A New Parallel Framework of SPH-SWE for Dam Break Simulation Based on OpenMP
Abstract
:1. Introduction
2. Methodology
2.1. Governing Equations
2.2. Water Depth Solutions
2.3. Speed Solution
2.4. Time Integration and Boundary Processing
3. SPH-SWE Model Solution Framework
Algorithm 1. Calculation framework of the SPH-SWEs model. This algorithm is needed to read the particles data (include fluid particles/virtual particles/open boundary particle/riverbed particles). |
Read parameters Output initial data of the model Mesh riverbed particles and calculate fluid particles and the net water depth Search particles dototal_number_of_timesteps
|
- Calculation of fluid particles in the riverbed;
- Particle search, and calculation of the water depth;
- Calculation of the fluid particle depth and velocity gradient;
- Acceleration and riverbed gradient correction;
- Calculation of velocity and displacement rate.
3.1. Fluid Particle Riverbed Calculation
Algorithm 2. Computing fluid particle riverbed. |
1. Stage 1: , initialize to 0 |
2. !$OMP PARALLEL DO PRIVATE(private variable),& |
3. !$OMP& SHARED(shared variable), DEFAULT(none), SCHEDULE(static) |
4. do total_number_of_fluid particles |
5. if particle_i is valid then |
6. Calculate particles’ mesh locations based on the riverbed’s mesh |
7. ! is used to make shepard correction(CSPM) |
8. CALL PURE celij_hb |
9. endif |
10. enddo |
11. !$OMP END PARALEL DO |
3.2. Particle Search
- Before each time step, the temporary grid position was updated, and each grid was assigned to a unique number; the grid size can be set to a fixed size dx_grid/dy_grid;
- According to the position of the current SPH particles, all the SPH particles were allocated to the temporary mesh space, and the particle chain in the mesh was established;
- According to the range (2hi) of the tight support region of particle i, the search of other meshes (- xsize to xsize, - ysize to ysize) was completed in the tight support region of the mesh, storing the mesh number;
- All the SPH particles i and j in the mesh were searched (icell-xsize to icell + size, jcell-ysize to jcell + ysize) in the tight support domain.
Algorithm 3: The particle search. Read in the particles data (include fluid particles/virtual parti`cles/open boundary particle/riverbed particles). |
1. In each timestep 2. Mesh all particles based on fixed size dx_grid/dy_grid(generally select the maximum smooth length) and particles into nc array 3. ncx/ncy: total number of grids in x/y direction 4. iboxvv/iboxff/iboxob: store the virtual particles, fluid particles and open boundary particles within the affected region into two dimensional arrays 5. !$OMP PARALLEL DO PRIVATE (private variable),SHARED(shared variable),& 6. !$OMP& SHARED(shared variable),DEFAULT(none) 7. do total_number_of_fluid particles 8. if particle_i is valid then 9. Calculate mesh of the particle i:icell/jcell 10. Calculate search mesh range of particle i:xsize/yszie 11. do row -ysize,ysize 12. irow=jcell+1 13. do column -xsize,xsize 14. icolumn=icell+column 15. Calculate number of search grid: gridn 16. gridn=icolumn+(irow-1)*ncx 17. !Search for Virtual particles in the scope of I particle 18. do j nc(grindn,1) 19. if particle_i and particle_j are neighbours then 20. Write particle_j to iboxvv array 21. endif 22. enddo 23. !Search for Fluid particles in the scope of i particle 24. do j nc(grindn,2) 25. if particle_i and particle_j are neighbours then 26. Write particle_j to iboxff array 27. endif 28. enddo 29. !Search for Open boundary particles in the scope of i particle 30. do j nc(grindn,3) 31. if particle_i and particle_j are neighbours then 32. Write particle_j to iboxob array 33. endif 34. enddo 35. enddo 36. enddo 37. endif 38. enddo 39. !$OMP END PARALLEL DO |
3.3. Water Depth Calculation
Algorithm 4: Water depth calculation. |
1. Stage 1: Guess for density and smoothed length 2. !$OMP PARALLEL DO PRIVATE(private variable),& 3. !$OMP& SHARED(shared variable),DEFAULT(none),SCHDULE(static) 4. do total_number_of_fluid particles 5. if particle_i is valid then 6. 1a: 7. 1b: 8. endif 9. enddo 10. !$OMP END PARALLEL DO 11. CALL particle search() %Search particles 12. Stage 2: Calculate depth 13. do while ((maxval(resmax) .gt. Minimum error) .and. (Iterationtimes .lt. max) 14. !$OMP PARALLEL DO PRIVATE(private variable),& 15. !$OMP& SHARED(shared variable),DEFAULT(none),SCHEDULE(static) 16. do total_number_of_fluid particles 17. if particle_i is valid then 18. CALL PURE fluid particle(i,rhop_sum(i),alphap(i)) 19. CALL PURE virtual particle(i,rhop_sum(i),alphap(i)) 20. CALL PURE open boundary particle((i,rhop_sum(i),alphap(i)) 21. %Calculate next step’s water depth and the smooth length 22. 23. 24. 25. 26. endif 27. enddo 28. !$OMP END PARALLEL DO 29. enddo |
3.4. Velocity Calculations
Algorithm 5: Calculation of fluid particle velocity and water depth gradient. |
1. Stage 1: sum_f/alphap/grad_up/grad_vp/grad_dw=0, Initialize to 0 2. !$OMP PARALLEL DO PRIVATE(private variable),& 3. !$OMP& SHARED(shared variable),DEFAULT(none),SCHDULE(static) 4. do total_number_of_fluid particles 5. if particle_i is valid then 6. !First conduct matrix for gradient correction 7. CALL PURE celij_corr(i,sum_f (I,1:4)) 8. CALL PURE celij_alpha(i,alphap(i),grad_dw(i,1:2),grad_up(i,1:2), grad_vp(i,1:2)) 9. CALL PURE celij_alpha_vir(i,alphap(i)) 10. CALL PURE celij_alphap_ob(i,alphap(i),grad_dw(i,1:2),grad_up(i,1:2), grad_vp(i,1:2)) 11. endif 12. enddo 13. !$OMP END PARALLEL DO |
3.5. Calculation of Fluid Particle Acceleration, Riverbed Scouring, Speed, and Displacement
Algorithm 6: Calculation of acceleration, velocity, and position. The Lagrangian equation of motion for a particle i is d/dt ∂L/(∂v_i )-∂L/(∂x_i )=0, where the Lagrangian functional L is defined in term of kinetic energy K and potential energy π as L = K-π, where π is a function of particles position but not velocity. |
1. Stage 1: Calculate 2. !$OMP PARALLEL DO PRIVATE(private variable),& 3. !$OMP& SHARED(shared variable),DEFAULT(none),SCHDULE(static) 4. do total_number_of_fluid particles 5. if particle_i is valid then 6. 1a. use Riemann solution to calculate 7. 1b. use Numerical viscosity to calculate 8. ! ar(i) is used to calculate depth 9. CALL PURE fluid particle(i,ar(i),ay(i),ar(i)) 10. CALL PURE virtual particle(i,ar(i),ay(i),ar(i)) 11. CALL PURE open boundary particle(i,ar(i),ay(i),ar(i)) 12. endif 13. Stage 2: Calculate 14. Stage 3: Calculate 15. Stage 4: Calculate velocity and position of fluid particle 16. enddo 17. !$OMP END PARALLEL DO |
4. Applications
4.1. Validation 1: 2-D Dry Bed Dam Break with Particle Splitting
4.2. Validation 2: 2-D Dam Break with A Rectangular Obstacle Located in the Downstream Area
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Parameters | Artificial Viscosity | Lax Friedrichs Flux | Two-Shocks Riemann Solver | |
---|---|---|---|---|
Mean Absolute Error | Speed | 0.0617 | 0.0476 | 0.0351 |
WD | 0.0515 | 0.0482 | 0.0356 | |
Mean Relative Error | Speed | 0.0718 | 0.0667 | 0.0586 |
WD | 0.0142 | 0.0089 | 0.0092 | |
Standard deviation of error | Speed | 0.1042 | 0.0728 | 0.0568 |
WD | 0.0681 | 0.0633 | 0.0484 |
Parameters | 30 s | 50 s | |
---|---|---|---|
Mean Absolute Error | Speed | 0.0529 | 0.0603 |
Water Depth | 0.0567 | 0.0613 | |
Mean Relative Error | Speed | 0.0548 | 0.1037 |
Water Depth | 0.0075 | 0.0081 | |
Standard deviation of error | Speed | 0.1543 | 0.1514 |
Water Depth | 0.1257 | 0.1238 |
Case | Number of Fluid Particles | Number of Virtual Particles | Number of Riverbed Particles |
---|---|---|---|
Case 1 | 4374 | 1276 | 14,094 |
Case 2 | 9801 | 3300 | 31,581 |
Case 3 | 38,801 | 9424 | 125,561 |
Cases | ||||||
---|---|---|---|---|---|---|
Open Source Code (Case 1) | N/A | 1040.44 | 213.12 | 1253.56 | 1.0 | |
Parallel Operation Code | Single Core | 87.47 | 174.95 | 49.99 | 312.41 | 4.01 |
2000 | 60.27 | 118.38 | 36.59 | 215.24 | 5.82 | |
1000 | 36.43 | 70.34 | 18.84 | 125.61 | 9.98 | |
Open Source Code (Case 2) | N/A | 5892.29 | 1039.82 | 6932.11 | 1.0 | |
Parallel Operation Code | Single Core | 338.853 | 643.8207 | 146.8363 | 1129.51 | 6.14 |
2000 | 116.8514 | 218.6252 | 41.4634 | 376.94 | 18.39 | |
1000 | 75.2985 | 150.597 | 33.7545 | 259.65 | 26.70 | |
Open Source Code (Case 3) | N/A | 107,218.04 | 16,021.09 | 123,239.13 | 1.0 | |
Parallel Operation Code | Single Core | 5498.28 | 10,481.09 | 1202.75 | 17,182.12 | 7.17 |
2000 | 554.20 | 990.83 | 134.35 | 1679.38 | 73.38 |
Number of Single-Thread Particles (pcs) | ||||
---|---|---|---|---|
2000 | 20 | 1679.38 | 10.23 | 51.16 |
2500 | 16 | 1759.15 | 9.77 | 61.05 |
3000 | 13 | 1833.58 | 9.37 | 72.08 |
4000 | 10 | 1935.12 | 8.88 | 88.79 |
5000 | 8 | 2300.79 | 7.47 | 93.35 |
6000 | 7 | 2579.88 | 6.66 | 95.14 |
7000 | 6 | 2934.02 | 5.86 | 97.60 |
8000 | 5 | 3455.94 | 4.97 | 99.44 |
10,000 | 4 | 4312.39 | 3.98 | 99.61 |
20,000 | 2 | 8616.82 | 1.99 | 99.70 |
Case | Particle Spacing | Number of Fluid Particles | Number of Virtual Particles | Number of Riverbed Particles | T8 (s) | T (s) |
---|---|---|---|---|---|---|
Case 4 | 0.01 | 12,423 | 4798 | 129,645 | 1511.38 | 7.12 |
Case 5 | 0.005 | 51,858 | 9582 | 516,889 | 14,538.83 | 7.32 |
Case 6 | 0.002 | 323,145 | 23,934 | 807,111 | 108,868.42 | 7.46 |
Cases | R (t) | C (t) | A (t) | T8 (s) | t (s) |
---|---|---|---|---|---|
Case 4 | 468.53 | 876.60 | 166.25 | 1511.38 | 7.12 |
Case 5 | 4216.26 | 8432.52 | 1890.05 | 14,538.83 | 7.32 |
Case 6 | 34,837.91 | 66,409.72 | 7620.80 | 108,868.42 | 7.46 |
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Wu, Y.; Tian, L.; Rubinato, M.; Gu, S.; Yu, T.; Xu, Z.; Cao, P.; Wang, X.; Zhao, Q. A New Parallel Framework of SPH-SWE for Dam Break Simulation Based on OpenMP. Water 2020, 12, 1395. https://doi.org/10.3390/w12051395
Wu Y, Tian L, Rubinato M, Gu S, Yu T, Xu Z, Cao P, Wang X, Zhao Q. A New Parallel Framework of SPH-SWE for Dam Break Simulation Based on OpenMP. Water. 2020; 12(5):1395. https://doi.org/10.3390/w12051395
Chicago/Turabian StyleWu, Yushuai, Lirong Tian, Matteo Rubinato, Shenglong Gu, Teng Yu, Zhongliang Xu, Peng Cao, Xuhao Wang, and Qinxia Zhao. 2020. "A New Parallel Framework of SPH-SWE for Dam Break Simulation Based on OpenMP" Water 12, no. 5: 1395. https://doi.org/10.3390/w12051395