Model reduction of parametrized aerodynamic flows: stability, error control, and empirical quadrature. Masayuki Yano

Transcription

1 Model reduction of parametrized aerodynamic flows: stability, error control, and empirical quadrature Masayuki Yano University of Toronto Institute for Aerospace Studies Acknowledgments: Anthony Patera; NSERC WCCM 2018 New York, United States 26 July 2018

2 Motivation: parametrized aerodynamic flows Goal: rapid and reliable solution of parametrized conservation laws in many-query scenarios. Conservation laws: Euler and Reynolds-averaged Navier-Stokes (RANS) equations Scenarios: 1. parameter sweep (flight-envelope/design study) 2. uncertainty quantification 1

3 Parametrized steady conservation laws µpde: given µ D R P, find u V s.t. ( F (u; µ) + K(u; µ) u) = S(u, u; µ) }{{}}{{}}{{} advection flux diffusion flux source B(u, n K(u; µ) u; µ) = 0 on Ω }{{} boundary operator in Ω µpde (weak): given µ D, find u V s.t. O( ) ops r(u, v; µ) v F (u; µ) + dx = 0 v V. }{{} Ω residual form 2

4 Finite element (FE) method FE space: N -dim space e.g., piecewise polynomial V h V FE integral: K h -pt quadrature Q h {(x k, ρ k )} K h k=1 s.t. e.g., Gauss quadrature [ ]dx Ω (x k,ρ k ) Q h ρ k [ ] xk FE: given µ D, find u h V h s.t. O(N, K h ) ops r h (u h, v; µ) ρ }{{} k [ v F (u h ; µ) + ] xk = 0 v V h. FE residual form (x k,ρ k ) Q h 3

5 Reduced basis (RB) method RB space: N N -dim empirical space V N V h V for parametric manifold {u(µ) µ D}. RB (FE integ.): given µ D, find u N V N s.t. r h (u N, v; µ) }{{} FE residual form O(N, K h ) ops (x k,ρ k ) Q h ρ k [ v N F (u N ; µ) + ] xk = 0 v N V N. 4

6 RB method with empirical quadrature [Ryckelynck; An et al; Farhat et al;... ] EQP: K ν K h -pt quadrature for empirical (parametric) functions (x ν k, ρ ν k) Q ν s.t. [( )(µ)]dx ρ ν k[( )(µ)] x ν k Ω (x ν k,ρν k ) Qν RB-EQP: given µ D, find u ν N V N s.t. O(N, K ν ) ops r ν (u ν N, v; µ) ρ ν }{{} k[ v N F (u ν N; µ)+ ] x ν k = 0 v N V N. EQP residual form (x ν k,ρν k ) Qν 5

7 RB-EQP example: NACA0012 Euler (µ = (α, M )) RB space: V N span{φ i } N i=1 φ 1 φ 2 φ 3 φ 4 EQP: with element-wise grouping N = 4, K ν = 26 N = 10, K ν = 59 6

8 Objectives Goal: stable, accurate, and systematic hyperreduction 7

9 Objectives Goal: stable, accurate, and systematic hyperreduction 1. ensure stability of hyperreduced system energy-stability for linear hyperbolic systems symmetry and positivity for linear diffusion systems r ν (u ν N, v) = 0 v V N can be solved. 7

10 Objectives Goal: stable, accurate, and systematic hyperreduction 1. ensure stability of hyperreduced system energy-stability for linear hyperbolic systems symmetry and positivity for linear diffusion systems r ν (u ν N, v) = 0 v V N can be solved. 2. quantitatively control solution error due to hyperreduction u N u ν N V δ. 7

11 Objectives Goal: stable, accurate, and systematic hyperreduction 1. ensure stability of hyperreduced system energy-stability for linear hyperbolic systems symmetry and positivity for linear diffusion systems r ν (u ν N, v) = 0 v V N can be solved. 2. quantitatively control solution error due to hyperreduction u N u ν N V δ. 3. train hyperreduced system (i.e., identify RB and EQP) in systematic manner r ν (, ) is found systematically. 7

12 Formulation: stable decomposition Overview Hyperbolic systems Diffusion systems (digest) Summary

13 Formulation: stable decomposition Overview Hyperbolic systems Diffusion systems (digest) Summary

14 Discontinuous Galerkin (DG) method DG: given a discontinuous FE space [Reed & Hill; Cockburn & Shu;... ] V h = {v L 2 (Ω) v P p (κ), κ T h }, find u h V h such that, v V h, r h (u h, v) = v F (u h ) dx + v ˆF (u h ) ds = 0. κ T κ h }{{} σ Σ σ h }{{} element integral facet integral Features: stability for conservation laws unstructured meshes hp flexibility u h σ Σ h κ T h 8

15 DG reduced basis (RB) method DG-RB (FE integ.): given a RB space for {u h (µ) µ D}, V N V h, find u N V N such that, v V N, r h (u N, v) = v F (u N ) dx + v ˆF (u N ) ds = 0. κ T κ h }{{} σ Σ σ h }{{} element integral facet integral Hyperreduction: find a hyperreduced form r ν (u N, v N ) r h (u N, v N ) with reduced number of element and facet integrals. 9

16 Hyperreduction of DG residual Step 1: find an element-wise decomposition stability r h (w, v) = η }{{} κ (w, v) }{{} κ T DG residual h element-wise residual such that for any (sparse) weights {ρ ν κ 0} κ Th r ν (w, v) ρ ν }{{} κη κ (w, v) κ T DG-EQP residual h is stable. Step 2. find EQP weights {ρ ν κ} that provides 1. sparsity: nnz{ρ ν κ} T h speed 2. accuracy: u N u ν N V δ accuracy 10

17 Formulation: stable decomposition Overview Hyperbolic systems Diffusion systems (digest) Summary

18 Linear hyperbolic system PDE: for Ω R d and I (0, T ], u t Energy balance: u 2 t 2dx Ω }{{} + Au = 0 in Ω I, (n A) u = (n A) u b on Ω I, u(t = 0) = u 0 in Ω {0}, change in energy = 2 u(n A) u b ds Ω }{{} net energy entering Ω F (u) = Au, A sym. 11

19 DG discretization Steady DG residual: r h (w, v) v F (w)dx + [v] + ˆF (w +, w ; n + )ds Ω Σ }{{}} i {{} volume term interior numerical flux + v + ˆF b (w + ; n + )ds, u Σ h } b {{} boundary numerical flux Energy balance: u 2 t 2dx + Ω }{{} change in energy Σ i [u] + n A [u] + ds }{{} facet jump dissipation: 0 = 2 σ Σ h κ T h u(n A) u b ds Ω }{{} net energy entering Ω 12

20 DG discretization Steady DG residual: r h (w, v) v F (w)dx + [v] + ˆF (w +, w ; n + )ds Ω Σ }{{}} i {{} volume term interior numerical flux + v + ˆF b (w + ; n + )ds, u Σ h } b {{} boundary numerical flux Energy balance: u 2 t 2dx Ω }{{} change in energy σ Σ h κ T h 2 u(n A) u b ds Ω }{{} net energy entering Ω Key: numerical flux provides energy stability. 12

21 Element-wise decomposition Naive decomp.: apply elemental mask to discrete DG residual η κ (w, v) r h (w, v κ ) v F (w)dx + v + ˆF b (w + ; n + )ds κ κ }{{}} b {{} volume term boundary numerical flux + v + ˆF (w +, w ; n + )ds. κ } i {{} one-sided interior numerical flux Element-wise energy: in general, η κ (v, v) 0 no energy stability. 13

22 Energy-stable elemental decomposition Energy-stable decomp.: redistribute interior facet residuals η κ (w, v) v F (w)dx + v + ˆF b (w + ; n + )ds κ κ }{{}} b {{} volume term boundary numerical flux + 1 [vn F (w; µ)] ± 4 ds + 1 [v] + κ 2 ˆF (w +, w ; n + ; µ)ds } i κ {{}} i {{} energy redistribution term split interior numerical flux Element-wise energy: for any v V h, η κ (v, v) 0 (modulo BC) energy stability. 14

23 DG-EQP discretization Steady DG-EQP residual: for weights {ρ ν κ 0} κ Th, r ν (w, v) κ T h ρ ν κη κ (w, v). Energy balance: r ν (u ν N, uν N ) 0 (modulo BC) and hence ( ) u ν t N 2 2dx 2 ρ ν κ u ν + N (n A) u b ds. κ T h κ T h ρ ν κ κ }{{} EQP approx of change in energy σ κ Σ b h } {{ } EQP approx of net energy entering Ω σ 15

24 Formulation: stable decomposition Overview Hyperbolic systems Diffusion systems (digest) Summary

25 Linear diffusion system PDE: for Ω R d Σ i h K u = 0 in Ω, } {{ } interior flux consistency term [c.f., Farhat et al, Carlberg et al.] u = u b on Ω. DG residual r h (w, v) v K wdx + ds Ω Σ }{{} b h }{{} volume term BC term { v K} w + v {K w}ds v {Kl σ ( w )}ds.. SPD: the bilinear form r h (, ) is 1. symmetric: r h (w, v) = r h (v, w) Σ i h } {{ } lifting term 2. positive: r h (v, v) 0. 16

26 Element-wise decomposition of DG residual Naive decomp.: apply elemental mask to discrete residual η κ (w, v) r h (w, v κ ). In general, η κ (, ) is neither symmetric nor non-negative. Structure-preserving decomp.: redistribute interior facet residuals η κ (w, v) = v K wdx + ds κ κ }{{}} b {{} volume term BC term 1 v K w + v K wds 1 v Kl σ ( w )ds 2 κ 2 } i κ {{}} i {{} redistributed interior flux consistency term redistributed lifting term For any w, v V h, η κ (, ) is symmetric and non-negative. 17

27 Formulation: stable decomposition Overview Hyperbolic systems Diffusion systems (digest) Summary

28 Summary: stable decomposition Stable decomp.: for any (sparse) weights {ρ ν κ 0} κ Th the hyperreduced system is r ν (w, v) ρ ν }{{} κη κ (w, v), κ T DG-EQP residual h 1. energy stable for linear hyperbolic systems 2. symmetric and positive for linear diffusion systems Key: naive elemental-mask decomp. stable decomp. 18

29 Formulation: hyperreduction error control

30 RB approximation: discrete form Reduced basis: Z N [φ 1,..., φ N ] RB residual: r N ( w }{{} state RB-EQP residual: ; µ }{{} i = param.) κ T h energy-stable elem. res. {}}{ η κ (Z N w, φ i ; µ) }{{} DG res. form: r h (Z N w,φ i ;µ) r ν N(w; µ) κ T h ρ ν κη N,κ (w; µ), i = 1,..., N = κ T h η N,κ (w; µ) i where EQP weight {ρ ν κ} κ Th achieves 1. sparsity: nnz{ρ ν κ} κ Th T h 2. accuracy: u N (µ) u ν N (µ) V δ, µ D note: control accuracy of solution not quadrature 19

31 RB-EQP: linear program (LP) RB-EQP: set δ δ/ N R >0. Find basic feasible solution ρ = arg min ρ subject to κ T h ρ κ 1. non-negativity: ρ κ 0, κ T h 2. constant accuracy: Ω κ T h ρ κ κ δ 3. manifold accuracy: define g N,κ (µ) J N (u train N (µ); µ) 1 η N,κ (u train N (µ); µ) ; }{{} Jacobian-preconditioned element residual for Ξ train J {µ train j } J j=1 and UJ train {u train N g N (µ) κ T h ρ κ g N,κ (µ) δ (µ)} µ Ξ train, J µ Ξ train J 20

32 RB-EQP: linear program (LP) matrix-vector form RB-EQP: set δ R 0. Find basic feasible solution subject to 1. non-negativity: ρ 0 ρ = arg min ρ R K h 1 T ρ 2. constant accuracy: { κ } T κ T h ρ < > Ω ± δ 3. manifold accuracy: define g N,κ (µ) J N (u train N (µ); µ) 1 η N,κ (u train N (µ); µ) ; }{{} Jacobian-preconditioned element residual g N,1 (µ train 1 )... g N,Kh (µ train 1 ) ρ < > g N,1 (µ train J )... g N,Kh (µ train J ) ρ Kh }{{} (N J) K h g N (µ train 1 ). g N (µ train J ) ± δ 21

33 RB-EQP: properties 1. simple: solved by (dual) simplex method 2. sparse: nnz{ρ ν κ} κ Th T h intuition: l 1 minimization 3. error control: under mild assumptions, if J N (µ) 1 r N (µ) J N (µ) 1 r ν N(µ)) }{{} δ, manifold accuracy constraints I J N (µ) 1 J ν N(µ) max ɛ, then u N (µ) u ν N(µ) V δ. Proof: application of Brezzi-Rappaz-Raviart (BRR). 22

34 RB-EQP: properties 1. simple: solved by (dual) simplex method 2. sparse: nnz{ρ ν κ} κ Th T h intuition: l 1 minimization 3. error control: under mild assumptions, if J N (µ) 1 r N (µ) J N (µ) 1 r ν N(µ)) }{{} δ, manifold accuracy constraints I J N (µ) 1 J ν N(µ) max ɛ, then u N (µ) u ν N(µ) V δ. Proof: application of Brezzi-Rappaz-Raviart (BRR). 22

35 Formulation: automated training

36 Simultaneous FE, RB, and EQP greedy training Input: training set Ξ train J Output: RB V N ; EQP weights {ρ ν κ} κ Th D; EQP tolerance δ; FE tolerance ɛ FE For N = 1,..., N max 1. Find µ N that maximizes the truth dual-norm µ N = arg sup r h (u ν N 1(µ); ; µ) (V h ) µ Ξ train J 2. Solve truth problem: u h (µ N ); enrich FE space as necessary. 3. Update reduced basis: Z N = {Z N 1, u h (µ N )} 4. Update EQP rule repeat N smooth times bootstrap Set U train J {u train N (µ) uν N (µ)} µ Ξ train J Solve LP-EQP for {Ξ train J, UJ train } 23

37 Computational cost per iteration 1. truth residual sampling and J Ξ train J 4. EQP update by bootstrapping J RB-EQP solutions O(JN ) J truth residual evaluations O(JN ) 2. truth snapshot calculation 1 truth solution evaluation O(N ) Note: solution evaluation residual evaluation 24

38 DG-FEM anisotropic mesh adaptation Employ Solve Estimate Aniso Mark Refine. Solve: high-order DG-FEM Estimate: dual-weighted residual (DWR) method Anisotropy Mark: error-to-dof optimization via local solves Refine: anisotropic hanging-node refinement 25

39 Related work

40 Most relevant related work (1/2) Nonlinear MOR: interpolate-then-integrate Gappy POD, MPE, EIM, BPIM, GNAT,... [Everson & Sirovich; Astrid et al; Barrault et al; Nguyen et al; Carlberg et al;... ] Nonlinear MOR: direct integration Hyperreduction [Ryckelynck] Optimal cubature [An et al] Energy-conservative sampling and weighting (ECSW) [Farhat et al] Empirical cubature method [Hernández] EQP for continuous Galerkin Empirical quadrature (for polynomials on arbitrary domains): LP framework [Ryu & Boyd; DeVore et al] 26

41 Most relevant related work (2/2) Structure-preserving/stability-aware MOR: ECSW for solids [Farhat et al] Matrix gappy POD for solids [Carlberg et al] Stable Galerkin ROM for fluids [Barone et al; Kalashinova et al] MOR for parametrized aerodynamics: Euler [LeGresley and Alonso] Euler and RANS [Washabaugh et al] Euler [Zimmermann and Görtz] DG-RB-EQP: hyperreduction with (i) energy stability, (ii) direct error control, (iii) simple LP, and (iv) adaptive snapshots. 27

42 NACA0012 Euler ONERA M6 Euler RAE2822 RANS MDA high-lift RANS Example

43 NACA0012 Euler ONERA M6 Euler RAE2822 RANS MDA high-lift RANS Example

44 NACA0012 Euler Equation: Euler (entropy variables) Parameters.: 1. angle of attack: α [0, 3 ] 2. Mach number: M [0.3, 0.5] (M loc,max > 0.8) 28

45 DG-RB-EQP setup: DG setup: goal-oriented adaptive P 2 DG output (c l ) tolerance: δ DG = 10 4 RB-EQP setup: Ξ train J : 5 5 uniform grid. L 2 (Ω) tolerance: δ EQP = 10 4 Note: DG controls output error but RB-EQP controls L 2 error; full goal-oriented error control is ongoing work. 29

46 NACA0012 Euler: mesh (µ = µ centroid ) Initial: P 2, dof = 13440, c l (5.2% error) Adapted: P 2, dof = 47112, c l counts (0.1% error) 30

47 c l error NACA0012 Euler: spatial error convergence 10-1 best-practice (p=1) ani-h adapt (p=2) % error % error dof For 1% error level, 2nd-order method on best-practice mesh: 400k dof p = 2 anisotropic-h adaptation: 16k dof ( 25 reduction) 31

48 NACA0012 Euler: DG-RB-EQP Reduced basis 1 4 (x-momentum) φ 1 φ 2 φ 3 φ 4 Integration weights: N = 4, K = 26 N = 10, K = 59 32

49 NACA0012 Euler: DG-RB-EQP convergence Assessment: max error over Ξ test Ξ train of Ξ test = 30 (δ = 10 4 ) { RB vs RB-EQP }} { N K u N u ν N L 2 (Ω)

50 NACA0012 Euler: DG-RB-EQP convergence Assessment: max error over Ξ test Ξ train of Ξ test = 30 (δ = 10 4 ) RB vs RB-EQP {}}{ u N u ν N L 2 (Ω) FE vs RB-EQP {}}{ u h u ν N L 2 (Ω) N K

51 NACA0012 Euler: DG-RB-EQP convergence Assessment: max error over Ξ test Ξ train of Ξ test = 30 (δ = 10 4 ) RB vs RB-EQP {}}{ u N u ν N L 2 (Ω) FE vs RB-EQP {}}{ u h u ν N L 2 (Ω) FE vs RB-EQP {}}{ J h J ν h N K

52 NACA0012 Euler: DG-RB-EQP convergence Assessment: max error over Ξ test Ξ train of Ξ test = 30 (δ = 10 4 ) RB vs RB-EQP {}}{ u N u ν N L 2 (Ω) FE vs RB-EQP {}}{ u h u ν N L 2 (Ω) FE vs RB-EQP {}}{ J h J ν h N K Online timing (1 thread): N = 10: t rb = 0.23 sec ( 160 speedup wrt adaptive DG) reduced residual, linear solve, & Newton iterations 33

53 NACA0012 Euler ONERA M6 Euler RAE2822 RANS MDA high-lift RANS Example

54 ONERA M6 Euler Equation: Euler (entropy variables) Parameters.: 1. angle of attack: α [0, 3 ] 2. Mach number: M [0.3, 0.5] 34

55 ONERA M6 Euler: DG-RB-EQP convergence Assessment: max error over Ξ test Ξ train of Ξ test = 30 (δ = 10 4 ) RB vs RB-EQP {}}{ u N u ν N L 2 (Ω) FE vs RB-EQP {}}{ u h u ν N L 2 (Ω) FE vs RB-EQP {}}{ J h J ν h N K Online timing (24 threads ): N = 10: t rb = 0.81 sec ( 130 speedup) DG-RB-EQP implementation is not correctly load balanced. 35

56 NACA0012 Euler ONERA M6 Euler RAE2822 RANS MDA high-lift RANS Example

57 RAE RANS UQ: problem setup Goal: combined model and discretization error quantification Equation: RANS equations with Spalart-Allmaras turbulence model Parameters: [Spalart & Allmaras; Schaefer et al] 1. turbulent Prandtl number: σ [0.60, 1.00] 2. Kármán constant: κ [0.38, 0.42] 3. 2nd wall destruction constant: c w2 [0.10, 0.35] 4. 3rd wall destruction constant: c w3 [1.75, 2.50] Flow cond.: M = 0.3, Re c = , α =

58 RANS-SA equations RANS mean-flow equations: (ρu) = 0 (ρu u + pi) = (2(µ + µ t )τ) (ρuh) = (c p ( µ + µt ) T + τu) Pr Pr for τ = 1 2 ( u + ut ) 1 3 tr( u)i, H =.... SA model: µ t = ρ νf v1 for f v1 = χ3 χ 3 +c 3 v1 (ρu ν) = ρ(c b1 (1 f t2 ) S ν c w1 f w ( ν d )2 ), χ = ν ν, where 1 (ρ(ν + ν) ν) c b2 σ σ ρ( ν)2 + 1 (ν + ν) ρ ν σ and S = S + ν κ 2 d 2 (1 χ 1+χf v1 ) f w = g( 1+c w3 6 g 6 +c w3 6 ) 1/6, g = r + c w2 (r 6 r), r = ν Sκ 2 d 2 c w1 = c b1 κ c b2 σ 37

59 RAE RANS: mesh (µ = µ centroid ) Initial: P 2, dof = 15180, c d counts (27% error) Adapted: P 2, dof = 41250, c d 90.1 counts (0.3% error) 38

60 c d error RAE RANS: spatial error convergence best-practice (p=1) ani-h adapt (p=2) % error % error dof For 0.3% error level, 2nd-order method on best-practice mesh: 1M dof p = 2 anisotropic-h adaptation: 40k dof ( 25 reduction) 39

61 RAE RANS: DG-RB-EQP Reduced basis 1 4 (x-momentum) Integration weights: N = 4, K = 24 N = 7, K = 55 40

62 RAE RANS: DG-RB-EQP convergence Assessment: max error over Ξ test Ξ train of Ξ test = 30 (δ = 10 4 ) RB vs RB-EQP {}}{ u N u ν N L 2 (Ω) FE vs RB-EQP {}}{ u h u ν N L 2 (Ω) FE vs RB-EQP {}}{ J h J ν h / J h N K Online timing (1 thread): N = 7: t rb = 0.77 sec ( 25 speedup wrt adaptive DG) 41

63 RAE RANS UQ: mean c d estimation Drag coefficient distribution (1000 MC solves) c d (counts) E[c d ] 90.4 counts online timing: 13 min (for 1000 solves) error estimate: 0.3% FE }{{} spatial + 0.4% RB-EQP }{{} model reduction % MC }{{} statistical 42

64 NACA0012 Euler ONERA M6 Euler RAE2822 RANS MDA high-lift RANS Example

65 MDA high-lift RANS Equation: RANS equations with Spalart-Allmaras turbulence model Parameters: [Spalart & Allmaras; Schaefer et al] 1. Kármán constant: κ [0.38, 0.42] 2. freestream turbulence intensity χ ν/ν [3, 30] Condition: M = 0.2, Re c = , α = 16 (M loc,max > 0.75) 43

66 MDA high-lift RANS: mesh (µ = µ centroid ) Initial: P 2, dof = , c d counts (47% error) Adapted: P 2, dof = , c d counts (0.1% error) 44

67 c d error MDA high-lift RANS: spatial error convergence 10 0 best-practice (p=1) ani-h adapt (p=2) 1% error 0.1% error dof For 1% error level, 2nd-order method on best-practice mesh: 25M dof p = 2 anisotropic-h adaptation: 250k dof ( 100 reduction) 45

68 MDA high-lift RANS: DG-RB-EQP Reduced basis 1 4 (x-momentum) Integration weights: L 2 (Ω) control output control N = , K =

69 MDA high-lift RANS: DG-RB-EQP convergence Assessment: max error over Ξ test Ξ train of Ξ test = 30 (δ = 10 4 ) RB vs RB-EQP {}}{ u N u ν N L 2 (Ω) FE vs RB-EQP {}}{ u h u ν N L 2 (Ω) FE vs RB-EQP {}}{ J h J ν h / J h N K Online timing (24 threads ): N = 10: t rb = 1.8 sec ( 70 speedup wrt adaptive DG) DG-RB-EQP implementation is not correctly load balanced. UQ: c d = counts 47

70 Summary

71 Summary A model reduction framework for conservation laws based on adaptive DG + RB + empirical quadrature (EQP) DG-RB-EQP provides energy stability (linear hyperbolic systems) and symmetry & positivity (linear diffusive systems) solution error control systematic offline training RB, EQP, and FE errors Ongoing work online & goal-oriented error estimates entropy stability treatment of parameter-dependent singularities 48