A parametric ROM for the linear frequency domain approach to time-accurate CFD
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1 Platzhalter für Bild, Bild auf itelfolie hinter das Logo einsetzen A parametric ROM for the linear frequency domain approach to time-accurate CFD Ralf Zimmermann, AG Numerik, U-BS Model Reduction of Complex Dynamical Systems ModRed, MPI Magdeburg Dec
2 A parametric ROM for the LFD approach to CFD Motivation: Parametric reduced-order models Interpolating data on manifolds he Linear Frequency Domain (LFD approach to CFD Offline, semi-offline and online ROM stages A practical example Conclusions & Outlook Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 2
3 Motivation: Parametric reduced-order models Key idea of subspace based reduced order modeling: Construct low-dimensional subspace Restrict computations to low-dimensional subspace (e.g. via projection Examples: Reduced Bases methods Krylov subspace methods Proper Orthogonal Decomposition Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 3
4 Motivation: Parametric reduced-order models In this talk: Parameter dependent large-scale linear systems:,,,, Parameters determine operating points Parameters are considered as examination parameters Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 4
5 Motivation: Parametric reduced-order models Standard order reduction ansatz: use subspace of dim,, Determine coefficient vector via Orthogonal residual condition Minimum residual condition inner product:, = ( S( Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 5
6 Motivation: Parametric reduced-order models Standard order reduction ansatz: use subspace of dim,, Determine coefficient vector via Orthogonal residual condition Minimum residual condition inner product: A ortha = b orth, = ( S( Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 6
7 Motivation: Parametric reduced-order models Standard order reduction ansatz: use subspace of dim,, Determine coefficient vector via Orthogonal residual condition Minimum residual condition inner product: A mina = b min, = ( S( Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 7
8 Motivation: Parametric reduced-order models Suppose that at r operating points, we have computed bases,, Objective: Construct ROM at new operating point,, Required: Parametrized trajectory of projection bases : Idea: Interpolate given bases [1] [2] [1] D. Amsallem, C. Farhat, 28 [2] Nguyen.S., 212 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 8
9 Motivation: Parametric reduced-order models Challenges: ROM subspace spanned by orthogonal reduced-order basis (ROB Ensure that parametric variation always gives orthogonal bases Reduced bases matrices must be considered as points in the Grassmann manifold of k-dimensional subspaces or the Stiefel manifold of tall-skinny orthonormal matrices [3] [3] A. Edelman,.A. Arias, S.. Smith, 1998 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 9
10 A parametric ROM for the LFD approach to CFD Motivation: Parametric reduced-order models Interpolating data on manifolds he Linear Frequency Domain (LFD approach to CFD Offline, semi-offline and online ROM stages A practical example Conclusions & Outlook Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 1
11 Interpolating data on manifolds (A: basic principle Matrix manifolds Stiefel manifold,! Grassmann manifold ", # dim => each point is a subspace of dimension m, equivalence class of matrices spanning a given subspace may be represented by point in Stiefel manifold Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 11
12 Interpolating data on manifolds (A: basic principle G( n, m U ( q 3 ( q 1 U ( q 1 U ( q ( q 2 U ( q 2 U ( q 3 U ( q * U ( q 2 ( q 3 U ( q U q ( G( n, m U( q 1 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 12
13 Interpolating data on manifolds (B: Matrix Manifolds and here numerical linear algebra comes in Grassmann exponential mapping '( '( [ ] [ ] G( n, m, U = exp [ U WΣQ ] U ( [ ] = SVD = [ U Q cos( Σ m m : + W sin( Σ] G( n, m, Isometric mapping: tangent space to manifold Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 13
14 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 14 and here numerical linear algebra comes in Grassmann logarithmic mapping Interpolating data on manifolds (B: Matrix Manifolds ( 1 1 ] [ ] [ ( (, ( tan,, ( ] [ ] [ log :,, ( ] [ = Σ Σ = = = U U U U U I Q W W m n G U I U U m n G U n n SVD U U m m Isometric mapping: manifold to tangent space '( '(
15 Interpolating data on manifolds (C: A word of caution Challenges It cannot be taken for granted that the interpolation s result stays in the tangential space! here are interpolation schemes, that are not admissible in this regard! angential vectors are represented by * +-matrices: Entry-by-entry interpolation may require millions of interpolation procedures Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 15
16 Interpolating data on manifolds (C: A word of caution rivial example: Nearest-neighbor interpolation leaves the data subspace y(x = 5x x Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 16
17 Interpolating data on manifolds (C: A word of caution,,,,, Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 17
18 Interpolating data on manifolds (C: A word of caution Conclusion Let - -,,.. be vectors in a Grassmann tangent space, represented by matrices An admissible interpolator must be of the form * ( q = k l = 1 f l ( q l In 1D, e.g. linear interpolation and Lagrange interpolation fulfill this requirement In multiple D, radial basis function interpolation is feasible Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 18
19 A parametric ROM for the LFD approach to CFD Motivation: Parametric reduced-order models Interpolating data on manifolds he Linear Frequency Domain (LFD approach to CFD Offline, semi-offline and online ROM stages A practical example Conclusions & Outlook Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 19
20 he linear frequency approach to CFD Underlying assumption Flow to be simulated is periodic and approximately harmonic Valid for pitching motions of small amplitude Flow state vector given by mean flow + time-dependent fluctuations / 1 2 / 3, 4 #(flow vars #(grid points Same applies to grid coordinates and grid cell volumes Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 2
21 he linear frequency approach to CFD First-order aylor and transition to Fourier space give complex linear system: : ; < = 3 (: = temporal mean of grid cell volumes, 9 = frequency Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 21
22 he linear frequency approach to CFD First-order aylor and transition to Fourier space give complex linear system: : ; < = 3 9, : ; 9, : = 3 (: = temporal mean of grid cell volumes, 9 = frequency Parametric dependency: - Mach number determines operating point (mean value : - Frequency varies at each operating point 9 ; ; *9, :+ = 3 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 22
23 A parametric ROM for the LFD approach to CFD Motivation: Parametric reduced-order models Interpolating data on manifolds he Linear Frequency Domain (LFD approach to CFD Offline, semi-offline and online ROM stages A practical example Conclusions & Outlook Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 23
24 Offline, semi-offline and online stage Offline stage Select operating points (OPs -,, > At each OP, compute projection basis? via snapshot POD Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 24
25 Offline, semi-offline and online stage Offline stage Select operating points (OPs -,, > At each OP, compute projection basis? via snapshot POD For any OP of interest, assess the trajectory : via Grassmann Interpolation to obtain * + = 3 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 25
26 Offline, semi-offline and online stage Semi-offline stage At operating point, project full order system onto Exploit separable parameter 9, : : A 1 9 < " 1 89B* + 9, 1 9 < 1 * + = Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 26
27 Offline, semi-offline and online stage Semi-offline stage At operating point, project full order system onto Exploit separable parameter 9, : : A 1 9 < " 1 89B* + = Similar for orth res 9, 1 9 < 1 * + = Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 27
28 Offline, semi-offline and online stage Semi-offline stage At operating point, project full order system onto Exploit separable parameter dependency: precompute A, ", B* + = * + = Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 28
29 Offline, semi-offline and online stage Online stage For any frequency 9 compute reduced system and right hand 9, A 1 9 < " 1 89B* + 9, 1 9 < 1 * + Solve the (m x m 9, Effort: E D independent of full scale dimension = real time online stage! Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 29
30 A parametric ROM for the LFD approach to CFD Motivation: Parametric reduced-order models Interpolating data on manifolds he Linear Frequency Domain (LFD approach to CFD Offline, semi-offline and online ROM stages A practical example Conclusions & Outlook Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 3
31 A practical example Grassmann interpolation of ROBs for LFD approach to CFD NACA64A1 airfoil: Snapshots in transonic Regime Shocks 2-parameter model: Computational grid, size: 1,727 points Ma {.8,.82,.84, }.86,.88,.81, κ {.1,.3,.5,.7,.9} α, Re 7.5mio fixed Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 31
32 A practical example Snapshot and OP-bases sample plan (reduced frequency Sampled operating points defined by Mach number Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 32
33 Ma.85, κ =.2 metric: 1/vols prom LFD Ref proj LFD Ref Ma.85, κ =.4 metric: 1/vols prom global ROM LFD Ref proj LFD Ref.5.5 A practical example Freqdom_drho_re Freqdom_drho_re , * + :G- * + Freqdom_drho_re X Ma.85, κ =.6 metric: 1/vols prom LFD Ref Proj LFD Ref Freqdom_drho_re X Ma.85, κ =.8 metric: 1/vols prom LFD Ref proj LFD Ref Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page X X
34 Ma.87, κ =.2 metric: 1/vols prom LFD Ref proj LFD Ref Ma.87, κ =.4 metric: 1/vols prom global ROM LFD Ref proj LFD Ref.5.5 A practical example Freqdom_drho_re Freqdom_drho_re , * + :G- * + Freqdom_drho_re X Ma.87, κ =.6 metric: 1/vols prom LFD Ref Proj LFD Ref Freqdom_drho_re X Ma.87, κ =.8 metric: 1/vols prom LFD Ref proj LFD Ref Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page X X
35 Ma.85, κ =.2 metric: Euclid prom LFD Ref proj LFD Ref Ma.85, κ =.4 metric: Euclid prom LFD Ref proj LFD Ref.5.5 A practical example Metrics matter! Freqdom_drho_re -.5 Freqdom_drho_re X X, * + * + Freqdom_drho_re Ma.85, κ =.6 metric: Euclid prom LFD Ref Proj LFD Ref Freqdom_drho_re Ma.85, κ =.8 metric: Euclid prom LFD Ref proj LFD Ref Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page X X
36 A practical example Metrics matter! W.r.t. this metric, even the projection of the reference is quite off!, * + : * + Freqdom_drho_re Freqdom_drho_re Ma.85, κ =.2 metric: L2 prom LFD Ref proj LFD Ref X Ma.85, κ =.6 metric: L2 prom LFD Ref Proj LFD Ref Freqdom_drho_re Freqdom_drho_re Ma.85, κ =.4 metric: L2 prom LFD Ref proj LFD Ref X Ma.85, κ =.8 metric: L2 prom LFD Ref proj LFD Ref Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page X X
37 A practical example he orthogonal residual approach is non-competitive Ma.85, κ =.4 orthogonal res LFD Ref prom metric: 1/vols prom metric: Euclid prom metric: L2 Ma.87, κ =.8 orthogonal res LFD Ref prom metric: 1/vols prom metric: Euclid prom metric: L2.5.2 Freqdom_drho_re Freqdom_drho_re X X Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 37
38 A practical example he global 2D POD basis approach is non-competitive Ma.85, κ =.4 minimum res, metric:1/vols Global ROB Ma.85, κ =.4 orthogonal res, metric: 1/vols Global ROB.5.5 Freqdom_drho_re Freqdom_drho_re -.5 LFD Ref dim-3 grob dim-5 grob -.5 LFD Ref dim-3 grob X X, * + :G- * + Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 38
39 Ma.85, κ =.4 minimum res, metric:1/vols Global ROB he reduced global 2D POD basis approach is competitive, if only the 5 most dominant POD modes are kept, but Freqdom_drho_re LFD Ref dim-3 grob dim-5 grob X Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 39, * + :G- * +
40 1 Ma.84, metric: 1/vols grob prob 1 Ma.85, metric 1/vols grob prob.8.8 any global basis approach lacks the interpolation property! Error κ Ma.86, metric: 1/vols grob prob Error κ Ma.87, metric: 1/vols grob prob Error Error κ Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page κ
41 Computational effort Full order dimension: 5*2*#(grid points = 17,27 Reduced order dimension: 5(complex = 1 (real at each OP iming results Offline Mean flow Full-order LFD Complex POD total CPU time ~3s 5*35s ~.26s ~475s (7min Semi-offline Grassmann interp. Reading system Order reduction CPU time ~.66s ~1.s ~.15s ~11s Online Set up ROM Solve ROM Build flow state CPU time Below measure acc. Below measure.1s ~.1s Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 41
42 Computational effort Full order dimension: 5*2*#(grid points = 17,27 Reduced order dimension: 5(complex = 1 (real at each OP iming results Offline Mean flow Full-order LFD Complex POD total CPU time ~3s 5*35s ~.26s ~475s (7min Semi-offline Grassmann interp. Reading system Order reduction CPU time ~.66s ~1.s ~.15s ~11s Online Set up ROM Solve ROM Build flow state CPU time Below measure acc. Below measure.1s ~.1s Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 42 Online speed up: factor ~35
43 A parametric ROM for the LFD approach to CFD Motivation: Parametric reduced-order models Interpolating data on manifolds he Linear Frequency Domain (LFD approach to CFD Offline, semi-offline and online ROM stages A practical example Conclusions & Outlook Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 43
44 Conclusions & Outlook Parametric ROM for LFD solver introduced Parameters of major interest (Mach, frequency allow for a natural division into operation points and examination parameters ROM splits into three stages Offline (provide snapshot data, compute projection bases Semi-Offline Online (precompute frequency-independent system parts, separable parameter dependency renders an interpolation of reduced systems as in [4,5] unnecessary (solve reduced system, real-time performance Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 44 [4] Degroote J. et al., 21 [5] Panzer, H. et al., 21
45 Conclusions & Outlook Metric proves to be essential in order to obtain good results (requires minor technical modifications, not shown here Minimum residual approach outperforms orthogonal residual approach Reduced global basis approach gives satisfactory results, but underperforms when compared to Grassmann interpolation, (Necessarily lacks interpolation property Next step: geometrically optimized reduced order basis Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 45
46 hank you for your attention! Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 46
47 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 47 An admissible interpolation scheme: radial basis functions Radial basis functions (RBF in a nutshell Given: Sample points Sample values RBF interpolator: R p y y p y y R p p k k d k = = (,..., (,..., (,..., ( ( 1 1 * p y y p y k ψ Ψ = ( ( k k k j i j i R p p r = Ψ, ( ( k k R p p r p p r = M 1 ψ
48 An admissible interpolation scheme: radial basis functions Application to Grassmann tangent vectors Given: Sample points Sample values p 1 1,..., p k = ( p R 1 d,..., k = ( p k R n m Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 48
49 Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 49 ( ( ( = = = Ψ = Ψ Ψ = k l l l l m k l l l k l l p e p p 1 1 :, 1 1 :,1 1 1 ( (,, ( ψ ψ ψ K An admissible interpolation scheme: radial basis functions Application to Grassmann tangent vectors RBF interpolator: ( ( ( m j k j j m j n i k j i j i m j n i j i p p p Ψ = Ψ = (,..., ( (,..., ( ( 1 :, 1 :,, 1, 1,, *, ψ ψ = = k l l l p f 1 (
50 An admissible interpolation scheme: radial basis functions Conclusion RBF interpolation is admissible (so is e.g. linear interpolation Entry-by-entry RBF interpolation tangent matrices interpolating coefficients of linear expansion Associated computational effort: IJ interpolation procedures, each for K sample points vs. K interpolation procedures, each for K sample points Dec 12, 213 Ralf Zimmermann A parametric ROM for the LFD approach to CFD Page 5
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