12, 122 12, 12, 125 ,w 121 128 120 134 ,,I 131 111 IY 135 ',6 137 I38 IX 1.0 141. Figure 6: Element Number of Truss Structure. To simulate damage, the ...
Application of an Integrated Structural Optimization Code Wolfgang G. Luber EADS Deutschland GmbH Military Aircrafl MT24 P.O.Box801160 D-81663 Munich, Germany
ABSTRACT In structural design of aircraft optimization methods are applied to analyze complex structures with high accuracy. During the last decade considerable effort has been spent to develop modern multidisciplinary structural optimization procedures, using efficient mathematical optimization algorithms which satisfy all requirements simultaneously and to find an optimum of the design variables. An overview of the Multidisciplinary Optimization (MDO) techniques shows how these methods primarily support the preliminary - and detailed design phase. The different optimization methods are discussed and the architecture of the used program LAGRANGE is described. This program is designed as open as possible to incorporate all kinds of disciplines for actual and further projects. Features like shape and topology optimization are being added to the overall program to improve the quality of the products and reduce development phase of advanced systems. Different examples are discussed to show the applicability and to demonstrate the benefits of the exciting structural optimization software in design phases.
INTRODUCTION Modern military and transport aircraft are complex systems whose performance depends on the interaction of many different disciplines and parts. The complexity of the problems, the coupling of very large sets of governing equations, is handled traditionally by solving only one subset of the system, such as aerodynamics, structures, flight control system, flightmechanics, propulsion and so on. This is described by J. Sobieski ,Ref. 10 with the well known sentence: Everything infloences everything else. For each individual disciplines great advances were made due to theoretical, computational and methodology breaks through. However, the more sophisticated methods result in a decrease of the awareness of the influence of the specialists decision in the own area and other disciplines. On the other hand it became
more and more difficult to account strictly for all those couplings between these subsets only by parametric studies. In this investigations a small number of principle parameters were investigated to find out their influences on the design requirements - which were oflen contradictory-to improve the design. With this approach, working in a limited design space, the engineer may obtain good results, but very often it also leads to a penalty on the design objectives to make the initial concept feasible. A more efficient way to integrate the different disciplines and to balance their distribution in the early design phase is the multidisciplinary design optimization (MDO) approach. Mathematical optimization algorithms together with reliable analysis programs and the optimization model is the basis for MD0 calculations with a high rate of generality and efficiency. The main tasks are: . .
Achieve an optimal objective Find a design which meet requirements simultaneously
all
specified
Without time consuming and search for modification of the initial design. These conditions require new analytical tools to comply with the requirements and to exploit the full potential of a new generation of aircraft design. TECHNICAL APPROACH The computer aided structural optimization system EADS-LAGRANGE (Ref. 3,5,12) was used to perform the applications shown. This system is based on finite element methods as well as on nonlinear mathematical programming codes. For other optimization programs see Ref. 1,2,7,&l 1. Applications of the optimization method for detecting error in finite element models or for damage detection on structures and updating of dynamic models by means of ground resonance test results see Ref. 6,9) Design models are represented by their objective function, their design variables (parameters) and many different constraints (restrictions). Available
design variables are element thickness, cross sections, fibre angles, and concentrated masses. To handle strength or stiffness designs, restrictions on stresses, strains, buckling loads, natural frequencies, dynamic modes, dynamic responses, aeroelastic efficiencies, flutter speed and generalized displacements can be taken into account as constraints. Besides isotropic, orthotropic and anisotropic materials the design of composite structures is of main concern. The primary aim is to minimize the structural weight subject to the selected restrictions. But sometimes the problem in question is to get a feasible design and the weight is less important. The extended range of mathematical programming codes, the modular architecture and the possibility to take all selected constraints simultaneously into consideration are the highlights of the program system. The general program architecture is shown in Fig. 1 and Ref. 5 explains the system in more detail.
design elements is described by corresponding shape functions and controlled by so-called master-key points, which are varied during the optimization. Program Features: Static strength and strain (failure safety criteria) Static stability (buckling) Deformations (generalized displacements) Thermal stress Dynamic quantities (eigenvalues, eigenvectors, and frequency response with transient deformation, velocity or acceleration constraints) Manufacturing constraints (tape curvature, drop off angle, fibre shifting) Flutter speed and damping Static aeroelastic quantities (force and moment effectiveness, criteria with respect to loads, static aircraft stability criteria, maneuver performance control structural divergence, requirements, surface reversal) Description of Optimization Method The general formulation of structural optimization task can be stated as a nonlinear programming problem (NLP, see Ref. 7):
subject to a set of inequality constraints
g,(x) 2 0
j E3=
{l,Jn}
x, I x < x, The objective function l(x) is general the weight or volume of the structure which is linear with respect to the design variables. But it is also possible to use other objective functions for the updating with the aim of minimum changes of the FE model:
f(x) =p - x1( =@( xp- q2 ,=I Furthermore multi objective problems can be handled by what is called vector optimization.
Figure 1: General Program Architecture The design model as an important part of the optimization model describes the relation between the design variable x and the state variables y of the analysis model which is required for the calculation of the structural behavior. Applying shape optimization the usage of design elements is a suitable procedure. In this case the structure is divided into simple subelements like points, lines and surfaces. Each
The design variables can be considered as different types of structural parameters: 1. sizing variables like . 290
cross sectional areas for truss and beam elements.
.
A second useful method is a combination between the dynamic force method and the above described method of eigenvalue sensitivity. The error vector is defined as the difference between the theoretical stiffness matrix and the mass matrix multiplied by the measured eigenmodes. The gradient is given by:
wall thickness for membrane and shell elements.
.
laminate thickness for all single layers in composite elements. 2. concentrated masses to balance dynamic behavior. 3. angles of fibre directions for layered composite materials.
CJe, z=OL
The most important part of structural optimization tools is the variety of restrictions that can be imposed on the design model. A typical formulation of the constraint functions is given by the following dimensionless inequality equation:
g(x)=l-
aK,-dM,h. ( ax ax
Q
JM1 JM
This is the sensitivity of the error matrix which shows the Changes of the energy which is necessary if the analytical structure is forced to vibrate in the measured mode shape. For the purpose of damage localization it may be useful to evaluate the sum of all elements of the error sensitivity matrix.
r(x) -.-.-EL>0 ro,,ow
s,=gg!?F$
The satisfaction of constraints results in a feasible structure. In principle constraints have to be dealt with simultaneously as they restrict the design space in which the optimized structure will ultimately be found. Using gage constraints as preconditions for manufacturing are included. Which combination of constraints has to be applied depends on the physical problem statement. The state variables i. e. response of the structure are implicitly dependent on the design variables x.
,=o ,=I I
Optimization Algorithm Many structural optimization applications have shown that several different Mathematical Programming (MP) algorithms are required. These codes internally use appropriate approximations to meet the specific problem structure, which refer to mathematical characteristics such as: Linear, quadratic, separable, convex, homogeneous etc. An amount of experience is required when choosing the proper MP-code. The following algorithms are implemented in LAGRANGE:
Sensitivity Sensitivity analysis is necessary to localize the weakness of a structure. This method assumes that the eigenvalues are a function of stiffness, mass, damping and geometric data. The discrete eigenvalue problem associated with linear vibration is defined in terms of two symmetric matrices K and M. In general, the eigenvalue problem is guaranteed to have real eigenvalues if either K or M is positive definite and the discussion here is limited to this case. When the eigenvalues are distinct each derivative is given by (Ref. 8):
IBF MOM SLP SUMT SQP CONLIN GRG RQP QPRLT
Eigenvalue sensitivities are useful when resonance frequencies need to be restricted. Exact analytical expressions for eigenvalue sensitivity can readily be derived for the case of non repeated roots.
+ 2 A.--. QT.!?!-,aM.. ,=,,,s,h,- hl
r ( ax
Inverse Barrier Function Method of Multiplier Sequential Linear Programming Sequential Unconstraint Minimization Technique Sequential Quadratic Programming Convex Linearization Generalized Reduced Gradients Recursive Quadratic Programming Quadratic Programming with reduced Line search technique
Usually the best results are obtained by starting with simple, sublinear convergent algorithms and restart with second order method like SQR. Examples: The following examples shows the wide used capabilities of the used optimization program LAGRANGE by EADS Deutschland.
The eigenvector derivative is given by:
Ep:.?!&@,
1
'ax 1 1 ' 291
Frame of a Combat Aircraft Fuselage This frame is located in the intake of a supersonic combat aircraft. This is a typical example for a sizing problem of a very light airframe structure of an aluminum alloy. For the formulation of the optimization equations it is important to know that the manufacturing milling machine will be used to realize a variable thickness distribution. To calculate the optimal thickness of the frame a large number of design variables can be defined and introduced into the optimization process. The finite elements model is shown in Figure 3 as a front view. The finite element model involves 975 degrees of freedom by 930 elements and 97 defined load cases.
The initial design has infeasible buckling loads and stresses. The applied optimization algorithm SLP needs about 20 iterations to obtain convergence. Figure 4 depicts the optimization history for the buckling constraints, where respectively the most critical constraint of the corresponding iteration was taken. The optimal design met all static requirements of the 97 selected load cases and achieves a weight reduction of about 25 percent. Damage Detection on a Truss Structure This example is an application on the benchmark of the Group for Aeronautical Research in Europe (GARTEUR), The test structure is a plane clamped free truss system, fixed on the nodal points 5 (x and y single point constraint) and 6 (x single point constraint), whose members are characterized by E = 7500 Mpa, I = 0.0756 m4, and p = 2800 kg/m3, with Awrt,ca,elements = 0.006 m2, Adiagon3, elements = 0.003 m2 = 0.006 m . The initial model and &mmtal elements contains 78 nodal points, and 83 bar elements resulting in 216 DOF, of which 12 (6x and 6y) translation DOF are assumed to be measured. Figure 5 shows the grid points and Figure 6 the beam element numbers.
Figure 3: Finite Element Model of a Combat A/C Frame The objective function for this optimization example is the weight. The constraints ensure static requirements. For each element the feasibility of the stress values have to be performed. Stability constraints are taken into account to avoid buckling and lateral instability of the webs. Since the constraints have to be satisfied for each single load case we get a very large number of inequality constraints (mg = 92829). Together with the thickness design variables (n = 187) it is a real large scale optimization problem.
i8
1; I,
8 I,
I,
h w 4.14; 4;
20 2, 20 IO 35 3s 4, 4, 47 50 54 +a 62 @5 111 II
74
Figure 5: Nodal points of Truss Structure 142 ,,a 144 115 118 1.1 7,s 14s 150 151 152 153 1% 155 ,511157 1% 159 180 ‘8, 162
~~~
12, 122 12, 12, 125 ,w 121 128 120 134 ,,I 131 111 IY 135 ‘,6 137 I38 IX 1.0 141
Figure 6: Element Number of Truss Structure To simulate damage, the following modifications are introduced: Reduction of 50% of the cross sectional properties on the elements between nodal point 26 and 49. From earlier sensitivity analysis it was known, that the middle diagonal of the truss structure has negligible influence on the first mode shapes. This fact was used to prove the robustness of the method. To localize the damage, the first five eigenmodes were used with 12 displacements of each mode. The mode shapes were introduced with four digits without any interpolation. Two static load cases were introduced for improving the localization procedure on nodal point 73 in x and y direction. The first five eigenfrequencies
Number of Iterations
Figure 4: History of most critical buckling constraints 292
of the undamaged structure were used as constraints with f 0.5 deviations. The eigenfrequency difference of the undamaged and damaged structure was less than 5%.
Figure 8 depicts at the left hand side the conventional stiffener layout of the frame, which is idealized by membrane and bar elements. To distinguish between the different design cases, two main load cases were applied. it is known, that the wing interface loads coming from the maximum positive and negative wing moments. To get an idea how the optimal placements of the stiffeners should be placed, the frame model was adapted with the above mentioned software TopLESS by a truss structure on a design space by a grid points only. The grid points were placed on the outer skin flange and inner duct flange contour as well as inside the frame plane. Due to the expected heavy loads, the result is fully dominated by strength constraints and no buckling constraints for compressed bars were taken into account. Based on the load path resulting from the optimum truss topology shown in Figure 9 and 10, the new stiffer layout was designed to direct the loads by the new stiffeners and therefore a new corresponding FE-model was generated. The results of the new stiffener design are shown for comparison on the right hand side of Figure 8. It should be mentioned, that the original engineered layout was a sizing optimization run with stress constraints, performed with the EADS-LAGRANGE code.
Figure 7: Damage detection in % versus elements The results are shown in Figure 7. This localization stage allow us to locate damage elements 170 to 176. The error is calculated on the elements between 55 and 75 %. The reason for this overpredicted error margin can be explained by the above mentioned shape factor, because the simulated Lagrange program error is always higher than the real physical damage. The elements on the connection points are also marked with some damage. This is also an indicator for more detailed analysis. The elements 171 and 175 shows not the right damage. Topology Layout A main wing attachment fuselage frame of a combat Aircraft was redesigned with the external tool Topology Layout Element System Software (TOPLESS). This example demonstrates how truss topology optimization methods can be applied to a real designing task: the optimal position of stiffeners in the frame. A comparison of the topological stiffener design versus the existing engineered design was performed to assess the useful algorithm.
Figure 9: Conceptual stiffener design In the first sizing run all stiffeners of the original, engineered layout were linked to only one variable, i.e. only scaling up to the defined cross-sectional area gauges was allowed. This is required to freeze the proportions of the principal stiffener which are the crossing bars near the upper wing interference, see left hand side of Figure 8. In the second run, all stiffeners were chosen as individual variables, with the result that the crosssectional areas of the principal stiffeners approached the minimum gauges and the membranes took over the normal stiffness by adjusting their thickness. The minimized volume in the second run was about 30% lass than the minimum volume with the linked
Figure 8: Engineered versus topology stiffener layout 293
stiffeners. This results show the benefits of the TopLESS optimization, because the stiffeners of the original engineered design were not optimal placed with respect to strength (Figure 10).
mass balance for the control surfaces. This was an direct approach during the flight test phase of the aircraft, because the gearing ratio of the tabs were optimized for pilot stick forces and performance and handling characteristics. Taking into account also the flight control system the changes could be easily adapted to investigate the sensitivity effects on flutter stability. In Figure 11 the interaction between structural design, static aeroelastic effectiveness and structural loads are demonstrated. The first graph depicts the optimized structural weight with different constraints for the aileron roll moment effectiveness. The second graph demonstrates the effect of the constraint on the aileron hinge moment that is required to achieve the same roll rate for the different designs. Of course the hinge moment has a direct impact on structural loads and the required actuator power.
l~“I’w
Figure 10: Load path of the optimized structure In the last step a sizing optimization was done to determine the volume of the optimal strengths design with the stiffeners placements of the topological layout. The volume was now about 10% less than the minimum volume for the engineered design even with individual variables for the stiffeners (second run). To confirm this result, a finer meshed FE-model has to be used to ensure that this improvement is not only based on the inaccuracy of the FE-model. An additional function of the stiffeners is to support the membrane elements in plane. Consequently, appropriate buckling constraints for membranes elements should be taken into account.
g 5 08 ’ I 06 E ’ !iz 0,4 s ?$ 0,2
0.0 Cover Skin Weight [ kg ]
Aircraft Components design The optimization code LAGRANGE was applied for the wing design of the experimental Aircraft X31. Manufacturing reasons demands the final drawings almost before engineering could start. The result of the optimization analysis satisfied not only the requested target weight, but also the high boundaty of flutter stability. The wing skins were designed for pre designed fiber orientations with a balanced +45 lay-up. At that point it is worth to notice, that the design of composite structure the individual fibre direction is allowed to change the preselected minimum radius. For manufacturing the theoretical results could be adapted with minor modification. It should be mentioned, that the final weight of the skins were only 2% above the theoretical value. In an other study for an trainer aircrafl the optimization code was used to determine the minimum amount of 294
Figure 11: Optimized wing skin weight and resulting aileron hinge moment Aircraft overall design The optimization program was also applied for the pre design of a demonstrator aircraft. The structural concept is shown in Figure 12 and the Finite element model in Figure 13. In this case the dynamic model is a compromise between sufficient accuracy for buckling stability of the aluminum skins and an acceptable computing time for integrated optimization runs with strength, buckling stability, flutter and static control surface constraints for aerolelastic effectiveness. First of all the wing, tail and fuselage components were optimized separately. Then the design optimization was applied for the complete
model. In general the single steps for the component optimization converged very rapid.
Figure 12: Conceptual Aircraft
suitable completions and extensions of the structural and sensitivity methods as well as the optimization models (i.e. local and global stability, heat transfer, acoustics, thermal stress, flight mechanics, flight control and manufacturing) are required in a optimization code. Important factors for the efficient use of MD0 concepts is the fast, automated generation of analysis models, and the application of the approaches very early in the design phase. To obtain optimal results, none of the major aircraft design parameters must be fixed. MD0 should also be an integral part of the conceptual or pre-design phase to investigate the impacts from various mission and performance requirements on the design. Sometimes a small reduction of the requirements may save high costs or improve other performances.
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Figure 13: Finite Element Model for demonstrator Aircraft
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Conclusion This paper presents a of solving design tasks in the aircraft development process using structural optimization methods. As design criteria, requirements on the static, dynamic and aeroelastic behavior of aircraft are considered. The analysis procedure for the various state variables describing this behavior are based on the finite element Method. The application of this program demonstrates that the design process can be supported very efficiently by structural optimization method. An important advantage of the application of structural optimization programs is to achieve technically optimal design. In order to optimize real life structures a lot of important procedures and methods have to combine in the optimization programs. Since an optimal design has to fulfill all demands on structure simultaneously,
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