[PDF] PhD Thesis - Swansea University

107094_7PhDThesis_DarongJin_08022016.pdf Anhp-Finite Element Computational Framework for

Nonlinear Magneto-Fluid Problems including

Magnetostriction

by

Darong Jin BEng, MSc

Submitted to Swansea University

in fulfilment of the requirements for the Degree of

Doctor of Philosophy

at

Swansea University

June 2016

AbstractCoupled magneto-mechanic effects have become major field of interest in a large number of industrial applications. However, coupled magneto-fluids phenomena remain at an earlier stage of understanding. Nevertheless, with the increasing number of applications of the magnetic fluids, the coupled magneto-fluid mechanism is now growing with interest and has become highly topical. To understand the coupling mechanism cost-effectively, computational mechanics should be applied. The numerical approach is based on the coupling between the Navier-Stokes equa- tions, which govern the fluid flow; and the Maxwell"s equations, which are the basic equations for the electromagnetic field. Although much has been done for the numerical simulations of both fluid and magnetic fields separately, the research on coupling phe- nomena remains limited. In previous work done on coupled magneto-fluids problems, many assumptions and simplifications were made, e.g. neglecting the magnetostrictive effect, in order to couple the two different physics with ease. However, as new important applications emerge, there comes the need to add the magnetostrictive effect to provide a more complete and accurate simulation of coupled magneto-fluid problems. In this thesis, we have introduced a novel coupling mechanism, which includes mag- netostriction, and is applicable for both conducting and non-conducting magnetic fluids. For the aspect of the numerical simulation, we focus on the solution of coupled magneto- fluid problems where the fluid is Newtonian, incompressible and magnetostrictive. The presence of an electromagnetic field exerts stresses on the fluid field. Meanwhile, the strain rate of the fluid alters the magnetic properties of the fluid by magnetostriction, resulting in a non-linear fully coupled system. Further non-linearity is introduced by the constitutive behavior of a ferro-magnetic fluid. From the computational point of view, a novel framework is established for both conducting and non-conducting magnetic fluids. The finite element method is employed to discretise the problem and the resulting non- linear algebraic equations are consistently linearised via a Newton-Raphson strategy. The complexity of the equations requires spatial discretisation of the velocity, pressure and magnetic fields by different element types in order to satisfy the LBB constraint. High accuracy is achieved by using high order (orhp-) versions of these finite elements. From the implementation point of view, a novel numbering scheme is introduced, which allows the solution of coupled multi-physics problems cost-effectively. In order to study the effectiveness and accuracy of our implementation, a series of benchmark problems for both conducting and non-conducting fluids are performed. The benchmarked framework is then applied to a series of problems to investigate the influence of a magnetic field over the fluids with magnetostriction. DECLARATIONThis work has not previously been accepted in substance for any degree and is not being concurrently submitted in candidature for any degree. Signed ............................................................. (candidate) Date .........................................................................

STATEMENT1

This thesis is the result of my own investigations, except where otherwise stated. When correction services have been used, the extent and nature of the correction is clearly marked in a footnote(s). Other sources are acknowledged by footnotes giving explicit references. A bibliography is appended. Signed ............................................................. (candidate) Date .........................................................................

STATEMENT2

I hereby give consent for my thesis, if accepted, to be available for photocopying and for inter-library loan, and for the title and summary to be made available to outside organisations. Signed ............................................................. (candidate) Date .........................................................................

AcknowledgementsI would like to express my sincere gratitude to my main supervisor Dr. Paul D. Ledger,

who has guided me all along the course of this thesis with great enthusiasm and patience. As an international student with inadequate background knowledge of this field of science in conjunction with the natural language barrier, I have encountered a lot of troubles both in research and daily life during this course. Paul has been my best supporter, providing me with great knowledge and the best advices. I could never thank him enough for all the help he has been giving me. I also want to thank my co-supervisor Dr. Antonio J. Gil for providing me technical and theoretical guidance during the research. He taught me the way to become a good researcher and how to carry out good research, which will benefit me for my lifetime. Their rigorous attitude towards research and their strong responsibility will greatly influence my future career. It would be impossible for me to complete this work without their excellent supervision. I gratefully acknowledge the financial support from the Zienkiewicz scholarship. With this financial help, I was able to concentrate on my research without other worries. Many thanks to my friends in Swansea as well as my room-mates during the three years of the Ph.D. course, especially to Jiyuan Sui, Jin Zhang, Wanliang Jiao, Bowen Cheng and Liqi Liu. With them the life of monotony has been enlivened with much joy and happiness. Special thanks go to my wife Haiyan An for her continuous trust and support. You are the reason that I could stand the seemingly endless loneliness in a foreign country during the last five years. Last but not least, I would like to thank my parents (Chengzhu Jin and Huizi Jin). Thank you for rasing me to be healthy, happy and confident. Thank you for everything. Your unconditional love and continuous support are the greatest source of my motivation and persistence. I am also grateful to my parents in law (Tianyi An and Meiyu Bu), no matter how hard it is, they never stop supporting me. I really appreciate it. I also want to thank my brother (Dasheng Jin) who has been really concerned and helpful over the past few years. This work is completely dedicated to them.

To my brothers, sisters, my wife and my parents

for all their love and understanding

Contents

I Preliminaries 1

1 Introduction 3

1.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1 Non-conducting magneto-fluid applications . . . . . . . . . .

4

1.2.2 Conducting magneto-fluid applications . . . . . . . . . . . .

5

1.3 The coupling approach . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4 Computational methodologies . . . . . . . . . . . . . . . . . . . . .

7

1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.7 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.7.1 Journal publications . . . . . . . . . . . . . . . . . . . . . .

13

1.7.2 Conferences and poster sessions . . . . . . . . . . . . . . . .

13

II Underlying Physical Laws and Fields 15

2 Aspects of Fluid Mechanics 17

2.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.2 Eulerian/Lagrangian representations . . . . . . . . . . . . . . . . . .

18

2.2.1The relationship between derivatives in the Lagrangian and

Eulerian coordinate descriptions . . . . . . . . . . . . . . . . 20

2.2.2 The Reynolds transport theorem . . . . . . . . . . . . . . . .

21

2.3 The basic governing equations . . . . . . . . . . . . . . . . . . . . .

21

2.3.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . .

22

2.3.2 Conservation of momentum . . . . . . . . . . . . . . . . . .

23

2.3.3 Governing equations for incompressible fluid . . . . . . . . .

25

2.4 Constitutive relations for an incompressible Newtonian fluid . . . . .

26

2.5 Boundary value problem for incompressible Newtonian fluid . . . . .

27
xiiCONTENTS3 Aspects of Electromagnetism 29

3.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.2 The Maxwell"s equations . . . . . . . . . . . . . . . . . . . . . . . .

30

3.2.1 Gauss"s law . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.2.2 Faraday"s law . . . . . . . . . . . . . . . . . . . . . . . . . .

31

3.2.3 Magnetic Gauss"s law . . . . . . . . . . . . . . . . . . . . .

31

3.2.4 Ampère-Maxwell"s law . . . . . . . . . . . . . . . . . . . . .

32

3.2.5 The general form of Maxwell"s equations . . . . . . . . . . .

32

3.2.6 The static Maxwell"s equations . . . . . . . . . . . . . . . . .

33

3.3 Constitutive relations for electromagnetics . . . . . . . . . . . . . . .

34

3.3.1 The general form of constitutive relations . . . . . . . . . . .

34

3.3.2 The scaled constitutive relations . . . . . . . . . . . . . . . .

36

3.3.3 Constitutive relations for the ferromagnetic media . . . . . . .

37

3.4 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3.4.1 The total current densityJ. . . . . . . . . . . . . . . . . . .38

3.4.2 Governing equation for non-conducting media . . . . . . . .

39

3.4.3 Governing equations for conducting media . . . . . . . . . .

41

3.4.4 Magnetostatic boundary value problems for isotropic material

41
III Anhp-FiniteElementFrameworkfortheSimulationofNon-

Conducting Magnetostrictive Fluids 45

4 Coupling Approach 47

4.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

47

4.1.1 Electromagnetic body force . . . . . . . . . . . . . . . . . .

47

4.1.2 Extensions to electromagnetic constitutive behaviour . . . . .

49

4.1.3 Electromagnetic-fluid coupling . . . . . . . . . . . . . . . . .

50

4.2 Electromagnetic description . . . . . . . . . . . . . . . . . . . . . .

51

4.3 Fluid description . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.3.1 Two-dimensional compressible linearised elasticity . . . . . .

52

4.3.2 Incompressible Navier-Stokes flow . . . . . . . . . . . . . .

54

4.4 Conservative systems and magnetostrictive materials . . . . . . . . .

55

4.4.1 Newton-Raphson scheme for conservative materials . . . . .

58

4.5 Non-conservative systems and magnetrostrictive fluids . . . . . . . .

60

4.5.1 Newton-Raphson scheme for non-conservative fluids . . . .

62

4.6 Coupled magnetostrictive ferrofluid materials . . . . . . . . . . . . .

63

4.6.1 Newton-Raphson scheme for ferrofluid . . . . . . . . . . . .

65

CONTENTSxiii5 Numerical Implementation 67

5.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

67

5.2 Numerical discretisation . . . . . . . . . . . . . . . . . . . . . . . .

68

5.2.1 Conservative magnetostrictive materials . . . . . . . . . . . .

68

5.2.2 Non-conservative magnetostrctive Fluids . . . . . . . . . . .

69

5.2.3 Coupled magnetostrictive ferrofluids . . . . . . . . . . . . . .

70

5.3 The Constructions of basis functions for reference element . . . . . .

70

5.3.1 TheH1(

)vertex basis . . . . . . . . . . . . . . . . . . . .71

5.3.2 TheH1(

)edge basis . . . . . . . . . . . . . . . . . . . . .72

5.3.3 TheH1(

)interior basis . . . . . . . . . . . . . . . . . . . .73

5.3.4 TheL2(

)basis . . . . . . . . . . . . . . . . . . . . . . . .73

5.4 The construction and implementation of the linear system . . . . . . .

74

5.4.1 The general linear system of equations for coupled problems .

75

5.4.2 Global numbering . . . . . . . . . . . . . . . . . . . . . . .

78

5.4.3 Static condensation . . . . . . . . . . . . . . . . . . . . . . .

84

5.4.4 Elemental numbering . . . . . . . . . . . . . . . . . . . . . .

86

5.4.5 Assemble global system . . . . . . . . . . . . . . . . . . . .

89

5.5 Vectorised implementation . . . . . . . . . . . . . . . . . . . . . . .

93

5.6 Enhancing robustness of the Newton-Raphson scheme . . . . . . . .

95

5.6.1 Incremental approach . . . . . . . . . . . . . . . . . . . . . .

95

5.6.2 Picard iteration for initial guess . . . . . . . . . . . . . . . .

96

6 Numerical Examples 97

6.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

97

6.2 Examples for conservative systems . . . . . . . . . . . . . . . . . . .

97

6.2.1Infinite plate with rigid diamagnetic insert with small magne-

tostrictive effects . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.2 Infinite plate with rigid diamagnetic insert with large magne- tostrictive effects . . . . . . . . . . . . . . . . . . . . . . . . 99

6.3 Examples for non-conservative systems . . . . . . . . . . . . . . . .

101

6.3.1 Flow past a cylinder with a magnetic field . . . . . . . . . . .

103

6.3.2 Lid driven cavity with a magnetic field . . . . . . . . . . . . .

113

6.3.3 Multiphase ferrofluid flow problem . . . . . . . . . . . . . .

116
IV Anhp-FiniteElementFrameworkfortheSimulationofCon- xivCONTENTSducting Magnetostrictive Fluids 121

7 Derivation and Framework 123

7.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

123

7.2 Coupling approach . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

7.2.1 Simple conducting fluids . . . . . . . . . . . . . . . . . . . .

124

7.2.2 Magnetostrictive conducting fluids . . . . . . . . . . . . . . .

125

7.3 Linearised schemes for simple materials . . . . . . . . . . . . . . . .

127

7.4 Linearised schemes for magenostrictive materials . . . . . . . . . . .

130

8 Numerical Implementation 135

8.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

135

8.2hp-Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . . .136

8.2.1 Simple conducting fluids . . . . . . . . . . . . . . . . . . . .

137

8.2.2 Conducting magnetostrictive fluids . . . . . . . . . . . . . .

137

8.3 The construction of basis functions for reference element . . . . . . .

138

8.3.1 TheH(curl;

)conforming basis functions in two-dimensions138

8.3.2The extension of the basis functions for three-dimensional prob-

lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.4 The construction and implementation of the linear system . . . . . . .

142

8.4.1 The general linear system of equations for coupled problem .

143

8.4.2 Global numbering . . . . . . . . . . . . . . . . . . . . . . .

147

8.4.3 Static condensation . . . . . . . . . . . . . . . . . . . . . . .

150

8.4.4 Elemental numbering . . . . . . . . . . . . . . . . . . . . . .

150

9 Numerical Examples 153

9.1 Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . .

153

9.2 Two-dimensional problems . . . . . . . . . . . . . . . . . . . . . . .

153

9.2.1 Linearised L-shape domain smooth problem . . . . . . . . . .

154

9.2.2 Linearised L-shape domain singular problem . . . . . . . . .

155

9.2.3 Fully coupled non-linear square domain smooth problem . . .

158

9.2.4 The two-dimensional Hartmann flow problem . . . . . . . . .

159
9.2.5 Two-dimensional duct flow with cylinder shaped different me- dia inside . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

9.3 Three-dimensional problems . . . . . . . . . . . . . . . . . . . . . .

169
9.3.1 Thethree-dimensionallid-drivencavityproblemforpureNavier- Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

9.3.2 The three-dimensional Hartmann flow problem . . . . . . . .

172
CONTENTSxv9.3.3 The three dimensional multi-phase MHD flow problem . . . .176

V Conclusion 179

10 Conclusions and Further Work 181

10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181

10.2 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

Bibliography 198

xviCONTENTS

List of Figures

1.1Two way coupling mechanism for magnetic fields and fluids (The

parameters are described in detail in Chapter 4) . . . . . . . . . . . . 6

2.1 General motion of deformable continuum . . . . . . . . . . . . . . .

19

2.2 An arbitrary volume

enclosed by the surface@ . . . . . . . . . .22

2.3 Boundary conditions for fluid domain . . . . . . . . . . . . . . . . .

27
3.1

An arbitrary volume

enclosed by the surface@ under the electro- magnetic flied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Non-linearBHcurve . . . . . . . . . . . . . . . . . . . . . . . .37

3.3 Boundary conditions for material domain . . . . . . . . . . . . . . .

42

3.4 Overview of different types of materials considered in this work . . .

44

5.1 Reference elements for triangular . . . . . . . . . . . . . . . . . . . .

71
5.2 The low order vertex basis functions for reference element inH1space. (a)Vertex1,(b)Vertex2and(c)Vertex3forp= 1. . . . . . . . . .71 5.3 The high order edge basis functions for reference element inH1space. (a);(b);(c)are the basis for edge1;2and3at orderp= 2,(d);(e);(f) are the basis for edge1;2and3at orderp= 3,(g);(h);(i)are the basis for edge1;2and3at orderp= 4. . . . . . . . . . . . . . . . . . . .72 5.4 The high order interior basis functions for reference element inH1 space.(a)is the basis for interior at orderp= 3,(b)and(c)are the basis for interior at orderp= 4. . . . . . . . . . . . . . . . . . . . .73 5.5 TheL2basis functions for reference element.(a)is the basis at order p= 0which is piecewise constant,(b)and(c)are the basis at order p= 1,(d);(e)and(f)are the basis at orderp= 2,. . . . . . . . . . .74

5.6 The numbering of vertices, edges and elements. . . . . . . . . . . . .

79
5.7 The numbering of the continuous (vertex and edge) coefficients for the magnetic potential when the polynomial order isp= 4. . . . . . . . .79 xviiiLIST OF FIGURES5.8The numbering of the continuous (vertex and edge) coefficients for the velocity when the polynomial orderp= 4. . . . . . . . . . . . . . . .81 5.9 The numbering of the coefficients for pressure when the polynomial orderp= 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82 5.10 The numbering of the interior coefficients for magnetic potential when the polynomial orderp= 4. . . . . . . . . . . . . . . . . . . . . . . .83 5.11 The numbering of the interior coefficients for velocity when the poly- nomial orderp= 4. . . . . . . . . . . . . . . . . . . . . . . . . . . .84 5.12 The mapping from reference element to local element and to the global mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.13

TheconstructionoftheelementalbasisfunctionsforH1(

)conforming space forp= 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .87 5.14

The elemental numbering of basis functions forL(

)2conforming space and the example withp= 2. . . . . . . . . . . . . . . . . . . .87

5.15 The elemental numbering for DOFs withp= 4. . . . . . . . . . . . .90

5.16 The DOFs mapping between reference element to local element and local element to the global system withp= 4. . . . . . . . . . . . .91 5.17 The comparison between looping (black) and vectorized (red) imple- mentation for total computational cost (time in s) for orderp= 3;4;5;6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.18 The percentage of the stiffness calculation in the whole calculation time. Top row of pie charts shows the loop-based implementations and bottom row of pie charts shows the vectorized scheme.(a);(b);(c)and (d)corresponding to the orderp= 3;4;5;6, respectively. . . . . . . .94

5.19 The flow chart for incremental approach. . . . . . . . . . . . . . . . .

95

5.20 The flow chart for Picard for initial guess. . . . . . . . . . . . . . . .

96
6.1 Infinite magnetostrictive plate with rigid elliptical insert under plane strainconditionsshowing:(a)convergenceofk(R[m]u;R[m] #)k=k(R[0]u;R[0] #)k and(b)ku[M] hpuk0;

CE=kuk0;

CE,ku[M]

hpukE;

CE=kukE;

CEand ku[M]ukSNS;

CE=kukSNS;

CEfor the displacement formulation and

the corresponding results in(c)and(d)for the mixed formulation . .100 6.2 Infinite magnetostrictive plate with rigid elliptical insert under plane stressconditionsshowing:(a)convergenceofk(R[m]u;R[m] #)k=k(R[0]u;R[0] #)k and(b)ku[M] hpuk0;

CE=kuk0;

CE,ku[M]

hpukE;

CE=kukE;

CEand ku[M]ukSNS;

CE=kukSNS;

CEfor the displacement formulation and

the corresponding results in(c)and(d)for the mixed formulation . .101 LIST OF FIGURESxix6.3Infinite magnetostrictive plate with rigid elliptical insert under plane stressconditionsshowing:(a)convergenceofk(R[m]u;R[m] #)k=k(R[0]u;R[0] #)k and(b)the deviation ofku[M]ukSNS;

CE=kukSNS;

CEaway from

the analytical for the mixed formulation . . . . . . . . . . . . . . . . 102
6.4 Infinite magnetostrictive plate with rigid elliptical insert under plane strainconditionsshowing:(a)convergenceofk(R[m]u;R[m] #)k=k(R[0]u;R[0] #)k and(b)the deviation ofku[M]ukSNS;

CE=kukSNS;

CEaway from

the analytical for the mixed formulation . . . . . . . . . . . . . . . . 102
6.5 Flow past a circular cylinder with magnetic loading showing an illustra- tion of the geometry of the problem . . . . . . . . . . . . . . . . . . 103
6.6 Flow past a circular cylinder with magnetic loading showing the un- structured mesh of 448 elements . . . . . . . . . . . . . . . . . . . . 104
6.7 Flow past a circular cylinder atRe= 40without a magnetic field showing in(a)the convergence ofC^pwhenp= 2;3;4and in(b)the convergence ofC^pwhenp= 5;6. . . . . . . . . . . . . . . . . . . .105 6.8 Flow past a circular cylinder without a magnetic field showing the distribution ofC^pcomputed usingp= 6elements forRe= 7;10;20;40.106 6.9 Flow past a circular cylinder without a magnetic field showing a com- parison of elapsed CPU time for the computation ofCDforRe= 40 when bothh- andp-refinements are performed whereGFindicates the grading factor of the graded mesh. . . . . . . . . . . . . . . . . . 106
6.10 Flow past a circular cylinder atRe= 40with a magnetic field of magnitudeH1= 0:1with= 0whenr2= 2showing in(a)the convergence ofC^pwhenp= 3;4;5;6;7;8;9and in(b)the converged distribution ofC^pforRe= 40. . . . . . . . . . . . . . . . . . . . . .107 6.11 Flow past a circular cylinder atRe= 40with magnetic fields with magnitudesH1= 0:01;0:05;0:10;0:15with= 0whenr2= 2 computed usingp= 8elements showing the distribution ofC^p. . . . .108 6.12 Flow past a circular cylinder atRe= 40with a magnetic field with = 0whenr2= 2computed usingp= 8elements showing(a)CD with varyingH1and(b)CLwith varyingH1. . . . . . . . . . . . .108 6.13 Flow past a circular cylinder atRe= 40with magnetic fields with mag- nitudesH1= 0:01;0:05;0:10;0:15at= 0whenr2= 2computed usingp= 8elements showing(a)F^p

1with,(b)F^p

2with,(c)Fv1

with,(d)Fv2with,(e)FH1withand(f)FH2with. . . . . . . .109 xxLIST OF FIGURES6.14Flow past a circular cylinder atRe= 40with magnetic fields such that= 0andr2= 2computed usingp= 8elements showing the convergence ofkR[m]v;R[m] ^p;R[m] #k=kR0v;R[0] ^p;R[0] #k((a),(c)and (d)) and the streamlines for the flow patterns ((b),(d)and(f)) when H1= 0((a)and(b)),H1= 0:05((c)and(d)) andH1= 0:15((e) and(f)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 6.15 Flow past a circular cylinder atRe= 40with a magnetic field with H1= 0:05whenr2= 2computed usingp= 8elements showingC^p for= 0;=6;=3and=2. . . . . . . . . . . . . . . . . . . . . . .111 6.16 Flow past a circular cylinder atRe= 40with a magnetic field with H1= 0:05whenr2= 2computed usingp= 8elements showing(a) C Dwith varyingand(b)CLwith varying. . . . . . . . . . . . . .111 6.17 Flow past a circular cylinder atRe= 40with magnetic fields with magnitudeH1= 0:05at orientations= 0;=6;=3and=2when r2= 2computed usingp= 8elements showing(a)F^p

1with,(b)F^p

2 with,(c)Fv1with,(d)Fv2with,(e)FH1withand(f)FH2with.112 6.18 Lid driven cavity flow showing a mesh of 444 unstructured triangular elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.19 Lid driven cavity flow withRe= 0:01;10;100;400;1000without a magnetic field computed usingp= 5elements showing the distribution ofv1along the centrelinex= 0,1=2y1=2. The reference solution is given in [92]. . . . . . . . . . . . . . . . . . . . . . . . . 114
6.20 Lid driven cavity flow withRe= 1000without a magnetic field com- puted usingp= 5elements showing(a)kR[m]v;R[m] ^pk=kR[0]v;R[0] ^pkand (b)streamlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115 6.21 Lid driven cavity flow with a cylinder showing a mesh of 1174 unstruc- tured triangular elements. . . . . . . . . . . . . . . . . . . . . . . . . 115
6.22 Lid driven cavity flow with a cylinder of radiusR= 0:05mplaced at the centre of the cavity andRe= 0:01;10;100;400;1000showing the distribution ofv1along the centrelinex= 0,1=2y1=2for(a) H 1= 0 and(b)H1= 0:15, in each case= 0,r2= 5andp= 5 elements are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.23 Lid driven cavity flow with a cylinder of radiusR= 0:05mplaced at the centre of the cavity showing results forRe= 10where(a)H1= 0 and(b)H1= 0:15,Re= 400where(c)H1= 0and(d)H1= 0:15 andRe= 1000where(e)H1= 0and(f)H1= 0:15in each case = 0,r2= 5andp= 5elements are used. . . . . . . . . . . . . . .117 LIST OF FIGURESxxi6.24Two-dimensional multiphase ferrofluid problem showing: geometry and mesh with2910unstructured triangular elements . . . . . . . . .118 6.25 Two-dimensional multiphase ferrofluid problem showing:(a);(c);(e) the quadratic convergence for updatesv;^p;#andkRkof Newton- RaphsonimplementationforH= 100;200;400,(b);(d);(f)thestream- lines forH= 100;200;400, respectively. . . . . . . . . . . . . . . .119 8.1 QuiverplotsillustratingthebehaviourofthebasisfunctionsforH(curl; ) conforming space.(a);(b);(c)shows the lower orderp= 0edge basis functions associated with the three edges,(d);(e);(f)shows the addi- tional higher orderp= 1edge basis functions associate with the three edges,(g);(h);(i)shows the three interior basis functions required for p= 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .139 8.2 The reference tetrahedron. The circle indicates the vertex and the rectangular means edge. The face number is assigned by the number of the vertex opposite to the corresponding face. (i.e. face1is consist of vertices2;3;4, face2with vertices1;3;4, onwards.) . . . . . . . . .141 8.3 The edge orientation for (a) Type I and (b) Type II tetrahedron elements [5]141

8.4 The global numbering of DOFs for an MHD problem. . . . . . . . . .

148
8.5 The extension of the global numbering of velocity from two- to three- dimensional DOFs for MHD. . . . . . . . . . . . . . . . . . . . . . . 149
8.6 The elemental numbering of basis functions for triangularH(curl; ) conforming element. . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.1 Two-dimensional L-shape domain with a smooth solution showing: uniform mesh of 382 triangular elements . . . . . . . . . . . . . . . . 154
9.2 Two-dimensional L-shape domain with a smooth solution showing: (a)the convergence ofkv[0] v;hpkH1andkH[0]

H;hpkH(curl)with

p-refinement and(b)the streamlines[0] v;hp. . . . . . . . . . . . . . .155 9.3 Two-dimensional L-shape domain with a singular solution showing: unstructured grading mesh of(a)214 and(b)308 triangular elements .156 9.4 Two-dimensional L-shaped domain with with a singular solution show- ing:(a)the convergence ofkv[0] v;hpkH1andkH[0]

H;hpkH(curl)

withhp-refinement and(b)the streamlines[0] v;hp,(c);(d)contour of the xandycomponents of[0] H;hp. . . . . . . . . . . . . . . . . . . . . .156 9.5 Two-dimensional square domain with a smooth solution showing: un- structured uniform mesh of 2048 triangular elements . . . . . . . . . 157
xxiiLIST OF FIGURES9.6Two-dimensional L-shape domain with a smooth solution showing:(a) the convergence ofkvv[m] hpkH1;k^p^p[m] hpkL2andkHH[m] hpkH(curl) withhp-refinement and(b)the quadratic convergence for updatesv;^p andH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158 9.7 Two-dimensional rectangular domain with Hartmann flow showing: unstructured mesh of400triangular elements . . . . . . . . . . . . .160 9.8 Two-dimensional rectangular domain with Hartmann flow showing: (a)the typical quadratic convergence of the residualkRkfor Newton- Raphson method for uniformp= 6elements and(b)the convergence ofkv[M] v;hpkH1andkH[M]

H;hpkH(curl)for Newton-Raphson and

kv[0] v;hpkH1andkH[0]

H;hpkH(curl)for one step Picard withp-

refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.9 Two-dimensional rectangular domain with Hartmann flow showing: Center line profile for(a):vxand(b):HxforHa= 1;10;10p10with the mesh shown in Figure 9.7 forp= 7. . . . . . . . . . . . . . . . .161 9.10 Two-dimensional multiphase duct flow problem showing: geometry and mesh with2910unstructured triangular elements for duct flow problem167 9.11 Two-dimensional multiphase duct flow problem showing:(a);(c);(e) the quadratic convergence forkRkof Newton-Raphson implementation forH= 0:1;0:2;0:4,(b);(d);(f)the streamlines forH= 0:1;0:2;0:4, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.12 Three-dimensionallid-drivencavityproblemshowing:(a)thegeometry of the problem,(b)the uniform mesh with756tetrahedrons . . . . . .169 9.13 Three-dimensional lid-driven cavity problem showing:(a);(c);(e)the quadratic convergence forv;^pandkRkof Newton-Raphson im- plementation forRe= 100;400;1000withp= 6;6;8, respectively. (b);(d);(f)the centreline profiles forRe= 100;400;1000with p- refinement on(0:5;0;0:5)(x;y;z)(0:5;1;0:5)on a mesh with

756uniform elements, respectively and the comparison with reference

solution [129]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.14 Three-dimensional lid-driven cavity problem showing:(a);(b)the streamlines forRe= 100withp= 6,(c);(d)the streamlines for Re= 400withp= 6,(e);(f)he streamlines forRe= 1000withp= 8.171 9.15 Three-dimensional rectangular domain with Hartmann flow showing: (a)the geometry(b)the unstructured mesh with 125 tetrahedron elements.174 9.16 Three-dimensional rectangular domain with Hartmann flow showing: quadratic convergence(a);(b)forHa= 1;10forp= 6. . . . . . . .175 LIST OF FIGURESxxiii9.17Three-dimensional rectangular domain with Hartmann flow showing: centreline profile along(5;2;0)(x;y;z)(5;2;0)for(a):vxand (b):HxforHa= 0:01;1;10with125tetrahedra elements forp= 6. .175 9.18 Three-dimensional rectangular domain with Hartmann flow showing: centreline profile for(a):vxand(b):HxforHa= 10p10with125 elements forp= 7. . . . . . . . . . . . . . . . . . . . . . . . . . .176 9.19 Three-dimensional multiphase duct flow problem showing: the geometry.177 9.20 Three-dimensional multiphase duct flow problem showing:(a)and (b)are the quadratic convergence forkHk;kvk;k^pk;kRkwith the permeability for inside sphererin= 0:1;10. . . . . . . . . . . . . .177 9.21 Three-dimensionalmultiphaseductflowproblemshowing:(a);(b);(c);(d) streamlines for different material properties for inside sphere:rin=

0:01;0:1;1;10respectively . . . . . . . . . . . . . . . . . . . . . . .178

xxivLIST OF FIGURES

List of Tables

5.1The number of degrees of freedom needed for each entity (i.e. per

vertex, edge or interior) for theH1( )andL2( )conforming spaces with the orderpin two-dimensions. Note thatpstarts from 1 forH1( ) space while from 0 forL2( )conforming space. . . . . . . . . . . .74 5.2 The mapping table for the DOFs of the continuous magnetic potential e;C #.(L2G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90 6.1 Flow past a circular cylinder without a magnetic field showing the comparison ofC^p

D,CvDandCDcomputed usingp= 6elements for

Re= 7;10;20;40with the previous studies. . . . . . . . . . . . . . .105 6.2 Two-dimensional multiphase ferrofluid problem showing: The parame- ters for inside and outside fluids . . . . . . . . . . . . . . . . . . . . 118
8.1 The number of degrees of freedom needed for each entity (i.e. vertex, edge or interior) for theH(curl; )conforming spaces of orderpfor a triangular element in two-dimensions. Note thatpstarts from 0 for

H(curl;

)conforming space. . . . . . . . . . . . . . . . . . . . . .139 8.2 The number of degrees of freedom needed for each entity (i.e. vertex, edge or interior) for theH1( );H(curl; )andL2( )conforming spaces of orderpin three-dimensions. Note thatpstarts from1for the

H(curl;

)andL2( )conforming space. . . . . . . . . . . . . . . .142 9.1 Two-dimensional rectangular domain with Hartmann flow showing: the convergence withHa= 0:01. . . . . . . . . . . . . . . . . . . . . .162 9.2 Two-dimensional rectangular domain with Hartmann flow showing: the convergence withHa= 1. . . . . . . . . . . . . . . . . . . . . . .163 9.3 Two-dimensional rectangular domain with Hartmann flow showing: the convergence withHa= 10. . . . . . . . . . . . . . . . . . . . . . .164 9.4 Two-dimensional rectangular domain with Hartmann flow showing: the convergence withHa= 10p10. . . . . . . . . . . . . . . . . . . .165 xxviLIST OF TABLES9.5Two-dimensional multiphase duct flow problem showing: parameters for inside and outside MHD fluids . . . . . . . . . . . . . . . . . . . 167
9.6 Three-dimensional rectangular domain with Hartmann flow showing: the parameters for Hartmann flow problem with various Hartmann numbersHa= 0:01;1;10;10p10. . . . . . . . . . . . . . . . . . . .174

Nomenclature

Scalar function of position and time xCurrent position vector,(x1;x2;x3)or(xj;j= 1;2;3);(m) XInitial Position vector,(X1;X2;X3)or(Xj;j= 1;2;3);(m) dDimension of the problem,d= 2or3 tTime,(s) Mapping relation Basis functions vVelocity of fluid,v1;v2;v3or(vj;j= 1;2;3);(m s1) nUnit normal vector pointing outwards n ?Unit tangential vector

MMass of the whole domain(kg)

Domain(md)

@

Boundary of the domain(md1)

SOpen surface domain(m2)

CLine boundary enclosing open surface domainS(m)

Density of the domain,(kg m3) ^Dynamic viscosity for fluids,(N s m2) Kinematic viscosity for fluids,(m2s1) ~Poisson"s ratio ^pStatic fluid pressure,(N m2) Second coefficient of viscosity for fluid(N s m2) ~;~Lamé elasticity constants(N m2) F sStress force per unit surface(N m2) fBody force per unit volume,(N m3) uDisplacement for elasticity(m) v

DPrescribed velocity,(m s1)

tTraction vector,(N m2) f

EMPondermotive force per unit volume,(N m3)

rLagrangian multiplier xxviiiLIST OF TABLES[["]]Strain rate tensor(s1) [[]]Shear stress tensor or deviatoric stress tensor(N m2) [[]]Cauchy stress tensor(N m2) [[TL]]Total stress tensor(N m2) [[EM]]Total electromagnetic stress tensor(N m2) [[S]]Stress tensor for Stoke flow(N m2) [[F]]Stress tensor generated inside the fluid(N m2) [[NS]]Stress tensor for Navier-Stoke flow(N m2) [[]]Permittivity tensors,(A2s4kg1m3) [[]]Permeability tensors,(A2s2kg m) 

0Permittivity of free space,(08:8541012A2s4kg1m3)



0Permeability of free space,(0= 4107A2s2kg m)

cSpeed of light,(c= 2:998108m s1) ^vVolume charge density,(C m3)  vScaled volume charge density #Magnetic scalar potential [[ ^~]]Electric conductivity,(A V1m1) [[~]]Scaled electric conductivity [[r]]Relative permittivity tensors [[e]]Electric susceptibility tensors [[r]]Relative permeability tensors [[m]]Magnetic susceptibility tensors

EElectric field intensity vector,(V m1)

PPolarisation vector,(C m2)

DElectric flux intensity vector or electric displacement,(C m2)

HMagnetic field intensity vector,(A m1)

MMagnetisation vector,(A m1)

BMagnetic flux intensity vector or magnetic induction,(kg A1s2)

GElectromagnetic momentum,(kg m2)s1)

JElectric current density,(A m2)

AMagnetic vector potential

EScaled electric field intensity vector

PScaled polarisation vector

DScaled electric flux intensity vector

HScaled magnetic field intensity vector

MScaled magnetisation vector

BScaled magnetic flux intensity vector

LIST OF TABLESxxixGScaled electromagnetic momentum

JScaled electric current density

J exScaled external electric current density J sScaled surface electric current density ^ EMEnthalpy energy functional ^ TLTotal energy functional ^ SHelmholtz"s free energy density W

EMElectromagnetic energy functional

W

TLTotal energy functional

W

SElasticity energy functional

# ;v;^pDirection of directional derivatives  [m]v;[m] #;[m] ^pIncrements direction of directional derivatives # hp;vhp;^phpDiscretised direction of directional derivatives  [m] v;hp;[m] #;hp;[m] ^p;hpDiscretised increments direction of directional derivatives

Part I

Preliminaries

Chapter 1

Introduction

1.1 Introductory remarksIn this chapter, the background of the coupled magneto-fluid problems will be presented.

After outlining a series of significant and abundant applications based on the coupled magneto-fluid phenomena in the background, the computational approach and numerical methods as well as the engineering approach used for the simulation will be summarised. This will be followed by the details of the specific approaches that have been adopted in this research. Then, the objective of this thesis will be stated and the outline of this thesis will be listed.

1.2 Background

Until recently, the coupling phenomena between magnetic field and fluids have often been neglected as the coupling was thought to be too weak to recognize. Back in 1832, Faraday, who discovered the electromagnetic forces, placed two copper electrodes in the river Thames in London and hoped to obtain the measurement of a voltage created by the coupling between the earth"s magnetic field and the salt water of Thames. Although the principle shown by Faraday was correct, he had no luck in being able to detect any voltage at all as the coupling was too small to be detected by the poor device he had at that time [41]. However, the interaction between electromagnetic fields and fluids has been attract- ing increasing attention with growing number of applications. Nowadays, the coupling between electromagnetic fields and fluids can be found in various fields of science, engineering and industry. The coupling phenomena between magnetic fields and fluids can be divided into two categories depending on whether the fluid is conducting or not. The important modern applications for both of these two categories will be discussed in

41. Introductionthe following sections.

1.2.1 Non-conducting magneto-fluid applicationsAmongst non-conducting fluids, a ferrofluid is the most relevant magnetic fluid material,

which has great potential applications. A ferrofluid is a non-conducting fluid with small ferromagnetic particles suspended in its volume and the research related to ferrofluid has been termed "Ferrohydrodynamics" by Rosenweig in 1964 [97]. The important applications of ferrofluids are stated below. Ferrofluids have a wide range of applications, in particular, as magnetic seals and in loudspeakers. Ferrofluids, which consist of a high quality lubricating oil and ferromagnetic particles, can minimize the wear for the seal. The magnetic seal has become an important technique for hard disk drives, vacuum rotary devices, etc. The high-tech loudspeaker also has advanced performance thanks to the ferroflud, which will conduct heat away from the coil and provides more damping. Another potential application is using the ferrofluid to clean oil spills [9]. The oil pollution caused by the crash of a large crude oil carrier has become an important environmental issue. The oil that spills out of the carrier will spread to a large area of ocean surface in the form of a thin oil film that prevents light penetrating into the ocean, without which the energy needed for the photosynthesis is not available. This can destroy the ecological cycle of the ocean and cause a huge damage to the ecosystem of an area of the ocean. The Fermilab physicist Arden Warner has proposed a potential application of ferrofluids that is by spreading a ferromagnetic particle into the oil film making the oil a type of ferrofluid which then can be cleaned up with a magnet [9]. Ferrofluids can also be used in some medical applications. Ferrofluids can be used in the treatment of retinal detachment. Retinal detachment is an eye disorder caused by various diseases, in which the retina peels off from its underlying support tissue. Detachment usually initiates from a tiny hole in the retina. Without rapid and effective treatment, the entire retina may detach, resulting in permanent blindness. By injecting a tiny droplet of ferrofluid into the vitreous humour and controlling it by an applied magnetic field, it can moved to the position of the hole. The ferrofluid droplet will seal the hole and cure the retinal detachment. The computational simulation of this application can be found in [4]. The ferrofluid can also be used in the field of the drugs delivery [115, 91]. Another class of non-conducting fluids is a paramagnetic fluid, which has an impor- tant potential industrial applications such as the non-contact orientation technique using magnetic levitation. The technique has been proposed by Subramaniam et al.[121]. This technique has shown a significant importance in the field of advanced manufacturing.

1.3. The coupling approach51.2.2 Conducting magneto-fluid applicationsThe study of coupled conducting magneto-fluid behaviour is also called Magnetohy-

drodynamics (MHD). Since it was initiated by the Nobel laureate Hannes Alfvén in

1942 [6], the field of MHD has grown rapidly and is now widely developed and used

in a variety of research fields such as geophysics, plasma, astrophysics, etc. Some important applications are stated below. MHD is also applied in the field astrophysics. MHD is used to simulate or explain the behaviour of the solar winds or other stellar winds and it has become a very important tool for detecting and forecasting the space weather. For example, the forecast of a solar storm blast will be very helpful to prepare for the sudden interruption of the radio wave communications. The forecast of space weather will become more and more important in the future, where MHD will play a major role [104]. The nuclear fusion reactor is a major application of MHD. The plasma inside the reactor is well described by the MHD equations. As the stock of the fossil fuels is consistently decreasing, the importance of alternative energy sources, such as nuclear energy, is gradually being recognised by the public. The need to optimise and design more efficient and safer reactors is becoming ever more important. There are also abundant number of other engineering applications. These include metal casting [68], in particular the production of aluminium [53] is topical and involves the application of MHD. The basic theory behind the Lorentz force velocimetry or flowmeter involves MHD[43, 81]. In many continuous metallurgical processes, due to the high temperature of the liquid metal, the non-contacting flowmeter has significant importance. From the listed applications of the coupled magneto-fluids phenomena for conduct- ing and non-conducting cases, we could observe the importance and the necessity to consider the coupled magneto-fluid problem.

1.3 The coupling approach

Real life physics is full of coupled problems. As a result, multi-physics coupling has become a hot topic in the past decade. Much work has been undertaken on different multi-physics couplings, which includes fluid-structure interaction, e.g. [64,56,34,

36,35,125,55], electro-solid interaction(piezoelectric elasticity), e.g. [16,108,102],

thermoelectric coupling, e.g. [101,100,99] and so on. The topic of magneto-fluid coupling is a branch of multi-physics coupling problems and its importance has been introduced in previous sections. There are several different approaches in which the magnetic field and the fluids

61. Introductionphenomena may be coupled together. The coupling could be introduced by body force,

through constitutive behaviour or through boundary terms. Here, note that when dealing with multi-phase problems, such as ice melting [23], the main coupling occurs at the interfaces. While for other applications the treatment of the body force is more relevant. The coupling mechanism generally consists of two way coupling, which is illustrated in Figure 1.1. The magnetic field will apply a magnetic body force (ponderomotive force) onto the fluid domain, which will affect the fluid flow. Through the induced strain rate, the fluid flow will alter the magnetic properties of the fluid domain, which influences the magnetic field distribution. Based on whether the fluid is conductive or not, the phenomena can be divided into two categories. The way in which the fluid Figure 1.1: Two way coupling mechanism for magnetic fields and fluids (The parameters are described in detail in Chapter 4) properties depend on the velocity and magnetic field are described by the constitutive laws for the fluid. There are several different expressions for this magnetic body force, which can be found in literature [44,48,49,120,84,77]. However, most of them can be sum- marized as saying that the magnetic body force consists of the Lorentz force, the magnetostrictive force and the dimagnetophoretic force [44]. The Lorenz force is due to magnetic induction, the dimagnetophoretic force occurs due to the spatial gradients of the permeability and the magnetostrictive force is associated with the deformation of the magnetised material. We remark that there are some variations in the literature as to explicit definitions and names given to each of these contributions [44,48,49,120,84]. For instance, in Stratton [120], the dimagnetophoretic and magnetostrictive forces are

1.4. Computational methodologies7combined and simply called magnetostriction. This is also the approach we follow in

this thesis. In many applications, simplifying assumptions are usually made [44,48,49,8,

95,106,105,30,120,69,74]. Indeed, the magnetostrictive effects in fluids have

not been investigated to the same extent as in solids, but new important applications begin to emerge [66]. Examples of magnetostrictive effects include dipolar fluids [18], polycrystalline iron films [128], cylindrical type II superconductors [75] and magneto- rheological elastomers [59]. The magnetostrictive effects in ferrofluids have also become important [66,63] and have also been found to be important when associated with spark erosion is used to produce them [24]. For MHD, the Lorentz force is often considered as the dominant term of the body force [95,30]. However, when considering problems with different fluid phases, the magnetostrictive contributions are still needed to be included to describe the coupling more fully. Magnetic constitutive laws are available in a large body of literature [120,69,74,58]. Most of them deal with the simplest isotropic homogeneous materials. For ferrofluids, the non-linearity is expressed by the non-linear expression of the permeability [78,7,

82,4,66,3]. However, the magnetostrictive effect was not taken into consideration.

Knops [80] has introduced the electrostriction into the constitutive relationship for solids by adding additional terms, which depend on the deformation, i.e. strains. In [57], Ledger and Gil have adopted the constitutive relationship to solve the electrostriction of an infinite dielectric plate with a rigid circular elliptical dielectric insert [80,117,118]. In our previous work [71] and this thesis, the same assumptions are made to develop the constitutive relationship, not only electrostriction but also magnetostriction with a detailed explanation.

1.4 Computational methodologies

In order to solve the partial differential equations (PDEs) governing an engineering problem(herecoupledmagneto-fluidproblem)weneedanumericalmethodtodiscretise and approximate the PDEs. There are several methods, such as the finite difference method, the finite volume method, the boundary elements method and so on, which are well known and widely used for engineering problems. The finite difference method is the simplest of the listed methods, the approximation of a boundary value problem involves a regular grid of points to discretise the unknown variables in the volume and introducing Taylor's series approximations of the derivatives in the PDEs [88]. The operations involved are often intuitive and many problems leads to very efficient schemes. This method is efficient and robust to be used in a variety of

81. Introductionproblems, and extensions to higher order approximations are possible, but limited to

rectangular geometries. It can not solve problems with complex geometries [62,21,27] without staircasing of the grid. The finite volume method adds greater geometric flexibility over the finite difference method. Both methodologies can be established from variational statement by choosing different weighting functions. The schemes will turn into finite difference method when applied to linear problems, but without the grid restriction. The finite volume method is not suitable for the higher order reconstruction [62,22,127,89]. The finite volume method is widely used in the field of industrial CFD codes [133,27].To solve the fluid problem in such cases, stabilisation terms must be added [90, 46]. The boundary element method is popular for potential problems. It offers con- siderable advantage for acoustic and wave propagation problems set on unbounded domains [51,109,25]. As the discretisation is done on the boundaries, this avoids truncation and approximation of unbounded domains, as required for other methods. However, the boundary element method leads to fully populated matrices and their inversion and storage is prohibitive for realistic three dimensional problems [62], which will increase the computational cost considerably. The finite element method overcomes the shortcomings of the finite volume and finite difference methods [67,136,135,134]. It can be applied to complex geometries and extensions to higher order approximations are possible. Although the classical finite element methods based on low degree Lagrange interpolation polynomials often require dense meshes (h-refinement) in order to converge to accurate solutions [136].The finite element method leads to sparse matrices and is amenable to rapid inversion of the linear system. For smooth problems, it is known that increasing the polynomial degree (p-refinement) is more beneficial than performing mesh orh-refinement leading to exponential instead of algebraic convergence [124]. Problems with piecewise analytic functions and singularities due to sharp corners or edges, problems in elasticity, which exhibit locking, and problems with boundary layers are suitable for combiningh andprefinement to obtain the exponential convergence. A large body of literature is available for illustrating the advantages of thehprefinement over standard finite element techniques [71, 87, 79, 57, 31, 32, 114]. As an alternative to the classic continuous Galerkin finite element methods, discon- tinuous Galerkin method has been recently developed. This method differs from the continuous finite element method in that, it will duplicate degrees of freedom and the flux constraints will be applied on the edges. The advantage of this method is that it al- lows local mesh refinements and provides simplicity basis functions. The disadvantage over continuous methods is it could increase the total degrees of freedom [62]. For this

1.4. Computational methodologies9reason, we focus on the continuous version of the finite element method.

In order thatprefinement can be performed efficiently, hierarchic basis functionsshould be used instead of the standard Lagrange finite element family, because the

high order hierarchic basis functions are additive [134]. There are many different choices for the hierarchic basis functions with the lowest order always corresponding to simple finite element hat functions associated with the lowest order Lagrange finite element family. When choosing a finite element basis, it is important to bear in mind the condition number of the resulting stiffness and mass matrices as well as their ease of implementation. Some possible high order finite element basis functions include those described in [79, 5, 31, 32, 124, 131, 112]. A further important aspect in this work is the need to select the suitable FEM spaces for the different physical variables in the PDEs that make up the multi-physics problem. The complexity of the electromagnetic field and the fluid velocity and pressure means that it is necessary to use a range of finite element basis functions other than classical H1( )conforming finite element basis. In this context we shall requireL2( );H1( ) andH(curl; )conforming finite element basis functions. The basis functions prepared in [131,112] satisfy all the proposed requirements and they are also hierarchic basis functions we shall use throughout this thesis. The multi-physics coupled system result in a set of non-linear coupled PDEs, which should be solved. There are a number of different iterative algorithmic approaches that can be developed for their solution [38]. The Picard iteration leads to a simple implementation, but the convergence is linear [113]. The Newton-Raphson iteration requires consistent linearisation of the equations [17,34]. It is suitable for large non- linear systems and provides quadratic convergence of the residual. There also exists other approaches such as Gauss-Seidel approach [76,130]. For the computational approach for the magneto-mechanical problems, Kaltenbacher has provided several other approaches such as moving mesh method, coupled FE-BE method and alternative approaches for coupling the electromagnetic force term [77]. ThehpFEMhas beenappliedto awide class ofproblems, such asmechanics [124], fluids [79,114], elecromagnetics [31,32,86]. However, with the exception of the work of Gil and Ledger [57], which considers electrostriction in mechanics, very little has been done for coupled problems. This therefore motivates our work to apply the hp-FEM to solve the coupled non-linear magneto-fluid problem.

101. Introduction1.5 ObjectivesThe objective of this work is to develop a novelhp-finite element framework for the

fully coupled magneto-fluid problem including magnetostrictive effects. In particular, the general coupling mechanism will be adopted to describe conducting and non- conducting magnetic fluids. For computational efficiency, the consistent linearisation via Newton-Raphson strategy has been applied. The high orderhp-FEM has been adopted to ensure the accuracy of the solution. The framework will be implemented in a vectorized manner in two- and three-dimensions with the MATLAB software. The implementation will be benchmarked and applied to solve the example problems in order to give a better investigation of the coupled magneto-fluids phenomena. The magnetostrictive effect will also be investigated. In order to achieve this objective, the research is separated into different stages, which will be described in the outline of the thesis.

1.6 Outline of the thesis

This thesis is comprised of five parts and ten chapters. The main body of the work is contained in Parts II to IV, conclusions and further work are discussed in Part V. The detailed outline is given below.  Part I consists of Chapter 1 entitled "Introduction". This chapter provides a brief background, the objective and the scope of this thesis.  Part II consists of Chapters 2 to 3. This part presents the underlying physical laws forthefluidandelectromagnetism. Chapter2entitled"Aspectoffluidmechanics", derives the boundary value problem for the fluid mechanics from the basic theories andconservationlawsoffluids. Chapter3entitled"AspectsofElectromagnetism", derives the boundary value problem from the basic electromagnetic laws. These chapters form the preparation for the basic fundamentals of the following chapters. Part III consists of Chapters 4 to 6. This part is focused on the development of thehp-FEM framework for the non-conducting fluid with the magnetostrictive effects. Chapter 4 is entitled "Coupling Approach". The aim of this chapter is to provide the pre-requisites that are required for the implementation of the coupled schemes for non-conducting magnetostrictive fluids. The coupling mechanism between magnetic fields and fluids including magnetostrictive effect are fully discussed

1.6. Outline of the thesis11first. Then, once the coupling mechanisms and the underlying basic governing

equations have been summarised, which were already introduced in previous chapters for electromagnetic and fluid, the fully coupled governing equations for conservative and non-conservative magnetostrictive fluids are established. The non-linear schemes are monotonically linearised to obtain the corresponding Newton-Raphson schemes for conservative, non-conservative and ferrofluids by applying the directional derivative technique. In particular, the novel contribu- tions for this chapter are: first, introducing the expression of the pondermotive force as the divergence of stress tensor for describing the force exerted by the magnetic field; second, the constitutive relation between magnetic flux intensity and magnetic field intensity (i.e. the tensor form of magnetic permeability); third, extending the previous coupled schemes to include the magnetostrictive effects, which have until now been neglected. Chapter 5 is entitled "Numerical Implementation". The purpose of this chap- ter is to numerically implement ahp-finite element descretisation scheme in MATLAB. Firstly, the linearised schemes are discretised for conservative, non- conservative fluids and ferrofluids with magnetostrictive effect using the high order basis functions space. Then, the computational implementation of the linear system of equations is fully discussed by providing the numbering method, static condensation and assembly procedure. Finally, the techniques of vectorised implementation and the algorithm for the solution of the linearisation scheme are briefly presented. In particular, the novel contributions of this chapter are the num- bering scheme that allows the solution of coupled multi-physics problems using hp-finite elements and our new vectorised MATLAB implementation. The high order basis functions are visualised up to order 3 (i.e. the vertex, edge and interior basis functions). The matrix partitioning for the magnetic-fluid coupled system is provided and the speed comparisons between the new vectorised MATLAB code and Loop-based code are carried out. Chapter 6 is entitled "Numerical Examples". This chapter presents the results of the benchmark problems to verify the accuracy of the implementation as well as the convergence behaviour of the scheme for non-conducting fluids. The benchmarked implementation is also applied to several original examples to investigate the coupling behaviour of the magnetostrictive non-conducting fluids under the existence of magnetic field. In particular, the novel contributions of this chapter are the novel problems, which allow us to not only investigate the coupling behaviour between magnetic field and magnetostrictive fluids, but also the behaviour of multiphase magnetostrictive fluids.

121. IntroductionPart IV consists of Chapters 7 to 9. This part is focused on the development of the

hp-FEM framework for the conducting fluid (i.e. MHD) with the magnetostrictive effects. Chapter 7 is entitled "Derivation and Framework". This chapter is the analogue to Chapter 4. The coupling mechanisms including the mangetostrictive effects, which will be introduced in Chapter 4, are also adopted to derive the MHD scheme for simple conducting fluids and conducting fluids with magnetostrictive effects. The fully coupled system is again monotonically linearized via consistent Newton-Raphson linearization. In particular, the novel contribution of this chapter is that the magentostrictive effect is introduced into the MHD system, which to the best of our knowledge, is the first time this has been accomplished. The coupling mechanism allows the permeability and ponderomotive force to be expressed in the form of a tensor and the divergence of a stress tensor, respectively. The additional non-linearity introduced by the new coupling mechanism is linearised via consistent Newton-Raphson linearisation with directional derivatives. Chapter 8 is entitled "Numerical Implementation". This chapter is the analogue to Chapter 5, but mainly deals with the implementation for the MHD scheme for conducting fluids. This chapter will be focused on the important differences from the previous chapter. TheH(curl; )space is introduced for magnetic field intensity in two-dimensions as well as three-dimensions, although only the two-dimensionalH(curl; )conforming basis functions are visualized. Then, the computational implementation of the linear system of equations is fully discussed by providing the numbering method, static condensation and assembly procedure. Note that, here all the implementation is focused on three-dimensional case. In particular, the novel contributions of this chapter are the numbering scheme that allows the solution of coupled MHD problems usinghp-finite elements and our new vectorised MATLAB implementation in two- and three-dimensions. Chapter 9 is entitled "Numerical Examples". This chapter presents the results of the benchmark problems for MHD to verify the accuracy of the implementation as well as the convergence behaviour of the scheme. The benchmarked imple- mentation is also applied to several originally setup examples to investigate the coupling behaviour of the magnetostrictive conducting fluids under the existence of a magnetic field. In particular, the novel contributions of this chapter are the novel problems which allow us to not only investigate the coupling behaviour between magnetic field and conducting fluids with magnetostrictive effects, but also the behaviour of multiphase Magnetohydrodynamics with magnetostrictive effects.

1.7. Publications13Part V consists of Chapter 10 entitled "Conclusions and Further Work". This chap-

ter aims to draw conclusions of the whole thesis and provides several suggestions for the further work.

1.7 Publications

Here we present outputs of our work.

1.7.1 Journal publications

D. Jin , P.D. Ledger, A.J. Gil, "hp-Finite element solut