ITEM METADATA RECORD
Title: Soil Modeling, Dynamic Refinement, and Shared Memory Parallelization within SPH
Other Titles: Grondmodellering, dynamische verfijning en gedeeld geheugen parallellisatie in SPH
Authors: Reyes López, Yaidel
Issue Date: 24-Nov-2014
Abstract: The Smoothed Particle Hydrodynamics (SPH) method is an alternative to traditional mesh-based techniques such as the Finite Element Method (FEM). Due to its particle-based Lagrangian nature the method is very attractive to simulate and study complex problems involving large deformations, moving boundaries, free surfaces and multiple phases, which can be challenging for mesh-based approaches. In this thesis we extend the applicability of SPH in three ways: (a) the application and validation of the method for the 3D computation of cohesive soil, (b) the development of a dynamic particle refinement procedure, and (c) the parallelization of the method for shared memory computers. The model used for the 3D computation of soil is based on an elastic-perfectly plastic stress-strain relationship with the Drucker-Prager yield criterion. We apply the artificial stress method to avoid tensile instabilities, a numerical issue that often appears when simulating cohesive soil in SPH. The artificial stress method is extended to the 3D general stress state. The model is validated by comparing SPH results with simulations obtained with FEM. We present a dynamic particle refinement procedure based on particle splitting. This procedure allows to dynamically increase the resolution of the discretization by replacing a refined particle with new daughter particles. The optimal separation and smoothing distance of the newly introduced particles are determined by taking into account the error introduced by the refinement in the gradient of the kernel, and by reducing possible numerical instabilities. Finally, we develop a parallel implementation of SPH for shared memory computers. The presented approach is based on domain decomposition and space filling curves. This ensures per thread local storage of most frequently accessed data, avoids NUMA-unfriendly memory allocations, reduces data races and allows efficient calculation of symmetric inter-particle forces.
Table of Contents: Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Smoothed particle hydrodynamics . . . . . . . . . . . . . . . . . . . . 3
1.2 Aims and scope of this thesis . . . . . . . . . . . . . . . . . . . . . 4
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Smoothed Particle Hydrodynamics . . . . . . . . . . . . . . . . . . . . . 9
2.1 Approximation technique . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Kernel approximation . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Particle approximation . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Properties of the kernel function . . . . . . . . . . . . . . . . . . . 12
2.3 Discretization of the Navier-Stokes equations . . . . . . . . . . . . . 13
2.3.1 The continuity equation . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 The momentum equation . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 Free surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.2 Solid walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Elastic-perfectly plastic constitutive model for soil . . . . . . . . . . 21
3.1 Soil as a continuum . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Elastic-perfectly plastic behavior . . . . . . . . . . . . . . . . . . 22
3.2.1 Flow theory of plasticity . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2 Stress-strain relationship . . . . . . . . . . . . . . . . . . . . . 25
3.3 Drucker-Prager yield criterion . . . . . . . . . . . . . . . . . . . . 26
3.4 Correction of numerical errors in computational plasticity . . . . . . 29
3.4.1 Tension cracking treatment . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Scaling back procedure . . . . . . . . . . . . . . . . . . . . . . . 29

4 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Weakly compressible SPH (WCSPH) for free surface fluid flows . . . . . . 31
4.1.1 Quasi-incompressibility . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.2 Artificial viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2 Elastic-perfectly plastic soil model in SPH . . . . . . . . . . . . . . 33
4.2.1 Stress-strain relationship . . . . . . . . . . . . . . . . . . . . . 34
4.2.2 Tensile instability and the artificial stress method . . . . . . . . . 36
4.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.4 Initial stresses . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Validation of the elastic-perfectly plastic soil model in 3D . . . . . . 41
5.1 Oscillating elastic beam . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Compression test . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Slope stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Dynamic particle refinement . . . . . . . . . . . . . . . . . . . . . . . 53
6.1 Varying resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3 Dynamic refinement procedure . . . . . . . . . . . . . . . . . . . . . . 55
6.3.1 Refinement criterion . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.3.2 Particle refinement . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.4 Numerical implications of the refinement . . . . . . . . . . . . . . . . 58
6.4.1 Refinement error . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.4.2 Density refinement error using the continuity density approach . . . . 59
6.4.3 Numerical instabilities . . . . . . . . . . . . . . . . . . . . . . . 60
6.4.4 Refinement parameters for the square pattern . . . . . . . . . . . . . 63
6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.5.1 Water discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.5.2 Splash of a drop of water . . . . . . . . . . . . . . . . . . . . . . 66
6.5.3 Non-cohesive soil model. Aluminum bars test case . . . . . . . . . . 67
6.5.4 Non-cohesive soil failure in 3D . . . . . . . . . . . . . . . . . . . 71
6.5.5 General comments about the selection of the refinement parameters . . 72
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7 Parallelization for modern shared memory architectures . . . . . . . . . 75
7.1 Parallelization on Shared Memory . . . . . . . . . . . . . . . . . . . 76
7.2 General considerations . . . . . . . . . . . . . . . . . . . . . . . . 77
7.3 Data Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.4 NUMA-friendly memory allocation . . . . . . . . . . . . . . . . . . . . 79
7.5 Domain decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.6 Handling non-local data . . . . . . . . . . . . . . . . . . . . . . . . 81
7.7 Final algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.7.1 Precomputing particle interactions . . . . . . . . . . . . . . . . . 83
7.7.2 Processing Particle Interactions . . . . . . . . . . . . . . . . . . 84
7.7.3 Final algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.8.1 Strong scalability . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.8.2 Weak scalability . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.8.3 Analysis of the load balance . . . . . . . . . . . . . . . . . . . . 91
7.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.2 Directions for future research . . . . . . . . . . . . . . . . . . . . 99
8.2.1 Simulation of cohesive soil . . . . . . . . . . . . . . . . . . . . . 99
8.2.2 Dynamic adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.2.3 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A Kernel gradient error . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
ISBN: 978-94-6018-916-6
Publication status: published
KU Leuven publication type: TH
Appears in Collections:Numerical Analysis and Applied Mathematics Section
Division of Mechatronics, Biostatistics and Sensors (MeBioS)

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