WingOpt - 1 WingOpt - An MDO Tool for Concurrent Aerodynamic Shape and Structural Sizing Optimization of Flexible Aircraft Wings. Prof. P. M. Mujumdar, Prof. K. Sudhakar H. C. Ajmera, S. N. Abhyankar, M. Bhatia Dept. of Aerospace Engineering, IIT Bombay

WingOpt - 2 Aims and Objectives Develop a software for MDO of aircraft wing Aeroelastic optimization Concurrent aerodynamic shape and structural sizing optimization of a/c wing Realistic MDO problem

WingOpt - 3 Aims and Objectives Test different MDO architectures Influence of fidelity level of structural analysis Study computational performance Benchmark problem for framework development

WingOpt - 4 Features of WingOpt Types of Optimization Problems –Structural sizing optimization –Aerodynamic shape optimization –Simultaneous aerodynamic and structural optimization

WingOpt - 5 Features of WingOpt Flexibility –Easy and quick setup of the design problem –Aeroelastic module can be switched ON/OFF –Selection of structural analysis (FEM / EPM) –Selection of Optimizer (FFSQP / NPSOL) –Selection of MDO Architecture (MDF / IDF) –Design variable linking

WingOpt - 6 Architecture of WingOpt Optimizer Analysis Block I/P O/P I/P processor MDO Control O/P processor INTERFACE Problem Setup History

WingOpt - 7 Test Problem Baseline aircraft  Boeing Objective  min. load carrying wing-box structural weight No. of span-wise stations  6 No. of intermediate spars (FEM)  2 Aerodynamic meshing  12*30 panels Optimizer  FFSQP

WingOpt - 8 Test Problem Design Variables Skin thicknesses - S Wing Loading Aspect ratio Sweep back angle t/c root A

WingOpt - 9 Test Problem Load Case 1 (max. speed) Altitude = ft Mach no.= (*1.4) ‘g’ pull = 2.5 Aircraft weight = W to Load Case 2 (max. range) Altitude = ft Mach no.= ‘g’ pull = 1 Aircraft weight = W to

WingOpt - 10 Test Problem Constraints Stress – LC 1 fuel volume – LC 1 M DD – LC 1 Range – LC 2 Take-off distance Sectional C l – LC 1 A S -

WingOpt - 11 Test Problem Structural Optimization (with and w/o aeroelasticity) Aerodynamic Optimization Simultaneous structural and aerodynamic optimization without aeroelasticity Simultaneous structural and aerodynamic optimization with aeroelasticity (6 MDO architectures)

WingOpt - 12 Test Cases CasesD.V. & C.S.M.AEMDO 1SEPMNo- 2SEPMYesMDF1 3AEPMNo- 4S + AEPMNo- 5S + AEPMYesMDF1 6S + AEPMYesMDF2 7S + AEPMYesMDF3 8S + AEPMYesMDF-AAO 9S + AEPMYesIDF1

WingOpt - 13 Results Case Active ConstraintsObjective nfngtime Stresses Fuel volume M dd Range Take-off distance C lmax Weight (kg) (20.29)

WingOpt - 14 Results Case Skin thickness (mm)Wing loading (N/m 2 ) Sweep angle (deg.) t/c ratio Aspec t ratio

WingOpt - 15 Summary Software for MDO of wing was developed Simultaneous structural and aerodynamic optimization Focused around aeroelasticity Handles internal loop instability MDO Architectures implemented

WingOpt - 16 Future Work Further Testing of IDF Additional constraints –Buckling –Aileron control efficiency Extension to full AAO

WingOpt - 17 Thank You

WingOpt - 18

WingOpt - 19 Problem Formulation Aerodynamic Geometry Structural Geometry Design Variables Load Case Functions Computed Optimization Problem Setup Examples

WingOpt - 20 Aerodynamic Geometry Planform Geometric Pre-twist Camber Wing t/c y x single sweep, tapered wing divided into stations S, AR, λ, Λ c itp b/2 Λ c root AR = b 2 /S λ = c itp / c root Wing stations

WingOpt - 21 Aerodynamic Geometry Planform Geometric Pre-twist Camber Wing t/c y x constant α' per station α' i, i = 1, N

WingOpt - 22 Aerodynamic Geometry Planform Geometric Pre-twist Camber Wing t/c formed by two quadratic curves h/c, d/c c h d First curve Second curve Point of max. camber

WingOpt - 23 Aerodynamic Geometry Planform Geometric Pre-twist Camber Wing t/c linear variation in wing box-height t stations

WingOpt - 24 Structural Geometry Cross-section Box height Skin thickness Spar/ribs y A A A x A symmetric front, mid & rear boxes r 1, r 2 r 1 = l 1 /c r 2 = l 2 /c l1l1 c l2l2 Front box Mid box Rear box Structural load carrying wing-box

WingOpt - 25 Structural Geometry Cross-section Box height Skin thickness Spar/ribs linear variation in spanwise & chordwise direction h root, h' 1i, h' 2i ; where i = 1, N A y A A x A h front h rear h' 1 = h rear / h front

WingOpt - 26 Structural Geometry Cross-section Box height Skin thickness Spar/ribs Constant skin thickness per span t si, where s = upper/lower i = 1, N AA t upper t lower y A A x

WingOpt - 27 Structural Geometry Cross-section Box height Skin thickness Spar/ribs modeled as caps linear area variation along length A sjki, where s = upper/lower j = cap no.; k = 1,2; i = 1, N A 2 A upper12 1 yA A x rib front spar rear sparintermediate spar spar cap

WingOpt - 28 Design Variables Wing loading Sweep Aspect ratio Taper ratio t/c root Mach number Jig twist* Camber* Skin thickness* Rib/spar position* Rib/spar cap area* t/c variation* wing-box chord-wise size and position Aerodynamics Structures * Station-wise variables

WingOpt - 29 Load Case Definition Altitude (h) Mach number (M) ‘g’ pull (n) Aircraft weight (W) Engine thrust (T)

WingOpt - 30 Functions Computed Aerodynamics –Sectional C l –Overall C L –C D –Take-off distance –Range –Drag divergence Mach number Structural –Stresses (σ 1, σ 2 ) –Load carrying Structural Weight (Wt) –Deformation Function (w(x,y)) Geometric –Fuel Volume (V f )

WingOpt - 31 Optimization Problem Set Up Select objective function Select design variables and set its bound Set values of remaining variables (constant) Define load cases Set Initial Guess Select constraints and corresponding load case Select optimizer, method for structural analysis, aeroelasticity on/off, MDO method.

WingOpt - 32 Design Case – Example 1 tsitsi Wt σ ---VfVf W(x,y)--M dd V stall CLCL C Di ClCl F A sjki h'2ih'2i h' 1 h root r2r2 r1r1 d/ch/cα' i ΛλARS X Structural Aerodynamic ConstraintObjectiveDesg. Vars. Structural Sizing Optimization: Baseline Design

WingOpt - 33 Design Case – Example 2 ClCl C Di AR ---VfVf W(x,y)Wtσ--M dd V stall CLCL F A sjki tsitsi h'2ih'2i h' 1 h root r2r2 r1r1 d/ch/cα' i ΛλS X StructuralAerodynamic ConstraintObjectiveDesg. Vars. Simultaneous Aerod. & Struc. Optimization

WingOpt - 34 Optimizers FFSQP Feasible Fortran Sequential Quadratic Programming Converts equality constraint to equivalent inequality constraints Get feasible solution first and then optimal solution remaining in feasible domain NPSOL Based on sequential quadratic programming algorithm Converts inequality constraints to equality constraints using additional Lagrange variables Solves a higher dimensional optimization problem

WingOpt - 35 History Why ? –All constraints are evaluated at first analysis –Optimizer calls analysis for each constraints –!! Lot of redundant calculations !! HISTORY BLOCK –Keeps tracks of all the design point –Maintains records of all constraints at each design point –Analysis is called only if design point is not in history database

WingOpt - 36 History Keeps track of the design variables which affect AIC matrix Aerodynamic parameter varies  calculate AIC matrix and its inverse

WingOpt - 37 VLMEPM/ FEM {  α} str. stresses Aerodynamic meshAerodynamic mesh, M, P dyn Aerodynamic pressure Structural deflections ClCl Structural Loads Deflection Mapping Structural Mesh, Material spec., Pressure Mapping Analysis Block Diagram non.–aero LoadsLoads To MDO Control {α} rigid +{  α} str. Trim ( L-nW =   From MDO Control To MDO Control 

WingOpt - 38 Aerodynamic Analysis Panel Method (VLM)VLM Generate mesh Calculate [AIC] Calculate [AIC] -1 {p}=[AIC] -1 {  } Calculate total lift, sectional lift and induced drag

WingOpt - 39 Structures Loads –Aerodynamic pressure loads –Engine thrust –Inertia relief Self weight (wing – weight) Engine weight Fuel weight

WingOpt - 40 Inertia Relief Self-weight calculated using an in-built module in EPM Engine weight is given as a single point load Fuel weight is given as pressure loads Self-weight is calculated internally as loads by MSC/NASTRAN Engine weight is given as equivalent downward nodal loads and moments on the bottom nodes of a rib Fuel weight is given as pressure loads on top surface of elements of bottom skin EPMFEM

WingOpt - 41 Aerodynamic Load Transformation Transfer of panel pressures of entire wing planform to the mid-box as pressure loads as a coefficients of polynomial fit of the pressure loads Transfer of panel pressures on LE and TE surfaces as equivalent point loads and moments on the LE and TE spars Transfer of panel pressures on the mid-box as nodal loads on the FEM mesh using virtual work equivalence EPMFEM

WingOpt - 42 Deflection Mapping EPM  w(x,y) is Ritz polynomial approx. FEM  w(x,y) is spline interpolation from nodal displacements

WingOpt - 43 Equivalent Plate Method (EPM) Energy based method Models wing as built up section Applies plate equation from CLPT Strain energy equation:

WingOpt - 44 Equivalent Plate Method (EPM) Polynomial representation of geometric parameters Ritz approach to obtain displacement function Boundary condition applied by appropriate choice of displacement function Merit over FEM –Reduction in volume of input data –Reduction in time for model preparation –Computationally light

WingOpt - 45 Analysis Block (FEM) NASTRAN Interface Code Wing Geometry Meshing Parameters Load Transformation Input file for NASTRAN Output file of NASTRAN MSC/ NASTRAN Loads Transferred on FEM Nodes FEM Nodal Co- ordinates Aerodynamic Loads on Quarter Chord points of VLM Panels Max Stresses, Displacements, twist and Wing Structural Mass Nodal displacements Panel Angles of Attack Displacement Transformation (File parsing) (Auto mesh & data-deck Generation)

WingOpt - 46 Need for MSC/NASTRAN Interface Code FEM within the optimization cycle Batch mode Automatic generation –Mesh –Input deck for MSC/NASTRAN Extracting stresses & displacements

WingOpt - 47 Flowchart of the MSC/NASTRAN Interface Code

WingOpt - 48 Meshing - 1

WingOpt - 49 Meshing - 2 Skins – CQuad4 shell element

WingOpt - 50 Meshing - 3 Rib/Spar web – CQuad4 shell element

WingOpt - 51 Meshing – 4 Spar/Rib caps – CRod element

WingOpt - 52 Loads and Boundary Condition

WingOpt - 53 Deformation transformation w = displacements (know on nodal coordinates) w(x,y) = a 0 + a x x + a y y +  a i  i (Interpolation function) –where a i is interpolation coefficient –  i (x,y) are interpolation functions  are displacement function solution of the equation for a point force on infinite plate a i are calculated using least square error method

WingOpt - 54 Deformation Transformation (contd..) In matrix notation {w} = [C]{a} where [C] represents the co-ordinates where w is known. This gives {a}=[C] -1 {w} At any other set of points where w is unknown {w} u is given by {w} u = [C] u [C] -1 {w} ie. {w} u = [G]{w} where [G] = transformation matrix

WingOpt - 55 Deformation Interpolation (contd..) {w} a = [G] as {w} s Panel angle of attack calculated as:

WingOpt - 56 Load Transfer Method Transformation based on the requirement that –Work done by Aerodynamic forces on quarter chord points of VLM panels = Work done by transformed forces on FEM nodes

WingOpt - 57 {u a } = [G as ] {u s } {  u a } T {F a }= {  u s } T {F s } {  u a } T ([G as ] T {F a } - {F s }) = 0 {F s } = [G as ] T {F a } Load Transfer Formulation Displacement Transformation Virtual Work Equivalence Force Transformation [G as ]  Transformation Matrix obtained using Spline interpolation

WingOpt - 58 Load Transfer Validation - 1

WingOpt - 59 Load Transfer Validation - 2

WingOpt - 60 Load Transfer Validation - 3

WingOpt - 61 LE control surfaces TE control surfaces Wing box FEM model Wing span divided into 6 stations Wing Topology Aerodynamic pressure on the entire planform to be transferred to the load-carrying structural wing box

WingOpt - 62 Loads Transferred From VLM Panels of Entire Wing Planform to the FEM Nodes of the Wing-box Planform

WingOpt - 63 Loads Transferred From VLM Panels of Wing-box Planform to the FEM Nodes of the Wing-box Planform

WingOpt - 64 VLM – Elemental Panels and Horseshoe Vortices for Typical Wing Planform

WingOpt - 65 VLM – Distributed Horseshoe Vortices  Lifting Flow Field

WingOpt - 66 MDO Control Tasks Carry out aeroelastic iterations j = iteration number; i = node number; N = number of node while satisfying  = L – nW = 0

WingOpt - 67 MDO Control Issues Handling aeroelastic loop –Stable/unstable –Asymptotic/oscillatory behavior Ways of satisfying L=nW (also aerodynamics and structures state equations) Ways of handling inter disciplinary coupling 1. Six methods of handling MDAO evolved 2. Special instability constraint evolved

WingOpt - 68 Divergence Constraint Parameter

WingOpt - 69 MDO Architectures Analysis 1 Iterations till convergence Analysis 2 Iterations till convergence Multi-Disciplinary Analysis (MDA) Interface Optimizer Analysis 1 Iterations till convergence Analysis 2 Iterations till convergence Disciplinary Analysis Interface Optimizer Evaluator 1 No iterations Evaluator 2 No iterations Disciplinary Evaluation Interface Optimizer Individual Discipline Feasible (IDF) All At Once (AAO) 1. Minimum load on optimizer 2. Complete interdisciplinary consistency is assured at each optimization call 3. Each MDA i Computationally expensive ii Sequential 1. Complete interdisciplinary consistency is assured only at successful termination of optimization 2. Intermediate between MDF and AAO 3. Analysis in parallel 1. Optimizer load increases tremendously 2. No useful results are generated till the end of optimization 3. Parallel evaluation 4. Evaluation cost relatively trivial Iterative; coupled Multi-Disciplinary Feasible (MDF) Uncoupled Non-iterative; Uncoupled

WingOpt - 70 Variants of MDF

WingOpt - 71 MDF - 1 AerodynamicsStructures aeroloads To optimizer From optimizer {  (w)<  )}? Update  root Update Yes No displacement (w) Aerodynamics

WingOpt - 72 MDF - 2 AerodynamicsStructures aeroloads To optimizer From optimizer {(  = 0 ) and  (w)<  )}? Update  root Update Yes No displacement (w)

WingOpt - 73 MDF - 3 AerodynamicsStructures aeroloads To optimizer From optimizer  =0 ? Update Yes No displacement (w) Update  root  (w)<  ? No Yes

WingOpt - 74 AerodynamicsStructures aeroloads To optimizer From optimizer MDF - AAO  (w)<  ? Update No displacement (w) Yes

WingOpt - 75 IDF - 1 Aerodynamics Structures To optimizer From optimizer Update No Yes  = 0 ? Calculate {  panel Calculate   & ICCs

WingOpt - 76 IDF - 2 Aerodynamics Structures To optimizer From optimizer Calculate {  panel Calculate  k,ICCs, 

WingOpt - 77 Analysis v/s Evaluators * Solving pushed to optimization level Conventional approach: INTERFACE Solve OPTIMIZER 2. Calculates 3. Calculates Evaluator: Does not solve Evaluates residues for given Computationally inexpensive OPTIMIZER INTERFACE EVALUATOR A different approach * : Analysis: Conservation laws of system If nonlinear, iterative Multidisciplinary Time intensive 1. Generates 2. Calculates

WingOpt - 78 MDO Architectures Analysis 1 Iterations till convergence Analysis 2 Iterations till convergence Multi-Disciplinary Analysis (MDA) Interface Optimizer Analysis 1 Iterations till convergence Analysis 2 Iterations till convergence Disciplinary Analysis Interface Optimizer Evaluator 1 No iterations Evaluator 2 No iterations Disciplinary Evaluation Interface Optimizer Individual Discipline Feasible (IDF) All At Once (AAO) 1. Minimum load on optimizer 2. Complete interdisciplinary consistency is assured at each optimization call 3. Each MDA i Computationally expensive ii Sequential 1. Complete interdisciplinary consistency is assured only at successful termination of optimization 2. Intermediate between MDF and AAO 3. Analysis in parallel 1. Optimizer load increases tremendously 2. No useful results are generated till the end of optimization 3. Parallel evaluation 4. Evaluation cost relatively trivial Iterative; coupled Multi-Disciplinary Feasible (MDF) Uncoupled Non-iterative; Uncoupled

WingOpt - 79 Overview Aims and objective WingOpt –Software architecture –Problem setup –Optimizer –Analysis tool –MDO architecture Results Summary and Future work

WingOpt - 80 Inference History block reduces computational time to 1/10 th FEM requires substantially more time than EPM dcp constraint fails in some cases to give optimum results whenever aeroelastic iterations are oscillatory MDF-1 fails occasionally without dcp constraint MDF -3 fails to find feasible solution More robust method for load transfer is required