1. Structure-Preserving B-spline Methods for the Incompressible Navier-Stokes Equations John Andrew Evans Institute for Computational Engineering and Sciences, UT Austin Stabilized and Multiscale Methods in CFD Spring 2013

2. Motivation So why do we need another flow solver? Incompressibility endows the Navier-Stokes equations with important physical structure: Mass balance Momentum balance Energy balance Enstrophy balance Helicity balance However, most methods only satisfy the incompressibility constraint in an approximate sense. Such methods do not preserve structure and lack robustness.

3. Motivation Consider a two-dimensional Taylor-Green vortex. The vortex is a smooth steady state solution of the Euler equations.

4. Motivation Conservative methods which only weakly satisfy incompressibility do not preserve this steady state and blow up in the absence of artificial numerical dissipation. 2-D Conservative Taylor-Hood Element Q2/Q1 and h = 1/8

5. Motivation Methods which exactly satisfy incompressibility are stable in the Euler limit and robust with respect to Reynold’s number. Increasing Reynold’s number 2-D Conservative Structure-preserving B-splines k’ = 1 and h = 1/8

6. Motivation Due to the preceding discussion, we seek new discretizations that: Satisfy the divergence-free constraint exactly. Harbor local stability and approximation properties. Possess spectral-like stability and approximation properties. Extend to geometrically complex domains. Structure-Preserving B-splines seem to fit the bill.

7. The Stokes Complex The classical L 2 de Rham complex is as follows: From the above complex, we can derive the following smoothed complex with the same cohomology structure:

8. The Stokes Complex Scalar Potentials Vector Potentials Flow Pressures Flow Velocities The smoothed complex corresponds to viscous flow, so we henceforth refer to it as the Stokes complex.

9. The Stokes Complex For simply-connected domains with connected boundary, the Stokes complex is exact. grad operator maps onto space of curl-free functions curl operator maps onto space of div-free functions div operator maps onto entire space of flow pressures

10. The Stokes Complex Gradient Theorem: Curl Theorem: Divergence Theorem:

11. The Discrete Stokes Subcomplex Now, suppose we have a discrete Stokes subcomplex. Then a Galerkin discretization utilizing the subcomplex: Does not have spurious pressure modes, and Returns a divergence-free velocity field. Discrete Scalar Potentials Discrete Vector Potentials Discrete Flow Pressures Discrete Flow Velocities

12. The Discrete Stokes Subcomplex

15. Structure-Preserving B-splines Review of Univariate (1-D) B-splines: Knot vector on (0,1) and k-degree B-spline basis on (0,1) by recursion: Ξ = {0, 0, 0, 0.2, 0.4, 0.6, 0.8, 0.8, 1, 1, 1}, k = 2 Knots w/ multiplicity Start w/ piecewise constants Bootstrap recursively to k

16. k=2 Open knot vectors: Multiplicity of first and last knots is k+1 Basis is interpolatory at these locations Non-uniform knot spacing allowed Continuity at interior knot a function of knot repetition Review of Univariate (1-D) B-splines: Structure-Preserving B-splines

17. Review of Univariate (1-D) B-splines: Derivatives of B-splines are B-splines k=4k=3 onto Structure-Preserving B-splines

18. Review of Univariate (1-D) B-splines: We form curves in physical space by taking weighted sums. - control points- knots 0 1 2 3 4 5 Quadratic basis: “control mesh” Structure-Preserving B-splines

19. Multivariate B-splines are built through tensor-products Multivariate B-splines inherit all of the aforementioned properties of univariate B-splines In what follows, we denote the space of n-dimensional tensor- product B-splines as polynomial degree in direction i i th continuity vector Review of Multivariate B-splines: Structure-Preserving B-splines

20. We form surfaces and volumes using weighted sums of multivariate B-splines (or rational B-splines) as before. Review of Multivariate B-splines: Control mesh Mesh Structure-Preserving B-splines

21. Define for the unit square: In the context of fluid flow: and it is easily shown that: Two-dimensional Structure-Preserving B-splines Structure-Preserving B-splines

22. Two-dimensional Structure-Preserving B-splines: Mapped Domains On mapped domains, the Piola transform is utilized to map flow velocities. Pressures are mapped using an integral preserving transform. Structure-Preserving B-splines

23. We associate the degrees of freedom of structure-preserving B-splines with the control mesh. Notably, we associate: Structure-Preserving B-splines Two-dimensional Structure-Preserving B-splines

24. Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2: Structure-Preserving B-splines

25. Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2:

26. Structure-Preserving B-splines Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2:

27. Structure-Preserving B-splines Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2:

28. Structure-Preserving B-splines Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2:

29. Structure-Preserving B-splines Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2:

30. Structure-Preserving B-splines Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2:

31. Structure-Preserving B-splines Two-dimensional Structure-Preserving B-splines, k 1 = k 2 = 2:

32. Define for the unit cube: It is easily shown that: Structure-Preserving B-splines Three-dimensional Structure-Preserving B-splines

33. Flow velocities: map w/ divergence-preserving transformation Flow pressures: map w/ integral-preserving transformation Vector potentials: map w/ curl-conserving transformation Structure-Preserving B-splines Three-dimensional Structure-Preserving B-splines

34. We associate as before the degrees of freedom with the control mesh. Structure-Preserving B-splines Three-dimensional Structure-Preserving B-splines

35. Control points: Scalar potential DOF Control edges: Vector potential DOF Structure-Preserving B-splines Three-dimensional Structure-Preserving B-splines

36. Structure-Preserving B-splines Weak Enforcement of No-Slip BCs Nitsche’s method is utilized to weakly enforce the no-slip condition in our discretizations. Our motivation is three- fold: Nitsche’s method is consistent and higher-order. Nitsche’s method preserves symmetry and ellipticity. Nitsche’s method is a consistent stabilization procedure. Furthermore, with weak no-slip boundary conditions, a conforming discretization of the Euler equations is obtained in the limit of vanishing viscosity.

37. Structure-Preserving B-splines Weak Enforcement of Tangential Continuity Between Patches On multi-patch geometries, tangential continuity is enforced weakly between patches using a combination of the symmetric interior penalty method and upwinding.

38. Summary of Theoretical Results Well-posedness for small data Optimal velocity error estimates and suboptimal, by one order, pressure error estimates Conforming discretization of Euler flow obtained in limit of vanishing viscosity (via weak BCs) Robustness with respect to viscosity for small data Steady Navier-Stokes Flow

39. Summary of Theoretical Results Unsteady Navier-Stokes Flow Existence and uniqueness (well-posedness) Optimal velocity error estimates in terms of the L 2 norm for domains satisfying an elliptic regularity condition (local-in-time) Convergence to suitable weak solutions for periodic domains Balance laws for momentum, energy, enstrophy, and helicity Balance law for angular momentum on cylindrical domains

40. Spectrum Analysis Consider the two-dimensional periodic Stokes eigenproblem: We compare the discrete spectrum for a specified discretization with the exact spectrum. This analysis sheds light on a given discretization’s resolution properties.

41. Spectrum Analysis: Structure-Preserving B-splines Spectrum Analysis

42. Spectrum Analysis: Taylor-Hood Elements Spectrum Analysis

43. Spectrum Analysis: MAC Scheme Spectrum Analysis

44. Selected Numerical Results Steady Navier-Stokes Flow: Numerical Confirmation of Convergence Rates 2-D Manufactured Vortex Solution

45. Selected Numerical Results Steady Navier-Stokes Flow: Numerical Confirmation of Convergence Rates 2-D Manufactured Vortex Solution

46. Selected Numerical Results Steady Navier-Stokes Flow: Numerical Confirmation of Convergence Rates 2-D Manufactured Vortex Solution

47. Selected Numerical Results Steady Navier-Stokes Flow: Numerical Confirmation of Convergence Rates Re0110100100010000 Energy 1.40e-2 H 1 error - u 1.40e-2 L 2 error - u 2.28e-4 L 2 error - p 3.49e-4 1.98e-41.96e-4 Robustness with respect to Reynolds number k ’ = 1 and h = 1/16 2-D Manufactured Vortex Solution

48. Selected Numerical Results Steady Navier-Stokes Flow: Numerical Confirmation of Convergence Rates Re0110100100010000 H 1 error - u 6.77e-4 7.11e-42.26e-32.16e-2X L 2 error - u 6.54e-46.54e-66.79e-61.97e-51.86e-4X L 2 error - p 1.96e-4 X Instability of 2-D Taylor-Hood with respect to Reynolds number Q2/Q1 and h = 1/16 2-D Manufactured Vortex Solution

49. Steady Navier-Stokes Flow: Lid-Driven Cavity Flow H H U Selected Numerical Results

50. Steady Navier-Stokes Flow: Lid-Driven Cavity Flow at Re = 1000 Selected Numerical Results

52. Steady Navier-Stokes Flow: Lid-Driven Cavity Flow at Re = 1000 Methodu min v min v max k’ = 1, h = 1/32 -0.40140-0.391320.54261 k’ = 1, h = 1/64 -0.39399-0.382290.53353 k’ = 1, h = 1/128 -0.39021-0.378560.52884 k’ = 2, h = 1/64 -0.38874-0.377150.52726 k’ = 3, h = 1/64 -0.38857-0.376980.52696 Converged -0.38857-0.376940.52707 Ghia, h = 1/156 -0.38289-0.370950.51550 Selected Numerical Results

53. Unsteady Navier-Stokes Flow: Flow Over a Cylinder at Re = 100 L H U D L out Selected Numerical Results

54. Unsteady Navier-Stokes Flow: Flow Over a Cylinder at Re = 100 Patch 1 Patch 2 Patch 3 Patch 4Patch 5 Selected Numerical Results

55. Unsteady Navier-Stokes Flow: Flow Over a Cylinder at Re = 100 Simulation Details: - Full Space-Time Discretization - Method of Subgrid Vortices - Linears in Space and Time - D = 2, H = 32D, L = 64D - Time Step Size: 0.25 - Trilinos Implementation - GMRES w/ ILU Preconditioning - 3000 time-steps (shedding initiated after 1000 time-steps) - Approximately 15,000 DOF/time step Selected Numerical Results

56. Unsteady Navier-Stokes Flow: Flow Over a Cylinder at Re = 100 Selected Numerical Results

57. Unsteady Navier-Stokes Flow: Flow Over a Cylinder at Re = 100 Quantity of Interest: Strouhal Number: St = fD/U Computed Strouhal Number: 0.163 Accepted Strouhal Number: 0.164 Selected Numerical Results

58. Unsteady Navier-Stokes Flow: Three-Dimensional Taylor-Green Vortex Flow Selected Numerical Results 3-D Periodic Flow Simplest Model of Vortex Stretching No External Forcing

59. Unsteady Navier-Stokes Flow: Three-Dimensional Taylor-Green Vortex Flow Selected Numerical Results Time Evolution of Dissipation Rate Reproduced with permission from [Brachet et al. 1983]

60. Unsteady Navier-Stokes Flow: Three-Dimensional Taylor-Green Vortex Flow at Re = 200 Selected Numerical Results Enstrophy isosurface at time corresponding to maximum dissipation

61. Unsteady Navier-Stokes Flow: Three-Dimensional Taylor-Green Vortex Flow at Re = 200 Selected Numerical Results Convergence of dissipation rate time history with mesh refinement (k ’ = 1)

62. Unsteady Navier-Stokes Flow: Three-Dimensional Taylor-Green Vortex Flow at Re = 200 Selected Numerical Results Convergence of dissipation rate time history with degree elevation (h = 1/32)