Worst-Case Gust Prediction through Surrogate Models applied to SVD-reduced Gust Databases

Transcription

1 Worst-Case Gust Prediction through Surrogate Models applied to SVD-reduced Gust Databases EngD Student: Supervisors: Simone Simeone T. Rendall and J. E. Cooper DiPaRT

2 Motivation Certification specifications require that enough points, on or within the boundary of the design envelope, are investigated to ensure that the most extreme loads for each part of the aircraft structure are identified. The flight conditions and manoeuvres, which provide the largest aircraft loads, are not known a- priori but Gusts are often the most critical load case for structural design Gusts are the main fatigue loading source for the majority of the structure Hence, worst-case gusts need to be somehow identified DiPaRT

3 Background Source: Calculating time-correlated gust loads using matched filter and random process theories, A.S. Pototzky et al., AIAA Journal of Aircraft, Vol. 28, No. 5. Source: Determining Worst-Case Gust Loads on Aircraft Structures Using an Evolutionary Algorithm, Karr, Zeiler and Mehrotra, Applied Intelligence 20, , Source: Rapid Prediction of Worst Case Gust Loads, H.H. Khodaparast et al., AIAA DiPaRT

4 Contents Objectives Methodology Singular Value Decomposition Surrogate model and optimisation Application Gust database Aircraft model Preliminary Results Concluding Remarks & Future Work DiPaRT

5 Objectives To apply the singular value decomposition technique to decompose a gust matrix built over n reconstructed gust profiles To construct a surrogate model that associates the weighing coefficients of the SVD to the max and min values of the aircraft gust response To use the surrogate model into an optimisation framework to find out the combination of coefficients that generates the worst case gust DiPaRT

6 Methodology Overview W g U L L L Z g Gust time histories Coefficients that define each gust time history Reduced set of characteristic gust shapes (fixed!) A/C response to gusts in the form of time histories Surrogate Model ധZ g The SVD technique is used to decompose the gust matrix W g into a coefficient matrix and a subset of characteristic gust shapes A surrogate model is built on a training set where for each combination of coefficients correspond the max and min value of the A/C response The surrogate model is then used within an optimisation framework to find out which combination of coefficients maximise the response (e.g. bending moment, displacement, lift, etc.) The resulting coefficients are finally multiplied by the fixed set of characteristic shapes to find out what makes the worst case gust DiPaRT

7 Methodology Singular Value Decomposition (1/2) Singular Value Decomposition (SVD) is a matrix decomposition technique that can be used to identify trends, patterns or consistencies effectively and is widely used in many engineering fields It was recently used in a work between AIRBUS and Queen s University Belfast to obtain a concise representation of characteristic loads cases (S.H.M. McGuinness, 2011) Less than 50 SVD-derived loads were used to reconstruct the full A350 XWB loads envelope (M. Hockenhull, 2010) SVD was also used in conjunction with surrogate models to enable a fast computation of correlated loads envelopes in systems where the effects of variation of design parameters need to be considered (I. Tartaruga et al., 2016) The approach was also extended to efficiently quantify the effects of uncertainty in the system parameters DiPaRT

8 Methodology Singular Value Decomposition (2/2) W = U Σ V T W = m n (m m)(m n)(n n) W is the gust matrix, Σ is a diagonal matrix that contains on its diagonal the non-negative singular values and U and V are orthogonal matrices that contain singular vectors m is the number of gust events, n is the number of time steps (i.e. length of time histories) An approximation can be built such that: T W U L L where L = Σ L V L L is a matrix of characteristic gust shapes constructed over a reduced set of k singular values (i.e. first non-negligible terms of Σ) The U L matrix contains the coefficients needed to superimpose these characteristic gust shapes to reconstruct the original set DiPaRT m n, n m

9 Methodology Surrogate Model Surrogate models are defined as analytical models that approximate the input/output behaviour of a complex system They are constructed over a reduced set of training data that is obtained by running expensive simulations at particular points of interest in the design space Here, a surrogate model is constructed to associate each singular vector of the SVD-reduced U L matrix to the maxima and minima values of the aircraft response of interest (accelerations, bending moments, torque, shear, etc.) k inputs Input U L Output ധZ g (m (m k) Surrogate Model 2) 2 outputs The construction is carried out by an automatic algorithm* that selects the most suitable modelling technique available (e.g. Kriging or GP based methods, Neural Networks, Regression Tree, Polynomial RBFs and others) for the considered training dataset DiPaRT *Datadvance MACROS Generic Toolkit

10 Methodology Optimisation The design variables of the optimisation are the coefficients of the SVD-reduced matrix U L and the aim of the objective function is to maximise the aircraft response Generate new set of parameters Input new parameters to Surrogate Model Execute Surrogate Model no Objective reached? yes Evaluate Objective Function min ധZ g U L Multiply parameters by characteristic gust shapes U L L Worst Case Gust DiPaRT

11 Application DiPaRT

12 Gust Database Von Karman spectrum Φ Ω = σ w 2 L π LΩ LΩ y n t = m=1 Φ ω m Δω cos(ω m t + Ψ m ) 200 Hanning Window DiPaRT

13 Aircraft Model FFAST using RFA for GAF matrices A/C Parameters: mass t span = m chord = m U = m/s M = alt = 7,620 m ω 2 ഥM ሚξ + iωഥd i ሚξ + ഥK i ሚξ = Q e + F Aero F Aero = q dyn Q hh M, k ሚξ h + Q hx M, k ሚδ x + Q hg M, k w g N Poles Q i, j k, M Q 0 + ikq 1 + ik 2 Q 2 + l=1 ik ik + b l, b l = k max l Nodal displacements, velocities, accelerations and loads max & min DiPaRT

14 Preliminary Results SVD of gust matrix W g = U Σ V T ( )( ) W g U L L ( ) L = Σ L V L T (50 50)( ) Notice: σ diag(σ L ) σ diag(σ) = 69.4% DiPaRT

15 Preliminary Results Surrogate model Input U L (200 (200 2) 50) Surrogate Model Surrogate model dimensions: [200 52] Effective input size: 50 Effective output size: 2 Output നN z RMSE = σ i=1 m y i y 2 i m MAX = max y i y i, i = 1,, m RMSE = MAX = Total number of samples: 200 Training algorithm: High Dimensional Approximation with Gaussian Process (MACROS GT Approx) RMSE = MAX = DiPaRT

16 Preliminary Results Optimisation min U L നN z നN z = max min N z, max N z Find the highest possible acceleration obtainable by an arbitrary gust profile of constrained max and min amplitude m/s U L L 100.0m/s Optimisation algorithm: Sequential quadratic constrained programming (SQ 2 P) DiPaRT

17 Preliminary Results Worst-Case gust DiPaRT

18 Preliminary Results A/C Response to Worst-Case gust ~ 2.45g increment DiPaRT

19 Concluding Remarks & Future Work SVD has successfully been applied to a large matrix of gust profiles constructed over a range of Von Karman profiles approximated in time A subset of ¼ of the total number of singular values has been used to construct an approximation of the gust matrix The approximation was necessary to reduce the dimensions of the problem A surrogate model has been successfully built over a training dataset of 200 samples with 50 inputs (coefficients that multiply the characteristic gust shapes) and 2 outputs (max and min acceleration response at the A/C c.g.) The surrogate model has been successfully used within an optimisation framework to find out the combination of coefficients that generates the worst A/C response Extend this approach to cover the entire design space with the aid of an additional surrogate model Apply SVD to a database of gust profiles reconstructed from real events (i.e. using data from flight data recorders) DiPaRT

20 Thank you for your attention! Questions? The Industrial Doctorate Centre in Systems, EPSRC and Airbus are gratefully acknowledged. DiPaRT

21 Back-up slides DiPaRT

22 SVD for gusts? W = U Σ V T m n = (m m)(m n)(n n) W is the gust matrix m is the number of gust events n is the number of time steps (i.e. length of time histories) must be equal for all events An approximation can be built such that: W U L L where L = Σ L V L T Here, L is a matrix of characteristic gust shapes constructed over a reduced set of n singular values (i.e. first non-negligible terms of Σ) The U L matrix contains the coefficients needed to superimpose these characteristic gust shapes to reconstruct the original set 22 11th August 2017 SVD for gust - Overview

23 Use of SVD to determine worst case gust sequence (1/3) Assume that gust time histories are all the same length Start at the same point and append zeros if necessary Set up matrix of input time histories A = m n, n m Perform skinny SVD (may have to transpose everything to work with SVD(A, φ) command) Truncate (p) from singular values so that: A = U Σ V T m m m n n n A ഥU ഥΣ ഥV T m p p p p n ഥU L Basis of inputs Scaling to get particular gust inputs An example follows in the next slide 23 11th August 2017 SVD for gust - Overview

24 Use of SVD to determine worst case gust sequence (2/3) W g = U Σ V T = ( )( )( ) If only a relatively small number of singular values (Σ) are non-zero and non-negligible, then this reduced set can be used to recreate the W g matrix For example, if only the first 50 values of the set above are non-zero and non-negligible, an approximation of the W g matrix can be obtained using only these 50 singular values i.e. W g U L Σ L V L T (500 50)(50 50)( ) This is the same as setting all the negligible terms of Σ equal to 0 As a result, for gust shapes, the matrix W g = L 1,1 L 1,n T can be written as W g U L L where L = Σ L V L L m,1 L m,n The L matrix is thus a matrix of characteristic shapes as each row is analogous to a gust shape The U L matrix contains the coefficients needed to superimpose these characteristic shapes to reconstruct the original set The characteristic shapes can be extracted by choosing the first k singular values 24 11th August 2017 SVD for gust - Overview

25 Surrogate Model - Decision Tree 25 11th August 2017 SVD for gust - Overview