Framework For Multi Delity Aeroelastic Vehicle Design .

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Framework for Multifidelity Aeroelastic Vehicle DesignOptimizationDean E. Bryson and Markus P. Rumpfkeil†University of Dayton, Dayton, Ohio, 45469, USAandRyan J. Durscher‡Air Force Research Laboratory, Aerospace Systems Directorate, Wright-Patterson AFB, Ohio, 45433, USAA multifidelity aeroelastic analysis has been implemented for design optimization of alambda wing vehicle. The goal of the multifidelity, multidisciplinary approach is to capturethe effects of nonlinear, coupled phenomena on vehicle performance at a cost amenableto conceptual and preliminary design. The goal of the optimization is to maximize rangeat a supersonic flight condition under constraints on trim, wing deformation, and structural stresses. The design variables include planform shape, material gauges, and cruiseangle of attack. The low-fidelity model couples linear, finite-element structural analysiswith linear panel aerodynamics. The high-fidelity model couples structural modes withEuler computational fluid dynamics. A single, unified geometric representation is centralto the multifidelity, multidisciplinary process, ensuring compatibility between disciplinesand fidelities. Finite differences are used to calculate coupled, aeroelastic gradients. Goodsensitivities are obtained for the low-fidelity model. However, noise in the high-fidelityresponse is found to dominate some derivatives, and is an area for further work. Theoptimization is demonstrated using the low-fidelity simulation, motivating the use of multifidelity techniques to reduce the cost of high-fidelity �Vehicle fuel consumptionVehicle drag coefficientVehicle lift coefficientVehicle pitching moment coefficient about the center of gravityOptimization objective functionOptimization inequality constraintOptimization equality constraintDynamic pressureVehicle rangeVehicle planform areaVehicle velocityVehicle nominal cruise weightVehicle initial weightVehicle final weightKreisselmeier-Steinhauser weighting parameterStructural stresses Ph.D.Candidate, Dept. of Mechanical and Aerospace Engineering, brysond1@udayton.edu, Member AIAAProfessor, Dept. of Mech. and Aerospace Engineering, Markus.Rumpfkeil@udayton.edu, Senior Member AIAA‡ Research Engineer, Aerospace Vehicles Division, Member AIAA† Associate1 of 16American Institute of Aeronautics and Astronautics

I.IntroductionTraditional aircraft design in the conceptual and preliminary phases is based largely on historical data andlow-fidelity modeling. This approach is necessitated by the large number of configurations to be evaluated,the speed and cost of higher-fidelity predictions, and the level of detail to which the vehicle has been defined.However, these analyses, which are generally performed without inter-disciplinary couplings, may fail toaccurately predict vehicle performance, leading to missed opportunities for improvement as well as cost andschedule overruns to correct defects discovered late in design.1 Furthermore, decisions made prior to themilestone of down-selecting a vehicle concept drive three-quarters of an aircraft system’s life cycle cost2 whiledesign freedom diminishes quickly.Multifidelity design methods seek to address these issues by balancing the cost and accuracy of predictingvehicle performance. Lower-fidelity data may predict the overall trend of the design space and identifypromising regions, while higher-fidelity data provide a basis for correcting inaccuracies. These calibrationstypically implement surrogate models, such as polynomial expansions or kriging, that capture varying degreesof non-linearity in the error between high and low fidelity. Identifying cases where the calibration processbecomes inadequate remains a challenge. The trust region optimization framework3 is a popular approachto addressing this need. This methodology constrains the use of approximate models to a local region wherethe corrected low-fidelity data is believed to be adequate. The size of this region is adjusted heuristicallybased on the error between the approximate and true high-fidelity function.This paper considers the optimal design of a lambda wing vehicle to maximize its range. The aeroelasticvehicle deformation will be included to provide a more complete picture of performance, and multifidelitymethods will be used to provide high-fidelity predictions accelerated by leveraging low-fidelity data. Thelambda wing planform strives to balance sub- and supersonic performance. Initially, to demonstrate theoptimization framework, the vehicle will be optimized for supersonic conditions only. While the multidisciplinary couplings here are limited to aeroelasticity, the design methods are extensible to include otherdisciplines such as propulsion as well as stability and control.II.II.A.MethodologyOptimization Problem DefinitionThe baseline lambda wing vehicle configuration is depicted in Figure 1. To maximize the vehicle range, apreliminary design-level optimization is performed using static aeroelastic analyses. The multidisciplinaryproblem includes the planform geometry and structural gauges. The highest fidelity considered consists ofbody-fitted Euler Computational Fluid Dynamics (CFD) coupled with linear structural mode shapes. Thelowest fidelity implements panel aerodynamics tightly coupled with full, linear structural Finite ElementAnalysis (FEA).(a)(b)Figure 1. Outer mold line and internal layout of the baseline lambda wing vehicle.2 of 16American Institute of Aeronautics and Astronautics

The aircraft range may be calculated using the Breguet range equation,R V CLWiln.C C D Wf(1)The lambda wing planform design seeks to balance subsonic and supersonic performance. As a preliminarysimplification, the optimization is only considered here at Mach 1.2. A constant engine performance assumption further simplifies the problem. Thus, by rearranging Equation (1) and taking its negative, the objectivefunction to minimize becomesRCCLWff ln.(2)VCDWiFor this study, it is assumed that gross take-off mass is constant at 23000 kg based on data taken fromAlyanak et al.4 This assumption is enforced between different configurations by varying the fuel weight. Theaerodynamic performance is driven by the vehicle outer mold line coupled with the aeroelastic deformations,while the weight performance is driven by the structures discipline. The vehicle weight at the end of cruiseWf equals the weight of subsystems (estimated by textbook methods), payload, and structure.The optimal designs must satisfy several constraints to be valid. First, the vehicle must trim. At themost basic level, the lift must equal the aircraft weight at the nominal design condition,h1 qSCL 1 0.Wc(3)As a simplification to the aeroelastic analysis problem, the vehicle is defined without control surfaces, whichwould be used to trim the moments. For the design optimization, the resultant moments will be monitoredbut not constrained.The structure must bear the applied loads without failing. On an element-by-element basis, these constraints may be defined based on von Mises stresses σV M and failure stresses σF ,σV M,i 1 0.σF,i(4)However, this implementation introduces a large number of constraints. Using an aggregation technique suchas the Kreisselmeier-Steinhauser (KS) function5 reduces the burden on the optimizer by combining stressconstraints over patches of structural elements (such as regions of constant thickness skin or spars). Theaggregated inequality constraints are defined as(σ)N elemV M,i1ρ σ 1F,igj lne 0.ρi 1(5)In the absence of aeroelastic divergence analysis, constraints on the wing deflection are also prudent.Limiting the magnitude of the tip displacement, the constraint is expressed asgk (X) δtip (X) 2 1 0.δlim(6)The structural design variables may be defined under varying levels of assumptions to reduce the sizeof the optimization problem. At the most detailed level, the upper and lower skins would have separatematerial gauges, and would be further segmented into spanwise patches delineated by ribs. The wing sparswould exhibit a similar stepwise taper. The outboard wing ribs have uniform gauges, as do the inboardbulkheads, ribs, and kick spars.However, to reduce the size of the design problem for this initial demonstration, the skin gauges aresymmetric on upper and lower surfaces, and are constant on three spanwise patches demarcated by theleading edge breaks. The material gauges of the internal structure are also separated into patches. The spargauges are constant over two separate patches, the outboard wing and the centerbody, as are the ribs. Thecenterbody bulkheads also have a uniform gauge.To design the planform, the area is assumed to be constant at 230 m2 for the full vehicle, again based ondata from Alyanak et al.4, 6 The free design variables considered include aspect ratio, outboard sweep angle,and outboard taper ratio. The aspect ratio is a primary driver of the induced drag and structural weight.3 of 16American Institute of Aeronautics and Astronautics

The outboard sweep angle is expected to have a significant impact on wave drag. The centerbody is heldconstant for packaging considerations, though the sweep angle of the midboard leading edge varies slightlyas the chord of the outboard wing varies.From an overall vehicle perspective, the angle of attack is a design variable free to satisfy the trimcondition.In summary, the reduced-size optimization problem is formally defined as:()CL (X)Wf (X)f (X) minimizelnCD (X)Wiwith respect toX [α, AR, Λout , λout , tskin , tspar , trib ]subject toh1 (X) qSCL (X) 1 0Wc(σ)N skinsM,i (X)1ρ Vσ 1F,ig1 (X) lne 0ρi 1Nspars ρ( σV M,i (X) 1)1σF,ie 0g2 (X) lnρi 1(7)(σ)Nribs M,i (X)1ρ Vσ 1F,ig3 (X) lne 0ρi 1 δtip (X) 2 1 0δlimUxLi xi xig4 (X) The multifidelity optimizer currently implemented, as described in Section II.B.4, is an unconstrainedoptimization algorithm. Thus, to introduce the constraints into the problem, a quadratic penalty functionis used4 2f (X) f (X) wh1 h21 (X) wgi max (gi (X), 0) .(8)i 1The penalty weights were selected based on the relative magnitudes of the responses at the baseline configuration, and different weighting schemes are compared in Section III.B. The design variable bounds areprovided in Table 1. Note that the aspect ratio variable seems very limited, however, because the entirelifting body is included in the reference area, the variable produces spans ranging from 10 to 30 m.Table 1. Design variable bounds.VariableAngle of Attack αAspect Ratio ARSweep ΛTaper Ratio λSkin Gauge tskinSpar Gauge tsparRib Gauge tribII.B.Lower Bound-52350.20.0050.0050.005Upper Multifidelity, Multidisciplinary AnalysisThe responses for the optimization problem defined above are calculated from the multifidelity, multidisciplinary analysis depicted in Figure 2. The process begins with the selection of design variables via anoptimizer, parametric study, or other means. The model configuration and geometry generator, Computational Aircraft Prototype Syntheses (CAPS),7 interprets the parameters into the required aerodynamic andstructural analysis models. These models feed the subsequent disciplinary analyses.4 of 16American Institute of Aeronautics and Astronautics

Figure 2. Logical connections and fidelities in multifidelity, multidisciplinary analysis.The first analysis is the evaluation of the structural weights of the as-designed model using the AutomatedSTRuctural Optimization System (ASTROS).8, 9 At this point, it is reemphasized that the gross takeoffweight is defined to be constant across all configurations. The structural weights combine with fixed estimatesof subsystem weights yield the empty weight used in the range calculation. The margin between takeoff andempty weights indicates the available fuel weight, of which 50% is used for the design maneuver and trimweight.The structural analysis model is updated to reflect this design weight before running the maneuverloads analysis in ASTROS. This analysis determines the stresses and strains for the structural constraintsby trimming (angle of attack only) the vehicle to achieve a prescribed load factor. Since the simplifyingassumption is made that the upper and lower skins have the same material gauges, a 9-g pull-up maneuveris sufficient for the analysis.The same structural analysis model also feeds the subsequent multifidelity static aeroelastic evaluation.All fidelities implemented are required to produce lift, drag, and tip displacement, which will be used in theobjective and constraint evaluations. The lowest fidelity considered is ASTROS, which combines linear FEAwith panel aerodynamics. It generates structural deformations, lift, and induced drag. The code AWAVE,10which is a low-order code that estimates wave drag, was considered to produce a more complete dragestimate. However, parametric studies indicated that the predictions by AWAVE are too noisy to be usefulin the optimization. The highest fidelity considered is Fully Unstructured Navier-Stokes 3D (FUN3D)11, 12 inEuler mode with modal structures calculated again by ASTROS. The loads transfer and mesh deformationare handled internally by FUN3D.The aeroelastic optimization requires coupled gradients of the response functions. ASTROS already provides sensitivities of responses with respect to the material gauges, and FUN3D provides rigid aerodynamicsensitivities with respect to shape parameters. However the efficient calculation of coupled gradients is atopic of ongoing research. Here, finite differences are used as a demonstration. More efficient and accuratemethods will be considered as the tools become available.II.B.1.Parametric Geometry and Analysis Model GenerationA shared geometric representation of the vehicle is central to the multifidelity, multidisciplinary analysis andoptimization. Using a single source ensures that the inputs given to each analysis are consistent and aids inthe transfer of data between disciplines. This objective is achieved using CAPS.7Within CAPS exists a parametric, attributed model of the lambda wing vehicle. The attributes provide5 of 16American Institute of Aeronautics and Astronautics

logical information required for the generation of analysis inputs. For example, attributes identify thevehicle skins where aeroelastic data transfers take place, symmetry planes for the application of boundaryconditions, and bodies to which material properties should be applied. When a design parameter is changed,the geometry is regenerated, and analysis models (meshes, properties, etc.) may be requested for variousdisciplinary analyses. At the time of writing, analyses demonstrated include structural FEA using twodimensional elements and aerodynamics from empirical methods, through panel methods, up to NavierStokes.The analysis model generation proceeds as follows. Using the current design parameters, the airfoil crosssections and the planform shape are determined. Lofting these airfoils provides a solid body representing theouter mold line (OML). These same airfoils also provide the boundaries for defining mid-surface aerodynamicpanel models. The CFD domain is generated by subtracting the OML solid from a bounding box.The internal structure results from intersecting the OML body with a grid representing the structurallayout. The layout may have variable topology, though here the topology is held constant, and the shapefollows the planform parameterization. The wing skins are extracted from the outer surface of the OMLbody. Sample geometric entities used for building the analysis models are presented in Figure 3.(a) Airfoils for panel aerodynamics.(b) OML for FEA and CFD.(c) Internal structure.(d) Fluid domain for CFD.Figure 3. Representations of the same vehicle model with different geometric fidelities for multidisciplinary analyses.Size of CFD volume is reduced for illustrative purposes.II.B.2.Low-Fidelity Aeroelastic ModelingThe low-fidelity analysis is performed with the ASTROS8, 9 package. ASTROS performs static, modal,and transient linear FEA, and has an internal aerodynamics capability for static and dynamic aeroelasticanalyses. The aerodynamics model, Unified Subsonic and Supersonic Aerodynamic Analysis (USSAERO),13generates pressures based on the superposition of sources and vortices over a panel representation of thevehicle. USSAERO simulates both lifting surfaces and non-lifting bodies, though the latter capability is notrequired for the current configuration. The transfer of loads and displacements between the two disciplinesis handled using surface splines.14The optimization utilizes the ASTROS static aeroelastic capability, specifying the angle of attack, vehicle6 of 16American Institute of Aeronautics and Astronautics

shape, and structural gauges, and receiving the vehicle weight, finite element stresses and displacements, andlift and induced drag coefficients. A mesh convergence study for the aeroelastic cruise prediction is presentedin Table 2 for three different vehicle configurations. With the first configuration being the baseline, the secondconfiguration reduces the aspect ratio, increases sweep, and thickens the structure with the goal of creatinga very stiff wing. The third configuration, in contrast, increases aspect ratio and reduces the sweep andmaterial gauges to produce a relatively flexible wing. These configurations are illustrated in Figure 4 withsample structural and fluid meshes. The models selected for the optimization have approximately 2200 nodesand 1664 aerodynamic panels for the low-fidelity simulation.Table 2. Mesh convergence of ASTROS aeroelastic simulation. Errors are with respect to finest mesh.II.B.3.FEA NodesAero PanelsCLErrorCDErrorConfig �6.79E-046.60E-046.48E-044.8%1.8%—Config ��2.63E-032.58E-032.56E-033.0%1.1%—Config elity Aeroelastic ModelingThe high-fidelity analysis uses the internal aeroelasticity capability15 in the NASA FUN3D code.11, 12 Theresults presented here assume an inviscid fluid. FUN3D is a node-centered, implicit, upwind-differencingfinite-volume solver.11, 15 The integrated aeroelastic analysis utilizes a modal structural representation computed by an external solver, in this case ASTROS. While the coupled aeroelastic equations are implementedfor transient analysis,15 here the solution is time-marched to static equilibrium. The volume mesh deformation implements a linear elastic analogy driven by the surface mesh displacements transferred fromthe structure.15 The initial grids are generated using AFLR16, 17 for the surface and volume meshes. Arepresentative mesh is provided in Figure 5.The transfer of mode shapes from the structural mesh to the fluid surface mesh is handled by CAPSinternally. The structural and fluid surfaces are matched by tagging the parametric geometry model withattributes. The nodal displacements from the structural solution are read in through an analysis interface andmapped onto the source geometry, which has knowledge of both the structural and fluid meshes. Each modeshape is then transferred to the fluid surface mesh by interpolating the displacements from the structuralmesh through this shared geometry. This transfer is performed automatically with no further action requiredby the user.A mesh convergence study is provided in Table 3 for the same three configurations as ASTROS. Themeshes used for optimization have approximately 300,000 nodes for the CFD mesh and 2200 nodes for theFEA mesh. Eighteen mode shapes are included in the dynamic analysis. The finite element model is allowedto have rigid body motion about the wing box root to simulate free flight, though the rigid body modes areomitted from the CFD simulation. Inviscid wall boundary conditions are applied to t

Optimization Dean E. Bryson and Markus P. Rumpfkeily University of Dayton, Dayton, Ohio, 45469, USA and Ryan J. Durscherz Air Force Research Laboratory, Aerospace Systems Directorate, Wright-Patterson AFB, Ohio, 45433, USA A multi delity aeroelastic analysis has been implemented for design optimization of a lambda wing vehicle.

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