Divergence-Free Elements for Incompressible Flow on Cartesian Grids

Transcription

1 Divergence-Free Elements for Incompressible Flow on Cartesian Grids Tobias Neckel, Marion Bendig, Hans-Joachim Bungartz, Miriam Mehl, and Christoph Zenger, Fakultät für Informatik TU München

2 Outline The PDE Framework Peano Div-Free Elements Motivation Derivation Validation Variants of Div-Free Elements Enhancement Rotated Coordinate System Summary & Outlook

3 The PDE Framework Peano Cartesian grids (recursive adaptivity, full grid hierarchy) Low memory requirements Space-filling curves + stack data structures high cache-hit rates (>98%) flexible insertion/deletion of data (grid changes) Shared/distributed mem. parallelisation Software engineering aspects source: T. Weinzierl CFD component Incompressible flow (FEM, IDO) Explicit + implicit time-integration schemes (FE, RK4, BE, (adaptive) TR)

4 Motivation 1: Discrete Energy Conservation dy dx R. Verstappen & A. Veldman, J. Comput. Phys. 187, 2003

5 Motivation 1: Discrete Energy Conservation dy dx (skew-) symmetry guaranteed stable DAEs! R. Verstappen & A. Veldman, J. Comput. Phys. 187, 2003

6 Motivation 2: FEM Function Spaces for NSE min u=0 u = 0 Variational Functional NSE

7 Motivation 2: FEM Function Spaces for NSE min u h =0 globally ensured min u=0 u h = 0 u = 0 Variational Functional NSE

8 Motivation 2: FEM Function Spaces for NSE classical FEM Mu h = 0 u h 0 min u h =0 globally ensured min u=0 u h = 0 u = 0 Variational Functional NSE

9 Motivation 2: FEM Function Spaces for NSE classical FEM Mu h = 0 u h 0 min u h =0 globally ensured min u=0 u h = 0 u = 0 Variational Functional NSE

10 Motivation 2: FEM Function Spaces for NSE classical FEM Mu h = 0 u h 0 locally ensured globally ensured min u h =0 min u=0 u h = 0 u = 0 Variational Functional NSE

11 Summary of Motivations for Spatial Discretisations symmetry properties for operators C u h, D conservation of energy & momentum (mimic real physics) guaranteed stable DAEs!

12 Summary of Motivations for Spatial Discretisations FVM, FDM symmetry properties for operators C u h, D conservation of energy & momentum (mimic real physics) guaranteed stable DAEs!

13 Summary of Motivations for Spatial Discretisations FEM FVM, FDM (locally) continuously div-free elements symmetry properties for operators C u h, D Correct spaces u h = 0, no LBB conservation of energy & momentum (mimic real physics) guaranteed stable DAEs!

14 Derivation of Div-Free Ansatz Functions Cell-wise mass conservation: C. Blanke, Diploma Thesis, 2004

15 Derivation of Div-Free Ansatz Functions Cell-wise mass conservation: Split cell in triangles Linear velocities on triangles C. Blanke, Diploma Thesis, 2004

16 Derivation of Div-Free Ansatz Functions Cell-wise mass conservation: Split cell in triangles Linear velocities on triangles Conditions for u, v on center node C. Blanke, Diploma Thesis, 2004

17 Derivation of Div-Free Ansatz Functions II C. Blanke, Diploma Thesis, 2004

18 Derivation of Div-Free Ansatz Functions II φ(x, y) φ(x, y) N φ u h = u i i=1 φ + φ v i φ C. Blanke, Diploma Thesis, 2004

19 Validation Driven Cavity Re=1000

20 Validation Flow around a Cylinder # DoF Re = 20 Re = 100 c d c l C d,max C l,max St 88, ref

21 Outline The PDE Framework Peano Div-Free Elements Motivation Derivation Validation Variants of Div-Free Elements Enhancement Rotated Coordinate System Summary & Outlook

22 Div-free Elements Variant 1: Enhancement Additional DoF on faces: exact representation of fluxes on edges possible no checkerboarding T. Weinzierl, Diploma Thesis, 2005

23 Div-free Elements Variant 1: Enhancement Checkerboard Driven Cavity: u = 0: 1 in corner cells enhanced div-free (steady state) Re=1 Q1Q0 / div-free (step1, no convergence!)

24 Div-free Elements Variant 2: Rotated Coordinates 45

25 Div-free Elements Variant 2: Rotated Coordinates 45

26 Div-free Elements Variant 2: Rotated Coordinates φ(x, y) φ(x, y) N φ u h = u i i= v i φ

27 Div-free Elements Variant 2: Rotated Coordinates φ(x, y) φ(x, y) Advantages: u h = N φ u i simpler (derivation/representation) performance: ~20% less runtime for evaluation of operators D and C i= v i φ

28 Summary & Outlook Summary Div-free elements solenoidal on every point in cell symmetry properties for discrete operators conservation of discrete momentum AND energy Variants Enhanced div-free elements: DoF on faces 45 represenation advantageous Outlook Multigrid: Peano Framework designed for hierarchical applications Extension of div-free elements to 3D: no direct 45 trick :-(

29 Thanks for your attention!

30 The PDE Framework Peano Regular (lexicographic) Adaptive (spacetree)

31 The PDE Framework Peano Cartesian grids (arbitrary dimensions) Plug-in concept for applications Space-filling curves, spacetrees, and stack data structures Strictly element-wise access Low memory demands Dynamical load balancing Moving geometries, dynamical adaptivity, geometric multigrid Software Engineering automatic tests, continuous integration, OO, design patterns,... CFD component Incompressible flow (FEM, IDO) Explicit + implicit time-integration schemes (FE, RK4, BE, (adaptive) TR)

32 Backup I Low memory requirements (FEM + adap.): bytes 2D bytes 3D FE RK TR adap. sundance

33 Numerical Results FEM Q1Q0

34 Numerical Results - Performance 2D FEM 2 1,8 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 0 level 6 12,676 level 7 116,061 level 8 1,051,253 ratio adaptive vs. regular 3D FEM Overhead adaptive vs. regular < 3% 2D IDO Overhead Peano vs. Aoki (regular):

35 CFD Extensions Preparation for turbulent flows LES model Mesh stretching for regular grids Data averaging (2x in space, 1x in time) Restart + checkpointing Parallelisation Regular grids 2D/3D Restart + checkpointing Moving geometries Update of data + grid (regular + adaptive) Divergence correction joint work with Kristof Unterweger

36 CFD Extensions joint work with Janos Benk, Bernhard Gatzhammer, Miriam Mehl, Kristof Unterweger, and Tobias Weinzierl Moving geometries Update of data + grid (regular + adaptive) Divergence correction