An Improved Incompressible Smoothed Particle Hydrodynamics ...

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Chapter 2. The SPH Methodology, Existing and New ISPH Methods instability may occur due to highly distorted particle spacings. It is partially concluded here: • The wall boundary condition is generally built up with mirror particle or dummy particle method. The dummy particle method is more ef- ficient without updating the dummy particles during the simulations. Full comparison of these two boundary methods will be presented in the next chapter. • By projecting the intermediate velocity field onto the divergence-free and curl-free fields, the projection algorithm highly improves the accu- racy of the SPH method [63], taking the original incompressible SPH method presented by Cummins and Rudman [17] as an example. How- ever, the instability is set in the simulations when the particle spacing is highly distorted. <strong>An</strong>other approach, called ISPH_DI here, was intro- duced by Shao and Lo [100], and the incompressibility is achieved by maintaining the particle volume constant. In 2007, Hu and Adams proposed another ISPH method, called ISPH_DFDI here, in which both constant particle volume and divergence-free velocity field are kept, and the stability is well kept by this method. In this chapter, a new approach called ISPH_DFS is also proposed. By slightly shifting the particle position and correcting the corresponding hydrodynamic field, the instability problem is overcome with ISPH_DFS method. 65
Chapter 3 The <strong>Incompressible</strong> SPH Code THe incompressible SPH program is coded based on the original Weakly Compressible SPH (WCSPH) code SPHysics [121]. In SPHysics, the pressure field is calculated from a state equation without solving the pressure Poisson equation. This code is an open-source SPH code that has been released in 1 Aug 2007, developed jointly by researchers at the Johns Hop- kins University (U.S.A.), the University of Vigo (Spain), the University of Manchester (U.K.) and the University of Rome La Sapienza (Italy) [121]. Version 1.4 and below contains the basic SPH formulation for free-surface flows. The 2-D & 3-D code has been developed specifically for free-surface hydrodynamics. The structure of the weakly compressible code SPHysics can be referred to Appendix A and [31]. Based on the code SPHysics, an ISPH code is programmed. Both Laplacian operators shown in §2.1.2 are built. Two linear solvers, namely Conjugated Gradient (CG) and Bi-Conjugated Gradient (Bi-CGSTAB), are programmed. The predictor-corrector time marching scheme is applied. 66
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An Improved Incompressible Smoothed
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CONTENTS 2.1.2 SPH operators . . .
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CONTENTS 5 Validation by Benchmark
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List of Figures 1.1 Density and vel
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LIST OF FIGURES 4.8 The geometry an
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LIST OF FIGURES 6.6 Geometry of dam
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List of Tables 4.1 Computing costs
Page 15 and 16: Abstract attempts to improve the ac
Page 17 and 18: Declaration No portion of the work
Page 19 and 20: Copyright 4. Further information on
Page 21 and 22: Acknowledgments Armando and other f
Page 23 and 24: Chapter 1. Introduction and Thesis
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Page 71 and 72: 3.2.3 Velocity at time n+1 u n+1 Th
Page 75 and 76: 3.5 Wall boundary tests Chapter 3.
Page 77 and 78: P 0 -0.1 -0.2 -0.3 -0.4 -0.5 Chapte
Page 79 and 80: 3.7 Code structure Chapter 3. The I
Page 83 and 84: Chapter 4. ISPH Method Comparisons
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Page 113 and 114: Chapter 5. Validation by Benchmark
Page 115 and 116: u/U 1 0.8 0.6 0.4 0.2 0 -0.2 0 0.2
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P/(ρU 2 ) 2.5 2 1.5 1 0.5 0 Chapte
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P/(ρU 2 ) 1 0.8 0.6 0.4 0.2 0 -0.2
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Chapter 5. Validation by Benchmark
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Y/d Y/d Y/d 4 2 0 4 2 0 4 2 0 Chapt
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y/d 6 5 4 3 2 1 0 Chapter 5. Valida
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Cd Cd 4.7 4.6 4.5 4.4 4.3 Chapter 5
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y/d y/d 5 4 3 2 1 Chapter 5. Valida
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C d max / C d min 2.85 2.8 2.75 2.7
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5.3 Partial conclusion Chapter 5. V
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Chapter 6. ISPH_DFS Method in Free-
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Relative Error 6e-05 5e-05 4e-05 3e
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Y/a Y/a Y/a 3 2 1 Chapter 6. ISPH_D
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y (m) 0.5 0.45 0.4 0.35 0.3 0.25 0.
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Chapter 7. Conclusions and Future W
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7.2 Future work and suggestions Cha
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Chapter A. The SPHysics Code In SPH
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Chapter A. The SPHysics Code SPS mo
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Chapter A. The SPHysics Code Also t
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n+1 i = rn i +∆tun i 184 Chapter
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P0 = r0 Chapter B. Linear Solvers (
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If Xi is accurate enough then quite
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Conference Publications: Chapter C.
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BIBLIOGRAPHY [7] D. L. Brown, R. Co
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BIBLIOGRAPHY [24] J. W. Dold and D.
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BIBLIOGRAPHY [39] D. Hietel, K. Ste
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BIBLIOGRAPHY [55] J. Klapp, L. D. G
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BIBLIOGRAPHY [71] J. J Monaghan. Sm
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BIBLIOGRAPHY [88] R. K. Price, The
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BIBLIOGRAPHY [105] D. Teleaga and J
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BIBLIOGRAPHY [121] SPHysics, open-s
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