Hp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme

Transcription

1 Hp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme Aravind Balan, Michael Woopen and Georg May AICES Graduate School, RWTH Aachen University, Germany 22nd AIAA Computational Fluid Dynamics Conference, Dallas, Texas June 23rd, 2015 Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

2 Introduction Background and Motivation Theme of Present work: Adjoint-based anisotropic (h and hp) adaptation Context: Discontinuous Galerkin method for convection dominated flows Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

3 Introduction Scope of Present Work Anisotropic adaptation based on a continuous mesh model for triangular grids Use two error estimators: Density of continuous mesh is determined by adjoint-based estimate For triangle of given area, anisotropy follows from analytic minimization of the interpolation error in the L q -norm The latter is based on recently proposed error model and estimates 1 For hp adaptation, choose the configuration - polynomial degree and the anisotropy - with smallest error estimate. 1 V. Dolejsi., Applied Numerical Mathematics. 82, Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

4 Outline Proposed Adaptation Method 1 Proposed Adaptation Method 2 Numerical Results Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

5 Proposed Adaptation Method Mesh-Metric Duality The discrete setting: Metric tensors encode information about mesh elements e 1 e 2 h 2 h 1 θ The triangular element is equilateral w.r.t the norm induced by the metric. e k 2 M = ek T Me k = C, k = 1, 2, 3, The element is unit with respect to M Continuous Mesh: Reconstruct continuous metric Unit Mesh generated using length constraint in Riemannian geometry. Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

6 Proposed Adaptation Method Defining Size and Anisotropy Eigenvalue decomposition of the metric gives the size and shape of the triangle. ( ) T ( ) ( cos θ sin θ λ1 0 cos θ sin θ M = sin θ cos θ 0 λ 2 sin θ cos θ ). λ 1 = 1 h 2 1 λ 2 = 1 h2 2, The area of the triangle is given for C = 3, I = h 1h 2 We define the aspect ratio of the mesh element as The triangle is described by {I, β, θ} β = h 2 /h 1, Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

7 Proposed Adaptation Method Algorithm for h-adaptation 1 Find the implied metric of the current mesh. 2 Determine new area, I of mesh elements using adjoint-based error estimate. If the elemental error > the cut off error, then refine I = I c /r f If the elemental error < the cut off error, then coarsen I = I c c f 3 Determine anisotropy {β, θ} by minimization of error model in the L q norm. 4 {I, β, θ} is used to evaluate the desired metric on the current mesh and is fed to a metric-conforming mesh generator. 5 Do computation on new mesh 6 Done? Otherwise go to step 1 Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

8 Proposed Adaptation Method Error Model and Error bound Define the error function E, of degree p, as 1 p+1 ( ) p + 1 p+1 u( x) E x,p (x) = (p + 1)! l x l y p+1 l (x x)l (y ȳ) p+1 l l=0 Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

9 Proposed Adaptation Method Error Model and Error bound Define the error function E, of degree p, as 1 p+1 ( ) p + 1 p+1 u( x) E x,p (x) = (p + 1)! l x l y p+1 l (x x)l (y ȳ) p+1 l l=0 The error model in radial coordinates (x, y) = ( x cos φ, x sin φ) E x,p (x) = l=0 1 (p + 1)! u(p+1,φ) x x p+1 (1) where the directional derivative of order k is given by k ( ) k u (k,φ) k u = l x l y k l (cos φ) l (sin φ) k l Error model for interpolation of u(x) is that of Taylor series π x,p u(x) u(x) π x,p u(x) E x,p (x) Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

10 Proposed Adaptation Method Interpolation error function and bound A bound for the interpolation error function E x,p (x) can be written, in terms of 3 parameters {A p, ρ p, φ p } as ) p+1 E x,p (x) A p ((x x) T Q φp D ρp Qφ T 2 p (x x) x Ω where, [ cos φp sin φ Q φp = p sin φ p cos φ p ] [ 1 0, D ρp = 0 ρ 2 p+1 p ] Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

11 Proposed Adaptation Method Interpolation error function and bound A bound for the interpolation error function E x,p (x) can be written, in terms of 3 parameters {Ãp, ρ p, φ p } as E x,p (x) Ãp ( ) p+1 (x x) T Q φp D ρp Qφ T 2 p (x x) x Ω where, Q φp = [ ] [ cos φp sin φ p 1 0, D sin φ p cos φ ρp = p 0 ρ 2 p+1 p ] Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

12 Proposed Adaptation Method Interpolation error function and bound A bound for the interpolation error function E x,p (x) can be written, in terms of 3 parameters {Ãp, ρ p, φ p } as E x,p (x) Ãp ( ) p+1 (x x) T Q φp D ρp Qφ T 2 p (x x) x Ω where, and Q φp = [ ] [ cos φp sin φ p 1 0, D sin φ p cos φ ρp = p 0 ρ 2 p+1 p à p = 1 (p + 1)! u(p+1,φp) ρ p = u(p+1,φp) u (p+1,φp π 2 ) ] φ p = argmax φ u (p+1,φ) (2) Note: {A p, ρ p } modified from {Ãp, ρ p }, such that bound is safe (and sharp) Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

13 Proposed Adaptation Method Triangle with minimum interpolation error For given density, we can analytically find the optimal anisotropy. done by minimizing the bound in L q -norm taken over metric ellipse. The optimal anisotropy of the triangle is β = ρ 1 p+1 p, θ = φ p π/2, The minimum bound is given as Eˆx,p L q (K) ω where ω = c p,q A p ρ 0.5 p I ( p q ) and c p,q depends only on p and q. Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

14 Proposed Adaptation Method Hp-Adaptation Algorithm for Hp-Adaptation 1 Set the area I, of the triangle, κ using the adjoint error estimate. 2 do a dry run : Compute optimal triangle p = p κ 1, p κ, p κ Evaluate the error bound ω using {A p, ρ p, φ p } for the above three cases 4 Select, for each element κ, the p and the corresponding metric {I, β, θ}, which gives the smallest error bound ω. Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

15 Outline Numerical Results 1 Proposed Adaptation Method 2 Numerical Results Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

16 Solver details Numerical Results HDG scheme, with adjoint based estimator Damped Newton time integration GMRES, ILU(3) from PETSc as linear solver Netgen 2 as (isotropic) mesh generator, Ngsolve as FE library BAMG 3 as anisotropic mesh generator 2 Schöberl, Comput. Visual Sci., 1, F. Hecht, Technical report, Inria, France, 2006 Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

17 Verification of the bound The bound for the interpolation error is given as, Scalar boundary layer Eˆx,p (x) A p ((x ˆx) T Q φp D ρp Q T φ p (x ˆx) w(x, y) = ) p+1 2 ( ) ( ) x + ex/ɛ 1 y + ey/ɛ 1 1 e 1/ɛ 1 e 1/ɛ x Ω Figure: ɛ = 0.1 Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

18 Verification of the bound Interpolation Error and Numerical Bound with {Ãp, ρ p } Uniform Isotropic Refinement, ɛ = p = 1 error p = 1 bound L 2 norm Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

19 Verification of the bound Interpolation Error and Numerical Bound with {Ãp, ρ p } Uniform Isotropic Refinement, ɛ = p = 2 error p = 2 bound L 2 norm Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

20 Verification of the bound Interpolation Error and Numerical Bound with {Ãp, ρ p } Uniform Isotropic Refinement, ɛ = p = 3 error p = 3 bound L 2 norm Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

21 Verification of the bound Interpolation Error and Numerical Bound with {Ãp, ρ p } Uniform Isotropic Refinement, ɛ = p = 4 error p = 4 bound 10 5 L 2 norm Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

22 Verification of the bound Interpolation Error and Numerical Bound with {Ãp, ρ p } Uniform Isotropic Refinement, ɛ = 0.1 L 2 norm p = 1 error p = 1 bound p = 2 error p = 2 bound p = 3 error p = 3 bound p = 4 error p = 4 bound Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

23 Verification of the bound Interpolation Error, Numerical Bound with {Ãp, ρ p } and Modified Bound with {A p, ρ p } Adaptive Anisotropic Refinement, ɛ = p = 2 error p = 2 bound p = 2 modified bound L 2 norm Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

24 Verification of the bound Interpolation Error, Numerical Bound with {Ãp, ρ p } and Modified Bound with {A p, ρ p } Adaptive Anisotropic Refinement, ɛ = p = 3 error p = 3 bound p = 3 modified bound L 2 norm Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

25 Verification of the bound Interpolation Error, Numerical Bound with {Ãp, ρ p } and Modified Bound with {A p, ρ p } Adaptive Anisotropic Refinement, ɛ = p = 4 error p = 4 bound p = 4 modified bound L 2 norm Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

26 Verification of the bound Interpolation Error, Numerical Bound with {Ãp, ρ p } and Modified Numerical Bound with {A p, ρ p } Adaptive Anisotropic Refinement, ɛ = 0.1 L 2 norm p = 1 error p = 1 bound p = 1 modified bound p = 2 error p = 2 bound p = 2 modified bound p = 3 error p = 3 bound p = 3 modified bound p = 4 error p = 4 bound p = 4 modified bound Number of elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

27 Verification of the bound Element-wise distribution of interpolation error and numerical bound (sorted) Adapted Anisotropic Mesh, 1581 elements, ɛ = 0.1, p = L 2 norm 10 9 L 2 norm ,000 1,500 Element number Error vs. Numerical Bound with {Ãp, ρp} Error Function E x,p Bound ,000 1,500 Element number Error Function E x,p Modified Bound Error vs. Modified numerical bound with {A p, ρ p} Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

28 Verification of the bound Element-wise distribution of interpolation error and numerical bound (sorted) Adapted Anisotropic Mesh, 1450 elements, ɛ = 0.01, p = L 2 norm L 2 norm ,000 1,200 1,400 Element number Error vs. Numerical Bound with {Ãp, ρp} Error Function E x,p Bound ,000 1,200 1,400 Element number Error Function E x,p Modified Bound Error vs. Modified numerical bound with {A p, ρ p} Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

29 Verification of the bound Element-wise distribution of interpolation error and numerical bound (sorted) Adapted Anisotropic Mesh, 1402 elements, ɛ = 0.005, p = L 2 norm L 2 norm ,000 1,200 1,400 Element number Error vs. Numerical Bound with {Ãp, ρp} Error Function E x,p Bound ,000 1,200 1,400 Element number Error Function E x,p Modified Bound Error vs. Modified numerical bound with {A p, ρ p} Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

30 Scalar convection-diffusion 2D viscous Burger s equation with source We take the solution as (w, w) ɛ w = s (x, y) Ω = [0, 1] 2 w(x, y) = w(x, y) = 0 (x, y) Ω ( ) ( ) x + ex/ɛ 1 y + ey/ɛ 1, 1 e 1/ɛ 1 e 1/ɛ The target functional of interest is the weighted total boundary flux i.e. J = ψ(w ɛn q)dσ where the weighting function, ψ = cos(2πx) cos(2πy). Ω Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

31 Scalar convection-diffusion h-adaptation, p = 2, ɛ = Initial and adapted mesh Initial mesh elements h-adapted mesh, 2nd adaptation step elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

32 Scalar convection-diffusion h-adaptation, p = 2, ɛ = Solution on initial and h-adapted mesh Initial mesh elements Adapted mesh elements Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

33 Scalar convection-diffusion Comparison of isotropic and anisotropic adaptations p = 2 iso p = 2 J(w) J(wh) ndof Error Vs. Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

34 Scalar convection-diffusion Comparison of isotropic and anisotropic adaptations hp iso hp J(w) J(wh) ndof Error Vs. Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

35 Scalar convection-diffusion Comparison of h- and hp-adaptation, ɛ = Solution at same error level of 10 9 H-adapted mesh elements, p = 2 Hp-adapted mesh elements, p = 2,.., 6 Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

36 Flow over NACA0012 airfoil Target functional - Drag coefficient Flow cases : Subsonic viscous flow, M = 0.5, α = 1.0, Re = 5000 Subsonic inviscid flow, M = 0.5, α = 2 Transonic inviscid flow, M = 0.8, α = 1.25 Supersonic inviscid flow, M = 1.5, α = 0 Initial Mesh elements, Far field at a radius of 1000 chords. Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

37 Subsonic viscous flow over NACA0012 h-adaptation, p = 2 Adapted mesh and Mach number Adapted mesh elements Mach number Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

38 Subsonic viscous flow over NACA0012 h-adaptation, p = 2 Capture of flow recirculation Adapted mesh Streamline Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

39 Subsonic viscous flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations. For hp-adaptation, p = 1,.., p = 2 iso p = Cd ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

40 Subsonic viscous flow over NACA0012 hp-adaptation Capture of flow recirculation Adapted mesh elements Streamline Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

41 Subsonic viscous flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations. For hp-adaptation, p = 1,.., hp iso hp 10 4 Cd ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

42 Subsonic viscous flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations. For hp-adaptation, p = 1,.., 6 Cd p = 1 iso p = 2 iso p = 3 iso hp iso p = 1 p = 2 p = 3 hp ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

43 Inviscid subsonic flow over NACA0012 Hp-Adaptation, p = 1,..., 6 Adapted mesh elements Mach number Polynomial map Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

44 Inviscid subsonic flow over NACA0012 Comparison of Anisotropic and Isotropic Hp-adaptation Polynomial map Anisotropic adaptation, using Interpolation error Isotropic adaptation, using Jump indicator Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

45 Inviscid subsonic flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations. For hp-adaptation, p = 1,.., 6 Cd p = 1 iso p = 2 iso p = 3 iso hp iso p = 1 p = 2 p = 3 hp ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

46 Inviscid transonic flow over NACA0012 H-Adaptation, p = 2 Adapted mesh and Mach number Adapted mesh elements Mach number Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

47 Inviscid transonic flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations p = 2 iso p = 2 Cd ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

48 Inviscid transonic flow over NACA0012 Hp-Adaptation, p = 1, 2,..., 6 Adapted mesh, 4258 elements Mach number and Polynomial map Mach number Polynomial map Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

49 Inviscid transonic flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations. For hp-adaptation, p = 1,.., 6 hp iso hp 10 3 Cd ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

50 Inviscid transonic flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations. For hp-adaptation, p = 1,.., 6 Cd p = 1 iso p = 2 iso p = 3 iso hp iso p = 1 p = 2 p = 3 hp ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

51 Inviscid supersonic flow over NACA0012 H-Adaptation, p = 2 Adapted mesh and Mach number Adapted mesh elements Mach number Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

52 Inviscid supersonic flow over NACA0012 H-Adaptation, p = 2 Adapted mesh near bow-shock Adapted mesh Adapted mesh Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

53 Inviscid supersonic flow over NACA0012 Hp-Adaptation, p = 1,..., 6 Mach number and Polynomial map (8267 elements) Mach number Polynomial map Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

54 Inviscid supersonic flow over NACA0012 Comparison of error convergence for isotropic and anisotropic adaptations. For hp-adaptation, p = 1,.., 6 Cd p = 1 iso p = 2 iso p = 3 iso hp iso p = 1 p = 2 p = 3 hp ndof Error in drag coefficient Vs Degrees of freedom Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

55 Conclusions Numerical Results Interpolation bound and verification Anisotropic refinement - superior to isotropic refinement Hp-refinement - superior to pure h-refinement Future work : Extension to 3D Global optimization Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15

56 Acknowledgement Numerical Results Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111 is gratefully acknowledged Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, / 15