Nonlinear Aeroelastic Analysis using ROM/ROM Methodology: Membrane-on-Wedge with Attached Shock

Transcription

1 Nonlinear Aeroelastic Analysis using ROM/ROM Methodology: Membrane-on-Wedge with Attached Shock Danny D. Liu Zhicun Wang Shuchi Yang Chunpei Cai Marc Mignolet Presented at the Bifurcation and Model Reduction Techniques for Large Multi-Disciplinary Systems Meeting at the University of Liverpool June, E. Ironwood Square Drive, Scottsdale, AZ , Tel (48) , Fax (48) ,

2 Acknowledgement This present work is under the support of a NASA SBIR Phase I contract, with Dr. Robert Bartels as the Technical Monitor. 2

3 Different Ballute Types Clamped Ballute Trailing/Torroid Ballute Reentry aeroshell A conceptual design for ballute Inflatable ballute entry 3

4 Earth/Mars Entry Profiles *Parameters: ballute mass =22 lb, ballute diameter: 92 ft, Initial Mach number: 2; Gravity acceleration: Mars:1.4 ft/s 2, Earth ft/s 2. Initial altitude: Mars= 66 kft, Earth: 9 Kft Earth Entry* Martian Entry* Knudson number: Kn M Re, a GasKinetic parameter 4

5 Overview Ballute aeroelastic problem requires Gaskinetic (microscopic) aerodynamics in the rarefied hypersonic flight regime Boltzmann/BGK method (time accurate) is adopted. 1 Ballute is an inflatable (nonlinear) structure Nonlinear structural ROM (ELSTEP) is adopted. 2 Ballute flutter/lco computation procedure needs to be expedited ZONA s nonlinear/linear ROM-ROM procedures are adopted. 3 Membrane-on-Ballute with Bow-Shock is modeled first by a 2D membrane-on wedge with attached shock- thus the present study Supported by: 1. AFOSR/Schmisseur; 2. NASA/Rizzi; 3. AFOSR/Fahroo. 5

6 Outline Introduction: Ballute Systems Nonlinear Structural ROM (ELSTEP) Boltzmann Unsteady Aerodynamics: Time- Accurate BGKX Nonlinear Aeroelastic Static Deformation Analysis Aerodynamic ROM (Sys. Id. + ARMA) ROM/ROM Time-Domain Flutter Analysis for Undeformed/Deformed Mean Configuration Concluding Remarks 6

7 Modeled Ballute System Rigid ring x r Mean shock Oscillatory shock Reflected wave trains Vibrating membrane Rigid nose M Modeled (Axisymmetric) ballute system with nonlinear structural-aerodynamic interactions. Oscillatory Shock Characteristics (Mach Waves) Mean Shock θ Vibrating membrane M Wedge angle L Membrane-on-wedge in hypersonic/supersonic flow. 7

8 Present Nonlinear Aeroelastic Methodology NL Structural Solver Flutter/LCO Analysis NL Structural ROM Aerodynamic Solver Around Mean Config. NL AE Static Analysis Aerodynamic ROM 8

9 Generation of Nonlinear Structural ROM ZTRAN In-Plane Static Response to Transverse Loads SOL 16 Linear Transverse Modes (Nastran) SOL 13 Linear Flutter Modes Aerodynamic Forces NONLINEAR ROM STRUCTURAL MODEL Structural Response 9

10 About ELSTEP/FAT ELSTEP/FAT = Equivalent Linearization Stiffness Evaluation Procedure/Fatigue (due to) Acoustics and Thermal Gradients An advancement of ELSTEP code previously developed by Steve Rizzi/NASA Langley and Alex Muravyov/MSC ZONA/ASU R&D efforts of ELSTEP/FAT supported under several AF/SBIR s and NASA/SBIR s from 1999 ~ 25 1

11 Nonlinear ROM Procedure: ELSTEP Large Deformation Mechanics; Lagrangian Formulation EXACT FORMULATION Assumed Displacement Field, (m) U i Basis Funct ( ) M n u ) n u( X ( ), t, t) = qn t( tu) U ( X ) X i n= 1 n= 1 n i ( ) ( ) F Lin + F NL : Run Sol #16 + Sol #13 ij j ij j (1) ij j (2) ijl j l (3) ijlp M q&& + C q& + K q + K q q + K q q q = F j l p (t) i Linear stiffness Quadratic stiffness Cubic stiffness GAF 11

12 ELSTEP Nonlinear Structural ROM FEM model (Nastran) Runs in static nonlinear Sol13 Sol16 Evaluate coefficients of the ROM model (1) K ij (2) K ijl,, (3) K ijlp Solve for Nonlinear Equation of Motion & Boundary Conditions ij j ij j (1) ij j (2) ijl j l (3) ijlp M q&& + C q& + K q + K q q + K q q q = F j l p (t) i Time Histories (Displacements) Flutter/LCO Analyses Stresses/Fatigue Life Prediction 12

13 Procedure to Evaluate Nonlinear Stiffness Terms Impose a series of static deflections and determine (e.g. from finite element model) the forces required and the stresses. Then, identify the coefficients of the reduced order model. Cond. (a) F = K q + K q + K q ( j ) (1) (2) 2 (3) 3 u= q j U ( i ) a ij j ijj j ijjj j Cond. (b) u =q j U ( j) (1) (2) 2 (3) 3 ( F ) = K q + K q K q i b ij j ijj j ijjj j K (2) ijj = ( F) + ( F) i a 2 q 2 j i b Cond. (c) u= qˆ j U ( j) (1), K ij (3) K ijjj Inner terms j ( j) ( l) u = q U + q U u = q U q U j ( j) ( l) l u = q U + q U + q U l ( j) ( l) ( p) j l p (2), K ilj (3), K iljj Cross terms (3) K iljp 13

14 PSD (DISP^2/Hz) PSD Excitation ELSTEP Past Validation Example of Application: Fully clamped; no temperature Acoustic excitaiton 135dB 1.E-8 1.E-9 1.E-1 Nastran NL 2 Linear Modes 5 Linear + 5 Duals Frequency (Hz) 1.E-11 1.E-12 1.E-13 1.E-14 1.E-15 Significant nonlinearity first peak = 154 Hz first nat. freq. = 11 Hz 1.E Hz 14

15 NL Structural ROM for the Membrane-on-Wedge Trans. Disp./Thickness p = p p Assuming: u p l = l p p = p p = q C u l p u 15.1 The NL structural ROM for the flexible panel uses the first 6 transverse modes and 11 dual modes Deformation solutions under constant pressure agree excellently with Nastran nonlinear solutions NASTRAN ELSTEP (6T11D) Pressure (Pa)

16 ZONA Capability of Hypersonic Flow Solvers High level CFD methods CFL3D BGK NS level Burnett Level(Kn>) Time Accurate Aerothermodynamics Unsteady Motion CFL3D Deforming mesh FUN3D -- Deforming mesh BGKX Moving mesh = yes; -- = not available * Under development 2D Inviscid methods ZPEC (Zona Perturbed Euler Characteristics) Piston Theory Local Features Embedded Mesh Adaptive Mesh Adaptive Mesh * * Compu. Speed Faster Slowest Slower 16

17 Rarefied Hypersonics: Microscopic versus Macroscopic Approaches Macroscopic approaches (Continuum) Flow parameters: Mach no., Reynolds no. All continuum methods: Euler, N-S, etc. Microscopic approaches (Gaskinetic) Flow parameter: Knudsen number DSMC (direct simulation Monte Carlo) First-principal particle collision approach; no governing PDE Boltzmann Eqn. (Integro-Differential Eqn.) t + u f = I f1 f 2, f 1 f2 x BGK Eqn. (approximation of Boltzmann) g f u + f = t x 17

18 Boltzmann/BGK vs. Classical Eqns. Boltzmann/BGK uses distribution function (f) as a single dependent variable with (7) independent variables (t, x i, v i ) Euler/N-S have 5 prime variables (P, U, V, W, ρ) with (4) independent variables (t, x i ) Potential flow uses velocity potential function (Φ) as a single dependent variable with (4) independent variables (t, x i ). To recover solution from f and from Φ to prime variables (P, U, V, W, ρ) requires respectively to integrate f and to differentiate Φ. 18

19 BGK and BGKX Equations For BGK equation, the right hand side (RHS) of collision terms is simplified as one relaxation term between equilibrium state, g, and instantaneous distribution, f, and is the characteristic relaxation time: f f g f + u = t x For BGKX equation, Xu adopts modern CFD kinetic flux for the left handside (LHS) terms, for the RHS, Xu replaces the relaxation time τ by a strained relaxation time τ*, which allows for extended Knudsen number (Kn) range from towards 1., thus covering the continuum to transient flow regime up to the order of BGKX-Burnett. Note that tackling this flow regime with DSMC would overburden its computing cost and with continuum CFD would be pushing its capability; whereas the BGKX Burnett is a proper one. 19, 1 Kn

20 Merits of the BGKX Method BGK eqn. is a higher level one than continuum Euler/N-S eqns. BGKX covers wide range of Knudsen number (Kn); it unifies continuum flow (Kn~) with transition flow (<Kn<1.). BGK solver is time-accurate, hence most suitable for unsteady aerodynamic applications. One-step computational procedure for pressure and heat flux solutions. Single gas distribution function, f, simplifies the flux algorithm. Consistent and unified procedure to handle equilibrium, equilibrium and chemically reacting flows. ZONA has been supported by AFOSR/STTR on the BGKX Solver development since 24. For publications see: Cai, C., Liu, D.D., and Xu, K., A One-Dimensional Multi-Temperature Gaskinetic BGK Scheme for Planar Shock Wave Computation, AIAA Journal, Vol. 46, No. 5, May 28. 2

21 Shock-Shock Interaction by BGKX Shock-shock interaction. Surface pressure and heat flux distributions. 21

22 Heat-Rate Reduction by BGKX/MHD MHD actuator effects on the shock stand off distance. MHD actuator effects on heat flux from the cylinder. 22

23 BGKX for Blunted Cone/Cylinders: Cp and Heat flux along the Surface Ma =1.6, Re =1.1e5, T = 85R, R= 1.1 inch, Pr=.72, Tw =54R. BGK simulation results of Cp, heat flux over a spherically headed 15 degree cone. Surface C p and Mach number contours. Surface heat flux and pressure contours. 23

24 BGKX Simulation Results of a Hypersonic Flow over a Paraboloid M=16.3, T = K, T w =294.4 K Pressure Heat Flux a) b) c) d) a) Pressure, Re = , b) Pressure, Re= c) Heat Flux, Re = , d) Heat Flux, Re =

25 NL Aeroelastic Static Deformation Start Initial Configuration Run CFD Solver GAF CFL3D BGK Solve NL Structural ROM Equation Deformed Configuration No Converge End 25 Yes

26 Membrane-on-Wedge Rigid ring x r Mean shock Oscillatory shock Reflected wave trains Vibrating membrane Rigid nose M Oscillatory Shock Characteristics (Mach Waves) Mean Shock θ Vibrating membrane M Wedge angle L 26

27 NL AE Static Def. Solutions for Membrane-on-Wedge Cp GAF q y a x a Yy a Cp (CFL3D) Cp (BGKX) Comparison of Cp distribution along the undeformed wedge surface by BGKX and CFL3D (Mach = 5).1.5 Numerical simulation process (Alt = 1 Kft, Mach = 5) Mach= 5, Altitude= 1 ft.1 q1 q2 q3 q4 q5 -.1 q6 q7 q8 -.2 q9 q1 -.3 q11 q12 q q14 q15 q q17.35 t(s) 1 GAF1 GAF2 5 GAF3 GAF4 GAF5 GAF6 GAF7-5 GAF8 GAF9 GAF1-1 GAF11 GAF12 GAF13-15 GAF14 GAF15 GAF GAF17.35 t(s) 27

28 NL AE Static Deformed Shapes for Undeformed Alt = Kft Alt = 2 Kft Alt = 5 Kft Alt = 7 Kft Alt = 9 Kft Alt = 1 Kft Membrane-on-Wedge NL Aeroelastic Static Deformed Shape Nonlinear aeroelastic static deformed shapes for the flexible membrane at various altitudes represented in the structural coordinate system.2 NL Aeroelastic Static Deformed Shape y a z s x a Nonlinear aeroelastic static deformed shapes for the flexible membrane at various altitudes represented in the aerodynamic coordinate system Undeformed Alt = Kft Alt = 2 Kft Alt = 5 Kft Alt = 7 Kft Alt = 9 Kft Alt = 1 Kft x s 28

29 Cp Solution Comparison Using Different Aerodynamic Solver Mach = 5; Alt = ft Comparison of Static Deformed Shape Undeformed ZPEC CFL3D BGKX Undeformed-BGKX ZPEC CFL3D BGKX Comparison of Cp Distribution Alt= ft z s x s x a Comparison of the statically deformed shapes using various aerodynamic solvers (represented in the structural coordinate system) Comparison of Cp distributions along the statically deformed wedge surface using various aerodynamic solvers 29

30 y BGKX Solutions on the Statically Deformed Wedge Cp distribution along the deformed wedge surface (Mach =5) Undeformed Alt = Kft Alt = 2 Kft Alt = 5 Kft Alt = 7 Kft Alt = 9 Kft Alt = 1 Kft Alt = ft, Mach = 5 P C p x x Pressure contour plot on deformed configuration (Alt = ft; Mach =5) 3

31 Aerodynamic ROM System Inputs ( ) q t Aerodynamic System (CFD Solver) System Outputs GAF t ( ) ROMs 31

32 Unsteady BGK Solver 32

33 Aerodynamic ROM Approaches Various aerodynamic ROM methods: POD/ROM-HB: Dowell (1998) Volterra: Silva (1993) ERA: Kim (24) POD/ROM-Time Domain: Beran (23) ARMA: ZONA* (28) NNet: ZONA Present approach: System Identification technique, specifically, Auto-Regressive-Moving-Average (ARMA) model. * Z. Wang, et al., Flutter Analysis with Structural Uncertainty by Using CFD-based Aerodynamic ROM, presented in 49 th AIAA/ASME/AHS/ASC SDM, 7-1 April, 28, Schaumburg, IL 33

34 Aerodynamic ROM Training Excitations Filter Impulsive Method (FIM) signals are chosen as excitation signals for their Broader range of frequency with concentration on the frequency of interest. Symmetric about zero axis. A FIM signal is given by: ( ) ( ) ( ) 2 a t t u t = Ae sin t t when t t = when t t u(t) PSD Input FIM Signals t(s) FFT of the Input Freq(Hz) A staggered sequence FIM input of modal coordinates is employed. Each mode uses its own natural frequency as the ω. The lowest order goes first. 34

35 Aerodynamic ROM: ARMA With the prescribed staggered FIM excitations to the modal coordinates, a special run of CFD soler is carried out at the specified Mach number. The time histories of the normalized (nondimensionalized) generalized aerodynamic forces are recorded. Therefore, the complete set of training data is available. ROM are sought to define the relationship between modal coordinates (serving as the System Inputs, u) and GAFs, (serving as the System Outputs, y). If m structural modes are used, there will be m ROMs identified. Auto-Regressive-Moving-Average (ARMA) model is used: n n = i + bu j + i= 1 j= 1 a b ( ) ( ) ( 1 ) y t a y t i t j nk where na-nb-nk, the so-called delay order are found by a trial and error procedure 35

36 Aerodynamic ROM: NNet Nonlinear aerodynamic ROM is represented by neural network model. The modeled plant output at time t by the neural network would be given in a concise notation as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) y ( t) = a = W tanh W U + W y p + b + b 2 2,1 1,1 1,2 1 2 Inputs Input Layer Output Layer U. w ij (1,1) b 1 (1) b 2 (1) n 1 (1) n 2 (1) a 1 (1) a 2 (1) w i (2,1) n (2) a (2) y p.... b (2) w ij (1,2) b S (1) n S (1) Two-layer Feed-Forward Neural Network Using the training data, an optimization procedure is implemented to search for the best parameters ( 1,1 ) ( 1 ) ( 2,1 W, b, W ) and ( 2) b by minimizing the mean square of the error between model output and targeted output or the generalized mean square error. 36 a S (1)

37 Linearized Equation of Motion around Statically Deformed Configuration q = q + q (1) M q + K q + F = F NL a ( ) 1 Static : K q + F q = V GAF q 2 ( ) ( ) 1 2 NL ( ) F ( ) 1 NL 2 Dynamic : M q + K + q = V GAF q q q

38 N. GAF 6 N. GAF 5 N. GAF 4 N. GAF 3 N. GAF 2 N. GAF 1 q Aero ROM Training for Undeformed Mean/Wedge Configuration 5 x 1-3 q1 q2 q3 q4-5 q q x x CFL3D ROM Sim. CFL3D ROM Sim CFL3D ROM Sim x x 1-3 CFL3D ROM Sim CFL3D ROM Sim Nondimensional Time CFL3D ROM Sim. Aero ROMs are developed for the first 6 transverse modes using CFL3d Staggered FIM signals are shown in the first sub-figure ARMA models for the 6 normalized GAF are obtained by optimization procedure using the training data set GAFs for the other 11 dual modes are assumed zero Aero ROM predictions agree well with direct CFL3D outputs

39 N. GAF6 N. GAF5 N. GAF4 N. GAF3 N. GAF2 N. GAF1 q Validation of Aero ROMs for Undeformed Mean/Wedge 5 x 1-3 q1 q2 q3 q4-5 q q x 1-3 CFL3D ROM Sim x 1-4 CFL3D ROM Sim x 1-3 CFL3D ROM Sim x x 1-3 CFL3D ROM Sim x 1-5 CFL3D ROM Sim. CFL3D ROM Sim Nondimensional Time 39 The first sub-figure is the time histories of the modal coordinates providing inputs to both the aerodynamic ROMs and the direct CFL3D solver Specifically, only the second modal coordinate assumes a sinusoid time history while others are kept zero. Aero ROM predictions agree well with direct CFL3D outputs The exceptions are for the fourth and sixth GAFs, but these two are very small, two-order smaller the others

40 q q q q q q ROM-ROM Flutter Analysis: Undeformed Mean/Wedge x Dynamic Simulation Alt= ft q 1 q 2 q 3 q 4-2 q 5 x q Dynamic Simulation Alt= 2 Kft q 1 q 2 q 3 q 4-5 q x 1 5 q 6 1 Dynamic Simulation Alt= 5 Kft q 1 q 2 q 3 q 4-1 q q Dynamic Simulation Alt= 75 Kft q 1 q 2 q 3 q 4-1 q q x Dynamic Simulation Alt= 9 Kft q 1 q 2 q 3 q 4-1 q q x Dynamic Simulation Alt= 1 Kft q 1 q 2 q 3 q 4-5 q q t(s) 4 Conventional type of flutter analysis: linear structural EOM unchanged as altitude change Under our dynamic simulations, the first modal coordinate is given a small initial value; all the other initial conditions are zeros By varying the altitude (consequently, the freestream speed and the dynamic pressure, i.e., the match-point methodology), one explorer the decaying, near neutral, and diverging time responses.

41 Freq. (Hz) Linearized Stiffness of Deformed Mean/Wedge Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode Altitude (Kft) Change of the natural frequencies around the deformed mean wedge configuration at various altitudes 41

42 q q q q q q ROM-ROM Flutter Analysis: Deformed Mean/Wedge 1 x 1-3 Dynamic Simulation Alt= ft q1 q2 q3 q4-1 q q x 1-3 Dynamic Simulation Alt= 2 Kft q1 q2 q3 q4-1 q q x 1-3 Dynamic Simulation Alt= 5 Kft q1 q2 q3 q4-1 q q x 1-3 Dynamic Simulation Alt= 75 Kft q1 q2 q3 q4-2 q q x 1-3 Dynamic Simulation Alt= 9 Kft q1 q2 q3 q4-2 q q x 1-3 Dynamic Simulation Alt= 1 Kft q1 q2 q3 q4-1 q q6.25 t(s) 42

43 Conclusions Ballute aeroelastic problem requires Gaskinetic (microscopic) aerodynamics in the rarefied hypersonic flight regime. Boltzmann/BGK method (time accurate) is adopted Ballute is an inflatable (nonlinear) structure Nonlinear structural ROM (ELSTEP) is adopted Ballute flutter/lco computation procedure needs to be expedited ZONA s nonlinear/linear ROM-ROM procedures are adopted. Membrane-on-Ballute with Bow-Shock is modeled first by a 2D membrane-on-wedge with attached shock-- thus the present study For a wedge with a mean deformed membrane, its stiffness increases with decreasing altitude, thus it becomes dynamically more stable contrary to the outcome of undeformed membrane Axisymmetric membrane-on-ballute model aeroelastic study is in progress 43