An Optimization-based Framework for Controlling Discretization Error through Anisotropic h-adaptation

Transcription

1 An Optimization-based Framework for Controlling Discretization Error through Anisotropic h-adaptation Masayuki Yano and David Darmofal Aerospace Computational Design Laboratory Massachusetts Institute of Technology Acknowledgement: Dr. Steven Allmaras, ProjectX Team, The Boeing Company March 24, 2011 M. Yano (MIT) FEF 2011 March 24, / 22

2 Outline 1 Introduction 2 Optimization Framework : Anisotropic h-adaptation 3 Results : L 2 Error Control 4 Results : Functional Output Error Control 5 Conclusion M. Yano (MIT) FEF 2011 March 24, / 22

3 Objective and Approach Objective: Realize the full potential of high order methods through automated mesh adaptation and error control Our Choice of technologies: Discontinuous Galerkin (DG) method hp-flexible on unstructured meshes Regularity limited convergence - shocks, geometric singularities, turbulence models Adjoint-based error estimator Error estimate / localization for output of interest (e.g. lift, drag) Simplex mesh adaptation Complex geometries Arbitrarily oriented anisotropy M. Yano (MIT) FEF 2011 March 24, / 22

4 Functional Error Estimation and Localization Problem: Find J(u) with u V s.t. R(u, v) = 0 v V Error representation formula [Becker, 2001] E J h,p (u h,p ) J(u) = R h,p (u h,p, ψ), Local error indicator η K R h,p (u h,p, ψ h,p+1 K ) Error indicator assigns 1 value to each element Only sufficient for 1 parameter adaptation ψ = adjoint type n param parameters η K sufficient isotropic h 1 size isotropic hp 2 size, poly. order anisotropic h 3 (6) size, shape anisotropic hp 4 (7) size, shape, poly. order M. Yano (MIT) FEF 2011 March 24, / 22

5 Review of Anisotropic h-adaptation Need additional info to make anisotropy decision Previous work on output-based anisotropic h-adaptation Method p > 1 ψ ani. simplex Mach Hessian [Venditti, 2003] Mach Hessian [Fidkowski, 2007] Mach jump [Hartmann, 2008] Edge comp. ref [Park, 2008][Sun, 2009] Comp. ref. [Georgoulis, 2009][Ceze, 2010] Flux reconstruction + C.V. [Loseille, 2010] Anisotropic error estimate [Richter, 2010] Contribution : Anisotropic h-adaptation framework for simplex meshes that can be used with any error indicator M. Yano (MIT) FEF 2011 March 24, / 22

6 Outline 1 Introduction 2 Optimization Framework : Anisotropic h-adaptation 3 Results : L 2 Error Control 4 Results : Functional Output Error Control 5 Conclusion M. Yano (MIT) FEF 2011 March 24, / 22

7 Problem Definition / Example Objective: Develop optimal anisotropic mesh Example problem: L 2 -projection error control of boundary layer ( E D (T h ) = u u h,p 2 L 2 (Ω) where u = exp x ) 1 δ C D (T h ) = DOF(T h ) u = exp( x 1 /δ) optimal isotropic mesh optimal anisotropic mesh (δ = 0.01) E 1 2 = E 1 2 = *each mesh has 200 elements (p = 3) M. Yano (MIT) FEF 2011 March 24, / 22

8 Continuous Relaxation (Intractable) discrete optimization problem T h,opt = arg min T h E D (T h ) s.t. C D (T h ) = DOF target Approximability of T h encoded in metric tensor field M(x) Sym + d Continuous relaxation (d d SPD matrix) M opt = arg min M E(M) s.t. C(M) = DOF target Mesh Gen. Metric Field M Implied Metric Mesh T h M. Yano (MIT) FEF 2011 March 24, / 22

9 Continuous Optimization Problem Statement: Seek optimal metric field M opt = arg min M E(M) s.t. C(M) = DOF target Assume locality of error and cost functionals E(M) = e(x, M(x))dx and C(M) = Ω local error function : e(, ) : R d Sym + d R+ local cost function : c(m) = C p det(m) First order optimality Ω c(m(x))dx e c (x, M(x)) λ (x, M(x)) = 0 M M x Ω Need to estimate behavior of e(m) M. Yano (MIT) FEF 2011 March 24, / 22

10 Local Sampling Split edges of K 0 to form K i s, i = 0,..., m Obtain metric M Ki implied by split config. Obtain η Ki Freeze neighbor states Solve local problem Associate η Ki e(m Ki ) K i original edge 1 split edge 2 split edge 3 split uniform split {M K0, η K0 } {M K1, η K1 } {M K2, η K2 } {M K3, η K3 } {M Ku, η Ku } M. Yano (MIT) FEF 2011 March 24, / 22

11 Manipulation of SPD Tensor Space Original problem on Sym + d e c (x, M) λ (x, M) = 0 M M M Sym+ d SPD tensors define cone in Euclidean space Hard to manipulate SPD tensors directly Consider substitutions M(Σ) = M 1/2 0 exp(σ)m 1/2 0 M(Σ) Sym + d Σ Sym d e(f ) = e 0 exp(f ) e(f ) R + f R c(g) = c 0 exp(g) c(g) R + g R New problem on Sym d f e 0 Σ (x, Σ) λc g 0 (x, Σ) = 0 Σ Σ Manipulation of symmetric tensors Approximate f (Σ) and move toward optimality Sym d M. Yano (MIT) FEF 2011 March 24, / 22

12 Local Error Model / Parameter Identification Construct linear approximation of f (Σ) f (Σ) = tr(rσ), R Sym d = rate tensor R is generalization of convergence rate to tensor space tr(r) : sensitivity to uniform scaling R tr d (R)I : sensitivity to shape change Identify R from local samples {M Ki, η Ki } m i=1 Σ K1 R f (Σ K1 ) 11. 2R 12 =. Σ Km R 22 f (Σ Km ) Σ(K i ) = log(m 1/2 0 M Ki M 1/2 0 ) and f (Σ Ki ) = log(η Ki /η K0 ) M. Yano (MIT) FEF 2011 March 24, / 22

13 Discrete Optimality Condition Discrete Optimality Conditions (Anisotropic) Global isotropic error balance : λ R s.t. η K tr(r K ) λ d 2 dof(k ) = 0 Local anisotropic optimality : R K I K T h K T h e.g. R I : more reduction by increasing M 11 than M 22 decrease h 1 and increase h 2 R K,11 R K,22 M. Yano (MIT) FEF 2011 March 24, / 22

14 Algorithm 1 Obtain primal solution u h,p on T (i) h,p 2 Obtain adjoint solution ψ h,p+1 on T (i) h,p+1 (for DWR) 3 Solve local problems and obtain {M Ki, η Ki } m (i) i=0, K T h 4 Construct local error function e(m) ẽ(m; {M Ki, η Ki } m i=0 ) 5 Update metric field M (i+1) 6 Generate new mesh T (i+1) from M (i+1) and go to 1 Adapt 0 : E 1 2 = Adapt 2 : E 1 2 = Adapt 7 : E 1 2 = *each mesh has 200 elements (p = 3) M. Yano (MIT) FEF 2011 March 24, / 22

15 Outline 1 Introduction 2 Optimization Framework : Anisotropic h-adaptation 3 Results : L 2 Error Control 4 Results : Functional Output Error Control 5 Conclusion M. Yano (MIT) FEF 2011 March 24, / 22

16 L 2 Error Optimization Objective: Test algorithm for L 2 -projection error control: E D (T h ) = min u h,p V h,p (T h ) u u h,p 2 L 2 (Ω) Key features Error locality assumption holds Analytical solution obtainable via calculus of variation Functions considered r α -type singularity boundary layer u = r α sin [ α ( )] θ + π 2 (α = 2/3) u = exp( x1 /δ) (δ = 0.01) M. Yano (MIT) FEF 2011 March 24, / 22

17 L 2 Problem 1 : r α -type Singularity Optimal mesh consists of isotropic elements Analytical optimal h distribution h opt (r) = Cr k opt where k opt = 1 α + 1 p + 2 Stronger grading as α decreases or p increases h 10 1 k=0.43 k= k=0.44 k= dof=500 dof=1000 dof=2000 dof= r h k=0.67 k=0.67 k=0.67 k=0.68 dof=500 dof=1000 dof=2000 dof= r p = 1 (k opt = 0.44) p = 3 (k opt = 0.67) M. Yano (MIT) FEF 2011 March 24, / 22

18 L 2 Problem 2 : Boundary Layer Optimal mesh consists of anisotropic elements Analytical optimal h distribution h,opt = C exp ( k,opt x ) where k,opt = Weaker grading for higher p 1 δ(p + 3/2) 10 1 k perp =43.2 k perp = k perp =43.5 k =42.8 perp 10 1 k perp =24.9 k perp = k perp =22.9 k =22.7 perp h h dof=500 dof=1000 dof=2000 dof= x dof=1000 dof=2000 dof=4000 dof= x p = 1 (k,opt = 40.0) p = 3 (k,opt = 22.2) M. Yano (MIT) FEF 2011 March 24, / 22

19 Outline 1 Introduction 2 Optimization Framework : Anisotropic h-adaptation 3 Results : L 2 Error Control 4 Results : Functional Output Error Control 5 Conclusion M. Yano (MIT) FEF 2011 March 24, / 22

20 RAE2822 Transonic Airfoil RANS-SA : M = 0.729, Re c = , α = 2.31 ; p = 2 Objective: Application to transsonic turbulent flow Functional : Drag Comparison : Mach-based anisotropy ( p+1 M) Mach Drag error 10 3 mach ani. unified 10 4 Mass adjoint J J ref dof M. Yano (MIT) FEF 2011 March 24, / 22

21 RAE2822 Transonic Airfoil RANS-SA : M = 0.729, Re c = , α = 2.31 ; p = 2 Anisotropic resolution of features Primal: BLs, wake, shock Adjoint: BLs, stagnation streamline, shock-induced perturbations Unified (p = 2, dof = 60k) Mach anisotropy (p = 2, dof = 60k) M. Yano (MIT) FEF 2011 March 24, / 22

22 NACA0006 Shock Propagation Euler : M = 2.0; p = 2 Objective: Application to sonic-boom prediction Functional : Pressure 50c below airfoil, J = l (p p ) 2 ds Pressure Pressure line output error mach ani. unified Mass adjoint J J ref / J ref dof M. Yano (MIT) FEF 2011 March 24, / 22

23 NACA0006 Shock Propagation Euler : M = 2.0; p = 2 Unified (p = 2, dof = 40k) Mach anisotropy (p = 2, dof = 40k) M. Yano (MIT) FEF 2011 March 24, / 22

24 Three-Element MDA Airfoil RANS-SA : M = 0.2, Re c = , α = 8.10 ; p = 2 Objective: Application to complex turbulent flow Functional : Drag Mach Drag error 10 3 mach ani. unified Mass adjoint J J ref dof M. Yano (MIT) FEF 2011 March 24, / 22

25 Three-Element MDA Airfoil M = 0.2, Re c = , α = 8.10 ; p = 2 Unified (p = 2, dof = 90k) Mach anisotropy (p = 2, dof = 90k) M. Yano (MIT) FEF 2011 March 24, / 22

26 Outline 1 Introduction 2 Optimization Framework : Anisotropic h-adaptation 3 Results : L 2 Error Control 4 Results : Functional Output Error Control 5 Conclusion M. Yano (MIT) FEF 2011 March 24, / 22

27 Conclusion and Ongoing Work Conclusion: Presented optimization based framework for anisotropic h-adaptation Verified optimality of algorithm for L 2 error control Compared performance of algorithm against primal derivative based anisotropy detection Ongoing Work: Extension to hp-adaptation Detailed analysis of error locality assumption Improving robustness of error estimator for highly nonlinear problems M. Yano (MIT) FEF 2011 March 24, / 22