Model reduction of parametrized aerodynamic flows: discontinuous Galerkin RB empirical quadrature procedure. Masayuki Yano

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1 Model reduction of parametrized aerodynamic flows: discontinuous Galerkin RB empirical quadrature procedure Masayuki Yano University of Toronto Institute for Aerospace Studies Acknowledgments: Anthony Patera; NSERC MoRePaS 2018 Nantes, France 13 April 2018

Motivation: parametrized aerodynamic flows Goal:

aerodynamic flows in real-time/many-query scenarios.

2 Motivation: parametrized aerodynamic flows Goal: rapid and reliable solution of parametrized aerodynamic flows in real-time/many-query scenarios. Challenges: nonlinearity limited stability wide range of scales long-time integration parameter-dependent irregularities providing geometric/topological flexibility 1

3 Motivation: parametrized aerodynamic flows Goal: rapid and reliable solution of parametrized aerodynamic flows in real-time/many-query scenarios. Challenges: nonlinearity limited stability wide range of scales long-time integration parameter-dependent irregularities providing geometric/topological flexibility 1

4 Parametrized steady conservation laws µpde: given µ D R P, find u(µ) V s.t. v F (u(µ); µ)dx + = 0 v V. Ω O( ) 2

5 Parametrized steady conservation laws µpde: given µ D R P, find u(µ) V s.t. v F (u(µ); µ)dx + = 0 v V. Ω O( ) 2

6 Parametrized steady conservation laws µpde: given µ D R P, find u(µ) V s.t. v F (u(µ); µ)dx + = 0 v V. Ω O( ) FEM: 1. space: N -dim space V h V 2. integral: K h -point quadrature K h k=1 ρh k [ ] x h k Ω [ ]dx; given µ D, find u h (µ) V h s.t. O(N, K h ) K h k=1 ρ h k[ v h F (u h (µ); µ)] x h k = 0 v V h. 2

7 Model reduction for parametrized systems RB space: N N -dim empirical space for parametrized functions V N V h V; RB (FE quad.): given µ D, find u N (µ) V N s.t. O(N, K h ) K h k=1 ρ h k [ v N F (u N (µ); µ)] x h }{{} k = 0 v N V N. parametrized integrand g(µ;x h k ) Goal: find sparse empirical quadrature for parametrized integrals with quantitative error control. 3

8 Formulation Empirical quadrature procedure (EQP) RB-EQP method Offline training DG-RB-EQP method Related work

9 Formulation Empirical quadrature procedure (EQP) RB-EQP method Offline training DG-RB-EQP method Related work

10 Empirical quadrature procedure (EQP): idea Exact: given µ D, find I(µ) Gauss quadrature: I h (µ) Empirical quadrature: where {x ν l, ρν l } achieves K h k=1 Ω g(µ; x)dx. ρ h k }{{} weights g(µ; x h k }{{} points K ν I ν (µ) ρ ν l g(µ; x ν l ) l=1 1. sparsity: {x ν l }Kν l=1 {xh k }Kh k=1, Kν K h 2. accuracy: I h (µ) I ν (µ) δ, µ D. ). general functions parametric manifold {g(µ)} µ D 4

11 EQP for manifolds: linear program (LP) Idea: keep {x h k }, but assign {ρ k } that is mostly zero. LP-EQP. Set δ R 0. Find basic feasible solution subject to ρ = arg min ρ R Kh 1. non-negativity: ρ k 0, k 2. constant accuracy: Ω K h k=1 ρ k K h ρ k k=1 δ 3. manifold accuracy: for all µ {µ train j } J j=1, K h I h (µ) ρ k g(µ; x h k) δ; k=1 extract K ν K h nonzero weights {x h k, ρ k }Kh k=1 {xν l, ρν l }Kν l=1. 5

12 EQP for manifolds: LP matrix-vector form Idea: keep {x h k }, but assign {ρ k } that is mostly zero. LP-EQP. Set δ R 0. Find basic feasible solution subject to 1. non-negativity: ρ 0 ρ = arg min 1 T ρ ρ R Kh 2. constant accuracy: 1 T ρ < > Ω ± δ 3. manifold accuracy: g(µ train 1 ; x h 1)... g(µ train 1 ; x h ) K..... } g(µ train J ; x h 1)... {{ g(µ train J ; x h ) K h } J K h ρ 1. ρ K h < > I h (µ train 1 ) ± δ. I h (µ train J ) ± δ 6

13 EQP for manifolds: properties 1. simple: solved by (dual) simplex method 2. sparse: K ν K h intuition: l 1 minimization 3. error control: I h (µ) I ν (µ) δ + 2 Ω L g min µ µ train j 2 j }{{} 0 as J (sharper estimate available) 7

14 Formulation Empirical quadrature procedure (EQP) RB-EQP method Offline training DG-RB-EQP method Related work

15 RB approximation Reduced basis: Z N [φ 1 V h,..., φ N V h ] RB residual: i = 1,..., N R N ( w }{{} state ; µ }{{} param RB: find u N (µ) R N s.t. K h ) i set u N (µ) = Z N u N (µ) V N. k=1 ρ h k [ φ i F (Z N w; µ)] x h k }{{} integrand: r N (w; µ; x h k ) i R N (u N (µ); µ) = 0 in R N ; 8

16 RB-EQP: idea RB residual: RB-EQP residual: where {x ν l, ρν l } achieves K h R N (w; µ) ρ h kr N (w; µ; x h k) in R N. k=1 K ν RN(w; ν µ) ρ ν l r N (w; µ; x ν l ) l=1 1. sparsity: {x ν l }Kν l=1 {xh k }Kh k=1, Kν K h 2. accuracy: u N (µ) u ν N (µ) V δ, µ D FE quadrature {x h k, ρh k }Kh k=1 in R N note: control accuracy of solution not quadrature 9

17 RB-EQP: linear program (LP) RB-EQP. Set δ δ/(2 N) R >0. Find basic feasible solution K h ρ = arg min ρ i ρ R Kh subject to 1. non-negativity: ρ k 0, k 2. constant accuracy: Ω K h 3. manifold accuracy: for Ξ train J k=1 ρ k {µ train j } J j=1 and UJ train i=1 δ {u train N (µ)} µ Ξ train, J J N (u train N (µ); µ) }{{} Jacobian R N N ρ k r N (u train N (µ); µ; x h k) δ 1 Kh k=1 } {{ } residual R N µ Ξ train J. 10

18 RB-EQP: properties 1. simple: solved by (dual) simplex method 2. sparse: K ν K h intuition: l 1 minimization 3. error control: under mild assumptions, if then J N (u train N (µ); µ) 1 RN(u ν train N (µ); µ) }{{} δ, manifold accuracy constraints J N (u train N (µ); µ) 1 JN(u ν train N (µ); µ) I max ɛ, u N (µ) u ν N(µ) V δ 1 Nɛ δ. Proof: application of Brezzi-Rappaz-Raviart (BRR). 11

19 RB-EQP: properties 1. simple: solved by (dual) simplex method 2. sparse: K ν K h intuition: l 1 minimization 3. error control: under mild assumptions, if then J N (u train N (µ); µ) 1 RN(u ν train N (µ); µ) }{{} δ, manifold accuracy constraints J N (u train N (µ); µ) 1 JN(u ν train N (µ); µ) I max ɛ, u N (µ) u ν N(µ) V δ 1 Nɛ δ. Proof: application of Brezzi-Rappaz-Raviart (BRR). 11

20 Formulation Empirical quadrature procedure (EQP) RB-EQP method Offline training DG-RB-EQP method Related work

21 Simultaneous RB and EQP greedy training Input: training set Ξ train J D; EQP tolerance δ Output: RB V N ; EQP weights {x ν k, ρν k }Kν k=1 For N = 1,..., N max 1. Find µ N that maximizes the truth dual-norm µ N = arg sup R(u ν N 1(µ); ; µ) (V h ) µ Ξ train J 2. Solve truth problem: u h (µ N ) 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 } 12

22 Formulation Empirical quadrature procedure (EQP) RB-EQP method Offline training DG-RB-EQP method Related work

23 Adaptive discontinuous Galerkin (DG) method [Reed & Hill; Cockburn & Shu;... ] DG: find u h V h (discontinuous) s.t. v F (u h (µ); µ)dx + v F (u h ; µ)ds = 0 v V h. κ T h κ }{{} σ Σ h σ }{{} element facet stability for conservation laws unstructured meshes high compute-to-memory ratio y x Adaptive DG: u h (x, y) (dual-weighted) residual error estimate [Becker & Rannacher;... ] anisotropic/hp adaptive mesh refinement T h 13

24 DG-RB-EQP: overview DG-RB-EQP residual: R ν N(w; µ) i φ i F (Z N w; µ)dx + ρ ν κ ρ ν σ κ T ν T h κ }{{} σ Σ ν Σ h element fast DG kernel leveraged by element/facet grouping φ i F (Z N w; µ)ds σ } {{ } facet energy stability: u ν N (t > 0) E u ν N (t = 0) E + (B.C.) symmetrized split of convection term with Robin conditions EQP LP finds {ρ ν κ} κ T ν T h and {ρν σ} σ Σ ν Σ h 14

25 Formulation Empirical quadrature procedure (EQP) RB-EQP method Offline training DG-RB-EQP method Related work

26 Most relevant related work 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 APHR ( φ i as indicator) [Ryckelynck] ECSW (l 0 sparse NNLS) [Farhat et al] Empirical quadrature (for polynomials on arbitrary domains): l 2 framework [An et al] LP framework [Ryu & Boyd; DeVore et al] EQP/RB-EQP: LP framework (i) for parametric manifolds (ii) with error control. 15

27 Example

Reliable UQ: problem setup Goal: quantify effects of turbulence model error. Gov. eq.: Reynolds-averaged Navier-Stokes (RANS) equations with Spalart-Allmaras (SA) turbulence model Parameters: 1.

28 Reliable UQ: problem setup Goal: quantify effects of turbulence model error. Gov. eq.: Reynolds-averaged Navier-Stokes (RANS) equations with Spalart-Allmaras (SA) turbulence model Parameters: 1. turbulent Prandtl number: σ [0.60, 1.00] 2. Kármán constant: κ [0.35, 0.42] 3. 2nd wall constant: c w2 [0.10, 0.35] 4. 3rd wall constant: c w3 [1.75, 2.50] [Spalart & Allmaras; Schaefer et al] Flow condition: M = 0.3, Re c = , α =

29 Steady compressible RANS-SA equations (fully turbulent) 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 17

30 Reliable UQ: requirements Sources of discretization error: 1. spatial discretization error anisotropic adaptive high-order DG method 2. model reduction error (reduced basis + quadrature) DG-RB-EQP 3. statistical sampling error large Monte-Carlo sample by the rapid solver (more efficient approaches exist; e.g., PC, MLMC) 18

31 Spatial adaptivity: mesh Initial: P 2, dof = 15180, c d counts (27% error) Adapted: P 2, dof = 39450, c d 90.1 counts (0.3% error) 19

32 c d error Spatial adaptivity: 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 speedup) 20

Model reduction: DG-RB-EQP Convergence: #dof #elem #facet rel. output error 1 7 3 2.16 10 2 reduced 4 22 19 6.

33 Model reduction: DG-RB-EQP Convergence: #dof #elem #facet rel. output error reduced adaptive DG Reduced basis 1 4 (x-momentum) #dof = 7, #elem = , #facet =

34 Model reduction: DG-RB-EQP Convergence: #dof #elem #facet rel. output error reduced adaptive DG Integration weights: #dof = 7, #elem = , #facet =

35 Model reduction: DG-RB-EQP Convergence: Timing: #dof #elem #facet rel. output error reduced adaptive DG t RB 1.3 CPU-sec/solve 25 reduction wrt adaptive DG 600 wrt 2nd-order method on best-practice mesh 21

36 Reliable UQ: mean c d estimation Drag coefficient distribution (1000 MC solves) E[c d ] 90.4 counts c d (counts) online timing: 20 min (for 1000 solves) error estimate: 0.3% FE }{{} spatial + 0.3% RB-EQP }{{} model reduction (offline informed not online bounds/estimates) % MC }{{} statistical 22

37 Summary

38 Summary Empirical quadrature procedure (EQP) for manifolds that provides simplicity: LP solved by (dual) simplex method sparsity: K ν K h error control: integral (EQP) RB solution (RB-EQP, DG-RB-EQP) systematic offline training RB, EQP, and FE errors limitation: no online error bounds/estimates 23

39 Supplementary materials

40 DG: energy stability Convection: split element-facet integrals as follows: ηκ conv 1 v F (w) + 2 v+ n + F (w)ds κ κ η conv σ σ ( v ˆF (w +, w ; ˆn) v+ n + F (w + ) v n F (w ) ) ds; for ρ R T h 0 and τ R Σh 0, DG-RB-EQP provides energy stability: u(t > 0) A u(t = 0) A (modulo BC) Diffusion: natural element/facet splitting results in semi-positivity. 24

41 Problem setup: neo-hookean beam with a hole Rubber-like neo-hookean beam subjected to self-weight Parameters: 1. Poisson ratio: ν [0.35, 0.45] 2. gravity angle: θ g [ 90, 90 ] Quantity of interest: strain energy density on Ω annulus 25

42 Problem setup: neo-hookean beam with a hole Rubber-like neo-hookean beam subjected to self-weight Parameters: 1. Poisson ratio: ν [0.35, 0.45] 2. gravity angle: θ g [ 90, 90 ] Quantity of interest: strain energy density on Ω annulus linear neo-hookean

43 EQP: setup Simultaneous RB-EQP greedy training parameter set: Ξ train, 31 5 uniform points EQP tolerance: δ = 10 3 RB residual tolerance: ɛ = 10 2 Integration problems: 1. RB-EQP for solution field: µ u ν N (µ) 2. EQP for output integral: u ν N (µ) sν N (uν N (µ)) s ν N(u ν N) = Ψ(u ν N(µ))(x)dx Ω annulus 26

44 EQP: quadrature points Residual Output N = 7, K ν = 44 K h = 6878 N = 7, K ν = 14 K h =

45 EQP: error control & convergence Assessment: maximum relative error over Ξ test = 50 (δ = 10 3 ). Field: Output: N N K ν K o,ν truth integ vs EQP integ {}}{ s(u ν N) s ν (u ν N) / s(u ν N)

46 EQP: error control & convergence Assessment: maximum relative error over Ξ test = 50 (δ = 10 3 ). Field: Output: N truth RB vs RB-EQP {}}{ N K ν u N u ν N V / u N V K o,ν truth integ vs EQP integ {}}{ s(u ν N) s ν (u ν N) / s(u ν N)

47 EQP: error control & convergence Assessment: maximum relative error over Ξ test = 50 (δ = 10 3 ). Field: Output: N truth RB vs RB-EQP {}}{ u N u ν N V / u N V FE vs RB-EQP {}}{ u h u ν N V / u h V N K ν K o,ν truth integ vs EQP integ {}}{ s(u ν N) s ν (u ν N) / s(u ν N) FE output vs RB-EQP output {}}{ s(u h ) s ν (u ν N) / s(u h ) Speedup: 60 in wall-clock time. 28

48 Computational cost per iteration truth dual-norm residual sampling J RB-EQP solutions O(JN ) J truth residual evaluations O(JN ) J Ξ train J truth snapshot calculation 1 truth solution evaluation O(N ) note: solution evaluation residual evaluation EQP update by bootstrapping J RB-EQP solutions O(JN ) J truth residual evaluations O(JN ) 29

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

Model reduction of parametrized aerodynamic flows: stability, error control, and empirical quadrature Masayuki Yano University of Toronto Institute for Aerospace Studies Acknowledgments: Anthony Patera;