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Spectral/^ Methods on Polymorphic

Multi-Domains: Algorithms and Applications

by

Tim othy Warburton

B.A., Oxford University, 1993 Sc.M., Brown University, 1994 M.A. Oxford University, 1998

Thesis

Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy

in the Division of Applied Mathematics at Brown University

May 1999


UMI Number: 9932499

UMI Microform 9932499 Copyright 1999, by UMI Company. All rights reserved.

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© Copyright

by

Timothy W arburton

1999


This dissertation by Tim othy Warburton is accepted in its present form by the Division of Applied Mathematics as satisfying the

dissertation requirement for the degree of Doctor of Philosophy

Date \ I f ^1 1 &______ G -*—"George Em Karniadakis, Director

Recommended to the Graduate Council

Date J - j - Z z k #

Date 7 ^ , 7 f ? S'

Approved by the Graduate Council

D a t e VWH 7 / ] .Sd-Peder J. Estrup Dean of the Graduate School and Research

ISUa SjlZ 'Spencer Sherwin, Reader

Chi-Wang Shu, Reader


The V ita of Timothy Warburton

Bom 5 October, 1971, Ipswich, United Kingdom

E ducation

B.A., University of Oxford, 1993

Sc.M, Brown University, 1994

M.A., University of Oxford, 1998

P u blication s

• T.C. W arburton, S.J. Sherwin, and G.E. Karniadakis, Spectral Basis Functions

for 2D Hybrid hp Elements, Submitted to SIAM J. Num. Anal., (1997).

• T.C. W arburton, I. Lomtev, Y. Du, S.J. Sherwin and G.E. Karniadakis, Galerkin

and Discontinuous Galerkin Spectral/hp Methods, CMAME Spectral, Spectral

Element and hp Methods in CFD, 1998.

M eetin gs and C onferences

• G.E. Karniadakis, T.C. W arburton and I. Lomtev, Spectral/hp Element Meth

ods on Unstructured Meshes, Fourth U.S. National Congress on Computational

Mechanics. August 6-8, 1997, San Francisco, California.

• T.C. W arburton and G.E. Karnidakis, Discontinuous Galerkin Methods for

MHD, 1998 SIAM Annual Meeting Ninth SIAM Conference on Discrete Math-iii


ematics.

• T.C. W arburton, I. Lomtev, R.M. Kirby and G.E. Karniadakis, A Discontinu

ous Galerkin Method for the Compressible Navier-Stokes Equations on Hybrid

Grids. Tenth Int. Conf. on Finite Elements in Fluids, January 5-8, 1998, Tuc

son, Arizona.

• S.J. Sherwin, T.C.E. Warburton and G.E. Karniadakis, Spectral/hp Methods for

Elliptic Problems on Hybrid Grids, Tenth Int. Conf. on Domain Decomposition

Methods, August 10-14, 1997 Boulder, Colorado.

• T.C. W arburton and G.E. Karniadakis, Spectral Simulations o f Flow Past a

Cylinder Close to a Free Surface, ASME paper FEDSM97-3389.

• T.C. W arburton and G.E. Karniadakis The Wake of an Oscillating Cylinder

Close to a Free Surface, APS 49th Annual Meeting of the Division of Fluid

Dynamics, Syracuse, NY 24-26 November 1996.

• T.C. W arburton, S.J. Sherwin and G.E. Karniadakis, Hierarchical refinement

using spectral/hp triangles and prisms, Sixth Int. Conf. on CFD, September

4-8, 1995, Lake Taho, Nevada.

• T.C. W arburton, S.J. Sherwin and G.E. Karniadakis, Unstructured hp/spectral

elements: Connectivity and optimal ordering, Int. Conf. on Comp. Engin.

Science, Ju ly 30-August 3, 1995, Mauna Lani, Hawaii.

iv


A cknowledgm ents

In a large scale project it is impossible to carry on a lone quest and work in glorious

isolation. I therefore owe a debt of thanks to many people who contributed ideas,

thoughts, technology and most valuably criticism.

Professor George Karniadakis has been a very understanding mentor, helping

with his keen physical insight and providing many good ideas to keep the A /s k T ocr

project growing and developing. In some ways it was Dr. Spencer Sherwin who

gave me the greatest initial boost by bringing me into the ways of simulation science

with the original M etzT o c r code, and I will always be thankful for his insistance on

structured thought and code.

When I moved from developing the incompressible package, Igor Lomtev provided

a great source of information and help. His methods are a credit to him and he has

made my progress speed along.

M e n T o c r would not have taken on a life and style of its own if it had not received

so much challenging attention from the CRUNCH group. Dr. James Trujillo and Dr.

Ali Beskok have both been incredibly helpful with their expertise in fluids and their

garbage detectors. Dr. Ma Xia and Constantions Evangelinos have both pushed

and tested the code to great limits, which I initially thought were beyond its scope.

Robert (Mike) Kirby recently joined CRUNCH, yet in the last nine months he has

helped so much with mesh technology that I cannot see how we ever managed without

his spirited assistance. I feel tha t we have all benefited from cooperation and I hope

J\f s k T ocr matures with the next generation of CRUNCH.v


Dr. Chi-Wang Shu and Dr. Jan Hesthaven have been very helpful answering my

( sometimes silly ) questions with patience and u n d e rs ta n d in g . They and the rest of

the faculty of Applied M ath have been so accommodating that I would like to thank

them all.

The N e k T cscr code does not stand on its own. It has been boosted by the support

of the METIS domain decomposition package[l], Henry Tufo’s GS communications

package [2], NASA’s FELISA mesh generation package [3] and Dr Tim B arth’s SIM

PLEX mesh generation package [4]. I cannot stress the importance of these codes

enough, they have been vital for rapid development.

This work was funded under the following grants:

• AFOSR: F49620-97-1-0185

• DOE: DE-FG02-95ER25239

• ONR: N000014-95-1-0256

• NSF: CTS-9417520

vi


C ontents

1 In trod u ction 1

1.1 O b je c tiv e s ........................................................................................................ 7

1.2 O u tlin e ............................................................................................................... 8

2 T h e E lem en ts 10

2.1 The T riangle..................................................................................................... 11

2.2 T he Q uadrila teral........................................................................................... 13

2.3 T he T e tra h e d ro n ........................................................................................... 15

2.4 T he P y r a m id ................................................................................................. 17

2.5 The P r i s m ........................................................................................................ 18

2.6 The Hexahedron ........................................................................................... 20

2.7 Global A sse m b ly ........................................................................................... 20

2.7.1 Algorithm 1 ......................................................................................... 23

2.7.2 Algorithm 2 ......................................................................................... 25

2.7.3 Algorithm For Connecting Prisms And Tetrahedra ................ 27

3 B asis F unctions 29

vii


3.1 One-Dimensional Orthogonal P o ly n o m ia ls ................................................. 29

3.2 Singular Sturm-Liouville Problems on a Simplex in d-Dimensions . . 31

3.3 Orthogonal (Modal) B a s e s ............................................................................. 34

3.3.1 The Orthogonal Triangle Basis ........................................................ 37

3.3.2 The Orthogonal Quadrilateral B asis .................................................. 38

3.3.3 The Orthogonal Prism B a s is .............................................................. 38

3.3.4 The Orthogonal Hexahedral B a s is ..................................................... 39

3.3.5 The Decay of Basis C oeffic ien ts....................................................... 40

3.4 Semi-Orthogonal B a se s ...................................................................................... 41

3.4.1 Modal b a s i s ........................................................................................... 42

3.4.2 Nodal and Mixed Bases .................................................................... 51

3.5 Comparison of the B a s e s .................................................................................. 56

4 Operators 62

4.1 Elemental Spatial O p e ra to rs ........................................................................... 62

4.1.1 Integration ........................................................................................... 63

4.1.2 Flux In teg ra ls ........................................................................................ 64

4.1.3 Differentiation....................................................................................... 65

4.1.4 Implementation and P ro g ra m m in g ................................................ 67

4.2 Global Projection O p e r a t o r ........................................................................... 71

4.2.1 Global Convergence in Skew E le m e n ts .......................................... 72

4.3 Convective O p e ra to r ........................................................................................ 74

4.3.1 Accuracy of the Convective O p e ra to r ............................................. 78

viii


4.3.2 Spectrum of the Galerkin 2D Convective O perator.................... 79

4.3.3 Three-Dimensional Galerkin Convective Operator Spectrums . 83

4.3.4 Spectrum of the Galerkin 3D Convective O perator.................... 85

4.4 Diffusion O p e r a to r ......................................................................................... 88

4.4.1 The Schur Complement M ethod of Inverting Global Operators 89

4.4.2 Optimal Numbering for the Schur Complement Boundary M atrix 90

4.4.3 Numerical Results ............................................................................. 92

4.4.4 Convergence for the Galerkin Helmholtz O p e ra to r ...................... 96

5 Incom pressible F low Sim ulations 100

5.1 Incompressible Navier-Stokes E q u a t io n ..................................................... 101

5.1.1 Form ulation .......................................................................................... 101

5.1.2 Summary of Scheme, Boundary Conditions and Implementation 101

5.1.3 Wannier F lo w ...................................................................................... 103

5.1.4 Kovasznay F lo w ................................................................................... 105

5.1.5 Cylinder F lo w ...................................................................................... 106

5.1.6 The R obo tuna ...................................................................................... 110

5.2 ALE Incompressible Navier-Stokes ........................................................... 115

5.2.1 Form ulation .......................................................................................... 118

5.2.2 Temporal D isc re tiza tio n ................................................................... 119

5.2.3 Rayleigh Sliding Plate P r o b le m ..................................................... 121

5.2.4 Design of a Micro-Pump via S im ulation ........................................ 124

5.2.5 Micro-Pump Geometric Specifications........................................... 125

ix


5.2.6 Efficiency A n a ly s is ............................................................................ 128

5.2.7 Numerical S im ulation......................................................................... 129

6 Incom pressible V iscous M agnetohydrodynam ics 138

6.1 F orm ula tion ...................................................................................................... 139

6.2 Numerical Scheme ......................................................................................... 141

6.3 Magnetic Pearson’s V o rte x ............................................................................ 142

6.4 Orszag-Tang V ortex......................................................................................... 145

6.5 Cylinder F lo w ................................................................................................... 147

7 C om pressible F low Sim ulations 152

7.1 Numerical F orm ulation ................................................................................... 153

7.1.1 Discontinuous Galerkin for D if fu s io n ............................................ 154

7.2 Compressible Navier-Stokes Simulations ................................................... 156

7.2.1 Convergence......................................................................................... 158

7.2.2 Cylinder F lo w ..................................... 158

7.2.3 Flow Past a Two-Dimensional NACA0012 A ir f o i l ....................... 164

7.2.4 Flow Past a Multi-Body A i r f o i l ..................................................... 165

7.2.5 Flow Past A NACA0012 W ing W ith E n d p la te s ........................ 171

8 C om pressible V iscous M agnetohydrodynam ics Sim ulations 174

8.1 F orm ula tion ..................................................................................................... 175

8.2 The V • B = 0 Constraint ........................................................................... 176

8.3 Implementation of the Euler Flux T e r m s .................................................. 178

x


8.4 Implementation of the Viscous T e rm s ....................................................... 183

8.5 Summary of the Compressible MHD Code Im p lem en ta tio n ................ 183

8.6 Two-Dimensional M agnetohydrostatic E x a m p le ................................... 185

8.7 Three-Dimensional M agnetohydrostatic Example ................................. 186

8.8 Compressible Orszag-Tang V o rte x .............................................................. 190

8.9 Flow Past a C y lin d e r..................................................................................... 197

9 Sum m ary 200

A Continuous E xpansion B asis 204

A .l Jacobi P o lynom ials........................................................................................ 204

A.2 Modal B a s is ..................................................................................................... 205

A.2.1 Two-Dimensional E x p a n s io n s ........................................................ 205

A.2.2 Three-Dimensional Expansions .............................................. 210

A.3 Nodal B a s i s ..................................................................................................... 218

A.3.1 Two-Dimensional E x p a n s io n s ........................................................ 218

A.3.2 Three-Dimensional Expansions .............................................. 220

xi


List o f Tables

4.1 G LL implies Gauss Lobatto Legendre which is the Gauss quadrature

for a constant weight function with both x = ±1 points endpoints

included. G R Ja,p implies Gauss Radau Jacobi quadrature with (a, /?)

weights and the endpoint x = — 1 included................................................. 64

4.2 Timings for the Schur complement obtained on a single node SGI R8000. 93

5.1 Simulation param eters for the Kovasznay flow........................................... 106

5.2 Simulation param eters for the incompressible cylinder flow simulation. 109

5.3 Variation of Strouhal frequency with Reynolds number for incompress

ible flow past a two-dimensional c y lin d e r ................................................. 109

5.4 Variation of Cd with Reynolds number for incompressible flow past a

two-dimensional cylinder ........................................................................... 110

5.5 Simulation parameters for the Rayleigh problem....................................... 122

5.6 Simulation parameters for the micro-pump simulation............................ 131


5.7 Volumetric flow rate per unit width W as a function of Reynolds num

ber for closed-position piston-wall gap of g = 0.025L. The value for

v ~ 1 x 10-5 for both air and water are used. T he mass flow rate is

~M = pQ ..................................................................................................................................

6.1 Param eters for the equations of incompressible MHD..............................

6.2 Simulation parameters for the incompressible MHD Pearson’s vortex

sim ulation...........................................................................................................

6.3 Simulation parameters for the incompressible MHD Orszag-Tang vor

tex simulation....................................................................................................

6.4 Simulation parameters for incompressible MHD flow past a cylinder. .

7.1 Simulation parameters for compressible flow past a cylinder..................

7.2 Comparison of Strouhal frequency of incompressible against compress

ible (Mach=0.5) flow past a cylinder............................................................

7.3 Comparison of Strouhal frequency, drag and lift coefficients from hy

brid A fsK 'Tccr with results from other methods. Re=100, M=0.7 . .

7.4 Simulation parameters for compressible flow past a NACA0012. . . .

7.5 Simulation parameters for compressible flow past a m ultibody airfoil.

7.6 Simulation parameters for compressible flow past a NACA0012 airfoil

with endplates....................................................................................................

8.1 Variables and parameters used in the equations of compressible MHD.

8.2 Simulation parameters for the compressible Orszag-Tang Vortex. . . .

xiii

133

140

144

146

147

161

161

162

165

168

171

177

191


8.3 Simulation parameters for compressible flow past a cylinder with a

magnetic field. ............................................................................................... 197

xiv


List o f F igures

1.1 Top: Unstructured prism mesh, Bottom: p-type convergence for the

Helmholtz problem with A = 1 and solution sin(y)sin(z). The face

geometry was downloaded from the web-site: http://www.3dcafe.com/ 3

1.2 Hybrid discretization around a multi-element airfoil. Only part of the

domain is shown................................................................................................ 6

1.3 Hierarchy of the M en T 'o cr code .............................................................. 9

2.1 The tensor coordinates of the triangle.................................................. 12

2.2 The tensor coordinates of the quadrilateral......................................... 14

2.3 The tensor coordinates of the tetrahedron........................................... 16

2.4 The tensor coordinates of the pyramid................................................ 18

2.5 The tensor coordinates of the prism..................................................... 19

2.6 The tensor coordinates of the hexahedron........................................... 21

2.7 Diagram of the local top and base vertex of the tetrahedron.................. 22

xv


2.8 Setting up the required connectivity for the discretization of a box as

shown in a ). Vertex A is given as the first local top vertex as shown in

b). In c) vertex B is then given as the local base vertex and the local

base vertices from group one are aligned to satisfy connectivity and the

final element orientation is shown................................................................ 24

2.9 Connectivity summary. Solid arrows imply we can connect elements

together with no extra mesh refinement, dashed arrows mean that there

are some constraints on allowable configurations...................................... 28

3.1 Mode shapes for the triangle modal basis (T rH H ) with N = 5. . . . 44

3.2 Top: Geometric ordering for modes, Bottom: Stiffness m atrix (V&, V<f)j)

for the continuous basis for triangles (N = 15) with the geometric or

dering of the rows and columns.................................................................... 46

3.3 Top: Polynomial degree ordering for modes, Bottom: Stiffness matrix

(V0i, V(j>j) for the continuous basis for triangles (N = 15) with modes

ordered by polynomial order......................................................................... 47

3.4 Mode shapes for the quadrilateral modal basis (Q uH H ) with N = 5. 48

3.5 Mode shapes for the quadrilateral mixed modal basis Q u H N with N = 5. 50

3.6 Mass matrix for the quadrilateral mixed modal basis (Q uH N ) with

N = 15............................................................................................................... 51

3.7 Mode shapes for the quadrilateral nodal basis (Q uN N ) with N = 5. . 53

3.8 Mode shapes for the triangle mixed basis (T rN H ) with N = 5............. 57

xvi


3.9 Comparison of the elemental mass m atrix for (a) TrHH, (b) TrNH

using collocation, and (c) TrN H using exact integration for N = 15. 58

3.10 Comparison of the elemental stiffness m atrix for (a) TrHH, and (b)

T rN H bases for N = 15.................................................................................. 59

4.1 A piece of Jsf e tiT o c r code demonstrating how a list of elements can

be constructed.................................................................................................... 69

4.2 A piece of A fen 'T 'o tr code demonstrating how operations can be made

on a list of elements.......................................................................................... 70

4.3 Meshes (A-H) are consisted of three quadrilaterals and two triangles

which are progressively skewed by shifting the interior vertex............... 72

4.4 Convergence in the L2 norm for modal projection of the function u =

sin(rcx)sin{'Ky) on meshes A — H ................................................................. 72

4.5 Convergence in the L2 norm for mixed projection of the function u =

sin(irx)sin(7ry) on meshes A — H ................................................................. 73

4.6 Illustration of local and global numbering for a domain containing one

quadrilateral and one triangular elements. Here the expansion order is

N = 3, and we only show the boundary modes.......................................... 76

4.7 Z-matrix m ap from global to local degrees of f re e d o m ......................... 77

4.8 Exponential accuracy is achieved for the wave equation with u =

sin(7rcos(xx)) as initial condition................................................................. 78

4.9 Spectral radius of the weak convective operator on a periodic domain,

N = 12................................................................................................................. 79

xvii


4.10 Growth of the spectral radius of the Galerkin convective operator with

expansion order.................................................................................................. 80

4.11 pu{6) is an exact fit for the spectral radius of the Galerkin convective

operator on a periodic domain tiled with regular quadrilaterals and

w ith N = 12....................................................................................................... 81

4.12 p&.(9) is a close fit for the spectral radius of the Galerkin convective

operator on a periodic domain tiled with triangles and with N = 12. . 81

4.13 Spectral radius of the Galerkin convective operator on a periodic do

main discretized with non-regular elements................................................ 82

4.14 Left: Exploded spectral element mesh, Right: Variation of the spectral

radius of the convective operator in a periodic box discretized with six

tetrahedra and fourth-order expansion......................................................... 86


radius of the convective operator in a periodic box discretized with six

pyramids and fourth-order expansion........................................................... 87


radius of the convective operator in a periodic box discretized with two

prisms and fourth-order expansion................................................................ 87


radius of the convective operator in a periodic box discretized with

eight hexahedra and fourth-order expansion............................................... 88

xviii


4.18 Form of the global operator m atrix. Notice the sparsity of A and the

block nature of B and C.................................................................................. 90

4.19 Here we show the domain tha t we solved the Helmholtz equation in the

following forcing function, / = — (3.47T2) sin (7r.x)cos(jry) cos (0.27r.z)

and boundary conditions u = sin (ir.x)cos(7r.y) cos with pe

riodic boundary conditions at the spiral ends. We see a dramatic

bandwidth reduction because of the aspect ratio of the mesh and the

exponential decay in the error with increasing expansion order............. 92

4.20 The exponential decay in the error with increasing expansion order for

the Helmholtz equation solved on the spiral domain................................ 94

4.21 Convergence test for the H elm h o ltz equation using quadrilaterals and

triangles, with Dirichlet boundary conditions. The exact solution is

u = sin(irx)cos(iry) and forcing function / = —(A + 2ir2)sin(Trx)cos(iTy) . 95

4.22 Convergence test for the Helmholtz, using a triangle and a quadrilat

eral, with Dirichlet boundary conditions: u = sm (7rcos(7rr2))............... 96

4.23 Convergence for the Helmholtz problem on a mesh of hybrid elements.

Dirichlet boundary conditions u = sin (x)s in (y)sin (z) and A = 10000. 97

4.24 Convergence is guaranteed for the Helmholtz problem even on very

skewed tetrahedra. Dirichlet boundary conditions u = s in ( y ) s m ( ^ ) s m ( y )

and A = 1........................................................................................................... 99

xix


5.1 Discretized solution domain for the Wannier-Stokes flow using 68 ele

ments. Left: Graph of exponential spatial convergence, Top Right: Hy

brid spectral element mesh, Bottom Right: Streamlines of the steady

state Stokes flow............................................................................................... 104

5.2 Steady state solution for the Kovasznay flow a t Reynolds number Re =

40 using a discretization using K = 24 elements. Left: The spectral

element mesh used, Right: Steady sta te streamlines................................ 105

5.3 Convergence in the L00 and Hi norms as a function of expansion order

for the steady state Kovasznay flow a t a Reynolds number Re = 40. . 107

5.4 Top Left: Three-dimensional spectral element mesh used to solve Ko

vasznay flow on, Top Right: Iso-contours of streamwise component of

velocity, Bottom: Convergence to exact solution with increasing order. 108

5.5 Domain and mesh for the two-dimensional incompressible flow past a

circular cylinder simulations................................................................. I l l

5.6 Instantaneous contours of vorticity for incompressible flow past a two-

dimensional cylinder at Reynolds numbers Re = 50,100 and 150. . . . 112

5.7 Instantaneous contours of vorticity for incompressible flow past a two-

dimensional cylinder at Reynolds numbers Re = 200 and 250..... 113

5.8 Top: The faces of the tetrahedra a t the nose of the fish. Bottom: The

spline surface on the body of the Robotuna..................................... 116

5.9 Robotuna simulation. Re=1000 based on height of tuna. N=5. Iso

contours of pressure are shown. Flow is from left to right 117

xx


5.10 Top: Spectral element mesh. K th = 50. Lower Left: Time convergence

history for zero initial conditions N = 5. Lower Right: Convergence

to exact solution with increasing N .............................................................. 123

5.11 Sketch of the micro-pump operating between two micro-channel sys

tems. Inlet and exit valves open and close periodicly with maximum

gap of Qrnax = 0.125L and minimum gap of gmin = 0.025L..........................125

5.12 Top: Deflection of the membrane y(x, t) = asiD.(nx/L) sin(o;£) Bottom:

Loci of the valve tips y (t) = ± ta n h (4 c o s (u ;t) ) ........................................ 126

5.13 Spectral element mesh used for the discretization of the micro-pump

system a t ejection-stage (top, membrane is moving up). The discretiza

tion of the flow domain during suction stage is shown a t the bottom

(membrane is moving down). The bottom figure also shows elemental

discretization obtained by 7th order modal expa n s ions used in M eKl'ar.1Z0

5.14 Non-dimensional volumetric flow rate variation w ith in a period of the

micro-pump, as a function of Reynolds number, R e = 9l l (a /L = 1/10,

a /h = 1 /3 ).......................................................................................................... 131

xxi


5.15 Vorticity contours for Re = 30 simulation. Top figure at t u j = 0.28,

corresponds to the beginning of the suction stage. Start-up vortices

due to the motion of the inlet valve can be identified. Middle figure

is a t t u j = 0.72, corresponding to the end of the suction stage. A

vortex jet pair is visible in the pump cavity. Bottom figure t u j = 84,

corresponding to early ejection stage. Further evolution of the vortex

je t and the start-up vortex of the exit valve can be identified...... 134

5.16 Close up of the vorticity contours for Re = 30 simulation at the left

valve (meshes shown on right side). Top: t u j = 0.28, corresponds to

the beginning of the suction stage. Start-up vortices due to the motion

of the inlet valve can be identified. Middle: t u j = 0.72, corresponding

to the end of the suction stage. A Vortex jet pair is visible in the

pump cavity. Bottom: t u j = 84, corresponding to early ejection stage.

Further evolution of the vortex je t and the start-up vortex of the exit

valve can be identified............................................................................. 135

5.17 Three-dimensional micropump simulation Re = 3. Top: An instant

during the suction stage. Bottom: An instant during the ejection stage. 137

6.1 Incompressible magnetic Pearson’s vortex (t = 2, instantaneous fields).

Top: Periodic spectral element mesh, Bottom Left: Pressure field,

Bottom Right: Magnetic streamfunction............................................ 143

xxii


6.2 Incompressible magnetic Pearson’s vortex. Convergence plot for L2

error in x-component of magnetic field at time t = 2 versus expansion

order.................................................................................................................... 144

6.3 Incompressible magnetic Pearson’s vortex. Time accuracy plot for the

simulation run with N = 10. Convergence plot for L2 error in x-

component of velocity a t time t = 1 versus time step A t ............................. 145

6.4 Incompressible Orszag-Tang vortex ( t= l, instantaneous fields). Top

Left: x component of velocity, Top Right: y component of velocity,

Middle: Pressure, Bottom Left: x component of magnetic field, Bottom

Right: y component of magnetic field.......................................................... 148

6.5 Incompressible Orszag-Tang vortex ( t= l, instantaneous fields). Top

Left: Velocity streamlines, Top Right: Magnetic streamlines, Middle

Left: Vorticity, Middle right: Divergence of velocity, Bottom Left: Curl

of magnetic field, Bottom Right: Divergence of magnetic field.................... 149

6.6 Incompressible flow past a cylinder with inflow magnetic fields. From

the top: (1) x component of the velocity field, (2) y component of the

velocity field, (3) x component of the magnetic field, (4) y component

of the magnetic f i e l d ..................................................................................... 150

6.7 Incompressible flow past a cylinder with inflow magnetic fields aligned

with inflow velocity. Top: Pressure field, Middle: Vorticity, Lower:

Stream function of the magnetic f i e ld ....................................................... 151

xxiii


7.1 Density contours obtained on a hybrid grid for an inviscid M = 0.3 flow

(Top), on a triangle grid (Middle). The bottom plot shows exponential

convergence of the error for the unstructured (triangles) and the hybrid

(squares) grid...................................................................................................... 159

7.2 Inviscid M = 0.3 flow past a bump in a three-dimensional (periodic in

the spanwise direction) domain. From the top: (1) domain, (2) spectral

element mesh used in the convergence test K = 120, (3) Iso-Contours

of density and (4) convergence of entropy to zero with increasing order. 160

7.3 Instantaneous iso-contours for the simulation of compressible flow past

a cylinder {Re — 100, M = 0.5) From the top: (1) density, (2) pressure

field, (3) x component of the velocity field, (4) y component of the

velocity f ie ld ...................................................................................................... 163

7.4 Instantaneous iso-contours of vorticity for the simulation of compress

ible flow past a cylinder {Re = 100, M — 0 .5 ) ........................................... 164

7.5 Top: Mesh of full domain for simulation of compressible, Mach 0.5,

Re=10,000 flow past a NACA 0012 airfoil at zero angle of attack to

the m ainstream flow, Middle: Mesh around body and wake, Bottom

Left: Close up of the airfoil, Bottom Right: Close up of part of the

wake region......................................................................................................... 166

xxiv


7.6 The wake region (from f = 2 to f = 5) of a NACA 0012 airfoil a t

zero angle of attack to the mainstream flow. Mach 0.5, Re = 10,000,

3307 triangles, 4180 quadrilaterals, N = ll . Instantaneous iso-contours

of the density are shown................................................................................... 167

7.7 The wake region (from jr = 2 to £ = 5) of a NACA 0012 airfoil at



of the divergence of momentum are shown.................................................. 167

7.8 The wake region (from f = 2 to f = 5) of a NACA 0012 airfoil a t



of the curl of momentum are shown.............................................................. 167

7.9 Hybrid Mesh and close ups for the simulation of compressible flow past

a multi-body wing ......................................................................................... 169

7.10 Iso-Mach contours and streamlines for M = 0.5 flow past a two-

dimensional multi-component wing............................................................... 170

7.11 Skeleton mesh for flow past a three-dimensional NACA0012 airfoil with

endplates.............................................................................................................. 172

7.12 Iso-contours for x-component of momentum for M = 0.5 flow past a

three-dimensional NACA0012 airfoil with endplates.................................. 173

xxv


8.1 Magnetohydrostatic test case for the compressible code. Top Left:

Mesh (K rri = 38, K qU(11i = 22) Top Right: Magnetic streamlines of

steady solution at N — 12, Bottom: dependence of steady state error

on expansion order........................................................................................... 187

8.2 Three-dimensional magnetohydrostatic test case for the compressible

code. Iso-contours of the magnetic and energy fields are shown a t time

t = 1. Top Left: Tetrahedral mesh used, Top Right: x-component of

the magnetic field, Middle Left: y-component of the magnetic field,

Middle Right: z-component of the magnetic field, Bottom: Conver

gence plot showing exponential decrease in L 2 error with increasing

expansion order................................................................................................. 189

8.3 Mesh used for the compressible Orszag-Tang vortex simulations on a

structured mesh................................................................................................ 191

8.4 Compressible Orszag-Tang Vortex ( t= l , instantaneous fields, Mach=0.5).

Top left: density, Top right: energy, Middle left: x-component of mo

mentum, Middle right: y-component of momentum, Bottom left: x

component of magnetic field, Bottom right: y component of magnetic

field...................................................................................................................... 192

xxvi


8.5 Compressible Orszag-Tang Vortex ( t= l , instantaneous fields, Mach=0.5).

Top left: curl of momentum, Top right: divergence of momentum, Mid

dle left: curl of magnetic field, Middle right: divergence of magnetic

field, Bottom left: momentum streamlines, Bottom right: magnetic

field streamlines................................................................................................. 193

8.6 Compressible Orszag-Tang vortex triangle mesh with K=132............... 194

8.7 Compressible Orszag-Tang Vortex ( t= l, instantaneous fields, Mach=0.2,

K=132). Top Left: Curl of momentum along the diagonal(N=4), Bot-

ton Left: Iso-contours of curl of momentum(N=4), Top Right: Curl of

momentum along the diagonal(N=6), Bottom Right: Iso-contours of

curl of m om entum (N = 6)............................................................................... 195

8.8 Compressible Orszag-Tang Vortex ( t= l, instantaneous fields, Mach=0.2,

K=132). Top Left: Curl of momentum along the diagonal(N=10), Bot-

ton Left: Iso-contours of curl of momentum(N=10), Top Right: Curl

of momentum along the diagonal(N=16), Bottom Right: Iso-contours

of curl of m om entum (N = 16)........................................................................ 196

8.9 Instantaneous iso-contours of the simulation fields for flow past a cylin

der (Sv = 100, Sr = 100, A = 0.1) with a magnetic field. From the top:

(1) density, (2) pressure, (3) x component of the velocity field, (4) y

component of the velocity field ................................................................. 198

xxvii


8.10 Instantaneous iso-contours of the simulation fields for flow past a cylin

der (Sv = 100, Sr = 100, A = 0.1) with a magnetic field. Top: x com

ponent of the magnetic field, Bottom: y component of the magnetic

f i e l d ................................................................................................................... 199

A .l Vertex labelling for the standard quadrilateral r e g io n .......................205

A.2 Mass m atrix for the continuous basis for quadrilaterals (N = 15). . . 207

A.3 Vertex labelling for the standard triangular re g io n ............................. 207

A.4 Mass m atrix for the continuous basis for triangles (N = 15)..............209

A.5 Vertex labelling for three-dimensional standard r e g io n s ................... 210

A.6 Mass m atrix for the continuous basis for hexahedra (N = 15)...........213

A.7 Mass m atrix for the continuous basis for prisms (N = 15)..................215

A.8 Mass m atrix for the continuous basis for tetrahedra (N = 15).......... 218

A.9 Vertex labelling for the (a) standard quadrilateral and (b) standard

Hexahedral reg ion ............................................................................................ 219

xxviii


C hapter 1

Introduction

In spectral m ethods the question as to what basis functions (f>n(x) to use to approxi

mate a smooth function f ( x ) « Z)£=o fn4>n{x) in a separable domain is relatively easily

answered. Typically, boundary conditions, fast transforms, and numerical quadrature

dictate this choice depending on the particular partial differential equation considered.

In multi-domain spectral methods which are used in complex-geometry computational

domains the choice of the subdomain shape is an im portant factor as well. One can

choose arbitrary polygons, for example, to accommodate geometric complexity but

such a choice may prevent the use of polynomial basis functions and lead to prohibitive

computational difficulty in maintaining interfacial continuity among subdomains.

For the hp version of the finite element method the choice of subdomain shape

and associated functions bases dictates the efficiency of the method. Starting with

the pioneering work of B.A. Szabo in the mid-seventies several versions of this ap

proach have been formulated for both solid mechanics as well as fluid dynamics in

1


2the founding papers of Babuska et al [5, 6] and by Patera et al Patera84, KBP85.

Similar formulations were developed to allow greater flexibility in handling geometric

complexities and local refinement requirements in the works of [9], [10], [11] and [12].

In figure 1.1 we demonstrate how a complex three-dimensional volume can be formed

using prisms. We solved the Helmholtz problem within this volume, and show that as

we increased the polynomial order of our representation we obtained an exponential

decrease in the approximation error.

The basis or shape functions employed in the aforementioned formulations typi

cally involve either Legendre or Chebyshev one-dimensional polynomials, and multi

dimensional expansions are constructed using tensor products for quadrilateral or

hexahedral elements [13], [14]. A basis directly associated with boundary and inte

rior elemental nodes usually lacks hierarchy and in addition a variable p-order per

element is not readily achievable [7]. However, a modal basis can be constructed that

consists of vertex modes, edge modes, and interior modes in two-dimensions that are

hierarchical and allow for a variable order within as well as along the boundary of the

element [13]. In addition to the flexibility in handling more easily nonuniform reso

lution requirements, hierarchical bases can lead to better conditioning of mass and

stiffness matrices [14], and thus fast iterative algorithms can be effectively employed

in the solution algorithms.

The need for constructing p-type bases in triangular domains became evident

from the beginning. A. Peano [15] constructed a hierarchical triangular basis based

on area (barycentric) coordinates by selecting as nodal variables high-order tangen-


3

1 cr*

10" 10"

i "-iio"

iff7icr* riff* - ■11» ■ •1 ■ »■11 ■«* ■1 • ■111 ■

4 5 6Expansion-Order

1 » * » • 1 ■ ■ » « 1

Figure 1.1: Top: Unstructured prism mesh, Bottom: p-type convergence for the Helmholtz problem with A = 1 and solution sin (y )s in (z). The face geometry was downloaded from the web-site: http://www.3dcafe.com /


http://www.3dcafe.com/

4tial derivatives along each side evaluated at the midside and imposing appropriate

continuity constraints for C° and C l elements. A variation of this construction was

later developed [13] th a t introduces Legendre polynomials to avoid round-off error for

high-order p-expansions. Both approaches however require special integration rules

which are quite complicated a t high polynomial order. Dubiner [16] first developed

an alternative hierarchical basis for triangular domains, which unlike Peano’s basis is

based on cartesian coordinates and preserves the tensor product property that leads

to sum factorization and low operation count, a crucial property for p-type finite

elements and spectral methods.

Dubiner’s basis was implemented in [20] using a Galerkin formulation of the

Navier-Stokes equations, and it was found to be competitive in cost with the nodal ba

sis on quadrilaterals employed in the spectral element m ethod [7]. This new modal ba

sis employs Jacobi polynomials of mixed weight to automatically accommodate exact

numerical integration using standard Gauss-Jacobi one-dimensional quadrature rules.

In particular, exploiting the tensor product property of the basis, multi-dimensional

integrals can be evaluated efficiently as a series of one-dimensional integrals; similarly

the cost of evaluating derivatives or squares of a function is m aintained at operation

count 0 ( N d+1) where N is the number of modes per direction as in the quadrilateral

or hexahedral elements and d is the dimension of the element.

Recent trends in mesh generation have emphasized the use of hybrid discretiza

tions consisting of triangular, quadrilateral, tetrahedral, prismatic, pyramidal and

hexahedral subdomains in order to provide greater flexibility in refinement and dere


finement procedures. For high-order discretizations this implies tha t suitable basis

functions have to be developed to conform to the subdomain shapes while a t the same

time preserve the good approximation properties and efficiency of the aforementioned

bases. In this thesis, we develop a unified description of such hybrid basis functions

following earlier developments in [16] and [21]. We also develop five types of basis

functions which are either modal or nodal or mixed and which may or may not be

hierarchical.

There has been recently an interesting debate as to what are the relative advan

tages of structured and unstructured grids, with a renewed interest in cartesian grids

for complex-geometry aerodynamic flows [22]. The current confusion stems from the

fact that most finite element and finite volume formulations in use today produce

solutions, which depend strongly on the quality of the mesh. Specifically, for highly

distorted grids convergence is questionable, and in most cases convergence rates are

typically less than second-order. To this end, some - relatively few - efforts have

addressed the development of hybrid grids, i.e. a mixture of structured and unstruc

tured grids in order to combine the merits of both discretizations in the context of

complex-geometry aerodynamic flows [24], [25], [26]. Such methods lead to more flex

ible geometric discretization and resolution placement, but they are still of low-order

accuracy, i.e. a t most second-order.

We also develop a new formulation for compressible Navier-Stokes and compress

ible magnetohydrodynamics solutions employing high-order spectral/hp element dis

cretization on mixed element grids. A discontinuous Galerkin formulation is devel


6oped both for the convective as well as the diffusion contributions that allows multi

domain representation with a discontinuous (i.e. globally L 2) trial basis. This dis

continuous basis is orthogonal, hierarchical, and maintains a tensor-product property

(even for non-orthogonal elements), a key property for the efficient implementation

of high-order methods. Due to this basis orthogonality the resulting mass m atrix is

diagonal, and thus the proposed method is matrix-free given that an explicit time-

stepping is involved. The conservativity property is maintained automatically by the

discontinuous Galerkin formulation, and monotonicity is controlled by varying the

order of the spectral expansion, around discontinuities.

Figure 1.2: Hybrid discretization around a multi-element airfoil. Only part of the domain is shown.

An example of a hybrid discretization is shown in figure 1.2 for simulation of

flow past a multi-element airfoil. This problem was studied by Barth [4] who used

unstructured disretizations, i.e. only triangles. There are two regions of structured

discretizations as shown in these two plots: First, around the four airfoils where

boundary layers are present and where all the vorticity is generated. Second, in

the wake region where one single family of square elements is embedded within the

unstructured discretization. This uniform meshing minimizes dispersion errors of the


7traveling vortex street and facilitates maximum storage efficiency. We will revisit this

example in section 7.2.1 in order to present simulation results a t subsonic speeds.

1.1 Objectives

The goals of this thesis are:

• To show how polymorphic elements can be built and interfaced to enhance

efficiency of the unstructured spectral element method (USEME).

• To demonstrate how Object-Oriented Programming can be used with minimal

overhead to provide a flexible toolkit for the polymorphic USEME.

« To provide theoretical and numerical evidence that USEME retains good ap

proximation properties even when very distorted meshes are used.

• To implement a semi-implicit Galerkin USEME scheme for the equations of

incompressible Navier-Stokes and magnetohydrodynamics (MHD) in the com

puter code f s fe n T ’a r .

• To investigate the effectiveness of USEME applied to moving domain problems

using the ALE formulation.

• To implement a fully explicit discontinuous Galerkin m ethod (DGM) USEME

scheme for the equations of compressible Navier-Stokes and magnetohydrody

namics in the computer code A/’e/cT 'ocr .

• To demonstrate accuracy for the above methods.


81.2 Outline

The thesis will be laid out in the following way. In Chapter 2 we describe the elements

that we build domains with. In Chapter 3 we will outline the polynomial basis

functions th a t we use to approximate functions defined on these elements. There we

will also show how these elements can be connected together so th a t their coordinate

sysems line up. Chapter 4 details how we build operators on the elemental level and

on the global level (i.e. over connected elements). In Chapters 5 and 6 we show

how these building blocks can be applied to solving the incompressible Navier-Stokes

equations and the equations of viscous magnetohydrodynamics. In Chapters 7 and

8 we show how the Discontinuous Galerkin Method, and its local operators can be

used to solve the compressible Navier-Stokes equations and the compressible viscous

Ma.gnetohydrodynamic equations.

We have begun the unification of the Galerkin and Discontinuous Galerkin meth

ods under the umbrella code M e n T 'cxr . An outline of the current capability of this

code is dem onstrated in figure 1.3. All of the tree-root applications use the same set

of code library routines and the same data structures.

In the applications chapters of this thesis we will discuss a number of benchmark

tests and simulations using N skTTa r . For each simulation we will provide a table

of paxameters in order to reduce the amount of repetition involved. These tests are

very im portant in determining th a t the code is correct and demonstrating that the

underlying methods are robust and accurate when implemented.


9

NeKtccrI

Galerkin

2d 2.5d1

3d

Steady . . F Steady Steady , PDomain Domain Domain

IDiscontinuous Galerkin

1

2d 3d

Incompressible Navier-Stokes Compressible

Figure 1.3: Hierarchy of the M eK/Tcxr code


C hapter 2

The E lem ents

We consider a number of element types that can be used to discretize two- and three-

dimensional spaces. In the hp community there has been a tendency to choose only

one element type to discretize a given domain. This is due to the widely conceived

idea that using more than one type of element is prohibitively expensive in terms

of implementation complexity. It is believed th a t this introduces a large number of

conditional statem ents. To counter these problems we have taken advantage of the

object-oriented language C-i—h- Object-oriented programming allows us to replace a

set of conditional “if” statements with a look up table for each element type which is

more computationally efficient. We will discuss the implementation aspects in section

4.1.4.

We show th a t it can be advantageuous to use both triangles and quadrilaterals

together to solve a problem set in a two-dimensional domain. In three-dimensions

we demonstrate th a t it is beneficial to use a combination of tetrahedra, pyramids,

10


11prisms, and hexahedra.

The most natural description of an element is as a subset of Rd, where d is the

dimension of the problem. Each element is then m apped to a reference element of its

own type. Though we only allow invertible mappings, these mappings can be quite

complex and can be represented as polynomial mappings in terms of the reference

elements coordinate system. This is commonly referred to as an isoparametric map

ping. Finally, we map the element to a tensor coordinate system attached locally to

the reference element.

In this chapter we develop new algorithms th a t enable the connection of three-

dimensional elements (i.e. tetrahedra, prisms and hexahedra) that have constraints

on the ways they can be connected together. Specifically we provide algorithms th a t

have an operation count that is proportional to the number of elements in a mesh.

2.1 The Triangle

We first consider the triangular shaped two-dimensional element. We will use the

formulation of Sherwin [20] to describe the element.

The natural coordinate systems will be given in term s of the ordered-pair x = (x, y ).

The coordinate system of the reference elements will be given in terms of (r, s ) . Finally

the local tensor space for each element will be given in terms of (a, b) coordinates.

The reference triangle, shown in figure 2.1, is described as the set of points:

T r ref = {(r, s)\ - 1 < r, s < 1; r + s < 0}


12

0.8

0.6

0.4

>.

-0.4

•0.6

•0.8

v*■ I ■ 1

r 1 * ' -1 i i I ■ • i i t • i t > I i • »-1 -0.5 0 0.5

Figure 2.1: The tensor coordinates of the triangle.

It is mapped to a straight-sided physical triangle with the following mapping:

(r + s) , (1 + r) 2 (1 + s) 3x = — —— v + ------ -v 2 + ------- -v 3

where the v 1^ 2, and v 3 are the physical coordinates of the vertices of the triangle

labelled in counter-clockwise manner.

The Jacobian for this mapping is:

§ ^ 4 = l(v2 - v l ) x ( v 3 - v I)|d{r, s)

The tensor (square) element is the set of points:

Qfensor = { ( a , b ) \ - l < a , b < l }


The tensor element is mapped to the reference triangle by:13

r =_ (1 + a) (1 - b) - 1

s = b

The Jacobian for this m apping is:

d(r, s) _ ( 1 - 6 ) d{a, 6) 2

We notice tha t this mapping is singular at 6 = 1. We will demonstrate that this

singularity will not present any numerical problems in chapter 4.

2.2 The Quadrilateral

Next we describe the quadrilateral. The reference quadrilateral, shown in figure 2.2,

is mapped to a straight-sided quadrilateral by the following:

( i - d t l - s ) l (l + r)(l-s) 2 (l+r)(l + s) 3 (l-r)(l + s) 4X — V " T " ------------------ " “ V “P ” V V

Again v 1 is an arbitrary vertex of the quadrilateral and v 2,v3,v4 are the vertices

labeled counter-clockwise from this vertex. The Jacobian for this mapping is:


14

0.8

0.6

0.4

0 2

0>•-02

- 0.4

-0.6

-0.8

-1

Figure 2.2: The tensor coordinates of the quadrilateral.

d{x, y) 11 (v3 — v 1) (v4 — v 2) (v2 — v 1) (v3 — v4) (v4 — v 1) (v2 — v 3) ,= 2 2 X 2 + r 2 X 2 +S 2 X 2 1

This Jacobian includes three terms. If the quadrilateral is rectangular then the last

two terms are zero and the first term is ju s t the ratio of the area of the quadrilateral

and the reference quadrilateral. If the quadrilateral is not rectangular then the last

two terms are proportional to the sine of the angles between the two sets of edges:

t = + 1 , - 1 and s = + 1 ,-1 . The reference quadrilateral and the tensor element

coincide so the mapping between (a, b) and (r, s) is the identity mapping. Thus,

f^fj- is identically one. When we compare the overall Jacobian of the mapping from

physical element to tensor element we see th a t the triangle Jacobian is dependent on

b whereas the quadrilateral Jacobian is both a function of a and b. The reference


: v* i1 1 1 » » » > ->- » 1 -t- > « > » t I . l . r I

*1 -0.5 0 0.5 1X

15quadrilateral is the set of points Quref = {(r, s)| — 1 < r ,s < 1} like the reference

element for the triangle.

We next consider a set of elements commonly used to discretize a three-dimensional

volume. There are now four elements: the tetrahedron, the pyramid, the prism and

the hexahedron. Again we will consider the physical, reference and tensor represen

tations for each element which were proposed in [27].

2.3 The Tetrahedron

The natural coordinate systems will be given in terms of the ordered triplet x =

(x ,y ,z ) . The coordinate system of the reference elements will be given in terms of

(r, s, £). Finally the local tensor space for each element will be given in terms of

(a, 6, c) coordinates.

The reference tetrahedron is described as the set of points:

Teref = {(r, s, £)| — 1 < r, s, t < 1; r + s -1-1 < 1}

The reference tetrahedron is mapped to a straight-sided physical tetrahedron with

the following mapping:

(1 t + s -f- £) . ( 1 + r ) 2 (1 + s) 3 (1 + £) 4X = ------------------V1 + ^ v + 2 + 2

where the v 1^ 2^ 3 and v 4 are the physical coordinates of the vertices of the tetahe-


16dron.

The tensor element is the set of points:

QtensoT = {(a.6 .c)| - 1 < a ,b ,c < 1}

The tensor element is mapped to the reference tetrahedron by:

r =

s =

(1 + a ) (1 — b) 2 2

(1 + 6)( ! — c) — 1

t = c

We notice that this mapping is singular a t b = 1 and c = 1.

V'

Figure 2.3: The tensor coordinates of the tetrahedron.


2.4 The Pyramid

The reference pyramid is described as the set of points:

17

Pyref = {(r, s , t ) | — 1 < r, s , t < 1; r + t < 0 ; s + t < 0 }

The pyramid will be useful for making bridges between tetrahedra and hexahedra or

between prisms and hexahedra. The reference pyram id is mapped to a straight-sided

physical pyramid by:

_ (r + t) (s + t) 2 x ^ (1 + r ) ( s + t) 2 22 2 (1 - t ) 2 2 ( 1 - t )

(1 + r) (1 + s) 2 3 _ (r + 1) (1 + s) 2 4 (1 + f) 52 2 ( 1 - t ) 2 2 ( 1 - t ) 2

The tensor element is mapped to the reference pyramid by:

r =

t = c

Clearly, this mapping has a first order singularity in the r, t and s , t coordinates

making it have a second-order singularity a t r = s = — 1, t = 1. However, this


18is only a singularity due to the choice of coordinates and this singularity will be

removed by the volume Jacobian in a volume integral. Also, we will use quadrature

for the pyramid that does not include the top vertex, hence avoiding the calculation

of derivatives at this point.

Figure 2.4: The tensor coordinates of the pyramid.

2.5 The Prism

The reference pyramid is described as the set of points:

Pfref = (O', S, £)| - 1 < r, s, t < 1; r + 1 < 0}

The reference prism is m apped to a straight-sided physical prism by:


(r + t) (l-s)„ i , (l+ r)(l 2 2

(r + t) (1 4- s)2 -v«

«/<* a

ettj,ePtsJ a

a)3 " v * ( i

Is PinPpetf to3 v *i~ (1 +

tb e te&*)

eroPce Ptis ,'*bIK:

r - . f t +^ Y -Z

- aPlS ? i s sin

S t

e

’W e*£■

6e tejj,s0tC°°r<]l'*abes

P tiPerfr>is-

Stti.s/or7

0ftt)e c°pyir'9ht °Wn{'Or

er rePr(° c/,c%0'’'■“a*®eC/ U,:,with,outPormr.rP'Ssi,Sion.

202.6 The Hexahedron

The reference hexahedron is described as the set of points:

Heref = {(r, M ) | - 1 < r ,s , t < 1}

The reference hexahedron is mapped to a straight-sided physical hexahedron by:

_ (1 — r ) ( l - a) ( 1 , (1 + r ) (1 ~ s) (1 ~ t) __2

2 2 2 2 2 2 | (1 + r) (1 + 5) ( l - i )v 3 + (1 — r) (1 + s) ( l - * ) v 4

(1 ~ r ) ( 1 - 5 ) ( l + *)v s + (1 + r) ( 1 - s ) (l + ^)v a

(1 + r) (1 + s) (1 + 1) + (1 — r) (1 + s) (1 + 1) g

The tensor element is mapped to the reference hexahedron by the identity.

2.7 Global Assembly

The elements we have described will be tessellated together to construct a continuous

solution domain. We shall only permit elements to connect by sharing common

vertices, complete edges and/or complete faces. Such a connectivity is commonly

referred to as a set of conforming elements. In this section we shall discuss issues

related to the process of globally assembling the elemental bases described in chapter

3.


21

V'

Figure 2.6: The tensor coordinates of the hexahedron.

When two elements share an edge it is im portant for them to be able to determine

if their local coordinate system a t the shared edge are aligned in the same direction.

For instance, the mode shapes used in the C° methods must be continuous across

element boundaries. If the local coordinate systems are not aligned in the same

direction then edges modes of odd polynomial order will have different signs and the

odd edge modes of one of the edges will need to be multiplied by —1.

This condition becomes more complicated in three-dimensions when two elements

share a face. In this case it is not automatic that their coordinate systems on the com

mon face will line up. Considering the triangular faces of the tetrahedron, pyramid

and prism we see th a t there is a vertex on each triangular face th a t the coordinate

system for that face radiates from. We will call this vertex the face origin as it is

similar to a polar coordinate origin. The alignment constraint necessitates tha t when

two triangular faces m eet their origin vertices must coincide. Initially, it is not obvi-


22

v4 (local top vertex)

v

v3 (local base vertex)

Figure 2.7: Diagram of the local top and base vertex of the tetrahedron.

ous how to satisfy this constraint for a mesh consisting of just tetrahedral elements.

We outline two algorithms tha t will satisfy this constraint. The first is based on the

topology of the mesh. We will only use the connections between elements to deter

mine how we should orientate elements. In the second method we will assume that

each unique vertex in the mesh will have been given a number. This second method

works under some loose conditions but is extremely easy to implement and is very

local in its nature.

It is useful to observe that one of the vertices of a tetrahedron is the face origin

vertex for the three faces sharing tha t vertex. We will call this the local top vertex.

Then there is one more face origin vertex on the remaining face which we call the

local base vertex. These are shown in figure 2.7.


232.7.1 A lgorith m 1

Given a conforming discretization we can generate the local orientation of the tetra-

hedra using the following algorithm. We assume th a t we have both a list of vertices

and a list of elements which touch each vertex. This list of elements will be called a

vertex group and gill elements are assumed to have a tag of zero.

For every vertex in the list:

• Orient all elements with a tag of one in this vertex group so that their local base

vertex points at this vertex. Then set their tags to two.

• Orient all elements with a tag of zero in this vertex group so that their local top

vertex points at this vertex. Then set their tags to one.

This algorithm visits all vertices in the mesh and if this is the first time the

elements in the vertex group have been visited the local top vertex is orientated to

this vertex. If this is the second time the elements in the vertex group have been

visited then set the local base vertex to this vertex. To see how this works we can

consider the example shown in figure 2.8.

Here we assume that we are given a discretization of a box using six tetrahedra

as shown in figure 2.8a. Starting our algorithm we begin with vertex A. Since all

elements have a tag of zero at this point we go straight to the second part of the

algorithm and orientate all elements that touch this vertex so that their local top

vertices point to A. Therefore tetrahedra HBDA and BHEA are orientated as shown

in figure 2.8b and now have a tag set to one. Continuing to the next vertex B we


24

DA.

c) Final d ement orientation a t cod o f algorithm

Figure 2.8: Setting up the required connectivity for the discretization of a box as shown in a). Vertex A is given as the first local top vertex as shown in b). In c) vertex B is then given as the local base vertex and the local base vertices from group one are aligned to satisfy connectivity and the final element orientation is shown.

see tha t all elements belong to this vertex group. The first part of the algorithm is

to orientate the elements with a tag of one to have their local base vertex pointing

at B. So the tetrahedra HBDA and BHEA are ro tated as shown in figure 2.8c and

their tags are set to two. The second part of the algorithm then orientates all the

other tetrahedra to have their local top vertex pointing a t B. The connectivity is

actually satisfied at this point since the orientation the faces have on the boundaries

is irrelevant. However, if we continue the algorithm looping through the vertices

consecutively we end up with the tetrahedra orientated as shown in figure 2.8c.

Clearly, the connectivity is not unique since any elements that have their local top

vertex pointing at E can be rotated about E. We have demonstrated however tha t it

is possible to satisfy the connectivity requirements imposed by the co-ordinate system


and thereby im ply tha t the requirement is non-restrictive.25

2.7.2 A lg o r ith m 2

Assuming th a t every global vertex has a unique number, then for every element we

have four vertices with unique global numbers:

• Place the local top vertex at the global vertex with the lowest global number.

• Place the local base vertex at the global vertex with the second lowest global

number

• O rientate the last two vertices to be consistent with the local rotation of the

element (typically anti-clockwise).

It has been stated before that since the coordinate systems on the faces of the

tetrahedra are not symmetric, it is too difficult to use these coordinate systems [19].

We have shown, however that it is possible in linear or even constant time to satisfy

this constraint for any given tetrahedral mesh. This algorithm is local to each element

and should be implemented at a pre-processing stage.

We now extend this approach to include meshes consisting of tetrahedra, prisms,

and hexahedra. Unfortunately, in this case we find counter-examples where it is not

possible to satisfy the origin alignment constraint. We have isolated the problematic

cases, and they are unlikely to come up when using a mesh generator.

First we deal with the case when a quadrilateral face is shared by two elements.

In this instance it is sufficient to simply make the coordinate directions agree by


26reversing either face coordinate if necessary.

We now investigate over-constrained meshes. These cases can occur when prisms

and tetrahedra are used together in a mesh. We will use these examples to motivate

the actual algorithm we propose. The cost of this algorithm also depends linearly on

the number of elements in the mesh.

It is instructive to construct a chain of prisms. This is simply a long prism,

with equilateral triangular faces, divided a t intervals along its length into a set of

prisms connected at their triangle faces. The connectivity constraint requires that

the coordinate origins of the triangular faces must meet at every prism-prism interface.

This condition enforces that the collapsed edge m ust run in a continuous line through

the one edge of the original long prism. Now we twist the chain around in a loop

and connect its triangle ends. The chain now forms a closed loop of prisms. The

orientation of the end faces of the original chain must also satisfy the connectivity

constraint when they meet. However, we can make the chain into a Mobius band by

twisting it around the axis along its length. In this case the connectivity cannot be

satisfied without changing the mesh.

We can construct a second counter-example, this time involving one tetrahedron

and two chains of prisms. We construct two chains of prisms as outlined above, and

we join the tetrahedra into the prism chain by connecting two of its faces to the prism

chain triangular end faces. We repeat this operation, again connecting the remaining

two faces of the tetrahedron to the end faces of the second chain of prisms. We can

now repeat the twisting of the prism loops. This over-constrains the tetrahedron so


27that it cannot be oriented to satisfy the connectivity condition.

These two cases indicate that we cannot allow prism chains to reconnect into

closed loops and still satisfy our constraints. Also, we should not allow a prism chain

to connect to a tetrahedron with more than one of its triangular faces. If we only

consider meshes tha t satisfy these two constraints, then the following algorithm will

satisfy the connectivity constraints:

2.7.3 A lg o rith m For C onnecting P rism s A nd T etrahedra

• Find all prism chains in the mesh.

• Create a virtual connection between the faces of the tetrahedra that meet the

triangular faces at each end of the chain.

• Proceed with Algorithm 2 to connect the tetrahedral mesh treating the virtual

links as real connections.

• Orient each prism in each chain with the same orientation as the triangular

faces of the tetrahedra at the ends of the virtual link. Thus, the orientation

propagates through the chain.

In figure 2.9 we summarize which elements can be connected using the above

algorithms. A solid line between elements implies we can connect a given mesh made

of these two element types with no problems. If the line is dashed then the mesh may

have to be changed to meet the connectivity constraints.


28

£

ir

Figure 2.9: Connectivity summary. Solid arrows imply we can connect elements together with no extra mesh refinement, dashed arrows mean that there are some constraints on allowable configurations.


C hapter 3

B asis Functions

This chapter is devoted to outlining the polynomial bases tha t we use to represent a

function defined on an element. A desirable basis has three properties: (i) it should

be orthogonal or nearly orthogonal under a convenient norm, (ii) well behaved a t

high polynomial order, and (iii) it should be computationally efficient to take inner

products w ith this basis if it is to be used in a Galerkin framework. The bases we

present are designed to satisfy these conditions. The computational complexity of

taking inner products has been minimized for these hierarchical bases by insisting

that each basis function is a tensor product of one-dimensional shape functions.

3.1 One-Dimensional Orthogonal Polynomials

In the following sections we use make considerable use of the Jacobi polynomials.

These are defined as the polynomial solutions of the following Sturm-Liouville prob-

29


30lem:

(1 - * ) ( ! + x ) ^ ^ - + p - a) - (2 + a + /3)x\ 5y(x)dx dx= -A ny(x)

or

d_dx

(1 -a : )1+Q( H - 2r)1+ dy(x)dx = —A„(l — x )a{l + x)py(x)

An = n(n + a + /? + 1)

y(«) =

They have the following orthogonality:

/ ( 1 — rr)Q(l + x )^ P ^ ,0( x ) P ^ ( x ) d x = 0 n ^ m

J \ 1 - x)a(l + x)<3P ^ { x ) P ^ { x ) d x =

qcc+p+i r(n + a + l)r(n + 0 + 1)2n + a + ft + 1 n !r(n -+- a + j3 + 1)

As the following sections were just being completed I found th a t work along exactly

the same lines had just been subm itted to the SIAM Journal on Numerical Analysis

by W ingate and Taylor [28]. I include the work here for completeness noting that the

results have already been independently obtained and presented there, except for the

results based on prisms.


313.2 Singular Sturm-Liouville Problems on a Sim

plex in d-Dimensions

The following section was inspired by the work of Owens [29]. In th a t paper he

derived an orthogonal basis for approximating functions on a triangle from a basic

premise. We will take an alternative route and show that it is possible to find a

singular Sturm-Liouville operator for a d-dimensional simplex and present its eigen

function/eigenvalue pairs explicitly. The eigenfunctions for the cases d = l (a segment),

d=2 (a triangle), and d=3 (a tetrahedron) will prove to be useful in the context of

polynomial-based approximation on these simplices. As stated before these results

have been proposed in [28].

A d-dimensional simplex S d can be defined as a set of constraints on the entries

of a d-dimensional vector:

S d = {r 6 Rd\0 < < 1; j = 1 ,2...,d}i=1

We define an operator on a d-dimensional space of at least twice differentiable func

tions of d variables:

L t = J 2 dn (rA - ~ ri S rA J )i=i j =i

We also define a (d + 1) vector, s = [(1 — u))ri, (1 — uj)t2 , (1 — w )^ , n>], where


u) € [—1,1]. Using the identities:

dsi — - dr. i = l , d1 — CJ

1 i=ddsd+i = Y v j d ri 4- dy

L ~ u i=i

Sid Si = Tidri i = 1, di=d+1 i i=dY S*dsi = i----- Y Tidn + Udwi=l 1 ~ W i=l

it is straightforward to show the following relationship:

From this we can repeat the recurrence relation ending up with the operator in terms

of the a coordinates:

L* = £ rTj=d r [5«i(oi(l - ai)d<n) ~ ~ 1) 0,-]1=1 Uj'=i+l(l

where a is defined by the canonical transform:

r i = a i( l - a2)( l - a 3)...( l - ad)

r2 = a2( l - a 3) . . . ( l - a d)

rd — ad


33This form of the operator shows tha t L \f is self-adjoint in the inner product taken

over the simplex S d. This is because S d maps to a d-dimensional unit box Ud = {r G

Rd|0 < Ti < 1 i = 1 , 2 and using integration by parts we notice that all the

surface integral terms are zero.

For example, we consider the ith operator in the sum for L d:

ud

ud

(u,(LiYv)st = (u, n jg i1 - O . W - (i -

, — a k ) k~ l f a I M ^= (W> ( t t jW ------- r ~ ) L { “ a*') v h do-j)

= (^ (O ifl - a,.)%)] u, at).<:~1 )v)u-

= ((^r Yu,v)s<l

We can also use the definition of the Jacobi polynomials to show that Ld has

eigenfunctions:

</>i = H S f ( l - Gi)*Pd?*'~L'°(Oi)

where d. € Nd, a = dj. The eigenvalues for these eigenfunctions are:

= cd+i (cd+i + d)

In summary, the operators we have defined in the d-dimensional simplex have simple

tensor product eigenfunctions when mapped to a d-dimensional unit box. Also, their

eigenvalues are n(n + d) where n is the total degree of the eigenfunction in both the


34mapped and original coordinates a and r.

3.3 Orthogonal (Modal) Bases

The results we now present for the triangle and the tetrahedron are special cases of

the previous section, but for clarity we will outline some of the steps used to find

these results.

We first consider the tetrahedron as this has the most complex mapping between

the local cartesian coordinates and the tensor coordinates. Using the scaled coordi

nates (a, b, c) E [0,1] x [0,1] x [0,1] for the tetrahedron (for simplicity) we can express

the local orthogonal coordinates (r, s, t) E {0 < r, s, t, r + s -f- £ < 1} as:

r = a ( l — 6)(1 — c)

s — 6(1 — c)

t = c

We consider the operator:

Lret ■= dr[r{ 1 — r)dr - rsds - rtdt) +

ds(s (l — s)ds — srdr — stdt) +

dt(t( l — t)dt — trdr — tsd3)

Using the following identities it is straightforward to express the operator with respect


35to the (a, b, c) coordinates:

rdr = ad,a

sd3 = ~ ab da + bdb (1 - b)

idt = 7= jtt:------r<9a + ■>- ~- - vdb + ed c( l - 6 ) ( l - c ) (1 - c )

rdr 4- sds + tdt = T 7771 + 77— r d 6 + cdc( l - 6) ( l - c ) (1 - c )

After some m anipulation the operator can be expressed in terms of the (a, 6, c) coor

dinates and is:

= (1 - i) ( 1 1 c) [ d M l - + ( ! --->) (1 - c) [at(6(1 - 6)2ft)]

+ ( 1 3 ^ 2 f c M 1 - c)3 ) ]

This demonstrates tha t the operator maintains tensor form in the (a, b, c) coordinate

system. It is now trivial to show that this is a self-adjoint operator by applying one

dimensional integration by parts to each of the three tensor parts. Also, by using the

definition of the Jacobi polynomials we can show that the orthogonal basis is a set of

eigenfunctions of L r et and find their eigenvalues.

We will now show th a t the polynomial functions 4>ijk defined by:

4>ijk = P , '0 (a) (—t — (b) (—r — P^ t+J'+2,0 (c)

are eigenfunctions of the Lret operator. We consider the first part of the operator.


36The definition of the Jacobi polynomial directly implies the following relationship:

da{a{ 1 - a)da(f>ijk) = -* (t + 1 )& #

We now consider the first two terms of the operator. Using the previous result we can

remove the dependency on a and then use the definition of the Jacobi polynomials

with non-zero (a, (3) to show that the polynomials are indeed eigenfunctions of the

first two terms of the operator:

Lbfiijk — bdb(f>ijk

i(i + 1)

1 - 6O.0/ ^ D2(*+i)+2,0= ( i - b y p r ( a ) P { (c)

b ( l — + (^ — (3 + + (—*(2 -1- z))

= 4>ijk [~ j ( j + (2z + 1) 4- 1) — z’(2 + *')]

Pi2i+l,0

(6)

Applying the same technique again we come to the relationship:

Lret&jk = dc(c( 1 - c)dc<f>ijk) ~ 2cdc<t>ijk - + + 2l(pijk1 — c

= Kjk4>ijk

Thus, the tensor product of Jacobi polynomials <j>ijk are eigenfunctions of the total


37operator L ret with eigenvalues:

Aijk = —{i + j + k)(i + j + k 3)

This approach clearly applies to the triangular, quadrilateral, prismatic and hexahe

dron elements as well.

3.3 .1 T he O rth ogon al Triangle B asis

The triangle is a special case of the tetrahedron with c = — 1. As before, we can

specify a self-adjoint operator tha t has the orthogonal basis, proposed by Proriol [17]

and later by Koomwinder [18] then Dubiner [16], as its eigenfunctions. We set out

the scaled coordinate system, operator and eigenfunction/eigenvalue pairs:

r = a(l — b)

s = b

Lth = dr (r( 1 - r)dr - rsds) + ds(s( 1 - s)d s ) - rsd r )

= r ^ [a“ (a(1_a)a<-)1 + ( r = T ) 56(i’( 1 - i ’)2s‘)

= Pi'Q(a)(l — b)'Pji+lfi(b)

A ij = — (i + j ) (i + j + 2)


383.3.2 T h e O rthogonal Q uadrilateral B asis

The straight tensor product property of the quadrilateral gives a very simple form

for the operator, it is the sum of two one-dimensional operators. Similarly, each

eigenfunction is a straight tensor product of one-dimensional eigenfunctions and their

eigenvalue is a sum of the one-dimensional eigenvaiues.

r = a

s = b

LQuad = dr{r{ 1 - r)dr) + ds (s( 1 - s)3,))

= da{a{ 1 - a)da) + db(b( 1 — b)db)

toj = Pi'°(a)Pj'a(b)

Xij = —i(i + 1) - j ( j + 1)

3.3.3 T h e O rthogonal P rism B asis

The prism is simply a tensor product of a triangle and a uniform third direction.

Hence, we sum the operator for the triangle in th e (r, t) directions and the one

dimensional operator in the s direction. Likewise, we can obtain the eigenfunc

tion/eigenvalue pairs from the above analysis for th e triangles.

r = a ( l — c)

s = b


39t = c

Lprism = dr(r( l — r)dr - rtdt) + d3(s ( l - s)ds) + dt {t(l — t)dt — rtdr)

= 3^ [8 «(o(l - a)a„)] + [%(6 ( 1 - 6 )%)] + 3^ [ScWl - o)2Sc)]

<hk = P i - ° ( a ) P ^ ( b ) ( l - c Y P ^ +l’a(c)

Xijk = — (i + k) (z + k + 2) — j ( j -I- 1)

3.3.4 T h e O rthogonal H exah ed ral B asis

The hexahedral analysis is trivial since, like the quadrilateral, the hexahedron is

a straight tensor product of three one-dimensional directions. The operators and

eigenfunction and eigenvalue pairs are:

r = a

s — b

t = c

Ttfex = dr(r( l - r)dT) + ds(s ( l - s)ds) + dt(t(l - t)dt)

= da(a( 1 - a)da) + db(b( 1 — b)db) + <9c(c(l — c)dc)

<f>ijk = P °’° (a) Pj'° (b) P °’° (c)

Xijk = + 1) - j ( j + 1) - k(k + 1)


403.3.5 T h e D eca y o f B asis C oefficien ts

Thus, each basis is a set of eigenfunctions of a singular Sturm-Liouville operator which

leads us to the following observations:

&ijk =" (u, 4>ijk)

= (w> -r— Hfajk))■ijk

''■ijk

ijk

Hence, if the function is infinitely smooth we see tha t the coefficients Uijk must de

crease faster than any polynomial power of i , j , k. Thus, the sum:

N N N

«* = E E E « ijkfajki j k

must converge exponentially fast to u as N increases for all infinitely smooth u.

It is im portant to notice that since straight-sided tetrahedra and triangles have

constant geometric mapping Jacobians these results hold for arbitrarily stretched

tetrahedra and triangles. This does not follow for the other elements since their

geometric Jacobians are quadratic for non-perpendicular elements. This backs up the

findings th a t the simplical elements handle deformation better than the other types.

Unfortunately, it does not appear th a t this method generalizes to the pyramid in


41a straightforward way. However, a suitable orthogonal basis is known for a pyramid:

r = a (l — c)

s = 6(1 — c)

t = c

frjk = Pi'°{a)Pj'°{b)(l - c)i+jP 2k{i+j]+2'\c )

This basis is only appropriate for supporting P n since the ‘c’ component is of the

order i + j + k. Thus, if each i, j , k < N but i + j + k > N then it is necessary to use

high-order quadrature to integrate these modes exactly. Also, if i + j + k < N the

function is a polynomial in r, s and t.

3.4 Semi-Orthogonal Bases

In appendix A there is a summary of the modal triangle basis proposed by Dubiner

[16] and the tetrahedral basis proposed and investigated by Sherwin [30]. Also, we

list the modal bases for the prism, pyramid and hexahedron suggested by Sherwin in

[27] and with Karniadakis in [31]. In the following subsections we will examine the

properties of the bases for the triangle and quadrilateral. We wall also propose some

modifications to the shape functions that will enhance efficiency for the quadrilateral

and provide compatibility between a triangle and a nodal quadrilateral.


423.4.1 M od a l basis

A modal basis for triangle elements has been presented in [16] and [21], and it was

applied to to fluid dynamics problems in [20], and to geophysical fluid dynamics

problems in [32]. We now consider a basis for quadrilaterals which is compatible with

the triangle basis and thus they can be used together. This combination was first

proposed in [27] for a set of three-dimensional polyhedra.

T riang le B asis: T rH H

In this and the following sections we will define a set of bases for the triangle and the

quadrilateral. For clarity we will label them with the first two letters of the element

they will be used on (i.e. Tr or Qu for triangle and quadrilateral respectively). The

next letter will denote the type of mode shapes used on the boundary of the element.

H will denote hierarchical and N will denote non-hierarchical. The last letter will

denote the type of mode shapes used for the interior modes, again using N and H as

before. For example, T rN H will denote a triangle T r with non-hierarchical boundary

modes N and hierarchical interior modes H.

We present here a basis which is a set of tensor products with respect to the tensor

product coordinates for the triangle and polynomials with respect to the reference

element coordinates. It maintains numerical linear independence up to high orders

due to the construction of the interior modes from Jacobi polynomials with carefully

chosen (a, /3) coefficients to ensure that mode shapes do not become too similar.

Increasing a shifts the roots of the Jacobi polynomials away from the coordinate


43singularity a t 6 = 1 as dem onstrated in [21] and hence the modes are prevented from

having the same shape a t this vertex.

The form of the T rH H basis is:

Vertex Modes:

T — a \ ( l — biv e r te x i _ f ^ V 2 J \ 2

(pvertex2 ^ ^

<f>vertex 3

2 J \ 2 '1 + 6'

Edge Modes (2 < m; 1 < n, m < M\ m + n < N ):

1 - b \ m= ( ^ ( ^ ) ^ ( ° > ( 2 ;

= ( ^ ) ( ^ ) ( ^ ) *£!,(»)

«*- - ( ^ ) ( ^ ( ^ ) p"->>

Interior Modes (2 < m; 1 < n, m < M] m + n < N):

= (^ ) (^ ) («) ( ^ ) ”“ (^ ) (»)

We represent this basis graphically for N = 5 in figure 3.1; the highest mode is

quartic.

In the next chapter we will be considering the variational form of elliptic operators.


44

\.o050.0

Vertex ModeInterior M<

Figure 3.1: Mode shapes for the triangle modal basis (T rH H ) with N = 5.

At this point it is useful to note tha t the T rH H basis has a very useful property that

the bandwidth of its stiffness m atrix is N + l if the modes are ordered in a specific

way. The stiffness m atrix is defined as the inner product of the the gradients of the

modes i.e.

where the 0t- represent the modes indexed in a given way. This indexing is important.

If we list the vertex modes, then the edge modes and then the interior modes we see

in figure 3.2 that the stiffness m atrix is partially banded. The interior modes in this

case have bandwidth 2N + 1 as shown in [30]. However, if we list the modes lowest

order first, i.e. linear modes, then quadratic modes, then cubic and so on, we see in

figure 3.3 tha t the stiffness m atrix becomes strongly banded with bandwidth N + 1.

Sij = (V0t-, V0j)


45This will become im portant later on when we need to invert matrices of this type

since reducing the bandwidth reduces the storage and number of operations required.

Q u a d rila te ra l B ases: Q u H H an d Q u H N

For quadrilateral elements Q l is a bijection so we do not need to worry about the

coordinate singularity as in the triangle case. We are free to choose a set of modes that

are C° compatible with the triangle expansion. An obvious choice th a t guarantees a

high degree of orthogonality is the Q u H H basis:

Vertex Modes:

Edge Modes (2 < n , m < N):


46

Figure 3.2: Top: Geometric ordering for modes, Bottom: Stiffness m atrix (V<fo, Vdj) for the continuous basis for triangles (iV = 15) with the geometric ordering of the rows and columns.


47

! I W { S|§lfe|fei>

Vertex Modes □* Edge 1 Modes

= Edge 2 Modes

= Edge 3 Modes

□ : Interior Modes

10

2030

40

50

60

70

8090

100

11 0

50 1000

Figure 3.3: Top: Polynomial degree ordering for modes, Bottom: Stiffness matrix (V0,-, V4>j) for the continuous basis for triangles ( N = 15) with modes ordered by polynomial order.


48Interior Modes (2 < m, n < N):

in teriorm n

We represent this basis for N = 5 in figure 3.4.

.Vertex Mode

Figure 3.4: Mode shapes for the quadrilateral modal basis (Q uH H ) with N = 5.

Alternatively, we could also choose a second set of modes with an even better

orthogonality relationship. To this end, we use the Legendre interpolant functions,

which we will investigate more thoroughly in the next section. These nodes are used

in a collocation manner to maximize the discrete orthogonality of the modes in the

interior region. More specifically, the Legendre quadrature points are used as nodal

points as well. We also modify the vertex modes and replace the edge modes with

tensor products of the one-dimensional modal basis and the Legendre basis. Similarly,

we replace the interior modes with tensor products of only the Legendre basis. This


49means th a t edge modes from one edge are orthogonal to edge modes from another edge

and all the interior modes. The interior modes are m utually orthogonal and hence we

can perform inner products and backwards transforms in 0 ( N 2) operations. Before

we write the basis we need to define an appropriate Lagrange interpolant in terms of

the Legendre polynomial Pn(x) = P%'° as:

M r) = d - ’- W ON ( N + l)Pjv(rn)(r — rn)

where r t- denote location of roots of (1 — r2)P'N(r) = 0 in the interval [—1,1]. Also,

by definition we have th a t hn(rm) = Smn. This construction destroys the hierarchy

of the basis in the interior of the element. More specifically, this Q uH N basis is

defined as follows:

Vertex Modes:

< r rteI = hda) +ft,(6) ( i ^ ) - hi (a) hi (b)

<Tr‘“ s = h„(a) - hN(a)hi(b)

= hH(a) + M &) ( V p ) - h „ ( a j h N(b)

= ft,(a) + ft„(f,) (L Z _?) - h M h s i b )


50Edge Modes (2 < n, m < N):

V S T = ( ^ p )

V*™ = hN(a) pM ^J)

« s=s = (^p) ( ^ ) P ^- i (a )h N (b)

t i d" ' = M&) ( ^ ) ( ^ p ) p» i W

Interior Modes (2 < m , n < N):

= hm(a)hn(b)

i.o

os 0.0

Figure 3.5: Mode shapes for the quadrilateral mixed modal basis Q u H N with N = 5.

Note tha t according to our convention we have th a t hi(—1) = 1 and h ^ ( l ) = 1,

etc. In figure 3.5 we see that the edge modes now have behavior localized to the edge,


Vertex M ode

51and the same is true for the vertex modes. In figure 3.6 we present graphically the

inner product (0mn, <ppq) which is the mass m atrix for this basis. Here, we order the

vertices first (around the origin), followed by the edges, and the interior contributions.

We verify tha t indeed there exists great sparsity in this m atrix indicative of the strong

discrete orthogonality between modes.

100

150

200

0 200100

Figure 3.6: Mass m atrix for the quadrilateral mixed modal basis (Q uH N ) with N = 15.

3.4 .2 N o d a l and M ixed B ases

We have so far presented a hybrid basis derived from Dubiner’s [16] C° basis for the

triangle. This dictates the shapes of the edge and vertex modes for the quadrilateral.

We will now consider an alternative starting point. In the spectral element method

[7], quadrilateral elements are commonly used with a nodal basis, constructed from

the Legendre interpolant polynomials. This method benefits from a fast transform to

the polynomial space, due to the collocation property of the modes. Similarly, inner


52products can be evaluated efficiently.

Q uadrilateral Basis: Q uN N

This basis is constructed based on the Gauss-Lobatto-Legendre interpolant of order

N already defined as {hn(r)}. It was first used for spectral element discretizations in

[33]. This is a nodal non-hierarchical basis.

For consistency we present the QuN Nbasis in the same format as the hierarchical

basis:

Vertex Modes:

^ v e r te x , =

= h ^ h ^ b )

0 vertex3 = ^ ( a) ^ ( 6)

^ v e r te x , = h ^ a ) h [ f ( h )

Edge Modes (1 < n ,m < N):

^ = h M h ^ b )

t f 9*2 = hN(a)hn(b)

t f e3 =


= hi(a)hn(b)53

Interior Modes (1 < n, m < N):

C r ,0r = hmWhnib)

We represent this basis for N = 5 in figure 3.7; all modes are fourth degree polyno

mials.

,Vertex Mode

Figure 3.7: Mode shapes for the quadrilateral nodal basis (Q uN N ) with N = 5.

T rian g le B asis : T rN H

We will discuss four ways that the triangle basis can be constructed, but eliminate

three of them due to undesirable properties of the resulting basis. A first approach

might be to construct the triangle basis in the same way as the quadrilateral, using a

complete set of N 2 tensor products of the Legendre interpolant functions {/in}. This


54basis suffers from over-resolution a t the singular vertex of the coordinate system as

shown in [16]. To avoid this overresolution we could choose a subset of the Legendre

interpolant functions on T2, as suggested in [34]. This leads however to a basis that

suffers from numerical linear dependency as noted in [16]. Lastly, we could construct

a two-dimensional nodal basis in the manner of [35] or [36] by explicitly specifying

the node distribution, but these bases do not have the tensor product property thus

making spatial derivatives expensive 0 ( N 4) operations. Recently, however, fast polar

derivative techniques have been developed [37] that reduce the scaling factor in the

asymptotic cost. These advances have made the cost comparable to the tensor product

cost for N < 10. It is still true tha t a modal approach provides a natural way to to

use exact integration and thus avoid aliasing.

Instead, we propose a basis th a t is compatible w ith the nodal quadrilaterals

(Q uN N ) and numerically similar to the modal triangle basis (T rH H ). The C°

continuity condition requires that the boundary modes of the triangle must have the

same shape as the {hn} modes of the quadrilaterals they can share an edge with.

However, this condition does not determine how the modes should be shaped in the

interior of the triangle, which leaves us a certain amount of freedom. We require

that the modes axe polynomials in (r, s) and that the basis spans the polynomial V m-

We complete the requirement th a t the bases be numerically similar by ensuring the

new basis is numerically linearly independent. This is guaranteed by using the same

interior modes as the modal triangle basis. As previously discussed, the choice of

(a , /?) for the Jacobi polynomials is key in ensuring th a t the interior modes do not


55

degenerate in the (r, s) coordinate system but are still tensor products in the (a, b)

coordinates.

We constructed the vertex and edge modes of the new basis in the (r, s) coordi

nates. The vertex 3, edge 2, edge 3, and interior modes still m aintain tensor-product

in the (a, b) coordinates but the remaining boundary modes become two-dimensional

functions in this frame. These new modes will increase the constant factor in the

asymptotic cost of inner products and transforms compared to the modal triangle.

The new basis is presented in closed form, but we note th a t it could also be derived

as a linear combination of modes from the modal basis, and this is how we guarantee

th a t it will share all properties that the modal basis has for linear operations.

So we now present the nodal-compatible T rN H basis for the triangle in local

cartesian coordinates: Vertex Modes:

^vertex i = ^ ^ + g + ^

^vertex* = hff ^

0 vertex3 = ^

Edge Modes (2 < m, n < N):

/ r + s') U - r j

) hm_ i( r )

(j dge 2 _ f l +U - s y ) hn—i (s)

C *363 = - i (S ) 1 ^n—1 (^)


In standard triangular coordinates the vertex Modes are:56

^ ertexi =

(a ( 1 ^ ) - (1 ± * ) )

4>verteX3 = htf(b)

Edge Modes (2 < m , n < N):

= (1 - q )(l - ft) , ( J l ^ ± \ _ (4 — (1 + a) (1 — 6) m_1(, 2 J 2 ,

^ = ( —J —) ^>.-1 M

The interior modes are unchanged from the modal triangle basis. We represent this

basis for IV = 5 in figure 3.8; the interior modes still have the bubble shape and thus

they are zero at the boundaries.

3.5 Comparison of the Bases

We have presented two classes of bases in sections 3.4.1 and 3.4.2 that give the same

approximations and share the same spectral properties. We are left with the decision

of which basis should we use for a given problem. In this section we summarize their

properties and suggest possible selection criteria.

First, we consider the structure of the mass and stiffness matrices encountered in


57

Vertex Mode

Vertex *3*

Vertex *2*

Vertex *1

Figure 3.8: Mode shapes for the triangle mixed basis (T rN H ) with N = 5.

the convection and diffusion equations, respectively. We concentrate on the triangular

elements only, and we examine the different structures corresponding to bases T rH H

and T rN H . The elemental mass m atrix is the m atrix which has entries defined by:

Bmn — (jfimi < n) •

Figure 3.9 shows in (a) the mass m atrix for the T rH H basis, and in (b) the mass

m atrix for T rN H and in (c) the mass m atrix corresponding to T rN H but calculated

using exact integration. In the latter case the number of quadrature points needed

is larger than the nodal points unlike the case in (b) where the quadrature points

coincide with the nodes a t the edges. We notice that the sparsity of the mass matrices

corresponding to bases represented in (b) and (c) is reduced compared to the T rH H

modal basis, but this is to be expected because each boundary mode in T rN H is a


58linear combination of all of the T rH H boundary modes.

(a)

(c)

Figure 3.9: Comparison of the elemental mass m atrix for (a) T rH H , (b) T rN H using collocation, and (c) T rN H using exact integration for N = 15.

We can also compare the elemental stiffness matrix for the TrH H m odal basis and

the T rN H mixed basis. The stiffness matrix is the m atrix which has entries defined

by:

Smn = (V0771) .

Figure 3.10 shows the two stiffness matrices for N = 15. Again the matrix corre

sponding to T rN H basis has a denser structure.

We now summarize the properties of the two classes of bases. The modal basis

(T rH H and Q uH H ) properties are:


59

(a)

: w xsss

Figure 3.10: Compaxison of the elemental stiffness matrix for (a) TrHH, and (b) TrNH bases for N = 15.

• The basis is hierarchical.

• Gaussian integration order is not dictated by the basis order.

• We are able to vary locally the number of modes per edge or interior.

• Elemental transforms and inner products are 0 ( N 3) operations.

• The interior-interior mass matrix is banded.

• The interior-interior stiffness matrix is banded.

In addition, this basis can be enhanced with other properties by varying the form of

Jacobi polynomials. For example, an interesting version proposed in [16], [32] uses

(a, @) = (2,2) and (2 m + 3 ,3) for the Jacobi constants. It has the following properties:

• The interior-interior mass matrix is diagonal.

• The interior-interior stiffness matrix is full.

There are many different choices in choosing a mixed basis. Their main properties

are:


60• The basis is non-hierarchical (Q uH N ,TrN H , and Q uN N ).

• Gaussian integration order is dictated by the basis order (Q uN N , TrN H a t

the edges, and Q uH N in the interior).

• The expansion order is (practically) fixed in nodal quadrilaterals (Q uN N ).

• Quadrilateral transforms and inner products cost 0 ( N 2) operations for Q u N N

and 0 ( m 2) for Q uH N .

• Triangle transforms and inner products are 0 ( 2 N 3) operations (TrNH).

• The interior-interior mass matrix is banded (TrNH); it is extremely sparse for

Q uH N and diagonal for QuNN.

• The interior-interior stiffness matrix is banded only for T rN H , otherwise it is

full.

These properties suggest situations where each type of basis is appropriate: For

example, the m odal basis can lead to high com putational efficiencies, if:

• The solution has local regions of interesting behaviour.

• The solution benefits from non-steady regions of interesting behaviour - this

can be captured by local p-refinement.

• The domain is highly irregular and needs a high ratio of triangles to quadrilat

erals.

On the other hand, the mixed basis can also lead to high efficiencies, if:


61• The solution, has a uniform variability.

• The domain is locally irregular, and thus it does not require a large number of

triangles to complement the quadrilaterals.

More importantly, it is the specific application th a t we consider and the dynamic

refinement procedure th a t ultimately decides which basis functions are the best choice.

If it becomes apparent tha t an initial choice of basis becomes inappropriate, it is not

an expensive operation to transform between bases.


Chapter 4

O perators

Now that we have set up the basic geometric properties of the elements, and how

to represent functions on them the next step is to show how their data can be ma

nipulated economically. In this chapter we develop a method for efficiently imple

menting the family of hybrid elements on a computer. Specifically we present an

efficient method for evaluating surface integrals on the elements. Also we examine

the approximation and wave propagation properties of these elements experimentally.

Finally, we provide a simple algorithm for minimizing the bandwidth of the matrices

that result from the construction of discrete linear operators for a domain.

4.1 Elemental Spatial Operators

In this section we consider the two basic operations that will be performed on a

function represented on an element. These are taking the spatial derivatives and

integrals of functions represented on each element type.

62


634.1 .1 In tegration

We take advantage of the tensor product element coordinate systems to perform

integration. The integrations over each element can be performed as a set of one

dimensional integrals using Gauss quadrature. If we used the reference coordinate

systems this would be very difficult since the limits of the “collapsed” elements are

not constant.

We first describe the choice of quadrature type for integrating each direction. We

will then motivate the inclusion of quadrature with non-constant weights in order to

reduce the number of points we use.

In two-dimensions we consider integrals of the form :

f f ( x )d x d y = f / ( x ( r ) ) ^ y c £ r d sJ P hysical J R eference u ( T J

= f /(x (r (a ) ) )9(-*> i f ^ dadbJ T e n s o r ^ ( T /

In three-dimensions:

/L f (* )d xd yd z = f / ( x ( r ) ) ^ ^J P h y s ic a l J R e fe r e n c e

drdsdt

= [ /(x (r(a )))JT ensor

d(x) d(v) dadbdcJTensor

We use the Gauss weights th a t will perform the discrete integral of a function as a


64

Element ‘a ’ ‘b ’ ‘c’Triangle G LL GRJifi -

Quadrilateral G LL GLL -Tetrahedron G LL GRJi'O GRJ2to

Pyramid G LL GLL GRJ2,oPrism G LL GLL G R J1jo

Hexahedron G LL GLL GLL

Table 4.1: G LL implies Gauss Lobatto Legendre which is the Gauss quadrature for a constant weight function with both x = ± 1 points endpoints included. G RJQip implies Gauss Radau Jacobi quadrature with (a, (3) weights and the endpoint x = — 1 included.

sum:

(1 - z)"(l + z)Bf ( z ) d z = Y ,J~l i=0

This will be used in each of the d directions in the d-dimensional elements. In table

4.1 we show the type of Gaussian quadrature we use in each of the ‘a ’,‘b ’ and ‘c’

directions.

4.1.2 F lu x Integrals

In the following chapters we will outline the Discontinuous Galerkin Method. In this

method it is necessary to evaluate terms of the form:

where dQ is the boundary of an element f2, for all the (f>n test functions in the elemental

basis. There are test functions for a triangle so the boundary integral is an


650 ( N 3) operation. This means that the flux integration is as expensive as the volume

integral. We can reduce the cost of this integral by examining the discrete sum form:

f M = Y^4>n(ai,0)fl (ai)wfJl (ai )+'^2(f)n( l ,bi ) f2(bi)WiJ2(bi)J a n i= 0 i= 0

t = 0

where J n and f n are the Jacobian and flux function for the nth edge.

We can rewrite the edgei flux as:

A , /&> = EJedgei j =Q i=Q W 0

where:

0 if j5{j —

1 if i = j

The fluxes for the other edges can be constructed in similar ways. Using this summa

tion representation we can now evaluate the surface flux integral by adding the edge

fluxes scaled by weight and Jacobians to the F field and then evaluating one volume

integral.

4.1 .3 D ifferen tia tion

A function represented at the quadrature points can be considered as a sum: if /,-j is

the (i , j ) th physical value of the function then the interpolating polynomial is:


66

f M = ' t ' t f ijht(a)hbj(b)i j

Here /if is the ith interpolating polynomial for the quadrature chosen in the ‘a ’ direc

tion. Thus, we can evaluate the derivative of this interpolating polynomial in each of

the (a, b) directions.

daf(a ,b ) = E E W M ) ( a)/lK i )n m

We can now evaluate the (i , j ) th quadrature value of the derivative as a m atrix vector

product:

daRai, bj) = J 2 f n i da(hn)(ai)n

Repeating the above we can evaluate the ‘b ’ derivative (and the ‘c’ derivative in

three-dimensions). To calculate the x ,y ov z derivatives we simply apply the chain

rule. In implementation we store the geometric factors for all the elements. This is

more memory efficient for straight-sided triangles and tetrahedra since the geometric

factors are a set of constants (except for (1 — b) and (1 — 6)(1 — c)2 factors).


674 .1 .4 Im p lem en tation and P rogram m in g

So fax we have outlined specifications for each element type. Each description includes

three coordinate systems, a set of basis functions, a set of operators and a set of surface

panels. By carefully defining each element type this way we can implement each type

using a unified methodology. We used an object oriented programming approach.

The first task is to define a “generic” element that has a sufficiently flexible da ta

structure. The computer language C+-F allowed us to define a base Element class.

The class was given storage for the following:

• The geometry of the element (i.e. the vertex positions and curvature informa

tion).

• The connections the element makes to other elements, through vertices, edges

and faces.

• Storage for the values of the field at the element’s quadrature points.

• Storage for the basis shape function coefficients.

• Parameters determining the expansion order for edges, faces and interior modes.

• A look up table for the functions (operators) defined on the element and its

data.

The last item is im portant for efficiency. If we were to create a class th a t did not

know how to operate on its own data, then we would need a list of conditional “i f

statements every time such an operation needed to be done in the code. Clearly this is


68not an efficient approach. The object oriented programming (OOP) approach replaces

the set of i f statements with a memory access that determines which operator should

be used for the element in question. Also this method is efficient for progra m m in g

purposes. Once an operator is defined for all the elements, any piece of code that

requires this operator only has to call the one interface to the operator regardless of

the element type. For instance, say we would like to differentiate a function held in

an array of values at quadrature points {field) then in pseudo code:

r y-r j- j.t t- u d f ie ld d f ie ld d fie ld .element— > Gradienti field, — ---- , —----- , — -----)ox ay oz

would take the spatial derivatives of field and place the x, y and z derivatives in the

arrays named S i m , and

A more complicated example might be to create two elements, set the data values

at the quadrature points of the elements, differentiate the fields and compare the

differentiation errors on each element. The pseudo code to set up the elements is

shown in figure 4.1. We would like to point out th a t the implementation of the

element constructors is provided in the library routines of A/’s k 'T'ocr . In figure 4.2

we show the code for the next stage of applying the derivative operators to the list

of elements we have created. Notice in this section th a t no explicit mention of the

types of each element is made when we use the operator on them.

In the next section we consider the discretization of the projection, the convection,

and the diffusion linear operators. They are formulated as discrete operators in each


69

/ / Make an array to hold the elements Element *elements[2];

/ / Set up the coordinates for a triangle Coordinate tri-coordinates; tri_coordinates.x = {-1, 1,-1}; tri_co ordinates.y = {-1,-1, 1};

/ / Set up the degree vector for the triangle int tri_degree_vector[4] = {2,2,2,!};

/ / Create the triangleTri elements[0] = new Tri(tri_degree_vector, tri_coordinates);

/ / Set up the coordinates for a quadrilateral Coordinate quad_coordinates; quad_coordinates.x = {-1, 1, 1, -1}; quad_coordinates.y = {-1,-1, 1, 1};

/ / Set up the degree vector for the quadrilateral int quad_degree_vector[5] = {2,2,2,2,1};

/ / Create the quadrilateralQuad elements[l] = new Quad(quad_degree_vector, quad_coordinates);

Figure 4.1: A piece of M e n T a r code demonstrating how a list of elements can be constructed.


70

/ / Set the quadrature point values of both elements for( i = 0; i < 2; i=i-Fl )

elements [i] — > Set-field ( “sin (PI*x) *cos (PI*y) ” );

/ / Create some storage for the derivatives of the fieldsdouble* data[2];for( i = 0; i < 2; i= i+ l ){

dx[i] = elements[i]— >Duplicate_data-space(); dy[i] = elements[i]— >Duplicate_data_space();

}/ / Take the spatial derivatives of the fields we just set for( i = 0; i < 2; i=i-f-l )

elementsjj]— >Gradient(elements[i]— >data, dx[i], dy[i])

/ / Calculate the error norm of the ‘x’ derivatives and the exact answer for( i = 0; i < 2; i= i+ l )

elements[i]— >Error(dx[i], “PI*cos(PI*x)*cos(PI*y)”);

Figure 4.2: A piece of A fe tt 'T c tr code demonstrating how operations can be made on a list of elements.


element and then globally assembled over the entire domain.71

4.2 Global Projection Operator

We construct the Galerkin projection that minimizes the L2 error for approximation

of a function by the C° basis. F irst we define the elemental Jacobi transform from

the polynomial space to the physical space:

N

U ~ ^ 'U-n<Pn

where un is the coefficient of the n th basis function <f>n. From this we can obtain an

approximation for the coefficients un. If we take the inner product of both side with

<j)m we obtain:

N

(0 rrn w) = ^ I ^ m i

which we can solve for un since (<£m, 4>n) is positive definite [20]. Explicitly, we can

write the projection coefficients as:

= B mn (0m w)

where B is the m atrix with entries B mn = (<£m, 4>n)- This approximation minimizes

the residual:


72

N

r — 11 y i UTnfimWL? ■

4.2.1 G lob a l C onvergence in Skew E lem en ts

A: D:

G: H:

Figure 4.3: Meshes (A-H) are consisted of three quadrilaterals and two triangles which are progressively skewed by shifting the interior vertex.

ofcW

10* 10*

10*

10* 10*

10* 10* 10* r

10'

10"10"10"10"

11 13

Expansion Order

Mesh A Mesh B MeshC Mesh D MeshE Mesh F Mesh G Mesh H

15 17 19

Figure 4.4: Convergence in the L2 norm for modal projection of the function u sin('Kx)sin(jvy) on meshes A — H.


73

10"

10"

10"

10"

10"

o 10'Tf r l 1 0 "

1 0 ™

1 0 "

1 0 ' “ r

10"'10’ ’

Mesh A Mesh B MeshC Mesh D Mesh E Mesh F MeshG Mesh H

7 9 11 13

Expansion Order

-1—015 17 19

Figure 4.5: Convergence in the L 2 norm for mixed, projection of the function u = sin(irx)sin(Try) on meshes A — H.

We now consider the effect of the skewness of the physical elements on the accuracy

of the projection operator. In figure 4.3 we examine 8 different meshes consisted

of triangles and quadrilaterals. We start by projecting sin(-Kx)sin{-Ky) onto a square

domain covered with standard elements. Figures 4.4 and 4.5 show results for the modal

and mixed bases that demonstrate exponential convergence is achieved. Subsequently,

we make the elements covering the domain progressively more skew in the meshes B-

H. In each case we see tha t exponential convergence is achieved, even when one of the

triangular elements has a minimum angle of about 10-3 degrees. Hence, the accuracy

of the method is extremely robust to badly shaped elements. Also, we note that

the similarity of the convergence curves demonstrates that the rate of exponential

convergence is unaffected by the skewing.

The independence of the rate of skewing might be expected based on our previous

observations about the accuracy of projection using the orthogonal basis. We saw


74that the coefficients in the expansion of an infinitely smooth function are bounded

by exponentially decaying functions of the polynomial order which are scaled by the

Jacobian of the geometrical mapping. So we infer tha t as an element becomes more

skew the energy in each mode should reduce in proportion to the area of the element.

4.3 Convective Operator

We now consider the two-dimensional linear convective equation for u(x, y, t):

This has been formulated for triangular elements in [20], and we note that using

quadrilateral elements requires very few changes to this method. We consider the

weak form of this equation in £lk, th a t is:

Find u 6 such th a t for all w G H l (£i)

( du du du \ , . ,(a t - a d i ~ % ’w) = 0 ’

Following a Galerkin formulation so tha t the trial and test spaces are spanned by the

same basis we obtain:

dx J V dy

We now define two local operators Bk and Lk{6):

ZLfi


75

B k = (<£*,<&)

Lk{0) =COS(0) i t

L ' + s i< e) U d- § -

where we have normalized the convection velocity appropriately so th a t its direction

6 is defined by the scalars a, b. Based on these definitions we now construct two new

operators for the entire domain:

and similarly:

m =

L 1 0 ... 0

0 L i . . . 0

0 0 ... L k

B =

B i 0 . . . 0

0 B 2 ... 0

0 0 ... B k

We can now assemble these elemental operators into a global operator, by means of

the Z operator [40] tha t assembles the local coefficients into the global coefficients

and ensures C° continuity.


76

Element 2

Element 1

Element 2

Element 1

Local Numbering Global Numbering

Figure 4 .6 : Illustration of local and global numbering for a domain co n ta in ing one quadrilateral and one triangular elements. Here the expansion order is N = 3, and we only show the boundary modes.

To illustrate this global assembly procedure, we consider a global domain made

up of two elements as shown in figure 4.6. The expansion order shown here is N = 3

which means there are six boundary modes on the triangle, and eight boundary modes

on the quadrilateral.

The to ta l number of local degrees of freedom is therefore Niocai = 14. Since three

modes meet along the connecting edge the number of global degrees of freedom is

eleven and so for this case Z is a 14 x 11 matrix as shown in figure 4.7.

The superscripts denote the local or global nodal number and the subscripts denote

the element number. The absolute column sum gives the multiplicity of a mode and

we see th a t columns 5,6 and 7 all have a multiplicity of 2. We also note that the

absolute row sum is always 1 since there is only one value of each local mode. It is

also possible to have a (—1) entry if we have two elements where: (1) edge 1 meets

edge 1, (2) edge 1 meets edge 2, (3) edge 2 meets edge 2, (4) edge 3 meets edge 3, (5)

edge 3 meets edge 4, or (6) edge 4 meets edge 4, then the local co-ordinates at the


77

1t-H

t

1

“ 3 1 u 9

u \ 1 « 2

1 U9

1 <

U j 1

u l = 1 u f

... u 97

“ 1 1 u 9

« 2 1 u f

1 f.Su 10

u \ 1 1~l9 _ U11

*5 1

. “ 6 . 1

Figure 4.7: Z-matrix map from global to local degrees of freedom

edges are in opposite directions. If we are using a hierarchical basis then we need to

negate odd modes along one of the edges.

Having defined the assembly operation, we can now write the global system as:

du9— = G(SK ,

where G(6) = (Z tB Z )~ 1(Z tL(9)Z).

The behaviour of this operator is very important with regards to determining the

maximum time step we will be able to use for any problem which involves an explicit

treatment of convective contributions.


784.3.1 A ccu racy o f th e C onvective O p erator

We tested the accuracy of the Galerkin convective operator using a third-order Adams-

Bashforth temporal scheme, and a periodic domain as shown in figure 4.8. We started

with the initial condition u = sm(7rcos(7ra;)) and examined the error a t t = 0

and t = 2. The convection velocity was constant. We chose a time step small enough

so that the time-stepping error is small compared to the initial projection error for

N < 16. We see th a t exponential convergence is m aintained after one time period.

10 ' ' Error at t=0 Error at t=2

L.10'

.<10

10'

10'

Expansion Order

Figure 4.8: Exponential accuracy is achieved for the wave equation with u = sin(7rcos(7rx)) as initial condition.


794 .3 .2 S p ectru m o f th e G alerkin 2D C on vective O perator

We can now examine the behaviour of this operator by examining its eigenspectrum.

The distribution of the spectral radius, p{9), shows us the level of directional inhomo

geneity of wave frequency supported in a given domain. We first consider a periodic

box that is discretized with essentially standard elements, and then we examine how

deforming these elements within the box affects wave propagation.

Ca)

Figure 4.9: Spectral radius of the weak convective operator on a periodic domain, N = 12.

In figure 4.9 we show three discretizations of the periodic box. The first one (a)

employs only quadrilaterals, the second one (b) a mix of quadrilaterals and triangles,

and the last one (c) only triangles. G{9) was constructed using a 12th order expansion

and the spectral radius of the G{9) for each mesh is shown as a function of 9. In the

upper right quadrant we see tha t the spectral radii are very similar for all three cases,

but in the lower right quadrant we see that there is a marked difference between the

spectral radii of the quadrilateral mesh and the triangle mesh with the hybrid mesh


80between these two cases.

Theoretically, we do not have to consider the spectral radius of the mixed basis, as

we have shown that since it is numerically similar to the modal basis it will share the

same spectral properties for linear operators. However, we did run this test for the

quadrilaterals using a nodal basis and obtained the same spectral radius to machine

precision, confirming the theory.

i121111101

<r

Mesh (a) Mesh (b) Mesh (c)

15105Expansion Order

Figure 4.10: Growth of the spectral radius of the Galerkin convective operator with expansion order.

So far we have examined the spatial variation at a given expansion order. In figure

4.10 we now demonstrate tha t supfl p{G{9)) grows as 0 ( N 2) for the modal basis used

on all three meshes, and again we note that the mixed basis has exactly the same

property to machine precision.

We have an exact fit for the numerical spectral envelope of the Galerkin convective

operator on a periodic square domain discretized with regular quadrilaterals. This

can be represented as (see also figure 4.11):


81

■°s?

OS

41 •

Figure 4.11: pa(&) is an exact fit for the spectral radius of the Galerkin convective operator on a periodic domain tiled with regular quadrilaterals and with N = 12.

i>’ -.1

.10 -OJ 00 OJ to

Figure 4.12: Pa(#) is a close fit for the spectral radius of the Galerkin convective operator on a periodic domain tiled with triangles and with N = 12.


82

(c)

Figure 4.13: Spectral radius of the Galerkin convective operator on a periodic domain discretized with non-regular elements.

Pn{Q) =

\/2/?n (O)szn(0 + f ), if 6 > 0

y/2pa(Q) sin{6 + ^ ) , otherwise.

We also have an approximate fit for the numerical spectral envelope of the Galerkin

convective operator on a periodic square domain discretized with regular triangles.

This is described by (see also figure 4.12):

Pa (0 )\/2(pA(O)sm(0 -1- f ) — sin(2Q)), if 9 > 0

y/2(p^(Q)sin(9 + + ( p a ( 0 ) - pa(O))sin{20)), otherwise

We have not so far indicated how the operator G{6) depends on the skewness

of the physical elements. We will now consider the same periodic box discretized


83with deformed elements, so tha t these mappings are not simply scaled identities. In

figure 4.13 we have created deformed elements by simply shifting the vertex in the

middle of the quadrilateral and triangle meshes we have just considered. In addition,

we created a new triangle mesh by taking the Delaunay triangulization of the given

vertices. This time we see th a t the deformation of the triangle mesh has increased the

spectral inhomogeneity of the operator, but th a t this can be ameliorated by choosing

the Delaunay triangulization.

4.3 .3 T h ree-D im en sion a l G alerkin C onvective O perator Spec-

trum s

We will now consider the same problem posed in three-dimensions. Each element

type will be shown to have different spectral properties. The three-dimensional linear

convective equation for u ( x , y, z, t ) is:

d u (x ,y ,z , t ) _ du . .d u . . .d u . .d u— a t-----+ L u s a t + a{x' y ' z ) d i + b(x ’v ' z ) a i + c(x' v ' z )T z = ° '

We consider the variational form of this equation in f2fc, that is:

Find u G H l {Vt) such th a t for all w € H l (Q)

( du du ,d u \ . u r r l .9 t + a d i + % + cd-z'wj = 0 '

Applying the Galerkin formulation so that the trial and test spaces are spanned by


84the same basis we obtain:

(^ni 0m)dumdt { ^ n,a d x ) + (^ " ’6 d y ) + (? n''

90m : dz

UT

We now define two local operators Bk and LkiO):

B k = (0E.0&)

Lk(0) = cos(0)cos{9) (<#;, + cos(<t>)sin(0) + sm(0)9y dz

where we have normalized the convection velocity appropriately so that its direction

9 is defined by the scalars a, b.

Based on these definitions we now construct two new operators for the entire

domain:

L(9) =

L x 0 ... 0

0 Lo ... 0

0 0 ... Lfc

and similarly:


85

B =

B l 0 ... 0

0 Bo ... 0

0 0 ... B K

As in the two-dimensional case (section 4.3) we assemble the boundary degrees

of the freedom to make the operator act continuously across the element interfaces.

This time the Z operator maps the globally numbered modes to the locally numbered

vertex, edge and face modes.

4.3 .4 S p ectru m o f th e G alerk in 3D C on vective O perator

As before we examine the spectral radius of the convective operator as a function of

the direction of the wave vector. This is now a vector in 3-space so the variation of

the spectral radius is a two-dimensional surface.

In three-dimensions we have four element types to consider. The observation we

wish to make is th a t the magnitude of the spectral radius of the Galerkin convective

operator scales as N 2 and the directional variation is a function of the element length

in a given direction. This “length” can be defined as the minimum length between a

vertex and an opposing face in the given direction.

We first consider a box discretized with tetrahedra. The spectral radius surface is

shown in 4.14. There is clearly a drop in support at the top rightm ost octant and the


86bottom leftmost octant. It is evident tha t the elements all have their longest edges

aligned in the direction between these octants.

Figure 4.14: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with six tetrahedra and fourth-order expansion.

Next we consider a box discretized with pyramids. The spectral radius surface is

shown in 4.15. The mesh has a six-way symmetry that is reflected in the symmetry

of the surface. The compression of the distance from apex to base causes a slight

increase in resolution in that direction.

Next we consider a non-unit box discretized with prisms. The spectral radius

surface is shown in 4.16. This is a simple two element mesh showing the effect of

compressing the uniform direction of the prisms. The surface shows overresolution in

the vertical direction compared to the horizontal directions. If we scale the vertical

direction to match the horizontal lengths the we would see tha t the surface is quite

uniform except for underresolution in the direction aligned with the longest triangular

face edges.


87

k .

-5

. * p £ 2 * « i *»• ‘iV iv iX

"« * '

{ j r - . t a i i *--4 •

. t c ,

Figure 4.15: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with six pyramids and fourth-order expansion.

k '

Figure 4.16: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with two prisms and fourth-order expansion.


88Lastly we repeated the experiment with a periodic box tiled with eight hexahedra

shown in figure 4.17. The surface of the spectral radius reflects the symmtries and the

same form of over resolution in the diagonal directions as we saw in the quadrilateral

mesh case.

Figure 4.17: Left: Exploded spectral element mesh, Right: Variation of the spectral radius of the convective operator in a periodic box discretized with eight hexahedra and fourth-order expansion.

4.4 Diffusion Operator

We now consider the two-dimensional elliptic Helmholtz equation:

Z

(v 2 - A ) u = f , A > 0.

Again using the Galerkin formulation and integrating by parts we obtain:


We will define a new set of K operators:89

L t = [ ( V ^ ,V ^ ) + A ( A n,A.)]

and a new set of K vectors:

Then repeating the process to assemble the weak convective operator we can assemble

the subdomain operators into a global operation to obtain:

u 9 = (Z tL Z )~ 1Z tF,

where the bold letters denote vectors.

4.4.1 T h e Schur C om p lem ent M eth o d o f Inverting G lobal

O perators

We are left with a m atrix inversion problem. The m atrix A = Z lL Z has to be

inverted. The Z m atrix groups together the degrees of freedom for the boundary-

mode shapes for all the elements. The remaining interior degrees of freedom are

collected in one group per element. The m atrix now has the form shown in figure

4.18.

We can take advantage of the bandedness of the A boundary-boundary matrix,


90Boundary d.o.f. Interior djj.f.

A B

Bt C

A = B =

C =

Figure 4.18: Form of the global operator matrix. Notice the sparsity of A and the block nature of B and C.

and the decoupled C matrix. The system can be rewritten as:

A - B C ~ lB T 0 06 f b - B C - l f {

B T C 0i f i

where 0& are an array of the unknown global boundary coefficients and 0,- are the

unknown local interior coefficients.

4.4 .2 O p tim al N um bering for th e Schur C om p lem en t B ound

ary M atrix

To solve the global matrix system the boundary system is solved first . This is done

by using the Schur complement. A direct solve on this system involves processing

a large da ta structure. Here we present a method for applying the Cuthill-McKee

algorithm [38] to reduce the bandwidth of the boundary solve matrix.


91Although the boundary system matrix is large it is sparse. By reordering the

global numbering of the unknowns we take advantage of the sparcity and pack the

data into a band along the diagonal of the matrix. We define the bandwidth of

the m atrix as the width of this band. The bandwidth is equivalent to the largest

elemental bandwidth taken of all the elements.The elemental bandwidth is defined as

the largest difference between the global identity of the unknow ns in a single element.

All the unknowns in a single edge (or face) are grouped together and numbered

consecutively because they will inevitably be closely associated in an optimal renum

bering. We define an approximate bandwidth as the maximum difference between

the global numbering of any two facets in an element. M inim iz in g the approximate

elemental bandwidth over all the elements is almost equivalent to the original mini

mization problem.

We start this re-ordering at a facet on the boundary of the region and consider all

the elements in which it lies. We order all the facets in these elements in ascending

order of their degree. The degree of a node is defined as the number of facets with

which it shares an element. The new facets are collectively defined as a layer. We

apply this procedure to all the facets in the layer, in their new order, to form a new

layer. We order the facets in the new layer in order of their degree. We apply this

ordering repeatedly to create a sequence of layers that span the mesh.

Because any element will contain facets from two layers the resulting bandwidth is

equivalent to the maximum number of facets in any single layer. We choose the initial

facet so that each of the layers contains the least number of elements possible. Thus,


92we start the algorithm on the boundary to restrict the directions tha t the layers can

expand. Because we do not know which boundary facet will give us the best result,

we start the algorithm from a number of the boundary facets in the order of their

degree. This number can be set for speed of the overall algorithm.

The described re-ordering leads to a distinct structure in the final boundary solve

matrix. The banded region of the matrix is formed from intersecting “squares” that

show how the different layers formed by the “wave-front” interact.

i Ui

t' k®' i t*■ n' IB

S g i lBfiSSS?7

ii!

Figure 4.19: Here we show the domain that we solved the Helmholtz equation in the following forcing function, / = — (3.47T2) sin (7r.a:)cos(7ry) cos (0.2ir.z) and boundary conditions u = sin (7 r .a :)c o s (7 r .y ) cos (0.27r..z), with periodic boundary conditions at the spiral ends. We see a dram atic bandwidth reduction because of the aspect ratio of the mesh and the exponential decay in the error with increasing expansion order.

We could have performed an O(Nbl) search where N b is the number of unknowns

but this would be prohibitive in all but the simplest meshes.

4.4.3 N u m erica l R esu lts

In this section we aim to examine the effect on computational efficiency of reducing

the bandwidth of the Schur complement matrix. Both optimized and unoptimized or-


93

p Bandwidth User solve time in secondsinitial final unbanded direct banded direct iterative

3 361 59 0.53 0.50 0.824 1501 244 1.14 0.98 2.415 3441 558 2.59 1.94 6.196 6181 1001 5.7 3.70 17.47 9721 1573 10.95 6.08 39.358 14061 2274 20.45 10.37 76.099 19201 3104 — 16.77 141.66

Table 4.2: Timings for the Schur complement obtained on a single node SGI R8000.

derings have been implemented in the spectral element code M e k T o r . We present

two examples: the first is a solution to a scalar elliptic Helmholtz equation, while the

second is a solution to incompressible Navier-Stokes equations. In order to obtain a

sensible measure of this effect we consider the CPU time for the banded direct solve

and the iterative solve of the boundary system resulting from the Helmholtz equation

( V 2u — u = f ) against the CPU time taken by J ^ e k “T o l t to perform a direct solve

on the unbanded boundary system. However, because of memory requirements we

run into problems w ith the unbanded system becoming too large to run while the

other methods are still very manageable.

We have performed the optimization analysis on the mesh inside a spiralling pipe

as shown in figure 4.20. In this case we solved the Helmholtz equation with a forcing

function designed so th a t we could measure the numerical error. In each of the three

methods the L<i and the errors were identical for a given expansion order.

In table 4.2 we have compared the timings for the backsolve routine only for the

unoptimized and optimized direct solves versus a diagonally preconditioned conjugate


94gradient iterative solve. The improvement in CPU time for the banded computations

compared with the unbanded computations is substantial, where an improvement of

an order of magnitude is obtained compared to the iterative solve. The preconditioner

used in the iterative solve is based on the diagonal; in current work we are using the

hierarchies of the expansion basis to obtain more effective preconditioners. The CPU

savings reported here are coupled with an extreme reduction in storage requirements

making this method quite competitive with the iterative method. Of course the

iterative method wins for very high orders p due to memory restriction, but for these

kinds of orders the banded method looks superior given a reasonable memory capacity.

Krror

864

Expansion ordar

Figure 4.20: The exponential decay in the error with increasing expansion order for the Helmholtz equation solved on the spiral domain.


95

IJ

is i« a s

I

rtf’

(a) (b)

(c) <d)"H

Figure 4.21: Convergence test for th e Helmholtz equation using quadrilaterals and triangles, with Dirichlet boundary conditions. The exact solution is u = sin(irx)cos(7ry) and forcing function / = —(A + 2'K2)s in ( t t x ) c o s (jry).


96

4 .4 .4 C onvergence for th e G alerk in H elm h oltz O perator

In figure 4.21 we demonstrate convergence to the exact solution with p-refinement

and /i-refinement for the Helmholtz equation with A = 1; the exact solution is:

u = sin(ncos(nr2))

and forcing function / = —(A + 47r4r 2szn(7rr2)2)sin(7r(cos(7rr2))) — 4ir2 (nr2 cos (irr2) -f-

sin(nr2))cos(ncos(7rr2)), where r 2 = x 2 + y 2 for Dirichlet boundary conditions.

10“

Iff*26 61 36 41 4 6 51 S 6 6116 2111

zo

to

OS

ao

1000

Figure 4.22: Convergence test for the Helmholtz, using a triangle and a quadrilateral, with Dirichlet boundary conditions: u = sin(ncos(nr2)) .

In figure 4.22 we show p — type convergence for a more complicated exact solution.

This example demonstrates that the method is stable to a t least N = 64.

C onvergence in 3D H y b rid E lem en ts

We constructed a blocky fish shape domain to test the Helmholtz solver with all the

three-dimensional element types connected together. In figure 4.23 we show th a t the


97

solution converges to the correct solution for the Helmholtz problem with increasing

N.

Pyramid

Hexahedron

l<r

10°

• 10“

10-

10'7 r

Expansion Order

X u.

Figure 4.23: Convergence for the Helmholtz problem on a mesh of hybrid elements. Dirichlet boundary conditions u = sin(x)sin(y)sin(z) and A = 10000.


98Convergence in 3D Skew Elem ents

We have shown in chapter 3 that the approximation properties of the orthogonal

bases should not be adversely affected by the shape of triangles or tetrahedra. We

constructed a twelve element tetrahedral mesh, shown in figure 4.24, for a box. All

the tetrahedra share a vertex at the center or the box. For the first test of the

Helmholtz solver this vertex was at the center of the box. We then moved the vertex

progressively closer to one of the comers of the box and ran the solver for each

position. In figure 4.24 we show that the convergence of the L error norm was only

minim ally effected by the element skewing. This is very good news because we can

use unstructured meshes without worrying too much about mesh quality for accuracy.

We should however still be careful about using elements with small angles because we

have already shown tha t these elements have over-resolution in their shortest vertex

to edge direction. This can lead to a very restrictive Courant-Friedrichs-Lax (CFL)

condition in wave propogation problems.


99

L _ E r r o r . C e n te r V e r te x = < 0 ,0 ,0 )

L _ E r r o r . C e n te r V e r te x = < 0 .4 ,0 .4 ,0 .4 )

E rr o r , C e n te r V e r t e x = ( 0 .8 ,0 .8 .0 .8 )

L „ E r r o r . C e n te r V e r t e x = ( 0 .9 5 .0 .9 5 ,0 .9 5 )

L . E rr o r . C e n te r V e n e x = < 0 .9 9 .0 .9 9 ,0 .9 9 )

5 10Expansion Order

1 5

Figure 4.24: Convergence is guaranteed for the Helmholtz problem even on very skewed tetrahedra. Dirichlet boundary conditions u = sm (^ )sz n (^ )sm (^ p ) and A = 1.


C hapter 5

Incom pressible Flow Sim ulations

In this chapter we show the results from applying the hybrid grid technology to solve

the Navier-Stokes equations. We modified an existing version of the N e t t T o t r code

[30] to allow the use of quadrilaterals in two-dimensions and hexahedra, prisms, and

pyramids in three-dimensions in addition to the original support for triangles and

tetrahedra. The solvers needed minor modifications due to the general nature of the

original code and the compatibility of the new element representation with the older

C-based representation. We tested the new version of A T e k . 'T otr on simple physical

models in order to verify the robustness of interfacing different element types together.

We then extend the algorithm to be able to solve the Navier-Stokes equations in

moving domains using the ALE formulation. We show tha t the numerical scheme is

accurate for a moving domain test problem.

100


1015.1 Incompressible Navier-Stokes Equation

5.1.1 F orm ulation

The two-dimensional incompressible Navier-Stokes equations are:

-ST + (v • V )v = —Vp 4- i/V2v + F, at

V • v = 0,

where v denotes the velocity of the fluid with components v = \u(x, y, t ), v(x, y, t)]T

in the x and y directions; p(x, y, t ) is the pressure; F (x, y, t) is a forcing function and

v is the kinematic viscosity. To discretize these equations in time we use a high-order

splitting scheme [39]. A variation of this scheme using the spatial operators we have

already described is explained and demonstrated in [30].

5.1.2 S um m ary o f Schem e, B ou n d ary C onditions and Im ple

m en ta tion

The splitting scheme involves the following four substeps:

v = ^ ocq-vn- q + /3ffN (v"“») + F n+l), (5.1)q = 0 9 = 0

—/£ 1/?,N(v"-'0 - v ^ 13,[V x (V x v -« )] (5.2)g=0 9=0

dpn+l= n

dn

V p = V - ( A ) , (5.3)


V2v n+1 - ————v n" 1 = ---- — v - A £Vp"+1.i/At i/At

102

(5.4)

This has been implemented in the J\fs k T ctr code [30] for triangular and tetrahedral

elements and in PRISM [40] for quadrilateral (nodal) elements. M erC T a r has now

been recoded in an object oriented way so th a t it now handles all the element types

we have described in chapter 2. The first step we call the non-linear step. We assume

that the velocity fields axe stored in terms of coefficients of the shape functions we

outlined in 3. We then evaluate the velocity fields at quadrature points for each

elements using a transform based on tensor product summation of the modes; this

has been detailed in [30]. We now have a “physical” representation of the velocity at

a set of points. For the non-linear step we take the spatial derivatives of these fields

at the quadrature points and form the the non-linear convection term s by multiplying

the velocity and its derivatives together at these points. We then take the current

values of the non-linear terms and either one, two or three non-linear terms from

previous times steps and add them together w ith a set of coefficients detailed in [39].

We now have the intermediate velocity as a field at the quadrature points.

For the second step we go through the list of boundary conditions for the elements

and calculate the Neumann boundary condition for pressure using the compatability

condition from equation 5.3.

For the th ird step we take the divergence of the intermediate velocity, using the

same spatial element-wise derivative matrices as before. We then take the inner

product over each element with that element’s basis functions. We now have a set


103of coefficients for the right-hand-side of the pressure equation. Next we calculate

the pressure field using the implicit solver we outlined in chapter 4, using the Schur

complement of the global discrete Laplace operator.

For the last step we need to evaluate the pressure field at the quadrature points

and then take its gradient also. From this we calculate the right-hand-side for the

velocity solves. We take the inner product of the righthand side with the set of

elemental basis functions. Again we use the implicit solvers to invert the Helmholtz

equations for the velocity.

5.1 .3 W annier F low

The first test we consider is Stokes flow past a rotating circular cylinder next to a

moving wall. The exact solution due to Wannier [41] allows us to evaluate the error

in a domain involving curvilinear elements. The exact solution can be written:

u(x, y i = y + d) 2(A + F Vl) r, K x l('

b r o ~ ~ k [ s + 2 y i ~

2yi(s + y i)(s + yQ ATi

v { x , y i = y + d) = (A + F yx){K2 - K x) -2I 1A 2

2 y i ( s - y i ) ( s - y i ) l _ DK 2 J

rxs Tf \ ^ B x y x{s + yx) 2Cxyx(s - yx)

- D

where we define:


104

D = —U, F = Uln (T )’

K i = x 2 + ( s + y i)2, k2 = x 2 + (s — 7/i)2, s2 = d 2 + R 2, F —d + s d — s'

Here we have used a cylinder of radius R = 0.25 which is a distance d = 0.5 from the

moving wall. The wall is moving with a velocity of U = 1 and the cylinder is rotating

in a counter clockwise sense with an angular velocity of uj = 2. The domain was split

into 65 elements and the discretized domain is shown in figure 5.1.

to 4

tff4TO*

Iff* -

i<r'to*nr*

to"*

L.

u

tj

u :

-a*

11 13 IS 17

gxpmsioo O rder

Figure 5.1: Discretized solution domain for the Wannier-Stokes flow using 68 elements. Left: Graph of exponential spatial convergence, Top Right: Hybrid spectral element mesh, Bottom Right: Streamlines of the steady state Stokes flow.

The steady state solution with Dirichlet boundary conditions using an expansion

basis of L = 11 is shown in figure 5.1. This figure shows iso-contours of velocity,

streamlines of the steady flow and the exponential convergence to the exact solution

with increasing order.


5 .1 .4 K ovasznay F low105

1.5

0.75

- 0.1 0.5X

0.75

0.25

-0.I0.5

x

Figure 5.2: Steady state solution for the Kovasznay flow a t Reynolds number Re = 40 using a discretization using K = 24 elements. Left: The spectral element mesh used, Right: Steady state streamlines.

The first Navier-Stokes solution we shall consider is the Kovasznay flow. This is a

laminar flow behind a two-dimensional grid, the exact solution of which is due to

Kovasznay [42]. This solution can be written as a function of Reynolds number Re

in the form:

u ( x ,y ) = 1 —eAlcos(27ry)

v (x ,y ) = ^ - e Alsin(27ry)Z7T

where

x Re (R e 2 A 2\ *~ T ~ ( “ i " )


106

Parameter ValueDimension 2d

Re 40A t 0.0001

N-Range 5 to 11K th 12

HQuad 12Method Galerkin

Table 5.1: Simulation parameters for the Kovasznay flow.

Using the exact solution as Dirichlet boundary conditions, a steady state solution

was obtained using the discretization shown in figure 5.2. Also shown in this figure

are the steady state streamlines and iso-contours of the rr-component velocity a t a

Reynolds number of Re = 40 using an expansion basis of N = 7.

Using the exact solution allows us to calculate the error, in the L00 and H i norms,

with expansion order as is shown in figure 5.3. A summary of the simulation param

eters is given in 5.1.

We repeated this experiment with a three-dimensional mesh. Again in 5.4 we see

tha t as we increase the expansion order we obtain exponentially increasing accuracy.

5.1 .5 C ylinder F low

Flow past a cylinder provides a good way to verify the M en.'Totr code. For R e > 40

vortex shedding occurs a t the cylinder and a von Karman street of vortices forms in

the wake of the cylinder. This shedding process causes the forces on the cylinder to

oscillate with a distinct frequency. This is known as the Strouhal frequency (S t). The

Strouhal frequency will be used as a measure to compare the results from the hybrid


107

L_ Error in H, Error in L_ Error in H, Error in

10'

10 '

10'

10'

6 115 7 8 9 10

Expansion Order

Figure 5.3: Convergence in the L ^ and H\ norms as a function of expansion order for the steady state Kovasznay flow at a Reynolds number Re = 40.

A /s k T a r code and other codes. For Reynolds numbers up to approximately 190

the flow structures remains two-dimensional. Above this number three-dimensional

instabilities occur, causing the two-dimensional approximation to be increasingly in

accurate for higher Re. Thus, we will consider a range of Re up to 250 in order to

investigate how difficult it will be to resolve high gradient fields.

For this test of the hybrid incompressible code we consider two-dimensional flow

past a circular cylinder. The cylinder has unit diameter, and the domain surrounding

the cylinder is a rectangle [—22,69] x [—22,22]. Uniform velocity boundaries are used

a t the inflow, upper and lower boundaries. Zero Neumann boundary conditions are

used for velocity and the pressure is set to zero at the outflow. Figure 5.5 shows the

domain, mesh and boundary conditions.


108

10 ' '

icr

10

1CTui ^

w 10-6

10"

1CT

1 0 '

10- ’

J L . 1 1 ' 1 ' 1 1 ' 1 ' i L10 12

Expansion Order

Figure 5.4: Top Left: Three-dimensional spectral element mesh used to solve Kovasznay flow on, Top Right: Iso-contours of streamwise component of velocity, Bottom: Convergence to exact solution with increasing order.


109


Re 100 to 250A t 0.002

N-Range 10 and 12K th 448

R-Quad 332Method Galerkin

Table 5.2: Simulation parameters for the incompressible cylinder flow simulation.

Resolution Re=100 Re=150 Re=200 Re=250Hybrid N e k T a r K = 780, N = 10

K = 780, N = 120.16590.1662

0.18510.1853 0.1972 0.2054

M s n 'T a r [30]) K = 173, N = 11 0.1667 NA 0.1978 NAPrism [12] 14000 dof 0.1664 0.1854 0.1969 0.2051

Table 5.3: Variation of Strouhal frequency with Reynolds number for incompressible flow past a two-dimensional cylinder

The mesh has 332 quadrilaterals and 448 triangles. It was designed to direct res

olution around the cylinder, have regular resolution in the wake behind the cylinder,

and block out with large elements to push the farfield boundaries away from the

cylinder. We ran the simulation at N = 10 for Re = 50,100,150,200 and 250.

In the following table we show good agreement between Strouhal numbers obtained

by Hybrid M e k 'T a r , M e k T a r [30] and Prism [40]. A summary of the simulation

parameters is given in table 5.2.

In figures 5.6 and 5.7 we show iso-contours of vorticity of the wake region of the

cylinder at Re = 50,100,150,200 and 250. We can see that at Re = 50,100 and

150 the vorticity is smooth and the vortex structure is well resolved. At R e = 200


110

Resolution Re=100 Re=150 Re=200 Re=250Hybrid M s k T a r K = 780, N = 12 1.34468 1.33136 1.34473 1.36339

Prism [12] 14000 dof 1.35000 1.33336 1.34116 1.35769

Table 5.4: Variation of C& with Reynolds number for incompressible flow past a two-dimensional cylinder

and more so a t Re = 250 we see th a t the bulk shape of the vortices is correct but

the features are a little unresolved. There are also streaks of spurious vorticity at

Re = 250. These effects will be reduced by increasing resolution.

M e n T a r has been taken to higher Reynolds number flows past a cylinder with

the two-dimensional module and with the Fourier module I wrote based on the model

in [40]. The Fourier module uses two-dimensional planes connected with a homoge

neous third direction for which we use a Fourier expansion. The results from these

simulations have been published by Ma Xia [45] and Evangelinos et al. [46]. The

numerical method was adapted from [40] to include both triangles and quadrilaterals

in the two-dimensional planes and allow variable expansion order from element to

element. The Fourier module is also being adapted by Evangelinos [47] to include a

time dependent spatial m apping in the third direction. T hat work is an adaption of

the work by [48] where only nodal quadrilaterals were used in the two-dimensional

planes.

5.1.6 T h e R o b o tu n a

So far we have demonstrated th a t Ss/ s k 'T a r can correctly simulate simple flows in

relatively simple geometries i.e. flow past a cylinder or flow in a box. Now we push


I l l

**=(1,0)

20

p = 0

-10

•206 0-20 -10 0 20 3 0 4 0 5010

ao(l,0)

\ / \ / \ / \ /___ ------------1 >t----------

- / \ / \ / \ / \ iiiiiii

i ; i(

D * ^ V \ Ai ii i

i

i i I !!

I /,\ 1 A l A 1 XX . , . .L0 5 1 0

X

Figure 5.5: Domain and mesh for the two-dimensional incompressible flow past a circular cylinder simulations.


112

Re 50

o

10

Re 100

' '

Re 150

■SSe'i ®8i

Figure 5.6: Instantaneous contours of vorticity for incompressible flow past a two- dimensional cylinder at Reynolds numbers Re = 50,100 and 150.


Figure 5.7: Instantaneous contours of vorticity for incompressible flow past a two- dimensional cylinder at Reynolds numbers Re = 200 and 250.


114the envelope by considering the flow past a Robotuna. Triantafyllou et al. at the

towtank facility at MIT have constructed a robotic tuna tha t can swim up and down

the towtank. They kindly provided the surface definition of the Robotuna device.

W ith the help of PhD student Mike Kirby [82] and FELISA [3] we constructed a

tetrahedral mesh around the surface definition. Ideally we would have liked to use

a thin layer of prisms around the fish surface, but as a first pass it was instructive

to use an off the shelf tetrahedral mesh generator to produce the mesh. As mixed

element mesh generators come on line to meet demand we expect that this task will

be as straightforward as tetrahedral meshing is today.

We did come across some problems in generating the unstructured mesh. The

actual robotuna is covered by Lycra, so we used a spline to approximate the surface

curvature. However, in doing so we found tha t this isoparametric represention [23]

of the surface caused a small minority of surface elements to become tangled. The

tangling is caused by the curved face projecting through one or more of the other faces

of the tetrahedron. To overcome this problem we identified the tangled elements and

replaced the curvature w ith linear blending of the neighbouring elements’ curvature.

In future work the curvature problems will be removed by the use of prismatic

elements at the surface of the fish. Only extreme deformation of a prism’s triangle

face will cause the prism to become tangled.

In figure 5.8 we show the mesh and surface spline on the surface of the tuna and

in figure 5.9 we show the instantaneous pressure on the surface of the Robotuna. The

high pressure at the nose of the Robotuna and at the front sections of the tail are


115as we would expect for a blunt body flow. The pressure on the wider parts of the

body is slightly rough. This may be due to the slight roughness of the surface there

introduced by the surface fixes we have already mentioned.

5.2 ALE Incompressible Navier-Stokes

The Arbitrary Lagrangian Eulerian (ALE) method has been developed by [49, 50, 51,

52, 53] and applied in to quadrilateral spectral elements in [54].

So far we have considered domains that are unchanging in time. The next step

is to allow parts of the domain to move in a time dependent manner. This is not

a trivial generalization because we now have to discretize time dependent operators

as well as time dependent fields. We have made an initial step here, however the

implementation we have used is far from ideal. The principle we wish to demonstrate

here is that the ALE approach using unstructured spectral elements is robust and has

great potential. We must adm it tha t a lot of work (and questions) remains in order

to efficiently implement the time dependent operators. For the moment we reform

the local elemental operators a t every time step. We do this because as yet we do

not have an effective preconditioner for the full implicit solver. By experiment we

have determined that it is more efficient in two-dimensional simulations to use the

fully direct Schur complement method for low (N < 6) simulations and the iterative-

boundary direct-interior Schur complement method [30] for higher order. Bearing in

mind the aforementioned caveats we will demonstrate the possible advantages of using

the ALE method with the spectral element method. Namely, the usual advantages of


u°!ssi,lUJJdcj

^ N o j d u°!lOn'POJddJ^ JdUMo

W'Mdoo

drJZ :-?ss

^ ^ - ^ Z ^ o J p o q

Op OS

0 T

" ~ * ho

M issio n

118minimal dispersion problems, high accuracy and the novel fact that since the spectral

elements are larger and less numerous than finite element or finite volume elements

they can support larger deformations without becoming entangled. We have also

shown that the triangles and tetrahedra can support deformation without loosing

excessive resolution.

5.2.1 Formulation

We now consider the incompressible Navier-Stokes equations in a tim e-dependent

domain f2(t)

Ui,t + UjUij = —{p5ij)j + v u ij j + f i in 0(£) (5.5)

Ujj = 0 in f2(£), (5-6)

where v is the viscosity and /,- is a body force. We assume for now homogeneous

boundary conditions; specific boundary conditions will be presented in a later section.

We present the variational form of the momentum equation following the derivation

in [54]. To this end, we consider test functions in the space where all functions and

their derivatives are square-integrable, i.e. i l 1 [£!(£)]. Multiplying equation (5.5) by

test functions and integrating by parts we obtain

J V i fa t + UjUij) dx = j f ^ VijipSij - v u i j + Vifi)dx. (5.7)


119The next step is to define the reference system on which time-differentiation takes

place. This was accomplished in [54] by use of the Reynolds transport theorem and

by using the fact that the test function z/t- is following the material points; therefore,

its time-derivative in th a t reference frame is zero, i.e.

dvi ,-jjr\xp = v i,t + w j viJ = 0

where Wj is a velocity th a t describes the motion of the time-dependent domain Q(t).

Amd X p denotes the m aterial point. The final variational statem ent then becomes

4 f ViUidx + f [viiuj - Wj)uij - ViUiVJjtj\dx = f [vidp5i:j - uvidu^u + Vifi]dxdt Jn(t) 7n(t) 7n(t)

(5.8)

This is the ALE formulation of the momentum equation. It reduces to the familar

Eulerian and Lagrangian form by setting Wj = 0 and Wj = Uj, respectively. However,

Wj can be chosen arbitrarily to minimize the mesh deformation.

5.2.2 T em poral D iscretiza tion

In order to describe in simple terms the solution algorithm, it is convenient to write

equation (5.8) in m atrix form. Thus, we introduce the mass m atrix M = (<£(x), 0(x)),

the derivative m atrix D = (0 ,0X), and the stiffness (Laplacian) m atrix L = <t>x),

where the basis functions <p(pc) are defined in section 3. We also denote the nonlinear

contributions by N (U , W ) to denote dependence on the two velocities, i.e. the flow


120velocity field U and the mesh velocity W . Using this notation, equation (5.8) becomes

+ AT<(U, W ) = D j P - LijUj + Ft. (5.9)

and the incompressibility condition (5.6) is

DtUi = 0. (5.10)

The mesh velocity is defined based on the mesh coordinates from

IT = w > - (5'n)

In the above equations we have denoted the fields with capital letters to denote the

discrete version of the continuous field, e.g. ut- £/,•, etc.

Before we proceed with the boundary conditions, we discretize equations (5.9,

5.10) and (5.11) in time using a splitting scheme and a third-order stiffly stable

integration scheme (see reference [55]).

We solve for time step (n+1); first treating explicitly the nonlinear and the mesh

velocity terms:

= ~ Y . ^ N { U n~ \ W n- q) + F n+l9

= E / W 4-9

Ui - 'E .q<xq{M U i)n- qA t

to x r i - n Qa qx r qA t


121Next we trea t the elliptic terms implicitly

L P n+1

7o(M U j)n+J- - &j A t

LW \1+1

where we compute the intermediate field from

b i = M U i 4- D j P n+l.

The constants a q, (3q are integration weights and are defined in [55]. The mesh velocity

is in general arbitrary, and it can be specified explicitly or be obtained from a Laplace

equation as done here following [54]. More recent work in [53] suggests a modified

approach where a variable coefficient is used in the Laplacian in order to provide

enhanced smoothing and thus prevent sudden distortions in the mesh.

5.2.3 R ay le igh S lid ing P la te P rob lem

We verified the ALE code using the Rayleigh problem. The domain is a box with a

sliding plate at the lower side. The plate slides horizontally with velocity cos(cdt). The

plate has a periodic length 2. Exact boundary conditions are imposed at y = —0.5,1.5.

The exact solution to the Navier-Stokes equations is:

= vLU?+1

= 0,


122

Param eter ValueDimension 2d

Re 40A t 0.01 and 0.001

N-Range 2 to 9K th 50

Method ALE Galerkin

Table 5.5: Simulation parameters for the Rayleigh problem.

u = e vcos(y — ut)

v = 0

p = 0

(y + 0.5)

First we tested convergence to the exact solution using zero velocity initial condi

tions. Figure 5.10 shows that a t N = 5, A t = 0.01 the simulation converges to 10~5

accuracy. Also we show that with A t = 0.001, starting from the exact solution and

checking errors at t = 2, we obtain exponential convergence with increasing N. A

summary of the simulation parameters is given in table 5.5.


123

1.5

1.25

1

0.75

0.5

>0.25

0-0.25

-0.5

-0.75

N r r f aa S 4 V . j s « a a 5 * '■»

Sliding Plate1 1 ' 1 1 ' 1 1 ' I i i J-L.

-1 -0.5 0x

0.5

56r

Iff’ -SO 100t

10* *.10- -!

10--3

£-j'tff'

Expansion Order

Figure 5.10: Top: Spectral element mesh K th = 50. Lower Left: Time convergence history for zero initial conditions N = 5. Lower Right: Convergence to exact solution with increasing N.


1245.2 .4 D esign o f a M icro-P um p v ia S im u lation

The following section is a summary of a paper subm itted to ASME [56]. The research

was done with Ali Beskok of CFM, Brown University. I t is included to demonstrate

that the ALE component of ft/en'T'ccr has been used as a conceptual design tool.

The goal of this project was to produce a potential design for a pump that can be

fabricated in micro-dimensions.

The performance of the micro-pump is evaluated as a function of the Reynolds

number and the geometric parameters. The volumetric flow rate is shown to increase

as a function of the Reynolds number. However, the efficiency of the micro-pump

decreases with increased Reynolds number, due to the increased leakage effects.

Micro-pump systems, delivering volumetric flow rates in the order of 10-8 ~

10~12m 3/ s can be used in many bio-fluidic, drug delivery, mixing and flow control

applications. Most of the micro-pump systems are actuated by a vibrating membrane

in a chamber with hanging-beam-type (Cantilever beam) inlet and exit micro-valves

[57], [58], [59]. These are uni-directional micro-pumps, since the Cantilever-type

micro-valves only open in a preferred flow direction.

In this study, we present a bi-directional (reversible) micro-pump utilizing a vi

brating membrane and piston-type (moving) inlet and exit valves (See figure 5.11).

Since the inlet and the exit valves are simple micro-pistons, oscillating in between

open and closed positions with a prescribed motion, it is possible to control the per

formance of the micro-pump with control of the micro-valves. The design is flexible,

so tha t the pumping direction can be easily reversed. This design has the advantage of


125

Micro-Pump0.2L Inlet Valve Exit Valve 0.2L

0.2L ®min — 0.025L Smax — 0-125L

H=0.4L _m__a=0.1 L

Membrane

L[ OutOut

Figure 5.11: Sketch of the micro-pump operating between two micro-channel systems. Inlet and exit valves open and close periodicly with maximum gap of gmax = 0.125L and minimum gap of gmin = 0.025L.

reversibility, yet it can maintain its performance for relatively high Reynolds number

applications.

5.2.5 M icro -P u m p G eom etric S p ecification s

The micro-pump geometry is presented in terms of the length of the membrane “L” ,

in figure 5.11. This allows us to interpret the results using geometric similarity in

determining the optimum pump dimensions.

The micro-pump is ideally placed in between two reservoirs. However, numer

ical simulation of such a system is difficult due to large reservoir size. Any finite

size reservoir would require inflow and outflow numerical boundary conditions, and

these must be carefully imposed, in order to avoid a preferred flow direction in the

micro-pump. This difficulty is overcome by placing the micro-pump in between two

symmetric micro-channel flow systems, where equal amount of liquid flow is main-


126tained from top to bottom direction. Therefore, the flow conditions are symmetric

and there is no net flow from one channel to another, when the pump is not actuated.

We verified this by simulations.

MembraneDeflection

o 0.6 0.8 10 .4T T T T

Exit ValveClosedPos.

- EJECTION EJECTIONSUCTION

Inlet ValveOpenPos.

0 02 0 .4 0.6 0.8 1

tcj

Figure 5.12: Top: Deflection of the membrane y(x ,t) = asm(jrx/L) sin(tut) Bottom: Loci of the valve tips y (t) = ± tanh(4cos(cu£))

Oscillation of the membrane with a specified frequency ui and amplitude a excites

the fluid within the micro-pump cavity. For our simulations we have used a = L j 10,

and uj = jj-, where c2 = T / M , T being the uniform tension, and M being the mass per

unit area of the membrane. The first-mode of vibration of the membrane is used to

determine the position (See figure 5.12, top), velocity and the acceleration a function

of time


127

y(x ,t) = asin(irx/L) sin(irct/L),

v(x ,t) = (5.13)

(5.12)

(5.14)

The pump geometry is symmetric. Therefore, active pumping can not be achieved

by only actuating the membrane. We have included inlet and exit valves, which open

and close periodicly, in order to break the flow symmetry. The inlet and the exit

valves are located a t the mean height of y0 — 0.325L from the membrane, and the

position of the valves are specified as a function of time

The valve motion is designed to be close to a step function, oscillating between open

valve tips and the top wall of the micro-pump are 0.125L and 0.025L, during open

cycle of the micro-pump is presented in figure 5.12, (bottom). The phase difference

y{t) = ya ± tanh [Acos(irct/L)\. (5.15)

The prescribed velocity of the valve tips are

v (£) = [l — [tanh(4cos(7rc£/L))]2j sin(wct/L).

and closed positions with finite velocity and acceleration. The gaps between the

and closed positions, respectively. The positions of the inlet and exit valves during a


between the inlet and the exit valves is 7r.128

5.2.6 E fficien cy A nalysis

The performance of our design is based on the following factors: the membrane length

L , the membrane width W , the pump-cavity height H, the amplitude of vibration of

the membrane a, the frequency of vibrations ui, the minimum valve clearance (the gap

between the closed-valve and the top wall) g, the time-lapse in between the opening

and the closing of the valves (see equation (5.15), and figure 5.12, bottom ) £-1, the

dynamic viscosity of the fluid p , and the fluid density p. There are nine variables

associated w ith the performance of the micro-pump, with-dimensions of length, time

and mass. This corresponds to six non-dimensional variables: j-, and2 _

esjfL- In this study, we have fixed 5 — 0.15a;. The geometric length-scales are set as

H = 0.4L and g = 0.025L. The parameters ^ and and es L are varied.

2The magnitude of the membrane velocity u — uia. Therefore the parameter es L —

^ is the Reynolds number. Since, the ratio of the dynamic viscosity to the fluid

density is the kinematic viscosity v = p./p, the Reynolds number can be simplified as

The volumetric flow rate per channel width (W ) can be calculated by using equa

tion (5.13)

(D r-k— = / v(x, t)dx = — 2accos(irct/L). (5.16)w Jo


129The suction stage of the micro-pump happens while — < t < Therefore, average

volumetric flow for a given period T = w~l is

— = —2 a c W c o s ( 7 r c t / L ) A t

4 aL W7T

The average flow rate is

— 4 aLWuj 4 LQ = ---------- = ------W uR e. (5.17)

7r 7r a

This simple analysis indicates that the volumetric flow rate is proportional to the

Reynolds number, the width of the micro-pump membrane W , and the ~ ratio. Our

analysis has assumed no leaks from both the inlet valve during ejection stage and

from the exit valve during the suction stage. Therefore, equation (5.17) gives the

maximum theoretical volumetric flow rate of the micro-pump system. This value will

be used in the next section in determining the efficiency of the micro-pump, while

leakage effects due to the imperfect motion of inlet and exit valves will be considered.

5 .2 .7 N u m erica l S im ulation

Full closure of the inlet and exit valves require annihilation of the elements trapped

in between the valves and the top wall, requiring re-meshing of the computational

domain. Re-meshing is avoided by allowing a gap in between the valves and the top

wall. This gap was g = 0.025L a t closed-valve position.


130

7 N

Figure 5.13: Spectral element mesh used for the discretization of the micro-pump system at ejection-stage (top, membrane is moving up). The discretization of the flow domain during suction stage is shown at the bottom (membrane is moving down). The bottom figure also shows elemental discretization obtained by 7th order modal expansions used in M en T a r .


131


Re 0.3, 3 and 30A t 0.0025

N-Range 5,6 and 7K Tri 222

Method ALE Galerkin

Table 5.6: Simulation parameters for the micro-pump simulation.

The mesh used in this study is presented in figure (5.13) at various time instants,

corresponding to suction and ejection stages. The flow domain in represented with

222 triangular (unstructured) spectral elements. Each element utilized T^-order poly

nomial modal expansions. We used successive p-refinements to verify convergence.

The results at 5</l-order and 7^-order modal expansions show that we are resolved

well beyond the scientific accuracy of 1%. A summary of the simulation parameters

is given in table 5.6.2

Re=30i

•S o

-20.6—r~ O.B—ro 0.4—r

R e=3.0

tcj

Figure 5.14: Non-dimensional volumetric flow rate variation with in a period of the micro-pump, as a function of Reynolds number, Re = (a /L = 1/10, a /h = 1/3).


132In figure 5.14, we present the instantaneous flow rate variation for two different

Reynolds numbers (Re = 3,30). The volumetric flow rate of fluid entering through

the inlet valve is taken as positive and leaving flow rate is taken as negative. The

sum of the two is the rate of change of the control volume due to the oscillation of

the membrane

d V _ ■q = Qin Qout-

The membrane’s motion is periodic. Therefore, the net amount of fluid displaced by

the membrane in a period is zero

j> dV = 0 = J Qindt - J Qoutdt. (5.18)

In other words, Qin = = Q. Numerically integrating the curves under inlet and

exit valves in figure 5.14, we determined the effective flow rate in our micro-pump for

g = 0.025L. The ratio of the numerical values of the flow rate to the maximum flow

rate given by equation (5.17) defines the efficiency (77) of the micro-pump. The results

are presented in table 5.7. It is clearly seen tha t the efficiency of the pump decreases

with increasing the Reynolds number (Re = ^ ) - The average flow rate of the pump

increases with the Reynolds number, as predicted by equation (5.17). This is either

due to the increase in the size of the pump (increase in a), or it is due to increase

in the frequency u . For fabrication of our conceptual design, actual-dimensions of

the micro-pump can be determined by either selecting the admissible amplitude of


133

Re 0.3 3 30Q fm2]W . * . TrrF.ORRTTdAT. 3.82 x 10"5 3.82 x 10~4 3.82 x 10"3

Q m2W . S . NTTMP.mr.AT. 3.51 x 10"5 3.36 x 10“4 2.86 x 10"3EFFICIENCY (77) 92% 88% 75%

Table 5.7: Volumetric flow rate per unit width W as a function of Reynolds number for closed-position piston-wall gap of g = 0.025L. The value for v ~ 1 x 10-5 for both air and water are used. The mass flow rate is M = pQ

vibration a or the actuation frequency to. For example, if we select a frequency range

1 k H z < to < 100k H z , we can determine the values of the amplitude a and hence,

the rest of the pump-dimensions can be determined.

There is a dominant vortical structure near the inlet valve through out the simu

lations, although its strength varies during the cycle.

The vorticity contours are presented for Re = 30 flow a t various time instants in

figure 5.15. The top figure is a t too = 0.28. This corresponds to the beginning of the

suction stage. S tart-up vortices due to the opening of the inlet valve can be identified

as a vortex pair just a t the top of the inlet valve. The middle figure corresponds to

the end of the suction stage (at too = 0.72). A vortex jet pair is visible in the pump

cavity. The flow pattern at too = 84, corresponding to early ejection stage is given in

the bottom figure. The exit valve has just open, and the start-up vortex due to its

motion is visible at the top of the exit valve. The vortex pair in the pump cavity has

evolved further. The negative vortex is trapped in the pump cavity, and the positive

vortex jet hits the membrane and bounces back, towards the middle of the pump

cavity. Presence of the vortex pair also creates strong vorticity on the membrane. In


134

Figure 5.15: Vorticity contours for Re = 30 simulation. Top figure at t u = 0.28, corresponds to the beginning of the suction stage. Start-up vortices due to the motion of the inlet valve can be identified. Middle figure is at t u = 0.72, corresponding to the end of the suction stage. A vortex je t pair is visible in the pump cavity. Bottom figure r u = 84, corresponding to early ejection stage. Further evolution of the vortex jet and the start-up vortex of the exit valve can be identified.


135

Figure 5.16: Close up of the vorticity contours for Re = 30 simulation a t the left valve (meshes shown on right side). Top: t u = 0.28, corresponds to the beginning of the suction stage. Start-up vortices due to the motion of the inlet valve can be identified. Middle: t u = 0.72, corresponding to the end of the suction stage. A Vortex je t pair is visible in the pump cavity. Bottom: t u = 84, corresponding to early ejection stage. Further evolution of the vortex jet and the start-up vortex of the exit valve can be identified.


136figure 5.16 we have repeated these time slices but showing a close up of the left valve

of the pum p and the mesh there. When the valve is closed we can see that the mesh

is very tightly packed yet the fields still look convincingly smooth around this area.

After demonstrating the performance of the micro-pump with two-dimensional

simulations, we also tested the design with three-dimensional simulations. We have

extended the two-dimensional geometry of the micro-pump uniformly, using W =

0.4L. We observed boundary layer growth on the side walls, which reduces the flow

rate slightly. Nevertheless, the micro-pump performance is also acceptable in three-

dimensions. The stream-wise velocity contours during the suction and the ejection

stages are presented in figure 5.17.

The algorithm is fast enough to test two-dimensional conceptual design on a work

station. Low Reynolds number (Re < 5) simulations require about 222 triangular

spectral elements with 5th order expansions. The simulation takes about 0.6 CPU

seconds per time step on a 195 MHz Silicon Graphics Onyx2 work station. Three-

dimensional simulations however require large memory and CPU times. A parallel

version of the algorithm is used to test three-dimensional effects on several design

cases. For three-dimensional runs, we have used 444 spectral elements prisms, utiliz

ing 5th order polynomial expansions per element. There are 90 calculation points per

element. The computational domain is divided into 8 sub-domains and the parallel

simulations are performed on the IBM SP2 (thin-2nodes) at Brown University, Center

for Fluid Mechanics. We have observed 18 CPU seconds per time-step for the parallel

run.


Figure 5.17: Three-dimensional micropump simulation Re = 3. Top: An instant during the suction stage. Bottom: An instant during the ejection stage.


Chapter 6

Incom pressible V iscous

M agnetohydrodynam ics

The magnetohydrodynamic (MHD) equations are a model for the dynamics of an

overall electrically neutral fluid, called a plasma, tha t is made up of moving charged

particles. These moving particles react to magnetic fields and their relative motion,

as charged particles (i.e. a current), creates magnetic fields too. Thus, the system is

non-linear and a good target for spectral methods. There is interest in the dynamics

of a plasma confined in a toroidal geometry such as a fusion reactor. Clearly, it is

desirable to be able to also consider plasm a in more complicated geometries, thus

spectral elements appear to be an ideal tool for this type of simulation.

The model as stated is actually a two fluid problem with a positively charged fluid

of ions and a negatively charged fluid of electrons. The mass of an electron is very

small compared to the mass of an ion, thus we can represent the system as a single

138


139fluid th a t reacts with magnetic fields.

For this section we will assume th a t the plasma is incompressible. This approx

imation greatly reduces the complexity of the system of equations for the model.

Using a simplified model th a t is easily verified provides a good benchmark for the

more complex compressible model th a t we investigate in chapter 8.

The MHD equations have been treated numerically by many including [61] and

[62, 63] using finite differences and [64] using Fourier collocation. There has been some

disagreement about how to deal with the constraint th a t no magnetic monopoles can

exist (i.e. th a t V • B = 0 where B is the magnetic field). However, it has been shown

in [65] th a t the smallest errors in satisfying this constraint can cause a significant

cumulative effect as a simulation progresses. We will test a magnetic streamfunction

formulation for the incompressible equations as this provides a formulation th a t fits

naturally into the set of operators tha t we developed in previous chapters. A simple

modification was needed to make the incompressible Navier-Stokes code handle the

extra field and interactions.

6.1 Formulation

The non-dimensionalized equations for incompressible magnetohydrodynamics can be

expressed as:

< V 1— + v • V v — (V x B) x B = —Vp + — V2vdi

^ + V x ( B x v ) = —J - V x ( V x B )


140

Variable Description1 Magnetic resistivityP- Viscosity

O _ PoVa O— u Viscous Lundquist numberSr = ^ Resistive Lundquist number

Alfven wave speed

II

§5

|

Alfven Number

L 0 Characteristic length scaleV0 Characteristic velocityPo Characteristic density

Table 6.1: Parameters for the equations of incompressible MHD.

V - v = 0

V B = 0

where v denotes the velocity of the fluid with components v = [u{x, y, t) ,v(x , y, t)]T

in the x and y directions; p(x, y, t ) is the pressure; and B denotes the magnetic field

of the fluid with components B = [Bx(x, y, t), B y(x, y, £)]T in the x and y directions.

The non-dimensional parameters are shown in table 6.1.

In two-dimensions we used a magnetic streamfunction <j> to represent the magnetic

field:

B = V x (<£k)

This will automatically satisfy the magnetic divergence condition. The two magnetic

fields will now be evolved by the equation:


6.2 Numerical Scheme

The numerical splitting scheme we chose is:

g=0 9=0

2 JLTl+1 TO'S'r in+1A t A V '

J i - l Jc- 1

v = Y l a ,v n- ? + A ^ /?,Nvv n- q,B n9= 0 9=0

V2v n+1 - ^ 2 ° v n+1 = v - AtVpn+1. A t A t F

B n+1 = V x (<£n+1k)


where:142

= v x B

N v = —v • V v + (V x B) x B.

The constants a q, Pq and 70 are integration weights and are defined in [55] for the

stiffly stable splitting schemes of first-, second- or third-order depending on the set of

coefficients chosen. The convective terms are treated explicitly and the viscous terms

are treated implicitly. The boundary conditions for the magnetic streamfunction can

be set to 4> = UooV f°r uniform magnetic field at a boundary or <f> = 0 for an infinitely

conducting wall.

6.3 Magnetic Pearson’s Vortex

In order to test the code we tried a simple exact solution to the above equations. It

is a simple extension of the Pearson’s vortex solution for the incompressible Navier-

Stokes equations. We used a periodic box, length 1, and set the initial conditions at

t = 0:

u = — sin(2'Ky)e~ui‘!t2t

v = sin(2TTx)e~‘/A7r2t

B x = —sin(2'Ky)e~v*n2t


B y = sin( 2Trx)e-vi*H

4> = (cos(27ry) + cos(2nx))e~t'4w2t

143

We discretized the domain with a four-by-four array of quadrilaterals as shown

in figure 6.1. Also provided are the pressure and magnetic streamfunctions a t time

t = 2. In figure 6.3 we show th a t the error at time t = 2 decays exponentially with

increasing expansion order. The parameters used are shown in table 6.2.

I t H I itiai it h i

ininin0.8

0.7; I t 911uniliill0.6

uni liin it til

0.4

it an nailII H I I t H I

0.1

x

nit ' ; ' '

'O —* . ----.N

0.0515717

>sQ.545.012*44145.0257236

41.0314971

0.12646

00961313

410I1063I410361315410541973•0072263•O09Q32SS-0101393•012646

\ \ :

Figure 6.1: Incompressible magnetic Pearson’s vortex (t = 2, instantaneous fields). Top: Periodic spectral element mesh, Bottom Left: Pressure field, Bottom Right: Magnetic streamfunction.


1 1 , 14 6 8 10 12

Expansion Order

Figure 6.2: Incompressible magnetic Pearson’s vortex. Convergence plot for L2 error in x-component of magnetic field a t time t = 2 versus expansion order.


s v 100Sr 100A t 0.001

N-Range 10 to 25K-Quad 64

Method Galerkin

Table 6.2: Simulation parameters for the incompressible MHD Pearson’s vortex simulation.


145We also used this test case to determine the time accuracy of the splitting scheme.

We fixed the expansion order a t N = 10 and run the simulation to t = 1 for different

values of At. The maximum A t was dictated by the CFL condition. In figure 6.3 we

show that using the first-order coefficients resulted in a unit slope in the log-log plot of

L 2 error versus At. For the second-order coefficients we see second-order convergence.

■ n=10. 1“ order coeff.s ♦ 11= 10, 2“ order coeff.s

10 '

10: ♦

IQ-10'd t

Figure 6.3: Incompressible magnetic Pearson’s vortex. Time accuracy plot for the simulation run with N = 10. Convergence plot for L2 error in x-component of velocity a t time t = 1 versus time step At.

6.4 Orszag-Tang Vortex

The Orszag-Tang vortex is an initial value problem in a square periodic domain

(length L). It has been investigated by [66] and [64] amongst others . It demonstrates


146


100s r 100A t 0.001

N-Range 10 to 25K Q u a d 64

Method Galerkin

Table 6.3: Simulation parameters for the incompressible MHD Orszag-Tang vortex simulation.

that turbulent scales can result from a coherent initial condition with only two spatial

frequencies. The initial conditions are:

• r2ny\L/

■ ( 2 t x \ v = 52n(—— )L/

B x = -sin (y )

_ . ,4irx.By = szn(— )

An appropriate choice of initial magnetic stream potential is:

L , 2 t t y L f 4 t t x ^

* = ~

As the solution evolves in time we see that the initial vortex splits into two vortices.

The primitive variable fields are shown in figure 6.4 a t t = 1. We see that sharp

gradients build up and it took 64 quadrilateral elements a t N = 25 to resolve these


147


sv 100Sr 100VA 0.1A t 0.005

N-Range 10K m 460

Method Galerkin

Table 6.4: Simulation parameters for incompressible MHD flow past a cylinder.

features well. We see in figure 6.4 that the vorticity is smooth and well resolved,

and that the velocity divergence is small. The vorticity exhibits complex small scale

structures. Also in figure 6.4 we see tha t the divergence free conditions for the velocity

field is met to 10-4 and the magnetic field is divergence free to 10~8. A summary of

the simulation param eters is given in table 6.3.

6.5 Cylinder Flow

We now consider the effect of a uniform magnetic field that is aligned with the ve

locity field a t the inflow to flow past a cylinder. We have already verified that the

incompressible code obtains the correct answer for the case without magnetic field.

This provides us with a baseline for comparison. The farfield boundary conditions

are straightforward to provide for the magnetic streamfunction. The condition at the

wall was chosen (somewhat arbitrarily) so th a t the magnetic streamfunction is set to

zero there. This condition sets the normal component of the magnetic field to zero

a t the cylinder. A summary of the simulation parameters is given in table 6.4.


148

mm 1_ t I wlc

Figure 6.4: Incompressible Orszag-Tang vortex ( t= l , instantaneous fields). Top Left: x component of velocity, Top Right: y component of velocity, Middle: Pressure, Bottom Left: x component of magnetic field, Bottom Right: y component of magnetic field.


149

m m m| r ' r . . X • I !(

ama

<.mm•Ofnesr•last*•L<sm•Lacs•uiora•r7«nj•LMW

[£=1

§H

I aooorroti aaaoi«Mts aooonaw9 .m C E -6S

I 4.WT77&Q3I rsiot&asJ ZMJ23&47 j -Z**SU£M I -4.1C91E40 1.7.| -amiuui I ■aaeoiaaM l-aaoorm u

Lotnxa4.L4M 74& 0* 3 .7.4137IB4V 1 - U M 7 B 4 B f -LfJI77&OB 1 - r s « 7 & o * I • 1 Q M E 4 I I -1717G6&0* I -<JC1A&Oi

Figure 6.5: Incompressible Orszag-Tang vortex ( t= l , instantaneous fields). Top Left: Velocity streamlines, Top Right: Magnetic streamlines, Middle Left: Vorticity, Middle right: Divergence of velocity, Bottom Left: Curl of magnetic field, Bottom Right: Divergence of magnetic field.


150

Lt*ns

Figure 6.6: Incompressible flow past a cylinder with inflow magnetic fields. From the top: (1) x component of the velocity field, (2) y component of the velocity field, (3) x component of the magnetic field, (4) y component of the magnetic field


151The instantaneous primitive variable fields are shown in figures 6.6 and 6.7. We

see from figure 6.7 th a t the shedding pattern at the cylinder is only slightly perturbed

but that the vortex street has been destabilized downstream from the cylinder. The

wake has also become more complicated with more small scale structures appearing

than before.

I a x tm I a n u sI (tIMM

j l«.«R 7 I I4«Z3»

M JIZS

4Miim4ixm•444*447•Last•15*071- lm0.2101>10421•ttJOIl•14.1412■17 4314•I* H U

&i*ou01)147

4 im so ■01X771

Figure 6.7: Incompressible flow past a cylinder with inflow magnetic fields aligned with inflow velocity. Top: Pressure field, Middle: Vorticity, Lower: Stream function of the magnetic field


Chapter 7

Com pressible Flow Sim ulations

The Discontinuous Galerkin method (DGM) that we outline in the next sections is a

variant of the method proposed by Cockburn and Shu et al in a series of papers[68, 69,

70, 67, 71]. Lomtev implemented the method for triangular and tetrahedral spectral

hp elements in [72, 73, 74]. We have extended it to the polymorphic family of spectral

elements. Using the toolkit of element objects we have developed thus far this was

a straightforward procedure because the DGM is a local method. At each stage

of a computation, the only extra data that an element needs to access is from its

neighbouring elements or domain boundary conditions.

We have applied the DGM to the Euler equations, the compressible Navier-Stokes

equations and the Magnetohydrodynamic equations in two and three-dimensions us

ing the array of spectral elements described in chapter 2. In this chapter we will

consider the Euler and compressible Navier-Stokes equations, and in the next chapter

we outline our progress with the compressible MHD equations.

152


153We only dem onstrate explicit methods in. this chapter and the next chapter, which

will force A t to be small if the problem is dominated by viscous effects. Oden, Babuska

and Baumann presented a formulation for an implicit solver using DGM in a series of

papers [75, 76, 77, 78, 79]. Their m ethod shows promise if the iterative solver can be

preconditioned well. Their method does not require the use of auxiliary methods but

it does involve a large number of derivatives to be evaluated to compensate for this

saving. It will be interesting to see if the implicit m ethod is cheaper than the explicit

method when the number of iterations per time step axe factored into the total cost.

7.1 Numerical Formulation

For simplicity we first consider an convection equation for a state vector v.

d v dFx dF* dFz n frr xd t + ~ t e + W + ( ]

where F (d) is a function of the state vector.

In the discontinuous Galerkin formulation we consider an approximation space S

which may contain discontinuous functions. The discrete S 5 contains polynomials

within each “element” but zero outside the element. Here the element may be any of

the elements we have described and we denote this element by E{. Thus, the com

putational dom ain Q = (Ji Ei, and i?,-, E j overlap only on edges. Consequently, each


154element is treated separately, corresponding to the following variational statement:

Sfio.vlir. dT?x dl?y dFz ~= ~ (w’ (F * - F * K + ( p ’ - F,')» .+ (* “ - F ') n . ) Sa

(7.2)

where w E S 5.

Computations on each element are performed separately, and the connection be

tween elements is a result of the way boundary conditions are applied. Here, boundary

conditions are enforced via the flux F (v ) tha t appears in equation (7.2). Because this

value is computed at the boundary between adjacent elements, it may be computed

from the value of v given at either element. These two possible values are denoted

here as v 1 (internal) and v E (external), and the boundary flux written F^v7, v E).

Upwinding considerations dictate how this flux is computed. For the case of a hyper

bolic system of equations, an approximate Riemann solver would be used to compute

a value of F based on v 1 and v E.

7.1 .1 D iscontinuous G alerk in for D iffusion

We consider as a model problem the parabolic equation with variable coefficient v to

demonstrate the treatment of the viscous contributions:

ut = V • (j/Vu) -F / , in ft, u E L2(Q)

u = <7(x, t), on


We then introduce the flux variable155

q = —uV u

and re-write the parabolic equation

ut = —V - q -I- / , in Q.

1 „ . ^—q = —V u, in il v

u = <7(x, t), on d£2,

The variational formulation of the problem is then as follows: Find (q, u) G H (div ; Q) x

L2(Q) such tha t

(u t , w )Ei = (q, Vt«)£i- < w, q B - n >aEi + ( / , w)Ei, Vw G L2{0)

^(q>v) = Vu)Bi- < u B, v n >aEi, Vv G H(grad; fi)

u = g(x.,t), on dQ

where the parentheses denote standard inner product in an element (E{) and the angle

brackets denote boundary terms on each element, with n denoting the unit outwards

normal and:

H(div, f2) = { v G L 2( f i ) | V - w G L 2(fi)}


H(grad-,Q) = {u € £ 2(ft)|Vu € L2(fi)}156

The surface terms contain weighted boundary values of v B ,qB, which can be chosen

as the arithmetic mean of values from the two sides of the boundary, i.e. v B = ,

and qB = (qE+qI) .

Integrating by parts once more, we obtain an equivalent formulation which is

easier to implement and it is actually used in the computer code. The new variational

problem is

(ut , w )Ei = ( - V • q, w)Ei- < w , (qB - q1) • n >Ei + ( / , w)Ei, Vm E L 2(Q)

i ( q , v ) = ( -V u , v)Ei— < (u B — u1)n, v >dEi, Vv E H{grad; f2)

u = g (x ,t) , on d£2

7.2 Compressible Navier-Stokes Simulations

The compressible Navier-Stokes equations in conservative form are:

c?S QFEuler dFEuler QYEuler QjpVisc Q~pVisc gj?Visc I s | y | z — x | y j zdt dx dy dz dx dy dz

with following definitions:


157

Variable Descriptionp(x, t) Density

v (x ,t) = (u, v ,w ) (x,t) VelocityE - j ^ + ^ ( p v v ) Total energy

P Pressure

4 II P3 Tem peratureP r = SEHK Prand tl number

P Dynamic viscosityA Bulk viscosityK Therm al conductivity

p Euler _ px 2 _|_ p? pmu, (E + p)u)T

p Euler _ (pv,f)VU,pv2 + p ,f)V W ,(E + p )v )T

TpEuler _ (pyj: pwU, pwv, /9U/2 + P, (i? + p)w)T

T-.vi5C „ du 3u dv du dwF * = (0’ 2^ + A V - v ' ^ + ^ ’^ + f , & -

p d v 2 l d T . T A ( V . v ) u + M ( v - V ) « + ¥ — + - ^ )

dv du „ dv dv dwF » “ (0- ^ + ^ v 2^ + A V ' v ’f‘& + f ‘ v

A ( V . v ) „ + (1( v . V ) . + f | - + i | ) r

du/ du dw dv n dw , _F - = (0' ^ + ^ ^ - + ^ > 2' i 9 7 + AV ' v ’

A(V • V)w + „ ( v • V)w + f

The formulation of the Riemann solver for the Euler fluxes is identical to tha t

proposed by Lomtev [80]. We will omit them here for brevity. We will go into more

detail for the compressible MHD equations tha t we develop in the next chapter. The


implementation of the upwinding routine is independent of the element type.158

7.2.1 C onvergence

We first consider the convergence rate of this new formulation by solving for an

inviscid and isentropic flow problem in the geometry shown in figure 7.1. Low-order

methods erroneously produce entropy from inlet to outlet for this problem. Here we

show in figure 7.1 (bottom) that the entropy error converges exponentially fast to

zero with p-refinement. A comparison is shown on the right plot between a fully

unstructured and a hybrid discretization; more elements are used in the unstructured

grid.

We repeated this test with the three-dimensional compressible code. In figure

7.2 we show the three-dimensional domain that we extruded from a two-dimensional

bump mesh with K = 120. Large hexahedra are used at the inlet and outlet, whereas

smaller prisms and hexahedra are used around the bump. We also show th a t the

entropy again decreased exponentially fast with increasing expansion order.

7.2.2 C ylinder F low

For this test of the hybrid incompressible code we consider two-dimensional flow past

a circular cylinder. The cylinder has unit diameter and the domain surrounding the

cylinder is a rectangle [—22,69] x [—22,22]. Uniform upwind boundary conditions

are employed at the box. No slip and wall tem perature are imposed on the cylinder.

Figure 5.5 shows the domain, mesh and boundary conditions. This is the same mesh


159

2

1.5

>» 1

0.5

0

0.5

10 '* r

1°'* r

r rcUl~104 - J

lff* r

io-' r

Figure 7.1: Density contours obtained on a hybrid grid for an inviscid M = 0.3 flow (Top), on a triangle grid (Middle). The bottom plot shows exponential convergence of the error for the unstructured (triangles) and the hybrid (squares) grid.

5 10Expansion order

X


160

Expansion Order

Figure 7.2: Inviscid M = 0.3 flow past a bump in a three-dimensional (periodic in the spanwise direction) domain. From the top: (1) domain, (2) spectral element mesh used in the convergence test K = 120, (3) Iso-Contours of density and (4) convergence of entropy to zero with increasing order.


161


Re 100, 150, 200 and 250Mach 0.5 and 0.7

A t 0.05 to 0.0002N-Range 8 to 10

K th 332KQuad 448

Method Discontinuous Galerkin

Table 7.1: Simulation param eters for compressible flow past a cylinder.

Strouhal FrequencyRe Incompressible N s k 'T ocr Compressible M skTTocr (Mach 0.5)100 0.1662 0.1611150 0.1853 0.1812200 0.1972 0.1934250 0.2054 0.2020

Table 7.2: Comparison of Strouhal frequency of incompressible against compressible (Mach=0.5) flow past a cylinder.

that was used for the incompressible cylinder flow tests.

The mesh has 332 quadrilaterals and 448 triangles. It was designed to direct

resolution around the cylinder, have regular resolution in the wake behind the cylin

der, and block out with large elements to push the farfield boundaries away from

the cylinder. We ran the simulation at N = 8 for Re = 100,150, 200 and 250. A

summary of the simulation param eters is given in table 7.1. In table 7.2 we show that

the Strouhal number is only changed by about three percent by the compressibility

at Mach 0.5.

We also ran the cimulation a t Mach 0.7 in order to compare with the results of


162

Code Method St Q Outflow (-§)dgL [73] Discontinuous Galerkin 0.158 1.846 0.246 35

//flow [81] Explicit Galerkin 0.159 1.843 0.248 35AIe kT ocr Discontinuous Galerkin 0.157 1.802 0.255 69

Table 7.3: Comparison of Strouhal frequency, drag and lift coefficients from hybrid A /e k T ocr with results from other methods. Re=100, M=0.7

Lomtev [73] who used Discontinuous Galerkin with spectral element triangles and

Beskok [81] who used Galerkin on quadrilaterals. We show good agreement in table

7.3 for the Strouhal frequency, however the drag (Cd) and lift (Ci) coefficients are

different by about three percent. This difference can be accounted for by the different

dimensions of the meshes used. For the A /e k 'T'ocr run the outflow was 69 diameters

from the cylinder, whereas for the other methods the outflow was only 35 diameters

from the cylinder. When we ran with the smaller mesh it was obvious tha t there

was a higher level of noise induced when the von Karman street exited the domain.

The comparision with Lomtev shows that using quadrilaterals has not adversely af

fected the polymorphic implementation of the Discontinuous Galerkin method. The

comparison with Beskok shows that the Discontinuous Galerkin method on unstruc

tured polymorphic grids compares well with the more traditional Galerkin methods

on semi-unstructured quadrilateral meshes using collocation.

In figure 7.3 we show instantaneous iso-contours for compressible flow past a

cylinder at Mach=0.5. Also in figure 7.4 we show vorticity which is quite similar to

the incompressible case shown in figure 5.6.


163

Figure 7.3: Instantaneous iso-contours for the simulation of compressible flow past a cylinder {Re = 100, M = 0.5) From the top: (1) density, (2) pressure field, (3) x component of the velocity field, (4) y component of the velocity field


164

8

0

•s

•to

*

Figure 7.4: Instantaneous iso-contours of vorticity for the simulation of compressible flow past a cylinder {Re = 100, M = 0.5)

7.2 .3 F low P a st a T w o-D im en sion a l N A C A 0012 A irfoil

We simulated compressible flow at Mach=0.5, Re=10,000 (based on chord length)

past a NACA0012 airfoil a t zero angle of attack. This airfoil needs to have good res

olution at the leading and trailing edges and to have good boundary layer resolution.

We used thin quadrilaterals on the body, and small elements a t the nose and tail then

blocked out using triangles in the fax domain and a regular array of quadrilaterals in

the wake. It is im portant to keep the resolution smoothly varying in the wake since

abrupt changes in the resolution there can lead to noise generation.

4180 quadrilaterals and 3307 triangles were used in the mesh which was created

by Kirby [82] using SIMPLEX2D [83] and an advancing front/bioeking algorithm. A

summary of the simulation parameters is given in table 7.4.

In figures 7.6, 7.7 and 7.8 we show instantaneous iso-contours of density, divergence

of momentum, and curl of momentum, respectively, in the near wake of the airfoil.

The divergence of momentum is equal to minus the rate of change of density due to

the conservation of mass equation. Hence, we should expect to see noise generated


165


Re 10,000 based on total chord lengthMach 0.5

A t 0.001 to 0.00001N-Range 1 to 11

K th 3307K qucuI 4180


Table 7.4: Simulation param eters for compressible flow past a NACA0012.

by the m ethod appearing in this field. The plot shows th a t this calculated quantity

is smooth showing tha t at this resolution we are well resolved by this measure. The

vorticity plot, however, shows some noise near the vortices in the wake, so in this

case the vorticity is probably a good measure of accuracy and a small increase in the

resolution should reduce this noise to acceptable levels.

7.2.4 F low P a st a M u lti-B o d y A irfoil

We simulated compressible flow a t Mach=0.5, Re=10,000 (based on total chord

length) past a multi-body airfoil a t zero angle of attack. The small scale features

(i.e. nose and tails of each component) need to be resolved as well as the wake region

and boundary layers. We used thin quadrilaterals on the components, and small ele

ments at the noses and tails then blocked out using triangles in the far domain and

a regular array of quadrilaterals in the wake.

1678 quadrilaterals and 3699 triangles were used in the mesh which was created

by Kirby [82] using SIMPLEX2D [83] and his own advancing front/blocking routines.


166

10

5

0

•5

-1030-10 0 10 20

X

Figure 7.5: Top: Mesh of full domain for simulation of compressible, Mach 0.5, Re=10,000 flow past a NACA 0012 airfoil at zero angle of attack to the mainstream flow, Middle: Mesh around body and wake, Bottom Left: Close up of the airfoil, Bottom Right: Close up of part of the wake region.


167

Figure 7.6: The wake region (from f = 2 to ^ = 5) of a NACA 0012 airfoil at zero angle of attack to the mainstream flow. Mach 0.5, Re = 10,000, 3307 triangles, 4180 quadrilaterals, N = l l . Instantaneous iso-contours of the density are shown.

Figure 7.7: The wake region (from ^ = 2 to f = 5) of a NACA 0012 airfoil at zero angle of attack to the mainstream flow. Mach 0.5, Re = 10,000, 3307 triangles, 4180 quadrilaterals, N = l l . Instantaneous iso-contours of the divergence of momentum are shown.

Figure 7.8: The wake region (from ^ = 2 to ^ = 5) of a NACA 0012 airfoil at zero angle of attack to the mainstream flow. Mach 0.5, Re = 10,000, 3307 triangles, 4180 quadrilaterals, N = l l . Instantaneous iso-contours of the curl of momentum are shown.


168


Re approx. 10,000 based on to ta l lengthMach 0.5

A t 3e-5N-Range 4

Krri 1678KQuad 3699


Table 7.5: Simulation parameters for compressible flow past a m ultibody airfoil.

A summary of the simulation parameters is given in table 7.5.

The final two-dimensional example is a simulation of flow past a multi-element

airfoil shown in figure 7.9. The hybrid grid consists of 1638 quadrilaterals and 3739

triangles. The grid was constructed in three phases. Quadrilaterals were created using

an advancing front method from each airfoil. A block of structured quadrilaterals was

used to cover the wake region. The rest of the domain was covered with unstructured

triangles using SIMPLEX2D. This is a Delauney-based grid generator with Steiner

triangulation for guaranteed quality triangulation [4]. The Reynolds number is Re =

10,000 based on the to tal length of the multi-element airfoil, and the freestream

Mach number is M — 0.5. A third-order expansion was used uniformly in each

element everywhere in this problem. In figure 7.10 we plot instantaneous iso-Mach

contours and streamlines around the airfoils. The flow is unsteady as it is evident

from the vortex street present in the near-wake.


169

0 0 .0 5 0.1 0 .1 5 02x

Figure 7.9: Hybrid Mesh and close ups for the simulation of compressible flow past a multi-body wing


170

0.65846505544970.4505290.3465610.2425930.1386250.0346565

0.2

0

- 0.2

-0 .40 0 . 5 1

x

Figure 7.10: IsoMach. contours and streamlines for M = 0.5 flow past a two- dimensional multi-component wing.

0.658465

0.4505290.3465610.2425930.1386250.0346565


171


Re 2000 base on chord lengthMach 0.5

A t le-4N-Range 1 to 4K P rism s 1960

K Hex 2095Method Discontinuous Galerkin

Table 7.6: Simulation parameters for compressible flow past a NACA0012 airfoil with endplates.

7.2.5 F low P a st A N A C A 0012 W in g W ith E nd p lates

The first three-dimensional problem we consider is flow past a NACA 0012 airfoil

with plates attached to each end. We impose uniform upwind boundary conditions

a t inflow and outflow, and the domain is periodic from one end of the airfoil to the

other.

This is a simple model of a wing between an engine and fusalage. The domain was

meshed with SIMPLEX2D [83] and extruded using 1960 prisms and 2095 hexahedra.

A thin boundary layer of hexahedra was used on the surface of the wing and the

simulation was run with upto 4</l order expansion and Re=2000 (based on chord

length). A summary of the simulation parameters is given in 7.6.

At low-order the simulation ran to steady state. This is due to a reduced effective

Reynolds number achieved because of numerical dissipation. As we increased the

order we saw unsteadiness developing in the wake of the wing, and what appears to

be oblique shedding. This is only a marginally three-dimensional domain but it does


172

demonstrate the ability of f s f stz'T'c l t to direct resolution into boundary layers and

to fill out a domain w ith larger elements.

Figure 7.11: Skeleton mesh for flow past a three-dimensional NACA0012 airfoil with endplates.


173

Figure 7.12: Iso-contours for x-component of momentum for M = 0.5 flow past a three-dimensional NACA0012 airfoil with endplates.


C hapter 8

Com pressible V iscous

M agnetohydrodynam ics

Sim ulations

We have already described an incompressible plasma in chapter 6. In this chapter

we generalise the equations to describe a compressible plasma. We implemented the

algorithms in the J ^ b k 'T olt code so that simulations can be run on a broad range

of parallel computers using the Message Passing Interface (MPI) [60]. We tested the

code on some simple magnetohydrostatic test cases and on the Orszag-Tang vortex.

Also, we sim ulated flow past a cylinder with an imposed far-field magnetic field.

174


1758.1 Formulation

The equations for compressible magnetohydrodynamics can be expressed in conser

vative form in compact notation as:

- V • (pv)

- v • ( p w ‘ - B B ' + (p + i |B |2) / - -~ r )

—V x (B x v 4- -^-V x B)Or

- V • ((£ + p )v + ( i |B |2I - B B ‘) . v - i y - r

+ j -(B ■ VB - V ( | lB |2)) - ~ ^ T )

0

(djVi + diVj) - | v • vSi:i

Alternatively, in flux form with the explicitly stated fluxes as:

dXJ dF Euier dF Euler dFEuler dJ?Visc dFVisc Q~pVisc~dt = dx dy dz + dx + dy + dz

V -B = 0

U = (p ,p u ,p v ,p w ,B x,B y,B z, E )

FEuler = pu2 - B l + p, puv - B xB y, puw - B XB Z, 0, uBy - vB x, u B z - wBx,

(E + p )u — (v • B )BX)T

FEuler = (pv, pvu - B yB x, pv2 - B 2 + p, pvw — B yB z, vBx - uB y, 0, v B z - wBy,

dp_dt

d(frv)dtdBdtd Edt

V B =


(E + p)v — (v • B )By)T176

■pEuier _ /JWU _ b zB x, pyjy _ B zB y, pw2 — B 2 + p, wBx — uB z, w B y — vB z, 0,

(E + p)w — (v ■ B)BZ)T

f 1'*” = <0,— <— - - v - V) - i — + ? y - ) — (— + — )x ’ S v dx 3 ’ Sv dy dx S v dz dx

0 L (^ y _ dB-) 1 (dB * dB ^’ 5 / dx d y U Sr Kdx d z J’1 / 2 , _ , „ l d v 2 1 dT. I T d f l B l 2) „ . , T

^ ( - 3 (V - v)u -F v • Vu + 4- - — ) + - ( _ ---------B - V B,))

F w,c _ ,n _i_f 9v_ du 2. r f t L _ I v 7 f 1 r d v , d w \y S v ^dx + d y ^ S ^ d y 3 S v d z dy

1 d B x dBy 1 dBz dBySr dy dx ’ 5 / dy d z h1 f 2 r ^ t~7 l ^ v 2 1 d T , 1 ,1 0 ( |B |2) „5 ( - 5 (V • v ) . + v ■ V , + - - g - + - w ) + - ( — B • V B ,) )

revise rn 1 ,dw du 1 dw dv 2 dw 1p - = (0’ i ; {^ + 5 I ) ' ^ ( ai7 + a ; )' ^ (^ - 3 V v ) '

1 d B x dB z I dBy dB z Sr K d z d x h STKdz d y }’ '1 / 2 ,t7 vr 1 d v 2 1 dT . l , l d ( | B ] 2) „^ ( - 3 (V • v )« + v - Vw + - w + j - r ^ ) + B • V S .))

with the variables and parameters defined in table 8.1.

8 .2 The V • B = 0 Constraint

The presence of the V -B = 0 constraint implies th a t the equations have a semi-elliptic

character. It has been shown in [65] that even a small divergence in the magnetic

fields can dramatically change the character of results from numerical simulations.

An alternative approach to the magnetic stream function which we used for the

incompressible MHD simulations was developed in [62]. This modified the equations


177

Variable Descriptionp(x, t ) density

v(x, t) = (u,v,w)(-x,t) velocityB(x, t) = (B x , B y, Bz)(x, t) magnetic fields

E — + 2 Cpv - v + B - B) total energyP Pressure

P = P + • B Pressure plus magnetic pressureT ----E_Rp Temperature

P r = ^K Prandtl NumberR Ideal gas constantV Magnetic resistivityP Viscosity

C _ Po vA L.0v p. Viscous Lundquist numberSr = ^ Resistive Lundquist number

% Specific heat at constant pressure

CS II •S Alfven wave speed

ii Alfven Number

Table 8.1: Variables and parameters used in the equations of compressible MHD.


178by adding a source term proportional to V • B:

Spoweii = - ( V • B )(0 ,B x,B y,B z, u , v , w , v - B ) T

to the right hand side of the evolution equation. In [62] this term was incorporated

into the Riemann solver for the Euler flux terms. We evaluate this term as a source

term, without modifying the Riemann solver. The divergence of the magnetic field is

calculated using the DGM derivatives of the magnetic fields.

8.3 Implementation of the Euler Flux Terms

We now apply the same techniques tha t we used for the Navier-Stokes equations.

We evaluate these fluxes and their derivatives in the interior of the elements and add

correction terms for the discontinuities in the flux between any two adjacent elements.

In order to evaluate the Euler flux a t an element interface we use a one-dimensional

Riemann solver to supply an upwinded flux there. At a domain boundary we expect

the user to provide far field conditions and treat the exterior boundary as the bound

ary of a “ghost” element. This way we can use the same Riemann solver at all element

boundaries.

We linearize the one-dimensional flux F f “/er in the normal direction to a shared

element boundary using the average of the state vector at either side of the element

boundary. That is, since F&uler is a nonlinear function of the state vector we use the

average state to form an approximation to the Jacobian of the flux vector A c.


179The Jacobian m atrix for the flux vector for the evolution equations expressed in

primitive variables is simpler than the conserved form. Thus, we will perform the

linearization about the primitive form and transform to the conserved form. The

primitive Jacobian m atrix A p has the form:

u p 0 0 0 0 0 0

0 u 0 0 Br Ex B~ 1

P p P p

0 0 u 0 - E x B r 0P P p

0 0 0 u B zP

0 B zP

0

0 0 0 0 0 0 0 0

0 By ~ B X 0 — V u 0 0

0 B z 0 ~ B X —w 0 u 0

0 IP 0 0 — (7 — l)u • B 0 0 u

The left and right eigenvectors of the primitive Jacobian m atrix A p, due to Powell

et al [63], are:

Entropy wave:

Ae = u

le = ( 1 ,0 ,0 ,0 ,0 ,0 ,0 , - 4 )a1

r e = (1, 0, 0, 0, 0, 0, 0, 0)‘

Alfven Waves:


Fast waves:

A f = u ± c /

l, = 5ij(0,±a/c/,Ta,c,/3I ,T«,c.A,i8„0,2^5,2^2,2i)2a y/P y/P P

Tf = ( p a f , ± a f C f , ^ f a scsp xP y , T a scsPxPz, 0 , a sPya y / p , a sPzay/p,af 'yp) t

Slow waves:

As = u ± cs

l. = ± ( 0 , ± a , c , , ± a / cf /3x/3s, ± a / cf & & A - ? ^ , - ? , - )Z a V P V P P

rs = (pots, ± a scs, ± a f cf pxPy, ± a f cf /3x/3z, 0, - a f j3yaVp, ~<^fPzaVp, Ois'yp)t

Where:

_ 7p -F B • B(a*)2


181

a 2 a 2

7 C /_ c ?2 ^ ~ g2

Px = sgn(Bx)

PyBy

Z

We can transform between the primitive and conserved variables with the following

transform:

<9U „ d WA c — d W p d V

where:

U = (p, pu , pv, pw, B x, By, B z, E)

are the conserved variables and:

W = (p, u, v, w, B x, By, B z,p)

are the primitive variables. This gives:


182

d U

d W

1 0 0 0 0 0 0 0

u p 0 0 0 0 0 0

V 0 P 0 0 0 0 0

w 0 0 p 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

v - v2 p u p v p w B x B y B z 1

7-1 .

and

a wa u

1 0 0 0 0 0 0 0

_XL

p

1p

0 0 0 0 0 0

_y_

P0 I

p0 0 0 0 0

_w

p0 0 I,

p0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

f v - v —7 u — 7 u — ■yw - 7 B x -7 B y ~ l B z 7

where 7 = 7 — 1.

We are now in a position to evaluate the upwinded flux at the element boundaries.

We use following formulation for the upwinded flux:


183

1 9 U k=?F(Uj, U s) = ~ (F (U r) + F ( U £ ) - —

d Wo* = Ik * - ^ j ( U s - U r)

Here the lk and r k are the ordered left and right eigenvectors of the primitive Jacobian

matrix. We have to apply the operator to the right eigenvectors to calculate the

conserved flux. The A* are the wave speeds associated with the eigenvectors.

8.4 Implementation of the Viscous Terms

The viscous terms are evaluated in two steps. First, we obtain the spatial derivatives

of the primitive variables using the discontinuous Galerkin approach. Then, we repeat

the process for each of the viscous fluxes using these derivatives. If the user supplies

Dirichlet boundary conditions for the momentum and energy variables we set these

terms explicitly after the fluxes have been evaluated and then project the result using

the orthogonal basis.

8.5 Summary of the Compressible MHD Code Im

plementation

A possible implementation of the algorithm is:


184• Step 1: Read in initial conditions U(x, 0), evaluate the fields at the element

quadrature points. Set n = 0.

• Step 2: Calculate the upwinded fluxes Fn at the Gauss quadrature points on

the element interfaces. At domain boundaries use the prescribed boundary

conditions for the exterior values of the fields.

• Step 3: Calculate the Euler flux terms: F ^uier, F ^ “ier and F f uler at the element

quadrature points.

• Step 4: For each component of the state vector Uk = (U(x, tn))k calculate

.Q - p E u U r g p E u / e r g p E u l e r

1 d x + d y ”*■ d z >k ■

• Step 5: Interpolate the upwinded flux to the correct quadrature for the ele

ment interface. Scale the fluxes with the edge Jacobian divided by the volume

Jacobians. Add this to the divergence of the Euler fluxes calculated in Step 4.

• Step 6: Take inner product of the fluxes with the orthogonal basis. Evaluate

the resulting polynomials at the quadrature points.

• Step 7: Calculate the spatial derivatives of the primitive fields. For example,

we will need We evaluate this as described in section 7.1.1 using the DGM

to compensate for the derivative across element interfaces, and the average of

u at the element interface.

• Step 8: Use the derivatives of the primitive fields to construct the Viscous flux

terms: F^'sc, F^'5C and F ^ isc.


185• Step 9: Take the divergence of the viscous flux terms.

• Step 10: Subtract the divergence of the Euler fluxes obtained in Step 6 from

the divergence of viscous fluxes obtained in Step 9.

• Step 11: Take inner product of the result from Step 10 with the orthogonal

basis. Evaluate the resulting polynomials at the quadrature points and place it

in Uf(x, £n" 9)

• Step 12: Update the state vector U(x, £n+1) = U(x, tn) + A t 5D?/39Uf(x, tn~q)

using an Adams-Bashforth integration scheme.

• Step 13: Increase n by one. If tn is less than the term ination time return to

Step 2.

• Step 14: O utput final values of the state vector U(x, tend) .

8.6 Two-Dimensional Magnetohydrostatic Exam

ple

A simple test for the compressible MHD component of the M s k T ctr code is to

consider a steady irrotational magnetic field and zero velocity. The test was performed

as an initial value problem and the following exact solution:

p = 1


186u = 0

v — 0

e(-2*ry)£ = 19.84 + ---------

2

Bx = — cos(‘irx)e(- ~ ^

By = sin(7ra;)e^-iry^

was used as the boundary conditions and as the initial condition. This solution was

derived by Priest [84]. The irrotational magnetic field implies tha t the Lorentz force

is zero so the momentum equations are trivially satisfied. The magnetoviscous term

is zero and the v x B term is also zero. Thus, the compressible MHD equations are

satisfied.

The domain and discretization we used is depicted in figure 8.1. We also show

that the approximation decreases exponentially with increasing expansion order.

8.7 Three-Dimensional Magnetohydrostatic Exam

ple

We modified the two-dimensional test case, used in the previous section, to be a

three-dimensional test case for the compressible MHD component of the N ek'Totr

code. Again we used a steady irrotational magnetic field and zero velocity solution

to the MHD equations. The test was performed as an initial value problem and the

following exact solution:


187

0 .7

0.6

0 .4

O.t

X

0.8

0 .7

0.6

0 .4

0.1

X

10"

10"

O m f \-7 fc10w

10 s

10 '

1 1 1 ' ' 1 ' 1 1 ' ' 1 1 ' ' ' ' 1 ' ' ' ' I ' '4 6 8 10 12

Expansion Order

Figure 8.1: Magnetohydrostatic test case for the compressible code. Top Left: Mesh {Kth = 38,KQuad = 22) Top Right: Magnetic streamlines of steady solution at N = 12, Bottom: dependence of steady state error on expansion order.


188

1

0

0

0

sin(ax)(e~az — e_ay)

—sin(ay)e~ax — cos(ax)e~ay

(cos(ax) — cos(ay))e~az

5 + 0 .5(B | + f l J + B j )

27rT

was used as the boundary conditions and as the initial condition. By construction,

the magnetic field is irrotational and the Lorentz force is zero so the momentum

equations are trivially satisfied. The magnetoviscous term is zero and the v x B term

is also zero.

The domain and discretization we used is depicted in figure 8.2. We also show

that the approximation decreases exponentially with increasing expansion order.

P =

u =

v =

w =

B x =

By =

B z =

E =

a =


189

: a s x j m0357647 0307315 00*64137

4X113345 -0173576

i -0434006 •0544337 •0734661 -0314999 -L07333

Expansion Order

Figure 8.2: Three-dimensional magnetohydrostatic test case for the compressible code. Iso-contours of the magnetic and energy fields axe shown at time t = 1. Top Left: Tetrahedral mesh used, Top Right: x-component of the magnetic field, Middle Left: y-component of the magnetic field, Middle Right: z-component of the magnetic field, Bottom: Convergence plot showing exponential decrease in L 2 error with increasing expansion order.


1908.8 Compressible Orszag-Tang Vortex

The Orszag-Tang vortex is an initial value problem in a squaxe periodic domain

(length L). It has been investigated by [66] and [64] amongst others . It demonstrates

that turbulent scales can result from a coherent initial condition with only two spatial

frequencies.

The initial conditions we used were:

p = 1

u II 1 Co <s>,

to «ci

V = sm { L )

B x = sm{ L )

By. .47TX.

= sm { )

p ^ 1 A= C + -cos(- 4

S-k x . 4 Arrx. 2iry. .27ra;. .27ry. 1 , 47n/.-) + -c o s (-^ -)c o s (— ) - cos(— )cos(— ) + -cos{— )

where C fixes the initial background mach speed and p is the instantaneous pres

sure for the equivalent incompressible flow. A summary of the simulation parameters

is given in table 8.2, and figure 8.3 shows the mesh used for this case.

In figure 8.4 we show iso-contours of the conserved variables at t = 1. We see tha t

using an expansion order of nineteen shows very smooth and symmetric fields.

We now consider the effect of expansion order and using an unstructured grid on

accuracy. Vorticity is a good indicator of noise in the solution of unsteady simulations.


191

6

5 ts:isi

4

tic198!issi3

2

1

00 2 53 4 6X

Figure 8.3: Mesh used for the compressible Orszag-Tang vortex simulations on a structured mesh.


5 , 100Sr 100

A (Alfven Number) 1.Mach 0.5

N 19KQuad 64


Table 8.2: Simulation parameters for the compressible Orszag-Tang Vortex.


192

14.101713.650811212.7491121982111474IL396510.945710.494810.0449193119.142268.69148.24054

L 27227114041108531.17665L 144781.1129LOB 1(231.049161.017280.9854080.9535340.921660.8897860857912

(.06259

•0.104175

■0.437537•0.604218

• 1.10426

'II, W: ..HUis: iJM A ' S . i .

1.414751.192830.9709060.7489850J270630.3051420.0832205

■0.1387014X3606224X5825444X804465•1.02639

114831•1.47023

'&r mmm nw .•' - <: ■'P l i - /d i r w

MUrr1 P>l -

1.414411.192540.9706710.748803

•0.138667

•1.02614•114801•1.46987

(.1414

i iS S R0.783313

014618400671411

•0.111902

•0.469988•0.649031•0.823073

.00712• I .18616

Figure 8.4: Compressible Orszag-Tang Vortex ( t= l , instantaneous fields, Mach=0.5). Top left: density, Top right: energy, Middle left: x-component of momentum, Middle right: y-component of momentum, Bottom left: x component of magnetic field, Bottom right: y component of magnetic field.


193

OaXr)M0615

0.7130160.139732

L3S0122.13341

>172669-319997•317326

02434310.1926420139030.0870244

-00714026•0124212-0177021

-

6.97376347611

3.076811.777160477314

-112179-3.42144-4.72109

-7J2039

-9.91969

■ i ■ ■ i ■ ' ■ *■

Dhrffl)I 0.00404916 I 000344398 j 00028313

000223362 | OOOI62U4

000102326 j 000041108

3 >000011707 1-000079227 I -0.00139746 R -0.00200264 1-0.00260782 I >0.003213 I -0.00381118

>»

X

Figure 8.5: Compressible Orszag-Tang Vortex ( t= l , instantaneous fields, Mach=0.5). Top left: curl of momentum, Top right: divergence of momentum, Middle left: curl of magnetic field, Middle right: divergence of magnetic field, Bottom left: momentum streamlines, Bottom right: magnetic field streamlines.


194Low resolution will show up as non-smoothness in vorticity since it is the derivative

of the computed fields. In figure 8.7 we compare the vorticity at time t = 1 for a 132

triangle mesh (shown in figure 8.6) run w ith N = 4 and N = 6. The top figures show

how the vorticity varies across the lower-left to top-right diagonal. The vorticity in

this direction should be symmetric about the mid-point. We see that a t N = 4 the

profile is noisy, the peeks are not well resolved and the symmetry is not very well

represented. At N = 6 the symmetry is better but the profile is still quite noisy. In

the next figure 8.8 we see at N = 10 th a t the symmetry is very good but the slopes

are still a little rough. Finally, at N = 16 we see tha t the sym m etry is perfect and the

profile is very smooth. We also see that the smoothness of the iso-contours improve

with each increase in expansion order.

6

5

4

3

2

1

oo 1 2 4 5 63X

Figure 8.6: Compressible Orszag-Tang vortex triangle mesh with K=132.


195

x x

>%

X X

Figure 8.7: Compressible Orszag-Tang Vortex ( t= l , instantaneous fields, Mach=0.2, K=132). Top Left: Curl of momentum along the diagonal(N=4), Botton Left: Iso- contours of curl of momentum(N=4), Top Right: Curl of momentum along the diagonal(N=6), Bottom Right: Iso-contours of curl of momentum(N=6)


196

x x

x

>•

X

Figure 8.8: Compressible Orszag-Tang Vortex ( t= l , instantaneous fields, Mach=0.2, K=132). Top Left: Curl of momentum along the diagonal(N=10), Botton Left: Isocontours of curl of momentum(N=10), Top Right: Curl of momentum along the diagonal(N=16), Bottom Right: Iso-contours of curl of momentum(N=16)


197


s v 100Sr 100A 0.1

Mach 0.5A t le-4

N-Range 1 to 8K Tri 490


Table 8.3: Simulation parameters for compressible flow past a cylinder with a magnetic field.

8.9 Flow Past a Cylinder

As with the other applications we have developed using the spectral element toolkit

we resort to cylinder flow as a test case. We see that the fields in figures 8.9 and 8.10

closely resemble the equivalent fields in the incompressible version of this simulation in

section 6.5. This is no surprise since we chose a fairly low Mach number (Mach=0.5)

to run this test a t. We can also compare this to the compressible case without a

magnetic field 7.2.2. We see th a t up-down symmetry of the x-component of velocity

has been broken in the wake and the regular patterm of the y-component of velocity

also breaks down ten diameters from the cylinder. This is clear evidence tha t the

magnetic fields are indeed causing the von Karmann street to become unstable with

time.

A summary of the simulation param eters is given in 8.3.


198

to

Figure 8.9: Instantaneous iso-contours of the simulation fields for flow past a cylinder (Sv = 100, Sr = 100, A = 0.1) with a magnetic field. From the top: (1) density, (2) pressure, (3) x component of the velocity field, (4) y component of the velocity field


199

Figure 8.10: Instantaneous iso-contours of the simulation fields for flow past a cylinder (St, = 100, Sr = 100, A = 0.1) with a magnetic field. Top: x component of the magnetic field, Bottom: y component of the magnetic field


Chapter 9

Sum m ary

In this thesis we have achieved the following goals:

• Developed a natural approach to implementing polymorphic elements ( hp finite

mixed-elements) using an object oriented language.

• Developed efficient algorithms to align the coordinate systems of polymorphic

elements. (Reducing NP hard problems to linear algorthms)

• Quantified theoretically and numerically the approximation properties of the

orthogonal bases on these elements.

• Applied polymorphic element methods to the following physical problems:

— Incompressible Navier-Stokes in two- and three-dimensional domains.

— Incompressible Navier-Stokes in two- and three-dimensional moving do

mains.

— Incompressible Magnetohydrodynamics in two-dimensional domains.200


201— Compressible Euler in two- and three-dimensional domains.

— Compressible Navier-Stokes in two- and three-dimensional domains.

— Compressible Magnetohydrodynamics in two- and three-dimensional do

mains.

The Navier-Stokes solvers are direct extensions of the work of Sherwin [30] for

the incompressible code and Lomtev [72, 73, 74] for the compressible code. The new

development here is to allow all the polymorphic element types to be used as well as

the triangles and tetrahedra reported there. This is a significant advance th a t will

facilitate the move to even higher resolution because thin boundary layers can now

be resolved. This advance did not come in terms of a loss in efficiency. We applied

object oriented programming methodology to make the management of varied element

types as transparent as possible, with no perfomance hit. Also, since the companies

tha t make compilers are now concentrating on optimizing C-H- code, Afetz'Tocr will

benefit from their new developments.

We think that this is the first time high-order polymorphic elements have been

applied to the equations of conservation laws, and especially the MHD system. The

equations are quite complex and without the natural programming approach we de

veloped, it would have been difficult to construct a robust code for them.

Now that N e k 'T clt has been developed to be a powerful tool the question re

mains as to what kind of physical problems can be simulated with it. We have shown

it capable of handling simulations with millions of degrees of freedom when run on

parallel super-computers. The discontinuous Galerkin modules only utilize pairwise


202communication, i.e. no piece of data needs to be communicated between more than

two processors. This form of communication can be efficiently implemented and good

scaling can be achived, i.e. the overhead per extra processor does not increase rapidly

as the number of processors is increased. This means th a t a discontinuous Galerkin

simulation can be spread over many processors efficiently and therefore A fe n 'T olt

should be capable of solving simulations with several orders more degrees of freedom

as processor power increases and many-processor machines are built. We predict that

two- and three-dimensional simulations of flow past complex two-dimensional bodies

at up to Re = 106 should be possible a t the turn of the millenium.

For three-dimensional simulations the potential to reach these goals is there, how

ever some advances will have to be made to ease the data jam that results from the

huge numbers of data points created. Now that the simulation code A fe n 'T a r has

been written, simulations can be started once the domain has been meshed. However,

the data mining of results will be a significant challenge. In future work the use of in

telligent software agents for extracting relevant data on the fly should be investigated.

These same agents will also be necessary to direct the available resolution towards

interesting parts of the flows. This data migration in itself will need new approaches

to balancing the simulation over multiple processors. This task will be helped by

the OOP nature of A fe tzTo tr , since each element can be enabled to replicate itself,

delete itself or move itself between processors through a simple interface.

For example, an agent could “tell” an element it is no longer needed. Then it will

communicate to its neighbours th a t it is going out of existence, transmit any data


203that will be needed to them and then delete itself. This would have been quite a

difficult task with plain C code, and even harder with Fortran code but C + + should

make this a m atter of planning the design of the agent wisely.


A ppendix A

C ontinuous Expansion B asis

A .l Jacobi Polynomials

(1 - * ) ( ! + x) + [(/? - a) - (2 + a + 0)x] dy{x)dx= -A ny(x)

or

d_dx

(1 — ar)1+tt(l 4- x )1+&dy(x)dx

= -A n( l - x ) Q(l + x)py(x)

An = n (n -+- a + j3 + 1)

y ( x ) = Pn P(x )

In the following appendix we document the complete polynomial expansions for

both the m odal and nodal basis discussed in chapter 3. The bases are defined in terms

of three orders P i, P2, P3 corresponding to the local coordinate system. For the modal

expansions we have indicated the indices over which the bases should be assembled

204


205in order to achieve the maximum polynomial space although it is possible to reduce

the interior, face and edge expansion orders to produce a variable order expansion as

discussed. The nodal basis must be assembled over all the indicated indices for the

expansion to spam a complete polynomial space.

A.2 Modal Basis

In the following section refers to the ith Jacobi polynomial defined in section

A.

A .2.1 T w o-D im ension al E xpansions

Quadrilaterail D om ain

sJ L

c D

0 r

A B

Figure A .l: Vertex labelling for the standard quadrilateral region


Using the labelling given in figure A .l the quadrilateral basis is:206

Vertex A : (1=») (1=5)

Vertex B : (l± s) (1=5)

Vertex C : ( » ? ) ( “ )

Vertex D : (l± s) (1±4) ,

Edge AB : (1=4) (l±e) fW (a) (1=4) (0 < t < P i)

Edge CD : ( l= s ) (1±») p W (o) (1±4) (0 < i < P ,)

Edge AC : (1=4) (1=5) (l±it) pM l(6) (0 < j < P2)

Edge BD : (1±4) (1=5) (I±5) pM (6) (0 < j < P 2),

Interior : (l=*) (l± s) P t\ (a ) (1=5) (1±5) P ^ ( b )

(0 < i , j ; i < P i , j < P2)-

Triangular D om ain

Using the labelling given in figure A.3 the triangular basis is:


207

50

100

150

200

500 100 150 200

Figure A.2: Mass m atrix for the continuous basis for quadrilaterals ( N = 15).

Figure A.3: Vertex labelling for the standard triangular region


208

Vertex A : ( V ) (V )

Vertex B : ( l± s ) (1=4)

Vertex C : (*§^) i

Edge AB : (l= e) (l±e)/> W (0) (l=S)i+l (0 < i < A )

Edge AC : (fa s) (l= i) ( l f i ) p /J p ) (0 < j < P2)

Edge BC : ( l± t ) ( l f t ) ( l i t ) pM (6) ( 0 < i < A ) ,

Interior : ( l= t ) (1 ft) p W (0) ( l f t ) ‘ (1 ft) P * ? ' \ b )

(0 < i, j; i < P i ; i + j < P2, Pi < P2)■

The mass matrix (<&, <f>j) is shown graphically in A.4. In this figure we have

ordered the modes in the geometric sense, i.e. vertex modes then edge modes then

interior modes last. We see that the interior-interior matrix is strongly banded.


209

Figure A.4: Mass m atrix for the continuous basis for triangles (N = 15).


A .2.2 T h ree-D im en sion a l E xpansions210

a) H exahedral Domain b) Prismatic Dom ain

c) Pyram idic Domain d) Tetrahedral Dom ain

Figure A.5: Vertex labelling for three-dimensional standard regions


H exahedral D om ain

Using the labelling given in figure A.5 (a) the hexahedral basis is:

Vertex A : (1=*) ( t o ) ( t o ) Vertex B : ( t o ) ( t o ) ( t o )

Vertex C : ( t o ) ( t o ) ( t o ) Vertex D : ( t o ) ( t o ) ( t o )

Vertex E : ( t o ) ( iz i ) ( t o ) Vertex F : ( t o ) ( t o ) ( t o )

Vertex G : ( l = £ ) ( t o ) ( t o ) Vertex H : ( t o ) ( t o ) ( t o ) ,

Edge AB : (»=*) ( t o ) ( t o ) ( t o ) (0 < i < P l)

Edge CD : ( t o ) ( t o ) p u (a) ( t o ) ( t o ) :

Edge EF : ( t o ) ( t o ) (o) ( t o ) ( t o )

Edge GH : ( t o ) ( t o ) ^ t ( a ) ( t o ) ( t o ) i

Edge AC : ( ¥ ) ( ¥ ) ( ¥ ) Pi - d b) ( ¥ ) (o < j < p l)

Edge BD ■■ ( ¥ ) ( ¥ ) ( ¥ ) p }-\(b) ( ¥ )

Edge EG : ( t o ) ( t o ) ( t o ) p V (j) ( t o ) :

Edge FH : ( t o ) ( t o ) ( t o ) p U (&) ( t o ) :

Edge AE : ( t o ) ( t o ) ( t o ) ( t o ) pW l(c) (o < k < p 3)

Edge BF : ( t o ) ( t o ) ( t o ) ( t o ) pM l(c) :

Edge CG : ( t o ) ( t o ) ( t o ) ( t o ) p tU (C) :

Edge DH : ( t o ) ( t o ) ( t o ) ( t o ) p « l(c) :


212

Face ABCD : (1=*) ( i f* ) P h \(a) ( i f ) ( M ) P b\(b) ( i f )

Face EFGH : ( i f ) (l±») * ( » ) (If* ) (1±S) P / i ( 6 ) ( i f s )

(0 < i, j ; * < P i , j < P2)

Face ABEF : ( i f ) (I f* ) P « ( 0) (1=4) (If*) ( i f ) F " ( c )

Face CDGH : (1=*) { i f ) P " ( a ) ( lf4 ) { i f ) ( if* ) P & ( c)

(0 < i, A;; i < Pi; k < P3)

Face ACEG : ( 1 ^ ) (1 ^ ) (1±4) p U (j) (1 ^ ) (If*) P « l(c)

Face BDFH : (l±a) (1=4) (1±») p / ' \ (4) (I js ) (J±s) pWl(c)

(0 < j , k; j < P2; k < P3)

Interior : { i f ) ( I f ) P & W ( f ) ( i f ) P & ( b ) { f ) ( f ) P " ( c )

(0 < i , j , k; i < P i ; j < P 2; k < P 3).

Prism atic D om ain

Using the labelling given in figure A.5(b) the prismatic bases is

( 1 - c )

Vertex A : (1=*) (1=4) (1=*) Vertex B : ( i f ) ( i f ) ( f )

Vertex C : ( i f ) ( i f ) ( i f ) Vertex D : ( if* ) ( i f ) ( f )

Vertex E : (±=a) (±±s) Vertex F : (±±a) (±±s)


Reproduced withpeimjssion of (he copyright owner. Further reproduce

reproduction prohibited without permission.

214

Edge AB : ( t o ) ( t o ) (a) ( t o ) ( t o ) (0 < i < Pt )

Edge CD : ( t o ) ( t o ) f " ( a) ( t o ) ( t o ) i

EdgeEF : ( t o ) ( t o ) P # (a) ( t o ) !

Edge AC : ( V ) ( ¥ ) ( ¥ ) ( # ' ( .0 < j< P 2)

Edge BD : ( t o ) (1=4) ( t o ) p W (6) (!? )I+1 :

EdgeA E : ( t o ) ( t o ) ( t o ) ( t o ) P U l(c) (0 < fc < P3)

Edge BF : ( t o ) ( t o ) ( t o ) ( t o ) P u , (c) :

EdgeCE : ( t o ) ( t o ) ( t o ) ( t o ) P « ( C) :

Edge DF : ( t o ) ( t o ) ( t o ) ( t o ) :

Face ABCD : ( t o ) ( t o ) P t\(a ) ( t o ) ( t o ) p f y b ) ( t o ) J+1

(0 < i , j ; i < PX’j < P2)

Face ABEF : ( t o ) ( t o ) P u (o) ( t o ) ( t o ) ( t o ) PjU (c)

Face CDEF : ( t o ) ( t o ) P M (o) ( t o ) ( t o ) ( t o ) f u (c)

(0 < i, k\ i < P i;k < P3)

Face ACE : ( t o ) ( t o ) ( t o ) P M,(6) ( t o ) ’+1 ( t o ) P ^ \ c)

Face BDF : ( t o ) ( t o ) ( t o ) ^ l( j) ( t o ) >+1 ( t o ) P V ? - \c )

(0 < j , k] j < P2; j + k < P3; p 2 < p 3)

Interior :

(0 < i , j , k; i < Px- j < P2; j + k < P3; P2 < P3.)


215

Figure A.7: Mass m atrix for the continuous basis for prisms (N = 15).

P yram idic D om ain

Using the labelling given in figure A. 5(c) the pyramidic basis is :

r = 2 i i ± 4 - i , s = 2 ^ 4 - i ,(1 - c ) ( 1 - c )

t = c.

Vertex A : ( fca ) ( i j i ) (l=s) Vertex B : (1±») (i=S) (l=s)

Vertex C : ( V ) ( ¥ ) ( ¥ ) Vert<=* D : ( ¥ ) (* ? )

Vertex E :


216

Edge AB : ( 1 f t ) ( i f a ) p h \ ( a) (1=1) (i= s)i+1 (0 < i < P t)

Edge CD : (i= e) (I± s) ^ \ {a) (i± i) ( I je y +1 :

Edge AC : (1=*) (l= i) (1±4) P " ( b ) ( i= t) i+l (0 < J < P2)

Edge BD : (If®) (1=4) ( l± t) p £ ( b) ( t s ) * 1 I

EdgeA E : ( I f* ) (1=4) (1 ft) (if* ) f « ( c) (0 < k < P3)

Edge BE : (1±S) ( l= i) (J=£) (l±e) p u i(c) :

EdgeCE : ( I f a ) ( l f i ) (1=«) (J±s) pW l(c) :

EdgeD E : (l± a) (1±S) (l=e) (i±£) P Ml(c) :

Face ABCD : (1=*) O f* ) P & ifi) ( i f t ) (1 ft) p U (4) (i= *)i+)+1

(0 < 2, i ; 2 < P u j < P2)

Face ABE : ( i f* ) ( if* ) J * ‘ (a ) (1=*) (l=*)i+1 ( if* ) P “ « . ‘(c)

Face CDE : ( i f* ) ( if* ) P « (a) (1 ft) ( if* ) i+I ( if* ) P ^ ( c)

(0 < i, k; i < P i; i + k < P3; Pi < P3)

Face ACE : ( i f* ) ( i f t ) ( l± t) ph \(b ) ( i f * ) * 1 ( if* ) ^ . ‘(c)

Face BDE : ( I f* ) (1=4) (1 ft) P « (4) (i=*)J+1 ( i p ) P?_*'-\c)

(0 < j , k; j < P2- j + k < P3; P2 < P3)

Interior :

(V) (‘f2) **.(«) (¥) (¥) pm (*?)**" Of*) p™lw(c) (0 < i , j , k; i < Px; j < P2; i + j + k < P3; P L, P2 < P3.)


217T e tra h e d ra l D o m a in

Using the labelling given in figure A.5(d) the tetrahedral basis is

(l + q) 2 (1 + 6) 1 t =( - b - c ) ’ (1 — c) ’

Vertex A : ( i p ) ( i p ) ( i p ) Vertex B : ( i p ) ( i p ) ( I p )

Vertex C : ( I p ) ( i p ) ( i p ) Vertex D : ( i p )

Edge AB : ( i p ) ( i p ) i ^ ) ( i p ) '+I ( i p ) i+1 (0 < i < f t )

Edge AC : ( I p ) ( i p ) ( i p ) p / i f t ) ( i p ) J+1 (0 < j < P2)

Edge BC : ( I p ) ( i p ) ( I p ) pM l(6) ( i p ) J+1 :

Edge AD : ( I p ) ( I p ) ( I p ) ( I p ) P “ (C) ( 0 < k < P 3)

Edge BD : ( I p ) ( I p ) ( I p ) ( I p ) Pt‘i ( c ) I

Edge CD : ( I p ) ( i p ) ( i p ) pW l(c) ;

Face ABC : ( I p ) ( i p ) P « ( a ) ( i p ) i+I ( I p ) p “ +U(i,) ( l p ) ' ^ +1

(0 < i , j; i < Pi; i + j < P2\ Pi < P2)

Face ABE : ( I p ) ( I p ) f £ i ( a ) ( lp ) * +‘ ( I p ) ‘+1 ( I p )

(0 < i, fc; i < P i \ i + k < P 3; P\ < P3)

Face ACE : (-

Face BCE :

(0 < j, k ; j < P2; j + k < P3; P2 < P3)

(0 < i, k ; i < Pi, i + k < P 3; P i <

( i p ) ( i p ) ( i p ) p ji\(b ) ( i p ) J+1 ( i p ) i f 5 u (c)

( Ip ) ( I p ) ( I p ) Pjl\(.b) ( i p ) ’+1 ( i p ) p £ p ‘(c)


218

Interior :

0 ? ) (* ?) ( W * 1 ( ¥ ) p?~t ww (i? ) '+i+111? ) pI-? * 1' \c)(0 < i j , k ; i < Px; i + j < P 2; i + j + k < P3; Px < P2 < P3.)

3K3

I

100

200

300 -

400

500

GOO

300 400 500 600 700

Figure A.8: Mass m atrix for the continuous basis for tetrahedra (N = 15).

A.3 Nodal Basis

A .3.1 T w o-D im ension al E xpansions

In the following section we define the notation hp (£) to be the Lagrange polynomial of

order P through the ( P + l ) zeros of the Guass-Lobatto-Legendre integration scheme.


219

(b)(a)

Figure A.9: Vertex labelling for the (a) standard quadrilateral and (b) standard Hexahedral region

Q uadrilateral D om ain

Using the labelling given in figure A.9(a) the quadrilateral bases is:

Vertex A : /if1 (a)/if2 (6)

Vertex B : /i%(a)/if2(6)

Vertex C : /if1 (a) hp2 (b)

Vertex D : hpl(a)hp2(b)

Edge AB : /if1 (a)/if2(b) (0 < i < Pi)

Edge CD : /if1 (a)/if2 (6) (0 < i < Px)

Edge AC : /if1 (a)/if2 (6) (0 < j < P2)

Edge BD : /if[(a)/if2(6) (0 < j < P2),

Interior : /if1 (a)/if2(6) (0 < i ,j; i < Px; j < P2).


220A .3.2 T hree-D im en sional E xpan sion s

H exahedral D om ain

Using the labelling given in figure A.9(b) the hexahedral basis is:

Vertex A : hPi (a) hP2 (b) hPz (c) Vertex B : hp\(a)hP2 (b)hPz {c)

Vertex C : hPl (a)hPp2(b)hPz(c) Vertex D : hp\(a)hPl(b)hP3(c)

Vertex E : hPl {a)hP2 (b)hp3 (c) Vertex F : hp[(a)h^2(6)/ip^(c)

Vertex G : hPl (a) hpl (b) hp* (c) Vertex H : hp\ (a) hp2 (6) hPpz (c)

Edge AB : h f l (a) Hq2 (b) hPz (c) (0 < * < Px)

Edge CD : h f l (a)h£2(b)h£3(c) :

Edge EF : hp'(a )hP2{b)h%{c) ;

Edge GH - h f l (a )h % m p l(c ) :

Edge AC : hPl (a) hPi (6) hPs (c) ( 0 < j < P 2)

Edge BD : hPl (a) hPi {b)hPz (c) :

Edge EG : hp'{a)hP2{b)h%{c) :

Edge FH : tf (a )/» ? (6 )A § (c )

Edge AE : hp'(a )hP2(b)hPz(c) (0 < k < Pz)

Edge BF : hp (a)hP2 {b)hPz (c) '■

Edge CG : h ^(a )h ^ l(b )h ^(c ) \

Edge DH : hp\{a)h%{b)hPz{c) :


Face ABCD : /if1 (<z)/if2 (6)/if3 (c)

Face EFGH : h[l (a)hf2(b)h^{c)

(0 < i , j ; i < Px; j < P2)

Face ABEF : /if1 (a) /if2 (b) /if3 (c)

Face CDGH : /if1 (a)/if2 (6)/if3 (c)

(0 < i, k; i < P i ,k < P3)

Face ACEG : h ? (a )h f2(b)h?(c)

Face CDGH : /ig (a )/if2(6)/if3(c)

(0 < j, k; j < P i ,k < P3)

Interior : /if1 (a) /if2 (6) /if3 (c)

(0 < i, j , k; i < Pi; j < P2;k < P3).


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