OPTIMIZATION OF A SIMPLE AIRCRAFT WING

Transcription

1 OPTIMIATION OF A SIMPLE AIRCRAFT WING Susana Angélica Falco and Alfredo Rocha de Faria Fibraforte Engenharia, São José dos Campos, SP, Brazil. R. José Alves dos Santos 281/306, sufalco@uol.com.br; arfaria@directnet.com.br Phone: ; fax: ABSTRACT A wing-like structure consisting of spars, ribs, reinforcements and skin is optimized considering two cases: (i) weight minimization, and (ii) critical load maximization. The wing carries an elliptically distributed load along the span. Positioning of spars and ribs as well as dimensions of different parts of the structure are the design variables. Results indicate that significant improvement in terms of objective function has been achieved through the optimization procedures. KE-WORDS Structural optimization, finite elements, aerospace designs, modal and buckling analysis

2 INTRODUCTION Structural and multidisciplinary optimization have been gaining increasing attention in recent years for their acknowledged contributions made to design enhancement, specially in early stages of product development. Pioneer applications can be found in the aerospace industry where lightweight structural components are required for either cost reductions or payload increase. Grihon and Mahé (1999) presented a study where a simplified expression of the operating cost combining cruise drag and wing weight was minimized. Their work has the outstanding advantage of dealing directly with wing plan form, which enables designers to assess their product efficiency very early in the development process, and modeling the entire wing structure through finite elements. Another approach (Garcelon and Balabanov, 1999; Garcelon, Balabanov and Sobieski, 1999) breaks the optimization problems into structural optimization and aerodynamic optimization and a strategy is devised which solves one or another at a time, iterating until a given convergence criterion is met. The difficulty here is that interface issues come into play since, usually, wing deformation results in aerodynamic load variation. All the optimization techniques described above are heavily based on state-of-the-art algorithms and codes that require an enormous computational effort to be solved. The MSC.Nastran commercial structural optimization code is based on traditional continuous methods such as feasible directions, sequential linear programming and sequential quadratic programming (Vanderplaats, 1999). The reduction in weight and the increase in critical loads, quality and reliability of structures are considered to be major aspects in the design of aerospace structures. In these cases, structural optimization turns out to be an adequate procedure, aiming at designing efficient structural components. To solve the non-linear constrained optimization problem, the Sequential Quadratic Programming algorithm is used (Powell, 1978). This method was chosen among others available in commercial codes because it gave better results for the present investigations. Although design optimization has been acknowledged as an effective procedure, it is not widely used as a standard design tool for various reasons. Non-technical reasons include difficulty in creating design data and lack of time to educate engineers to perform optimization. Those engineers spend most of their time modifying and analyzing their designs and making iterative changes, rather than modifying designs using optimization techniques. However, since there is enormous pressure to shorten the time to market, while enhancing the quality of product at the same time, those attitudes are gradually changing and/or are under pressure to be changed. It is the researchers and software developers responsibility to keep refining the state of the art of optimization technologies and to provide easy to use, user friendly optimization tools, so that as many engineers as possible turn their attention and curiosity to it. SIMPLE AIRCRAFT WING MODEL A real commercial aircraft wing contains thousands of structural components, ranging from small bolts and rivets to large, heavyweight skin panels and spars. Creation of a detailed wing model incorporating, simultaneously, all the wing features is virtually impossible either because of budget constraints or tight time schedules. Thus, engineers rely on simplified models that provide a fairly accurate approximation of the real wing structure behavior. Moreover, it is seldom to find a design procedure that starts off with a detailed approach. Most commonly, the 2

3 design process is a multi-step procedure where initial steps contemplate simplified configurations and each step inherits properties from the previous ones. Therefore, the importance of obtaining initial optimized designs is to guarantee, or at least favor, that its best features will pass on to subsequent steps. The simplified aircraft wing model depicted in Fig. (1) has 6000 mm total length, 1500 mm width and 275 mm height at the chord mid point. The material properties of machined aluminum are oung modulus N/mm 2, mass density Kg/mm 3 and Poisson coefficient The model has 4 spars, 7 ribs, 36 rectangular skin panels and beams on the edges of all these parts as seen in Fig. (1) V1 Figure 1. Geometric model of the wing-like structure. The skin panels, spars and ribs have been discretized into quadrilateral plate elements (CQUAD4) and the beam into bar elements (CBAR). The boundary conditions are fully clamped at one end and free at the other. The wing supports an elliptical distributed load as shown in Fig. (2) of N/mm at the clamped end dropping to at the free end. The load is equally distributed on the upper loft surface and lower loft surface. Chordwise the load is assumed to have a constant distribution E E E E E E E E E E E E E E E E E E E E Carga com forma elíptica Figure 2. Wing load distribution. 3

4 OPTIMIATION PROBLEM DEFINITION In a design optimization problem, the objective is to find the values of a set of n given design variables x R n which minimize or maximize a function f(x), denoted objective-function, while satisfying a set of equality (h k (x)=0) as well as inequality (g r (x) 0) constraints which define the viable region of the solution space. The constraints x lj x j x uj are called side constraints. R is the set of the real numbers (Vanderplaats, 1999). Objective-function: f (x) Variable Vector: x R n Constraints : h k (x) = 0 k=1,..., q (1) g r (x) 0 r=1,..., m (2) x lj x j x uj j = 1,..., n (3) The wing model is optimized considering two optimization problems: PROBLEM I: Objective-function to minimize: Wing mass Constraint: Maximal critical load PROBLEM II: Objective-function to maximize: Critical load Constraint: Wing mass Generally stated, careful selection of the search parameters, scaling of the design variables and selection of a good initial design are often indispensable. Furthermore, it is observed that the choice of a representative set of design variables is a decisive factor for a successful optimization procedure as shown in the remainder of this work. THE CHOICE OF THE WING DESIGN VARIABLES MSC.Nastran commercial code can simultaneously solve both member dimension (sizing) and coordinate location (shape) optimization problems and a wide range of options are available to define the design variables. For example, design variables may be individual member dimensions and/or grid locations, or may be linear or nonlinear combinations of these. The allowable shapes are defined using shape basis vectors. The engineer uses these to describe how the structure is allowed to change. The optimizer determines how much the structure can change by modifying the design variables. It is used the Direct Input Shape method to describe these shape basis vectors. With this method, externally generated vectors are used to define shape basic vectors and an auxiliary model analysis provides these externally generated vectors. A total of 43 design variables, including sizing and shape, are defined for the wing model. Tables (1) and (2) present the shape and sizing variables, respectively, used in the calculations. Figures (3), (4) and (5) show the location of the shape, the thickness sizing and the cross section area sizing variables, respectively. Observe that BEAM37 to BEAM40 comprise the entire wing span, from root to tip. Also, BEAM30 to BEAM36 are present in all ribs. Moreover, full geometric symmetry about plane is assumed. 4

5 Table 1. Shape design variables definition VARIABLES DEFINITION rib 1 position rib 2 position rib 3 position rib 4 position rib 5 position spar 1 position spar 2 position Table 2. Sizing design variables definition VARIABLES DEFINITION COLOR PLA11 first plate (free end, left, Fig. 4) dark yellow PLA12 second plate light green PLA13 third plate dark green PLA14 fourth plate yellow PLA15 fifth plate light yellow PLA16 sixth plate (clamped end) emerald green PLA21 first plate (free end, center, Fig. 4) red PLA22 second plate dark orange PLA23 third plate light orange PLA24 fourth plate light pink PLA25 fifth plate magenta PLA26 sixth plate (clamped end) dark pink PLA31 first plate (free end, right, Fig. 4) celestial blue PLA32 second plate light blue PLA33 third plate dark blue PLA34 fourth plate celestial dark blue PLA35 fifth plate blue PLA36 sixth plate (clamped end) purple RIB1 first rib portion (left) navy blue RIB2 second rib portion (center) dark purple RIB3 third rib portion (right) pink SPAR1 first spar (left, Fig. 4) yellow SPAR2 second spar light red SPAR3 third spar dark red SPAR4 fourth spar (right, Fig. 4) blue BEAM30 horizontal beam 1 red BEAM31 horizontal beam 2 green BEAM32 horizontal beam 3 blue BEAM33 vertical beam 1 black BEAM34 vertical beam 2 black BEAM35 vertical beam 3 black BEAM36 vertical beam 4 black BEAM37 spanwise beam 1 orange BEAM38 spanwise beam 2 dark blue BEAM39 spanwise beam 3 magenta BEAM40 spanwise beam 4 brown 5

6 Figure 3. Location of the shape design variables for the wing model Figure 4. Location of the thickness sizing design variables BEAM37 BEAM38 BEAM39 BEAM40 BEAM33 BEAM34 BEAM30 BEAM31 BEAM32 BEAM35 BEAM36 Figure 5. Location of the cross section area sizing design variables 6

7 PROBLEM I: MASS MINIMIATION WITH BUCKLING CONSTRAINT The wing model is optimized for minimum weight, considering an elastic buckling constraint. Defining P cr as the critical buckling load and P a the actual applied load, the buckling problem is to find the minimum λ which will trigger loss of stability, where P cr =λp a. If the minimum λ is less than 1.0 the structure has buckled. Therefore, the elastic buckling constraint requires, effectively, the solution of an eigenvalue problem. Five cases of increasing complexity are considered. In case I only thicknesses of spars, ribs and skin panels are design variables, i.e., the basic geometry is unchanged. Case II gives more freedom to the structure to minimize its mass; although the spars sweep angle is fixed 0 o. Ribs are allowed to change their positions in case III but they must remain parallel to the flight direction and spars are maintained fixed. Case IV considers that both spars and ribs position vary. Finally, case V incorporates beam cross sectional areas in the set of design variables. All cases are summarized in table (3). Table 3. Definition of cases I II III IV V Sizing variables Shape variables (Spars position) 6 (Ribs position) 8 (Spars and ribs ) 8 (Spars and ribs) Total variables Inspection of table (4), reporting the optimization results, reveals that, for cases I-IV rib and spar thicknesses where kept as low as possible whereas in case V mass reduction is achieved chiefly because of beam cross sectional area reduction. This suggests that the beam elements played a crucial role in terms of load carrying capacity for cases I-IV. On the other hand, in case V, a significant portion of the load is transferred to the panels. Figure (6) presents the optimal designs geometry obtained. An interesting pattern is observed for case III. The tendency of two ribs ( and ) to coalesce is striking and they perhaps would have if the side constraints were relaxed. This peculiar behavior leads to the conclusion that yet another kind of design variables should be considered in the aircraft wing optimization, namely, the number of structural components such as ribs or stringers. Even tough the load carrying mechanism changes from cases I-IV to V mass reduction is consistently achieved. In the most complex case investigated a reduction of 18% was obtained. Notice that even the simplest case results in 9% mass savings what is by itself a significant accomplishment. 7

8 Table 4. Optimal results for problem I Variable Lower bound Initial value Upper bound I II III IV V PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA RIB RIB RIB SPAR SPAR SPAR SPAR BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM Mass (Kg) Initial:

9 INITIAL SHAPE II III IV V Figure 6. Optimum design obtained with MSC.Nastran program 9

10 PROBLEM II: CRITICAL LOAD MAIMIATION WITH MASS CONSTRAINT In this case, the wing model is optimized for maximum critical load, considering the initial mass of the structure (390 Kg) as a constraint. Table (5) presents the results obtained by MSC.Nastran. The optimum shape is shown in Figs. (7). Differently from problem I, table (5) shows that the most important design variables for buckling load maximization are spar positions. This becomes evident when cases II and III are compared. Although case III has more design variables (30), it delivers a poorer result (λ=0.40) when compared to case II (λ=0.56) that has fewer design variables (27). The noticeable difference is precisely the inclusion or exclusion of variables and. Table (5) indicates that the beam cross sectional areas and panel thicknesses do not play a role as significant as in problem I. They do not vary dramatically from case to case. However, shape design variables are the driving force behind optimal designs for buckling load maximization. The optimal skin panel thicknesses shown in table (5) are in accordance with the load distribution presented in fig. (2). It can be observed that, from the clamped to the free end, the thickness decreases. This is expected since the compressive stresses in the upper skin panels are higher in the neighborhood of the wing root. Rib and spar web thicknesses seem to benefit from the inclusion of shape design variables in the optimization procedure as seen in case II and on. In real applications an aircraft wing is never subjected to only one load case. Indeed, it is not uncommon to have structures (wings, horizontal stabilizers, control surfaces) that experience many distinct load cases during operation, including but not limited to maneuver loads, gust loads, landing loads, etc. The load distribution depicted in fig. (2) and used in the present simulations, reflect cruising aerodynamic loads. However, a more comprehensive optimization should consider the envelope of maximum loads rather than the loads associated with a particular load case. Elegant techniques exist to handle buckling load maximization in a multiple load case situation (de Faria and Hansen, 2001). 10

11 Table 5. Optimal results for problem II Variable Lower bound Initial value Upper bound I II III IV V PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA PLA RIB RIB RIB SPAR SPAR SPAR SPAR BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM BEAM Parameter l Initial l:

12 INITIAL SHAPE II III IV V Figure 7. Optimum design obtained with MSC.Nastran program CONCLUSIONS Two optimization problems were investigated in this paper. It is noticed that even though the basic geometry and the same sets of design variables are used, both problems have radical 12

13 differences with respect to design sensitivity. Problem I possesses high sensitivity with respect to sizing design variables whereas in problem II the highest sensitivity is observed with respect to shape design variables. Generally, it is not possible to determine beforehand which subset of design variables will be dominant in terms of sensitivity. Nevertheless, a preliminary sensitivity analysis could certainly be helpful to identify relevant subsets. In problem I, case III, the tendency of ribs to coalesce was detected. This kind of situation suits better in topology optimization algorithms, a feature not yet available in MSC.Nastran, SO00. With the resources currently implemented the best procedure would be to start off with a large number of ribs and try to identify coalescence tendencies. Subsequently, a finer optimization would be conducted with the right number of ribs. The only structural constraint involved in the previous optimization calculations was elastic buckling. Nonetheless, other constraints are equally important, including but not limited to, allowable stress/strain, fatigue, crippling and rivet bearing. The best finite element model generated for optimization purposes will not be useful unless all appropriate constraints are taken into account. If they are not, the resulting optimal design will be vulnerable to precisely the constraint left aside. A simplified model such as the one studied herein provides an initial design but more realistic optimizations must be performed, perhaps in the component level, before these structures become operational. ACKNOWLEDGEMENTS The authors wish to acknowledge FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) for the financial support provided to this research. REFERENCES 1. Grihon, S. and Mahé, M., Structural and multidisciplinary optimization Applied to Aircarf Design, 3 rd World Congress of Structural and Multidisciplinary Optimization, Buffalo, N, May 17-21, Garcelon, J.H. and Balabanov, V., Integrating VisualDOC and GENESIS for Multidisciplinary Optimization of a Transport Aircraft Wing, 3 rd World Congress of Structural and Multidisciplinary Optimization, Buffalo, N, May 17-21, Garcelon, J.H., Balabanov, V. and Sobieski, J., 1999, Multidisciplinary Optimization of a Transport Aircarft Wing using VisualDOC, Optimization in Industry-II, Banff, Alberta, Canada, June Vanderplaats, G.N., Numerical Optimization Techniques for Engineering Design, 3 rd edition, McGraw-Hill Book Company, Powel M. J. D., Algorithms for Nonlinear Constraints that use Lagrangian Functions, Math, Moore, G.J., MSC.Nastran Design Sensitivity and Optimization, User s Guide, Version 68, The Macneal-Schwenler Corporation, de Faria, A.R. & Hansen, J.S., On Buckling Optimization under Uncertain Loading Combinations, Structural and Multidisciplinary Optimization Journal, Vol. 21, No. 4, 2001, pp