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  • School of Mathematical SciencesQueensland University of Technology

    Modelling Sea Water Intrusionin Coastal Aquifers UsingHeterogeneous Computing

    Benjamin CummingBachelor of Applied Science (Mathematics)

    Bachelor of Applied Science (Hons I)Masters of Applied Science (Mathematics)

    A thesis submitted for the degree of Doctor of Philosophy in the Faculty ofScience and Technology, Queensland University of Technology according to

    QUT requirements.

    Principal Supervisor:Associate Supervisors:

    Professor Ian TurnerDr Timothy MoroneyAssociate Professor Malcom CoxAssociate Professor Les DawesProfessor Vo Anh

    2012

  • Abstract

    The objective of this PhD research program is to investigate numerical meth-ods for simulating variably-saturated flow and sea water intrusion in coastalaquifers in a high-performance computing environment. The work is dividedinto three overlapping tasks: to develop an accurate and stable finite volumediscretisation and numerical solution strategy for the variably-saturated flowand salt transport equations; to implement the chosen approach in a highperformance computing environment that may have multiple GPUs or CPUcores; and to verify and test the implementation.

    The geological description of aquifers is often complex, with porous materialspossessing highly variable properties, that are best described using unstruc-tured meshes. The finite volume method is a popular method for the solutionof the conservation laws that describe sea water intrusion, and is well-suitedto unstructured meshes. In this work we apply a control volume-finite ele-ment (CV-FE) method to an extension of a recently proposed formulation(Kees and Miller, 2002) for variably saturated groundwater flow. The CV-FEmethod evaluates fluxes at points where material properties and gradientsin pressure and concentration are consistently defined, making it both suit-able for heterogeneous media and mass conservative. Using the method oflines, the CV-FE discretisation gives a set of differential algebraic equations(DAEs) amenable to solution using higher-order implicit solvers.

    Heterogeneous computer systems that use a combination of computationalhardware such as CPUs and GPUs, are attractive for scientific computing dueto the potential advantages offered by GPUs for accelerating data-parallel op-erations. We present a C++ library that implements data-parallel methodson both CPU and GPUs. The finite volume discretisation is expressed interms of these data-parallel operations, which gives an efficient implementa-tion of the nonlinear residual function. This makes the implicit solution ofthe DAE system possible on the GPU, because the inexact Newton-Krylovmethod used by the implicit time stepping scheme can approximate the actionof a matrix on a vector using residual evaluations. We also propose precon-ditioning strategies that are amenable to GPU implementation, so that allcomputationally-intensive aspects of the implicit time stepping scheme areimplemented on the GPU.

  • Results are presented that demonstrate the efficiency and accuracy of theproposed numeric methods and formulation. The formulation offers excellentconservation of mass, and higher-order temporal integration increases bothnumeric efficiency and accuracy of the solutions. Flux limiting producesaccurate, oscillation-free solutions on coarse meshes, where much finer meshesare required to obtain solutions with equivalent accuracy using upstreamweighting. The computational efficiency of the software is investigated usingCPUs and GPUs on a high-performance workstation. The GPU version offersconsiderable speedup over the CPU version, with one GPU giving speedupfactor of 3 over the eight-core CPU implementation.

  • Statement of Original Authorship

    The work contained in this thesis has not been previously submitted for adegree or diploma at any other higher educational institution. To the best ofmy knowledge and belief, the thesis contains no material previously publishedor written by another person except where due reference is made.

    Signed:

    Date: 18 September 2012

    QUT Verified Signature

  • Contents

    1 Introduction 1

    1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 3

    1.2 Objectives of The Thesis . . . . . . . . . . . . . . . . . . . . . 11

    1.3 Contribution of The Thesis . . . . . . . . . . . . . . . . . . . . 15

    1.4 Overview of The Thesis . . . . . . . . . . . . . . . . . . . . . 15

    2 Problem Formulation 19

    2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 19

    2.2 Closure Of The System . . . . . . . . . . . . . . . . . . . . . . 22

    2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 26

    2.4 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    3 Computational Techniques 35

    3.1 The Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    3.1.1 The Finite Element Mesh . . . . . . . . . . . . . . . . 36

    iii

  • CONTENTS iv

    3.1.2 The Dual Mesh . . . . . . . . . . . . . . . . . . . . . . 37

    3.1.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . 42

    3.2 The Control Volume-Finite ElementMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    3.2.1 Accumulation Terms . . . . . . . . . . . . . . . . . . . 48

    3.2.2 Source Terms . . . . . . . . . . . . . . . . . . . . . . . 51

    3.2.3 Surface Fluxes . . . . . . . . . . . . . . . . . . . . . . . 51

    3.2.4 Discretised Equations . . . . . . . . . . . . . . . . . . . 63

    3.3 Temporal Solution . . . . . . . . . . . . . . . . . . . . . . . . 68

    3.3.1 Solving the Linear System . . . . . . . . . . . . . . . . 71

    3.3.2 Preconditioner . . . . . . . . . . . . . . . . . . . . . . . 72

    3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    4 Algorithms and Data Structures 77

    4.1 Time Stepping with IDA . . . . . . . . . . . . . . . . . . . . . 78

    4.2 Evaluating the Residual:The CV-FE Discretisation . . . . . . . . . . . . . . . . . . . . 80

    4.2.1 Time Step Preprocessing . . . . . . . . . . . . . . . . . 83

    4.2.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . 88

    4.2.3 Edge-Based Weighting . . . . . . . . . . . . . . . . . . 89

    4.2.4 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . 90

    4.2.5 Flux Assembly . . . . . . . . . . . . . . . . . . . . . . 94

    4.2.6 Residual Assembly . . . . . . . . . . . . . . . . . . . . 96

    4.3 Domain Decomposition . . . . . . . . . . . . . . . . . . . . . . 97

  • CONTENTS v

    4.4 Preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    4.4.1 The Global Matrix . . . . . . . . . . . . . . . . . . . . 100

    4.4.2 Finding the Local Block . . . . . . . . . . . . . . . . . 101

    4.4.3 Preconditioning the Local Block . . . . . . . . . . . . . 103

    4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

    5 GPU Implementation 107

    5.1 Using GPUs as Computational Accelerators . . . . . . . . . . 108

    5.1.1 Fermi Architecture . . . . . . . . . . . . . . . . . . . . 110

    5.2 The vectorlib Library . . . . . . . . . . . . . . . . . . . . . . . 113

    5.3 Sparse Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 116

    5.4 Domain Decomposition . . . . . . . . . . . . . . . . . . . . . . 118

    5.4.1 Sub-Dividing the Computer Into Processes . . . . . . . 118

    5.4.2 Mesh Generation and Domain Decomposition . . . . . 119

    5.4.3 Communication Between Processes . . . . . . . . . . . 120

    5.5 The IDA Library . . . . . . . . . . . . . . . . . . . . . . . . . 124

    5.6 The CV-FE Discretisation . . . . . . . . . . . . . . . . . . . . 126

    5.6.1 Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 126

    5.6.2 Edge-Based Weighting . . . . . . . . . . . . . . . . . . 129

    5.6.3 Interpolation . . . . . . . . . . . . . . . . . . . . . . . 132

    5.6.4 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . 132

    5.6.5 Flux Assembly . . . . . . . . . . . . . . . . . . . . . . 134

    5.6.6 Residual Assembly . . . . . . . . . . . . . . . . . . . . 136

  • CONTENTS vi

    5.7 Mesh Renumbering To Optimise Indirect Indexing . . . . . . . 138

    5.7.1 Indirect Indexing in Computing Relative Permeability . 140

    5.8 Implementation of Preconditioners . . . . . . . . . . . . . . . 141

    5.8.1 Forming the Preconditioner . . . . . . . . . . . . . . . 142

    5.8.2 Applying the Preconditioner . . . . . . . . . . . . . . . 143

    5.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

    6 Model Verification 148

    6.1 Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

    6.1.1 Richards Equation: Infiltration Into Dry Heteroge-neous Soil The dry infiltration Test Case . . . . . . . 150

    6.1.2 Richards Equation: Transient Water Table Experi-ment The water table Test Case . . . . . . . . . . . . 152

    6.1.3 Transport Model: Flow and Transport in UnsaturatedSoil The unsaturated transport Test Case . . . . . . . 153

    6.1.4 Transport Model: Flow Tank Experiments The tank steady ,tank plume and tank tidal plume Test Cases of Zhang . 155

    6.1.5 Transport Model: Leaching of a Contaminant Plumein a Shallow Aquifer The heap leaching Test Case . . 159

    6.2 Richards equation: the dry infiltration test case . . . . . . . . 161

    6.3 Richards equation: the water table test case . . . . . . . . . . 173

    6.4 Transport Model: Unsaturated flow and transport . . . . . . . 180

    6.5 Transport Model: Zhangs Flow Tank Experimen

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