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Accurate Methods for Computing High Mach Number Reactive Flow Matthew J. Zahr and Joseph M. Powers Department of Aerospace and Mechanical Engineering University of Notre Dame, USA Lawrence Livermore National Laboratory Livermore, California

  1. Reaction zone structure normal to shock 3 ρ (kg/ m ) λ 1 4 0.8 a) 3 b) 0.6 2 0.4 1 0.2 X (m) 4 X (m) 0.0 0 -1 0 1 2 3 4 -1 0 1 2 3 T(K) MX 600 3 500 c) 2 d) 400 1 300 X (m) X (m) 200 0 -1 0 1 2 3 4 -1 0 1 2 3 4 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 18 / 62

  2. Exact solution: streamlines y (m) straight oblique 4 shock, β = π / 4 3 Curved streamlines identical to wedge contour. 2 9 . 0 ~ λ Streamline curvature approaches zero as reaction 1 curved completes. wedge 0 x (m) 0 1 2 3 4 Powers and Aslam, AIAA Journal , 2006. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 19 / 62

  3. Reaction zone structure normal to shock 3 ρ (kg/ m ) λ 1 4 0.8 a) 3 b) 0.6 2 0.4 1 0.2 X (m) 4 X (m) 0.0 0 -1 0 1 2 3 4 -1 0 1 2 3 T(K) MX 600 3 500 c) 2 d) 400 1 300 X (m) X (m) 200 0 -1 0 1 2 3 4 -1 0 1 2 3 4 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 20 / 62

  4. Verification of WENO shock capturing Algorithm of Xu, Aslam, and Stewart, 1997, CTM . Uniform Cartesian grid. Embedded internal boundary with level set representation. Nominally fifth order weighted essentially non-oscillatory (WENO) discretization. Non-decomposition based Lax-Friedrichs solver. Third order Runge-Kutta time integration. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 21 / 62

  5. Exact versus WENO solution y (m) 2.0 ρ = 2.9 kg/m 3 exact ρ = 2.6 kg/m 3 1.5 numerical ρ = 2.3 kg/m 3 256 × 256 uniform numerical grid. 1.0 ρ = 2.0 kg/m 3 good agreement in picture 0.5 norm. 0.0 x (m) numerical solution stable. 0.0 0.5 1.0 1.5 Powers and Aslam, AIAA Journal , 2006. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 22 / 62

  6. Iterative convergence to steady state: various grids L (kg/m) 1 10 64 x 64 Coarse grids relax quickly; fine 128 x 128 256 x 256 1 512 x 512 grids relax slowly. 1024 x 1024 All grids iteratively converge 0.1 to steady state. Iterative convergence is 0.01 t (s) 0 0.002 0.004 0.006 0.008 0.01 distinct from grid convergence. Powers and Aslam, AIAA Journal , 2006. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 23 / 62

  7. Grid convergence L (kg/m) 1 1 ∆ x 0 . 779 � � Convergence rate: O . 0.1 Both shock capturing and embedded boundary induce the low convergence rate. 0.01 ∆ x (m) 0.001 0.01 0.1 Powers and Aslam, AIAA Journal , 2006. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 24 / 62

  8. Conclusions Modeling flows with thin zones and surfaces of discontinuity is challenging. Most common shock-capturing methods converge at less than first order. Heroic measures are often needed to achieve high order convergence Implicit shock tracking...... LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 25 / 62

  9. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  10. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  11. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  12. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  13. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  14. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  15. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  16. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  17. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  18. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  19. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  20. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  21. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  22. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  23. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  24. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  25. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  26. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  27. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  28. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  29. High-order implicit tracking: reacting inviscid flow Original mesh: 100 triangular elements (unstructured) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 26 / 62

  30. Reacting inviscid flow: density ( ρ ) p = q = 1 p = q = 2 p = q = 3 p : polynomial degree of solution q : polynomial degree of mesh LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 27 / 62

  31. Reacting inviscid flow: density ( ρ ) p = q = 1 p = q = 2 p = q = 3 p : polynomial degree of solution q : polynomial degree of mesh LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 27 / 62

  32. Reacting inviscid flow: reaction progress ( λ ) p = q = 1 p = q = 2 p = q = 3 p : polynomial degree of solution q : polynomial degree of mesh LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 28 / 62

  33. Isolines of density indicate excellent agreement with analytical solution on coarse ( 100 element) mesh for p > 1 2 shock ρ = 2 . 9 Analytical ( ) ρ = 2 . 6 p = 1 ( ) ρ = 2 . 3 1 . 5 p = 2 ( ) p = 3 ( ) x 2 ρ = 2 1 0 . 5 0 − 0 . 20 0 . 5 1 1 . 5 1 . 9 x 1 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 29 / 62

  34. Implicit shock tracking recovers optimal O ( h p +1 ) convergence rates; compares favorably to WENO 10 − 1 L 1 error ( ρ ) 10 − 2 10 − 3 10 − 4 10 − 2 10 − 1 mesh size parameter ( h ) WENO ( ), p = 1 ( ), p = 2 ( ), p = 3 ( ) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 30 / 62

  35. Implicit shock tracking Goal: Align element faces with (unknown) discontinuities to perfectly capture them and approximate smooth regions to high-order DG discretization: inter-element jumps, high-order Discontinuity-aligned mesh is the solution of an optimization problem constrained by the discrete PDE = ⇒ implicit tracking Simultaneously converge solution and mesh to ensure solution of PDE never required on non-aligned mesh LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 31 / 62

  36. Implicit shock tracking Goal: Align element faces with (unknown) discontinuities to perfectly capture them and approximate smooth regions to high-order DG discretization: inter-element jumps, high-order Discontinuity-aligned mesh is the solution of an optimization problem constrained by the discrete PDE = ⇒ implicit tracking Simultaneously converge solution and mesh to ensure solution of PDE never required on non-aligned mesh LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 31 / 62

  37. Discontinuous Galerkin discretization of conservation law Inviscid conservation law: ∇ · F ( U ) = 0 in Ω Test V h,p ′ and trial V h,p spaces, where h is the mesh size and p/p ′ is the polynomial degree, to define the finite-dimensional DG residual: � � ψ + h,p ′ · H ( U + r K h,p , U − h,p ′ ( U h,p ) := F ( U h,p ) : ∇ ψ h,p ′ dV h,p , n ) dS − ∂K K Introduce basis for polynomial spaces to obtain discrete residuals ( p ′ = p ) , ( p ′ = p + 1) , r ( u , x ) R ( u , x ) where u is the discrete state vector and x are the coordinates of the mesh nodes. The “standard” DG equations are: r ( u , x ) = 0 . LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 32 / 62

  38. Discontinuous Galerkin discretization of conservation law Inviscid conservation law: ∇ · F ( U ) = 0 in Ω Test V h,p ′ and trial V h,p spaces, where h is the mesh size and p/p ′ is the polynomial degree, to define the finite-dimensional DG residual: � � ψ + h,p ′ · H ( U + r K h,p , U − h,p ′ ( U h,p ) := F ( U h,p ) : ∇ ψ h,p ′ dV h,p , n ) dS − ∂K K Introduce basis for polynomial spaces to obtain discrete residuals ( p ′ = p ) , ( p ′ = p + 1) , r ( u , x ) R ( u , x ) where u is the discrete state vector and x are the coordinates of the mesh nodes. The “standard” DG equations are: r ( u , x ) = 0 . LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 32 / 62

  39. Discontinuous Galerkin discretization of conservation law Inviscid conservation law: ∇ · F ( U ) = 0 in Ω Test V h,p ′ and trial V h,p spaces, where h is the mesh size and p/p ′ is the polynomial degree, to define the finite-dimensional DG residual: � � ψ + h,p ′ · H ( U + r K h,p , U − h,p ′ ( U h,p ) := F ( U h,p ) : ∇ ψ h,p ′ dV h,p , n ) dS − ∂K K Introduce basis for polynomial spaces to obtain discrete residuals ( p ′ = p ) , ( p ′ = p + 1) , r ( u , x ) R ( u , x ) where u is the discrete state vector and x are the coordinates of the mesh nodes. The “standard” DG equations are: r ( u , x ) = 0 . LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 32 / 62

  40. Implicit shock tracking: constrained optimization We formulate the problem of tracking discontinuities with the mesh as the solution of an optimization problem constrained by the discrete PDE (DG discretization) minimize f ( u , x ) u , x subject to r ( u , x ) = 0 . We propose an objective function that balances the tracking objective with maintaining a high-quality mesh 2 R ( u , x ) T R ( u , x ) + κ 2 f ( u , x ) = 1 2 R msh ( x ) T R msh ( x ) . LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 33 / 62

  41. Implicit shock tracking: constrained optimization We formulate the problem of tracking discontinuities with the mesh as the solution of an optimization problem constrained by the discrete PDE (DG discretization) minimize f ( u , x ) u , x subject to r ( u , x ) = 0 . We propose an objective function that balances the tracking objective with maintaining a high-quality mesh 2 R ( u , x ) T R ( u , x ) + κ 2 f ( u , x ) = 1 2 R msh ( x ) T R msh ( x ) . LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 33 / 62

  42. Implicit shock tracking: SQP solver Define z = ( u , x ) and use interchangeably. To solve the optimization problem, we define a sequence { z k } updated as z k +1 = z k + α k ∆ z k . The step direction ∆ z k is defined as the solution of the quadratic program (QP) approximation of the tracking problem centered at z k g z ( z k ) T ∆ z + 1 2∆ z T B ( z k , ˆ minimize λ ( z k ))∆ z ∆ z ∈ R N z subject to r ( z k ) + J z ( z k )∆ z = 0 , where B z ( z , λ ) ≈ ∂ 2 L g z ( z ) = ∂f J z ( z ) = ∂ r ∂ z ( z ) T , ∂ z ( z ) , ∂ z ∂ z ( z , λ ) , λ ( z ) = ∂ r ∂ u ( z ) − T ∂f L ( z , λ ) = f ( z ) − λ T r ( z ) , ∂ u ( z ) T . ˆ LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 34 / 62

  43. Implicit shock tracking: SQP solver Define z = ( u , x ) and use interchangeably. To solve the optimization problem, we define a sequence { z k } updated as z k +1 = z k + α k ∆ z k . The step direction ∆ z k is defined as the solution of the quadratic program (QP) approximation of the tracking problem centered at z k g z ( z k ) T ∆ z + 1 2∆ z T B ( z k , ˆ minimize λ ( z k ))∆ z ∆ z ∈ R N z subject to r ( z k ) + J z ( z k )∆ z = 0 , where B z ( z , λ ) ≈ ∂ 2 L g z ( z ) = ∂f J z ( z ) = ∂ r ∂ z ( z ) T , ∂ z ( z ) , ∂ z ∂ z ( z , λ ) , λ ( z ) = ∂ r ∂ u ( z ) − T ∂f L ( z , λ ) = f ( z ) − λ T r ( z ) , ∂ u ( z ) T . ˆ LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 34 / 62

  44. Practical considerations: element collapse Despite measures to keep mesh well-conditioned, best option may be to remove element from the mesh: tag elements for removal based on volume, collapse shortest edge, trivial DG solution transfer Only meshing operation required by implicit shock tracking Simplices: arbitrary dimensions, arbitrary polynomial degree LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 35 / 62

  45. Practical considerations: element collapse Despite measures to keep mesh well-conditioned, best option may be to remove element from the mesh: tag elements for removal based on volume, collapse shortest edge, trivial DG solution transfer Only meshing operation required by implicit shock tracking Simplices: arbitrary dimensions, arbitrary polynomial degree LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 35 / 62

  46. Linear advection, straight shock p = 0 space for solution, q = 1 space for mesh L 1 error = 3 . 84 × 10 − 11 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 36 / 62

  47. Newton-like convergence when solution lies in DG space Linear advection with straight shock 10 0 Solver convergence 10 − 4 10 − 8 10 − 12 10 − 16 0 2 4 6 8 10 Iteration � � � ∇ x L ( z , ˆ � r ( z ) � ( ), � R ( z ) � ( ), λ ( z )) � ( ) � � LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 37 / 62

  48. Linear advection, trigonometric shock p = 0 space for solution, q = 2 space for mesh L 1 error = 1 . 15 × 10 − 3 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 38 / 62

  49. Linear advection, trigonometric shock, 3D LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 39 / 62

  50. Linear advection, trigonometric shock, 3D LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 39 / 62

  51. Inviscid Burgers’ equation: space-time formulation p = 0 space for solution, q = 3 space for mesh LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 40 / 62

  52. Temporal snapshots show discontinuity perfectly represented, smooth solution well-approximated elsewhere 2 1 . 5 U ( x, t ) 1 0 . 5 0 − 1 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 x Temporal snapshots of inviscid Burgers’ equation from p = 4 implicit tracking simulation LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 41 / 62

  53. Supersonic flow past NACA0012 airfoil ( M = 1 . 5 ) Initialization p = 1 tracking p = 2 tracking e H = 1 . 30 × 10 − 3 e H = 6 . 73 × 10 − 5 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 42 / 62

  54. Supersonic flow past NACA0012 airfoil ( M = 1 . 5 ) Initialization p = 1 tracking p = 2 tracking e H = 1 . 30 × 10 − 3 e H = 6 . 73 × 10 − 5 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 42 / 62

  55. Spatial slices show both discontinuities are tracked and solution well-approximated elsewhere 2 . 5 1 . 6 2 1 . 4 2 M 1 . 5 ρ P 1 . 2 1 . 5 1 1 1 0 . 8 − 0 . 5 0 0 . 5 1 1 . 5 − 0 . 5 0 0 . 5 1 1 . 5 − 0 . 5 0 0 . 5 1 1 . 5 x x x Spatial slice of supersonic flow past NACA0012 airfoil ( M = 1 . 5 ) from p = 3 implicit tracking simulation LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 43 / 62

  56. High-order, implicit shock tracking Primary benefit : highly accurate solutions on coarse meshes, recover optimal convergence rates Traditional barrier to tracking (explicitly meshing unknown discontinuity surface) replaced with solving constrained optimization problem Future extensions: 3D, viscous, time-dependent, model reduction , hypersonics Zahr, Shi, Persson, Journal of Computational Physics , 2020. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 44 / 62

  57. Implicit shock tracking for hypersonics applications Strong local features : shock waves and contact discontinuities Improved solver robustness (time-stepping, vanishing viscosity) Viscous effects : thin boundary layer, critical to predict heating Extension to viscous problems straightforward Expect r -adaptivity near boundary layer to improve approximation Interacting features : shock/shock and shock/boundary layer Handled by optimization formulation and element collapse Unsteady : prediction of combustion stability Slab-based space-time formulation (moving features) Method-of-lines-based formulation (stationary features) Multi-scale/physics : turbulence, ablation, conjugate heat transfer Clip objective function to avoid tracking all turbulent features Integrate in partitioned multiphysics setting LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 45 / 62

  58. Reduction of parametrized discontinuities Fundamental issue: linear subspace approximation ill-suited for advection- dominated features (slowly decay Kolmogorov n -width) apply parameter-dependent domain mapping to align features use linear subspace in reference domain to reduce dimension push forward to physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 46 / 62

  59. Reduction of parametrized discontinuities Fundamental issue: linear subspace approximation ill-suited for advection- dominated features (slowly decay Kolmogorov n -width) apply parameter-dependent domain mapping to align features use linear subspace in reference domain to reduce dimension push forward to physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 46 / 62

  60. Reduction of parametrized discontinuities Fundamental issue: linear subspace approximation ill-suited for advection- dominated features (slowly decay Kolmogorov n -width) apply parameter-dependent domain mapping to align features use linear subspace in reference domain to reduce dimension push forward to physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 46 / 62

  61. Reduction of parametrized discontinuities Fundamental issue: linear subspace approximation ill-suited for advection- dominated features (slowly decay Kolmogorov n -width) apply parameter-dependent domain mapping to align features use linear subspace in reference domain to reduce dimension push forward to physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 46 / 62

  62. Reduction of parametrized discontinuities Fundamental issue: linear subspace approximation ill-suited for advection- dominated features (slowly decay Kolmogorov n -width) Proposed solution apply parameter-dependent domain mapping to align features use linear subspace in reference domain to reduce dimension push forward to physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 46 / 62

  63. Reduction of parametrized discontinuities Fundamental issue: linear subspace approximation ill-suited for advection- dominated features (slowly decay Kolmogorov n -width) Proposed solution apply parameter-dependent domain mapping to align features use linear subspace in reference domain to reduce dimension push forward to physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 46 / 62

  64. Reduction of parametrized discontinuities Fundamental issue: linear subspace approximation ill-suited for advection- dominated features (slowly decay Kolmogorov n -width) Proposed solution apply parameter-dependent domain mapping to align features use linear subspace in reference domain to reduce dimension push forward to physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 46 / 62

  65. Numerical experiment: parametrized linear advection The governing parametrized conservation law is in Ω := (0 , 1) 2 ∇ · ( β µ U ) + σ µ U = f µ U = ˆ U µ on Γ i := { x ∈ ∂ Ω | β µ · n < 0 } µ 1 : shock angle ( β µ ) µ 2 : magnitude of source term ( σ µ ) µ 3 : magnitude of boundary condition ( ˆ U µ ) LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 47 / 62

  66. Optimization-based alignment of discontinuities in reference domain (similar to implicit shock tracking) Snapshot 1 Physical domain Reference domain The blue line indicates the true orientation of the discontinuity in the physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 48 / 62

  67. Optimization-based alignment of discontinuities in reference domain (similar to implicit shock tracking) Snapshot 5 Physical domain Reference domain The blue line indicates the true orientation of the discontinuity in the physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 48 / 62

  68. Optimization-based alignment of discontinuities in reference domain (similar to implicit shock tracking) Snapshot 6 Physical domain Reference domain The blue line indicates the true orientation of the discontinuity in the physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 48 / 62

  69. With feature alignment, POD modes do not have to resolve moving discontinuity since it is (mostly) fixed Mode 1 Mode 2 Mode 3 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 49 / 62

  70. Without feature alignment, POD modes must resolve moving discontinuity, which leads to slowly decaying Kolmogorov n -width Mode 1 Mode 2 Mode 3 LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 50 / 62

  71. Alignment framework leads to reduction in error for discontinuities not encountered during training Test point 96 (discontinuity not in training) ROM (aligned) ROM (non-aligned) HDM The blue line indicates the true orientation of the discontinuity in the physical domain LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 51 / 62

  72. Alignment framework leads to reduction in error for discontinuities not encountered during training Testing set Maximum error Mean error discontinuity in training 0.92% 0.45% discontinuity not in training 26.5% 19.5% Classical minimum-residual model reduction Testing set Maximum error Mean error discontinuity in training 2.61% 1.22% discontinuity not in training 4.18% 3.31% Minimum-residual model reduction with feature alignment Parameters: 12 training, 125 testing LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 52 / 62

  73. Appendix LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 53 / 62

  74. Implicit shock tracking: SQP solver The solution of the QP leads to the following linear system B uu ( z k , ˆ B ux ( z k , ˆ J u ( z k ) T       λ ( z k )) λ ( z k )) ∆ u k g u ( z k ) B ux ( z k , ˆ B xx ( z k , ˆ  = −  , λ ( z k )) T J x ( z k ) T ∆ x k g x ( z k ) λ ( z k ))     η k r ( z k ) J u ( z k ) J x ( z k ) 0 where g � ( z ) = ∂f J � ( z ) = ∂ r ∂ � ( z ) T , ∂ � ( z ) , the approximate Hessian of the Lagrangian is partitioned as ∂ 2 L B � △ ( z , λ ) ≈ ∂ � ∂ △ ( z , λ ) , and η k are the Lagrange multipliers of the QP. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 54 / 62

  75. SQP solver: Levenberg-Marquardt Hessian approximation The proposed objective function takes the form of a residual norm � � f ( z ) = 1 R ( z ) 2 � F ( z ) � 2 2 , F ( z ) = κ R msh ( x ) and therefore the Hessian of the Lagrangian has the special structure ∂ z ( z ) + F i ( z ) ∂ 2 F i ∂ 2 r i H ( z , λ ) = ∂ F ∂ z ( z ) T ∂ F ∂ z ∂ z ( z ) − λ i ∂ z ∂ z ( z ) The last two terms are dropped: they are difficult to compute and neg- ligible if the � F � and � λ � are small (Gauss-Newton approximation) H ( z , λ ) ≈ ∂ F ∂ z ( z ) T ∂ F ∂ z ( z ) To guard against ill-conditioning, a regularization matrix is added T ∂ F T ∂ F T ∂ F B uu = ∂ F B ux = ∂ F B xx = ∂ F ∂ u , ∂ x , ∂ x + γ D . ∂ u ∂ u ∂ x LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 55 / 62

  76. SQP solver: Levenberg-Marquardt Hessian approximation The proposed objective function takes the form of a residual norm � � f ( z ) = 1 R ( z ) 2 � F ( z ) � 2 2 , F ( z ) = κ R msh ( x ) and therefore the Hessian of the Lagrangian has the special structure ∂ z ( z ) + F i ( z ) ∂ 2 F i ∂ 2 r i H ( z , λ ) = ∂ F ∂ z ( z ) T ∂ F ∂ z ∂ z ( z ) − λ i ∂ z ∂ z ( z ) The last two terms are dropped: they are difficult to compute and neg- ligible if the � F � and � λ � are small (Gauss-Newton approximation) H ( z , λ ) ≈ ∂ F ∂ z ( z ) T ∂ F ∂ z ( z ) To guard against ill-conditioning, a regularization matrix is added T ∂ F T ∂ F T ∂ F B uu = ∂ F B ux = ∂ F B xx = ∂ F ∂ u , ∂ x , ∂ x + γ D . ∂ u ∂ u ∂ x LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 55 / 62

  77. SQP solver: Levenberg-Marquardt Hessian approximation The proposed objective function takes the form of a residual norm � � f ( z ) = 1 R ( z ) 2 � F ( z ) � 2 2 , F ( z ) = κ R msh ( x ) and therefore the Hessian of the Lagrangian has the special structure ∂ z ( z ) + F i ( z ) ∂ 2 F i ∂ 2 r i H ( z , λ ) = ∂ F ∂ z ( z ) T ∂ F ∂ z ∂ z ( z ) − λ i ∂ z ∂ z ( z ) The last two terms are dropped: they are difficult to compute and neg- ligible if the � F � and � λ � are small (Gauss-Newton approximation) H ( z , λ ) ≈ ∂ F ∂ z ( z ) T ∂ F ∂ z ( z ) To guard against ill-conditioning, a regularization matrix is added T ∂ F T ∂ F T ∂ F B uu = ∂ F B ux = ∂ F B xx = ∂ F ∂ u , ∂ x , ∂ x + γ D . ∂ u ∂ u ∂ x LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 55 / 62

  78. Practical considerations: regularization matrix D The mesh regularization matrix D is taken as the stiffness matrix of the linear elliptic PDE ∇ · ( k ∇ v i ) = 0 in Ω for i = 1 , . . . , d . The coefficient is constant over each element and in- versely proportional to the element volume | K ′ | min K ′ ∈E h,q k ( x ) = , x ∈ K | K | for each element K in the mesh T : critical to maintain well-conditioned search directions for meshes where element size varies significantly. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 56 / 62

  79. Practical considerations: regularization matrix D The mesh regularization matrix D is taken as the stiffness matrix of the linear elliptic PDE ∇ · ( k ∇ v i ) = 0 in Ω for i = 1 , . . . , d . The coefficient is constant over each element and in- versely proportional to the element volume | K ′ | min K ′ ∈E h,q k ( x ) = , x ∈ K | K | for each element K in the mesh T : critical to maintain well-conditioned search directions for meshes where element size varies significantly. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 56 / 62

  80. SQP solver: step length ( α k ) The step length, α k ∈ (0 , 1] , is selected using a backtracking line search to ensure sufficient decrease of a merit function ϕ k : R → R ϕ k ( α k ) ≤ ϕ k (0) + cα k ϕ ′ k (0) , c ∈ (0 , 1) . We use the ℓ 1 merit function ϕ k ( α ) := f ( z k + α ∆ z k ) + µ � r ( z k + α ∆ z k ) � 1 � � � ˆ where µ > λ ( z k ) ∞ because it is “exact”, i.e., any minimizer of the � � � original optimization problem is a minimizer of ϕ k . LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 57 / 62

  81. SQP solver: step length ( α k ) The step length, α k ∈ (0 , 1] , is selected using a backtracking line search to ensure sufficient decrease of a merit function ϕ k : R → R ϕ k ( α k ) ≤ ϕ k (0) + cα k ϕ ′ k (0) , c ∈ (0 , 1) . We use the ℓ 1 merit function ϕ k ( α ) := f ( z k + α ∆ z k ) + µ � r ( z k + α ∆ z k ) � 1 � � � ˆ where µ > λ ( z k ) ∞ because it is “exact”, i.e., any minimizer of the � � � original optimization problem is a minimizer of ϕ k . LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 57 / 62

  82. Practical considerations: termination criteria The termination criteria for the solver is based on the Karush-Kuhn- Tucker (KKT) conditions: z ⋆ is a solution if there exist Lagrange multi- pliers λ ⋆ such that ∇ u L ( z ⋆ , λ ⋆ ) = 0 , ∇ x L ( z ⋆ , λ ⋆ ) = 0 , r ( z ⋆ ) = 0 Our choice for the Lagrange multiplier estimate ˆ λ ( z ) ensures ∇ u L ( z , ˆ λ ( z )) = 0 and therefore termination is based on the remaining KKT conditions � � � ∇ x L ( z , ˆ λ ( z )) � < ǫ 1 , � r ( z ) � < ǫ 2 , � � where ǫ 1 , ǫ 2 > 0 are convergence tolerances. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 58 / 62

  83. Practical considerations: termination criteria The termination criteria for the solver is based on the Karush-Kuhn- Tucker (KKT) conditions: z ⋆ is a solution if there exist Lagrange multi- pliers λ ⋆ such that ∇ u L ( z ⋆ , λ ⋆ ) = 0 , ∇ x L ( z ⋆ , λ ⋆ ) = 0 , r ( z ⋆ ) = 0 Our choice for the Lagrange multiplier estimate ˆ λ ( z ) ensures ∇ u L ( z , ˆ λ ( z )) = 0 and therefore termination is based on the remaining KKT conditions � � � ∇ x L ( z , ˆ λ ( z )) � < ǫ 1 , � r ( z ) � < ǫ 2 , � � where ǫ 1 , ǫ 2 > 0 are convergence tolerances. LLNL Virtual Seminar Accurate CFD for Shocks 24 June 2020 58 / 62

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