Performance and Quality Analysis of Interpolation Methods for Coupling

Transcription

1 Performance and Quality Analysis of Interpolation Methods for Coupling Neda Ebrahimi Pour, Sabine Roller

2 Outline Motivation Coupling Approaches Interpolation/Evaluation Results of Investigation Summary 11/10/17 2

3 Motivation More and more complex devices à Better understanding of the physical phenomena à Improvements in product design in the engineering field Important: Applications involve multi-physics and multi-scales 11/10/17 3

4 Strategy Simulating whole problem with one solver à too expensive Partitioned coupling à splitting whole domain in subdomains - according physical phenomenon à Solving each domain according to its requirements - equations, element size etc. 11/10/17 4

5 Partitioned Coupling APES Navier-Stokes equations FV 2 nd order Euler equations DG 4-8 th order Linearized Euler equations DG 8-64 th order Coupling different domains with different numerical methods (Finite Volume, Discontinuous Galerkin) e.g. FSA-Interaction Coupled simulation with spatial discretization 11/10/17 5

6 Coupling Types Matching Non-matching à Data mapping from one side to the other Op2on 1: Interpola2on (scheme independent) Op2on 2: Evalua2on of polynomial (scheme dependent, e.g. DG) à Matching coupling à Non-matching coupling - same grid size - different grid size - same scheme order - different scheme order à same number of coupling points à different number of coupling points 11/10/17 6

7 Two coupling approaches 1) precice An external library Treats solver as a black-box Different coupling strategies Parallel/ serial explicit, Implicit quasi newton methods Different data-mapping methods Nearest-Neighbor, Nearest-Projection, Radial-Basis-Function B. Gatzhammer, B. Uekermann (München), F. Lindner, M. Mehl (Stuttgart) 11/10/17 7

8 Interpolation Methods 1a) Nearest-Neighbour 1b) Nearest-Projection A A B B + nearest No computation neighbour ma + 2 nd order accuracy - 1 st order accuracy - provide neighborhood - not suitable for non-matching information coupling 11/10/17 8

9 1c) Radial-Basis-Function s(x) = NX i=1 i '(kx x i k)+q(x) φ : basis func2on x i : point, which has to be evaluated q(x) : global linear func2on s(x) : interpolant + No neighborhood information needed + Different basis-functions available - Linear equation system has to be solved - Condition number depends on point distribution (equidistant/non-equidistant) e.g. basis function: Gaussian function p ln(10 9 ) shape-parameter: a = m h m : average number of points to cover h: average distance between points 11/10/17 9

10 Two coupling approaches 2) APESmate Integrated approach in APES Evaluation of polynomial Data-mapping according to scheme order Accuracy depends on scheme order Aotus Configuration with Lua Shepherd Deployment Scripts Seeder Mesh Generation APES Adaptable Poly-Engineering Simulator TreElM Octree Mesh Infrastructure Ateles Discontinuous Galerkin Muriqui Space-Time DG APESmate Coupling of APES solvers Musubi Lattice Boltzmann K. Masilamani, V. Krupp, H. Klimach, S. Roller (Siegen) Harvester Post-Processing Analysis 11/10/17 10

11 Results: Testcase 11/10/17 11

12 Results: Convergence Study O(2) O(4) O(8) O(12) O(14) 10 1 O(3) O(6) O(10) O(16) O(32) Monolithic simula2on - Varia2on of the element size and scheme order 10 2 L2error Calcula2on of the L2error from simula2on and analy2cal solu2on Elementsize Chosen element size / L2error for coupling two domains 11/10/17 12

13 Testcases: Testcase 1 Inner: Outer: Testcase 2 Inner: Outer: Testcase 3 Inner: Outer: Testcase 4 Inner: Outer: Scheme Order Number of Points O(3) O(4) O(3) O(6) O(3) O(8) O(3) O(14) Number of Elements /10/17 13

14 Testcase: Gaussian Pressure Pulse 11/10/17 14

15 Testcase: Gaussian Pressure Pulse Domain: 5 x 5 x 5 Halfwidth: 0.25 Inner domain: 1 x 1 x 1 Amplitude: /10/17 15

16 Results: Interpolation/Evaluation method NP RBF AP ESmate L2error Highest error with NP Can we reduce the L2error by choosing a different Increasing error with point distribu2on? RBF Constant error with APESmate Scheme Order 11/10/17 16

17 Results: Equidistant Points for Radial-Basis- Function Providing equidistant points for the interpola2on 11/10/17 17

18 Results: Radial-Basis-Function RBF RBF RBF EQ EQ Computational L2error T ime [s] Scheme Order Increasing error for RBF with non-equidistant points Decreasing error with equidistant point distribu2on Increasing computa2onal cost for RBF with equidistant pointd 11/10/17 18

19 Results: Oversampling for Nearest-Projection Oversampling factor of 2 11/10/17 19

20 Results: Nearest-Projection NP NP factor=2 NP EQ NP EQfactor= Increasing error, when using oversampling for non-equidistant points L2error Decreasing error with increasing oversampling factor for equidistant points Scheme Order 11/10/17 20

21 Results: L2error for all methods NP NP EQfactor=3 RBF EQ RBF AP ESmate Highest error with NP and non-equidistant points L2error Error APESmate and NP almost constant over scheme order RBF with equidistant points à decreasing error Scheme Order 11/10/17 21

22 Results: Computational cost 10 3 NP RBF NP EQfactor=3 RBF EQ AP ESmate RBF with equidistant points à highest cost Computational T ime [s] 10 2 NP with non-equidistant points à lows cost APESmate à increasing cost with increasing scheme order Scheme Order 11/10/17 22

23 Results: Performance (Weak Scaling) Nearest-Projec2on Radial-Basis-Func2on APESmate Testcase 1 Testcase 2 Testcase 3 Testcase 4 Testcase 1 Testcase 2 Testcase 3 Testcase 4 Testcase 1 Testcase 2 Testcase 3 Testcase Computational Time Computational Time 10 3 Computational Time np rocs np rocs np rocs 11/10/17 23

24 Results: Performance (Weak Scaling) NP RBF AP ESmate 10 4 Computational T ime np rocs 11/10/17 24

25 Summary Coupling Tool precice Method Error Computa2onal cost RBF non-equidistant Increase with scheme order points Increasing with Decrease when using scheme order When RBF equidistant coupling different points solvers equidistant à precice points with Nearest- Projec2on interpola2on NP non-equidistant Increase when using Increasing with When points using our solvers à oversampling APESmate provides factor very good scheme results order NP equidistant points Decrease with oversampling factor Increasing with scheme order and oversampling factor APESmate Evalua2on of the polynomial Scheme dependent Increasing with scheme order 11/10/17 25

26 Thank you for your attention 11/10/17 26