.Equation (24) indicates the importance of the performance parameters together with overall evaluation criteria formulation.( ) OEC is an indicator of the desirability level of design requirements.Table 11 shows the outputs which are included in the optimization process and the importance levels of them.

Design of experiment (DOE)
Design of Experiments is a mathematical methodology used for planning and conducting experiments as well as analyzing and interpreting data obtained from the experiments.It is a multipurpose method that can be used in various situations such as design for comparisons, variable screening, transfer function identification, optimization, and robust design (Durakovic 2017).To benefit from these advantages of the statistical methods in the design process, DOE methodologies are utilized.In this scope, JMP is used to implement statistical applications to the overall design process, which is an interactive data visualization and analysis tool.In this study, physics-based analyses are performed by MAD environment and the result of the analyses are used to feed JMP.As a result, MAD software and JMP software are used to conduct design and optimization processes collaboratively.
The screening study aims to determine the most effective input set in the design process.Thus, in the early stages of the experimentation usually a large number of potential factors (design variables) are investigated to discover a few important factors (Khan et al 2015).From this standpoint, the two-level fractional factorial design is used for screening to identify the important factors that can then be investigated more deeply in subsequent experiments.
The screening steps which are followed in the process, are stated as follows: • Chose potential design input parameters.
• Select factorial type: two-level fractional factorial design.
• Select resolution: two Factor Interaction.
• Perform the analysis for each design variable combination.
• Fit model for related analysis outputs.
• Check the level of residual (error norm).
• Reduce the number of factors to investigate with Response Surface Method.
In the study initially 15 design variables are investigated with screening methodology as can be seen in table 12.In this process, instead of 2 15 design points, 2 15−8 points are analyzed with the MAD environment by using the fractional factoring model.The analysis outputs are scored for each input combination.In order to fit a model which creates a connection between design variables and OEC values, the first-order linear fitting model The figure shows that fuel weight, wing aspect ratio, wing incidence angle, wing reference area, maximum engine power, wing area, horizontal tail area, and wing location have an important effect on the resultant value of the OEC.These parameters are regarded as main inputs of the OEC equation, due to their high impact level on the cost function.
In order to simplify the optimization process, as a result of the screening study the most effective 8 factors are obtained.Analysis intervals of the selected design variables are indicated in table 13.
In the study, the Response Surface Method (RSM) is utilized to predict system response to any design.In the response surface methodology, the central composite design experiment is used by building a second-order quadratic model without needing to use a complete three-level factorial experiment.Although the response surface is a very effective and useful method, it can be used effectively only within a limited envelope.In this context, response surface designs are available for only continuous factors, and they support up to eight factors.Thus, in order to use the response surface method the screening study is mandatory to reduce the number of design variables for the systems having the number of factors more than 8.In figure 5 the reduced factors and the related design space are presented.
According to figure 5, there are 273 unique vertical lines (273 different geometric alternatives) which are composed of different level of factors.These points can be classified as 2 8 corner points, 2×8 axial points, and 1 center point.In the figure, blue, red, and grey colors represent the maximum, minimum, and center values of the design variables respectively.After defining the design space, the coefficients of the quadratic response surface function can be obtained via the standard least square method.Considered factor interaction levels of the response surface method are tabulated in table 14.
The steps for the implementation process of the response surface method are stated as follows: • Determine important design variables according to the screening study.
• Select Factorial Type: Central composite design.
• Run the analyses for the points determined with the central composite design.
• Exclude the deviated points from the fitting model.
• Get the response surface function for each output separately.

Optimization process
The optimization stage of the design process is required to obtain the best design alternative which meets the needs optimally.In this scope, the main objective of the optimization process is determined to maximize the result of the overall desirability function.The desirability functions are used to combine the response surface functions that belong to each output.If the importance values are defined for each design output, then they can be integrated into the overall desirability function.The overall desirability function can be defined as a weighted geometric mean of the individual response surface functions as shown in equation (25).In the equation, the scaled importance values are denoted by w 1 , w 2 , K, and w k. (JMP 2023)  Table 13.Selected design input parameters and optimization intervals.

Selected design inputs Interval
Wing Reference Area (m 2 ) In the optimization process, the objective of each response surface function can be specified as maximizing, minimizing, or achieving the target value.In desirability profiling, the desirability functions can be defined as a function for each output separately or as an overall function for all design outputs.The overall desirability function is optimized by the gradient descent algorithm (JMP 2023).The overall design and optimization processes workflow are shown in figure 6.

Results of stability analysis
According to the specified methodology in previous sections, the flying and handling quality of the Cessna 172N aircraft is analyzed.System identification-based prediction (Çetin 2018) and MAD software prediction of eigenvalue responses of Cessna 172 are compared in figure 7.According to the results, the eigenvalues predicted by MAD software are coherent with system identification results that are specified in (Çetin 2018).
In order to get flight characteristics of the related points in the envelope, trim analyses are performed for each speed and altitude combination.Afterward, the system is linearized at the trim point by using a 6-DOF model.By using a linearized system matrix, longitudinal and lateral dynamic modes are obtained via MAD software.The dynamic modes are classified according to flying quality level requirements.
Figure 8 shows flying quality levels (MIL-F-8785C flying quality levels, Yechout et al 2003) of each point in the flight envelope for most aft and most forward center of gravity configurations.The flight envelope is dominated by the trim points having level 1 (flying qualities clearly adequate for the mission flight phase) and level 2 (flying qualities adequate to accomplish the mission flight phase, some increase in pilot workload or degradation in mission effectiveness, or both, exists) flying qualities.Level 3 implies that flying qualities such that the airplane can be controlled safely, but pilot workload is excessive, mission effectiveness is inadequate, or both.
From the handling quality point of view, the linear and nonlinear models are compared by perturbing the velocities in the body y and z axes.According to figure 9, the linear and non-linear model responses have the same frequency and 0.2 m s −1 amplitude difference.This difference can result from the nonlinearity of the model.

Model validation and optimization results
As a result of the validation process, the flight manual performance data of the Cessna 172N aircraft is compared with MAD software performance predictions.The results are tabulated in table 15.
According to table 15 the maximum error norm of the calculations in the MAD environment is observed as 13.39% for the Cessna 172N aircraft.The reasons for the error are engine thrust and drag force calculation errors which are based on theoretical and empirical formulations.Also, the error in the landing distance calculation is caused by the error in the estimation of the structural parameters of the landing gear such as damping ratio, stiffness, or friction coefficients.At the further steps of the design process, these errors will be decreased via highfidelity aerodynamic analyses and more realistic estimation of landing gear parameters.
In table 15, the average error value of all results is calculated as 6.16%.The improvement percentage values are observed as bigger than the error values for most of the cases except for landing distance and service ceiling.This situation is important because, when the improvement percentage is bigger than the error percentage, then it can be guaranteed that the performance of the aircraft will enhance despite errors.
In the optimization process, underestimation of the drag coefficient can cause calculation errors.However, the selection of the geometry with minimum drag is independent of the underestimation of the drag calculation.Because all drag coefficients of the alternative geometries are calculated with the same methodology.Optimum geometry will not be changed according to the error norm which is caused by underestimation.The main goal is the comparison between a variety of geometries.If the calculation tool underestimates the drag coefficient of both geometries which are compared with each other, then the result of the comparison will be the same.Nevertheless, in order to minimize error value, the fidelity of the data related engine, landing gear, and aerodynamics should be improved.However, this will cost more analysis time for the optimization process.
As a result of the optimization part of the study, the optimum design variables are obtained according to customer demands by following up a systematic methodology (table 16).The performance values which are calculated with MAD software for base geometry, are tabulated in table 17 as well as the optimized design outputs.In addition, the difference between the performance analysis predictions of response surface function and MAD software are included in table 17.The results indicate that the error norm percentage of fitting function is under 2% tolerance.
From tables 16 and 17, it is observed that the resulting design variables of the optimized geometry correspond well with the performance outputs that are obtained through optimization.The relation between stall speed and wing area, range and fuel weight, maximum engine power, and take-off distance exemplifies these conditions too.In table 17 the only parameter having a negative improvement percentage is the cost which denotes that the cost of the aircraft is increased for the optimized aircraft in comparison with the baseline case, Cessna 172N.This is a result of the customer importance rating in the QFD table where the weight of the 'Affordable Price' is selected as 3 out of 5 which shows a moderate importance of cost in comparison with the other parameters.The FPS parameter, which is a stability indicator, has %35.23 improvement percentage.This improvement results from the high importance rate of the stability parameter in the QFD table.In the optimization process, it is necessary to compromise on certain optimization objectives.

Conclusion
In this study, the design and optimization processes of the Cessna 172N aircraft are explained in view of a newly designed aircraft design tool (MAD).Model-Based Design (MAD) environment is elaborated to introduce its model structure.According to the literature study, it is seen that there is numerous alternative design software, however, any software apart from MAD does not have the capability of performing instant trim and simulation via the six degrees of freedom dynamic model.This capability provides the user to analyze any unconventional unique design without a pre-calculated database need.Analysis methodologies are validated with Cessna 172N aircraft, and optimization is carried out to improve the performance characteristics of Cessna 172N aircraft in view of a set of customer requirements.With the help of the QFD matrix, coefficients of the OEC equation are obtained.The Design of Experiment (DOE) concept is utilized for statistical analysis of the system via JMP.Initially screening study is conducted with 129 different geometric alternatives, by using the first-degree polynomial model that benefits from the fractional factorial design.This is sufficient to determine which explanatory variables affect the response variable of interest.Once significant explanatory variables are left, then a more complicated central composite design is implemented to estimate a second-degree polynomial model, which is named as Response Surface Method (RSM).In order to use response surface methodology, initially 273 different geometric alternatives are analyzed, and desirability functions are defined to obtain the maximum desirable point within the design space.In order to get the global optimum point, the optimization method is selected to maximize or minimize each design output according to the desirability concept.
According to optimization results, service ceiling and landing distance parameters are overestimated.The reasons for the calculation error are the underestimated drag coefficient, landing gear, and engine data insufficiency.
Underestimation of the drag coefficient can cause an error, but it cannot change the selection process of the optimum geometry.Because the analysis tool will underestimate the drag coefficients of all alternative geometries and this situation will not affect the comparison between geometries.
At the end of the optimization average performance parameters are calculated with %6.16 error.Apart from service ceiling and landing distance parameters, the improvement percentage value of most of the parameters is more than the error norms.This assures the performance of the aircraft is increased despite the calculation errors.In order to minimize the error norm of the calculations, high-fidelity analysis should be performed.This requires more analysis time when compared to low-fidelity analysis.Thus, by using basic analysis tools, the number of geometry alternatives should be reduced to prepare for high-fidelity analysis.In this study, only empirical and theoretical aerodynamic solvers are utilized for optimization.At the further loop of the optimization also CFD or wind tunnel data can be included in the process.In the optimization process, fitting function error is observed as less than %2.The only negatively affected parameter is cost.This is caused by the low importance rate of cost parameters in customer requirements.This means that in order to increase the performance of some desired parameters, it may be necessary to compromise on other parameters.At the end of the optimization process, the design variables of the optimized geometry accurately correspond to the resultant performance outputs.
As a result, the resultant design variables provide more desirable performance outputs when compared to the base performance of Cessna 172N.In the whole analysis and optimization process, the instant trim and simulation algorithm of the MAD environment works effectively.

Figure 2 .
Figure 2. MAD software structure and flowchart.

Figure 3 .
Figure 3. State conditions for trim search iterations.

Figure 5 .
Figure 5. Response surface method design space.

Figure 9 .
Figure 9. Linear and nonlinear model response to perturbation.

Table 1 .
Aerodynamic calculation tools capability comparison.

Table 3 .
Positive sign conventions of the coordinate system.

Table 4 .
MAD tool output parameters.

Table 5 .
Aerodynamic calculation tools capability comparison.

Table 6 .
Aerodynamic analysis results of Cessna 172N aircraft.

Table 7 .
Equations and unknown alternatives.

Table 9 .
Customer requirements and importance levels.

Table 10 .
Quality function deployment (QFD) analysis.By using the fitting model, the effects of each factor on OEC are figured out as a Pareto plot in figure 4.

Table 12 .
Design input list.

Table 14 .
Considered factor (design variable) interactions by response surface method.

Table 15 .
Comparison between MAD prediction and Cessna 172 performance data.

Table 16 .
Design variables of optimized Cessna 172N geometry.

Table 17 .
Performance enhancements of Cessna 172N aircraft.