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Optimization involves the pursuit of the "best" - or a significant "better". Better what? A better value of some defined "measure of merit" or "objective function". For aircraft conceptual design, the measure of merit is typically weight and/or cost for some specified capability, or capabilities such as range or payload at a specified weight or cost. This pursuit of better/best is limited by specified conditions involving real-world operational aspects or must-meet capabilities, which in mathematical terms are the "constraints" of the optimization. Fundamentally, we can define optimization as the determination of a minimum or maximum of one or more objective functions such that no constraints are violated. The first importance of the optimisation application was seen during the great cathedrals of Europe time. The disaster of the Swedish warship Vasa is instructive concerning the problems of attempting to optimize with insufficient analytical tools to assess the design constraints based on prior experience. Optimization by mathematical analysis became possible in the 1600's when Isaac Newton and Gottfried Leibniz independently developed calculus. About the same time, Pierre de Fermat defined a general approach to compute local minimums and maximums of functions by solving for the derivative and setting it to zero - the basis of most analytical optimization today. Fermat, along with Blaise Pascal, founded the theory of probability that is critical to Monte Carlo techniques and the recently developed evolutionary/genetic optimization algorithms. In the 1700's, Leonhard Euler developed methods to find the extreme values of functions, along with many other contributions to mathematics and physics including definition of a basic equation of hydrodynamics still used in computational aerodynamics. Joseph Lagrange, together with Euler, developed the calculus of variations. Lagrange also developed generalized equations of motion and developed the concept of partial differential equations which is widely used in the optimisation process. In the early 1800's, Adrien-Marie Legendre and Carl Friedrich Gauss developed the method of least-squares curve fit that is often used in optimization, especially the modern Response Surface method. In the mid-1800's, William Hamilton developed theorems concerning differential equations, dynamic analysis, and imaginary numbers which have great application for the solution of optimum design problems. Andrei Markov in the early 1900's developed the theory of stochastic processes. These are sequences of random variables in which the future value of the variable is determined by the present value but is independent of the way in which the present value was derived from its predecessors. Vilfredo Pareto, an economist in the early 1900's, developed the principle of multi objective optimization for use in allocation of economic resources. His concepts became known as "Pareto optimality", defined as a situation in which you cannot make someone better off without making someone else worse off. A graphical representation of Pareto optimality is widely used to depict two-objective optimality in present aircraft conceptual design. The Kuhn-Tucker Theorem (Albert Tucker and Harold Kuhn) of 1950 is considered to have launched the modern field of nonlinear programming. Kuhn-Tucker gives necessary and sufficient conditions for the existence of an optimal solution to a nonlinear objective in the face of constraints. Kuhn-Tucker is widely used in the proofs of analytical optimization methods. The classic aircraft design carpet plot is an excellent illustration of Kuhn-Tucker & widely used in the aircraft conceptual optimisation process.
The aim of this literature review is to represents the research activities performed in the past on the aircraft design at conceptual level using multidisciplinary optimisation(MDO) in order to satisfy the objectives like cost & weight optimisation, aircraft robust design optimisation. After reviewing different literature, the findings of different literature are considered in order to proceed for dissertation on the conceptual aircraft design using multidisciplinary optimisation methods. The initial stage of design is called as "concept development," during which the needs of the target market are identified, alternative product concepts are created and evaluated, and a single concept is selected for further development. The different MDO methods like Orthogonal Steepest Descent, Monte Carlo, a mutation-based Evolutionary Algorithm, and three variants of the Genetic Algorithm with numerous options etc were evaluated in terms of their ability to find the optimal aircraft, as well as total execution time, total cost, total gross weight of the objects, convergence history, tendencies to get caught in a local optimum, sensitivity to the actual problem posed, and overall ease of programming and operation.
Kristian Amadori, Christopher Jouannet & Petter Krus published a paper on "Aircraft conceptual design optimisation" with the use of CAD, framework & simulation methodology. The figure-1 represents the multiple disciplines with different design stages involved in the design of the aircraft. The aircraft design to be optimised at the conceptual level which will minimise possibility of the error in the design at preliminary & detail stage. During the conceptual design phase of a new aircraft designers will evaluate a large number of different concepts, searching for the one that meets the requirements in the best way. This means that they need to iteratively cycle through sketching a concept, analyze it and evaluate and compare its performances.
Figure-1: Different Disciplines in aircraft at different design phases
The framework, simulation tools (CFD or PANAIR) & CAD tools(GAMBIT) are intended to be a multidisciplinary optimization tool for defining and refining aircraft designs, with respect to its aerodynamics, performance, weight, stability and control. Figure 2 below describes how the complete framework will look like once all modules will be ready and connected.
Figure-2: The complete aircraft design framework
Conclusions: In this paper a framework architecture that focuses on flexibility of application, has been outlined. To avoid continuing using semiempirical or statistical equation during the conceptual phase of aircraft design it has been suggested to make a larger use of analytical tools. For the aerodynamics, a high order panel code - PANAIR - has been employed. PANAIR may not represent the most advanced tool for aerodynamic analysis, but it served the purpose of illustrating the process. Clearly any other panel code or CFD software could equally be used instead. A CAD model has also been included as one module in the framework, where geometric calculations, as well as structural analysis are performed.
Ruben E. Perez, Hugh H. T. Liu and Kamran Behdinan present the paper on "Evaluation of Multidisciplinary Optimization Approaches for Aircraft Conceptual Design". This paper presents the different MDO methods like Multi-Disciplinary Feasible (MDF), Individual Discipline Feasible (IDF), Collaborative Optimization (CO), Concurrent Subspace Optimization (CSSO) and Bi-Level Integrated Synthesis System (BLISS). This paper gives guideline when dealing with multidisciplinary optimization formulations which can be applied to aircraft conceptual design problems.
For better illustration, the Aircraft Conceptual Design for Supersonic Business Jet Example(figure-3) was considered using all MDO methods. The results obtained for this example was evaluated based on the proposed metrics & tabulated as shown in table-1 for MDO Comparative Summary.
Figure-3: Supersonic Business Jet Example
The goal was to maximize the range of a supersonic business jet subject to individual disciplinary constraints. Four coupled disciplinary systems were used representing structures, aerodynamic, propulsion, and performance. The first three disciplines were fully coupled since they shared common variables and exchange computed states. The fourth discipline (performance) received information from the others to evaluate the range performance of the design. Structures and weights were coupled to aerodynamic and propulsion. The aerodynamic loads caused changes in aircraft structural deflection that in turn changed the aerodynamics characteristics of the aircraft. Similarly, the propulsion and weights were coupled. The thrust required is dependent on the total aircraft weight, including the engine weight, which is also the function of thrust.
Table-1: MDO Comparative Summary
Conclusion: This paper presents an extended evaluation of MDO methods. The above results show that the MFD method is more sound while considering the metrics of accuracy, transparency & simplicity whereas IDF & CO are sound in Efficiency & portability respectively. The complex aircraft conceptual design example is applied to evaluate the five MDO methods. The evaluation is based on our proposed metrics, which take into account formulation and the algorithm considerations. The quantitative description of the metrics provides a systematic approach in evaluating the MDO methods. Simulation results demonstrate the effectiveness of the proposed metrics, and concur with the experience from practice. Much work still needs to be done, not only for its informative "systematic study", but also for its contribution to establishing standards or guidelines in MDO selection and testing. Work under investigation will include additional examples, involving variance in the formulation complexity and the number of coupling and global variables.
Mattia Padulo, Shaun A. Forth, and Marin D. Guenov present a paper on "Robust Aircraft Conceptual Design using Automatic Differentiation in Matlab" This paper shows the need for robust optimisation in aircraft conceptual design & for that the design parameters were assumed stochastic, was introduced. They highlighted two approaches, first-order method of moments (IMM) and Sigma-Point (SP) reduced quadrature, to estimate the mean and variance of the design's outputs. The method of moments requires the design model's differentiation and here, since the model is implemented in Matlab, is performed using the AD (Automatic Differentiation) tool MAD. Gradient-based constrained optimisation of the stochastic model is shown to be more efficient using AD-obtained gradients than finite-differencing. A post-optimality analysis, performed using AD enabled third-order method of moments and Monte-Carlo analysis, confirms the attractiveness of the Sigma-Point technique for uncertainty propagation.
They demonstrated the benefits of using AD in robust optimization of a Matlab implemented, industrially relevant, conceptual design test case. This conceptual design model determines performance and sizing of a short-to-medium range commercial passenger aircraft and makes use of 96 sub-models and 126 variables.
The original deterministic optimization problem was the following:
Objective: Minimize Maximum Take-Off Weight MTOW with respect to the design variables x (described in Tab. 1 together with their permitted ranges).
1. Approach speed: vapp < 120 Kts) g1 = vapp-120;
2. Take-off field length: TOFL < 2000 m) g2 = TOFL-2000;
3. Percentage of total fuel stored in wing tanks: KF > 0:75 ) g3 = 0:75-KF;
4. Percentage of sea-level thrust available during cruise: KT < 1 ) g4 = KT-1;
5. Climb speed: vzclimb > 500 ft/min ) g5 = 500-vzclimb;
6. Range: R > 5800 Km ) g6 = 5800-R.
Table A: Considered design variables for the deterministic problem.
Table B: Fixed parameters.
Table C: Results of the robust optimisations.
Table D: Relative error compared to Monte Carlo estimates of mean and variance at each optima.
Table E: Performance improvements yielded by AD to the optimisation problems.
This paper's results demonstrate the benefits AD may give to robust optimisation for aircraft conceptual design. They performed robust optimisations of an industrially relevant, Matlab-implemented aircraft sizing problem using the AD tool MAD.
Two robust design strategies:
i) First Order Method of Moments (IMM) - robust objective and constraints use AD-obtained first order derivatives; second order derivatives used for gradients.
ii) Sigma Point Method (SP) - reduced quadrature for robust objective and constraints; AD for their gradients.
For test case considered, a Monte-Carlo post-optimality analysis indicates that SP more accurate for estimation of the mean but IMM more efficient (with AD gradients).
In both cases AD gradients significantly reduced optimisation c.p.u. time compared to finite-differencing (FD).
Hence AD may benefit robust optimisation for aircraft conceptual design.
Rymer. D.(2002) presents a thesis on "Enhancing Aircraft Conceptual Design using Multidisciplinary Optimisations". The thesis on the "Aircraft conceptual design optimisation is the research into the improvement of the Aircraft Conceptual Design process by the application of Multidisciplinary Optimization (MDO). The thesis shows that Aircraft conceptual design analysis can be performed using a variety of optimization methods including Orthogonal Steepest Descent (full-factorial stepping search), Monte Carlo, a mutation based Evolutionary Algorithm, and three variants of the Genetic Algorithm with numerous options, hybrid methods, analysis methods, test run case methods Robust aircraft conceptual design using automatic differentiation in matlab, local Pareto approximation for multi objective optimisation etc. Each method has certain advantages & disadvantages how they use for particular aircraft concepts e.g. advanced multirole fighter aircraft, a commercial aircraft(either passenger or transport of goods), UAV(unmanned aerial vehicle), etc. To better stress the utilisation of optimisation methods, the different aircraft designs are deliberately modified for different case runs to reflect a very poor initial selection of design parameters including wing loading, sweep, aspect ratio, fineness ratio, bypass ratio etc & based on the reflection the best combination of these parameters considered to achieve the objectives . The recommended measures of merits are solely depending on the types of aircraft to be optimised. The recommended measures are weight based, cost based, revenue based & utility based. If the fighter aircraft conceptual design is taken into consideration then the utility based performance is more important rather than the cost whereas if the commercial aircraft conceptual design is considered then the cost, weight & revenue based performance becomes essential. After the lot of research activities over the period of years, more than a million parametric variations of these aircraft designs were defined and analyzed in the course of this research.
Recommended Design variables:
Recommended Design constraints:
Recommended Measures of Merits:
Instead of explaining all MDO methods, some methods are explained in very brief for better illustration.
Orthogonal Steepest Descent Full-Factorial Stepping Search:
No derivatives or finite differences are required to find the direction of maximum local improvement to the objective function because no attempt is made to find exactly the best "direction" to move - motion is always along the orthogonal axes of one or more variables. Each variable is parametrically varied by the selected step size (plus and minus), and the resulting aircraft are all analyzed.
Figure 1: Orthogonal Steepest Descent Full-Factorial Stepping Search
It is defined as a situation in which you cannot make someone better off without making someone else worse off. A graphical representation of Pareto optimality is widely used to depict two-objective optimality. An aircraft design example might be a requirements trade study in which you attempt to maximize both range and payload weight, and plot a curve showing the optimum trade off between the two.
Figure 2: Pareto Graph between two variables of aircraft.
In this thesis four notional aircraft concepts were designed as test cases for evaluation of MDO methods and options, namely an advanced fighter, a commercial airliner, an asymmetrical light twin, and a tactical UAV & made conclusion as below.
Research has been conducted into the improvement of the Aircraft Conceptual Design process by the application of Multidisciplinary Optimization (MDO). Aircraft conceptual design analysis codes were incorporated into a variety of optimization methods including Orthogonal Steepest Descent, Monte Carlo, a mutation-based Evolutionary Algorithm, and three variants of the Genetic Algorithm with numerous options.
The commercial airliner design was deliberately modified for certain case runs using poorly-chosen design parameters including wing loading, sweep, and aspect ratio, to see if the MDO methods could "fix it." MDO methods and options were evaluated using these notional designs in over a hundred case runs totally more than a million parametric variations of these designs. These variations included application of automatic redesign procedures to improve the realism of such computer-designed aircraft. Each design variation was completely analyzed as to aerodynamics, weights, performance, cost, and mission sizing, and evaluated as to performance and geometric constraints.
The key conclusion - aircraft conceptual design can be improved by the proper application of such Multidisciplinary Optimization methods. MDO techniques can reduce the weight and cost of an aircraft design concept in the conceptual design phase by fairly minor changes to the key design variables. These methods proved to be superior to the traditional carpet plots used in the aircraft conceptual design process for many decades.
Evaluation of the different MDO methods for aircraft design optimization indicated that all of the methods produce reasonable results.
For a smaller number of variables the deterministic Orthogonal Steepest Descent searching method provides a slightly better final result with about the same number of case evaluations.
For more variables, evolutionary/genetic methods seem superior.
The Breeder Pool approach defined herein seems to provide the best convergence in the fewest number of case evaluations.
The Net Design Volume approach defined herein to assure sufficient volume for fuel and internal equipment appears to work well and improves the design realism with little user effort. Other geometric constraints such as diameter, length, and span limits were also found to be useful for some design problems.
Remark on the literature review:
The main purpose of Literature review is to gather the information about the dissertation of conceptual aircraft design & to see what the other people did the research in the past for the same subject which would give guideline in the dissertation to proceed further.
In the dissertation of aircraft conceptual design, it is difficult to choose all the variable, constraints & measure of merits in conceptual aircraft design dissertation due to the time constraint & because of this, some of critical design variables, constraints & measures of merits will be considered for aircraft conceptual design project which will have greater impact on the design & accordingly the MDO methods will be considered.
The selection of the MDO methods for aircraft conceptual design depends on the nature of the problem i.e. if numbers of variables are small then Orthogonal Steepest Descent searching method provides a slightly better final result than other methods whereas the numbers of variables are more than evolutionary/genetic methods seem superior.