Shape Optimization of Wing Structures with Aeroelastic Constraints J. Schwochow, T. Klimmek Deutsches Zentrum f¨ ur Luft- und Raumfahrt, G¨ottingen, Germany e-mail:
[email protected],
[email protected] Key words: optimization
Aeroelastic wing design, aeroelastic trim, CAGD, MDO, structural
A strategy is presented to optimize wing structures under static aeroelastic constraints with respect to parameters defining wing shape and structural layout. The aerodynamic and structural models are based on one common parameterized geometry description. While the aerodynamic model includes the description of the three dimensional surface geometry, the structural model description is based on the Finite Element Method and includes a detailed wing layout concept with arbitrarily oriented spars and ribs. The aeroelastic toolkit ZAERO employs the modal formulation for solving the trim problem of the flexible aircraft and calculates the loads for selected trim cases. The optimization code LAGRANGE minimizes the mass of the structure sustaining the design loads by sizing the thicknesses of skins, spars and ribs. The optimization framework is integrated in the design system iSIGHT which finds the optimal wing planform under given mass and aeroelastic constraints. In two design studies the sweep and aspect ratio are varied and the convergence performance of the framework is shown. 1
INTRODUCTION
The design process of an aircraft can be broken down into three major phases: the conceptual, preliminary, and detailed design. These different phases mirror the different analysis methods with increasingly complex models in the advanced process. In the structural analysis the quantitive level of FE model idealization advances from coarse to fine. The aerodynamic part starts with panel methods and then switches to CFD applications. This tendency in the design process leads to the paradox that as the amount of knowledge rises, the freedom to act on this knowledge falls. Fundamental decisions on the design, e.g. the planform of the wing (sweep, aspect ratio etc.), that need to be made very early in the design process (conceptual design), are influenced by analysis results at a much more detailed level (preliminary or detailed design). To overcome this paradox, the analysis models of the preliminary design should be parameterized depending on variables normally used in conceptual design, for example the positions of spars and ribs depending on the sweep. Parameterized analysis models of the preliminary design should be generated automatically after varying a typical conceptual design parameter for example the sweep. In the past some programs have been developed for creating of such analysis models. For example, ASTRIGS is an input preprocessor and model generator with graphic capabilities for the ASTROS structural optimization program [8]. The model generator presented in this paper coordinates the execution of a number of aircraft design and geometry defintion modules to create the multiple discipline analysis 1
models. It is based on Computer Aided Geometric Design (CAGD). CAGD is the study and application of curves and surfaces to represent objects in a computational description [1]. Surfaces can be defined by implicit parametric-algebraic equations. Parametric equations have dominated CAGD because of their intrinsic simplicity for modeling complex objects. Inside the model generator the shape of the wing consists of geometric objects represented by parametric curves and surfaces, including the planform and the airfoils as the outer surface and the wing box with spars and ribs defining the inner shape. The generation of the geometric part of the analysis models is based on the parametric description. Combining the aeroelastic toolkit ZAERO and the structural optimization code LAGRANGE with the process integration tool iSIGHT it becomes possible to consider aeroelastic effects in the preliminary design phase of an aircraft. The example presented in this study is the Modified Intermediate Complexity Wing (MICW) examined in [9] for sweep variation. In this study some parameter variations were added with respect to the planform geometry aiming at the conceptual design phase. 2
PROCESS INTEGRATION
A two level optimization approach is proposed for conceptual and structural design. The conceptual design concerning the wing planform and structural layout is denoted as the upper level and the preliminary structural design is denoted as the lower level. At this level sizing variables are changed to reach minimum weight. For this purpose a design process has been established that allows for the creation of various wing planform and wing box variations which follow the generation of models for aerodynamic and structural analysis and optimization. The quality of the simulation models is determined by the analysis methods and tools which are used in the design process. The first step in the design process is the generation of the analysis models with the model generator. For the process of creating the wing and the wingbox geometry the following procedure is integrated in the model generator: • Draw planform of wing with the necessary dimensions to scale area, sweepback, aspect and taper ratio. • Define airfoils at certain locations of the planform. • Locate the front spar (at a constant percentage of the chord), from root to tip. • Locate the rear spar in the same manner, depending on control surface layout. • Define additional spars between the front and rear spar for a multispar configuration. • Define ribs from the front to rear spar with different orientations: in flight direction, perpendicular to one spar, or with a free orientation. Based on these geometrical objects the finite element model and the definition of the aerodynamic surface have been included (see Fig. 1a). As geometric objects (planform, profiles, spars, and ribs) are represented by parametric B-Spline curves or surfaces, they can be exported as IGES Entities (Initial Graphics Exchange Specification) to visualize 2
the geometry. The finite element idealization of the wing box has preliminary design accuracy, but it can be improved by refining the FE mesh. As the structural FE model is derived from the three-dimensional outer wing surface, the requirement that the outer wing skins should be located inside the determined wing volume has been met. The outer wing surface can be used to derive models for CFD calculations as an further extension of the presented framework. The analysis tools used in the design process are the structural optimization program LAGRANGE [10] and the aeroelastic toolkit ZAERO [6]. The system is embedded and controlled by the iSIGHT process integration tool with optimization capabilities (see Fig. 1b). The finite element wing box model, which is generated by the model generator in NASTRAN compatible conventions, consists of a number of different elements like rods, shear panels, and membranes. The grid points are generated by evaluation on the parametric surfaces of the skins, spars, and ribs. Aerodynamic planar surfaces for ZAERO are created which correspond to the wing planform. After the generation of the geometrical part of the analysis models, the first load generation process for the rigid aircraft is performed for certain trim conditions with ZAERO. The loads are used to size the skin thicknesses of the wing box in a first optimization run followed by a modal analysis to obtain eigenvalues, mode shapes, and generalized masses and stiffnesses. After generating the generalized aerodynamic forces, ZAERO uses this information to perform a trim analysis for the flexible aircraft based on a modal approach and the loads for the flexible aircraft are calculated, followed by a structural optimization run with LAGRANGE under fixed loads. The optimized thickness distribution is used later as input for the eigenvalue analysis in the next iteration. The loop is repeated until the optimization results converge in the lower level. Now the wing planform parameters can be changed in the upper level and new steps are started till there is no change in the aeroelastic response and weight. The architecture of the design process allows the analysis programs to be replaced by further tools such as CFD code for improved aerodynamic analysis. Based on the modularity, distributed analysis within a computer network is possible. The two analysis codes are running on different maschines. 3 3.1
ANALYSIS AND OPTIMIZATION MODELS Parametric geometry description
NonUniform Rational B-Splines (NURBS) were used to build up the curves and surfaces of the geometric model describing the wing shape and the internal wing box layout step by step. In order to realize a full parameterized description of the wing shape and the wing box the calculation of intersection points between curves and surfaces or intersection curves between surfaces is necessary. For this purpose the DT NURBS library is employed, which is a collection of geometrical evaluation, manipulation, and storing routines of NURBS curves and surfaces [5]. A NURBS surface is defined as a function of two parameters as
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Figure 1: a) Model generator flow, b) Process flow
T
T
R (U) = {X (U) , Y (U) , Z (U)} , U = {u1 , u2 } ∈ [(a, b) , (c, d)] ,
(1)
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R (U) =
m n P P
Ni,p (u1 ) Nj,q (u2 ) Wi,j Pi,j
i=0 j=0 m n P P
(2) Ni,p (u1 ) Nj,q (u2 ) Wi,j ,
i=0 j=0
where Pi,j are control points, Wi,j includes the weighting factor, and Ni,p and Nj,q are the p − th and q − th degree B-spline basis functions defined on the non-periodic and nonuniform knot vector. In a least square approach the control points Pi,j are determined by given points, for example airfoil coordinates. The procedure is shown in [5] and [7]. The NURBS representation is able to handle complex geometries and maintain smooth surface curvature. It is local in nature allowing the surface to be deformed locally, leaving the rest unchanged. A curve of order p is p − 2 differentiable. If a surface representation is available, the generation of a CFD or FE meshes can be controlled by adapting the subdivision of the surface parameters u1 and u2 . 4
3.2
Loads Calculation
The distributed flight loading results from distributed aerodynamic and inertial loads on a flexible aircraft undergoing a given maneuver condition. The trim analysis searches a balanced condition between the integrated aerodynamic forces and the integrated inertial forces of the six rigid-body degrees of freedom. A modal approach for the load analysis is used to determine the maximum loads within the flight envelope. The method is formulated by Karpel in [4]. The use of modal coordinates is especially effective when the static aeroelastic analysis is performed parallel to flutter analysis. ZAERO employs the modal formulation to solve the trim system of the flexible aircraft. The basic assumption in the modal approach is that elastic deflections of the structure can be represented as a linear combination of a limited set of rigid mode shapes Φr and elastic mode shapes Φe . Assuming there is no mass coupling between the rigid and elastic modes, the aircraft matrix equation of motion becomes ·
Mrr 0
with
0 Mee
¸½
q ¨r q ¨e
¾
+
µ·
0 0
M = ΦT MΦ K = ΦT KΦ Q = ΦT GT [AIC] GΦ
0 Kee
¸
· ¸¶ ½ ¾ Qrr Qre qr − q∞ = Qer Qee qe µ½ ¾ ½ ¾¶ Qrδ Qr (t) q∞ δ+ Qeδ Qe (t)
(3)
generalized mass matrix, generalized stiffness matrix, generalized aerodynamic force matrix.
The vector δ contains the trim variables and the deflections of the control surfaces. The vector q ¨r contains the generalized rigid-body accelerations. The load transfer between the aerodynamic panel model and the structural finite element model is carried out by a spline matrix G. For trim variables and control surface deflections the mode matrix Φ contains the corresponding rigid-body mode shapes. If a quasisteady motion is assumed with no elastic vibrations, then q ¨e disappears. With the provision of user-defined trim variables Eq. 3 can be solved for the modal deflection qe . In order to prepare the loads for structural optimization the resulting generalized forces have to be transferred back to the physical degrees of freedom for the clamped wing. The load vector for different trim conditions is used for structural optimization within LAGRANGE. If the generalized aerodynamic forces are extended to the unsteady domain, Eq. 3 can be rearranged and modified to perform flutter stability analyses by setting the right side to zero. If the righthand side is time dependent, gust and aeroelastic response analysis is possible. ZAERO offers all these capabilities to solve a wide range of aeroelastic problems based on a common model. 3.3
Optimization
In order to deal with optimization problems in the structural design process, a procedure following the ”Three-Columns-Concept” [2] is suitable. According to this concept, the problem is separated into three columns: analysis model, optimization model, and optimization algorithm. The analysis model supplies a mathematical description of the physical behaviour of the problem using appropriate state variables (e.g. deformations, 5
stresses etc.), which depend on the analysis variables (dimensions of components, crosssections, etc.). The optimization model prepares the analysis model for the optimization code. An optimization algorithm as a mathematical procedure for solving a possibly constraint minimization problem is defined as M in{f (x)| h(x) = 0; g(x) ≥ 0; x ∈ IRn }
(4)
with f as the objective function, x as the vector of n design variables, h as the vector of mh equality constraints, g as the vector of mg inequality constraints and IRn the n-dimensional design space. The structural optimization program LAGRANGE follows this ”Three-ColumnsConcept”. It provides the minimum structural weight with respect to the loadcase and fixed wing planform variables under structural constraints. An efficient Convex Linearization Method is employed. In the upper level the planform is varied under constraints of maximum weight and aeroelastic constraints like rolling moment of the flexible wing. This level is implemented in the optimization tool iSIGHT. Gradient based optimizers are used which employ a Sequential Quadratic Programming Method. The necessary sensitivity analysis for objective function and constraints is provided by forward finite differences calculations. Although central finite differences analyses are recommended for shape optimization applications they are avoided to save computer time. 4
DESIGN STUDIES
The MICW example (Modified Intermediate Complexity Wing), which is introduced in [9], is used to illustrate the effect of changing the geometric design variables. It is a simplified finite element model for MDO applications including a trapezoid planform for steady and unsteady aeroelastic analysis. The fuselage is simplified as a 11340 kg concentrated mass simulating one half of the aircraft and connected with the wing structure through rigid bars. The structural model divides the structural box into eight equally spaced spanwise bays and four equal chordwise segments. The skins on the upper and lower surface are modeled as isoparametric quadrilateral membrane elements. Eight ribs and five spars are modeled using shear panels with rod elements used as posts connecting all upper and lower surface nodes. This results in 45 rod elements, 64 quadrilateral membrane elements, and 72 shear panels. All elements use the material properties of aluminium having Young’s modulus of 72400N/mm2 , a Poisson’s ratio of 0.30, and a weight density of 2768kg/m3 . The wing skins are described by 45 thickness design variables which control the thicknesses of the wing skins, the spar webs and caps, and the wing ribs. The layout is shown in Fig. 2. The aerodynamic model represents the wing as a flat plate with 160 boxes. The unsteady aerodynamic model has 10 equally spaced spanwise boxes and 16 chordwise boxes. For the structural optimization within LAGRANGE the boundary condition cantilevers the wing root. The objective consists of finding the minimum weight design that satisfies the stress constraints appearing at two different loadcases. The stresses in the wing skins are constrained by 462N/mm2 for tension and compression, 269N/mm2 for shear. As no buckling constraints are considered, the minimum thicknesses of the skins are restricted. In ZAERO the boundary condition is modeled in such 6
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Figure 2: a) Structural model with definition of design areas, b) aerodynamic model a way that the simplified fuselage mass grid is allowed for pitch, plunge, and roll. The lowest ten normal modes of the optimized design are used to represent the structure in the load analysis based on the modal approach. Two different cases determine the external loads acting on the wing: The first is a 6g symmetric pull up maneuver and the second an antisymmetric roll acceleration maneuver forced by 2.5◦ aileron rotation combined with a 1g steady flight, both at Mach number 0.95 and an altitude of 3000m. The wing sweep of the leading edge is changed from 35◦ to -10◦ where a negative angle implies a forward sweep. The ribs remain parallel to the x-axis. In Fig. 3a the optimized masses are compared with the results from [9]. The results from this study show a more smoother behaviour of the mass depending on the sweep with a mininum range in the intervall from 20◦ to 25◦ . The differences between the curves are caused by the different optimization models. In [9] all thicknesses of every structural element are variables. As shown in Fig. 4 the thickness distributions change from a uniform distribution in chordwise direction between 20◦ and 25◦ to a more concentrated distribution on the upwind elements for the unswept design. It is remarkable that the lowest weight appears at balanced thickness distribution in the root area. In Fig. 3b the roll efficiency increases by decreasing the sweep angle because the wash-out effect is reduced.
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Figure 3: a) Optimum mass versus sweep, b) aileron efficiency versus sweep
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Figure 4: Optimum thickness distributions for the sweep variations
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Figure 5: Optimization histories of configurations A1 and A2 (Fig. a and b) and of configurations B1 and B2 (Fig. c, d, e and f) To check the convergence of the optimization framework two different initial designs with 15◦ (A1) and 35◦ sweep (A2) are chosen to minimize the mass. The optimization histories of the design variable and the resulting mass are shown in Fig. 5a and b. It should be noted that every second step is needed for the finite difference analysis. Starting with the more unswept wing the best result is found at 23.6◦ and 82.080 kg after 10 iterations. The results are shown in Table 1. The second run is finished after 17 steps with 23.1 ◦ and 82.043 kg. The deviation is about 2% for the design variable and 0.5% for the objective function. It is based on numerical influences in the process. The optimum is located at the minimum region of Fig. 3a. 8
A1 initial A1 optimized A2 initial A2 optimized B1 initial B1 optimized B2 initial B2 optimized
Sweep 15◦ 23.1◦ 35◦ 23.6◦ 15◦ 12.6◦ 35◦ 12.1◦
Aspect ratio 3.13 3.13 3.13 3.13 2.0 3.05 4.0 3.05
Mass 82.711kg 82.075kg 87.104kg 82.080kg 57.7kg 81.017kg 109.483kg 80.987kg
CM roll. flex. 0.00761 0.00719 0.00655 0.00737 0.00426 0.00749 0.00936 0.00748
Aileron eff. 0.850 0.810 0.798 0.831 0.881 0.865 0.811 0.866
Table 1: Optimization results for different initial configurations
An extended design study varies the sweep and aspect ratio simultaneously while the wing area and root chord are fixed. Again, two different initial designs are chosen: the first (B1) starts with an aspect ratio of 2 and a sweep of 15◦ and the second (B2) with 4 and 35◦ , respectively. Now the rolling moment of the flexible wing is to be maximized with a maximum mass constraint of 82 kg which is the optimum mass of the sweep only run. The results are shown in Table 1 and the history in Fig. 5c, d, e, f. The first initial design runs straightly forward to the area of the optimum design within 6 steps. The second optimization run jumps between upper and lower bound of the aspect ratio and reaches the optimum area after 26 iterations. Both optimum designs show good agreement, although the maximum possible mass is not utilized to increase the rolling moment. The two different design studies, each started with two different initial planforms, show the convergence properties of the optimization framework. The performance can be increased if the parameters of the optimization algorithm are adapted to the problem, the gradient steps, for example. 5
SUMMARY AND CONCLUSIONS
In this paper a two-level optimization approach is developed to vary the sweep and aspect ratio of the wing planform in the conceptual design phase. Based on a common geometric description in B-Spline formulation an aerodynamic and a structural model are created for the aeroelastic toolkit ZAERO and the structural optimization code LAGRANGE. At lower level the maneuver loads are calculated and the wing structure is optimized for minimum mass. ZAERO delivers the aeroelastic responses of the flexible wing. At the upper level the planform is changed to decrease wing mass or to increase certain aeroelastic quantities under user-specified constraints. The framework is integrated and controlled by the process integration tool iSIGHT. The proposed method is demonstrated using the MICW model. Two design studies one for sweep only variation and one for simultaneous sweep and aspect ratio variation show good convergence behaviour. While in this paper wing planform parameters are changed only, the developed model generator has further capabilities to vary the structural layout of wings like positions, orientations and numbers of spars and ribs. It is possible to tailor the structural layout with respect to aeroelastic response with fixed planform. As the framework is modular
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it can be extended by additional analysis tools or disciplines. ACKNOWLEDGEMENTS The authors would like to thank Dr.V.B Venkayya and V.A. Tischler from Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio (USA), for making available the input data of the MICW for the structural optimization program ASTROS. REFERENCES [1] W. Boehm, J. Hoschek, and H.-P. Seidel. Mathematical aspects of computer aided geometric design. In M. Artin, H. Kraft, and R. Remmert, editors, Duration and Change - Fifty Years at Oberwolfach, pages 106–138. Springer-Verlag, 1994. [2] H.A. Eschenauer, J. Geilen, and H.J. Wahl. SAPOP – An Optimization Procedure for Multidisciplinary Structural Design, pages 207–227. Volume 110 of International Series of Numerical Mathematics [3], 1993. ISBN: 3764328363. [3] H.R.E.M. H¨ornlein and K. Schittkowski. Software Systems for Structural Optimization, volume 110 of International Series of Numerical Mathematics. Birkh¨auser Verlag, Basel-Boston-Berlin, 1993. ISBN: 3764328363. [4] M. Karpel and L. Brainin. Stress considerations in reduced-size aeroelastic optimization. AIAA Journal, 33(4):716–722, Apr 1995. [5] N.N. DT NURBS, Spline Geometry Subprogram Library, Theory Document, Version 3.6, 1998. [6] NN. ZAERO Vers. 4.3 Theoretical Manual. ZONA Technology, www.zonatech.com, 2000. [7] J. A. Samareh. Use of CAD in MDO. In 6th Symposium on Multidisciplinary Analysis and Optimization, pages 88–93, September 4–6 1996. AIAA-96-3991-CP. [8] D. C. Stevens, A. G. Striz, and R. N Yurkovich. An interactive model Generator/Preprocessor for the multidisciplinary optimzation of aircraft. In 5th Symposium on Multidisciplinary Analysis and Optimization, pages 1310–1317, 1994. [9] V.A. Tischler, V.B. Venkayya, F.E. Eastep, and G. Bharatram. Desing interfaces and data transfer issues in multidisciplinary design. In 6th Symposium on Multidisciplinary Analysis and Optimization, pages 1212 –1222, 1996. [10] R. Zotemantel. MBB-LAGRANGE: A Computer Aided Structural Design System, pages 143–158. Volume 110 of International Series of Numerical Mathematics [3], 1993. ISBN: 3764328363.
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