A Comparative Study of Conforming and Nonconforming High-Resolution Finite Element Schemes

Transcription

1 A Comparative Study of Conforming and Nonconforming High-Resolution Finite Element Schemes Matthias Möller Institute of Applied Mathematics (LS3) TU Dortmund, Germany European Seminar on Computing Pilsen, Czech Republic June 25, 2012

2 Family of AFC schemes Kuzmin et al. M L u =[K + D]u + F( u, u) lumped mass matrix antidiffusive correction artificial diffusion operator

3 Family of AFC schemes Kuzmin et al. M L u =[K + D]u + F( u, u) F ( u, u) =[M L M C ] u Du F (u) = Du F 0 NL-GP NL-FCT NL-TVD Low-order nonlinear flux-correction Lin-FCT M L u = M L u L + F ( u L,u L ) linearized flux-correction

4 Family of AFC schemes Kuzmin et al. M L u =[K + D]u + F( u, u) F ( u, u) =[M L M C ] u Du F (u) = Du F 0 NL-GP NL-FCT NL-TVD Low-order nonlinear flux-correction AFC-LPT... Lin-FCT M L u = M L u L + F ( u L,u L ) linearized flux-correction

5 Review of design principles Jameson s Local Extremum Diminishing criterion IF m i u i = X j6=i ij(u j u i ) positive not negative THEN local solution maxima/minima do not increase/decrease

6 Review of design principles Jameson s Local Extremum Diminishing criterion IF m i u i = X j6=i ij(u j u i ) positive not negative THEN local solution maxima/minima do not increase/decrease Semi-discrete high-resolution scheme m i u i = X (k ij + d ij )(u j u i )+ i u i + X ij f ij j6=i j6=i m i = X j m ij not negative by construction of diffusion coefficient controlled by flux limiter

7 Review of design principles Jameson s Local Extremum Diminishing criterion IF m i u i = X j6=i ij(u j u i ) positive not negative THEN local solution maxima/minima do not increase/decrease Semi-discrete high-resolution scheme m i u i = X (k ij + d ij )(u j u i )+ i u i + X ij f ij j6=i j6=i m i = X j m ij not negative by construction of diffusion coefficient controlled by flux limiter Edge-based assembly of operators/vectors is feasible and efficient!

8 Finite Elements Bilinear Q1 FE nodal values at cell vertices Rotated bilinear ~Q1 FE nodal values at cell midpoints integral mean-values at edges

9 Finite Element properties Uniform 128x128 grid of unit square with 5% stochastic disturbance (A) MC ML NEQ NA Q1 8 16,641 82,680 mean value point value ~Q1 par 6 33, ,483 ~Q1 np 6 33, ,478 ~Q1 par 6-7.2E-07 ( ) 33, ,204 ~Q1 np 6-7.2E-07 ( ) 33, ,576

10 Finite Element properties Uniform 128x128 grid of unit square with 5% stochastic disturbance (A) MC ML NEQ NA Q1 8 16,641 82,680 mean value point value ~Q1 par 6 33, ,483 ~Q1 np 6 33, ,478 ~Q1 par 6-7.2E-07 ( ) 33, ,204 ~Q1 np 6-7.2E-07 ( ) 33, ,576

11 Finite Element properties

12 Finite Element properties Q1 FE nz = 369

13 Finite Element properties Q1 FE nz = 369 ~Q1 FE nz = 530

14 Finite Element properties Storage format: CRS, CCS Q1 FE nz 5 = Number of non-zero entries per matrix 10 row ~Q1 FE nz = Number of matrix row Storage format: ELLPACK Q1 ~Q1

15 Solid body rotation Pure convection problem u + r (vu) =0 in =(0, 1) 2 ~Q1 FE NL-FCT sim. time t=2π u = 0 on inflow Velocity field v(x, y) =(0.5 y, x 0.5) Grid size h =1/2 l,l =5, 6,... Stochastic grid disturbance 2 {0%, 1%, 5%} Time step in Crank- Nicolson t =1.28 h Initial = exact solution at t =2 k, k 2 N

16 SBR: L2-error (0% mesh disturbance) Low-order Lin-FCT NL-FCT NL-GP NL-TVD Q1 FE ~Q1 FE 1 1 0,1 0,1 0, , Refinement level Refinement level

17 SBR: L2-error (5% mesh disturbance) Low-order Lin-FCT NL-FCT NL-GP NL-TVD Q1 FE ~Q1 FE 1 1 0,1 0,1 0, , Refinement level Refinement level

18 Rotation of a Gaussian hill Convection-diffusion equation u + r (vu ru) =0 ~Q1 FE NL-FCT time t=5/2π in =( 1, 1) 2 Velocity field, diffusion coefficient v(x, y) =( y, x), =0.001 Analytical solution u(x, y, t) = 1 4 t e r 2 4 t r 2 =(x ˆx) 2 +(y ŷ) 2 Position of the peak ˆx(t) = 0.5 sin t ŷ(t) = 0.5 cos(t) Other parameters as before

19 RGH: L2-error (0% mesh disturbance) Low-order Lin-FCT NL-FCT NL-GP NL-TVD Q1 FE ~Q1 FE ,1 0,1 0,01 0,01 0, , Refinement level Refinement level

20 RGH: dispersion-error (0% mesh disturbance) Low-order Lin-FCT NL-FCT NL-GP NL-TVD Q1 FE ~Q1 FE 5,5 5,5 4,5 4,5 3,5 3,5 2,5 2,5 1,5 1,5 0,5 0,5-0, , Refinement level Refinement level

21 RGH: L2-error (5% mesh disturbance) Low-order Lin-FCT NL-FCT NL-GP NL-TVD Q1 FE ~Q1 FE ,1 0,1 0,01 0,01 0, , Refinement level Refinement level

22 RGH: dispersion-error (5% mesh disturbance) Low-order Lin-FCT NL-FCT NL-GP NL-TVD Q1 FE ~Q1 FE 5,5 5,5 4,5 4,5 3,5 3,5 2,5 2,5 1,5 1,5 0,5 0,5-0, , Refinement level Refinement level

23 Taxonomy of finite elements conforming3:3 non conforming good accuracy good small numerical diffusion small smaller #DOFs, #edges larger irregular sparsity pattern regular

24 Can nonconforming finite elements help to improve performance on many-core hardware? NVIDIA Kepler GK110 Die Shot (taken from:

25 Example: edge-based flux-assembly with Q1 FE 1 OMP 2 OMP 4 OMP 8 OMP 12 OMP Xeon X5680(2x6) CUDA C best GPU performance for large problem sizes Bandwidth (GB/s) best CPU performance if data fits into cache 0 0K 1K 10K 100K 1.000K #edges per color group

26 Example: edge-based flux assembly with Q1 FE OMP 6 on Core i7 EP CUDA on C2070 #edges per color group F i = NX colors k=1 X ij2cg k ij(u j U i ) Bandwidth (GB/s) GB/s Color groups K 375K 250K 125K 0K Accumulated time (ms) w/ transf. 55 ms 23 ms 5,7x 132 ms CUDA OMP 6

27 Example: edge-based flux assembly with ~Q1 FE Bandwidth (GB/s) OMP 6 on Core i7 EP CUDA on C2070 #edges per color group 6-7 GB/s Color groups K 1.125K 750K 375K 0K Accumulated time (ms) w/ transf. 95 ms 32 ms 5,9x 191 ms CUDA OMP 6

28 Summary Nonconforming ~Q1 finite elements can be used within the algebraic flux correction framework are comparable to conforming FEs (accuracy/numerical diffusion) increase number of DOFs as compared to conforming Q1 FEs lead to system matrices with regular structure favorable for HPC Future plans Apply nonconforming AFC schemes to systems of equations Exploit benefits of nonconforming FEs for many-core architectures: speed up parallel (edge-based) assembly loops, implement more efficient matrix structures (ELLPACK), reduce communication costs in parallelized code

29 Acknowledgements AFC schemes D. Kuzmin (University of Erlangen-Nuremberg) CUDA/GPU programming D. Göddeke, D. Ribbrock, M. Geveler (TU Dortmund) Featflow2 M. Köster, P. Zajac (TU Dortmund) Source code freely available at: