I would like to express my deepest appreciation and sincere gratitude to ... multigrid methods are proposed to solve different problems including: ...... instances, the mathematical formulations of the fundamental laws of fluid ...... useful point of view, then, is to identify the grid points with the indices of the unknown quantities.
Zagazig University Faculty of Engineering Physics and Engineering Mathematics Department
Efficient Multigrid Techniques for the Solution of Fluid Dynamics Problems A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy In Physics and Engineering Mathematics
Presented by
Norhan Alaa El Dain Mohamed Assistant Lecturer in Physics and Engineering Mathematics Department Supervised by Prof. Dr.
Prof. Dr.
Salwa Amin Mohamed
Ahmed Farouk Abdel Gawad
Physics and Eng. Mathematics Department
Mechanical Power Eng. Department
Faculty of Engineering
Faculty of Engineering
Zagazig University
Zagazig University
Prof. Dr.
Mohamed Saad Matbuly Physics and Eng. Mathematics Department Faculty of Engineering Zagazig University
2013
بِ ْس ِم الْلَّ ِه ال َّْر ْح َم ِن ال َّْرِح ْي ِم
ب ا ْشرح لِ ص ْد ِري ي قَ َ َ ال َر ِّ َ ْ س ْر لِي أ َْم ِري َويَ ِّ
واحلُل عُ ْق َد ًة ِمن لِ سانِي يَ ْف َق ُهوا قَ ْولِي َ ْ ْ ْ َ صدق اهلل العظيم
(سوره طه أيه 25إلى )28
Dedicated to: My dear parents ………… My dear husband Amr ………. My dear sons Alaa, Moustafa &Mohannad…. And …….To my great family ……..
ii
Acknowledgments First for most and always, I bow my head in gratitude to Great Allah, the merciful, the compassionate who gave me the ability to complete this work. I would like to express my deepest appreciation and sincere gratitude to Prof. Dr. Salwa Amin Mohamed, Prof. of Engineering Mathematics, Faculty of Engineering, Zagazig University, for suggesting the subject, grateful help, valuable advice and continuous encouragement throughout the performance of this work. My cordial thanks and deepest appreciation to Prof. Dr. Ahmed F. Abdel Gawad, Prof. of Mechanical Engineering, Faculty of Engineering, Zagazig University, for his continuous help during all stages of this work. Supervisors and I want to introduce special thanks to Prof. Dr. Mohammed S. Matbuly, Prof. and Head of Department of Physics and Eng. Mathematics for his kind help. Grateful thanks are extended to the staff of Department Of Eng. Physics and Mathematics, for their help during the course of the study.
iii
Abstract The multigrid technique (MG) is one of the most efficient methods for solving a large class of problems very efficiently. One of these multigrid techniques is the algebraic multigrid (AMG) approach which is developed to solve matrix equations using the principles of usual multigrid methods. In this work, various algebraic multigrid methods are proposed to solve different problems including: general linear elliptic partial differential equations (PDEs), as anisotropic Poisson equation, problems with steep boundary layers, as convectional dominant convection-diffusion equations, and nonlinear system of equations as Navier-stokes equations. In addition, a new technique is introduced for solving convection-diffusion equation by predicting a modified diffusion coefficient (MDC) such that the discretization process applies on the modified equation rather than the original one. For a class of one-dimensional convection-diffusion equation, we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. Extending the same technique to obtain analytic MDC for other classes of convection-diffusion equations is not always straight forward especially for higher dimensions. However, we have extended the derived analytic formula of MDC (of the studied class) to general convection-diffusion problems. The analytic formula is computed locally within each element according to the mesh size and the values of the associated coefficients in each direction. The numerical results for two-dimensional, variable coefficients, convection-dominated problems show that although the discrete solution does not coincide with the exact one, it provides stable and accurate solution even on coarse grids. As a result, multigrid-based solvers benefit from these accurate coarse grid solutions and retained its efficiency when applied for convection– diffusion equations. Many numerical results are presented to investigate the convergence of classical algebraic and geometric multigrid solvers as well as Krylovsubspace methods preconditioned by multigrid. Also, in this thesis, we were concerned with the channel flow, which is an interesting problem in fluid dynamics. This type of flow is found in many real-life applications such as irrigation systems, pharmacological and chemical operations, oil-
iv
refinery industries, etc. In the present work, the channel flow with one and two obstacles are considered. The methodology is based on the numerical solution of the Navier-Stokes equations by using a suitable computational domain with appropriate grid and correct boundary conditions. Large-eddy simulation (LES) was used to handle the turbulent flow with Smagorinsky modeling. Finite- element method (FEM) was used for the discretization of the governing equations. Adaptive time stepping is used and the resulting linear algebraic systems are solved by different methods including preconditioned minimum residual method, geometric and algebraic multigrid methods. The investigation was carried out for a range of Reynolds number (Re) from 1 to 300 with a fixed blockage ratio β = 0.25 and an artificial source of turbulence is introduced in the inflow velocity profile to ensure the turbulent nature of the flow. The finite element method is used in the present work to discretize many CFD problems and we have developed algebraic multigrid (AMG) approaches for anisotropic elliptic equation, problems with steep boundary layers, as convectional dominant convection-diffusion equations, and nonlinear system of equations as Navier-stokes equations. The conclusions which are obtained in the present work can be stated as: (i)
AMG can be used for many kinds of problems where the application of standard multigrid methods is difficult or impossible.
(ii)
Implementation of the proposed MDC technique produces the exact nodal solutions for the 1-D singularly-perturbed convection diffusion problems even on coarse grids with uniform or non-uniform mesh sizes.
(iii)
Numerical results show that extension of MDC to 2-D eliminates the oscillations and produces more accurate solutions compared with other existing methods.
(iv)
As a result, multigrid-based solvers retain its efficient convergence rates for singularly-perturbed convection diffusion problems.
(v)
Excellent convergence behavior is obtained for numerical solution of Navier-Stokes system for different values of Re in two cases, 1- and 2obstacles, when we used the proposed AMG algorithm as a solver or a preconditioner of GMRES.
v
CONTENTS Page iii
ACKNOWLEDGMENTS ABSTRACT………………...………………………………………
iv
CONTENTS……………..…………………………………………
vi
LIST OF FIGURES …….…………………………………………
ix
LIST OF TABLES ………………….…………………………….
xii
NOMENCLATURE ……………………………………………….
Xiii
CHAPTER (1) Introduction ……………..……………………..…………
1
1.1 Multigrid Method…………………………………………………................. 1.2 Computational Fluid Dynamics (CFD)…………………………..………….
1
1.3 Outlines of the Thesis …………………..………………………………..…..
5
3
CHAPTER (2) Literature Review …………………………………………
7
2.1 Survey of the Computational Fluid Dynamics (CFD)…………………...…...
7
2.2 Survey Of Convection – Diffusion Equation……………………....................
11
2.3 Survey Of Navier Stokes Equations ………………………………………….
15
2.4 Survey Of Mutigrid Method………………………………………………….
21
CHAPTER (3) Multigrid Methods………………………………………....
26
3.1 Introduction …………………………………………………………………...
26
3.2 Algebraic versus Geometric Multigrid ………………………………………..
27
3.3 Geometric Multigrid Method (GMG)………………………….......................
30
3.4 Algebraic Multigrid Method (AMG)………………………………………....
34
3.4.1 Theoretical Basis of AMG………………………………………………
34
3.4.2 Formal Algebraic Multigrid Components and Notations……………….
37
3.4.3 The Algebraic Multigrid Algorithm…………………………………….
37
3.4.3.1 The Coarsening Process………………………………………….
38
3.4.3.2 The Interpolation Operator……………………………………....
40
3.4.4 Examples for Coarsening Process and Interpolation…………………….
41
vi
3.5 Convergence Behavior of GMG and AMG……………………………………
46
3.6 Multigrid for Systems of Equations……………………………………………
49
3.6.1 Introduction…………………………………………………………….
49
3.6.2 AMG for Systems of Partial Differential Equations……………………
50
3.6.3 The Proposed Two Steps Iteration Method……………………………...
51
CHAPTER (4) The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation…………………..
53
4.1 Introduction……………….…………………………………………………..
53
4.2 Convection Diffusion formulation………………………………....................
55
4.3 The Behavior Of The Convection Diffusion Equation [129]…….....................
58
4.3.1 A Class Of Convection Diffusion Problem ……………………………
58
4.3.2 Interpretation Of The Oscillatory Performance ………………………..
62
4.3.3 Investigation Of Numerical Solution For High Peclet Number………
64
4.4 The Streamline Diffusion Method [130]………………………………………..
65
4.5 A Modified Diffusion Coefficient Technique [129]……………………………
68
4.5.1 Estimation Of The Modified Diffusion Parameter..…………..………..
68
4.5.2 Generalization Of The Diffusion Parameter ……………………….….
71
4.8 Numerical Results ………………………………………………....................
72
4.8.1 Problem 1 ...……………………………………………………………
72
4.8.2 Problem 2 ........…………………………………………………………
75
4.8.3 Problem 3 .......…………………………………………………………
80
4.8.4 Problem 4 ......…………………………………………………………
84
CHAPTER (5) Investigation of the Turbulent Flow Inside the 2-D Channel with Internal Obstacles…….………..………..…
87
5.1 Introduction …………………………………………………………………..
87
5.2 Our approach to solve unsteady Navier-Stokes equation…………………….
88
5.3 The variational multiscale method and its relation to turbulence theory.
90
5.4 Adaptive Time Stepping Of Navier Stokes Equation ……………………….
93
5.4.1 Time Integration ……………………………………………………….
95
5.4.2 Solving The Discrete Oseen System …………………………………..
96
vii
5.4.3 Used Finite Element Mesh ……………………………….....................
97
5.5 Our approach to solve steady Navier-Stokes equation………………………
98
5.6 Subgrid –Scale Modeling Within The Multiscale Environment …………….
103
CHAPTER (6)
The Numerical Results of the Flow inside the 2-D 105
Channel with Internal Obstacles 6.1 Introduction ……………………………………..........................................
105
6.2 Computational Domain And Boundary Conditions [139]................................
105
6.3 Turbulence Generation ……………………………………………………….
107
6.4 Results And Discussions [138]……………………………………………..
107
6.4.1 Flow Pattern Around A Single Square Obstacles ……………………...
108
6.4.2 Flow Pattern Around Two Square Obstacles……………………………
115
6.5 The Performance of AMG in Solving the Navier Stokes Flow……………….
128
6.5.1 The Performance of AMG of unsteady flow in a channel with 1-obstacle
128
6.5.2 The Performance of AMG of unsteady flow in a channel with 2-obstacles
130
6.5.3 The Performance of AMG of steady flow………………………………
132
CHAPTER (7) Conclusions and Future work………………………………… REFERENCES………………………………………………………………. ARABIC SUMMARY………………………………………………………..
viii
134 137
LIST OF FIGURES
Figure Figure Title No.
Page
(2.1)
The turbulence flow Past Circular Cylinder.
8
(2.2)
The flow Past Circular Cylinders at Low Speeds.
9
(2.3)
Some common examples for liquid flows.
16
(2.4)
Some common examples for gas (air) flows.
17
(3.1)
Algebraic versus geometric multigrid [92].
28
(3.2)
The nonzero structure of A, where X indicates a nonzero entry, is shown on the left. The resulting undirected adjacency graph appears on the right.
29
(3.3)
Influence of lexicographic Gauss-Seidel iteration on the error [92].
(3.4)
A sequence of coarse grids for the unit square starting with ℎ = 16 [92].
31
(3.5)
A fine and a coarse grid with the injection operator [92].
32
(3.6)
V- cycle of the multigrid with 4 grid levels.
33
(3.7)
The fine grid points and its numbering.
41
(3.8)
The stencil relation between any point (i) and its neighbors.
42
(3.9)
The coarse and fine point for case 1.
43
(3.10)
The coarse and fine point for case 2.
45
(3.11)
Convergence behavior for AMG and GMG for 𝑎 = 1 and 𝑏 = 1.
47
(3.12)
Convergence behavior for AMG and GMG for 𝑎 = 1 and 𝑏 = 10.
48
(3.13)
Convergence behavior for AMG and GMG for 𝑎 = 1 and 𝑏 = 1000.
49
(4.1)
The exact solution of equation (4.18) for different 𝜈.
60
(4.2)
Performance of Galerkin FEM, ( 𝑃𝑒 = 2𝜐 ,𝑟 = 1−𝑃𝑒 ).
(4.3) (4.4)
1
ℎ
1+𝑃𝑒
Oscillation of the discrete solution for large values of Pe for even and odd numbers of elements (n). Relation between the diffusion coefficient 𝜈 and the modified one 𝜈0 .
ix
30
61 65 70
(4.5)
(4.6) (4.7) (4.8) (4.9) (4.10)
Comparison of solutions of Problem 1 on a uniform grid by different numerical techniques Full agreement of the computed solution by MDC and the exact solution for different values of 𝜈 on a non-uniform grid. Exact Solution of Problem 2 (𝜖 = 0.05).
73
74 75
Galerkin finite-element solution of problem 2 with uniform grid size h=1/16 for different values of 𝜖. Finite-element solution using MDC of Problem 2 (𝜖 = 0.001, ℎ = 1/16 ). Distribution of the discretization error norm of the finite-element solution using MDC of Problem 2 (𝜖 = 0.001, ℎ = 1/16 ).
77 78 78
(4.11)
Convergence of GMG for different h for problem 2 using MDC.
79
(4.12)
Convergence of AMG for different h for problem 2 using MDC
80
(4.13) (4.14) (4.15)
(4.16)
Computed solution on a stretched grid of Problem 3 with 𝛽 = 1/20, 𝑛 = 32 and different values of 𝜖 = 0.1, 𝜖 = 0.01 Convergence behavior of GMRES for different preconditioners for problem 3 Convergence behavior of AMG for different stretching factors 𝛽 for problem 3 Computed solution of problem 4 on a uniform grid with different mesh sizes ℎ = 1/16, ℎ = 1/64.
81 82 83
85
(4.17)
Convergence behavior of AMG for different mesh sizes
86
(5.1)
Q2–Q1 element (
97
(6.1)
(6.2) (6.3) (6.4)
(6.5)
velocity components;
pressure).
a) The geometry and domain for a single square obstacle. b) The geometry and domain for two square obstacles. Streamline patterns around a single square obstacle for different values of Reynolds number. A comparison of the present results with the results of Yojina et al. Velocity contours around a single square obstacle for different values of Reynolds number. Time-series data of the flow velocity for a single obstacle at several positions, Re=85
x
107
109 110 113
114
(6.6) (6.7) (6.8)
(6.9)
Cross-wise distributions of the flow velocity at different positions (Nx), Re = 85. Streamline patterns around the two square obstacles for different values of Re A comparison of the present results of streamline patterns around the two square obstacles for different values of Re with Results of Yojina et al. Velocity contours around the two square obstacles for different values of Re and 𝑙o (𝑙o = 50 (upper), 𝑙o = 100 (middle) and 𝑙o = 150 (lower)).
115 119 121
127
(6.10)
The Performance of AMG around 1-obstacle at Re=30.
128
(6.11)
The Performance of AMG around 1-obstacle at Re=100.
129
(6.12)
The Performance of AMG of unsteady flow around 1-obstacle at Re=300.
129
The (6.13) efficiency of AMG at different grid size. (6.14)
(6.15)
130
The Performance of AMG of unsteady flow around 2-obstacles at Re=200 and 𝑙o =50. The Performance of AMG of unsteady flow around 2-obstacles at Re=200 and 𝑙o =100.
131
131
(6.16)
The Performance of AMG based solvers for steady flow around 1-obstacle.
132
(6.17)
The Performance of AMG of steady flow around 2-obstacles 𝑙o =100.
133
xi
LIST OF TABLES
Table No Table Title
Page
Maximum discretization error of problem 1 on uniform grid (4.1)
1
73
(ℎ = 16 ) Maximum discretization error on a non-uniform grid for
(4.2)
74
problem 1.
(4.3)
Convergence of MDC finite-element method for problem 2.
79
(4.4)
Convergence of MDC finite-element method for problem 3.
82
(4.5) (6.1)
Convergence of MDC finite-element method for problem 4. Parameters of the simulations.
86 106
xii
Nomenclatures Notation
Explanation
𝐴ℎ
Coefficient Matrix of velocities on the fine grid.
𝐴𝐻
Coefficient Matrix of velocities on the coarse grid.
𝐶𝑖 = {𝐶𝑖 }
Set of coarse-grid variables used to interpolate the fine-grid.
𝐶𝑠
Smagorinsky coefficient which takes the value 0.1.
𝐷𝑖𝑠 = {𝐷𝑖𝑠 }
Set of neighboring fine-grid points that have strong influence on i.
𝐷𝑖𝑤 = {𝐷𝑖𝑤 }
Set of points that have not strong influence on i.
𝑃𝑒
Peclet number.
𝑆
Rate-of-strain tensor of the resolved field.
𝑙o = 𝑛 𝑑
Inter-distance between the two square obstacles, n = 5, 10, 15.
𝑛𝜕
Number of Dirichlet boundary nodes.
𝑠𝑖 = {𝑠𝑖 }
Set of points that have strong influence on the point i.
𝑠𝑖𝑇 = {𝑠𝑖𝑇 }
Set of points that have strong dependence on the point i.
𝑡𝑛+1
Time level.
𝑢
Vector of unknown velocities.
𝑑2 𝑢 𝑢′′ = 2 𝑑𝑥 𝑑𝑢 𝑢′ = 𝑑𝑥
Second derivative of the velocity. First derivative of the velocity.
𝑤
Vector of convection coefficient.
A
Coefficient Matrix of velocities.
B
Coefficient Matrix of pressure.
C
Set of coarse points.
d
Obstacle diameter.
e
Computed error.
F
Set of fine points.
h
Mesh size.
L
Plane channel length.
l
Inflow length.
xiii
N
Coefficient Matrix of convection.
NT
Number of time step.
p
Scalar pressure.
P or 𝐼𝐻ℎ
Prolongation operator.
r
Residual component.
R or 𝐼ℎ𝐻
Restriction operator.
𝑅𝑒
Flow Reynolds number, 𝑅𝑒 =
Recrit
Critical Reynolds number.
ui and uj
Components of 𝑢 in x- and y-direction, respectively
v
Approximation to the exact solution.
𝑢 𝑚𝑎𝑥 𝑑 𝜈
.
Greek ∆𝑛+1
Time step.
𝜆𝑖
Measure of importance of each point.
𝜈𝑡
Turbulent (eddy) viscosity.
β
Blockage ratio, β = d/H.
Γ
Boundary of the domain.
Ω
Domain of PDE.
𝜃
Threshold value.
𝜈
Kinematic viscosity.
𝜓
Basis function of pressure.
𝜔
Interpolation weights.
𝜖
The diffusion coefficient.
𝜙
Basis function of velocity.
Subscripts a
Start point of the interval.
b
End point of the interval.
D
Dirichlet boundary.
i
Row number in the matrix.
j
Column number in the matrix.
N
Neumann boundary.
t
Time derivative.
x
x derivative. xiv
y
y derivative.
Superscripts ℎ0
Scheme on the finest grid.
2h
Scheme on the coarse grid.
h
Scheme on the fine grid.
s
Strong influence.
w
Weak influence.
Abbreviations AB
Adams–Bashforth method.
AMG
Algebraic Multigrid Method.
CFD
Computational Fluid Dynamics.
DNS
Direct Numerical Simulation.
FEM
Finite Element Method.
GMG
Geometric Multigrid Method.
GMRES
Generalized Minimum Residual Method.
LES
Large Eddy Simulation.
MDC
The Modified Diffusion Coefficient Technique
NS
Navier Stokes Equations.
PDEs
Partial Differential Equations.
RANS
Reynolds Averaged Navier-Stokes Simulation.
SUPG
Streamline-Upwind Petrov/Galerkin Finite Element Method.
TR
Trapezoid rule.
xv
Introduction
chapter (1)
Chapter (1) Introduction
1.1
Multigrid Method For large-scale calculations in engineering and physics, the multigrid (MG)
method potentially far surpasses other known methods as far as efficiency is concerned. A distinguishing feature of the multigrid algorithms is that their convergence rate does not deteriorate with increasing problem size. Although we frequently refer to the multigrid method, it has become clear that multigrid is not a single method or even a family of methods. Rather, it is an entire approach to computational problem solving, a collection of ideas and attitudes, referred to by its chief developer Achi Brandt [1] as multilevel methods. Originally, multigrid methods were developed to solve boundary value problems posed on spatial domains. Such problems are made discrete by choosing a set of grid points in the domain of the problem. The resulting discrete problem is a system of algebraic equations associated with the chosen grid points. In this way, a physical grid arises very naturally in the formulation of these boundary value problems. More recently, these same ideas have been applied to a broad spectrum of problems, many of which have no association with any kind of physical grid. The original multigrid approach has now been abstracted to problems in which the grids have been replaced by more general levels of organization. This wider interpretation of the original multigrid ideas has led to new powerful techniques, namely 'Algebraic Multigrid methods (AMG)' with a remarkable range of applicability. The efficiency of proper multigrid methods is due to the fact that error, only slightly affected by relaxation (smooth error), can be easily approximated on a coarser grid by solving the residual equation there, where it is cheaper to compute. This error approximation is interpolated to the fine grid and used to correct the solution. Generally, in classical (geometric multigrid GMG), uniform coarsening and linear
1
Introduction
chapter (1)
interpolation are used, so the key to constructing an efficient multigrid algorithm is to pick the relaxation process that quickly reduces error not in the range of interpolation. Computational Fluid Dynamics (CFD) gives rise to very large systems, requiring efficient solution methods. Multigrid has contributed to many applications in CFD, however, full multigrid efficiency has not yet been achieved in realistic engineering applications in CFD in general. An important reason for this is that in CFD we often have to deal with singular perturbation problems. This gives rise to grids with cells having high aspect ratios. Another reason is that the governing equations may show elliptic or parabolic behavior in one part of the domain and hyperbolic behavior in another part. This requires careful design of both the discretization and the solver. The methodology for efficient multigrid insists that each of the difficulties should be isolated, analyzed, and solved systematically using a carefully constructed series of model problems. It is well known that, one of the major obstacles to obtain better multigrid performance for convection-dominated flows, which is usually accompanied with singular perturbation behavior, is that the coarse grid provides only a fraction of the needed correction for smooth error components. Moreover, the coarse grid solutions usually exhibit oscillatory behavior. These difficulties are analyzed carefully in Chapter (4) in this thesis and a modification to the diffusion coefficient of the governing convection-diffusion equation is derived analytically. Using this modification, accurate non-oscillatory solutions are obtained on coarse grids and consequently multigrid efficiency is retained. With respect to the second difficulty that the governing equations may show elliptic, parabolic or hyperbolic behavior, this obstacle can be removed by designing a solver that effectively distinguishes between the elliptic, parabolic, and hyperbolic (advection) factors of the system and treats each one appropriately. In this thesis, the multigrid technique is developed to regain its standard efficiency to solve the Convection-Diffusion equation and the Navier-Stokes equations.
2
Introduction
1.2
chapter (1)
Computational Fluid Dynamics (CFD) The computational fluid dynamics (CFD) technique is one of the ways to
study the science of fluid motion (fluid dynamics). In many instances, analytical solutions of the equations of fluid dynamics are limited. So, they often are solved using numerical algorithms implemented in computer programs [2]. These methods applied for the solution of the fluid equations of motion are named computational fluid dynamics or simply CFD [3]. Due to the development of a variety of numerical acceleration techniques like multigrid, it was possible to compute inviscid flows past complete aircraft configurations or inside turbo-machines [4]. Convection-Diffusion Equation Convection-diffusion equations represent an important class of partial differential equations that arise in fluid mechanics, gas dynamics, and atmospheric modeling. In fluid dynamics, for example, the movement of a solute in ground water is described by such an equation. Since these equations normally have no closed-form analytical solutions, it is very important to have accurate numerical approximations. When diffusion dominates the physical process, standard finite-difference methods (FDM) and finite-element methods (FEM) work well in solving these equations. In most practical problems, the magnitude of convection coefficient is much greater than that of diffusion coefficient. So, these problems are called convectiondominated or singularly-perturbed. The numerical solution of these problems represents serious difficulties because the solution of these diffusion-convection problems possesses boundary layers that are small sub-regions, where derivatives of the solution are very large. These boundary layers make standard finite-element or finite-difference methods unsuitable for solving such problems. This is because numerical solutions produce non-physical oscillations and low-order of accuracy unless refined meshes are introduced in the boundary regions using an adaptive meshrefinement strategy. Another difficulty occurs when multigrid is used for solving convectiondominated problems using classical discretization methods. Even if the grid where the solution is computed provides suitable accuracy, the multigrid algorithm requires a sequence of coarser grids, and it is important that the discretizations on these grids 3
Introduction
chapter (1)
capture the character of the solution with a reasonable degree of accuracy. For these reasons, it is necessary to have a discretization strategy that does not have the deficiencies exhibited by the classical discretization method. To overcome these difficulties, various formulations were introduced including: unwinding techniques, streamline-upwind Petrov/Galerkin finite-element method (SUPG) [5,6], stream-line diffusion methods [7,8] and other special finite-element formulations [9,10]. As it is well-known, streamline-upwind Petrov/Galerkin method (SUPG) corresponds to modifying a weighting function in the Galerkin formulation to produce a small additional diffusion in the streamline direction. The amount of such additional diffusion is tuned by a parameter 𝜏 that must be chosen in a suitable way. The choice of 𝜏 is still considered as a major drawback of the method; a lot of numerical tests and several recipes have been proposed for the choice of 𝜏.
Navier-Stokes Equations With the mid 1980’s, the focus started to shift to the significantly more demanding simulation of viscous flows governed by the Navier-Stokes equations. The analytical point of view on the Navier-Stokes equations is deficient, in particular with regard to the turbulent flow regime. There exist no analytical solutions even to the simplest turbulent flow situations. There are 3 basic conceptual alternatives for the numerical simulation of turbulence: direct numerical simulation (DNS), large eddy simulation (LES), and simulations based on the Reynolds-averaged Navier-Stokes (RANS) equations. The three concepts for the numerical simulation of turbulent flows in its basic form struggle with different problems in terms of computational accuracy and efficiency. Furthermore, most of the numerical approaches to laminar flows are also far from being ideal. In view of this situation, the Variational Multiscale Method, which was introduced as a general concept for problems of computational mechanics, appears to be a valuable framework for developing improved numerical methods in fluid mechanics. The channel flow, which is described by the Navier-Stokes equations, is an interesting problem in fluid dynamics. This type of flow is found in many real-life applications such as irrigation systems, pharmacological and chemical operations, oilrefinery industries, etc. Usually, the flow velocity is low in such applications, which
4
Introduction
chapter (1)
leads to small values of Reynolds number (Re). However, due to practical reasons as flow in channel with obstacles, the flow may be completely turbulent in spite of the low Reynolds number. The turbulent nature of the flow is more assured if obstacles are present inside the channel. In this case, the obstacles provoke asymmetry and instability behind them in the flow until vortex shedding becomes a periodic event. The periodic vortex shedding with unsteady nature strengthens the mixing of the fluid particles. Thus, the process of mixing becomes more successful, which is important for many industrial applications where chemical mixing is a principal process.
1.3
Outlines of the Thesis In Chapter (2), the literature survey of the previous work of the other scientists
in computational fluid dynamics (CFD) is presented, especially for the ConvectionDiffusion equations, the Navier-Stokes equation and the Multigrid techniques as a best numerical solver for these equations. In Chapter (3), a general comparison between geometric and algebraic multigrid methods is presented with their basic concepts, algorithms and main components. The convergence behavior of the geometric and algebraic multigrid methods is examined through a model problem. One of the main contributions in this thesis concerning application of the algebraic multigrid technique to solve NavierStokes system is introduced. In Chapter (4), the Convection-Diffusion Equation is presented in details for the one- and two-dimensional problems. Also, we introduce a new contribution for solving convection-diffusion equation by predicting a modified-diffusion coefficient (MDC) such that the discretization process applies on the modified equation rather than the original one. For a class of 1-D convection-diffusion equation, we derived the modified-diffusion coefficient analytically as a function of the equation coefficients and mesh size, then, prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. The numerical results for 1-D and 2-D, variable coefficients, convectiondominated problems are also presented in this chapter.
5
Introduction
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In Chapter (5), the Navier-Stokes equations are presented in details for steady and unsteady flows. Large-Eddy simulation (LES) was used to handle the turbulent flow with Smagorinsky modeling. Finite-element method (FEM) was used for the discretization of the governing equations. In Chapter (6), the channel flow with 1- and 2-obstacles is considered. Artificial source of turbulence was introduced in the inflow velocity profile to ensure the turbulent nature of the flow. Using a suitable computational domain with appropriate grid and correct boundary conditions, the numerical results of the NavierStokes equations, are presented and compared with available similar ones in literature. Finally, in Chapter (7), the main conclusions that could be derived from the present study are presented. Also, recommendations for future work are given.
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Chapter (2) Literature review
2.1
Survey of the Computational Fluid Dynamics (CFD) The task of obtaining solutions for the governing equations of fluid dynamics
represents one of the most challenging problems in science and engineering. In most instances, the mathematical formulations of the fundamental laws of fluid dynamics are expressed as partial differential equations (PDE). Second-order partial differential equations appear frequently and, therefore, are of particular interest in fluid dynamics and heat transfer. Generally, the governing equations of fluid dynamics form a set of coupled, nonlinear PDEs which must be solved within an irregular domain subject to various initial and boundary conditions. In many instances, analytical solutions of the equations of fluid dynamics are limited. This is further restricted due to the imposed boundary conditions. For example, a PDE subject to a Dirichlet boundary condition (i.e., values of the dependent variable on the boundary are specified) may have an analytical solution. However, the same PDE subject to a Neumann boundary condition (where normal gradients of the dependent variable on the boundary are specified) may not have an analytical solution [3]. So they often are solved using numerical algorithms implemented in computer programs [2]. With the appearance of powerful and fast computers, new possibilities for solving the differential equations which describing fluid motion are appeared by using either a finite-volume or sometimes, but more rarely due to larger amount of CPU time required, a finite-element method. These methods applied for the solution of the fluid equations of motion are named computational fluid dynamics or simply CFD [3]. The computational fluid dynamics (CFD) is one of the ways to study the science of fluid motion (fluid dynamics). As mentioned in [11], the history of fluid dynamics started during the period from 17th to 19th century when a significant work
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was done trying to mathematically describe the motion of fluids. The beginning with Daniel Bernoulli (1700-1782) who derived Bernoulli‟s equation and Leonhard Euler (1707-1783) who proposed the Euler equations, which describe conservation of momentum for an inviscid fluid, and conservation of mass. Euler also proposed the velocity potential theory. Then, Navier (1785-1836) and George Stokes (1819-1903) introduced viscous transport into the Euler equations, which resulted in the NavierStokes equation and forms the basis of modern day CFD. Other key figures were Jean Le Rond d‟Alembert, Siméon-Denis Poisson, Joseph Louis Lagrange, Jean Louis Marie Poiseuille, John William Rayleigh, M. Maurice Couette, and Pierre Simon de Laplace. One of the greatest scientists in this field is Osborne Reynolds (1842-1912). He is most well-known for the Reynolds number, which is the ratio between inertial and viscous forces in a fluid. This governs the transition from laminar to turbulent flow [11]. In the first part of the 20th century, much work was done on refining theories of boundary layers and turbulence. Ludwig Prandtl (1875-1953) made his work in boundary layer theory, the mixing length concept, compressible flows, the Prandtl number, and more.
Fig. 2.1: The turbulence flow past circular cylinder [11]. As mentioned, in [11], Theodore von Karman (1881-1963) analyzed what is now known as the von Karman vortex street. Geoffrey Ingram Taylor (1886-1975) focused his work in statistical theory of turbulence and the Taylor micro scale, fig. 2.1. Andrey Nikolaevich Kolmogorov (1903-1987) introduced the Kolmogorov scales and the universal energy spectrum. George Keith Batchelor (1920-2000) presented contributions to the theory of homogeneous turbulence. 8
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As reported in [11], Lewis Richardson (1881-1953) developed the first numerical weather prediction system. His model's enormous calculation requirements led Richardson to propose a solution he called the “forecast-factory”. From 1930s to 1950s, earliest numerical solution was done for flow past a cylinder. Also, Thom studied the flow past circular cylinders at low speeds, in 1933, as in fig. 2.2. Kawaguti obtained a solution for flow around a cylinder, in 1953, by using a mechanical desk calculator.
Fig. 2.2: The flow past circular cylinders at low speeds [11]. During the 1960s, the theoretical division at Los Alamos contributed many numerical methods that are still in use today, such as the following methods [11]: – Particle-in-Cell (PIC). – Marker-and-Cell (MAC). – Vorticity-stream function Methods. – Arbitrary Lagrangian-Eulerian (ALE). – 𝑘𝜀 turbulence model. During the 1970s a group working under D. Brian Spalding, at Imperial College, London, developed [11]: – Parabolic flow codes (GENMIX). – Vorticity-Stream function based codes. – The SIMPLE algorithm and the TEACH code. – The form of the 𝑘𝜀 equations that are used today.
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– Upwind differencing [11]. One of the most important approaches appears to solve the fluid dynamics problems is computational fluid dynamics (CFD) which started in the early 1970‟s. Around that time, it became an acronym for a combination of physics, numerical mathematics, and, to some extent, computer sciences employed to simulate fluid flows. The beginning of CFD was triggered by the availability of increasingly more powerful mainframes and the advances in CFD are still tightly coupled to the evolution of computer technology. Among the first applications of the CFD methods was the simulation of transonic flows based on the solution of the non-linear potential equation. With the beginning of the 1980‟s, the solution of first two-dimensional (2D) and later also three dimensional (3-D) Euler equations became feasible. Due to the development of a variety of numerical acceleration techniques like multigrid, it was possible to compute inviscid flows past complete aircraft configurations or inside turbomachines. With the mid 1980‟s, the focus started to shift to the significantly more demanding simulation of viscous flows governed by the Navier-Stokes equations. Together with this, a variety of turbulence models evolved with different degrees of numerical complexity and accuracy [4]. Suhas V. Patankar published, in 1980, "Numerical Heat Transfer and Fluid Flow", probably the most influential book on CFD to date. Previously, CFD was performed using academic, research and inhouse codes. When one wanted to perform a CFD calculation, he had to write a program to specifically study a certain problem, but today, thanks to the rapidly increasing speed of supercomputers, which helps the most commercial CFD codes originated to be available, during the period 1980s and 1990s, such as: Fluent (UK and US), CFX (UK and Canada), Fidap (US), Polyflow (Belgium), Phoenix (UK), Star CD (UK), Flow 3d (US), ESI/CFDRC (US) and SCRYU (Japan) [11].
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Survey of Convection – Diffusion Equations Convection–diffusion equation describes the solute transport due to combined
effect of diffusion and convection in a medium. It is a partial differential equation of parabolic type, derived on the principle of conservation of mass using Fick‟s law. These equations are mathematically important because they arise in many problems in Science and Engineering. These equations are also important because they solve serious computational difficulties, especially when convection dominates the physical process. So, the convection–diffusion equation has drawn significant attention of hydrologists, civil engineers and mathematical modelers. Its analytical/numerical solutions along with an initial condition and two boundary conditions help to understand the contaminant or pollutant concentration distribution behaviour through an open medium like air, rivers, lakes and porous medium like aquifer, on the basis of which remedial processes to reduce or eliminate the damages may be enforced. It has wide applications in other disciplines too, like soil physics, petroleum engineering, chemical engineering and biosciences. In the initial works while obtaining the analytical solutions of dispersion problems in ideal conditions, the basic approach was to reduce the convection–diffusion equation into a diffusion equation by eliminating the convective term(s). It was done either by introducing moving coordinates (Ogata and Banks 1961; Harleman and Rumer 1963; Bear 1972; Guvanasen and Volker 1983; Aral and Liao 1996; Marshal et al 1996) or by introducing another dependent variable (Banks and Ali 1964; Ogata 1970; Lai and Jurinak 1971; Marino 1974 and Al-Niami and Rushton 1977). Then Laplace transformation technique has been used to get the desired solutions. [12] Also, as in [12], the unsteady flow through porous medium has been considered to obtain the analytical solutions (Banks and Jerasate 1962; Hunt 1978 and Kumar 1983). Some one-dimensional analytical solutions were given (Tracy 1995) by transforming the non-linear convection–diffusion equation into a linear one. Analytical solutions were developed for describing the transport of dissolved substances in heterogeneous semi-infinite porous media with a distance dependent dispersion of exponential nature along the uniform flow (Yates 1990, 1992). As mentioned in [12], the work of Yates was extended, by Logan and Zlotnik 1995 and Logan 1996, by including the adsorption and decay effects.
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Kumar et al., as stated in [12], mentioned that one-dimensional analytical solution was presented for the convection–diffusion equation for solute dispersion (Zoppou and Knight 1997). Time-dependent dispersion along uniform flow has been considered (Aral and Liao 1996) to solve two-dimensional convection–diffusion equation. An analytical solution has been obtained for two-dimensional steady state mass transports in a trapezoidal embankment (Tartakovsky and Federico 1997). The temporal moment solution for one-dimensional advective-dispersive solute transport has been applied to analyze soil column experimental data (Pang et al. 2003). An analytical approach was developed for nonequilibrium transport of reactive (Severino and Indelman 2004). Analytical solutions were presented for solute transport in rivers (Smedt 2006). In 2007, two-dimensional semi-analytical solution was presented by Kim et al. to analyze stream–aquifer interactions in a coastal aquifer where groundwater level responds to tidal effects [12]. In 2008, Li introduced exact solutions of constant and variable coefficient diffusion-convection equations [13]. In 2011, Abbasbandy et al. presented exact analytical solution of forced convection in a porous-saturated duct [14]. In 2012, the analytical solution is achieved by traveling wave analysis [15]. Convection-diffusion equations are an important class of partial differential equations (PDE) that arise in fluid mechanics, gas dynamics, and atmospheric modeling. In fluid dynamics, for example, the movement of a solute in ground water is described by such an equation. Since these equations normally have no closed form analytical solutions, it is very important to have accurate numerical approximations. When diffusion dominates the physical process, standard finite-difference methods (FDM) and finite-element methods (FEM) work well in solving these equations. For the convection-dominated case, this equation becomes hyperbolic and develops sharp features in the solution. Classical numerical methods for the convection–diffusion equation result in non-convergent elements. So, many specialized schemes have been developed to overcome the difficulties. As stated in [16], Pepper et al. (1979) and Okamoto et al. (1998) solve the one-dimensional convection equation by using a spline interpolation technique that they call a quasi-Lagrangian cubic-spline method. In (1976), Sastry used a cubic spline technique to approximate the solution of the one-dimensional diffusion 12
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equation and in (2001) Ahmad and Kothyari solve the one-dimensional convectiondiffusion equation by using cubic-spline interpolation for the convection component and the Crank-Nicolson scheme for the diffusion component. The one-dimensional convection-diffusion (convection dispersion) equation is solved by Thongmoon and McKibbin (2006) using the cubic-spline interpolation to obtain estimates for the convection and diffusion terms. The same problem is investigated using finitedifference schemes [16]. Kadalbajoo et al. (2008) solved the time-dependent linear convection–diffusion problem [17]. Bok Lee (2011) applied the nodal integral expansion method for one-dimensional time-dependent linear convection–diffusion equation [18]. Huang and Schaeffer (2012) used the finite-difference solution of onedimensional linear convection–diffusion equations on moving meshes [19]. One class of these methods usually referred to as the class of Eulerian methods. Among these methods are the Petrov - Galerkin FEM methods, as in [20] ( Barrett et al. 1984, Bouloutas et al. 1991, Celia et al. 1989), Nadukandi et al. 2012[21], Nissen et al. 2012[22] ) which there improvements are over the standard Galerkin FEM, the streamline diffusion FEM methods (SDM), as in [20] (Eriksson et al. 1993, Hansbo et al. 1990, Hughes 1995), Zhiyong et al. 2008[23], Chen et al. 2008[24] and Qian et al. 2012[25], the continuous and discontinuous Galerkin methods (CGM, DGM), as in [20] (Falk et al. 1992, Richter 1992, Cockburn and Dawson 1999, Houston et al. 2002, Zarin and Roos 2005, Gopalakrishnan and Kanschat 2003, Hughes et al. 2006), Nguyen et al. 2009[26], Ern et al. 2010[27], Zhu et al. 2011[28] and Wu et al. 2013[29]. The SDM improve over the standard spacetime Galerkin FEM by adding a multiple of the (linearized) hyperbolic operator of the problem considered to the standard test functions. Thus they add numerical diffusion only in the direction of the streamlines. The SDM formulations have a free parameter which determines the amount of diffusion applied and therefore has a great effect on the accuracy of these methods. In addition to the methods mentioned above, the class of Eulerian methods includes the high resolution methods in fluid dynamics such as the Godunov methods, the total variation diminishing methods (TVD), and the essentially non-oscillatory methods (ENO) (Sweby 1991 and Finlayson 1992),as in [20]. These methods, as well as the CGM and DGM, are well-suited for convection-diffusion equations with small diffusion coefficients and in general impose an extra stability restriction on the size of 13
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the time step taken based on the magnitude of this coefficient. Therefore, they are very sensitive to changes in the diffusion coefficient. Another class of methods, usually known as characteristic methods, makes use of the hyperbolic nature of the governing equation. These methods use a combination of Eulerian fixed grids to treat the diffusive component, and Lagrangian coordinates by tracking particles along the characteristics to treat the advective component. Included in this class are the Eulerian - Lagrangian methods (ELM), the modified methods of characteristics (MMOC), and the operator splitting methods, as in [20] (Wheeler and Dawson 1988, Dahle et al. 1990, Ewing 1991), Remešíková 2007 [30], Li and Yuan 2009 [31], Acosta and Mejía 2010 [32] and Roos and Zarin 2003 [33]. The Eulerian - Lagrangian localized adjoint methods (ELLAM) were developed as an improved extension of the characteristic methods that maintains their advantages but enhances their performance by conserving mass and treating general boundary conditions naturally in their formulations [20]. Also, as in [34], the differential transform (DT) method has been used to solve convection-diffusion equation in different situations such as eigen value problems (Chen and Ho 1996) and initial value problems (Jang et al. 2000). The technique concept was firstly introduced by Zhou (1986) for electrical circuits. More recently, Kurnaz et al., Patricio and Rosa (2007) used DT methods for approximating the solution of a system of ODEs, with good results for smooth profile solutions. In [35], John and Knobloch (2007) mentioned that two-dimensional upwind finite element discretizations were derived by Heinrich et al. in (1977) and by Tabata (1977). Like in the finite difference method, the upwind finite element discretizations remove the unwanted oscillations but the accuracy attained is often poor since too much numerical diffusion is introduced. A further important drawback is that these methods are not consistent, i.e., the solution of the equation is no longer a solution to the variational problem as it is the case for a Galerkin formulation. Consequently, the accuracy is limited to first order. So, a significant improvement came with the streamline upwind/Petrov–Galerkin (SUPG) method developed by Brooks and Hughes (1988), Ma et al. 2004 [36], Tejada and Kenneth 2005 [37], Park et al. 2003 [38], Nadukandi et al. 2010, 2012 [39, 21], Almeida and Silva 1997 [40], Ma and Sun 2011 [41]. In view of its stability properties and higher-order accuracy, the SUPG method is regarded as one of the most efficient procedures for solving convection14
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dominated equations. This technique turned out to be the fore-runner of a new class of stabilization schemes, called the Galerkin/least-square (GLS) stabilization methods, as Hughes et al. 1989 [42], Coronado et al. 2006 [43], Yano and Darmofal 2010 [44], Longo et al. 2012 [45]. GLS stabilization was followed by the „„unusual stabilized methods‟‟ introduced by Franca and coworkers (1995-2000), as mentioned in [42]. Concurrently, another class of stabilized methods that was based on the idea of augmenting the Galerkin method with virtual bubble functions was introduced, as mentioned in [42], by Brezzi and coworkers (1992-2000). In 1998, Hughes revisited the origins of the stabilization schemes from a variational multiscale approach. In the Hughes variational multiscale (HVM) method different stabilization techniques appear as special cases of the underlying sub-grid scale modelling concept [42]. variational multiscale method was introduced by Hughes and coworkers (1998) as stated in [42], Narayanan and Zabaras 2005 [46], John et al. 2006 [47], Song et al. 2010 [48] and Marras et al. 2012 [49]. Another numerical procedures like the Residual-Free Bubble (RFB) Finite Element Method (FEM) for solving of the linear convection-diffusion equation which was introduced by Brezzi and Russo (1994), Dolbow and Franca (2008) [50], Parvazinia and Nassehi (2010) [51].
2.3
Survey of Navier-Stokes equations The ultimate goal of the field of computational fluid dynamics (CFD) is to
understand the physical events that occur in the flow of fluids around and within designated objects. These events are related to the action and interaction of phenomena such as dissipation, diffusion, convection, shock waves, slip surfaces, boundary layers and turbulence. Fluids can be divided into liquids and gases with water and air being the most prominent representatives of these two groups, respectively. In Fig. 2.3, some everyday examples for flows of liquids like (i) the wake of a motorboat, (ii) the water flow of an alpine creek, (iii) the oil flow in a pipeline or (iv) the blood flow in an artery are displayed.
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(i) The wake of a motorboat
(iii) The oil flow in a pipeline
(ii) The water flow of an alpine creek
(iv) The blood flow in an artery are displayed.
Fig. 2.3: Some common examples for liquid flows [52]. In almost all cases, the flow of liquids like water may be treated as being incompressible. Under certain conditions like, for instance, flow governed by a low Mach number, air and other gases may also be dealt with based on the assumption of incompressibility. Fig 2.4 illustrates some well known examples for flows of gases like (i) the air flow around a plane, (ii) a motorbike air flow, or (iii) a bridge air flow. Of course, a flow around an airplane usually has to be viewed as being compressible. 16
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However, the respective flow parameters have to be considered in the other two cases, in order to judge the classification of the flow finally.
(i) The air flow around a plane.
(ii) The motorbike air flow.
(iii) The bridge air flow. Fig. 2.4: Some common examples for gas (air) flows [52]. The concept of incompressibility describes a presumed flow performance where the temperature field will exhibit no influence on the velocity and pressure field, if, in addition to the density, all further characteristic material properties like, for example, the viscosity are constant as well. The scope of the incompressibility assumption is broad and, thus, the area of application of the perceptions of this work.
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The Navier-Stokes equations are a mathematical model aimed at describing the motion of an incompressible viscous fluid, like many common ones as, for instance, water, glycerine, oil and, under certain circumstances, also air. As in [52], they were introduced in 1822 by the French engineer Claude Louis Marie Henri Navier and successively re-obtained, by different arguments, by a number of authors including Augustin-Louis Cauchy in 1823, Siméon Denis Poisson in 1829, Adhémar Jean Claude Barré de Saint-Venant in 1837, and, finally, George Gabriel Stokes in 1845. Interestingly, the last proposition is basically valid for two crucial states of flow, laminar and turbulent flows, although they are quite different from the physical point of view. The occurrence of one or the other state strongly depends on the so called Reynolds number associated with the flow. This has already been observed by Osborne Reynolds (1842--1912), the eponym of this dimensionless number, in the later half of the 19th century. In typical engineering applications, turbulent flows are surely prevalent due to its positive features like a more effective transport and mixing ability with respect to a comparable laminar flow. By dissecting Figs. 2.3 and 2.4, one may also verify that most of the depicted common examples represent turbulent flow situations or flow situations which may at least enter the turbulent regime temporarily. According to this, the interest in turbulent flow regime is very high. The analytical point of view is extremely deficient with regard to the turbulent flow regime in particular. There exist no analytical solutions even to the simplest turbulent flow situations. To that effect, all hope culminates in a numerical way of solving this ‟chief outstanding‟ problem. Currently, there are three basic conceptual alternatives for the numerical simulation of turbulence: (i) direct numerical simulation (DNS), (ii) large eddy simulation (LES), and (iii) simulations based on the Reynolds averaged Navier-Stokes (RANS) equations. DNS is the theoretically most straightforward concept for the numerical simulation of turbulent flows. It “merely” attempts to solve the NavierStokes equations numerically (i.e., without any additional modeling efforts), since it has been widely accepted in the meantime that the Navier-Stokes equations govern the turbulent state of flow as well as the laminar state of flow. DNS aims at a complete resolution of all scales contained in the respective flow. Hence, no unresolved scales would be left theoretically. The problem lies in the broad range of length and time scales appearing in a turbulent flow, which makes this approach unfeasible in most of the cases. As a result, DNS still is not a viable approach to simulate high or, for the 18
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most part, even moderately high Reynolds number flows of engineering interest with the currently available computer hardware. Even for the lower Reynolds number flows which have been simulated as by DNS, the Kolmogorov scale is seldomly resolved. [53] Thus, there are still unresolved scales in a DNS, although the theoretical claim of DNS assumes no unresolved scales to be left over. The basics of DNS and its impact on the numerical simulation of turbulence, including a historical review, are provided by Moin and Mahesh (1998) in [54], Simens et al. (2007) [55], Jin (2009) [56], Lin and Dengbin (2010) [57], Wei and Pollard (2011) [58]. As aforementioned, one usually faces the problem that adequate computational power to execute DNS of a turbulent flow is not available. A promising alternative then is LES. The strategy of LES consists of resolving the larger flow structures and modeling the effect of the smaller flow structures on the larger structures. It has been learned from Kolmogorov‟s (1941) in [52], hypotheses that the smaller scales exhibit a more universal character than the larger scales, which favors a more general validity of a once developed model for the smaller scales than for the larger scales. As a result, LES appears to be a promising approach in two respects. On the one hand, a coarser discretization, which is substantially coarser than a comparable DNS discretization in the majority of the cases, is sufficient for resolving the larger scales, and on the other hand, the universal character of the smaller scales simplifies the modeling process. The traditional way of performing LES is described, for instance, in the comprehensive articles by Lesieur and Metais (1998), Piomelli (1999), Fröhlich and Rodi (2002), in the book by Sagaut (2002) and some recent advances in LES for complex flows are reported in Moin (2002), as mentioned in [52]. A more mathematically oriented view on LES is provided by Layton (2003), Guermond et al. (2004) and John (2004), as stated in [54], Guermonda and Prudhomme (2005) [59], Gravemeier et al. (2005) [60], Young et al (2006) [61], Chang et al. (2007) [62], Meyers et al. (2007) [63], Bouffanais (2010) [64], Bazilevs and Akkerman (2010) [65], Gungor and Menon (2010) [66], Gravemeier et al. (2010) [67], Li et al. (2012) [68] and Ali (2012) [69] . The traditional LES relies on a filter to separate resolved and unresolved scales [54]. Of course, LES is not the ‟miracle cure‟. The computational effort required for LES may be less than for DNS, but it is still of substantial complexity. Hence, it is also
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still not possible to perform LES for most of the turbulent flows arising in characteristic engineering applications. However, it seems to be more auspicious to believe in LES than in DNS for the near future and there maybe some support in improving LES erupting from quite a different field. This redeemer potentially being instrumental in fighting the devil of turbulence has also a name as a matter of course: the Variational Multiscale Method. This theoretical framework has been established by Hughes (1995) in [52] and further developed as a powerful means for problems of computational mechanics having to deal with large scale ranges by Hughes and coworkers in Hughes and Stewart (1996) and Hughes et al. (1998) in [52]. The basic concept consists in differentiating scale groups, for example large (resolved) and small (resolved) scales or resolved and unresolved scales depending on the resolution requirements. This methodological framework has also been applied to the underlying problem, the incompressible Navier-Stokes equations, in order to facilitate numerical simulations in the sense of LES. The variational multiscale method was introduced in Hughes et al. (2000a), Hughes et al. (2001a, 2001b) [52], Rispoli et al. (2005) [70], John and Kaya (2005) [71], Gravemeier et al. (2006) [72], Zheng et al. (2009) [73], John and Kindl (2010) [74, 75], Masud and Calderer (2011) [76] and Yu et al. (2012) [77]. Also, the spectral difference (SD) method is a high-order approach which solves the strong form of Navier–Stokes equations without multiplying a weighted function. The SD method deals with the differential form of the governing equations without performing volume integrations. The SD method combines elements from finite-volume and finite-difference techniques. It is able to achieve optimal order of accuracy for various compressible flows in Kopriva (1998), Sun (2007), Liang et al. (2011) [78] Liang et al. (2009) [79], Zhou (2010) [80] and Chen et al. (2012) [81]. The method is particularly attractive because it is conservative, and has a simple formulation and easy to implement.
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Survey of Multigrid Method Multigrid (MG) methods in numerical analysis are a group of algorithms for
solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called multiresolution methods, very useful in (but not limited to) problems exhibiting multiple scales of behavior. For example, many basic relaxation methods exhibit different rates of convergence for short- and longwavelength components, suggesting these different scales be treated differently, as in a Fourier analysis approach to multigrid. MG methods can be used as solvers as well as preconditioners. The main idea of multigrid is to accelerate the convergence of a basic iterative method by global correction from time to time, accomplished by solving a coarse problem. This principle is similar to interpolation between coarser and finer grids. The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions. Multigrid methods can be applied in combination with any of the common discretization techniques. For example, the finite-element method may be recasted as a multigrid method. In these cases, multigrid methods are among the fastest solution techniques known today. In contrast to other methods, multigrid methods are general in that they can treat arbitrary regions and boundary conditions. They do not depend on the separability of the equations or other special properties of the equation. They have also been widely used for more-complicated non-symmetric and nonlinear systems of equations, like the Lamé system of elasticity or the Navier-Stokes equations. The combination of the following algorithmic ingredients should result in efficient multigrid algorithms: 1. Smoothing of rough error parts via classical iterative methods, 2. Approximation of smooth errors on a coarser grid, 3. Recurrent call of 1 and 2 on a sequence of coarser and coarser grids, 4. Nested iteration for producing good initial guesses. As mentioned in [82] Fedorenko's pioneering paper (1961) proposed the first two grid method for solving the finite-difference equations arising from the five-point 21
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approximation of the Poisson equation in a rectangular domain. Already the first numerical results presented in his paper showed the great potential of these methods. In his second paper from 1964, he gave a first convergence analysis and proposed to combine the first three steps in efficient multigrid algorithms that results in a complete multigrid cycle. In [82], Bachwalov (1966) provided the first rigorous convergence proof on the basis of the sum splitting technique and showed that the nested iteration technique with the multigrid method as nested iteration process allows us to produce approximate solutions that differ from the exact solution of the boundary value problem in the order of the discretization error with optimal complexity. In the Seventies, these techniques were generalized to variational finite difference equations and general finite element equations by Astrachanzev (1971) and by Korneev (1977), as reported in [82]. Also in the Seventies, Brandt (1977) [1] and W. Hackbusch (1976), as mentioned in [82], reinvented the multigrid method independently of the Russian school, generalized it to new classes of problems, and developed the theory. In particular, Hackbusch (1981), as mentioned in [82], proposed the so-called product splitting that reduces the two grid rate estimate to the proof of the approximation property and the smoothing property. At the end of the Seventies and at the beginning of the 80ies there was a real boom in research and in publication of research results on multigrid methods. The proceedings of the first European Conference on Multigrid Methods, as reported in [82], which was held in 1981, reflect the state-of-the-art in the research on multigrid methods, give the first interesting applications and present a multigrid bibliography with more than 200 entries. The first comprehensive treatment of multigrid methods was Brandt (1977)[1], which remains very useful. The review article of Stuben and Trottenberg (1982) contains detailed analyses of simple model problems, along with some interesting historical notes. Both of the above sources also contain listings of sample multigrid problems. Brandt (1982) is an invaluable practical guide to developing multigrid algorithms for a wide range of problems; an updated version (Brandt, 1984) contains additional introductory material plus detailed information on multigrid methods for systems of equations in fluid dynamics. Since then, other mathematicians extended Fedorenko's work to general elliptic boundary value problems with variable coefficients as in Wesseling (1982)[83], Chen and Strain (2008)[84], Coco and Russo(2013)[85]. However, the full 22
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efficiency of the multigrid approach was realized after the works of Brandt (1977)[1] and Hackbusch (1977). These authors also introduced multigrid methods for nonlinear problems like the multigrid full approximation storage (FAS) scheme in [1], Durga and Ramakrishna (2004)[86]. Another achievement in the formulation of multigrid methods was the full multigrid (FMG) scheme in [1] which is based on the combination of nested iteration techniques and multigrid methods. Now, Multigrid algorithms are applied to a wide range of problems, primarily to solve linear and nonlinear boundary value problems. Other examples of successful applications are eigenvalue problems as in Brandt et al. (1983)[87] and Feng (2001)[88]. Also, multigrid method is used to solve parabolic problems as in Borzı̀ (2003)[89], hyperbolic problems as in Mulder (1989) [90] and Nishikawa and Leer (2003)[91]. A successful commercial use of multigrid methods for solving hard practical problems depends not only on the level-independent convergence behavior that is typical for multigrid methods, but also on their robustness against bad parameters (e.g., geometrical parameters, large coefficient jumps, singularities, anisotropies, etc.) arising often in many practical applications. The construction of efficient and at the same time robust multigrid algorithms for problems with bad parameters is a hot topic in the current research activities. Geometrical multigrid method Geometrical multigrid method (GMG) is one of the multigrid methods which is based on solving a finely discretized problem, by using several levels of coarser discretizations for that same problem. The basic idea of multigrid is to use a projection of the fine grid problem on a coarser grid to remove the slowly varying components from the error, the so-called coarse grid correction. The quickly oscillating components in the error are then removed with an iterative technique, which is called the smoother. In order for this to work the coarse grid correction and the smoother must be each others complement, in the sense that modes that are not damped by the smoother are damped by the coarse grid correction and vice versa. The coarse grid correction uses a restriction operator (R), to restrict a fine grid residual to the next coarser grid. Also, an interpolation operator (P) is used, to interpolate a coarser grid error to the fine grid. Suppose we have to solve 𝐴𝑥 = 𝑏 and have some 23
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iterate x, we compute the residual 𝑟 = 𝑏 − 𝐴𝑥 and transfer it to the coarse grid using the restriction operator to get the coarse grid residual 𝑟𝑐 = 𝑅 𝑟. Then, we (approximately) solve the coarse grid system 𝐴𝑐 𝑒𝑐 = 𝑟𝑐 , where 𝐴𝑐 is the representation of A on the coarse grid. Since the dimensions of this coarse grid problem are small, this is much cheaper than solving the equations on the fine grid. Next, we prolongate the coarse grid error 𝑒𝑐 back to the fine grid and use this to update the iterate, 𝑥 = 𝑥 + 𝑃 𝑒𝑐 . For the coarse grid correction, we have to solve the projected coarse grid problem 𝐴𝑐 𝑒𝑐 = 𝑟𝑐 . This is a smaller system, but it can be still too big to solve directly, in which case we apply the same strategy to approximately solve this coarse grid problem, using a smoother for the coarse grid and an even coarser grid for the coarse grid correction. This strategy can be recursively applied which leading to the use of multiple levels of coarser grids, hence the name multigrid [92]. Algebraic multigrid The multigrid boom at the beginning of the Eighties of the previous century initiated many interesting research directions and many new applications of multigrid and related methods. Algebraic multigrid (AMG) methods are one of the most important issues at least from a practical point of view. In contrast to the geometrical multigrid method GMG, its algebraic version recovers algebraically the ingredients for a complete multigrid algorithm from the fine grid data only, we discus this method briefly in chapter 3. The first serious approach to AMG was made in 1982 by Brandt, McCormick and Ruge (1981), as reported in [82].This approach was afterwards improved in several papers by Brandt (1986) [93], Stuben(1999, 2009) [94, 95]. Since then a lot of new ideas have been published as in Okusanya et al. (2004) [96], Heys et al. (2005) [97], Gravemeier (2009)[98], Shitrit et al. (2011) [99] and Khelifi et al. (2013) [100]. There is an urgent need in efficient implementation of robust AMG methods that can be used in commercial codes. It is much easier to use an AMG code in an existing software package as solver than redesign this package in such a way that hierarchical data structures for implementing geometrical multigrid solvers are available. In particular, AMG has successfully been applied to various nonsymmetric (e.g., convection–diffusion) and certain indefinite problems as in Boyly and Silvester (2006)[101], Wu et al. (2006)[102] and Notay (2011)[103]. Moreover, important 24
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progress has been achieved in the numerical treatment of systems of PDEs (mainly Navier-Stokes and structural mechanics applications) [92]. However, major research is still ongoing and much remains to be done to obtain an efficiency and robustness comparable to the case of scalar applications.
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Chapter (3) Multigrid Methods
3.1
Introduction Nowadays the multigrid (MG) technique is one of the most efficient methods for solving a
large class of problems very efficiently. They work by approximating a problem on multiple overlapping grids with widely varying mesh sizes and cycling between these approximations, using relaxation to reduce the error on the scale of each grid. The focus in the application of standard (geometric) multigrid methods is on the continuous problem to be solved. With the geometry of the problem known, the user discretizes the corresponding operators on a sequence of increasingly finer grids. Each grid generally is a uniform refinement of the previous one. In addition, the user has to define transfer operators between the grids. The coarsest grid is sufficiently coarse to make the cost of solving the (residual) problem there negligible, while the finest is chosen to provide some desired degree of accuracy. The solution process, which involves relaxation, transfer of residuals from fine to coarse grids, and interpolation of corrections from coarse to fine levels, is a very efficient solver for the problem on the finest grid, provided the above "multigrid components" are properly chosen. Roughly, the efficiency of proper multigrid methods is due to the fact that error only slightly affected by relaxation (smooth error) can be easily approximated on a coarser grid by solving the residual equation there, where it is cheaper to compute. This error approximation is interpolated to the fine grid and used to correct the solution. Generally, uniform coarsening and linear interpolation are used, so the key to constructing an efficient multigrid algorithm is to pick the relaxation process that quickly reduces error not in the range of interpolation. The algebraic multigrid (AMG) approach is developed to solve matrix equations using the principles of usual multigrid methods. In contrast to "geometric" multigrid methods, the relaxation used in AMG is fixed. The coarsening process (picking the coarse "grid" and defining interpolation) is performed automatically in a way that ensures the range of interpolation approximates those errors not efficiently reduced by relaxation. From a theoretical point of view, the process is best understood in the context of symmetric M-matrices, although, in practice, its use is not restricted to 26
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such cases. The underlying idea of the coarsening process is to exploit the fact that the form of the error after relaxation can be approximately expressed using the equations themselves, so that the coarse grid can be chosen and interpolation defined if the equations are used directly. This makes AMG attractive as a "black box" solver. In this work, various algebraic multigrid methods are proposed to solve different problems including: general linear elliptic partial differential equations (PDEs), as anisotropic Poisson equation [Section 3.5], problems with steep boundary layers, as convection dominant of convectiondiffusion equations [Chapter 4], and nonlinear system of equations as Navier-Stokes equations [Section 3.6 and Chapter 5]. In each of these applications, efficiency is optimal: the computational work is proportional to the number of unknowns. In the rest of this Chapter, general comparison of geometric and algebraic multigrid methods is presented in Section 3.2. Basic concepts, algorithms and description of main components of the geometric and algebraic multigrid methods are given in Sections 3.3 and 3.4, respectively. The convergence behavior of the two multigrid methods is examined through a model problem in Section 3.5. Finally in Section 3.6, our new contribution which concerning application of the algebraic multigrid technique to solve Navier-Stokes system of equations is introduced.
3.2
Algebraic versus geometric multigrid As mentioned above, the most important conceptual difference between geometric and
algebraic multigrid is that all geometric multigrid (GMG) approaches operate on predefined grid hierarchies. That is, the coarsening process itself is fixed and kept as simple as possible. Fixing the hierarchy, however, puts particular requirements on the smoothing properties of the smoother used in order to ensure an efficient interplay between smoothing and coarse-grid correction. In contrast to this, AMG fixes the smoother to some simple relaxation scheme such as plain point Gauss-Seidel relaxation, and enforces an efficient interplay with the coarse-grid correction by choosing the coarser levels and interpolation appropriately. This is shown schematically in Fig. 3.1 [92].
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Geometric multigrid
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Requirement for any multilevel approach : Efficient interplay between smoothing + coarse-grid correction
Fix coarsening adjust smoother
Algebraic multigrid
Fix smoother adjust coarsening
Grid equations
Algebraic systems
𝐴 𝑢 = 𝑓
𝑎𝑖𝑗 𝑢𝑗 = 𝑓𝑖 𝑗
(hierarchy given)
(no hierarchy given)
Fig. 3.1: Algebraic versus geometric multigrid [92]. To clarify the main difference between AMG and GMG, let us begin by deciding what is meant by a grid. In the geometric case, the unknown variables 𝑢𝑖 are defined at known spatial locations (grid points) on a fine grid. Then, a subset of these locations is selected as a coarse grid. As a consequence, a subset of the variables 𝑢𝑖 is used to represent the solution on the coarse grid. For AMG, by analogy, we seek a subset of the variables 𝑢𝑖 to serve as the coarse-grid unknowns. A useful point of view, then, is to identify the grid points with the indices of the unknown quantities. Hence, if the problem to be solved is Au=f and 𝑢 = 𝑢1 , 𝑢2 , ⋯ , 𝑢𝑛 𝑇 , then the fine grid points are just the indices {1,2,. . . , n). Having defined the grid points, the connections within the grid are determined by the undirected adjacency graph of the matrix A. Letting the entries of A be 𝑎𝑖𝑗 , we associate the vertices of the graph with the grid points and draw an edge between the ith and jth vertices if either 𝑎𝑖𝑗 ≠ 0 or 𝑎𝑗𝑖 ≠ 0. The connections in the grid are the edges in the graph; hence, the grid is entirely defined by the matrix A. A simple example of this relationship is given in Fig. 3.2.
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Multigrid Methods
𝐴=
1
2
𝑋 𝑋
𝑋 𝑋 𝑋 𝑋
𝑋 𝑋
chapter(3) 3
𝑋 𝑋 𝑋 𝑋 𝑋
4
5
𝑋 𝑋 𝑋 𝑋
𝑋
6
1
6 5
𝑋
𝑋
𝑋 𝑋
𝑋 𝑋
4 3
2
Fig. 3.2: The nonzero (NZ) structure of A, where X indicates a nonzero entry, is shown on the left. The resulting undirected adjacency graph appears on the right.
Now, how do we select a coarse grid? With geometric multigrid methods, smooth functions are geometrically or physically smooth; they have a low spatial frequency. In these cases, first assume that relaxation smoothes the error and select a coarse grid that represents smooth functions accurately. Then, choose intergrid operators that accurately transfer smooth functions between grids. With AMG, the approach is different. First select a relaxation scheme that allows us to determine the nature of the smooth error. Because we do not have access to a physical grid, the sense of smoothness must be defined algebraically. The next step is to use this sense of smoothness to select coarse grids, which will be subsets of the unknowns. A related issue is the choice of intergrid transfer operators that allow for effective coarsening. Finally, select the coarse-grid versions of the operator A, so that coarse-grid correction has the same effect that it has in geometric multigrid: it must eliminate the error components in the range of the interpolation operator. Each of GMG and AMG approaches has its advantages and domain of applicability. The geometric multigrid (GMG) in many cases is more efficient in terms of storage and time, because it does not need to store the (sparse) matrix entries (both on fine and on coarser grids) and it needs no assembly of the coarse grid equations. However, in other cases this storage and work are still invested even though the geometric algorithm is used, for such cases and some others, AMG has several advantages: (1) The real practical advantage of AMG is that it can be applied directly to structured as well as unstructured grids. For example, finite element discretizations using irregular triangulation result in such problems. Again, AMG deals effectively with these problems, since uniform grids are not at all necessary. (2) AMG can be used as a black box solver. Since the algorithm is based on the given matrix. So, the operator-dependent interpolation and the Galerkin operator can be derived directly from the underlying matrices, without any reference to the grids. This is particularly
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important for problems where they cannot be solved with the usual geometric coarsening where the domain of the problem is complex enough so that any sensible discretization is too fine to serve as the coarsest grid. (3) Another case consists of problems caused by the operator itself, not the domain. In such cases, when uniform coarsening is used with linear interpolation, it may be difficult to find a relaxation process that smoothes the error sufficiently to admit a good coarse grid correction. For example, when we are solving diffusion problems with discontinuous coefficients, it is necessary to change the definition of interpolation across the discontinuities. (4) Also, some problems are purely discrete. It is even possible that such a problem has no geometric background. Examples of discrete problems arise in geodesy, structural mechanics and economics [92].
3.3
Geometric Multigrid Method (GMG) Error smoothing principle is one of the two basic principles of the multigrid approach. Many
classical iterative methods (e.g. Gauss-Seidel) if appropriately applied to discrete elliptic problems have a strong smoothing effect on the error of any approximation. Obviously, most iteration methods can be interpreted as an error averaging process. Fig.3.3 shows a classical smoothing effect of lexicographic Gauss-Seidel iteration on initial error of discrete elliptic problems [92,104-107].
Error of initial guess Error after 5 iterations
Error after 10 iterations
Fig.3.3: Influence of lexicographic Gauss-Seidel iteration on the error [92].
The other basic principle of the multigrid approach is the following: a quantity that is smooth on a certain grid can, without any essential loss of information, also be approximated on a coarser grid, say a grid with double the mesh size. In other words: if we are sure that the error of the approximation has become smooth after some iteration steps, this error may be approximated by a suitable procedure on a (much) coarser grid. This is the coarse grid principle. A coarse grid procedure is substantially less expensive (substantially fewer grid points) than a fine grid procedure.
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We can form the grid sequence Ω , Ω2 , Ω4 , ⋯ , Ω 0 just by doubling the mesh size successively. We assume that this sequence ends with a coarsest grid Ω 0 which may well be the grid consisting of only one interior grid point (see Fig. 3.4). As discussed above, one can apply an iterative scheme on the fine grid Ω until the error only consists of smooth, low frequent components. Then, this error is transformed to the coarse grid Ω2 where it looks more oscillatory. If we now relax on this grid, it should effectively eliminate additional error modes. This could be continued with more coarse grids.
Finest grid
Coarsest grid 1
Fig.3.4: A sequence of coarse grids for the unit square starting with = 16 [92]. Let the linear system of equations on grid Ω be given by 𝐴 𝑢 = 𝑓
(3.1)
where 𝑢 is the exact solution of the algebraic system (3.1), 𝐴 is a sparse (symmetric) matrix. Let 𝑣 denotes an approximation to the exact solution 𝑢 . There are now two important measures of 𝑣 as an approximation to 𝑢 :
Error 𝑒 given by 𝑒 = 𝑢 − 𝑣
(3.2)
The problem is that the error is inaccessible as the exact solution itself.
Residual 𝑟 ; is the amount by which the approximation 𝑣 fails to satisfy the original problem 𝐴 𝑢 = 𝑓 : 𝑟 = 𝑓 − 𝐴 𝑣
(3.3)
Substitution from Eq. 3.3 into Eq.3.1 and using Eq.3.2, the linear system of equations on grid Ω can be written as 𝐴 𝑒 = 𝑟
(3.4)
Now, if 𝑣 was obtained after performing some relaxation steps, then the error 𝑒 is smooth and hence Eq.3.4 can be solved on a coarser grid Ω2 with less computational work. Once 𝑒 is obtained, it works as a correction for 𝑣 according to Eq.3.2.
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As a simple implementation of the above ideas we use the residual equation of the fine grid and restrict it on the coarse grid and apply the following scheme, called the Two-Grid Correction Scheme:
Relax 𝐴 𝑢 = 𝑓 on Ω to obtain an approximation v
Compute the residual 𝑟 = 𝑓 − 𝐴v and restrict it to the coarser grid
Solve the residual equation 𝐴2 𝑒 2 = 𝑟 2 on Ω2 to obtain an
𝑟 2 = 𝐼2 𝑟
approximation to the
error 𝑒 2
Correct the approximation obtained on Ω with the error estimate obtained on Ω2 : v ← v + 𝐼2 𝑒 2
Relax 𝐴 𝑢 = 𝑓 on Ω starting by v The transformation of corrections from coarse to fine grids is denoted as 𝐼2 and called
interpolation or prolongation while the transformation of residuals from fine to coarse grid is denoted as 𝐼2 and is called restriction. The simplest example for a restriction operator is the "injection" operator which identifies grid functions at coarse grid points with the corresponding grid functions at fine grid points. A fine and a coarse grid with the injection operator are presented in Fig. 3.5. Another frequently used restriction operator is the full weighting (FW) operator. A very frequently used prolongation method is the bilinear interpolation [92].
Fig. 3.5: A fine and a coarse grid with the injection operator [92]. Now, it is possible to apply any multigrid scheme, for example the V-cycle which is based on the correction scheme. It is build up by arbitrary coarse grids and is called V-cycle because it goes from the finest grid down to the coarsest grid and then up again. In addition, Fig.3.6 represents a V-cycle using a sequence of four grid levels.
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The steps of the V-cycle can be summarized as: ● Relax on 𝐴 𝑢 = 𝑓 𝜈1 times to get approximation v . ● Compute the residual 𝑟 = 𝑓 − 𝐴v and restrict it as 𝑓 2 = 𝐼2 𝑟 . ● Relax on 𝐴2 v 2 = 𝑓 2 𝜐1 times. 4 2 ● Compute 𝑓 4 = 𝐼2 𝑟
● Relax on 𝐴4 v 4 = 𝑓 4 𝜐1 times. 8 4 ● Compute 𝑓 8 = 𝐼4 𝑟
⋮ ● Solve 𝐴 0 v 0 = 𝑓 0 (may be possible with direct solver) ⋮ 4 8 ● Correct v 4 ⟵ v 4 + 𝐼8 v
● Relax on 𝐴4 v 4 = 𝑓 4 𝜐2 times. 2 4 ● Correct v 2 ⟵ v 2 + 𝐼4 v
● Relax on 𝐴2 v 2 = 𝑓 2 𝜐2 times. ● Correct v ⟵ v + 𝐼2 v 2
● Relax on 𝐴 𝑢 = 𝑓 𝜐2 times with initial guess v . Ω
Finest grid
Ω2 Ω4 Ω8
(computational grid) Coarsest grid
Relaxation (Smoothing) Restriction Prolongation Direct Solver
Fig. 3.6: V- cycle of the multigrid with 4 grid levels.
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3.4
chapter(3)
Algebraic Multigrid Method (AMG) Geometric multigrid is very effective if a problem is solved on a structured mesh but for
many problems an unstructured mesh is necessary where it is difficult to define coarse grid based on geometric relations. Instead one wants to define coarse grids in an algebraic sense. While geometric multigrid essentially relies on the availability of robust smoothers, AMG takes the opposite point of view. It assumes a simple relaxation process to be given (typically plain Gauss– Seidel relaxation) and then attempts to construct a suitable operator-dependent interpolation 𝐼2 .
Note that it is not important here whether relaxation smoothes the error in any geometric sense. What is important, though, is that the error after relaxation can be characterized algebraically to a degree which makes it possible to automatically construct coarser levels and define interpolations which are locally adapted to the properties of the given relaxation. This is one of the motivations for AMG that reasonable operator-dependent interpolation and Galerkin operator can be derived directly from the underlying matrices, without any reference to the grids and the coarsening process is fully automatic. This automation is the major reason for AMG’s flexibility in adapting itself to specific requirements of the problem to be solved and is the main reason for its robustness in solving large classes of problems despite using very simple point-wise smoothers. The real practical advantage of AMG is that it can be applied directly to structured as well as unstructured grids in 2D as well as in 3D [108]. 3.4.1 Theoretical Basis of AMG In contrast to geometrically based multigrid, algebraic multigrid (AMG) does not require a given problem to be defined on a grid but rather operates directly on (linear sparse) algebraic equations 𝐴𝑢 = 𝑓
𝑜𝑟
𝑛 𝑗 =1 𝑎𝑖𝑗 𝑢𝑗
= 𝑓𝑖
𝑖 = 1,2,3, … , 𝑛 .
(3.5)
If one replaces the terms grids, subgrids and grid points by sets of variables, subsets of variables and single variables, respectively, one can describe AMG in formally the same way as a geometric multigrid method. In particular, coarse-grid discretizations used in geometric multigrid to reduce low-frequency error components now correspond to certain matrix equations of reduced dimension. Moreover, we mostly consider that A is a symmetric M-matrix: it is symmetric (𝐴𝑇 = 𝐴) and positive-definite ( 𝑢𝑇 𝐴𝑢 > 0 for all 𝑢 ≠ 0 ) and positive diagonal entries and non positive offdiagonal entries. Most of AMG rests on two fundamental concepts. The first concept is smooth error which gives us an important foothold in determining an interpolation operator. The second concept is that of strong dependence or strong influence which is important for the selection of the coarse grid. In AMG, the 34
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sense of smoothness must be defined algebraically because it is used to select the coarse grid. So, we will discuss briefly these two concepts to understand the automated processes of the coarse variables selection and interpolation. Algebraic Smoothness Having chosen a relaxation scheme, the crux of the problem is to determine what is meant by smooth error. If the problem provides no geometric information, then we cannot simply examine the Fourier modes of the error. Instead, we must proceed by analogy. In the geometric case, the most important property of smooth error is that it is not effectively reduced by relaxation. Thus, we now define smooth error loosely to be any error that is not reduced effectively by relaxation. We need to understand what this definition means. As we know, Gauss-Seidel and Weighted Jacobi relaxations have the property that after making great progress toward convergence, it stalls, and little improvement is made with successive iterations. At this point, the error is defined to be algebraically smooth. It is useful to examine the implications of algebraic smoothness because all of AMG is based on this concept. The condition of algebraic smoothness can be written as [108]: 𝐴𝑒 ≈ 0
(3.6)
and read it as meaning that smooth error has relatively small residuals. One immediate implication of Eq.3.6 is that 𝑟𝑖 ≈ 0, so 𝑎𝑖𝑖 𝑒𝑖 ≈ −
𝑎𝑖𝑗 𝑒𝑗 𝑖≠𝑗
that is, if e is a smooth error, then 𝑒𝑖 can be approximated well by a weighted average of its neighbors. This fact gives us an important foothold in determining an interpolation operator.
Influence and Dependence An important concept for the selection of the coarse grid is the definition of strong influence and dependence. Both definitions take into account that for diagonal dominant matrix the ith row is associated with the ith unknown. The job of the ith equation is to determine the value of 𝑢𝑖 . Nevertheless, our first task is to determine which other variables are most important in the ith equation; that is, which 𝑢𝑗 are most important in the ith equation in determining 𝑢𝑖 ? One answer to this question lies in the following observation: if the coefficient, 𝑎𝑖𝑗 , which multiplies 𝑢𝑗 in the ith equation, is large relative to the other coefficients in the ith equation, then a small change in the value of 𝑢𝑗 has more effect on the value of 𝑢𝑖 than a small change in other variables in the ith 35
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equation. Intuitively, it seems logical that a variable whose value is instrumental in determining the value for 𝑢𝑖 would be a good value to use in the interpolation of 𝑢𝑖 . Hence, such a variable (point) should be a candidate for a coarse-grid point. This observation suggests the following definition. Definition 1: Given a threshold value 0 < 𝜃 ≤ 1, the variable (point) 𝑢𝑖 depends strongly on the variable (point) 𝑢𝑗 if −𝑎𝑖𝑗 ≥ 𝜃 𝑚𝑎𝑥𝑘≠𝑖 −𝑎𝑖𝑘 This says that the variable i strongly depends on grid point j if the coefficient 𝑎𝑖𝑗 is comparable in magnitude to the largest off- diagonal coefficient in the ith equation. This definition can be stated from another point of view. Definition 2: If the variable 𝑢𝑖 depends strongly on the variable 𝑢𝑗 , then the variable 𝑢𝑗
has strong
influences on the variable 𝑢𝑖 . The sets which can be derived from these two definitions are: 𝑠𝑖 :
The set of points that have strong influence on i, that is the points on which the point i strongly depends 𝑠𝑖 = 𝑗: −𝑎𝑖𝑗 ≥ 𝜃𝑚𝑎𝑥𝑘≠𝑖 −𝑎𝑖𝑘
𝑠𝑖𝑇 :
The set of points that depend strongly on the point i. 𝑠𝑖𝑇 = 𝑗: 𝑖 ∈ 𝑠𝑗
𝐶𝑖 :
(3.7)
(3.8)
The set of coarse-grid variables used to interpolate the value of the fine-grid variable 𝑢𝑖 which called the coarse interpolatory set for i.
The formal structure of AMG, with all its components, is described in the next section and some additional notations are introduced [92].
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3.4.2 Formal Algebraic Multigrid Components and Notations Since the recursive extension of any two-level process to a real multilevel process is formally straightforward, we describe the components of AMG only on the basis of two-level methods with indices h and H distinguishing the fine and coarse level, respectively. In particular, we rewrite Eq.3.5 as 𝐴 𝑢 = 𝑓
𝑜𝑟
𝑗 ∈Ω
𝑎𝑖𝑗 𝑢𝑗 = 𝑓𝑖 (𝑖 ∈ Ω )
(3.9)
with Ω denoting the index set { 1,2, . . . , n}. We implicitly assume always that 𝐴 corresponds to a sparse matrix. The particular indices, h and H, have been chosen to have a formal similarity to geometric two-grid descriptions. In general, they are not related to a discretization parameter. In order to derive a coarse-level system from Eq.3.9, we first need a splitting of Ω into two disjoint subsets Ω = 𝐶 ∪ 𝐹 with 𝐶 representing those variables which are to be contained in the coarse level (C-variables) and 𝐹 being the complementary set (F-variables). Assuming such a splitting to be given and defining Ω𝐻 = 𝐶 , coarse-level AMG systems, 𝐴𝐻 𝑢𝐻 = 𝑓 𝐻
𝑜𝑟
𝑙∈Ω 𝐻
𝐻 𝐻 𝑎𝑘𝑙 𝑢𝑙 = 𝑓𝑘𝐻 (𝑘 ∈ Ω𝐻 )
(3.9)
will be constructed based on the Galerkin principle [92], i.e. the matrix 𝐴𝐻 is defined as the Galerkin operator 𝐴𝐻 ≔ 𝐼𝐻 𝐴 𝐼𝐻
(3.10)
where 𝐼𝐻 and 𝐼𝐻 denote interpolation (or prolongation) and restriction operators, respectively, mapping coarse-level vectors into fine-level ones and vice versa. We always assume that both operators have full rank.
3.4.3 The Algebraic Multigrid Algorithm The application of AMG to a given problem is a two-part process. The first part, a fully automatic setup phase, consists of recursively choosing the coarser levels and defining the transfer and coarse-grid operators. The second part, the solution phase, just uses the resulting components in order to perform normal multigrid cycling until a desired level of tolerance is reached (usually involving Gauss-Seidel relaxation for smoothing). The solution phase is straightforward and requires no explicit description. This section describes algorithmic components used in the setup phase. The construction of C/F-splitting (Coarsening) and the transfer operators 𝐼𝐻 and 𝐼𝐻 components, which forms the major task of AMG's setup phase, involves closely related processes and, whenever we talk about transfer 37
Multigrid Methods
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operators, we always implicitly assume a suitable C/F-splitting to be given. These components need to be selected so that an efficient interplay between smoothing and coarse-grid correction, and consequently good convergence, is achieved. It is equally important that the splitting and the transfer operators are such that 𝐴𝐻 is still reasonably sparse and much smaller than 𝐴 . Summarizing, the algorithm which defines the setup phase process can be listed as [109]: Algorithm 3.1: The setup phase 1. Given a fine grid matrix ( 𝐴 ). 2. Choose the coarse grid (coarsening) Ω𝐻 3.
Define the interpolation operator 𝐼𝐻 .
4. Set the restriction operator 𝐼𝐻 = 𝐼𝐻
𝑇
.
5. Define the coarse grid matrix𝐴𝐻 as in Eq. (3.10) 6. Repeat from step 2 to step 5 to define these components for all levels. 3.4.3.1 The Coarsening Process The proposed simple splitting algorithm corresponds to the “preliminary C-point choice”. Essentially, one starts with defining some first variable, i, to become a C-variable. Then all variables, j, which are strongly coupled to i (i.e. all 𝑗 ∈ 𝑆𝑖𝑇 as in Eq.3.8) become F-variables. Next, from the remaining undecided variables, another one is defined to become a C-variable and all variables which are strongly coupled to it (and which have not yet been decided upon) become Fvariables. This process is repeated until all variables have been taken care of. The only problem is that, in order to avoid randomly distributed C/F-patches and instead obtain reasonably uniform distributions of C- and F-variables, we need to perform this process in a certain order. In order to ensure that there is a tendency to build the splitting starting from one variable and continuing “outwards” until all variables are covered, we introduce a “measure of importance”, 𝜆𝑖 , of any undecided variable i to become the next C-variable. Define [1]: 𝜆𝑖 = 𝑠𝑖𝑇 ∩ 𝑈 + 2 𝑠𝑖𝑇 ∩ 𝐹
(3.11)
where U, at any stage of the algorithm, denotes the current set of undecided variables. (For any set P, 𝑃 denotes the number of elements it contains.) 𝜆𝑖 acts as a measure of how valuable a variable 𝑖 ∈ 𝑈 is as a C-variable, given the current status of C and F. Initially, variables with many others strongly coupled to them become C-variables, while later the tendency is to pick as Cvariables those on which many F-variables strongly depend.
38
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The complete algorithm is carried out like this [109]: 1. Set 𝐶 = 𝜙, 𝐹 = 𝜙, 𝑈 = Ω , and compute 𝜆𝑖 for all 𝑖 . 2. Pick an 𝑖 ∈ 𝑈 with maximal 𝜆𝑖 . Set 𝐶 = 𝐶 ∪ 𝑖 and 𝑈 = 𝑈 − {𝑖}. 3. For all 𝑗 ∈ 𝑆𝑖𝑇 ∩ 𝑈, perform steps 4 and 5. 4. Set 𝐹 = 𝐹 ∪ 𝑗 and 𝑈 = 𝑈 − {𝑗}. 5. For all 𝑙 ∈ 𝑆𝑗 ∩ 𝑈, set 𝜆𝑙 = 𝜆𝑙 + 1. 6. For all 𝑗 ∈ 𝑆𝑖 ∩ 𝑈, set 𝜆𝑗 = 𝜆𝑗 − 1. 7. If 𝑈 = 𝜙, stop. otherwise, go to step 2. We point out that the measure 𝜆𝑖 has to be computed globally only once, at the beginning of the algorithm. At later stages, it just needs to be updated locally. Another two important criteria for choosing the coarse grid points:
First criterion: F – F dependence - (C1) For each 𝑖 ∈ 𝐹 , each point 𝑗 ∈ 𝑠𝑖 should either be in C itself or should depend on at least one point in 𝐶𝑖 .
Since the value of 𝑢𝑖 depends on the value of 𝑢𝑗 , the value of 𝑢𝑗 must be represented on the coarsegrid for good interpolation. If j isn’t a C-point, it should depend on a point in 𝐶𝑖 so its value is “represented” in the interpolation.
Second Criterion: Maximal Subset
- (C2): C should be a maximal subset with the property that no C-point depends on another.
(C1) tends to increase the number of C-points. In general, the more C-points on coarse grid mean the better the h-level convergence. But more C-points mean more work for relaxation and interpolation. (C2) is designed to limit the size (and work) of the coarse grid. [92]
39
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3.4.3.2 The Interpolation (Prolongation)Operator Assume that we have already designated the coarse-grid points. This means that we have a partitioning of the indices {1, 2, …., n}= 𝐶 ∪ 𝐹, where the variables (points) corresponding to 𝑖 ∈ 𝐶 are the coarse-grid variables. These coarse-grid variables are also fine-grid variables; the indices 𝑖 ∈ 𝐹 represent those variables that are only fine-grid variables. Suppose that 𝑒𝑖 , 𝑖 ∈ 𝐶, is a set of values on the coarse grid representing a smooth error that must be interpolated to the fine grid, 𝐶 ∪ 𝐹. If a C-point j strongly influences an F-point i, then the value 𝑒𝑗 contributes heavily to the value of 𝑒𝑖 in the ith (fine-grid) equation. It seems reasonable that the value 𝑒𝑗 in the coarse-grid equation could therefore be used in an interpolation formula to approximate the fine-grid value 𝑒𝑖 . Thus, we have a justification for the idea that the fine-grid quantity 𝑢𝑖 can be interpolated from the coarse-grid quantity 𝑢𝑗 if i strongly depends on j. For each fine-grid point i, we define 𝑁𝑖 , where: 𝑁𝑖 = 𝑗 ∈ Ω : 𝑗 ≠ 𝑖, 𝑎𝑖𝑗 ≠ 0 .
(3.12)
These points can be divided into three categories:
𝐶𝑖 : the neighboring coarse-grid points that have strong influence on i; this is the coarse interpolatory set for i and can be defined by 𝐶𝑖 = 𝐶 ∩ 𝑆𝑖 .
(3.13)
𝐷𝑖𝑠 : the neighboring fine-grid points that have strong influence on i.
𝐷𝑖𝑤 : the points that have not strong influence on i, this set may contain both coarse- and finegrid points; it is called the set of weakly connected neighbors.
To determine the prolongation operator between the grids, we need to define the component (𝐼2 𝑒)𝑖 for each ith on the fine grid. The component (𝐼2 𝑒)𝑖 can be given by: (𝐼2 𝑒)𝑖
𝑒𝑖 =
𝑖𝑓 𝑖 ∈ 𝐶
𝑗 ∈𝐶𝑖 𝜔𝑖𝑗 𝑒𝑗
𝑖𝑓 𝑖 ∈ 𝐹
where the interpolation weights, 𝜔𝑖𝑗 , are given by [108]: 𝑎𝑖𝑗 + 𝜔𝑖𝑗 = −
𝑚 ∈𝐷𝑖𝑠
𝑎𝑖𝑖 +
40
𝑎𝑖𝑚 𝑎𝑚𝑗 𝑘∈𝐶𝑖 𝑎𝑚𝑘 𝑛∈𝐷𝑖𝑤
𝑎𝑖𝑛
(3.14)
Multigrid Methods
chapter(3)
3.4.4 Examples for coarsening process and interpolation To illustrate the coarsening process, we start with the following simple differential equation: 𝑎𝑢𝑥𝑥 + 𝑏𝑢𝑦𝑦 = 0,
(3.15)
in a unit square domain with Dirichlet boundary conditions. We use the central finite difference 1
method to discretize Eq.3.15 with mesh size = 6. The resulting coefficient matrix A has size (25 × 25). Although matrix A contains all required information in classical AMG, the used grid is shown in Fig 3.7 to simplify presentation.
(
) boundary point, (
) interior point, (bold segments) strong connections
Fig.3.7: The fine grid points and their numbering. In the following, we consider two cases for the values of coefficients 𝑎, 𝑏. (i)
Case1:
𝒂 = 𝟏 and 𝒃 = 𝟏
Substituting 𝑎 = 1 and 𝑏 = 1 in Eq.3.15, we get Laplace equation 𝑢𝑥𝑥 + 𝑢𝑦𝑦 = 0. A general row i in the coefficient matrix A can be written (see Fig.3.7) as: 4𝑢𝑖 − 𝑢𝐸 − 𝑢𝑊 − 𝑢𝑁 − 𝑢𝑆 = 0
Recall that the set ′ 𝑠𝑖 ′ of points that strongly influence i are defined by Eq.3.8: 𝑠𝑖 = 𝑗: −𝑎𝑖𝑗 ≥ 𝜃 𝑚𝑎𝑥𝑘≠𝑖 −𝑎𝑖𝑘 . 1
Assuming 𝜃 = 4, one can easily conclude that 𝑠𝑖 = 𝐸, 𝑊, 𝑁, 𝑆 .
41
Multigrid Methods
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N
1 -1
W
4
-1
-1
i
E
-1 S
Fig.3.8: The stencil relation between any point (i) and its neighbors. The coarsening process Adopting algorithm 3.1, we first define sets 𝑠𝑖𝑇 using Eq. 3.8 and compute the measuring values 𝜆𝑖 by Eq.3.11 for all points shown in Fig.3.7. 𝑠1𝑇 𝑠2𝑇 𝑠3𝑇 𝑠4𝑇 𝑠5𝑇 𝑠6𝑇 𝑠7𝑇 𝑠8𝑇 𝑠9𝑇
= {2,6} = {1,3,7} = {2,4,8} = {3,5,9} = {4,10} = {1,7,11} = {2,6,8,12} = {3,7,9,13} = {4,8,10,14}
𝑇 𝑠10 𝑇 𝑠11 𝑇 𝑠12 𝑇 𝑠13 𝑇 𝑠14 𝑇 𝑠15 𝑇 𝑠16 𝑇 𝑠17 𝑇 𝑠18
= {5,9,15} = {6,12,16} = {7,11,13,17} = {8,12,14,18} = {9,13,15,19} = {10,14,20} = {11,17,21} = {12,16,18,22} = {13,17,19,23}
𝑇 𝑠19 𝑇 𝑠20 𝑇 𝑠21 𝑇 𝑠22 𝑇 𝑠23 𝑇 𝑠24 𝑇 𝑠25
= {14,18,20,24} = {15,19,25} = {16,22} = {17,21,23} = {18,22,24} = {19,23,25} = {20,24}
Step 1: F= ∅, C= ∅, U= { 𝑈1 , 𝑈2 , 𝑈3 , … . . , 𝑈25 }. 𝜆1 𝜆2 𝜆3 𝜆4 𝜆5 𝜆6 𝜆7 𝜆8 𝜆9
=2 =3 =3 =3 =2 =3 =4 =4 =4
𝜆10 𝜆11 𝜆12 𝜆13 𝜆14 𝜆15 𝜆16 𝜆17 𝜆18
=3 =3 =4 =4 =4 =3 =3 =4 =4
𝜆19 = 4 𝜆20 = 3 𝜆21 = 2 𝜆22 = 3 𝜆23 = 3 𝜆24 = 3 𝜆25 = 2
Step 2: Since maxim value of 𝜆𝑖 = 4, and 𝑠7𝑇 = {2,6,8,12} one can choose 𝑖 = 7 as a coarse variable, and hence update the sets C and U as 𝐶 = 7 , U={1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}. Steps 3-4:
𝑠7𝑇 ∩ 𝑈 = {2,6,8,12}
42
Multigrid Methods
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The updated sets become: F={2,6,8,12}, U= {1,3,4,5,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24,25}. Steps 5: Since 𝑗 ∈ {2,6,8,12} and 𝑆2 = 1,3,7 , 𝑆6 = 1,7,11 , 𝑆8 = 3,7,9,13 , 𝑆12 = {7,11,13,17}, then, applying 𝜆𝑙 = 𝜆𝑙 + 1, then 𝜆1 = 4, 𝜆3 = 5, 𝜆9 = 5, 𝜆11 = 5, 𝜆13 = 6, 𝜆17 = 5. Steps 6: For all 𝑗 ∈ 𝑆7 ∩ 𝑈 = ∅, set 𝜆𝑗 = 𝜆𝑗 − 1 doesn't lead to any change. The updated 𝜆𝑖 values for 𝑖 ∈ 𝑈 becomes: 𝜆1 = 4 𝜆13 = 6 𝜆20 = 3 𝜆3 = 5 𝜆14 = 4 𝜆21 = 2 𝜆4 = 3 𝜆15 = 3 𝜆22 = 3 𝜆5 = 2 𝜆16 = 3 𝜆23 = 3 𝜆9 = 5 𝜆17 = 5 𝜆24 = 3 𝜆10 = 3 𝜆18 = 4 𝜆25 = 2 𝜆19 = 4. 𝜆11 = 5 The process is repeated from step2-6 with the new sets U, C, F and the updated 𝜆𝑖 values for 𝑖 ∈ 𝑈. The result of this sweep is given here briefly. Let: 𝐶2 = 13 then 𝐹2 ={8,12,14,18} and the updated sets are F={2,6,8,12,14,18}, C= 7,13 , U={1,3,4,5,9,10,11,15,16,17,19,20,21,22,23,24,25}. These sweeps continued until all points in set U become in C or F sets. The resulting sets F and C is shown in Fig.3.9, where C={1,3,5,7,9,11,13,15,17,19,21,23,25}, F={2,4,6,8,10,12,14,16,18,20,22,24}.
The coarse (C
), fine (F
) and boundary (
) points.
Fig. 3.9: The coarse and fine point for case 1.
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Multigrid Methods
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Construction of the prolongation operator The prolongation operator is defined algebraically by Eq.3.14. To illustrate this process we consider here, for example, the fine point 'i =2', using Eq.3.12 and Eq.3.13, N2= {1, 3, 7},
C2= {1, 3, 7}.
So, the error at i =2 will be obtained from Eq.3.14 as 𝑒2 = −
−1 4
𝑒1 + −
−1
𝑒3 + −
4
−1 4
1
𝑒7 = 4 (𝑒1 + 𝑒3 + 𝑒7 )
And this operation is repeated for all points in the fine grid. So, the prolongation operator (𝐼2 ) is
written as:
This row contains the coarse points numbering
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
1 3 5 7 1 1 4
1 4
1 1 4
11 13 15 17 19 21 23 25
1 4
1 4
1
1 4
9
1 4 1 4
1
1 4
1 4
1 4 1 4
1 4
1
1 4
1 4
1 4
1
1 4
1 4
1 4
1
1 4
1 4
1 4 1 4
1
1 4
1 4 1 4
1
1 4
1 4
1 4 1 4
1
1 4
1 4
1 4
1
1 4
23 24 25
1 4
1 4
1 4
1 4
1 1 4
1 4
1
44
Multigrid Methods
(ii)
Case 2:
chapter(3)
𝒂 = 𝟏 𝐚𝐧𝐝 𝒃 = 𝟏𝟎𝟎𝟎
Substituting a=1 and b=1000 in Eq.3.15, then 𝑢𝑥𝑥 + 1000𝑢𝑦𝑦 = 0 and upon discretization, the ith algebraic equation is given by (using notation given in Fig,3.8 ) 2002𝑢𝑖 − 𝑢𝐸 − 𝑢𝑊 − 1000𝑢𝑁 − 1000𝑢𝑆 = 0.This equation shows that variable i is strongly connected to only the two
variables N, S in contrast with case 1 (Laplace equation) where variable i was influenced by variables 𝐸, 𝑊, 𝑁, 𝑆. Then, we repeat the same steps that have been made in the previous example and get the sets F and C. These sets are shown in Fig.3.10 which is completely different compared with the sets resulted in case 1(Laplace equation).
The coarse (C
), fine (F
) and boundary (
) points.
Fig. 3.10: The coarse and fine point for case 2.
45
Multigrid Methods
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The prolongation operator (𝐼2 ) is computed as:
This row contains the coarse points numbering
6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
3.5
7
8
9
10
16
17
18
19
20
.5 . 5005 . 5005 . 5005 .5 1 1 1 1 1 . 4998
. 4998 .5
.5 .5
.5 .5
.5 . 4998
. 4998 1 1 1 1 1 .5 . 5005 . 5005 . 5005 .5
Convergence behavior of GMG and AMG
Model problem We consider the 2D boundary value problem governed by 𝑎𝑢𝑥𝑥 + 𝑏𝑢𝑦𝑦 = 𝑎 12𝑥 2 + 𝑏(30𝑦)
(3.16)
in a unit square domain with Dirichlet boundary conditions in three cases for the parameters a and b. The exact solution is given by 𝑢 = 𝑥 4 + 5𝑦 3 . So, the boundary conditions are given by: 𝑢 = 5𝑦 3 at 𝑥 = 0, 𝑢 = 𝑥 4 at 𝑦 = 0, 𝑢 = 5𝑦 3 + 1 at 𝑥 = 1 and 𝑢 = 𝑥 4 + 5 at 𝑦 = 1. 1
Finite difference method is used to discretize the problem on a uniform grid with mesh size = 26 . The resulting algebraic system is written as 46
Multigrid Methods
chapter(3)
𝐴𝑈=𝐹 where coefficient matrix A has dimension 3969 × 3969 and is spares; it contains 19593 nonzero elements. In each of the following cases, both GMG and AMG are implemented as solvers and the convergence behavior of 20 V-cycles is reported. In each V-cycle, one lexicographic point-wise Gauss-Seidel relaxation sweep is performed before restriction and after prolongation at every level. Case 1: 𝒂 = 𝟏, 𝒃 = 𝟏 (Poisson equation) Substituting 𝑎 = 1, 𝑏 = 1, Eq.3.16 is rewritten as: 𝑢𝑥𝑥 + 𝑢𝑦𝑦 = 12𝑥 2 + (30𝑦) The convergence behavior for AMG and GMG is shown in Fig.3.11 where the residual error norm 𝐴𝑈 − 𝐹
2
of the computed solution 𝑈 after each V-cycle is plotted. Convergence rates are
observed for both AMG and GMG in this symmetric case. 10
Multigrid Methods (a=1, b=1)
0
AMG
Residual Norm
GMG 10
10
10
-5
-10
-15
0
2
4
6
8 10 12 Number of iterations
14
16
18
Fig. 3.11: Convergence behavior for AMG and GMG for 𝑎 = 1 and 𝑏 = 1.
47
20
Multigrid Methods
chapter(3)
Case 2: 𝒂 = 𝟏, 𝒃 = 𝟏𝟎 Anisotropic equation
Substituting 𝑎 = 1, 𝑏 = 10 in Eq.3.16, yields 𝑢𝑥𝑥 + 10𝑢𝑦𝑦 = 12𝑥 2 + 10(30𝑦) In Fig. 3.12, the residual error norm after every V-cycle is plotted showing different performance for AMG and GMG. AMG still has the excellent convergence rate as in the case of Poisson equation while GMG fails even to converge.
Residual Norm
10
10
Multigrid Method (a=1, b=10)
0
AMG
-5
GMG
10
10
-10
-15
0
2
4
6
8 10 12 Number of iteration
14
16
18
20
Fig. 3.12: Convergence behavior for AMG and GMG for 𝑎 = 1 and 𝑏 = 10. Case 3: 𝒂 = 𝟏, 𝒃 = 𝟏𝟎𝟎𝟎 Severe anisotropic equation
To examine the performance of AMG in case of severe anisotropic problems, we substitute 𝑎 = 1, 𝑏 = 1000 in Eq.3.16, then 𝑢𝑥𝑥 + 1000𝑢𝑦𝑦 = 12𝑥 2 + 1000(30𝑦) In Fig. 3.13, the residual error norms of the solutions of AMG and GMG are plotted for this case. It is clear that AMG converges nearly at the same rates compared with the previous cases while GMG does not converge at all. It is important to mention that the efficiency of GMG is retained if the relaxation method is carefully chosen. In the present model problem, it is known that line Gauss-Seidel relaxation is the correct choice. However, choosing the best relaxation method for GMG is not straight forward in general problems. 48
Multigrid Methods
Residual Norm
10
10
chapter(3)
Multigrid Method (a=1,b=1000)
0
-5
AMG GMG
10
10
-10
-15
0
2
4
6
8 10 12 Number of iteration
14
16
18
20
Fig. 3.13: Convergence behavior for AMG and GMG for 𝑎 = 1 and 𝑏 = 1000.
3.6 Multigrid for systems of equations 3.6.1 Introduction Fluid Dynamics are usually governed by systems of partial differential equations (n differential equations in n unknown functions) requiring efficient solution methods. Classical multigrid methods may fail to solve such systems because these governing equations may show elliptic, parabolic or hyperbolic behavior. This difficulty can be removed by designing a solver that effectively distinguishes between the elliptic, parabolic, and hyperbolic (advection) factors of the system and treats each one appropriately. The application of MG to general systems of second order equations is not yet well developed, although some general considerations are given for GMG by Hackbusch [110] and Brandt [111]. Advanced geometric multigrid (GMG) flow solvers based on factorizable schemes have appeared and can be categorized into 3 approaches. One approach to separate the factors is the distributed relaxation method proposed in [111-115]. Distributed relaxation introduces a set of auxiliary local variables. Using these variables, the relaxation equations form a triangular matrix with simple
49
Multigrid Methods
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elliptic or hyperbolic terms on the main diagonal. Efficient relaxation of such factors is a much simpler task than relaxing the entire system of equations. Another approach was presented by Ta’asan [116]. This approach is based on a set of “canonical variables” which express the differential equations in terms of elliptic and hyperbolic partitions. Ta’asan used this partition to guide the discretization of the equations. A staggered grid was used, with different variables residing at cell, vertex, and edge centers. In the third approach [117], a projection operator is applied to the incompressible Euler system of equations resulting in a Poisson equation for the pressure. Because the elliptic and advection parts of the system are decoupled, ideal multigrid efficiency was achieved. Compared to the distributive relaxation and the canonical variables’ approaches, this method is extremely simple. In this section we discuss how to develop AMG for matrices arising from discretizing PDE systems. It is usually assumed that the AMG solver is supplied with more information than just the matrix. It can be assumed, in particular, that the unknowns and equations come with labels, i.e., they are classified so that the solver knows which continuous function or PDE equation is approximated by each of them. Any AMG user can easily supply this classification along with his matrix. A corresponding labeling can then be carried over to coarser levels, and the interpolation can be based on it, each function being separately interpolated. For example, in the case of Naveir-Stokes equations we distinguish the momentum equations from the continuity equation [118].
3.5.2 AMG for systems of partial differential Equations In the present Section, we introduce a new contribution for solving systems of PDEs by AMG. Navier Stokes equation is one of the most famous PDE system in CFD which is described by two equations, the momentum and continuity equations. The main difficulty appears when we try to solve the two equations together to get the unknowns, the velocity vector and the pressure, due to the different characteristic behavior of the two equations. In 2D, the incompressible flow is governed by a non-linear system of equations which contains 3 unknowns, two velocity components and the pressure and consists of two momentum equations and the continuity equation (see detailed description of the governing differential equations and their discretization by finite element in Chapter 5). We suggest an iterative scheme for the solution. First the momentum equation is solved with the pressure distribution from the last iteration step while in a second step the pressure is updated from a derived pressure-correction equation. The fast, robust linear algebraic multigrid solver is used to solve the resulting linear systems in both steps.
In Chapter 5, discretization of the steady and unsteady Navier-stokes equation by the finite element method is presented in details. In the steady case, linearization by Newton iteration produces the
50
Multigrid Methods
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linear algebraic system Eq.(5.43) in each iteration step. However, for the unsteady case, adaptive time stepping algorithm is applied and resulted in the algebraic system Eq.(5.64) in each time step. Both Eq.(5.43) and Eq.(5.64) can be put in the following partitioned matrix form: 𝐴 𝐵
𝐵𝑇
𝑢
0
𝑃
𝑓 =
𝑔
(3.17)
If 𝑛𝑢 is the number of velocity degrees of freedom in x- and y- directions and 𝑛𝑝 is the number of pressure degrees of freedom, then the dimension of A is 𝑛𝑢 × 𝑛𝑢 while the dimension of B is 𝑛𝑝 × 𝑛𝑢 . Equation (3.17) can be written as discretizations of the momentum equations: 𝐴𝑢 + 𝐵 𝑇 𝑃 = 𝑓 ,
(3.18)
and the continuity equation 𝐵𝑢 = 𝑔
(3.19)
3.5.3 The proposed two steps iteration method In the first step in an iteration 'n' we start by initial guess 𝑃𝑛 and solve the system 𝐴 𝑢𝑛 = 𝑓 − 𝐵 𝑇 𝑃𝑛
(3.20)
for 𝑢𝑛 as an approximation for the exact velocity vector 𝑢 = 𝑢𝑛 + 𝛿𝑢, where 𝛿𝑢 is the required correction. Let also the exact pressure vector be 𝑃 = 𝑃𝑛 + 𝛿𝑝. Then, substituting these expressions in Eq. 3.18, we get 𝐴 𝑢𝑛 + 𝛿𝑢 + 𝐵 𝑇 𝑃𝑛 + 𝛿𝑝 = 𝑓
(3.21)
Substitution of Eq. 3.20, into Eq. 3.21 yields 𝐴 𝛿𝑢 = −𝐵 𝑇 𝛿𝑝 from which 𝛿𝑢 is given by 𝛿𝑢 = −𝐴−1 𝐵 𝑇 𝛿𝑝
(3.22)
Now substituting 𝑢 = 𝑢𝑛 + 𝛿𝑢 = 𝑢𝑛 − 𝐴−1 𝐵 𝑇 𝛿𝑝 in Eq. (3.19), we get: 𝐵(𝑢𝑛 − 𝐴−1 𝐵 𝑇 𝛿𝑝 ) = 𝑔
(3.23)
Or, equivalently, the following algebraic system of the second step 𝑆 𝛿𝑝 = 𝐵 𝑢𝑛 − 𝑔 where, 51
(3.24)
Multigrid Methods
chapter(3)
𝑆 = 𝐵𝐴−1 𝐵 𝑇
(3.25)
It is clear that S is a square matrix of dimension 𝑛𝑝 . The solution of Eq.(3.24) produces the pressure correction 𝛿𝑝 that updates the pressure in the nest iteration step as 𝑃𝑛 +1 = 𝑃𝑛 + 𝛿𝑝. In the present implementation we make the approximation 𝐴−1 ≅ (𝑑𝑖𝑎𝑔(𝐴))−1 in the computation of S. The AMG algorithm 1. Compute matrix S using Eq.(3.25) 2. Apply the setup phase to generate automatically the coarser grids sequence for matrix A and matrix S. Iteration Set 𝑛 = 0, 𝑃𝑛 = {0} 3. Step 1: Apply 2 AMG-V-cycles to solve Eq. (3.20) to get 𝑢𝑛 . 4. Step 2: Apply 2 AMG-V-cycles to solve Eq. (3.24) to get 𝛿𝑝.Update the pressure with 𝑃𝑛+1 = 𝑃𝑛 + 𝛿𝑝 5. Repeat steps 3, 4 until 𝛿𝑝 < tolerance In chapter (6), this proposed AMG algorithm is applied for various applications in CFD, such as steady laminar Navier-stokes, steady and unsteady turbulent Navier-stokes, either as a solver or a preconditioner for generalized minimum residual method (GMRES). The efficiency of this algorithm is demonstrated by comparison with other solvers as GMG and GMRES.
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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Chapter (4) The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
4.1 Introduction: The convection-diffusion equations play an important role in many engineering and physics phenomena. In most practical problems, the magnitude of convection coefficient is much greater than that of diffusion coefficient. So, these problems are called convection-dominated or singularly perturbed, and the numerical solutions of these problems present serious difficulties because the solution of these diffusion-convection problems possesses boundary layers that are small subregions where derivatives of the solution are very large. These boundary layers make standard finite element or finite difference methods unsuitable for solving these problems because the numerical solutions produce non physical oscillations and low order of accuracy unless refined meshes are introduced in the boundary regions using an adaptive mesh refinement strategy. For this strategy to be effective, it is important that error do not propagate into regions where refinement is not needed. So, the computational costs increase to obtain satisfactory numerical results. Another difficulty occurs when multigrid is used for solving convection-dominated problems using classical discretization methods. Even if the grid where the solution is computed provides suitable accuracy, the multigrid algorithm requires a sequence of coarser grids, and it is important that the discretizations on these grids capture the character of the solution with a reasonable degree of accuracy. For these reasons, it is necessary to have available a discretization strategy that does not have the deficiencies exhibited by the classical discretization method. These difficulties have been the researchers' main motivation during the last two decades, to obtain discrete formulations that are, at the same time, accurate for smooth problems and stable for problems with boundary layers. High-order compact finite difference schemes have been formulated including polynomial [119, 120] or exponential [121, 122] schemes. The high order 53
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
compact difference techniques have also been generalized to three dimensions [123, 124]. However, those high order compact difference schemes are all based on uniform discretization. One existing strategy to deal with boundary layers is to first discretize the equation on a nonuniform (stretched) grid then a grid transformation technique is used to map the nonuniform grid to a uniform one, on which the fourth order compact scheme with uniform mesh size is applied [125]. This strategy has been proven successful to recover fourth order convergence rate in the presence of boundary layers. Two potential problems with the grid transformation strategy are that the transformed equation becomes more complex containing mixed terms in addition to the difficulty in constructing suitable grid transformation functions for general applications. Another strategy is to use high order compact difference scheme that can accommodate nonuniform mesh size [126, 127]. Within the finite element context the same difficulties arise when the standard Galerkin method is applied to convection-diffusion equations. To overcome these difficulties various formulations have been introduced including: upwinding techniques, see e.g. [128, Chapter 8] for a description of some of the most common of these schemes, streamline-upwind Petrov/Galerkin finite element method (SUPG) [5, 6], stream line diffusion methods [7, 8] and other special finite elements formulations see e.g. [9, 10]. As it is well known, streamline-upwind Petrov/Galerkin method (SUPG) corresponds to modifying a weighting function in the Galerkin formulation to produce a small additional diffusion in the streamline direction. The amount of such additional diffusion is tuned by a parameter τ that must be chosen in a suitable way. According to rule of a thumb arguments and a lot of numerical tests, several recipes have been proposed for the choice of τ. Nevertheless, the need for a suitable convincing argument to guide the choice of τ is still considered as a major drawback of the method by several users. One of our main contributions in this thesis is to compute directly the required modification to the diffusion coefficient of the convection diffusion equation that: in general, eliminates oscillatory behavior and for special classes of the equation produces the exact nodal solution [129]. The rest of this Chapter is organized as follows. In Section 4.2, the convection-diffusion weak formulation and the Galerkin finite element method are presented. The oscillatory behavior of classical finite-element methods for convection-dominated convection–diffusion problems is analyzed for one-dimensional model problem in Section 4.3. The well known approach; streamline diffusion method for solving convection-dominated-diffusion equations is reviewed in Section 4.4 since it is used as a 54
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
comparable method to our new approach. In Section 4.5, the proposed modified diffusion method is derived in details. Finally in Section 4.6, numerical results are presented to demonstrate the efficiency of the modified diffusion coefficient method in 1-D & extend its application to 2-D convection–diffusion problems with constant and variable coefficients. Also, numerical results concerning the behavior of multigrid based solvers are presented and discussed.
4.2 Convection-diffusion formulation: The convection-diffusion equation is −ϵ ∇2 𝑢 + w. ∇𝑢 = 𝑓 ,
(4.1)
where, scalar ϵ is the diffusion coefficient and w is the vector of convection cofficient. This equation arises in numerous models of flows and other physical phenomena. The unknown function u may represent the concentration of a pollutant being transported (or “convected”) along a stream moving at velocity w and also subject to diffusive effects. Alternatively, it may represent the temperature of a fluid moving along a heated wall, or the concentration of electrons in models of semiconductor devices. This equation is also a fundamental sub-problem for models of incompressible flow, where w is a vector-valued function representing flow velocity and ϵ is a viscosity parameter [130]. Typically, diffusion is a less significant physical effect than convection: on a windy day the smoke from a chimney moves in the direction of the wind and any spreading due to molecular diffusion is small. This implies that, for most practical problems, ϵ ≪ |w|. The boundary value problem considered is equation (4.1) posed on a two or threedimensional domain Ω, together with boundary conditions on 𝜕𝛺 = 𝜕𝛺𝐷 ∪ 𝜕𝛺𝑁 given by 𝑢 = 𝑔𝐷 𝑜𝑛 𝜕𝛺𝐷 ,
𝜕𝑢 𝜕𝑛
= 𝑔𝑁 𝑜𝑛 𝜕𝛺𝑁
(4.2)
where, f , 𝑔 are the forcing function in 𝛺 and the boundary function on Γ, respectively. It will be assumed, as is commonly the case, that the flow characterized by w is incompressible, that is, div w =∇. w =0.
55
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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Weak formulation of the convection-diffusion equation In the following we will use a simplified integration notation such that 𝑓
is equivalent to
𝑓 𝑑𝛺
𝛺
𝛺
To derive a weak formulation of a convection-diffusion problem equ.(4.1) , we require that for an appropriate set of test functions v, 𝛺
(−ϵ ∇2 𝑢 + w. ∇𝑢 − 𝑓)𝑣 = 0
Using the derivative of a product rule and the divergence theorem, 𝑣∇2 𝑢 = ϵ
−ϵ 𝛺
∇𝑢 . ∇𝑣 − ϵ 𝛺
=ϵ
(4.5) 𝛺
∇. 𝑢 =
𝜕𝛺
𝑢. 𝑛 , then
∇. 𝑣∇𝑢 𝛺
∇𝑢 . ∇𝑣 − ϵ
𝑣
𝛺
𝜕𝛺
𝜕𝑢 , 𝜕𝑛
So that, (4.5) can be written as: ϵ
𝛺
∇𝑢 . ∇𝑣 +
𝛺
(w. ∇𝑢)𝑣 =
𝛺
𝑓𝑣 + ϵ
𝜕𝑢
𝜕𝛺
𝑣 𝜕𝑛 .
(4.6)
For the boundary condition (4.2), the weak formulation (4.6) is as follows: Find u ∈ ℋ𝐸1 such that ϵ
𝛺
∇𝑢 . ∇𝑣 +
𝛺
(w. ∇𝑢)𝑣 =
𝛺
𝑓𝑣 + ϵ
𝜕𝛺 N
𝑣g N .
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣 ∈ ℋ𝐸10
(4.7)
where: ℋ𝐸1 ≔ 𝑢 ∈ ℋ 1 𝛺
𝑑
ℋ𝐸10 ≔ 𝑣 ∈ ℋ 1 𝛺
𝑢 = g D 𝑜𝑛 𝜕𝛺𝐷 ,
𝑑
𝑣 = 0 𝑜𝑛 𝜕𝛺𝐷 .
And, ℋ 1 ≔ 𝑢: 𝛺 → ℝ 𝑢,
𝜕𝑢 𝜕𝑢 , ∈ 𝐿2 𝛺 𝜕𝑥 𝜕𝑦
𝐿2 𝛺 ≔ 𝑢: 𝛺 → ℝ 𝑢 ≔(
𝑢2 < ∞ , 𝛺 Ω
𝑢 2 )1 2 .
(4.8)
Let 𝑎: ℋ 1 Ω × ℋ 1 (Ω) → ℝ denote the bilinear form on the left-hand side of (4.7) so that 𝑎 𝑢, 𝑣 : = ϵ
𝛺
∇𝑢 . ∇𝑣 + 56
𝛺
(w. ∇𝑢)𝑣 .
(4.9)
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chapter(4)
and introducing the linear functional 𝑙: ℋ 1 Ω → ℝ so that 𝑙 𝑣 :=
𝑓𝑣 + ϵ 𝛺
𝜕𝛺 N
𝑣g N .
The weak formulation (4.7) can be restated in the following shorthand form: Find u ∈ ℋ𝐸1 such that 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣 ∈ ℋ𝐸10
𝑎 𝑢, 𝑣 = 𝑙 𝑣
(4.10)
Approximation by finite element method In the formal aspects of finite element discretization of the convection–diffusion equation, the discrete weak formulation is defined using a discrete trial space 𝑆𝐸 ⊂ ℋ𝐸1 and test space 𝑆0 ⊂ ℋ𝐸10 , and implementation entails choosing bases for these spaces and constructing the finite element coefficient matrix and source vector. However, a significant difference for the convection–diffusion equation is the presence of layers in the solution that makes it more difficult to compute accurate discrete solutions [130]. The Galerkin finite element method The Galerkin method for the convection–diffusion problem (4.10) is Find 𝑢h ∈ 𝑆𝐸 such that 𝑎 𝑢h , 𝑣h = 𝑙 𝑣h
for all 𝑣h ∈ 𝑆0
(4.11)
To construct an approximation method, the test space 𝑆0 is a finite n-dimensional vector space of test functions for which {𝜙𝑗 }𝑛𝑗 =1 is a convenient basis, where n is the number of unknowns that corresponds to solution at both interior & natural boundary nodes. Then, in order to ensure that the Dirichlet boundary condition is satisfied, we extend this basis set by defining additional 𝑛+𝑛
𝜕 functions {𝜙𝑗 }𝑗 =𝑛+1 and select fixed coefficients 𝑢𝑗 , 𝑗 = 𝑛 + 1, … . . , 𝑛 + 𝑛𝜕 , so that the
function
𝑛+𝑛 𝜕 𝑗 =𝑛+1 𝑢𝑗 𝜙𝑗
interpolates the boundary data g D on 𝜕ΩD . Here 𝑛𝜕 is the number of
Dirichlet boundary nodes. The Galerkin system (4.11) is then equivalent to the problem Find {𝑢𝑗 }𝑛𝑗 =1 such that 𝑛 𝑗 =1
𝑎(𝜙𝑗 , 𝜙𝑖 ) 𝑢𝑗 = 𝑙 𝜙𝑖 , 57
𝑖 = 1, … , 𝑛,
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
where 𝑙 𝜙𝑖 =
Ω
𝑓𝜙𝑖 +
∂Ω 𝑁
𝑔𝑁 𝜙𝑖 −
𝑛+𝑛 𝜕 𝑗 =𝑛+1
(ϵ Ω
∇𝜙𝑗 . ∇𝜙𝑖 +
Ω
(w. ∇𝜙𝑗 )𝜙𝑖 )𝑢𝑗 . (4.12)
This system can be written in matrix notation as 𝐾𝑢 = 𝑓,
(4.13)
where, 𝐾 = 𝑘𝑖𝑗 , 𝑘𝑖𝑗 = 𝑎 𝜙𝑗 , 𝜙𝑖 ,
𝑓𝑖 = 𝑙 𝜙𝑖 .
(4.14)
The coefficient matrix K is the sum 𝐾 = 𝐴 + 𝑁,
(4.15)
where 𝐴 = 𝑎𝑖𝑗 ,
𝑎𝑖𝑗 =
𝑁 = 𝑛𝑖𝑗 ,
𝑛𝑖𝑗 =
Ω
Ω
∇𝜙𝑗 . ∇𝜙𝑖 , (w. ∇𝜙𝑗 )𝜙𝑖 .
are discrete diffusion and convection operators, respectively.
4.3 The behavior of the convection diffusion equation [129] Many pervious investigations show that the classical Galerkin method which is used to solve convection diffusion equation is depended on the mesh size (h) and other parameters like viscosity (𝜈) to get the accurate solution of this equation. So, in this section, the 1-D convection diffusion equation will be introduced as an example to show the behavior of this equation. 4.3.1 A Class of Convection–Diffusion Problem Consider the 1-D steady convection-diffusion equation −𝜖 𝑢′′ + 𝑤𝑢′ = 0 𝑖𝑛 Ω (0,1) 𝑢 0 = 0, 𝑢 1 = 1 𝜖
Assuming that both 𝜖 and 𝑤 are constants and using 𝜈 = 𝑤 , Eq.(4.16) becomes
58
(4.16)
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
−𝜈 𝑢′′ + 𝑢′ = 0 𝑖𝑛 Ω (0,1) 𝑢 0 = 0, 𝑢 1 = 1
chapter(4)
(4.17)
The exact solution of Eq.(4.17) is given by 𝑒 𝑥 /𝜈 −1
𝑢 = 𝑒 1/𝜈 −1
(4.18)
Discretization of Eq.4.17 with continuous, piecewise-linear finite-elements on a uniform grid, yields the scheme
−𝜈
𝑢 𝑖−1 −2𝑢 𝑖 +𝑢 𝑖+1
𝑢0 = 0, 𝑢𝑛 = 1
+
𝑢 𝑖+1 −𝑢 𝑖−1 2
= 0,
𝑖 = 1, ⋯ , 𝑛 − 1
(4.19)
where, n is the number of elements, h = 1/n , xi = ih, and ui is the finite-element solution at node i. In this particular case, the scheme given by Eq.(4.19) is identical to the finite-difference central
scheme when it is multiplied by 1/h. Introducing the mesh Peclet number defined as 𝑃𝑒 = 2𝜈 into Eq.(4.19), the discrete algebraic system is given by: 𝑢0 = 0 −𝑃𝑒 − 1 𝑢𝑖−1 + 2𝑢𝑖 + 𝑃𝑒 − 1 𝑢𝑖+1 = 0, 𝑖 = 1, ⋯ , 𝑛 − 1
(4.20)
𝑢𝑛 = 1 𝜖
Figure (4.1) shows that, as 𝜐 = 𝑤 decreases, the convection term becomes dominant and a boundary layer appears where the solution becomes nearly singular.
59
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
solution u
1
v nu=0.5 vnu=0.1 nu=0.01 v
0.5
0
-0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x Fig. (4.1): the exact solution of equation (4.18) for different 𝜐.
Also, figure (4.2) shows some characteristics of the general performance of the convectiondiffusion equations where two values for 𝜐 = 0.1, 𝜐 = 0.01 are considered. In this figure, both the discrete solution obtained by the classical Galerkin finite-element method (Eq. (4.20)) for different values of mesh sizes = 0.1, = 0.05, = 0.01 and the exact solutions are plotted. It is easy to conclude the following observations which are well-known as the general performance of the convection-diffusion equations.
𝜖
When 𝜐 = 𝑤 = 0.1, the numerical solution converges to the exact one even on a coarse grid. However, for a smaller value (𝜐 = 0.01), the discrete solution oscillates on coarse grids. On the coarsest grid (h=0.1), the oscillations spread away from the boundary layer and the oscillations become local in the boundary layer when (h=0.05) and then converges to the exact solution as h becomes sufficiently small.
Also, it can be noticed that accurate solutions can be obtained by classical Galerkin method on refined grids only when h is small enough such that the problem Peclet
number 𝑃𝑒 = 2𝜈 < 1. In the next section, we try to interpret this performance.
60
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
h=0.1
Solution u
1
0.5
Galerkin (Pe=0.5, r=3, exact (Pe=0.5, r=3, exact (Pe= 5, r=-1.5, Gaerkin (Pe=5, r=-1.5,
0
nu=0.1) v nu=0.1) v v vnu=0.01) nu=0.01) v
v v v
-0.5
-1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x h=0.05
Solution u
1 Galerkin (Pe=0.25, r = 1.667, nu=0.1) v Exact (Pe=0.25, r = 1.667, nu=0.1) v Galerkin (Pe=2.5, r = -2.333, vnu=0.01) v Exact (Pe=2.5, r = -2.333, vnu=0.01) v v v
0.5
0
-0.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
x h=0.01 1
Solution u
0.8
Galerkin (Pe=0.05, r = 1.1053, nu=0.1 ) v v Exact (Pe=0.05, r = 1.1053, nu=0.1) Galerkin (Pe=0.5, r = 3, nu=0.01) v vnu=0.01) Exact (Pe=0.5, r = 3, v v v
0.6 0.4 0.2 0 0
0.1
0.2
0.3
0.4
0.5
x
1+𝑃𝑒
Fig. 4.2: Performance of Galerkin FEM, (𝑃𝑒 = 2𝜐 , 𝑟 = 1−𝑃𝑒 ). 61
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
4.3.2 Interpretation of the Oscillatory Performance In this subsection we analyze the behavior of the discrete solution of the 1D convection– diffusion problem Eq.(4.17). First, we try to get the exact solution of the discrete system (4.20) which can be assumed in the following form: 𝑢𝑖 = 𝑐1 + 𝑐2 𝑟 𝑖 It is easy to show that r = 1 corresponds to a discontinuous solution: ui = constant at all interior points which may not satisfy the boundary conditions. Substituting with the boundary conditions in the assumed solution, the constants are obtained as 𝑎𝑡 𝑖 = 0 𝑎𝑛𝑑 𝑢0 = 0 → 0 = 𝑐1 + 𝑐2 → 𝑐2 = −𝑐1 i = n and un = 1 → 1 = c1 1 − r n → c1 =
1 1 − rn
So, the solution can be written as: ui =
1 − ri 1 − rn
To get the parameter r, we substitute this solution in Eq. (4.20) which gives: 1 1 − rn
−1 − Pe (1 − r i−1 ) + 2(1 − r i ) + −1 + Pe (1 − r i+1 ) = 0.
Simplifying the above equation (assuming 𝑟 ≠ 1) yields: −1 − Pe + 2r + −1 + Pe r 2 = 0, whose solution is r=
1 + Pe 1 − Pe
In general, the solution of Eq. (4.20) is given by: 1−r i
ui = 1−r n
i = 0, ⋯ , n
ui = 0
i = 0, ⋯ , n − 1,
if Pe ≠ 1 (4.22)
62
un = 1
if Pe = 1
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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It is important to notice that
r is undefined if 𝑃𝑒 = 1 and the discrete solution may be given by the second line in
(4.22), or generally as, 𝑢𝑖 = arbitrary constant , 𝑖 = 1, ⋯ , 𝑛 − 1, 𝑢0 = 0, 𝑢𝑛 = 1 . This can be easily derived from Eq.(4.20).
for 0 < 𝑃𝑒 < 1 → 𝑟 > 1, and hence the discrete solution is free of oscillations. Moreover, the discrete solution converges to the exact solution as h → 0. This can be proven directly by taking the limit of 𝑢𝑖 as → 0 1−𝑟 𝑖
lim 𝑢𝑖 = lim 1−𝑟 𝑛 = lim
→0
1−𝑟 𝑥 𝑖 /
→0 1−𝑟 1/
→0
.
But since, lim 𝑟
→0
1/
= lim
→0
2𝜈 1− 2𝜈
1+
1/
2
= lim 1 + 2𝜈− →0
1/
= 𝑒 1/𝜈 ,
then, lim→0 𝑢𝑖 =
1−𝑒 𝑥 𝑖 /𝜈 1−𝑒 1/𝜈
,
which coincides with the exact solution (4.18)
fo𝑟 𝑃𝑒 > 1 →
𝑟 < −1, and hence the numerical solution is accomplished with
oscillations.
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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4.3.3 Investigation of Numerical Solution for High Peclet Number We are interested in the behavior of the discrete solution, Eq.(4.22), for h fixed and large values
of Pe (i.e., 𝑃𝑒 going to +∞). First note that if 𝑃𝑒 = 2𝜈 ≫ 1 , then 0
1 𝑖𝑓 𝑃𝑘 ≤ 1.
(4.28)
Here, 𝑤𝑘 is the 𝑙2 norm of the wind at the element centroid, 𝑘 is a measure of the element length in the direction of the wind, and 𝑃𝑘 ≔ 𝑤𝑘 𝑘
2ϵ is the element Peclet number.
So, the coefficient matrix for the streamline diffusion discretization has the form K= A+N+S.
(4.29)
The additional matrix S is given by 𝑆 = 𝑠𝑖𝑗 ,
𝑠𝑖𝑗 =
𝑘
𝛿𝑘
∆𝑘
𝑤 . ∇𝜙𝑗
𝑤 . ∇𝜙𝑖 ,
(4.30)
which can be viewed as a discrete diffusion operator associated with the streamline direction defined by 𝑤 . It is symmetric and positive semi-definite, so that it enhances coercivity.
67
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
4.5
chapter(4)
A Modified Diffusion Coefficient Technique [129]
The main contribution in this thesis is a new technique for solving convection diffusion equation by predicting a modified diffusion coefficient (MDC) such that the discretization process applies on the modified equation rather than the original one. For a class of one-dimensional convection diffusion equation we derive the modified diffusion coefficient analytically as a function of the equation coefficients and mesh size and prove that the discrete solution of this method coincides with the exact solution of the original equation for every mesh size and/or equation coefficients. Extending the same technique to obtain analytic MDC for other classes of convection-diffusion equations is not always straight forward especially for higher dimension. However we have extended the derived analytic formula of MDC (of the studied class) to general convection diffusion problems. The analytic formula is computed locally within each element according to the mesh size and associated coefficients values in each direction. The numerical results for 2dimensional, variable coefficients convection-dominated problems show that, although the discrete solution does not coincide with the exact one, it provides stable and accurate solution even on coarse grids with boundary layers.
The Modified Diffusion Coefficient Method (MDC) In this section we prove that the discrete solution (Eq. (4.22)) of the class problem Eq.(4.17) coincides with the exact one (given by Eq.(4.18)) for every if Galerkin finite-element is applied to Eq.(4.17) with its original diffusion coefficient replaced by a modified coefficient.
4.5.1 Estimation of the modified diffusion parameter Let 𝜈 be replaced by 𝜈 + 𝛿 in Eq.(4.17) then, the discrete system becomes − 𝜈+𝛿
𝑢 𝑖−1 −2𝑢 𝑖 +𝑢 𝑖+1
+
𝑢 𝑖+1 −𝑢 𝑖−1
𝑢0 = 0, 𝑢𝑛 = 1
2
= 0,
𝑖 = 1, ⋯ , 𝑛 − 1
(4.31)
Introducing a parameter R, defined as 𝑅=2
68
𝜈+𝛿
(4.32)
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
we obtain the discrete algebraic system 𝑢0 = 0 −𝑅 − 1 𝑢𝑖−1 + 2𝑢𝑖 + 𝑅 − 1 𝑢𝑖+1 = 0, 𝑖 = 1, ⋯ , 𝑛 − 1 𝑢𝑛 = 1
(4.33)
Equation (4.33) is identical to Eq.(4.20) (just Pe is replaced by R). Now, define 1+𝑅
𝑟 = 1−𝑅
(4.34)
Analogous to Eq.(4.22), the exact solution of the discrete system (4.33) is 𝑢𝑖 =
1 − 𝑟𝑖 1 − 𝑟𝑛
𝑢𝑖 = 0
𝑖 = 0, ⋯ , 𝑛
if 𝑅 ≠ 1
𝑖 = 0, ⋯ , 𝑛 − 1,
𝑢𝑛 = 1
if 𝑅 = 1
Substituting 𝑖 = 𝑥𝑖 / , where = 1/𝑛 , then for 𝑅 ≠ 1, the discrete solution at 𝑥𝑖 is given by 𝑢𝑖 =
1−𝑟 𝑥 𝑖 / 1−𝑟 1/
=
1−𝜌 𝑥 𝑖 1−𝜌
, 𝜌 = 𝑟 1/
(4.35)
Since the exact solution Eq.(4.17) can be written as 𝑒 𝑥 /𝜈 −1
𝑢 = 𝑒 1/𝜈 −1 =
1−𝜇 𝑥 1−𝜇
,
𝜇 = 𝑒 1/𝜈 ,
(4.36)
It is easy to notice that discrete solution Eq.(4.35) coincides with the exact one Eq.(4.36) if 𝑟 1/ = 𝑒 1/𝜈 or equivalently, 2𝜈 +2𝛿+
from Eq.(4.32) yields
2𝜈 +2𝛿−
1+𝑅 1/ 1−𝑅
= 𝑒 1/𝜈 which leads to
1+𝑅 1−𝑅
= 𝑒 /𝜈 . Solving for 𝛿 yields 𝛿 =
= 𝑒 /𝜈 . Then, substitution
𝑒 /𝜈 2𝜈− − 2𝜈+ 2 1−𝑒 /𝜈
𝑃𝑒 = 2𝜈 , then 𝛿= Multiplying 𝛿 by
𝑒 2𝑃𝑒 2𝜈 − − 2𝜈 + 𝑒 2𝑃𝑒 1 − 𝑃𝑒 − 1 + 𝑃𝑒 = 2𝜈 , 2 1 − 𝑒 2𝑃𝑒 2 1 − 𝑒 2𝑃𝑒
𝑒 −𝑃 𝑒 𝑒 −𝑃 𝑒
and rearranging, we get
69
. Substituting
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
𝑒 𝑃𝑒 1 − 𝑃𝑒 − 𝑒 −𝑃𝑒 1 + 𝑃𝑒 𝛿=𝜈 𝑒 −𝑃𝑒 − 𝑒 𝑃𝑒
chapter(4)
𝑒 𝑃𝑒 − 𝑒 −𝑃𝑒 − 𝑃𝑒 𝑒 𝑃𝑒 + 𝑒 −𝑃𝑒 =𝜈 − 𝑒 𝑃𝑒 − 𝑒 −𝑃𝑒
Finally, 𝛿 is simplified as 𝛿 = 𝜈 −1 + 𝑃𝑒 coth (𝑃𝑒 )
(4.37)
Thus, we have proven the following theorem. Theorem 4.1 The Galerkin finite-element solution of Eq.(4.17) with the diffusion coefficient 𝜈 replaced by 𝜈0 = 𝜈 + 𝛿 = 𝜈 𝑃𝑒 𝑐𝑜𝑡 (𝑃𝑒 ) produces the exact solution of Eq.(4.17) at every node in the mesh. It is important to notice that this theorem is independent of the mesh size and more importantly is independent on the value of 𝑣. In fact, theorem 4.1 proves that the addition of the term −𝛿
𝑢 𝑖−1 −2𝑢 𝑖 +𝑢 𝑖+1
to the left-hand side of Eq.(4.19) compensates the truncation error
introduced by the numerical scheme. The relation between 𝜈 and 𝜈0 is shown in Fig.(4.3) for the value of = 1/50. It is noticed that, as 𝜈 → 0, 𝜈0 → /2, however, for greater values of 𝜈, the curves of both 𝜈0 and 𝜈 coincide. h/2=0.01 0.12
Modified vnu
0.1
Modified 𝜈nu 0 v nu
0.08
0.06
0.04
0.02
0
0
0.02
0.04
nu v
0.06
0.08
0.1
Fig. (4.4): Relation between the diffusion coefficient 𝜈 and the modified one 𝜈0 .
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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4.5.2 Generalization of the Diffusion Parameter First, a slight generalization of the class problem Eq.(4.16) is now considered as: −𝜖 𝑢′′ + 𝑤𝑢′ = 0 𝑖𝑛 Ω (𝑎, 𝑏) 𝑢 𝑎 = 𝑢𝑎 , 𝑢 𝑏 = 𝑢𝑏
(4.38)
Defining 𝜈 = 𝜖/𝑤 , the exact solution of Eq.(4.38) is given by 𝑢 = 𝑢𝑎 + (𝑢𝑏 − 𝑢𝑎 )
1−𝑒 (𝑥 −𝑎 )/𝜈
(4.39)
1−𝑒 (𝑏−𝑎 )/𝜈
To obtain the exact solution of the discrete algebraic system produced by applying classical 𝑢 −𝑢 𝑎
finite-element, Eq. (4.38) is first normalized to return to the form of Eq.(4.17). Let 𝑢 = 𝑢
𝑏 −𝑢 𝑎
𝑥−𝑎
and 𝑥 = 𝑏−𝑎 then Eq.(4.38) reduces to −𝜈 𝑢′′ + 𝑢′ = 0 𝑖𝑛 Ω (0,1) 𝑢 0 = 0, 𝑢 1 = 1 𝜖
1
𝑑𝑢
(4.40)
𝑑2𝑢
Where, 𝜈 = 𝑤 𝑏−𝑎 and 𝑢′ = 𝑑𝑥 , 𝑢′′ = 𝑑𝑥 2 . The exact solution of the algebraic system resulting by Galerkin finite-element on a uniformly divided domain can be obtained as 𝑢𝑖 = 𝑢𝑎 + 𝑢𝑏 − 𝑢𝑎
1−𝑟 𝑖 1−𝑟 𝑛
= 𝑢𝑎 + (𝑢𝑏 − 𝑢𝑎 )
1 − 𝑟 1/
𝑥 𝑖 −𝑎
1 − 𝑟 1/
𝑏−𝑎
(4.41) 1+𝑃
Where, 𝑢𝑖 is the discrete solution at 𝑥𝑖 = 𝑎 + 𝑖, where = 𝑏 − 𝑎 𝑛 and 𝑟 = 1−𝑃𝑒 , 𝑒
𝑃𝑒 = 2𝜈 ≠ 1 . Using procedures similar to that adopted in the previous section, it could be proven that the discrete finite-element solution will coincide with the exact solution Eq.(4.39) if 𝜈 is replaced in Eq.19 by 𝜈0 = 𝜈 𝑃𝑒 𝑐𝑜𝑡 𝑃𝑒 , where, 𝑃𝑒 =
2𝜈
, 𝑃𝑒 = 2𝜈 . Then, 𝜈0 = 2 coth (2𝜈 ) and
we have the following theorem.
Theorem 4.2 The Galerkin finite-element solution of Eq.(4.38), with the diffusion coefficient 𝜈 = 𝜖/𝑤 replaced by 𝜈0 = 𝑏 − 𝑎 𝜈 𝑃𝑒 𝑐𝑜𝑡 𝑃𝑒 , produces the exact solution of Eq.(4.38) at every node in the mesh. Other generalizations of theorem 4.1 are:
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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1. On non-uniform grids, 𝜈0 is computed locally within each element e using the element mesh size he. Using theorem 4.2, one can prove that exact solution is produced for the considered problem class Eq.(4.17) on non-uniform meshes. 2. For higher dimensional problems, 𝜈0 is computed differently in each direction according to the coefficients component in that direction. Although there is no guarantee that this process produces the exact solution, it is expected that it gives good approximate solutions. 3. For variable coefficients problems, 𝜈0 is computed locally within each element by approximating the coefficients by their corresponding values at the center of that element.[132]
4.6
Numerical results
In this section, we consider the linear and stationary diffusion-convection problem: −∇. 𝜖 ∇𝑢 + 𝒘. ∇𝑢 = 𝑓 𝑢=𝑔
in Ω on Γ
where, 𝜖 is the diffusive coefficient, 𝒘 is the transport advective field, 𝑓 is the volume source term, Ω ⊂ ℜ2 is a smooth convex domain and g is the boundary value prescribed for u on Γ.
4.6.1 Problem 1 [9] The domain Ω is the unit square 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1, with mixed boundary conditions. More precisely, 𝑢 = 0 on 𝑥 = 0, 𝑢 = 1 on 𝑥 = 1, and
𝜕𝑢 𝜕𝑛
= 0 on y=0 and y=1. The given transport
field is w = (w, 0) and f = 0. The importance of this problem is that it is of one-dimensional nature and is reducible to −𝜖 𝑢′′ + 𝑤𝑢′ = 0 𝑖𝑛 Ω (0,1) 𝑢 0 = 0, 𝑢 1 = 1 which is identical to Eq. (4.16) that was analyzed in the previous sections. 1
As a first experiment, the domain of the problem is discretized uniformly with = 16 and the 𝜖
1
coefficients are chosen such that 𝜈 = 𝑤 = 50 producing Peclet number 𝑃𝑒 = 2𝜈 = 1.5625. To examine the proposed modified diffusion coefficient (MDC), the problem is solved directly by classical Galerkin finite-element, using stream diffusion (SD) method as implemented in [130] 72
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
and using the proposed MDC method.
chapter(4)
In fig.(4.5), the solution of the three methods are
compared with the exact solution (Eq. 4.18). As expected, the MDC solution coincides with the exact one while the Galerkin solution oscillates near the boundary region and SD, although prevents oscillations, fails to approximate the solution. The previous experiment is repeated, but with different values of 𝑃𝑒 and the results of maximum discretization error computed as 𝑀𝑎𝑥𝑖
𝑢𝑒𝑥𝑎𝑐𝑡 − 𝑢𝑖 , where, 𝑢𝑖 is the computed solutions at
node i are summarized in Table (4.1). The results of MDC are in full agreement with the analysis of the previous section. It produces the exact solution (up to round-off error 10-15) for different values of Pe. Nu=1/50, h=1/16, Pe=1.5625 1
0.8
Galerkin Exact Solution SD Proposed MDC
Solution u
0.6
0.4
0.2
0
-0.2
-0.4 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Fig. 4.5: Comparison of solutions of Problem 1 on a uniform grid by different numerical techniques. 1
Table 4.1 Maximum discretization error of problem 1 on uniform grid ( = 16 ) Peclet number
Galerkin
SD
Proposed MDC
1.5625
0.2634
0.0439
4.0870e-015
2
0.3516
0.0183
3.8129e-015
3
0.5025
0.0025
3.9166e-015
5
0.6693
4.5400e-005
3.1866e-015
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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In the previous results, a uniform grid was used. In the next experiment, a non-uniform grid is used where the domain 0 < 𝑥 < 1 is subdivided into 16 intervals using a Shishkin grid [131]. Shishkin grids are specially designed to fit boundary layers where a transition line determines the boundary between the fine discretization that resolves the layer, and the coarse discretization outside of the layer. Using MDC on a piecewise uniform Shishkin grid, the results for different values of 𝜈 are plotted in Fig.(4.6) showing that the computed discrete solution coincides on the curve representing the exact solution. On a non-uniform mesh, element Peclet number is 𝑖 computed locally within each element i as 𝑃𝑒𝑖 = 2𝜈 , where, 𝑖 is the element length. Table (4.2)
summarizes the results. 1 MDC Nu=1/160 Exact Nu=1/160 MDC Nu=1/48 Exact Nu=1/48 MDC Nu=1/320 Exact Nu=1/320
0.8
u
0.6
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1
x Fig. 4.6: Full agreement of the computed solution by MDC and the exact solution for different values of 𝜈 on a non-uniform grid.
Table 4.2: Maximum discretization error on a non-uniform grid for problem 1. 𝑃𝑒𝑖
𝜈
𝑀𝑎𝑥 𝑢𝑒𝑥𝑎𝑐𝑡 − 𝑢𝑖 𝑖
1/48
2.8336
2.9976e-015
1/160
9.4455
3.0531e-015
1/320
18.8910
3.2058e-015
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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As expected by the analysis of Section 4.5.2, MDC method produces the exact solution for the 1D class of the convection-diffusion equation (Eq.4.16) even on non-uniform grids.
4.6.2. Problem 2 [125] We next consider a constant coefficient convection-diffusion equation −∇. 𝜖 ∇𝑢 + 𝒘. ∇𝑢 = 𝑓 𝑢=𝑔
in Ω on Γ
(4.42)
defined on the unit square 0 < x < 1; 0 < y < 1 with w = (1,0). The boundary condition is prescribed as 𝑢 𝑥, 0 = 𝑢 𝑥, 1 = 0; 𝑢 0, 𝑦 = sin 𝜋𝑦 ; 𝑢 1, 𝑦 = 2 sin 𝜋𝑦 The exact solution is [135] 𝑢= where, 𝜍 =
1 𝑒 𝑥/2𝜖 sin 𝜋𝑦 2 𝑒 −𝑥/2𝜖 sinh 𝜍𝑥 + sinh 𝜍 1 − 𝑥 sinh 𝜍
𝜋 2 + 1/4𝜖 2 . This problem represents a convection-dominated flow and was used
as one of the test problems by Gupta et al. [132]. The coefficient of the convective term is constant. Figure (4.7) shows the exact solution with 𝜖 = 0.05. For most part of the domain, the exact solution is smooth. But it has a steep boundary layer along the downstream edge at x = 1.
2 1 0 1 0.5 0
0.2
0
0.4
0.6
Fig. 4.7: Exact Solution of Problem 2 (𝜖 = 0.05).
75
0.8
1
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
Problem 2 is solved using classical Galerkin finite-element on a uniform grid (h=1/16) for 𝜖 = 0.001 and 𝜖 = 0.05. The solutions are plotted in Fig.(4.8) showing nonphysical oscillations that are spread along the whole domain for the lower value of 𝜖 = 0.001 and is localized near the boundary region as 𝜖 is increased to 0.05. This behavior is similar to that shown in Fig.(4.2) for the 1-D case. Next, Problem 2 is solved on the same uniform grid using MDC method and the solution for 𝜖 = 0.001 is plotted in Fig.(4.9). The absolute discretization error distribution is shown in Fig.(4.10). It is noticed that although extending MDC to this 2-D convection-diffusion problem does not produce the exact solution, it provides non-oscillatory accurate solution even on a coarse uniform grid. Table 3 reports the maximum discretization error norm of MDC finiteelement method for different mesh sizes.
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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2
1
0
-1 1 0.5 0 0
0.6
0.4
0.2
0.8
1
(𝑎) 𝜖 = 0.001
2 1.5 1 0.5 0 1
0.5
0
0
0.4
0.2
0.6
0.8
1
(b) 𝜖 = 0.05 Fig.(4.8): Galerkin finite-element solution of problem 2 with uniform grid size h=1/16 for different values of 𝜖.
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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2 1 0 1 0.5 0 0
0.2
0.6
0.4
0.8
1
Fig. 4.9: Finite-element solution using MDC of Problem 2 (𝜖 = 0.001, = 1/16 ).
x 10
Maximum Discretization error norm
-4
2 1 0 1
1
0.5
0.5 0 0
Fig. 4.10: Distribution of the discretization error norm of the finite-element solution using MDC of Problem 2 (𝜖 = 0.001, = 1/16 ).
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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Table (4.3): Convergence of MDC finite-element method for problem 2. H
1/16
1/32
1/64
1/128
Maximum discretization error
1.3084e-004
5.8178e-005
2.7241e-005
1.3175e-005
As expected, multigrid methods do not converge if classical Galerkin finite-element is used to discretize the convection-diffusion problems since it fails to provide good approximations on coarser grids. However, introducing the MDC-Galerkin discretization succeeded to retain the convergence of both geometric (GMG) and algebraic (AMG) multigrid methods as shown in Figs.(4.11,4.12), respectively. Convergence of AMG is slightly better than that of GMG but convergence rates are dependent on the mesh size h for both methods. GMG Convergence using MDC, nu=1/200 v
0
10
h=1/64 h=1/32 h=1/16
-2
10
-4
Residual norm
10
-6
10
-8
10
-10
10
-12
10
-14
10
-16
10
0
2
4
6
8
10
12
14
16
18
20
V-cycles
Fig. 4.11: Convergence of GMG for different h for problem 2 using MDC.
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The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
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v AMG Convergence using MDC, nu=1/200
2
10
0
h=1/16 h=1/32 h=1/64 h=1/128
10
-2
10
-4
Residual norm
10
-6
10
-8
10
-10
10
-12
10
-14
10
-16
10
2
4
6
8
10
12
14
16
18
20
V-Cycles
Fig. 4.12: Convergence of AMG for different h for problem 2 using MDC.
4.6.3 Problem 3 [125] In this test problem, we chose in Eq.(4.42) the following convection coefficients 𝒘 = (𝑥 𝑥 − 1 1 − 2𝑦 , −𝑦 𝑦 − 1 1 − 2𝑥 ) It is obvious that this test problem has a stagnation point at (0.5, 0.5). A stagnation point (x0; y0) is the one inside the computational domain where both convection coefficients vanish. Convection-diffusion equations with stagnation points in their domains are usually used to model recirculation flow problems. For small values of 𝜖, this type of problem is very hard to solve, especially when standard multigrid method is used as the solution technique. The exact solution is chosen similar to the one used [125] as 𝑢 =− 1−𝑎 𝑥
1−𝑎 𝑦
where, 𝑎 𝑥 =
𝑒
1−𝑥 /𝜖
𝑒 1/𝜖
−1 , −1
𝑎 𝑦 =
80
𝑒
1−𝑦 /𝜖
𝑒 1/𝜖
−1 −1
The Modified Diffusion Coefficient Technique for Convection-Diffusion Equation
chapter(4)
An exponential stretched grid in both x- and y-directions is used. The interval (0,1) is divided 𝑖
into n-sub-intervals by n+1 nodes such that 𝑥𝑖 =
− 𝑄 1−𝑒 𝑛
, 𝑖 = 0, ⋯ , 𝑛. As i changes from 0 to
1−𝑒 − 𝑄
n, 𝑥𝑖 gives the coordinates of nodes along the x-axis from 0 to 1. If the stretching parameter Q is chosen such that the ratio 𝛽 =
𝑙𝑒𝑔𝑛𝑡 𝑜𝑓 𝑓𝑖𝑟𝑠𝑡 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑙𝑒𝑔𝑛𝑡 𝑜𝑓 𝑙𝑎𝑠𝑡 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙
, then 𝑄 =
𝑛 ln 𝛽 𝑛−1
.
Computed solution epslon=0.1
0 -0.2 -0.4 -0.6 -0.8 -1 0
0 0.5 0.2
0.4
0.6
0.8
1
1
Computed solution epslon=0.01
0 -0.2 -0.4 -0.6 -0.8 0 -1 0
0.5 0.5 1
1
Fig. 4.13: Computed solution on a stretched grid of Problem 3 with 𝛽 = 1/20, 𝑛 = 32 and different values of 𝜖 = 0.1, 𝜖 = 0.01. The computed solution of problem 3 for 𝜖 = 0.1 , 𝜖 = 0.01 on a stretched grid is shown in Fig.(4.13) where the solution has steep boundary layers near x = 0 and y = 0. The width of these boundary layers becomes thinner as ϵ decreases and sharp variation in the solution is observed. It must be mentioned that the exact solutions are graphically indistinguishable from the shown
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computed solutions. It is clear that although the solution seems smooth for 𝜖 = 0.1, it becomes more steep as 𝜖 decreases and hence obtaining accurate solutions becomes more difficult. This is clear in Table 4.4 which reports the maximum discretization error for different values of 𝜖 on a grid with mesh size h=1/32. Table (4.4): Convergence of MDC finite-element method for problem 3. 𝜖
0.10
0.05
0.01
Maximum discretization error
3.654 × 10-4
5.431× 10-4
3.821× 10-3
The resulting algebraic linear system produced by applying the finite-element method with the modified diffusion coefficient of problem 3 is solved by the generalized minimum residual GMRES method [133] with and without preconditioners. The results in Fig.(4.14) show the effect of preconditioning on the convergence of the GMRES. For this problem, the best performance is attained by the algebraic multigrid (AMG) preconditioner; only 7 GMRES iterations were sufficient to reduce the residual error norm below 10-6 of its initial value. The convergence of the GMRES with geometric multigrid (GMG) and incomplete LU (ILU) preconditioners are slightly slower but are significantly better than GMRES without any preconditioner.
GMRES with different preconditioners
1
10
AMG GMG ILU non
0
10
Residual error norm
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
0
5
10
15
20
25
30
iterations Fig. 4.14: Convergence behavior of GMRES for different preconditioners for problem 3. 82
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It is important to mention that classical multigrid(GMG) does not converge when used to solve the resulting system. However, AMG works efficiently as observed from Fig.(4.15) where the convergence behaviour is shown for different stretching ratios 𝛽.
1
𝛽 AMG Convergene for problem 3 with different stretching ratios B
10
B=0.01 𝛽
0
10
B=0.1 𝛽 𝛽B=0.05
-1
Residual Error Norm
10
-2
10
-3
10
-4
10
-5
10
-6
10
2
4
6
8
10 12 AMG Cycles
14
16
18
20
Fig. 4.15: Convergence behavior of AMG for different stretching factors 𝛽 for problem 3.
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4.6.4 Problem 4 [120] Although the previous two problems are singularly perturbed, they seem to have quasi-1D character. Therefore, in this Section we consider a problem that has different variability in each direction and hence represents a real 2D convection diffusion problem. This problem is similar to Problem 3 but the boundary conditions and right hand side f x, y are computed such that the exact solution is u(x, y) = sin πx + cos 3πy + exy . The computed solutions: using MCD on uniform grids with different mesh sizes = 1/16, = 1/64 are plotted in Fig. (4.16). The algebraic systems produced by MCD were solved efficiently by AMG whose convergence rates are presented in Fig.(4.17). The discretization error versus mesh sizes are given in Table 4.5 which show that a reduction of the discretization error by a factor of 4 is attained when the mesh size is reduced by a factor of 2.
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Computed solution h=1/16
3.5 3 2.5 2 1.5 1 0.5 0 1
1 0.5
0.5 0
0
Computed solution h=1/64
3.5 3 2.5 2 1.5 1 0.5 0 1
1 0.8 0.6
0.5
0.4 0.2
0
0
Fig. 4.16: Computed solution of problem 4 on a uniform grid with different mesh sizes = 1/16, = 1/64.
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AMG convergence h=1/16 h=1/32 h=1/64 h=1/128
0
Maximum Error Norm
10
-5
10
-10
10
-15
10
2
4
6
8
10
12
14
AMG V-cycles Fig. 4.17: Convergence behavior of AMG for different mesh sizes.
Table (4.5) Convergence of MDC finite-element method for problem 4. H
1/16
1/32
1/64
1/128
Maximum discretization error
5.11× 10-4
1.27× 10-4
3.16 × 10-5
7.91 × 10-6
Error reduction factor
-
4.02
4.02
3.99
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Chapter (5) Investigation of the Turbulent Flow inside the 2-D Channel with Internal Obstacles 5.1
Introduction The Navier-Stokes equations, either incompressible or compressible, represent the
mathematical basis for two states of flow, laminar flow and turbulent flow, although these two states of flow are physically very different. The occurrence of one or the other state strongly depends on the Reynolds number (Re) associated with the flow. Turbulent flows are highly unsteady flows, where the main velocity field is superimposed by random velocity fluctuations. Because turbulent flows are prevalent in typical engineering applications, the general interest in the turbulent state of flow is very high with particular attention to the numerical simulation of turbulent flows. The analytical point of view on the Navier-Stokes equations is deficient, in particular with regard to the turbulent flow regime. There exists no analytical solution even to the simplest turbulent flow situations. There are 3 basic conceptual alternatives for the numerical simulation of turbulence: direct numerical simulation (DNS), large eddy simulation (LES), and simulations based on the Reynolds averaged Navier-Stokes (RANS) equations. The three concepts for the numerical simulation of turbulent flows in their basic form struggle with different problems in terms of computational accuracy and efficiency. Furthermore, most of the numerical approaches to laminar flows are also far from being ideal. In view of this situation, the variational multiscale method, which was introduced as a general concept for problems of computational mechanics, appears to be a valuable framework for developing improved numerical methods in fluid mechanics. This Chapter will try to illuminate the current situation of the variational multiscale method as a concept for the numerical simulation of laminar and turbulent flows of an incompressible fluid. So, this chapter is organized as following: the formulation of unsteady Navier-Stokes equation and the discrete weak formulation of this equation, are presented in Section 5.2. In Section 5.3, the variational multiscale method is introduced and extended to the problem of the incompressible Navier-Stokes equations in order to generate a new approach to LES of
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turbulent flows. To study the motion of the incompressible Navier-Stokes equation, we need to simulate this motion within time. So, adaptive time stepping of this equation is presented to overcome this problem in Section 5.4. Also in Section 5.5, we are concerned with the steady incompressible Navier-Stokes equation, so, the formulation and the discrete weak formulation of this equation are introduced. Finally in Section 5.6, to define the turbulence term, the Smagorinsky model was presented as a subgrid-scale modeling approach which is still a commonly used one due to its attractive simplicity to define the turbulence term.
5.2
Our approach to solve unsteady Navier-Stokes equation The Navier –Stokes system is the basis for computational modeling of the flow of an
incompressible fluid, such as air at low velocities or water. The set of unsteady incompressible Navier-Stokes equations is given as 𝜕𝑢 𝜕𝑡
− 𝜈 𝛻 2 𝑢 + 𝑢 . 𝛻𝑢 + 𝛻𝑝 = 𝑓
in Ω×(0,T)
∇. 𝑢=0
in Ω×(0,T)
(5.1)
with initial condition 𝑢 = 𝑢0
𝑖𝑛 Ω × {0}
where, ν>0 is a given constant called the kinematic viscosity and the initial velocity vector 𝑢0 is assumed to be divergence free which is characterized
by ∇. 𝑢0 = 0. The variable 𝑢
represents the velocity vector of the fluid and p represents the pressure. The convection term 𝑢. ∇𝑢 is simply the vector obtained by taking the convective derivative of each velocity component in turn, that is 𝑢. ∇𝑢 ≔ (𝑢. ∇)𝑢. The fact that this term is nonlinear is what makes boundary value problems associated with the Navier–Stokes equations can have more than one stable solution. Solution non-uniqueness presents an additional challenge for the numerical analysis of approximations to the system (5.1). The boundary value problem that is considered is the system (5.1) posed on a two- or threedimensional domain Ω, together with boundary conditions on 𝜕𝛺 = 𝜕𝛺𝐷 ∪ 𝜕𝛺𝑁 given by 𝜕𝑢
𝑢 = 𝑔 𝑜𝑛 𝜕𝛺𝐷 , 𝜈 𝜕𝑛 − 𝑛𝑝 = 0 𝑜𝑛 𝜕𝛺𝑁 ,
(5.2)
where 𝑛 is the outward-pointing normal to the boundary. If the velocity is specified everywhere on the boundary, that is, if 𝜕𝛺𝐷 ≡ 𝜕𝛺, then the pressure solution to the Navier– Stokes problem (5.1) and (5.2) is only unique up to a hydrostatic constant [130]. It is useful to have a quantitative measure of the relative contributions of convection and viscous diffusion. This can be achieved by normalizing the system (5.1) with respect to the size of the domain
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and the magnitude of the velocity. To this end, let L denote a characteristic length scale for the domain Ω. If points in Ω are denoted by 𝑥, then 𝜉 = 𝑥/𝐿denotes points of a normalized domain. In addition, let the velocity 𝑢 be defined so that 𝑢 = 𝑈𝑢∗ where U is a reference value, for example: the maximum magnitude of velocity on the inflow. If the pressure is scaled so that p (L𝜉 ) =U2 𝑝∗ (𝜉 ) on the normalized domain. The Weak Formulation The weak formulation of the Navier–Stokes flow problem (5.1) and (5.2) in d-dimensional space needs to define the solution and test spaces as ℋ𝐸1 ≔ 𝑢 ∈ ℋ 1 𝛺 ℋ𝐸10 ≔ 𝑣 ∈ ℋ 1 𝛺
𝑑 𝑑
𝑢 = 𝑔 𝑜𝑛 𝜕𝛺𝐷
(5.3)
𝑣 = 0 𝑜𝑛 𝜕𝛺𝐷
(5.4)
Then, the standard weak formulation is the following: Find 𝑢 ∈ ℋ𝐸1 and 𝑝 ∈ 𝐿2 𝛺 such that
Ω
𝑣∙
𝜕𝑢 𝜕𝑡
𝑑Ω − 𝜈
𝑣 ∙ 𝛻 2 𝑢 𝑑Ω +
Ω
Ω
𝛺
Ω
𝑣 ∙ 𝑢 ∙ 𝛻𝑢 𝑑Ω +
Ω
𝑣 ∙ 𝛻𝑝 𝑑Ω =
𝑣 ∙ 𝑓 𝑑Ω ,
for all 𝑣 ∈ ℋ𝐸10
(5.5)
𝑞(∇. 𝑢) = 0
for all 𝑞 ∈ 𝐿2 𝛺
(5.6)
The weighted residual formulation of the Navier-Stokes equations (5.5) and (5.6) can be written as 𝐵 𝑣, 𝑞; 𝑢, 𝑝 = (𝑣, 𝑓)Ω
(5.7)
(𝑣, 𝑓)Ω =
(5.8)
where, Ω
𝑣. 𝑓 𝑑Ω
and 𝜕𝑢
𝐵 𝑣, 𝑞; 𝑢, 𝑝 = 𝑣, 𝜕𝑡
Ω
+ 𝑣 , 𝑢 ∙ 𝛻𝑢
Ω
+ 𝑣, ∇𝑝
Ω
− (𝑣, 𝜈𝛻 2 𝑢)Ω + (𝑞, ∇. 𝑢)Ω
(5.9)
The discrete weak formulation is defined using finite dimensional spaces 𝑋0 ⊂ 𝐻𝐸10 and 𝑀 ⊂ 𝐿2 (𝛺). Specifically, given a velocity solution space 𝑋𝐸 , the discrete formulation of (5.9) is: find 𝑢 ∈ 𝑋𝐸 and 𝑝 ∈ 𝑀 such that: 𝐵 𝑣 , 𝑞 ; 𝑢 , 𝑝 = 𝑣 ,
𝜕𝑢 𝜕𝑡
Ω
+ 𝑣 , 𝑢 ∙ 𝛻𝑢
(𝑞 , ∇. 𝑢 )Ω
Ω
+ 𝑣 , ∇𝑝
Ω
− (𝑣 , 𝜈𝛻 2 𝑢 )Ω + (5.10)
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The Variational Multiscale Method and its Relation to Turbulence
Theory The variational multiscale method [134, 135] aims at problems with broad scale ranges, which often pose an unsolvable challenge for standard numerical methods. In the variational multiscale method, the scales of the underlying problem are separated into a predefined number of scale groups, and thus, a different numerical treatment of any of these scale groups is enabled. This theoretical framework was also extended to the problem of the incompressible Navier-Stokes equations in Hughes et al. [136], in order to generate a new approach to LES of turbulent flows. Define the weighting and solution function spaces as 𝑉𝑢𝑝 = 𝑉𝑢𝑝 ⊗ 𝑉𝑢𝑝 and 𝑆𝑢𝑝 = 𝑆𝑢𝑝 ⊗ 𝑆𝑢𝑝
respectively, where 𝑆 ⊗ 𝑇 is the tensor product. For example, if S and T are two dimensional 𝑎11 given by the matrices 𝑎 21
𝑎12 𝑏11 and 𝑎22 𝑏21
𝑏12 respectively, then the tensor product of 𝑏22
these two matrices is 𝑎11 𝑎21
𝑎12 𝑏11 𝑎22 ⊗ 𝑏21
𝑏12 = 𝑏22
𝑎11 𝑏11 𝑎11 𝑏21 = 𝑎21 𝑏11 𝑎21 𝑏21
𝑎11 𝑎21
𝑎11 𝑏12 𝑎11 𝑏22 𝑎21 𝑏12 𝑎21 𝑏22
𝑏11 𝑏12 𝑎12 𝑏21 𝑏22 𝑏11 𝑏12 𝑎22 𝑏21 𝑏22 𝑎12 𝑏11 𝑎12 𝑏12 𝑎12 𝑏21 𝑎12 𝑏22 𝑎22 𝑏11 𝑎22 𝑏12 𝑎22 𝑏21 𝑎22 𝑏22
𝑏11 𝑏21 𝑏11 𝑏21
𝑏12 𝑏22 𝑏12 𝑏22
Here, the resolved scales are indicated by the characteristic discretization length h and the unresolved scales by (ˆ). Let the weighting functions be composed as resolved scales and unresolved scales 𝑣 = 𝑣 + 𝑣,
𝑞 = 𝑞 + 𝑞 ,
(5.11)
𝑝 = 𝑝 + 𝑝
(5.12)
and the solution functions as 𝑢 = 𝑢 + 𝑢,
Since the weighted residual equation (5.7) is linear with respect to the weighting functions, it may be separated into a system of two equations as 𝐵 𝑣 , 𝑞 ; 𝑢 + 𝑢, 𝑝 + 𝑝 = (𝑣 , 𝑓)Ω
∀{𝑣 , 𝑞 } ∈ 𝑉𝑢𝑝
(5.13)
𝐵 𝑣, 𝑞 ; 𝑢 + 𝑢, 𝑝 + 𝑝 = (𝑣, 𝑓)Ω
∀{𝑣, 𝑞 } ∈ 𝑉𝑢𝑝
(5.14)
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The form 𝐵 in equation (5.13) for the resolved scales of the problem is linearized with respect to the convective term as 𝐵 𝑣 , 𝑞 ; 𝑢 + 𝑢 , 𝑝 + 𝑝 = 𝐵 𝑣 , 𝑞 ; 𝑢 , 𝑝 + 𝐵1 𝑣 , 𝑞 ; 𝑢 ; 𝑢 , 𝑝 + 𝐵 2 𝑣 ; 𝑢
(5.15)
where, 𝐵1 𝑣 , 𝑞 ; 𝑢 ; 𝑢 , 𝑝 = =
𝑑 𝑑𝜀
𝑑 𝐵 𝑣 , 𝑞 ; 𝑢 + 𝜀𝑢, 𝑝 + 𝜀𝑝 𝑑𝜀 𝑣 ,
𝜕(𝑢 + 𝜀𝑢) 𝜕𝑡
+ 𝑣 , ∇𝑝 + 𝜀𝑝
=
𝜕𝑢
𝑣 , 𝜕𝑡
Ω
Ω
𝜀=0
+ 𝑣 , (𝑢 + 𝜀𝑢) ∙ ∇(𝑢 + 𝜀𝑢)
− 𝑣 , 𝜈∇2 (𝑢 + 𝜀𝑢)
+ 𝑣 , 𝑢. ∇𝑢 + 𝑢 . ∇𝑢
Ω
Ω
Ω
+ 𝑣 , ∇𝑝
Ω
Ω
+ 𝑞 , ∇. 𝑢 + 𝜀𝑢
− 𝑣 , ν∇2 𝑢
Ω
+ 𝑞 , ∇. 𝑢
Ω
Ω
= 𝜀=0
(5.16)
and, 1 𝑑2
𝐵 2 𝑣 ; 𝑢 = 2 𝑑𝜀 2 𝐵 𝑣 , 𝑞 ; 𝑢 + 𝜀𝑢, 𝑝 + 𝜀𝑝 𝜀𝑢) ∙ ∇(𝑢 + 𝜀𝑢) 𝜀𝑢
Ω
+ 𝑣 , ∇𝑝 + 𝜀𝑝
Ω
1 𝑑2
𝜀=0
= 2 𝑑𝜀 2
− 𝑣 , 𝜈∇2 (𝑢 + 𝜀𝑢)
1 𝑑2
Ω
𝜀=0
𝑣 ,
= 2 𝑑𝜀 2 𝑣 , (𝑢 + 𝜀𝑢) ∙ ∇(𝑢 + 𝜀𝑢)
Ω 𝜀=0
Ω
𝜕 𝑢 +𝜀𝑢 𝜕𝑡
Ω
+ 𝑣 , (𝑢 +
+ 𝑞 , ∇. 𝑢 +
= 𝑣 , 𝑢. ∇𝑢
Ω
(5.17)
Using equations (5.13) and (5.15), then Eq. (5.15) can be written as: 𝐵 𝑣 , 𝑞 ; 𝑢 , 𝑝 = (𝑣 , 𝑓)Ω − 𝐵1 𝑣 , 𝑞 ; 𝑢 ; 𝑢, 𝑝 − 𝐵 2 𝑣 ; 𝑢
(5.18)
The last two terms on the right hand side of (5.18) may be identified as the influence of the unresolved scales onto the resolved scales and can be viewed as the projection of the unresolved scales onto the subspace of the resolved scales. A practical implementation of the above system needs to be supplied with the effect of the terms containing unresolved scales, 𝐵1 and 𝐵 2 in Eq. (5.16), (5.17), respectively. A relatively simple approach for representing the effect of unresolved scales in LES is the introduction of the Smagorinsky model. In fact, despite the well-known limitations of the Smagorinsky model within a traditional LES, it has proven its great potential in the context of variational multiscale LES, see [54] and many references therein. In general, Smagorinsky-type models provide the opportunity of reintroducing the insufficiently resolved dissipation in the form of an additional artificial 91
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viscosity. This is done by approximating the terms 𝐵1 and 𝐵 2 on the right-hand side of (5.18) by an additional viscous term, the so-called subgrid viscosity term 𝐵1 𝑣 , 𝑞 ; 𝑢 ; 𝑢, 𝑝 + 𝐵 2 𝑣 ; 𝑢 ≈ − 𝑣 , 𝜈𝑡 ∇2 𝑢
Ω
[137, 138], where 𝜈𝑡 is the subgrid viscosity parameter. As a
result, equation (5.18) can be read as 𝐵 𝑣 , 𝑞 ; 𝑢 , 𝑝 − 𝑣 , 𝜈𝑡 ∇2 𝑢
Ω
= 𝑣 , 𝑓
(5.19)
Ω
It is noticed that the second term in the left hand side of (5.19) is similar to the fourth term in the right hand side of (5.10). So, equation (5.19) is written as: 𝑣 ,
𝜕𝑢 𝜕𝑡
Ω
+ 𝑣 , 𝑢 ∙ ∇𝑢
(𝑞 , ∇. 𝑢 )Ω = 𝑣 ,
𝜕𝑢 𝜕𝑡
(𝑞 , ∇. 𝑢 )Ω = 𝑣 , 𝑓
Ω
Ω
+ 𝑣 , ∇𝑝
+ 𝑣 , 𝑢 ∙ ∇𝑢
Ω
− (𝑣 , 𝜈∇2 𝑢 )Ω − 𝑣 , 𝜈𝑡 ∇2 𝑢
Ω
+ 𝑣 , ∇𝑝
Ω
Ω
+
− 𝑣 , (𝜈 + 𝜈𝑡 )∇2 𝑢
Ω
+ (5.20)
Ω
As a result to (5.20), the subgrid viscosity term (𝜈𝑡 ∇2 𝑢) will add to the viscosity term (𝜈∇2 𝑢) in equation (5.1) and equations in (5.1) read as: 𝜕𝑢 𝜕𝑡
− 𝜈 + 𝜈𝑡 ∇2 𝑢 + 𝑢. ∇𝑢 + ∇𝑝 = 𝑓 ∇. 𝑢 = 0
𝑖𝑛 Ω × (0, T) 𝑖𝑛 Ω × (0, T)
(5.21)
To get the subgrid viscosity (𝜈𝑡 ), an appropriate modeling approach will be discussed in section 5.6.
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Adaptive time stepping of unsteady Navier stokes equation Simulation of the motion of an incompressible fluid remains an important but very
challenging problem. So, a new class of robust and efficient method is implemented to overcome this problem. The general solution strategy of this method has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule (TR) with an explicit Adams–Bashforth (AB) method for error control, and a robust solver for the spatially discretized system. The stabilized TR–AB is particularly well suited to long time integration of convection-dominated problems and is a very effective algorithm when faced with general convection-diffusion problems with different time scales governing the system evolution [137]. For simplicity, we started with taking f in the R.H.S of the equation (5.21) equal zero then rewrite it in the form: 𝜕𝑢 𝜕𝑡
− 𝜈 + 𝜈𝑡 ∇2 𝑢 + 𝑢. ∇𝑢 + ∇𝑝 = 0 ∇. 𝑢 = 0
𝑖𝑛 Ω × (0, T) 𝑖𝑛 Ω × (0, T)
(5.22)
And the boundary conditions on 𝜕𝛺 = 𝜕𝛺𝐷 ∪ 𝜕𝛺𝑁 are given by 𝜕𝑢
𝑢 = 𝑔 𝑜𝑛 𝜕𝛺𝐷 × (0, T) , (𝜈 + 𝜈𝑡 ) 𝜕𝑛 − 𝑛𝑝 = 0 𝑜𝑛 𝜕𝛺𝑁 × (0, T)
(5.23)
Time stepping algorithm is the well known, trapezoid rule (TR). Let the interval [0, T] be divided into N steps {t i }Nt=1 , and let v j denote v(𝑥, t j ). The semi-discretized problem is the following: Given (𝑢n , 𝑝n ) at time level 𝑡𝑛 and boundary data 𝑔n+1 at time level 𝑡𝑛+1 , compute (𝑢n+1 , 𝑝n+1 ) via 𝜕𝑢 𝜕𝑡
𝑛+1
− 𝜈 + 𝜈𝑡 ∇2 𝑢n+1 + 𝑢n+1 . ∇𝑢n+1 + ∇𝑝n+1 = 0
𝑖𝑛 Ω
∇. 𝑢n+1 = 0 And apply trapezoid rule (TR) to t n +1
tn
𝜕𝑢 n +1 𝜕𝑡
𝑖𝑛 Ω
, where:
𝜕𝑢 𝜕𝑢 n+1 n 𝑑𝑡 = 𝑢 − 𝑢 = ( t n+1 − t n ) 𝜕𝑡 𝜕𝑡
93
𝑛+1
+ 2
𝜕𝑢 𝜕𝑡
𝑛
(5.24)
Investigation of the turbulent flow inside the 2-D channel with internal obstacles
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Then, 𝜕𝑢 𝑛+1 𝜕𝑡
=
2
𝑢n+1 − 𝑢n −
t n +1 − t n
𝜕𝑢 𝑛
(5.25)
𝜕𝑡
Substitution of (5.25) in (5.24) yields, 𝑛
2 𝜕𝑢 ( 𝑢𝑛+1 − 𝑢𝑛 − ) − 𝜈 + 𝜈𝑡 𝛻 2 𝑢𝑛+1 + 𝑢𝑛+1 . 𝛻𝑢𝑛 +1 + 𝛻𝑝𝑛+1 = 0 𝑡𝑛+1 − 𝑡𝑛 𝜕𝑡 Defining the current time step as 𝐾𝑛+1 = 𝑡𝑛+1 − 𝑡𝑛 and rearrangement of the above equation, it becomes 2 K n+1
𝑢n+1 − 𝜈 + 𝜈𝑡 ∇2 𝑢n+1 + 𝑢n+1 . ∇𝑢n+1 + ∇𝑝n+1 = −∇. 𝑢n+1 = 0
2 K n+1
𝜕𝑢n 𝜕𝑡
𝑖𝑛 Ω
𝜕𝑢 n +1 𝜕𝑛
𝑖𝑛 Ω (5.26)
𝑢n+1 = 𝑔n+1 𝜈 + 𝜈𝑡
𝑢n +
𝑜𝑛 𝜕𝛺𝐷
− 𝑛𝑝n+1 = 0
,
𝑜𝑛 𝜕𝛺𝑁
(5.27)
Where, using Eq. (5.22) 𝜕𝑢n = 𝜈 + 𝜈𝑡 ∇2 𝑢n − 𝑢n . ∇𝑢n − ∇𝑝n 𝜕𝑡 From (5.26) it is evident that a numerical scheme for handling the nonlinear term 𝑢n+1 . ∇𝑢n+1 is needed at every time step. In this work an alternative approach is adopted by linearization of this term such that 𝑢n+1 . ∇𝑢n+1 ≈ 𝑤 n+1 . ∇𝑢n+1 , where 𝑢n+1 ≅ 𝑤 𝑛 +1 = 1 +
𝐾𝑛 +1
𝑢𝑛 −
𝐾𝑛
𝐾𝑛 +1 𝐾𝑛
𝑢𝑛−1 .
(5.28)
Where, 𝑢n+1 is interpolated from previously calculated 𝑢n , 𝑢n−1 . Results of this approach show that temporal stability is not compromised significantly. Let (. , .) denote the standard scalar or vector valued 𝐿2 inner product defined on 𝛺. Given the velocity solution space 𝐻𝑔1 ≔ 𝑢 ∈ ℋ 1 𝛺
𝑑
𝑢 = 𝑔 𝑜𝑛 𝜕𝛺𝐷 , the linearized semi discrete
problem can be formulated as a variational problem: given(𝑢n , 𝑝n ) ∈ 𝐻𝑔1n × 𝐿2 𝛺 , we seek (𝑢n+1 , 𝑝n+1 ) ∈ 𝐻𝑔1n +1 × 𝐿2 𝛺 such that: 94
Investigation of the turbulent flow inside the 2-D channel with internal obstacles 2 K n +1
chapter(5)
𝑢n+1 , 𝑣 + 𝜈 + 𝜈𝑡 ∇𝑢n+1 , ∇ 𝑣 + (𝑤 n+1 . ∇𝑢n+1 , 𝑣) − 𝑝n+1 , ∇. 𝑣 = K 𝜕𝑢 n
( 𝜕𝑡 , 𝑣)
2 n +1
𝑢n , 𝑣 +
𝑖𝑛 Ω
(∇. 𝑢n+1 , 𝑞) = 0
(5.29)
𝑖𝑛 Ω
(5.30)
for all (𝑣, 𝑞) ∈ 𝐻01 × 𝐿2 𝛺 . We get the fully discrete problem by using finite-dimensional approximation spaces 𝑋 ⊂ 𝐻01 and 𝑀 ⊂ 𝐿2 (𝛺) to find (𝑢 2
𝑢
K n +1
n+1
n+1
, 𝑝 n+1 ) ∈ 𝑋𝑔 × 𝑀 such that:
, 𝑣 + 𝜈 + 𝜈𝑡 ∇ 𝑢 2
n+1
, ∇ 𝑣 + (𝑤
n
K n +1
𝑢 , 𝑣 + (
(∇. 𝑢
n+1
𝜕𝑢
n
𝜕𝑡
n+1
. ∇ 𝑢
, 𝑣 )
, 𝑞 ) = 0
n+1
, 𝑣 ) − 𝑝 n+1 , ∇. 𝑣 =
𝑖𝑛 Ω
(5.31)
𝑖𝑛 Ω
(5.32)
for all (𝑣 , 𝑞 ) ∈ 𝑋 × 𝑀. The adaptive time-stepping algorithm has three ingredients: time integration, the time-step selection method and stabilization of the integrator.
5.4.1 Time integration To minimize potential round-off instability and inhibit subtractive cancellation, we compute the discrete velocity updates scaled by the time step rather than the velocity. Specifically, n 𝜕𝑢 𝜕𝑡
given 𝑢 ,
dn =
n
and the boundary update g = 1
t n +1 − t n
𝑢n+1 − 𝑢n = K
1 n +1
𝑔 n +1 −𝑔 n K n +1
, and using the substitution [137]
𝑢n+1 − 𝑢n n
in equation (5.31), we get the following equation to compute the pair (d , 𝑝 n+1 ) ∈ 𝑋𝑔 × 𝑀: n
n
2 d , 𝑣 + 𝜈 + 𝜈𝑡 K n+1 ∇ d , ∇ 𝑣 + K n+1 (𝑤 𝜕𝑢 𝜕𝑡
n
n
, 𝑣 − 𝜈 + 𝜈𝑡 ∇ 𝑢 , ∇ 𝑣 − (𝑤 (∇. d
n+1
n+1
, q ) = 0
95
n+1
n
. ∇ d , 𝑣 ) − 𝑝 n+1 , ∇. 𝑣 =
n
. ∇ 𝑢 , 𝑣 ) 𝑖𝑛 Ω
𝑖𝑛 Ω
(5.33)
(5.34)
Investigation of the turbulent flow inside the 2-D channel with internal obstacles
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for all 𝑣 , 𝑞 ∈ 𝑋 × 𝑀. Then update the velocity field and the acceleration (time derivative of the velocity) via 𝑢
n+1
n
= 𝑢 + K n+1 d
n
𝜕𝑢
,
n +1
𝜕𝑡
n
= 2d −
𝜕𝑢
n
(5.35)
𝜕𝑡
Equations (5.33 and 5.34) will subsequently refer as the discrete Oseen problem and the computed pressure field 𝑝 n+1 is not needed for subsequent steps and does not play a role in the time-step selection process. The time-step selection method and stabilization of the integrator will be done as in [137].
5.4.2 Solving the discrete Oseen system n
u Let {𝜙i }𝑖=1 define the basis set for the approximation of a function from the space 𝐻𝐸10 ≔
𝑣 ∈ ℋ1 𝛺
𝑑
n
𝑝 𝑣 = 0 𝑜𝑛 𝜕𝛺𝐷 , and let {𝜓j }𝑗 =1 define a basis set for the discrete pressure.
The fully discrete solution (𝑢
n+1
, 𝑝 n+1 ) corresponding to the Oseen problem (5.33)-(5.34)
is: 𝑢
n+1
=
n u x,n+1 i=1 αi
𝑝 n+1 =
nu y,n+1 𝜙i i=1 αi
𝜙i ,
np p,n+1 α k=1 k
+ 𝑔n+1 ,
𝜓k
(5.36)
where αx,n+1 , αy,n+1 , αp,n+1 represent vectors of coefficients. These are computed by solving the linear equation system defined below. Given the velocity basis set, we define the so-called velocity matrices Mv , Av and Nv , representing identity, diffusion, and convection operators in the velocity space, respectively: Mv = Mv Av = Av
ij
Nv 𝑢 = Nv
= 𝜙i , 𝜙j ,
(5.37)
= ∇𝜙i , ∇𝜙j ,
(5.38)
ij
ij
= 𝑢 . ∇𝜙i , 𝜙j .
(5.39)
Combining the three velocity matrices and using the linearization in (5.28) defines the velocity convection diffusion matrix at time t n+1 : 𝐾v n+1 ≔ K
2 n +1
Mv + 𝜈 + 𝜈𝑡 Av + Nv (𝑤
96
n+1
)
(5.40)
Investigation of the turbulent flow inside the 2-D channel with internal obstacles
Where, 𝑤
n+1
chapter(5)
is defined as in Eq. (5.28). In addition, given the pressure basis set, we can
define a discrete divergence matrix B = Bx , By Bx = Bx By = By
𝑘𝑖
= − 𝜓k ,
∂𝜙 i
∂𝜙 i
𝑘𝑖
= − 𝜓k ,
(5.41)
∂x
(5.42)
∂y
Finally, using the definitions (5.36) - (5.42), the discrete Oseen problem can be expressed as the following system: find αx,n+1 , αy,n+1 , αp,n+1 ∈ ℝn u ×n u ×n 𝑝 such that 𝐾v n+1
0
𝐵x T
αx,n+1
𝑓 x,n+1
0
𝐾v n+1
𝐵y T
αy,n+1 =
𝑓 y,n+1
Bx
By
0
αp,n+1
𝑓 p,n+1
(5.43)
The right hand side vector f is constructed from the boundary data 𝑔n+1 , the computed velocity 𝑢 at the pervious time level, and the acceleration
𝜕𝑢 𝜕𝑡
n
.
5.4.3 Used Finite Element Mesh In general, we have used rectangle domain which may be discretized either uniformly or nonuniformly by using rectangle elements. To get a stable finite-element method (FEM), the approximation of velocity needs to be enhanced relative to the pressure. In the present work, bi-quadratic approximation is used for the velocity components, whereas, bi-linear approximation is used for the pressure. The resulting mixed method is called Q2-Q1 approximation. The nodal positions of velocities and pressure are illustrated on a rectangular element in Fig. 5.1.
Fig.5.1: Q2–Q1 element ( velocity components;
pressure).
Numerical results for steady and unsteady Navier Stokes are presented in chapter (6).
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Investigation of the turbulent flow inside the 2-D channel with internal obstacles
5.5
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Our approach to solve steady Navier-Stokes equation
To get the steady state of Navier Stokes equation of turbulent flow we start with the system of equation (5.21) and take the term
𝜕𝑢 𝜕𝑡
= 0 to force the equation to be steady. Then equation
(5.21) will be rewrite as: − 𝜈 + 𝜈𝑡 ∇2 𝑢 + 𝑢. ∇𝑢 + ∇𝑝 = 𝑓
𝑖𝑛 Ω × (0, T)
∇. 𝑢 = 0
𝑖𝑛 Ω × (0, T)
(5.44)
The weak formulation equations (5.20, 5.21) will be written for steady state as: − 𝜈 + 𝜈𝑡
Ω
𝑣 ∙ 𝛻2 𝑢 +
Ω
𝑣 ∙ 𝑢 ∙ 𝛻𝑢 +
Ω
𝑣 ∙ 𝛻𝑝 =
Ω
𝑣 ∙ 𝑓 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣 ∈ ℋ𝐸10 (5.45)
𝛺
𝑞(∇. 𝑢) = 0
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑞 ∈ 𝐿2 𝛺
(5.46)
And equation (5.9) will be: 𝐵 𝑣, 𝑞; 𝑢, 𝑝 = 𝑣, 𝑢 ∙ ∇𝑢 𝑣, 𝑢 ∙ ∇𝑢
Ω
+ 𝑣, ∇𝑝
Ω
Ω
+ 𝑣, ∇𝑝
Ω
− (𝑣, 𝜈∇2 𝑢)Ω − 𝑣, 𝜈𝑡 ∇2 𝑢
Ω
+ (𝑞, ∇. 𝑢)Ω =
− (𝑣, (𝜈 + 𝜈𝑡 )∇2 𝑢)Ω + (𝑞, ∇. 𝑢)Ω
(5.47)
And the FE formulation of the unsteady equation (5.10) is written in the steady form as: 𝐵 𝑣 , 𝑞 ; 𝑢 , 𝑝 = 𝑣 , 𝑢 ∙ ∇𝑢 𝑣 , 𝜈𝑡 ∇2 𝑢 2
𝜈𝑡 )∇ 𝑢
Ω
Ω
Ω
+ 𝑣 , ∇𝑝
+ (𝑞 , ∇. 𝑢 )Ω = 𝑣 , 𝑢 ∙ ∇𝑢
Ω
Ω
− (𝑣 , 𝜈∇2 𝑢 )Ω −
+ 𝑣 , ∇𝑝
Ω
− 𝑣 , (𝜈 +
+ (𝑞 , ∇. 𝑢 )Ω
(5.48)
Solving the Navier–Stokes equations requires nonlinear iteration with a linearized problem being solved at every iteration step. Thus, given an “initial guess”(𝑢0 , 𝑝0 ) ∈ 𝐻𝐸1 × 𝐿2 (𝛺), a sequence of iterates
𝑢0 , 𝑝0 , 𝑢1 , 𝑝1 , 𝑢2 , 𝑝2 , … . ∈ 𝐻𝐸1 × 𝐿2 (𝛺) is computed, which
converges to the solution of the weak formulation. Two classical linearization procedures are briefly described. Newton iteration turns out to be a very natural linearization approach. Given the iteration 𝑢𝑘 , 𝑝𝑘 , start by computing the nonlinear residual associated with the weak formulation (5.45)–(5.46). This is the pair𝑅𝑘 𝑣 , 𝑟𝑘 𝑞 satisfying 𝑅𝑘 =
𝛺
𝑓 . 𝑣𝑑Ω + 𝜈 + 𝜈𝑡 𝑟𝑘 = −
Ω
𝑣 ∙ 𝛻 2 𝑢𝑘 𝑑Ω − 𝑐 𝑢𝑘 ; 𝑢𝑘 , 𝑣 −
𝛺
𝑞 ∇. 𝑢𝑘
Ω
𝑣 ∙ 𝛻𝑝𝑘 𝑑Ω
(5.49) (5.50)
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Investigation of the turbulent flow inside the 2-D channel with internal obstacles
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where 𝑐 𝑧; 𝑢, 𝑣 can be identified with a trilinear form 𝑐: 𝐻𝐸10 × 𝐻𝐸10 × 𝐻𝐸10 → ℝ defined as follows: 𝑐 𝑧; 𝑢 , 𝑣 ≔
𝑧. ∇𝑢 . 𝑣 .
𝛺
(5.51)
for any 𝑣 ∈ 𝐻𝐸10 and 𝑞 ∈ 𝐿2 (𝛺). With 𝑢 = 𝑢𝑘 + 𝛿𝑢𝑘 and 𝑝 = 𝑝𝑘 + 𝛿𝑝𝑘 being the solutions of (5.45) and (5.46), it is easy to see that the corrections 𝛿𝑢𝑘 ∈ 𝐻𝐸10 and 𝛿𝑝𝑘 ∈ 𝐿2 (𝛺) satisfy 𝐷 𝑢𝑘 ; 𝛿𝑢𝑘 , 𝑣 − 𝜈 + 𝜈𝑡
𝛺
𝑣 ∙ 𝛻 2 𝛿𝑢𝑘 − 𝛺
𝛺
𝛿𝑝𝑘 ∇. 𝑣 = 𝑅𝑘 (𝑣)
𝑞 ∇. 𝛿𝑢𝑘 = 𝑟𝑘 (𝑞)
(5.52) (5.53)
where: 𝐷 𝑢; 𝛿𝑢, 𝑣 = =
𝛺 𝛺
𝛿𝑢 + 𝑢 . ∇ 𝛿𝑢 + 𝑢 . 𝑣 − 𝛿𝑢. ∇𝛿𝑢 . 𝑣 +
𝛺
𝑢. ∇𝑢 . 𝑣
𝛿𝑢. ∇𝑢 . 𝑣 +
𝛺
𝛺
𝑢. ∇𝛿𝑢 . 𝑣
= 𝑐 𝛿𝑢; 𝛿𝑢, 𝑣 + 𝑐 𝛿𝑢; 𝑢, 𝑣 + 𝑐 𝑢; 𝛿𝑢, 𝑣
(5.54)
for all 𝑣 ∈ 𝐻𝐸10 and 𝑞 ∈ 𝐿2 (𝛺), where 𝐷 𝑢𝑘 ; 𝛿𝑢𝑘 , 𝑣 is the difference in the nonlinear terms, as in (5.54). Expanding 𝐷 𝑢𝑘 ; 𝛿𝑢𝑘 , 𝑣 and dropping the quadratic term 𝑐 𝛿𝑢𝑘 ; 𝛿𝑢𝑘 , 𝑣 , according to a standard linear stability analysis [see ch.7 in [130]], gives the linear problem: for all 𝑣 ∈ 𝐻𝐸10 and 𝑞 ∈ 𝐿2 (𝛺), find 𝛿𝑢𝑘 ∈ 𝐻𝐸10 and 𝛿𝑝𝑘 ∈ 𝐿2 (𝛺) satisfying 𝑐 𝛿𝑢𝑘 ; 𝑢𝑘 , 𝑣 + 𝑐 𝑢𝑘 ; 𝛿𝑢𝑘 , 𝑣 − 𝜈 + 𝜈𝑡
𝛺
𝑣 ∙ 𝛻 2 𝛿𝑢𝑘 −
𝛺
𝛿𝑝𝑘 ∇. 𝑣 = 𝑅𝑘 (𝑣)
𝑞 ∇. 𝛿𝑢𝑘 = 𝑟𝑘 (𝑞)
𝛺
(5.55)
The solution of (5.55) is the so-called Newton correction. Updating the previous iterate via 𝑢𝑘+1 = 𝑢𝑘 + 𝛿𝑢𝑘 and 𝑝𝑘+1 = 𝑝𝑘 + 𝛿𝑝𝑘 defines the next iterate in the sequence. The second approach for linearization is Picard’s method. In terms of the representation (5.52)–(5.53), the quadratic term 𝑐 𝛿𝑢𝑘 ; 𝛿𝑢𝑘 , 𝑣 is dropped along with the linear term 𝑐 𝛿𝑢𝑘 ; 𝑢𝑘 , 𝑣 , [see ch.7 in [130]]. Thus, instead of (5.55), we have the following linear problem: for all 𝑣 ∈ 𝐻𝐸10 and 𝑞 ∈ 𝐿2 (𝛺), find 𝛿𝑢𝑘 ∈ 𝐻𝐸10 and 𝛿𝑝𝑘 ∈ 𝐿2 (𝛺) satisfying 𝑐 𝑢𝑘 ; 𝛿𝑢𝑘 , 𝑣 − 𝜈 + 𝜈𝑡
𝛺
𝑣 ∙ 𝛻 2 𝛿𝑢𝑘 − 𝛺
𝛺
𝛿𝑝𝑘 ∇. 𝑣 = 𝑅𝑘 (𝑣)
𝑞 ∇. 𝛿𝑢𝑘 = 𝑟𝑘 (𝑞)
(5.56)
The solution of (5.56) is the Picard correction. Updating the previous iterate via 𝑢𝑘+1 = 𝑢𝑘 + 𝛿𝑢𝑘 and 𝑝𝑘+1 = 𝑝𝑘 + 𝛿𝑝𝑘 defines the next iterate in the sequence. If we substitute 𝛿𝑢𝑘 = 𝑢𝑘+1 − 𝑢𝑘 and 𝛿𝑝𝑘 = 𝑝𝑘+1 − 𝑝𝑘 into (5.56), then an explicit definition for the new iterate is obtained: 99
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for all 𝑣 ∈ 𝐻𝐸10 and 𝑞 ∈ 𝐿2 (𝛺), find 𝑢𝑘+1 ∈ 𝐻𝐸1 and 𝑝𝑘+1 ∈ 𝐿2 (𝛺) such that 𝑐 𝑢𝑘 ; 𝑢𝑘+1 , 𝑣 − 𝜈 + 𝜈𝑡
𝑣 ∙ 𝛻 2 𝑢𝑘+1 −
𝛺 𝛺
𝛺
𝑝𝑘+1 ∇. 𝑣 =
𝛺
𝑞 ∇. 𝑢𝑘+1 = 0
𝑓. 𝑣
(5.57) (5.58)
The formulation (5.57)–(5.58) is commonly referred to as the Oseen system. Comparing (5.57)–(5.58) with the weak formulation show that the Picard iteration corresponds to a simple fixed point iteration strategy for solving (5.45)–(5.46), with the convection coefficient evaluated at the current velocity. As a result, the rate of convergence of Picard iteration is only linear in general. Note that an initial pressure 𝑝0 is not needed if the iteration is coded in the alternative form (5.57)–(5.58).
Mixed finite element approximation There are two issues that arise in solving the Navier–Stokes equations. The first concerns the need to linearize the convection term. The second issue concerns the need to stabilize the approximation of the linearized convection–diffusion operator on coarse grids when multigrid is used to solve the discrete problem. A discrete weak formulation is defined using finite dimensional spaces 𝑋0 ⊂ 𝐻𝐸10 and 𝑀 ⊂ 𝐿2 (𝛺). Specifically, given a velocity solution space 𝑋𝐸 , the discrete version of (5.2)–(5.3) is: find 𝑢 ∈ 𝑋𝐸 and 𝑝 ∈ 𝑀 such that − 𝜈 + 𝜈𝑡
𝛺
𝑣 ∙ 𝛻 2 𝑢 +
𝛺
𝑢 . ∇ 𝑢 . 𝑣 − 𝛺
𝛺
𝑝 (∇. 𝑣 ) =
𝑞 (∇. 𝑢 ) = 0
𝛺
𝑓 . 𝑣 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑣 ∈ 𝑋0 (5.59)
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑞 ∈ 𝑀
(5.60)
Implementation entails defining appropriate bases for the finite element spaces, leading to a nonlinear system of algebraic equations. Linearization of this system using Newton iteration gives the finite-dimensional analogue of (5.55): find corrections 𝛿𝑢 ∈ 𝑋0 and 𝛿𝑝 ∈ 𝑀 (dropping the subscript k to avoid notational clutter) satisfying 𝑐 𝛿𝑢 ; 𝑢 , 𝑣 + 𝑐 𝑢 ; 𝛿𝑢 , 𝑣 − 𝜈 + 𝜈𝑡
𝛺
𝑣 ∙ 𝛻 2 𝛿𝑢 − 𝛺
𝛺
𝛿𝑝 ∇. 𝑣 = 𝑅𝑘 (𝑣 )
𝑞 ∇. 𝛿𝑢 = 𝑟𝑘 (𝑞 )
(5.61)
for all 𝑣 ∈ 𝑋0 and 𝑞 ∈ 𝑀 . Here, 𝑅𝑘 (𝑣 ) and 𝑟𝑘 (𝑞 ) are the nonlinear residuals associated with the discrete formulation (5.59)–(5.60). It is important to notice that dropping the term 𝑐 𝛿𝑢 ; 𝑢 , 𝑣 gives the discrete analogue of the Picard update (5.56).
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To define the corresponding linear algebra problem, we use a set of vector valued basis functions {𝜙𝑗 }, so that 𝑢 = with
𝑛𝑢 𝑗 =1 𝑢𝑗 𝜙𝑗
𝑛𝑢 𝑗 =1 𝑢𝑗 𝜙𝑗
+
𝑛 𝑢 +𝑛 𝜕 𝑗 =𝑛 𝑢 +1 𝑢𝑗 𝜙𝑗
,
𝛿𝑢 =
𝑛𝑢 𝑗 =1 ∆𝑢𝑗 𝜙𝑗
(5.62)
∈ 𝑋0 and fix the coefficients 𝑢𝑗 , 𝑗 = 𝑛𝑢 + 1, … . 𝑛𝑢 + 𝑛𝜕 so that the second
term interpolates the boundary data on 𝜕𝛺𝐷 . Pressure also is introduced as a set of basis functions 𝜓𝑘 , 𝑛𝑝 𝑘=1 𝑝𝑘
𝑝 =
𝜓𝑘 ,
𝑛𝑝 𝑘=1 ∆𝑝𝑘
𝛿𝑝 =
𝜓𝑘
(5.63)
Substituting into (5.61), gives a system of linear equations 𝜈 + 𝜈𝑡 𝐴 + 𝑁 + 𝑊
𝐵𝑇
𝐵
0
∆u
f =
∆p
(5.64)
g
Matrix A is the vector-Laplacian matrix in Eq.(5.38), matrix B is the divergence matrix in Eq.(5.41) and Eq.(5.42) and matrix N is the vector-convection matrix in Eq.(5.39). The new matrix in (5.64) is the Newton derivative matrix W which depends on the current estimate of the discrete velocity 𝑢 , and is given by W= [𝑤𝑖𝑗 ],
𝑤𝑖𝑗 =
𝛺
(𝜙𝑗 . ∇ 𝑢 ). 𝜙𝑖
(5.65)
for i and j = 1, . . . , 𝑛𝑢 . Notice that the Newton derivative matrix is symmetric. The righthand side vectors in (5.64) are the nonlinear residuals associated with the current discrete solution estimates 𝑢 and 𝑝 , expanded via (5.62) and (5.63): f = 𝑓𝑖 , 𝑓𝑖 =
𝛺
𝑓 . 𝜙𝑖 −
𝛺
𝑢 . ∇ 𝑢 . 𝜙𝑖 + 𝜈 + 𝜈𝑡
𝛺
g = 𝑔𝑘 , 𝑔𝑘 =
𝜙𝑖 ∙ 𝛻 2 𝑢 + 𝛺
𝜓𝑘 ∇. 𝑢
𝛺
𝑝 (∇ . 𝜙𝑖 )
(5.66) (5.67)
The system (5.64) is referred to as the discrete Newton problem. For Picard iteration, we omit the Newton derivative matrix to give the discrete Oseen problem: 𝜈 + 𝜈𝑡 𝐴 + 𝑁
𝐵𝑇
∆u
0
∆p
𝐵
101
f =
g
(5.68)
Investigation of the turbulent flow inside the 2-D channel with internal obstacles
chapter(5)
In general, the components of velocity are approximated using a single finite element space and it can be shown that the system (5.64) can be rewritten as 𝜈 + 𝜈𝑡 𝐴 + 𝑁 + 𝑊𝑥𝑥
𝑊𝑥𝑦
𝐵𝑥𝑇
∆𝑢𝑥
𝑊𝑦𝑥
𝜈 + 𝜈𝑡 𝐴 + 𝑁 + 𝑊𝑦𝑦
𝐵𝑦𝑇
∆𝑢𝑦 = 𝑓𝑦
𝐵𝑥
𝐵𝑦
0
∆p
𝑓𝑥 (5.69)
g
where matrix N is the n × n scalar convection matrix N= [𝑛𝑖𝑗 ],
𝑛𝑖𝑗 =
𝛺
𝑢 . ∇𝜙𝑗 . ∇𝜙𝑖
The n×n matrices 𝑊𝑥𝑥 , 𝑊𝑥𝑦 , 𝑊𝑦𝑥 and 𝑊𝑦𝑦 represent weak derivatives of the current velocity in the x and y directions; for example, defining the x component of 𝑢 by 𝑢𝑥 , we see that 𝑊𝑥𝑦 = [𝑤𝑥𝑦 ,𝑖𝑗 ],
𝑤𝑥𝑦 ,𝑖𝑗 =
𝜕𝑢 𝑥 𝛺
𝜕𝑦
102
𝜙𝑖 𝜙𝑗
(5.70)
Investigation of the turbulent flow inside the 2-D channel with internal obstacles
5.6
chapter(5)
Subgrid- Scale Modeling within the Multiscale Environment
The subgrid-scale modeling approach in this section will be based on the subgrid (or eddy) viscosity concept. The Smagorinsky model was the first subgrid-scale model historically and is still a commonly used one due to its attractive simplicity. Smagorinsky Model The resolved part of the velocity is defined by the discretization with characteristic length scale h, the subgrid viscosity can be expressed using Smagorinsky model which gave the coefficient of the eddy viscosity (𝜈𝑡 ) in the form [67]: 𝜈𝑡 = (𝐶𝑠 )2 𝑆
(5.71)
where: = 𝑚∗𝑙
(m, l are the grid sizes in x, y direction respectively.),
𝐶𝑠 : is the Smagorinsky coefficient which takes the value 0.1. and, 𝑆 : is the rate-of-strain tensor of the resolved field which is expressed in the form: 𝑆 =
2 𝑆𝑖𝑗 𝑆𝑖𝑗
where, 1 𝜕𝑢 𝑖
𝑆𝑖𝑗 = 2
𝜕𝑥 𝑗
𝜕𝑢 𝑗
+ 𝜕𝑥 , 𝑖
ui and uj are the components of 𝑢 in x- and y-direction, respectively.
and, 1 𝜕𝑢
𝑆11 = 2
𝜕𝑥
1 𝜕𝑣
𝑆21 = 2
𝜕𝑥
𝜕𝑢
𝜕𝑢
+ 𝜕𝑥 = 𝜕𝑥 , 𝜕𝑢
+ 𝜕𝑦
,
1 𝜕𝑢
𝑆12 = 2
𝜕𝑦
1 𝜕𝑣
𝑆22 = 2
𝜕𝑦
𝜕𝑣
+ 𝜕𝑥 , 𝜕𝑣
That, 𝑆𝑖𝑗 𝑆𝑖𝑗 = 𝑆11 𝑆11 + 𝑆12 𝑆12 + 𝑆21 𝑆21 + 𝑆22 𝑆22 So, 𝑆 =
2𝑢𝑥2 + (𝑢𝑦 + 𝑣𝑥 )2 + 2𝑣𝑦2
103
𝜕𝑣
+ 𝜕𝑦 = 𝜕𝑦
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The actual evaluation of Eq. (5.71) is performed in every element or control volume 𝛺𝑖 with the respective characteristic length 𝑖 , so that a value 𝜈𝑡 𝑖 in every element or control volume is obtained. We take in our consideration that in the first time step the initial velocity is used to estimate the 𝜈𝑡 while in the other time steps the linearization formulation of velocity 𝑤 (5.28) is used instead of 𝑢 to estimate 𝜈𝑡 [67].
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Chapter (6) The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
6.1
Introduction
In this chapter we study the two-dimensional flow around one and two square obstacles. Using the mathematical formulations given in Chapter 5 for Navier-Stokes equation with assistance of Finite Element method which is represented in ifiss3.0 code [138], flow patterns are described by streamline plots and velocity contours are presented for different values of Reynolds number (1 ≤ 𝑅𝑒 ≤ 300). To demonstrate the effect of the monitoring position on the periodic flow, the time-series data of the velocity values behind (downstream) the obstacle were recorded at several stream-wise positions. To illustrate the flow behavior behind the obstacle in the near wake and the far field, the velocity distributions in the cross-wise direction for different stream-wise positions are shown. In addition, different multigrid based methods are compared as solvers to the algebraic systems resulting from the linearization of unsteady and steady Navier Stokes discretizations [139]. The computational domain and boundary conditions of the channel and obstacles are shown in Section 6.2. In Section 6.3, the turbulence generation is described. Results concerning the characteristics of flow patterns around one and two obstacles are presented, discussed and compared in Section 6.4. To illustrate the efficiency of multigrid methods, the convergence performance of the algebraic multigrid method AMG either as solver or preconditioner for Krylov subspace methods are compared in Section 6.5 for steady and unsteady flow in a channel with one or two obstacles.
6.2
Computational Domain and Boundary Conditions [140]
In the present work, the two-dimensional (2-D) flow is studied around one and two square obstacles with side d, centered inside a plane channel (width H, length L). The inflow length is l and the distance between the two square obstacles is 𝑙o = 𝑛 𝑑, where, 105
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n = 5, 10, 15. The flow is simulated numerically on 𝑁𝑥 × 𝑁𝑦 grid (lattice) for 1≤ Re ≤ 300, where, Re is the flow Reynolds number 𝑅𝑒 =
𝑢 𝑚𝑎𝑥 𝑑 𝜈
, 𝜈 is the kinematic viscosity
and 𝑢𝑚𝑎𝑥 is the maximum flow velocity of the parabolic inflow profile. The computational domains are shown in Fig. 6.1. The blockage ratio, β = d/H, is fixed at β = 0.25. In the present simulation, we use the parameters that are shown in Table 6.1. Concerning the boundary conditions, we consider two types, namely: solid-wall and open boundary conditions. For the solid-wall boundary, the no-slip and no-penetration conditions are applied. The solid-wall lies exactly at grid nodes and it is assumed that all particles entering the boundary node leave with the same magnitude of speed but in the opposite direction of their incoming velocities. For the open boundary (such as inlet/outlet of the channel), it is common to assign a given velocity profile to the flow inlet, while either a given pressure or zero normal-velocity gradient is assigned to the flow outlet. The boundaries of the domain coincide exactly with the grid (lattice) points.
Table (6.1) Parameters of the simulations [140]. Parameter
Symbol 𝑁𝑥
Number of grid nodes in x-direction
Value 250 (One obstacle) 500 (Two obstacles)
𝑁𝑦
41
d
10
𝑢𝑚𝑎𝑥
0.02
Blockage ratio
β
0.25
Inflow length
𝑙
50
Number of grid nodes in y-direction Size of the obstacle Maximum velocity of inflow
𝑙o = 𝑛𝑑
Inter-distance between two obstacles
𝑛 = 5, 10, 15
106
50, 100, 150
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y u=v=0 H/2 umax
d d
H/2 x
u=v=0 l
L
(a) y u=v=0 l0 umax
H/2
d d
H/2 u=v=0
l
x
L
(b) Fig. (6.1) (a) The geometry and domain for a single square obstacle. (b) The geometry and domain for two square obstacles [140].
6.3
Turbulence Generation
In the present study, it is important to ensure that the flow is fully-turbulent. As the values of Reynolds number (1≤ Re ≤ 300) are small, it is not feasible to expect that the flow will develop to be fully-turbulent within the limited computational domain underconsideration. Thus, it is vital to introduce a source of turbulence (perturbation) in the velocity profile of the inflow. The instantaneous velocity u of the parabolic inflow profile at a certain position can be expressed as u = U u', where, U is the mean velocity and u' is the instantaneous turbulence (perturbation). In the present work, u' is introduced numerically by a mathematical random generator such that its values range between -1 and +1 [139].
6.4
Results and Discussions [139]
The numerical simulation was performed for a range of Re between 1 and 300. For all the considered cases, the size of the obstacle d × d = 10 × 10 of the grid units (grid stepsize). The square obstacle was positioned at l = 50 of the grid units downstream the entrance of the channel. The simulations were carried out until reached 100,000 seconds 107
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to ensure that complete convergence was occurred. The following sections describe the flow patterns (shown by streamline plots and velocity contours) with different values of Reynolds number (Re).
6.4.1 Flow pattern around a single square obstacle We study the flow around a single square obstacle, with size d =10 of the grid units, which is positioned inside the channel along its centerline as shown in Fig. 6.1(a). The streamline patterns and velocity contours for different values of Re are shown in Figs. 6.2 and 6.4, respectively, to demonstrate the flow characteristics. A comparison with the results of Yojina et al. [140] is carried out for validation of the present results, Fig. 6.3. For Re = 1, the flow pattern resembles that of steady laminar flow without separation, Fig. 6.2(a). For 30 Re < 85, the flow pattern is separated at the trailing edge of the obstacle. The length of the recirculation region increases with Re, Figs. 6.2(b)-(c). The results are very much symmetric with respect to the channel centerline. It seems that the turbulence effect is almost negligible for the values of Re below 85. The flow has a small amount of kinetic energy that is not enough to excite the turbulence in the flow. When Re increases, the symmetry of the flow starts to vanish gradually, Figs. 6.2(d)(e). The flow becomes eventually unstable with continuous vortex shedding, Figs. 6.2(f)-(h). This means that the effect of turbulence becomes dominant. This occurs when Re reaches 85, which is called the critical Reynolds number (Recrit) [140]. These results are supported by the flow velocity visualization, Fig. 6.4. When, Re < Recrit, the velocity contours resemble the steady flow without vortex shedding, Figs. 6.4(a)-(c). In Figs. 6.4(d)-(h), the flows become periodic and alternate the shedding of vortices into the stream. This is known as a von Karman vortex street, which exhibits an unstable flow pattern and performs a shedding pattern behind the obstacle. The present results compare very well to the results of Yojina et al. [140] as can be seen in Fig. 6.3.
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30
Ny
25
20
15
10
0
50
100
150
200
250
150
200
250
150
200
250
150
200
250
150
200
250
150
200
250
150
200
250
150
200
250
Nx
(a) Re=1 30
Ny
25
20
15
10
0
50
100
Nx
(b) Re=30 30
Ny
25
20
15
10
0
50
100
Nx
(c) Re=60 30
Ny
25
20
15
10
0
50
100
Nx
(d) Re=85 30
Ny
25
20
15
10
0
50
100
Nx
(e) Re=100 30
Ny
25
20
15
10
0
50
100
Nx
(f)
Re=160
30
Ny
25
20
15
10
0
50
100
Nx
(g) Re=200 30
Ny
25
20
15
10
0
50
100
Nx
(h) Re=300 Fig. 6.2: Streamline patterns around a single square obstacle for different values of Reynolds number. 109
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
(a) Re=1
(b) Re=30
(c) Re=60
(d) Re=85
(e) Re=100
(f) Re=160
(g) Re=200
(h) Re=300
Present Results
chapter(6)
Results of Yojina et al. [140]
Fig. 6.3: A comparison of the present results with the results of Yojina et al. [140]
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Veloc ity magnitude 40 0.025 35 0.02
30 25
0.015 20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(a) Re=1 Veloc ity magnitude 40 0.025 35 0.02
30 25
0.015 20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(b) Re=30
Veloc ity magnitude 40 0.025 35 0.02
30 25
0.015 20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(c) Re=60 Fig. 6.4: Velocity contours around a single square obstacle for different values of Reynolds number. 111
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Veloc ity magnitude 40 0.025 35 30
0.02
25 0.015 20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(d) Re=85 Veloc ity magnitude 40 0.025 35 30
0.02
25 0.015 20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(e) Re=100 Veloc ity magnitude 40 0.025
35 30
0.02
25 0.015 20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(f) Re=160 Fig. 6.4: Velocity contours around a single square obstacle for different values of Reynolds number.
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Veloc ity magnitude 40 0.025
35 30
0.02
25 0.015
20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(g) Re=200
Veloc ity magnitude 40 0.025
35 30
0.02 25 0.015
20 15
0.01
10 0.005 5 0
0
50
100
150
200
250
(h) Re=300 Fig. 6.4: Velocity contours around a single square obstacle for different values of Reynolds number. To demonstrate the effect of the monitoring position on the periodic flow, the timeseries data of the velocity values behind the obstacle were recorded at several streamwise positions (Nx = 120, 140, 180) at Re = 85, Fig. 6.5. It is shown that the position close to the obstacle (Nx = 120) generates periodic patterns quicker than the other two positions. Naturally, the amplitudes of these time-series data decrease as the position moves away from the obstacle. As can be seen in Fig. 6.5, the frequency amplitudes are approximately independent of the position (Nx = 120, 140) in the near wake behind the obstacle.
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Nx=120 Nx=140 Nx=180
0.018
0.017
0.016
U
0.015
0.014
0.013
0.012
0.011
0.01
0.009
0
1
2
3
4
Time/104
5
6
7 4
x 10
Fig. 6.5: Time-series data of the flow velocity for a single obstacle at several positions, Re=85.
To illustrate the flow behavior behind the obstacle in the near wake and the far field, the velocity distributions in the cross-wise direction (different values of Ny) for different stream-wise positions (Nx) are shown in Fig. 6.6. It is noticed that the velocity profile at Nx = 60 (the end of the obstacle) has values of zero at the back-face of the obstacle which corresponds correctly to the applied boundary conditions. At the outflow boundary of the computational domain (Nx = 250), the velocity profile takes the same shape and values of the inflow velocity profile. This gives confidence in the present results and shows that the extension of the present computational domain is suitable for the investigated problem.
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40 35 30
Nx=60 Nx=80 Nx=100 Nx=120 Nx=160 Nx=200 Nx=240 Nx=250
Ny
25 20 15 10 5 0
0
0.005
0.01
0.015 U
0.02
0.025
Fig. 6.6: Cross-wise distributions of the flow velocity at different positions (Nx), Re = 85.
6.4.2 Flow pattern around two square obstacles As a real-life application, we consider the flow pattern around two tandem square obstacles inside the channel. This application resembles the off-shore structures that are frequently found in marine channels. Moreover, such configuration may be found in many thermo-fluid industrial, chemical and technological applications such as microfluidic devices. The two square obstacles are modeled on a 41 × 500 grid with a fixed blockage ratio β = 0.25. The span-wise distance (inter-distance) between the two obstacles 𝑙o varies such that 𝑙o = 𝑛𝑑, where, n takes the values 5,10,15, respectively. To illustrate the flow phenomena, the flow characteristics are presented via the streamline patterns and velocity contours as shown in Figs. 6.7 and 6.9. Comparisons between the present results and the results of Yojina et al. [140] are also carried out in Fig. 6.8.
115
0.03
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As can be seen in Fig. 6.7, the flow of low Reynolds number (Re = 70) separates at the trailing edge of both obstacles, and the recirculation length does not increase when the distance 𝑙o increases, Fig. 6.7(a). The flow pattern is steady and symmetric with respect to the oncoming flow. When Re increases to greater than Recrit, the flow becomes unstable and breaks into asymmetry. We notice in Fig. 6.7(b) that asymmetrical flow occurs earlier in the case of 𝑙o = 50 before the other two values of 100, 150. Therefore, it is easy to conclude that the asymmetrical flow for 𝑙o = 50 appears at a Reynolds number below Recrit of the flow past one square obstacle (70 < Re 85, it is found that the flow wake shows asymmetry due to the vortex shedding behind the two obstacles for the three values of 𝑙o , Figs. 6.9(c)-(f). It is clear that there are two regions where the vortex shedding occurs, namely: (i) the inter-space between the two obstacles (behind the upstream obstacle), (ii) the region behind the second (downstream) obstacle. The absence of vortex shedding in the wake of the upstream obstacle is mainly due to the small inter-spacing (distance) between the two square obstacles. This observation points out that the upstream obstacle controls the unsteady wake of the downstream obstacle. Since the flow velocity in front of the downstream obstacle is mainly influenced by the inter-spacing (distance) between obstacles, the inter-spacing (distance) becomes a key parameter that governs the generation of the unsteady flow.
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30
Ny
25
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(b) Re=85 117
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(c) Re=100
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(f) Re=200 Fig. 6.7: Streamline patterns around the two square obstacles for different values of Re and 𝑙o ( 𝑙o = 50 (upper), 𝑙o = 100 (middle) and 𝑙o = 150 (lower)). 119
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
(a)Re=70
(b)Re=85
(c)Re=100
120
chapter(6)
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
chapter(6)
(d)Re=160
(e)Re=180
(f)Re=200 Present Results
Results of Yojina et al. [140]
Fig. 6.8: A comparison of the present results of streamline patterns around the two square obstacles for different values of Re with Results of Yojina et al. [140]
121
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(a) Re=70 Fig. 6.9: Velocity contours around the two square obstacles for different values of Re and 𝑙o (𝑙o = 50 (upper), 𝑙o = 100 (middle) and 𝑙o = 150 (lower)).
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(b) Re=85 Fig. 6.9: Velocity contours around the two square obstacles for different values of Re and 𝑙o (𝑙o = 50 (upper), 𝑙o = 100 (middle) and 𝑙o = 150 (lower)).
123
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(c) Re=100 Fig. 6.9: Velocity contours around the two square obstacles for different values of Re and 𝑙o (𝑙o = 50 (upper), 𝑙o = 100 (middle) and 𝑙o = 150 (lower)). 124
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
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(d) Re=160 Fig. 6.9: Velocity contours around the two square obstacles for different values of Re and 𝑙o (𝑙o = 50 (upper), 𝑙o = 100 (middle) and 𝑙o = 150 (lower)). 125
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126
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(f) Re=200 Fig. 6.9: Velocity contours around the two square obstacles for different values of Re and 𝑙o (𝑙o = 50 (upper), 𝑙o = 100 (middle) and 𝑙o = 150 (lower)). 127
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6.5
chapter(6)
The Performance of AMG in Solving the Navier Stokes Flow
In chapter 3, an iterative AMG algorithm was developed to solve the algebraic systems of equations resulting from discretization of Navier-Stokes equations. The main idea in this algorithm is to partition the coefficient matrix to separate the momentum operator. In this Section, we apply this algorithm (AMG) either as standalone solvers or preconditioners to the generalized minimum residual method (GMRES) in the solution of the channel flow problem with 1- and 2-obstacles. These results and its discussion are shown in Figs. 6.10 - 6.17. 6.5.1 The Performance of AMG of unsteady flow in a channel with 1-obstacle We solve the unsteady flow around 1-obstacle for different Re numbers and use AMG as a solver, GMRES as a solver with AMG as a preconditioner or GMRES as a solver without any preconditioners. In Figs. 6.10, 6.11 and 6.12, the residual norm of the algebraic system after each iteration step is plotted for unsteady flow at Re=30, 100 and 300, respectively. We noticed that, using GMRES without any preconditioner needs 20 iterations to achieve an error = 10−2 . However, use of AMG as a solver or as a preconditioner to GMRES speeds up the convergence; it converged very fast and get a high accuracy of error = 10−10 after only 15 iteration steps. It is important to notice that this excellent convergence behavior is independent of the value of Reynolds number.
10 10
Residual Norm
10 10 10 10 10 10
1-obstacle-NS unsteady-Re=30
0
-2
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-4
-6
-8
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0
5
10 15 Number of iterations
128
20
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
chapter(6)
Fig.6.10: The Performance of AMG around 1-obstacle at Re=30.
Residual Norm
10
10
10
10
1-obstacle-NS unsteady-Re=100
0
AMG(solver) GMRES(solver) GMRES(solver) and AMG(preconditioner)
-5
-10
-15
0
5
10 15 Number of iterations
20
Fig.6.11: The Performance of AMG around 1-obstacle at Re=100.
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10 10 10 10 10 10
1-obstacle-NS Unsteady-Re=300
0
-2
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-8
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0
5
10 15 Number of iteration
129
20
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
chapter(6)
Fig.6.12: The Performance of AMG of unsteady flow around 1-obstacle at Re=300. Next, we repeat the previous calculations on the same domain but on different grid 1
1
1
levels 𝑙 = 4 and 5 with mesh sizes ℎ = 2𝑙 = 16 and 32 , respectively. The convergence behavior of AMG as a solver (at Re=100) is plotted in Fig. 6.13. It is easily concluded that the convergence rates of the proposed AMG algorithm is slightly dependent of the mesh size.
10 10
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10 10 10 10 10 10
0
AMG at grid4 AMG at grid5
-2
-4
-6
-8
-10
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-14
0
5
10 Number of iteration
15
20
Fig. 6.13: The efficiency of AMG at different grid size.
6.5.2 The Performance of AMG of unsteady flow in a channel with 2-obstacles In this section, the unsteady flow around 2-obstacles for different Re number is considered. In Fig. 6.14, the residual norm of the algebraic system after each iteration step is plotted for the unsteady flow at Re=200 and 𝑙o =50 is shown. It is noticed that, GMRES without any preconditioners needs 30 iteration steps to reduce the error by only one order of magnitude, whereas the use of AMG as a solver or a preconditioner to GMRES results is very fast; residual error = 10−12 after 15 iteration steps. Fig.6.15 shows the results, if we increase 𝑙o to 100 units and solve the problem at Re=200. Applying GMRES without any preconditioner produces very slow convergence but when we used AMG as a preconditioner to GMRES, we need only 20 iteration to achieve the error = 10−12 . 130
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
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chapter(6)
lo 2-obstacles-NS Unsteady-lo=50-Re=200
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-8
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5
10 15 20 Number of iteration
25
30
Fig.6.14: The Performance of AMG of unsteady flow around 2-obstacles at Re=200 and 𝑙o =50. 10 10
Residual Norm
10 10 10 10 10 10
2-obstacles-NS Unsteady-lo=100-Re=200 lo
0
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GMRES(solver) and AMG(preconditioner) -6
-8
-10
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0
5
10
15
20
Number of iteration
Fig.6.15: The Performance of AMG of unsteady flow around 2-obstacles at Re=200 and 𝑙o =100. 131
The Numerical Results of the Flow inside the 2-D Channel with Internal Obstacles
6.5.3
chapter(6)
The Performance of AMG of steady flow
In this Section, we use the AMG as a preconditioner to the GMRES to solve various cases of steady Navier flow and compared the results with GMRES without any preconditioners. In Fig.6.16, results of the steady flow around 1-obstacle by GMRES with and without preconditioning are shown. GMRES without any preconditioners needs 100 iterations to achieve residual norm only one order of magnitude while it needs 50-70 iterations to achieve residual norm ≅ 10−6 when AMG is used as a preconditioner. In AMG, different smoothers were implemented; incomplete LU decomposition (ILU) and point damped Jacobi (PDJ).
10 10
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0
-1
GMRES(solver) GMRES(solver) and AMG(preconditioner) with ILU(smoother) GMRES(solver) and AMG(preconditioner) with PDJ(smoother)
-2
-3
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0
20
40 60 Number of iteration
80
100
Fig.6.16: The Performance of AMG based solvers for steady flow around 1-obstacle.
132
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chapter(6)
Also, we solve steady flow around 2-obstacles with 𝑙o =100. It is clear that convergence of AMG as preconditioner to the GMRES is very fast compared with GMRES without any preconditioners as shown in Fig.6.17.
10 10
Residual Norm
10 10 10 10 10 10
0
GMRES(solver) GMRES(solver) and AMG(preconditioner) with ILU(smoother) GMRES(solver) and AMG(preconditioner) with PDJ(smoother)
-1
-2
-3
-4
-5
-6
-7
0
20
40 60 Number of iteration
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Fig.6.17: The Performance of AMG of steady flow around 2-obstacles 𝑙o =100.
133
Conclusions and Future work
chapter(7)
Chapter (7)
Conclusions and Future work
Fluid dynamics are usually governed by systems of partial differential equations requiring efficient solution methods. The multigrid technique is one of the most efficient methods to solve elliptic partial differential equations. However, classical multigrid methods have not yet achieved full efficiency in realistic engineering applications in CFD in general. An important reason for this is that in CFD we often have to deal with singular perturbation problems. Another reason is that the governing equations are usually nonlinear and may show elliptic or parabolic behavior in one part of the domain and hyperbolic behavior in another part. This requires careful design of both the discretization and the solver.
The methodology for efficient
multigrid insists that each of the difficulties should be isolated, analyzed, and solved systematically using a carefully constructed series of model problems. In this dissertation, the finite element method is used to discretize many CFD problems and the resulting systems are linearized then solved by improved multigrid based methods including geometric multigrid, algebraic multigrid and some Krylov subspace methods preconditioned by multigrid. We have developed algebraic multigrid (AMG) approaches for anisotropic elliptic equation, problems with steep boundary layers, as convectional dominant convection-diffusion equations, and nonlinear system of equations as Navier-stokes equations. Main Contributions
In chapter 3, we extended the algebraic multigrid (AMG) approach to solve the nonlinear system of Navier-stokes equations, and obtained excellent results for various cases as presented in Chapter 6.
In Chapter 4, we investigated the idea that the discretization error introduced by a numerical scheme to solve boundary value problems can be compensated. For a class of singularly perturbed 1-D convection-diffusion problem, the analytic solution of the discrete system was computed and compared to the exact one to get the condition for being identical. We choose this condition in 134
Conclusions and Future work
chapter(7)
the form of a change in the normalized diffusion coefficient and get an analytic formula for the modified diffusion coefficient (MDC) as a function of the original coefficients and the mesh size.
The proposed MDC method is extended to convection-diffusion problems in higher dimensions with variable coefficients.
In chapters 5-6, the turbulent-flow characteristics and the mechanism of vortex shedding behind one and two square obstacles centered inside a 2-D channel is considered. The investigation was carried out for a range of Reynolds number (Re) from 1 to 300. The computations were based on the finite-element technique. Large-eddy simulation (LES) with the Smagorinsky method was used to model the turbulent flow.
Conclusions The following conclusions can be stated: (i)
AMG can be used for many kinds of problems where the application of standard geometric multigrid methods is difficult or impossible.
(ii)
Implementation of the proposed MDC technique produces the exact nodal solutions for the 1-D singularly-perturbed convection diffusion problems even on coarse grids with uniform or non-uniform mesh sizes.
(iii)
Numerical results show that extension of MDC to 2-D with constant and variable
coefficients
convection-diffusion
problems
eliminates
the
oscillations and produces more accurate solutions compared with other existing methods. (iv)
As a result, multigrid-based solvers retain its efficient convergence rates for singularly-perturbed convection diffusion problems.
(v)
The present results of turbulent-flow characteristics and the mechanism of vortex shedding behind one and two square obstacles compare very well to the results of other researchers.
(vi)
Excellent convergence behavior is obtained for numerical solution of Navier-Stokes system for different values of Re in two cases, 1- and 2obstacles, when we used the proposed AMG algorithm as a solver or a preconditioner of GMRES.
135
Conclusions and Future work
chapter(7)
Suggestions for future work Points for future work may be suggested as: 1. Extending AMG to solve applications in 3-D which seems to be straight forward. 2. Extending the proposed algorithm of AMG for systems of differential equations to solve CFD problems discretized with unstructured meshes on complex geometries. 3. More development is still needed to speed up the setup phase of AMG. 4. More research work is still needed to reveal the optimum operating conditions for turbulent-flow inside a channel with consideration of some parameters such as the number of obstacles (more than two), the obstacle shape, the boundary conditions, etc.
136
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ملخص الرسالة ذؼرثش ذمُٛح يرؼذداخ انشثكاخ أحذ أكصش انطشق كفاءج نحم انؼذٚذ يٍ يسائم انرطثٛماخ انُٓذسٛح
.احذ إَٔاع
يرؼذداخ انشثكاخ ْ ٙغشٚمح يرؼذداخ انشثكاخ انعثشٚح انر ٙذؼرًذ ػهٗ أساسٛاخ غشق يرؼذداخ انشثكاخ انؼادٚح ,إال أَٓا الذحراض يؼهٕياخ ػٍ ُْذسح انشثكاخ ,تم ذسرخهص انًؼهٕياخ يٍ لٛى ٔيٕاظغ ػُاصش يصفٕفاخ انًؼايالخ ,يًا ٔسغ يٍ يعال اسرخذايٓا ٔسفغ يٍ كفاءج انحم نثؼط انرطثٛماخ انرٗ لذ ذفشم غشق يرؼذداخ انشثكاخ انؼادٚح فٗ حهٓا. فْ ٙزِ انشسانح ذى انرشكٛض ػهٗ اسرخذاو غشٚمح يرؼذداخ انشثكاخ انعثشٚح نحم يشكالخ يرؼذدج يصم :حم انًؼادالخ انرفاظهٛح انعضئٛح انخطٛح ,كًؼادنح تٕاسٌٕ راخ انًؼايالخ انغٛش يرًاشهحٔ ,كزنك انًشاكم انرٙ ذحرٕ٘ ػهٗ غثماخ حذٚح حادج يصم يؼادالخ االَرشاس -انحًم ) (Convection-Diffusionانر ًٍٛٓٚ ٙػهٓٛا يؼايم انحًم اكصشٔ ,أخٛشا ذى حم يؼادالخ انُظاو انالخط ,ٙكًؼادالخ َافٛش سرٕكس ).(Navier-Stokes
َٕع اَخش يٍ انرطٕٚش ذى إظشاؤِ ف ٙانشسانح ٔرنك ترمذٚى يؼايم اَرشاس يؼذل ظذٚذ
) (MDCنحم يؼادالخ
االَرشاس -انحًم ٔرنك تاسرُراض انصٛغح انرحهٛهٛح نهًؼايم انًؼذل نًؼادالخ االَرشاس -انحًم يٍ انذسظّ األٔن. ٙ حٛس ذى ذطثٛك ػًهٛح انرعضئ ػهٗ انًؼادنح انًؼذنح تذالً يٍ انًؼادنح األصهٛح ٔ .تانفؼم ,ذى ذحهٛهٛالً إشثاخ أٌ حم انًؼادالخ انًؼذنح تؼذ ذطثٛك انًؼايم انًؼذل انعذٚذ ذُطثك ػهٗ انحم انفؼه ٙنهًؼادنح األصهٛح ٔرنك فٗ يسائم انثؼذ األحادٖ .كًا ذطاتمد انُرائط انؼذدٚح نحم انًؼادالخ انًؼذنح يغ انحم انفؼه ٙنكم حعى شثك ٙذى اسرخذايّ . كزنك ذى اخرثاس ْزا انًؼايم انًؼذل ) (MDCػهٗ يؼادالخ االَرشاس -انحًم ف ٙيسائم شُائٛح انثؼذ ٔذى انحصٕل ػهٗ َرائط ػذدٚحٔ ,إٌ كاَد غٛش يُطثمح ػهٗ انحم انفؼه ,ٙإال أٌ انُرائط انر ٙذى انٕصٕل إنٓٛا كاَد يسرمشج ٔراخ دلح ػانٛح ظذاً ,خصٕصا ً ف ٙانًؼادالخ راخ انًؼايالخ انًرغٛشج ٔ .لذ أشثرد يرؼذداخ انشثكاخ انعثشٚح كفاءذٓا ف ٙحم ْزا انُٕع يٍ انًسائم ,سٕاء ذى اسرخذايٓا كطشٚمح حم يُفشدج أٔ تذيعٓا ف ٙغشق حم أخشٖ يصم غشق انفشاغاخ انعضئٛح نكٛشٚهٕف ).(Krylov
يشكهح أخشٖ ذى اسرؼشاظٓا ف ٙانشسانح ْٙٔ ,انسشٚاٌ ف ٙانًعاس٘ انًائٛح انعٛمح انر ٙذؼرثش يٍ انًٕظٕػاخ انٓايح ف ٙدُٚايٛكا انًٕائغ حٛس أَّ غانثا ً تؼط انرطثٛماخ ذكٌٕ سشػح انسشٚاٌ فٓٛا يُخفعح يًا ٚؤد٘ إنٗ انحصٕل ػه ٙلٛى صغٛشج نًؼايم سُٕٚنذص ) .(Reنكٍ نثؼط األسثاب انرطثٛمٛح ًٚكٍ حذٔز سشٚاٌ يعطشب ف ٙحانح صغش يؼايم سُٕٚنذص)ٔ .(Reأحذ ْزِ األسثاب ٚشظغ إنٗ ٔظٕد ػٕائك داخم انًعشٖ انًائ ٙيًا ٚسثة حذٔز اظطشاب نهسشٚاٌ خهف يُطمّ انؼائك
َٔ .ظشاً ألًْٛح ْزا انرطثٛك فٗ انرطثٛماخ انحٛاذٛح فمذ ذى
اسرؼشاض حانر ٍٛنهذساسّ ًْأ :ظٕد ػائك أٔ ػائم ٍٛف ٙيُرصف انًعشٖ انًائٔ . ٙيٍ خالل خٕاسصو يُاسة
نحم ْزِ انًشكهح ,يغ اسرخذاو غشٚمح يرؼذداخ انشثكاخ انعثشٚح كطشٚمح نهحم ,حٛس ذى ذطٕٚشْا نحم انُظاو انالخط ٙانُاذط يٍ ْزا انُٕع يٍ انًسائم ٔ .كاَد انُرائط انر ٙذى انحصٕل ػهٓٛا ػُذ انحم نُطاق يٍ لٛى يؼايم سُٕٚنذص) (Reذشأحد ت 1 ٍٛإنٗ 300يرطاتمح تشكم كثٛش يماسَح يغ َرائط أتحاز أخشٖ .كًا أٌ اسرخذاو غشٚمح يرؼذداخ انشثكاخ انعثشٚح كطشٚمح نهحم حمك دلّ ػان ّٛظذاً يماسَح تطشق أخشٖ يصم ).(GMRES
ذركٌٕ انشسانح يٍ سثؼح أتٕاب: الباب األول :يمذيح ذى فٓٛا اسرؼشاض دُٚايٛكا انًٕائغ انحساتٛح ٔخاصح يؼادالخ االَرشاس -انحًم ٔيؼادالخ َافٛش سرٕكس )ٔ ,(Navier-Stokesيذٖ أًْٛرًٓا ف ٙانرطثٛماخ انُٓذسٛح انًخرهفح ٔانطشق انًرؼذدج انرٙ ذُأند حهٓا .كًا ذى ذُأل غشق يرؼذداخ انشثكاخ ٔتاألخص غشٚمح يرؼذداخ انشثكاخ انعثشٚح ٔيذٖ أًْٛرٓا ف ٙانرطثٛماخ انُٓذسٛح انًخرهفح.
الباب الثاني :اخرص تانثحس انراسٚخ ٙػٍ األػًال انساتمح ف ٙيعال دُٚايٛكا انًٕائغ انحساتٛح ٔخاصّ يؼادالخ االَرشاس -انحًم ٔيؼادالخ َافٛش سرٕكس ) .(Navier-Stokesكزنك غشق يرؼذداخ انشثكاخ تاػرثاسْا أفعم انطشق انؼذدٚح نحم ْزِ انًؼادالخ.
الباب الثالث :ذى ف ّٛذمذٚى أساسٛاخ غشق يرؼذداخ انشثكاخٔ ,دساسّ غشٚمح يرؼذداخ انشثكاخ انعثشٚح تشكم يفصمٔ ,ذمذٚى أيصهّ يخرهفح إلظٓاس يذٖ كفاءذٓا ف ٙحم إَٔاع يؼُٛح فشهد انطشق األخشٖ ف ٙحهٓا تذلح .كزنك ذى ذطٕٚش ظذٚذ نطشٚمح يرؼذداخ انشثكاخ انعثشٚح السرخذايٓا ف ٙحم يسائم انُظاو انالخط.ٙ
الباب الرابع :اخرص بيؼادالخ االَرشاس -انحًم ٔذمذٚى يؼايم اَرشاس يؼذل ظذٚذ ٔ .ذى ػشض انؼذٚذ يٍ األيصهّ ف ٙانثؼذ ٍٚاألٔل ٔانصأَ , ٙاسرؼشاض انُرائط انًخرهفح إلظٓاس يذٖ كفاءج غشٚمح يرؼذداخ انشثكاخ انعثشٚح نحم ْزا انُٕع يٍ انًسائم يماسَح تانطشق األخشٖ.
الباب الخامس :ذى ف ّٛذمذٚى يؼادالخ َافٛش سرٕكس )ٔ (Navier-Stokesانرحهٛم انشٚاظ ٙنٓا ف ٙحانرٙ انسشٚاٌ انًعطشب ٔانصفائحٔ ٙكزنك انسشٚاٌ انًعطشب انًسرمش ٔغٛش يسرمش.
الباب السادس :ذى ف ّٛذمذٚى انُرائط انخاصّ ب حم يؼادالخ َافٛش سرٕكس ) (Navier-Stokesنعًٛغ انحاالخ انًذسٔسح ف ٙانثاب انخايس ػهٗ انسشٚاٌ ف ٙانًعشٖ انًائ ٙانًحرٕ٘ ػهٗ ػٕائك ٔلٛى صغٛشج نًؼايم سُٕٚنذص ٔيماسَرٓا تُرائط ػهًاء آخش .ٍٚكًا ذى ذمذٚى َرائط إلشثاخ يذٖ َعاغ انرطٕٚش انز٘ ذى ف ٙانثاب انصانس ػهٗ غشٚمح يرؼذداخ انشثكاخ انعثشٚح نحم يسائم انُظاو انالخط ٙنهحصٕل ػهٗ دلح ػانٛح ف ٙظًٛغ انحاالخ انًذسٔسح.
الباب السابعٚ :حرٕ٘ ػهٗ االسرُراظاخ انؼايح يٍ انشسانح ٔااللرشاحاخ انًسرمثهٛح نهرطٕٚش. ذحرٕ٘ انشسانح أٚعا ً ػهٗ يهخص تانهغر ٍٛانؼشتٛح ٔاإلَعهٛضٚحٔ ,لائًح تانًحرٕٚاخ ٔانشسٕياخ ٔانعذأل ٔاالخرصاساخ ٔانشيٕص ٔ ,لائًح تانًشاظغ. ٔلذ ذى لثٕل َٔشش تحص ٍٛفٗ يٕظٕع انشسانح : [1] S. A. Mohamed, N. A. Mohamed, A. F. Abdel-Gawad, M. S. Matbuly, "A modified diffusion coefficient technique for the convection diffusion equation", Appl. Math. Comput., 219, 9317–9330, 2013. [2] A. F. Abdel Gawad, N. A. Mohamed,, S. A. Mohamed, M. S. Matbuly, "Investigation of the channel flow with internal obstacles using large eddy simulation and finite-element technique", Appl. Comput. Math., 2(1), pp. 1-13, 2013.
كلية الهٌذسة جاهعة السلازيك لسن الفيسياء والرياضيات الهٌذسية
االستخذام االهثل لتمٌيات هتعذدات الشبكات لحل بعض هسائل ديٌاهيكا الوىائع رسالة مقدمة من
ًىرهاى عالء الذيي هحوذ المدرش المساعد بقسم الفيسياء والرياضيات الهندسية كجسء من المتطلبات للحصىل علً درجة دكتو ارهالفمسفة في العموم الهندسية فً الفيسياء والرياضيات الهندسية
اإلشراف
أ.د /.سلىي اهيي هحوذ
أ.د /.أحوذ فاروق عبذ الجىاد
قسم الفيسياء والرياضيات الهندسية كلية الهندسة -جامعة السقازيق
قسم هندسة القىي الميكانيكية كلية الهندسة -جامعة السقازيق
أ.د /.هحوذ سعذ هتبىلي قسم الفيسياء والرياضيات الهندسية كلية الهندسة -جامعة السقازيق 2013
