finite element...2 convection dominated problems - finite element appriximations to the...
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![Page 1: Finite Element...2 Convection dominated problems - finite element appriximations to the convection-diffusion.....equation.....2.1 Introduction 2.2 the 2.3 The steady-state problem](https://reader031.vdocuments.site/reader031/viewer/2022022008/5ad98b3f7f8b9a6d318b75b9/html5/thumbnails/1.jpg)
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Preface to Volume 3.......................................................Acknowledgements....................................................................
1 Introduction and the equations of fluid dynamics...1.1 General remarks and classification of fluid mechanicsproblems discussed in the book.................................................1.2 The governing equations of fluid dynamics..........................1.3 Incompressible (or nearly incompressible) flows..................1.4 Concluding remarks.............................................................
2 Convection dominated problems - finite elementappriximations to the convection-diffusionequation...........................................................................
2.1 Introduction...........................................................................2.2 the steady-state problem in one dimension..........................2.3 The steady-state problem in two (or three) dimensions.......2.4 Steady state - concluding remarks.......................................2.5 Transients - introductory remarks.........................................2.6 Characteristic-based methods..............................................2.7 Taylor-Galerkin procedures for scalar variables...................2.8 Steady-state condition..........................................................2.9 Non-linear waves and shocks..............................................2.10 Vector-valued variables......................................................2.11 Summary and concluding...................................................
3 A general algorithm for compressible andincompressible flows - the characteristic-basedsplit (CBS) algorithm......................................................
3.1 Introduction...........................................................................3.2 Characteristic-based split (CBS) algorithm..........................3.3 Explicit, semi-implicit and nearly implicit forms....................3.4 ’Circumventing’ the Babuska-Brezzi (BB) restrictions..........3.5 A single-step version............................................................3.6 Boundary conditions.............................................................
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3.7 The performance of two- and single-step algorithms onan inviscid problems...................................................................3.8 Concluding remarks.............................................................
4 Incompressible laminar flow - newtonian andnon-newtonian fluids......................................................
4.1 Introduction and the basic equations....................................4.2 Inviscid, incompressible flow (potential flow)........................4.3 Use of the CBS algorithm for incompressible or nearlyincompressible flows..................................................................4.4 Boundary-exit conditions......................................................4.5 Adaptive mesh refinement....................................................4.6 Adaptive mesh generation for transient problems................4.7 Importance of stabilizing convective terms...........................4.8 Slow flows - mixed and penalty formulations.......................4.9 Non-newtonian flows - metal and polymer forming..............4.10 Direct displacement approach to transient metalforming.......................................................................................4.11 Concluding remarks...........................................................
5 Free surfaces, buoyancy and turbulentincompressible flows.....................................................
5.1 Introduction...........................................................................5.2 Free surface flows................................................................5.3 Buoyancy driven flows..........................................................5.4 Turbulent flows.....................................................................
6 Compressible high-speed gas flow...........................6.1 Introduction...........................................................................6.2 The governing equations......................................................6.3 Boundary conditions - subsonic and supersonic flow...........6.4 Numerical approximations and the CBS algorithm...............6.5 Shock capture......................................................................6.6 Some preliminary examples for the Euler equation..............6.7 Adaptive refinement and shock capture in Eulerproblems.....................................................................................
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6.8 Three-dimensional inviscid examples in steady state..........6.9 Transient two and three-dimensional problems...................6.10 Viscous problems in two dimensions.................................6.11 Three-dimensional viscous problems.................................6.12 Boundary layer-inviscid Euler solution coupling.................6.13 Concluding remarks...........................................................
7 Shallow-water problems.............................................7.1 Introduction...........................................................................7.2 The basis of the shallow-water equations............................7.3 Numerical approximation......................................................7.4 Examples of application.......................................................7.5 Drying areas.........................................................................7.6 Shallow-water transport........................................................
8 Waves...........................................................................8.1 Introduction and equations...................................................8.2 Waves in closed domains - finite element models...............8.3 Difficulties in modelling surface waves.................................8.4 Bed friction and other effects................................................8.5 The short-wave problem.......................................................8.6 Waves in unbounded domains (exterior surface waveproblems)...................................................................................8.7 Unbounded problems...........................................................8.8 Boundary dampers...............................................................8.9 Linking to exterior solutions..................................................8.10 Infinite elements.................................................................8.11 Mapped periodic infinite elements......................................8.12 Ellipsoidal type infinite elements of Burnnet and Holford...8.13 Wave envelope infinite elements........................................8.14 Accuracy of infinite elements..............................................8.15 Transient problems8.16 Three-dimensional effects in surface waves......................
9 Computer implementation of the CBS algorithm.....
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9.1 Introduction...........................................................................9.2 The data input module..........................................................9.3 Solution module....................................................................9.4 Output module......................................................................9.5 Possible extensions to CBSflow...........................................
Appendix A Non-conservative form ofNavier-Stokes equations................................................
Appendix B Discontinuous Galerkin methods inthe solution of the convection-diffusion equation......
Appendix C Edge-based finite element forumlation...
Appendix D Multigrid methods......................................
Appendix E Boundary layer-inviscid flow coupling....
Author index....................................................................
Subject index..................................................................
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& %% + L % + &%%= %& & ! + L %%= + .9 # % K 7.'48 % %%=> %
'
.'(
Pe = Uh/2k = 0
= =(All exact)
Pe = 1.0
Pe = 2.5
=(Exact)
Pe = ∞
L
h
1.0
Standard Galerkin α = 0Petrov–Galerkin α = 1.0 (full upwind difference)Petrov–Galerkin α = αoptExact
Exact
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+ L %%= K & K 7.'98
. ' ' . . ' .
B ..B
* L %%= 7 8 & & & =% , .. & & ' = > & % , .. H1 + = % L % L & +
2 %= E # %& >- K + = > %%=> &
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% F
%& & %+ 7 8A
.
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& %%= K K 7.'98 &
' ' ' . . ' ' ' .
B ../
B H1 %%= SK 7.'98T ' % K 7..B8 & = % L % %
% '
..9
= & # % 4 % >
''
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# , .. B H1 %
' ..4, ./ %
6 % % % % N 2 = & % % > , .9'B & N 2 % H1 % " & =% 3H1 &%% K 7 % + L % 8 & & + L , .) '' % # L %% K 7../8 & D
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B
B
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1.0
0.8
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α
αopt = coth Pe–1/Pe (optimal)
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# & K % K 7.''8
B ..(
& =% K 7../8 % H1 %
15
10
5
01 x
φ x 104
– d2φdx2 + 200 dφ
dx= x2
0 < x < 1φ = 0, x = 0, x = 1
1.00
0.95
0.90
0.85
1.002.0 x
φ
– d2φdx2 + dφ
dx= x2
1 < x < 2φ = 1, x = 1φ = 0. x = 2
60x
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% % 3H1 % % % 3H1 % + K =% , .) % + L % %%=> # % $ % K + L
L &1 & -+ & L %%= %%
K 7 H1 %8 & > % K # C < L & % &
' !!
K 7.''8 >- & % % &H '. % % % 7 >- K 8 % %% & & # 1 K 7.''8
B
B .;B
2
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Exact and Petrov–Galerkin(α = 1) solution
Finite differenceupwind solution
x
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# & %B
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B .;.
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%%= H1
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% %% L %% 7 8 %%= % % % & H1 K %%= '9')
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# & =% %%% % ' . % & 2 & & + , .* & & K
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L % 2 # & L 7 8 - # & + K 7./)8 7 8
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2 C& L < S& + & K 7./*8T &
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h
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# % & & & @ ;B ' % H1 % % % & K 7./(8, .( % K 7./)8
C> L < %% D & ;'
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z
yx
z
yx
(a)
T (y ) T = 0
T = 0
T = 0x
y
Boundary conditions for test problem
U
θ
(b) Solutions for θ = 45° (top) andθ = 60° (bottom)
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& & 3H1 %
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& %% C < 3H1 K 7./*8 & K H1 % ../ = > K % = K 7.;)8 7./.8 K 7./*8= H: 7
8
B .9;
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% # & > &2 % %-
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+ & K >- ../ % 6$ % % & % = % % %& & % >%% % > % L
% % & % = % & % &2 > % % 7 8
1 & & % & + % & & > $ % K
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! B .94
K % K 7.'B8 & % , % %
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B .9(
& %=# 3H1 % K 7.948 7.9(8
& % %% ../
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7&8 %% % 7 ..;8 3 % %%
6 & % K %& C % < %& >%% % K % K # % A
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. %+ & B K % 2 D= K %& H1 K &
&' (( '
0< ' 9 ' ' ''
$
# &- % % %%& & >%% K 7.'8
A
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% > & % % & % K SK 7.'B8T
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K % & 2 >- % % & % % % F '# K 7.).8 &
& 6 B B %
B
7.)98
S & & T# % K %% , .'B # %& > 7L 8 2
" 2 !
% % %%& > K # % L > # 3H1 %% & >1 &K -+ & > % # H1 K 7H:8 % & % K K & % & & 1& & & 3H1 & # 1 & + 7,!8 %& > K , & + %% &
% >
φ(x ); t = 0 φ(x – Ut)
x
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% & %> + # %% %& & 5 ;. ! @ ;;;/ @ 1;B;9;) ;4;* %> =% =% % & %E & %%
& E # K % L %% & -+ % + L & %& 3H1 & K # & A 7'8
% & > H1 .)P 7.8 % %% %& > %%= #3H1 " %% + &
% & & & # K & & & %%
% # & D K !% ; & =% % % & K
L & %% D K > 3L K & % 7K .'8 # %% % #3H1 % & K >% %& D %& % %% + % %& % % K %& 1 % # & %% H H: %& F '!% '. % H: % &2 N %& & # - % & !% ;, % % &
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0= '';/
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.))
t
t
Characteristic
Updated nodeposition
Initial nodeposition
h
∆t
∆t
(a) Forward
x
x
tn+1 = tn + ∆t
tn–1 = tn – ∆t
tn
tn
(b) Backward
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% %
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6 $&&/9 '(4/ % K # % & 2 % & > 3H1 /)
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% >- S T$ & %%= & =%
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B .4. H1 2 L K ' &
+= & K
% . .4;
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%& , .'. % & % S K 7.4'8T & # &
' & .4/6 K ' & %%= & %
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%' &#& .4)
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% &#& .4)&
# & %= % % = # % & %
N (y )
x
N (x )
1
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tn + ∆ty = x + u∆t
tn
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, ' ! (! $# !
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t
x
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φnφn(x – δ)
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6 % =% %% '. L 7 8 % &
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1.0
0.8
0.6
0.4
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00 1 2 3 4 5 6 7
Pe
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Stability limit C = 1/Pe + 1 – 1/Pe
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% % > D !$ & !% ; % K %% & $$ & !% '. F ' %& & = % > % % & % %
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1 F ' % & %% +
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u1 = u∞, u2 = 0
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u1 = u2 = 0p = 0
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& K 5 & 5 & 'B BBB , /(
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6 !% ; % =% %> 1 % N > & & =% % & & & %& L = = # & 'B =% % L %
88 ';:
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–0.01–0.02–0.03–0.04–0.05–0.06–0.07–0.08–0.09–0.10–0.11
0 0.2 0.4 0.6 0.8 1.0x1
p
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(b) Re = 400
–0.02–0.03–0.04–0.05–0.06–0.07–0.08–0.09–0.10–0.11
0 0.2 0.4 0.6 0.8 1.0x1
p
'CBS'
(c) Re = 1000
–0.02
–0.03
–0.04
–0.05
–0.06
–0.07
–0.08
–0.09
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p
'CBS'
(d) Re = 5000
# 8< 6 * " , B* # $
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Streamlines Pressures
(a) Stokes flow, viscosity 1, 9600 iterations
(b) Re = 400, 4400 iterations
(c) Re = 1000, 6100 iterations
(d) Re = 5000, 48000 iterations
# 8= 6 * - B* # $
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% % & 1 % 6 % & =% !% ; ;)
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% + !% '/ '9 F ' & & K # % & = D % %& % + % % # % & & % %& D 5 ''3)( & + % % D
)+ .
L %%
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40x_non-uniform'
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x
u
u
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% D %& =% > %%# & >
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u
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& 7 .. 8
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+
2 % %& .. # 1 & =
h
h
hh
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% 7 , /'.8
= '
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5 %
5 ''.B
..
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=% 7/..8 7/.;8 = 6 F ' % % K & & > %& K & # 1 &
..
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# %% & =% 7/./8 % %& % %
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. /.9
X >%+ % &
N K # 2 +> > % & % $ %
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n1 n2
emax
Exact φ
Linear φ
hn1, n2, nodesh, element size
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# = % %& 7 >8
.
/.)
+ K # % % =% # 7 8
& 7 % 8 %
& /.4 2 & + & # & %%= L $ !% '/ F ' & %
+ % % % & % % > % % % % & %% K + % % & %%= % , K + & % & # % K !% '/ F ' & % & & & % & % K" % +
K & 6 = & % % %
% = %%=
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# %%= & K %%= &
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& % .
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.
%'
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% = &
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.=
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% & % % & =
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& % %+ A
' , . & % & =
; ! 2 = K 7/;.8
/ ! = 2 7K /;;8 & = &
9 5 & 2
# & % N K % &1 % %.. =% % # % 2 % &1 & % %
7% %& D8 = &2 % # = & %% & 1 2 D & % % 2# % - & & %%
% % & 1 = % # & >
& &1 & & >% K & % %
x1
X 1
X 2
hmin
s hmin = hmax
α
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/ ) + .
# D D %& % & > & & K & %& , %& %& %& =% %% % &
/;/
= 2 7 28 # & =% & 2 % = K 6 % = 2 & " + 6 % = 2 && 0 12 9B =
= /;9
& B ' & = 2
1$
& + % & & + # K & & % & %% # % % & N % & & %& %& D %& %& D + % 7F %& D & K & 8" & & & &
% % & ! % & & %& & 9().)( =% &
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(!
=% %& D %& &> % % + %& D % = /*' **B , & & & & % , /'/ > & % & + %7, /'/8 & + % & + 7, /'/&8 $ = % & 1 H 9 7, /'/8
(a) Final adapted mesh, Nodes: 1514, Elements: 2830
(b) Streamlines
(a) Final adapted mesh, Nodes: 1746, Elements: 3293
(b) Streamlines
(c) Pressure contours
II Gradient based refinement
I Curvature based refinement
(c) Pressure contours
# 8.< 2! ! " B > " "
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6 = & D &1 % % , /'9 + & & % 6 & & % % &
+ & % N F ' & % K %&
8= 1 -) #' ' ' -'/
% % % % 1 "& + & 7 ) + 8 % & % %& %" K L % & /B & A = & % , /') %& +=
& !% '9 F ' % & &
1/'/.
8> (-' /# )) '
% L &2 & !$ 5 & D # %&D % %% L %& D %& # D &2 D L 5 < &'BB 9BBB % , /'4 & 5 & # L &2 5 'BB 5 9BBB % & &2 # % &2 S %% . K 7;.;8 7;./8T %& 5 < & %& % &2
85 " 3 9 : - '
: ' %$ !
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" 3 9 : - '
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#8.=
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K 7/'8 7/.8
B /;)
( B /;4
# & % &
.
;
/;*
%& %& %A
78 % &
With stabilization Without stabilization
With stabilization Without stabilization
(a) Re = 100
(b) Re = 5000
# 8.> + ," * B*&
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7&8 & 78 & %
K !% ' ;
: ( ! 2
# 2 & / % %%=>
& &# /;( & % K 7/;)8 & % %
'#
B //B
& % % %%& K = 2 & & & & *) 7 !% '. F ' 86 % =
% % % % 2 & = 2 # % D '(4B*4*(
(B(.
# 2 K &
/ *
*# .$
#
,
//'
% 2 $ =
/ #$B &
* &#&
&#
&#
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% & % # 1 (B(.
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+
+
O (h ) (T3B1/3C)
O (h2) (T6/3C)
+ O (h2) (Q9/4C)
(a) Continuous p interpolation
+ O (h2) (T6 B1/3D)*
+ O (h) (Q4/1D)*
+ O (h2) (Q9/4D)*
+ O (h2) (Q9/3D)
+ O (h) (T6/1D)
(b) Discontinuous p interpolationVelocity nodePressure nodeDenotes elements failingBabuska–Brezzi test butstill performing reasonably
*
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% (; D (/() & % % & K B %
% D % % % % % %
1 %& % # & = %& 1
D & + + D %%
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; 5% % ! !
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B B 3 % ! % & + K7;;/8 7&8 % # & & & % &
K 7;;;8 %+ & %%% % , /'( #& =% K 7//;8 1 " , /'(7&8 & % %
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# % % % D % B %
//9
" & % D
//)# 7 D8 & & &
% K 7//'8 & & = /& %
/ / / / //4 & %
σ
ε•
µ
(a)Linear, newtonian, fluidε•
σ ∝ εm
σ
ε•
µ
(b) Non-newtonian polymers
σ
ε
|m | < 1
µ = σ/3ε
ε•
σ
ε•
µ
(c) Viscoplastic-plastic metals
σ ∝ (σy + γεm)(Bingham)
γ = 0(Ideal plasticity)
σy γ = 0
γ > 0
ε•
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%% N 2 & D & & % 'B) % = 7 % + 8 -+& # & & % % & )* % %> =% # % & C>%<'B9'B) %= %# % %
% %% & # ''9 '.* '.( % +
; 5 !
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C< D %& , /.' % = % & #& /' , = = & ';B & & D L , /'* (; # + % %& > D % &
% 7 % 8 % & %& K # & %%% K K 7/)8 K + , 1 & %&
%+ %
/9B %+ & 1 %
Prescribedvelocity
Prescribedtraction
Extrusion
Rolling
(a) Steady rate
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Moving mesh
Extrusion
Rolling
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Sheet forming(deep drawing)
(b) Transient
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& &
/9B&
& K 7'(8
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62 !% % = , /.' % => %& 7 % 8(;
' 78 . 7+ 8
% = , Z !78 = , Z !78
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= .9 *BBB BBB [ .9 *BBB BBB [
CL
MESH I
CL
MESH I
v = 0u = 1specified
Slip boundary
Free boundary
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% L % K 7/9B8 & & & %&
7 K /)8
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# % %& & &1 & % 1 D + &1 % 1 K % % !% . % + '(4;
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& & 1 N % % '') # % % = N %
; - ! %$ $
# % %&& %& #%=% % & , /.; /./ % %% # & %
+ % % & 1 # + & & & & & =%
/9. % % % &
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9 1x
= 2.
25cm
9 1x
= 3.
25cm
9 1x
= 4.
9575
cm
9 1x
= 5.
915
cm
9 1x
= 6.
15cm
T = 400K
T =
460
K
Tem
pera
ture
con
tour
s fo
r ent
ry T
= 4
00K
at in
terv
als
∆T =
4.0
K
T = 700K
T =
722
K
Tem
pera
ture
con
tour
s fo
r ent
ry T
= 7
00K
at in
terv
als
∆T =
1.5
K
y,ν
30.2
ν tan
g =
ν tro
ll = 2
8.73
cm/s
k∂T
/∂n
= α 2
(T–3
22)
k∂T
/∂n
= α 1
(T–2
95)
T =
T1
0.88
9
2.25
3.66
52.
25ν
= 0
∂T/∂
n =
0x,
u
0.633
∂T/∂
n =
0
All d
imen
sion
s in
cm
k∂T
/∂n
= α 2
(T–2
95)
C L
Hor
izon
tal
Verti
cal
(a) G
eom
etry
Nodal points
20
301.
0 –1
.0–3
.0cm
/s(b
) Vel
ocity
pro
files
(c) T
empe
ratu
re d
istri
butio
n fo
r diff
eren
t ent
ry te
mpe
ratu
res
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& % 1 % & % %& =% , /.;*( %
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% % % %& & 2 %+ & % %& % %&
!% '9 F ' % =% & % ';;';/ , /./ %
U
(a) t = 0
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U
(b) t = 15∆t
30%
U
(c) t = 30∆t
60%
U
(d) t = 45∆t
90%
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Initi
al m
esh
Upd
ated
mes
h 63
6 D
OF
η =
15%
Upd
ated
mes
h 12
00 D
OF
η =
18%
Adap
tive
refin
emen
t 808
DO
F η
= 11
%Ad
aptiv
e re
finem
ent 1
242
DO
F η
= 10
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t = 0
t = 1
.1s
t = 7
.7s
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% % & # %% & =
&- # % % % " % > 6: 7& 3 8';9';( %& < + & K
644
CL‘Effective’ strain (ε) t = 2.9 s
CLTemperature (T ) t = 2.9 s
0.4
0.3
0.2
0.1
2 4 6 8 10 12 14 16Time (s)
Load
(MN
)
(c) Load versus time
(b) Contours of state parameters at t = 2.9 s
0.20.2 0.8 0.8
624
623624624 624643
649
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r b =
1.9
625
in
Punc
h
r c =
1.9
25in
Blan
k ho
lder
1.94
375
in
(Initi
al) 3
.60
in
0.035inRA =
1.
875
inµ 1
=
0.20
r d =
0.2
5in
Die
Geo
met
ry a
nd fi
nite
ele
men
tdi
scre
tizat
ion
of th
e bl
ank
usin
g lin
ear e
lem
ents
H =
445
T
µ 2 =
0.2
0
60 50 40 30 20 10 0
Stressσy (kips/in2)
0.05
0.15
0.25
0.35
0.10
0.20
0.30
Stra
inε
Dis
cret
ized
stre
ss-s
train
cur
ve
100 80 60 40 20 0
Punch load (kN)
20
40
60Pu
nch
trave
l (m
m)
0 Punc
h lo
ad –
pun
ch tr
avel
Num
eric
al
Expe
rimen
tal
0.5
0.3
0.1
–0.1
–0.3
–0.5
–0.7
Strain
20
40
60
800 Orig
inal
radi
al d
ista
nce
(mm
)St
rain
dis
tribu
tions
for p
unch
Trav
el =
60
mm
Expe
rimen
tal
Rad
ial s
train
Thic
knes
s st
rain
Circ
umfe
rent
ial
stra
in
z =
9.20
mm
z =
17.9
mm
z =
28.7
mm
z =
34.7
mm
z =
48.5
mm
The
arro
ws
indi
cate
the
velo
city
vec
tor
at e
ach
noda
lpo
int
µ 1 =
µ2
= 0.
20
Flat
bot
tom
pun
ch. D
efor
mat
ion
atdi
ffere
nt s
tage
s
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K 7/9.8 %
% 7 %% P % % %8 & D K & %%
& % % + %%& $ %& %& & % # %% & & '/B3'9; %% & & & , /.9 /.) % %&
(a) Mesh of 856 elements for sheet idealization
X
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; !
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t = 900 s
t = 2400 s
t = 4280 s
(c) Deformed shapes of various times
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(a) (b)
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(a)
g
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50
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1 unit of U/Uo = 1 unit of X/h
(a) Velocity profiles downward of the step
k-ε model (CBS)Exp.
(b) Streamline pattern
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& % & >% D & % % 6%%> / )* D )* % & 6 % =% %& % =% % + L + '(*B % L & + 2 + =% %" + %%=
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& % & % & & & + L %% & % N % %& 2 & & & + % % 7 + 8 !% ' ; & K D
%& %& D !% / %& %% %+ %& D >% D & = B9 D 1 , > =% !% . ; % 1 + < :Q
& 1 ';* 1 % %
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= ' 9 / -' 3
# K & & %& 31 K %& D & & & 12 /B D = K 31 %& % K & !%
; %%= & % % & % %&
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F + %%= & %& D %& # + #3H1 % >% >% & !% . .'B %%& !$ % !% ; & % N K & %>& % %& %& D % D %& % #3H1 =% !% ; % K & 7, ;/8 !$ %% % & =% >% >%
!$ $ %& &
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& >% % =% >% 6 1 > %&
2 % % % & # % % % ;;/ !% ; & =% % ;;/
%% % & = %& 1 L % % # %& %& D
1 &- =
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! + %%= & >% B = % 1 %& + % 1 % # & % & >+ 1 & %%= % K , + % % & 1 % %& % + 1 1 %
& 1 =% + 1 # % % N & & > %& & = , ' $ /'/.
# % L % 1 + & 5/; '(9B # &2 & & & + % & 1 + % :%// /9 !1 $ /) @ /4 6 + & % & % & ') /* + % # + % % &>K %& D 5 % 1 % /( & L # % + L % 1 +
% + %%= K
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& =% >% K & # % % K ' & &
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,, 9 $ !
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0.425
t = 0
ρ
0
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2.5
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e
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# =0 0 B 8E 0 " E # 8$ 6 6
" -'' :- ' ' 2
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' 9*
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00 1.0 2.0 3.0 4.0
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ρ
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2.0
1.5
1.0
0.5
00 1.0 2.0 3.0 4.0
x
u
2.0
1.5
1.0
0.5
00 1.0 2.0 3.0 4.0
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ρ
(b) Supersonic inflow and outflow
2.0
1.5
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0.5
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u
2.0
1.5
1.0
0.5
00 1.0 2.0 3.0 4.0
x
ρ
(c) Supersonic inflow–subbsonic outflow with shock
2.0
1.5
1.0
0.5
00 1.0 2.0 3.0 4.0
x
u
ExactCL = 2.0CL = 1.0
0.75 0.50.0 5.0
a/2u' u2
C1C1
# = 9 4! " ,, 2* ( ,
-'/ #;- # 3
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9BB % , % D & D 1 :%>% + L , );78 C < N :%
(a) Structured uniform mesh
3.00.6
1.0
Inflow
2016 elements1089 nodes
y
x
t = 0.5
t = 1.5
t = 2.5
t = 4.0
(b) Solution – contours of pressure at various times
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" -'' :- ' ' 2
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,, -%5 ! % !
# + =% % D % # %& + & !9( / % %& , )/
N % :% :% .B & % 1
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,3 -$ 5. ! $ $
" %%= & & 2 7 8 & > , %% & % # % /9!% /# % & # &
% = & C< + , )978 & & & N %% # &
-'/ #;- # 3
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+ 6 % % , )97&8 & % + & ( & %% % & %& % &1 % L 2 7 % 8
1 , )) , )) % 1
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K + & % & %% )'
(a) Triangle subdivision
(b) Restoration of connectivity
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1 -) '4 -' ' -'/
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,3 5. $ %5 !
N & & & $ & &- %% + &- /9 !% / # %& 2 & & %% & ''') N > %& 7 & % 8 %&
=% = % % , )4 % =%'' 1 D
% C>< %%%
20°CL
Initial configuration Density after 100 steps
20°CL
After 101 steps Density after 200 steps
20°CL
After 201 steps Density after 250 steps
Exactsolution
Q
# == - 1 / 4! !" + * - *
-'/ #;- # 3
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(a)
(d)
(b)
(e)
(c)
(f)
Anal
ysis
dom
ain
Wal
l
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% # # , )* %= =% )* &
> & A
(c) The corresponding pressure contours
(b) The corresponding density
(a) Sequence of meshes employed
Analysisdomain
22° Flowvelocity
# =5 A* 4! * 1 8 " 9 ' 8=E ' >E<) ' /</ ' ?>?) ' < ' 8E=
-'/ #;- # 3
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(c) (d)
(a) (b)
10
5
1
M = 3r = 1
u1 = 1u2 = 0
# =? - 4! *8? / #$ "* * #$ ' ?8 ' = >E> #$ 1 " #$1 "
1 -) '4 -' ' -'/
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1.80
1.60
1.40
1.20
1.00
0.80
0.60
0.40
0.20
0
–0.20–3.00 –2.00 –1.00 0 1.00 2.00 3.00
X
Gp
(a)
4.00
3.00
2.00
1.00
0–3.00 –2.00 –1.00 0 1.00 2.00 3.00
X
Mac
h nu
mbe
r
(b)
MUSCLCBS: Anisotropic Shock CaptureCBS: Second Derivative Shock Capture
# =.@ - 4! *8? / #$ #$ 1 " " *
-'/ #;- # 3
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' 6 1 &. 6 N 1 % D C+ < & = 2 %%
; , 2 & + %
, % %& % & 5 ).3)/ % K % >% D , )( )'B % ; D %
9) # 7, )(7&88 % 1 1 # & + % , )(78 )(78 & !$ & 1 %
Analysis domain
# =.. 9 "" ! !E
1 -) '4 -' ' -'/
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& 1 % % , )'B N % & & > % & !:). % 1 % 6 % = % % 7>&8 , )'' % =% % 1
& 1 6 = + & % =% '4 % %
=5 '; ) :-
#> %& D > %& & % %%=> %& > !% ; = > %& %% + , & % > %& =% & > & & % 6 % & & 1 % %& & 1 % L % % %
% = %% %
& & & % % & .) & % 2 6%% = !
% & % & % K + 6 & + % + + &K + %%= + % +
-'/ #;- # 3
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% % &K # % & % % 6%% = K )9)( % & & & C <
+ # & %% % L
$ # % %2 % % % % %& &-;);*
> %% =% %& )'B > 31K
,: $ % ! !
%% % D + L % & # & % : % %& + % >'(*B + 7 + + & @ 4B8 # + % & # %& & = 14' F % % & ') '(*4, )'. ') % +
% . # % % 7 8 + 2 1 '.9 BBB %%= 4B BBB
;9B BBB & # % & & % D % &6 % % &1
& D
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% %% L + & .) =% & 1 K ' )') BBB + & %
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% # % '9 "& '((4 1 %& & % & % 7 & 8 %& >= # % D % & % 6 K & % = 49B , %&# % &
% % & >% % &
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1 % %& 1 K + % & & # &- % & & % > % & & & # % %
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% =% #+ =% % 4; > , )'4 % & > % % 1 6 % , )'*
> %&4; " > % &
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NE = 7377NP = 3867
NE = 6847NP = 3580
NE = 8459NP = 4379
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‘Body’ layer subdivision in three dimensions joining a tetrahedral mesh
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(a)
Density contours(b)
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Initial mesh, 1804 points, 3487 elements Mach number
(a)
Mach number
Mach number
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% & 1 % 1 % 1 ! & %& & L K & % & 31 K
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% %%& %& & % & % D(/() 3 % K 6%% =
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=. # ''
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*) $ # & & & D 2 , 2 59 '443(' '()(
*4 @ H # % & & % %& D2 , 2 52 49;34* '()(
** $ K 6 ? # %+
+ " +2 8>> '(;4*( 5 1 55 ! 5 # D
5 & BB!:>? @ '(44
(B @! : $ ! % % R K= K R + < % 0 + 3@ '(*'
(' @ 2 6 F D > %% 0 34 % % >
F $ 7'((/8(. @ 2 F % K &1 "#
"#!=:!82B24 '(()
(; @ 2 6 % ==!?44? @ '((((/ @! : $ F3 % %& D
5+-;,- . 0 5+-!"484
$ L ! "& '((.(9 @! : $ ! > % D
3 4 2 #2
,* : $ !6 768 @ '((.() @! : $ H=>: ! & 3
%>& .# 48 , - , #G 6 '((9
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";' -'/
>. ('
# D & % % # % > % % K % # % D + D 2 %
% %& & & %%= + K 2 & > K % %A
*
+ , ' . 4'
K & 1 %& D 3 % % %& D 78 A
' # % % % %%&. # % % 7 1 8 %>& D
& % L % #&- % & K %%= K & =%# %%= D > &
& D < &- % % # + &1 = &- ' .
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6 % 2 >K % % & % = % %% ;
>0 / ;' 2
% % 31 K % %& % & %& K A
B 4.
A
'
'
( B 4.&
& ' . ; D , 4'
; ; % > & # K &
H
A
A
h
U1
u1
us1
Average velocity
η
x3
x2
x1
τs13 , wind drag
Hh
A
A
τh13 , bed ‘friction’
Free surfacep = pa
(atmosphere)
Mean waterlevel
(datum)
(a) Coordinates
(b) Velocity distribution
η
# >. 0 !! :
/ ;' 2
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'
; ( B 4;
(; ( 6
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7 9; !% 98
; &
'
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.49
&
&; &
&'
' &.
.49&
% , D %
&' &. B 4) &
&; B %%= & K &
% ; ! ( K 74.8 ;
;;
;
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;
..
; B 44
6 ' . 1 , 4'7&8 +
; 4*
' . :& 2 ) #
-
#) # )
#
-
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- # -#
4(
& K 7448 K 74)8 &
;
4'B
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' . # + K 7448 & %
;;
; ; 4''
7498 &
;;
;
4'.
6 K 74'B8 74'.8 %> +
B 4';
% % K 2
'
'
(
; B 4'/
' . & + & % >
%> K &
'
.(. . '
;
' ; &; ( (
-
B 4'9
& & %& = # & K =% & & !R 2 =%
&; (-
.4')
-
' .
!R 2 N K 74'98 ( ! % >
%& +
(' (. (. (' 4'4 ( ! %# K + N
& 1
.;
4'*
/ ;' 2
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6%%= K 74'98 &
'
; '
.;
4'(
K 74';8 74'98 > %& K 74'8 %%% + &# ' .
-
'
.
4.B
' ''. (. .
. .'. (. .
4.B&
* B
' .
4.B
74'(8
+
B
(. ( '
-'
' ;'
('-.
(' ( .
-.
' ;.
(.-.
4.B
# & > K + % / 9 %%& > K = %%= % >%%
& K % > %% > & & & % & %% % =% D % & # % > %% & & 2 K 74';8 74'98
> & %%= K
B 4.'
(
B 4.'&
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& &
B 4..
(
B 4..&
.
.
(
B 4.;
2 K , % = %# > K %>
D % % % & %& & % %& > & & # = & & D
L # + 2K % > %%= 6 L & & L C < & & 0 12 ) 2 & % % =
> %' --':
$ + L + % & > K # %%= & %% 4 * %% % % % 2 & & >%% %(')
+ %% + &1 '(4. % '4 %% & /9'*/'
6 & =% %& D K & %> > K & & A
% % '
. (. .
%' --':
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# >&>% % % % %& D & >K /./;
# = L + >% D > %& & #3H1 /9 >% !$ % % % D - 7 % 8A
- 4./
2 %
(
A
- 4.9
% =% #3H1 %%= /9;. % > %% , & %&% % & &
% % .B 7=%8 % L % D & =% & = /;//
=% %& & !$% #3H1
>8 :- --
3 - 5 ! !
% % =% % %%& # + , 4. % /9
& # K /)/4 %% & & %# =% , 4; > C &1< %&
% % # %& 1 & %& D & K + L# + =% , 4/ 2
C&< % % % & %
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D % C&< 7 (
8 % % >
1 %
3 -%5 !
# = % % % & %& 6
40 elements
80 elements
160 elements
(b) Solution for 40, 80 and 160 elements(b) at various times
(a) Problem statement
Wavepropagation
Hl
a = 0.1
10 1040
η = a sech2 1/2 (3a)1/2 (x–α–1)
u = –(l + 1/2a) η/(αx + η), a = 0.1, g = 1.0, α = 1/30
Initialconditions
# >0 -" !
:- --
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% ' . # L > % > %& 1 % N % & = & + + % , % 3 + N + & % N # + =% , 49 % %&
2 = 1 %% /* % # % % & & % + % & 6 %& > & K 1& =% %% /9/.///(
C &< % & $ ! 1 , 4) %&# &- =
7 %& % &K D & & & 8 $
h = 2
H = 1
t = 0
t = 2.5t = 5.0t = 7.5
h = H = 1L
0
40 elements in L
η
t = 0
t = 2.5t = 5.0t = 7.5
0
u
# > " ! 6 = *
8
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= & & & & " & %# >%& & &
% D # L & % > % & 71 % % 8 , % & & %% =% & ;9 ;)# , 44
# % % = & L & & % + # 2 . 9 1 2 + &>
# % %%= 9B & &% % % # $ ! % & %
% & =% >=% /; # % & & " % 7"8 +.
/( % %
1.0
0.5
0
–0.5
u
2
0
40 elements
Hho uo = 1 A
Prescribedwater levelhistory
η
# >8 %& ! ! #$ 6 / / #/ $ 6 8 8
:- --
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10
5
0
Surfa
ce e
leva
tion
η (c
m)
x = L
Analyticaly = 0, ±2 ∆yy = 0, ± ∆y
Computed ∆t = T/40
10
5
0
–5
Velo
city
u (c
m/s
)
x = L
Analyticaly = 0, ±2 ∆yy = 0, ± ∆y
Computed ∆t = T/40
t = T/2
t = 3T/4
∆x
L = 22∆x
∆y
y x
Inlet
h (x)
# >< -* " " 6 /
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7=%8 % 79B 8 / 7.BB 8 * 7/BB 8 % 6 N 7 8 BB;* % ! # % !L % % #& 4' % %> & & % % % L % % % 7, 4)8 L % & & # = % & '/Z % /BB % % 9B & % 7';8 9B 7''8 # & 1& >=% , 4* & L 7=% 8 % 5 %
7 , 4) 4(788 C&< % && 6 %% % & %+ = % % %%= & % H 7H8 7449 1 6 8 && % L & % $ 7 8 % , 4(78 % 6 $ D 6 %% & & & % D
StockpoleQuay
Tenby Swansea
WormsHead
Port Talbot
Porthcawl
Barry
Cardiff
NewportBeachley
Avonmouth
East
ern
limit
Clevedon
Weston-Super-Mare
Hinkley PtWatchet
MineheadLynmouth
Ilfracombe
Port Isaac
Wes
tern
lim
it 1 W
este
rn li
mit
2
0 50 km
# >= 6 . - + *
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25
6
CL:
164
nod
es, 2
51 e
lem
ents
810
9
74
3
1
FL: 5
78 n
odes
, 100
4 el
emen
ts2
5
68
10
9
74
3
1
FR: 3
82 n
odes
, 664
ele
men
ts
5
68
10
9
74
3
CR
: 111
nod
es, 1
61 e
lem
ents
5
68
10
9
74
3
Stat
ion
Loca
tion
1 2 3 4 5 6 7 8 9 10
Har
tland
Poi
ntTe
nby
Ilfra
com
bePo
rtloc
kSw
anse
aPo
rthca
wl
Wat
chet
Barry
Wes
ton-
Supe
r-Mar
eAv
onm
outh
#>>
2
.
-
+ *
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&& 7# D = .9 8
3 - %
6 %& & K1 2 > # & <
Time = 0 Time = 3 hours
Time = 6 hours Time = 12 hours
1 m/s
# >5 G* #26 $
72 $ ! 3 & , % 7,: 8 >% 7 'B.8
: "& ,
# & .). .)B 7'Z8 ;'9 ;B9 7;Z8!L /B( /'' 7 BZ8 ;'4 ;.4 7;Z8$ ;*. ;(/ 7;Z8 #& ;') ;') 7'Z8% /'; /.B 7.Z8& ;B* .** 7)Z8 ;9* ;). 7'Z8
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& = # % N % %> % % & # % %& - 7 %& 1 + 8
P1P2
P3
Finite element mesh including river
(a)A
BE
E
A
B
B
G
(b)
(c)
# >? -
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& & ) % 7,:8 # % , 4'B # %>% 3 " % % & #
& & 5 >% % & & % 1 # & % %% & # , 4'B +
L %
15
10
5
0
–5
–100 5 10 15 20 25 30
t (h)
Elev
atio
n (m
)
MWL
8
6
4
2
0
–2
12 14 16 18 20 22 24 26t (h)
Elev
atio
n (m
)
MWL
ComputedMeasured
10
9
8
7
6
512 14 16 18 20 22 24 26
t (h)
Elev
atio
n (m
)
ComputedMeasured
Point B Point EComputed and measured elevations
ABE
Water elevations for points A, B, E
Severn Bore
(d)
# >?
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Time = 0 Time = 5 min
Time = 10 min Time = 20 min
Time = 30 min Time = 40 min
# >.@ - " " 7 " # "$
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3 5
" > & & % D & =%% % L % %& D & % # + >
, 4'' % =% & &1 % > );B9B "& % % & % % & # C< &1 % # + L &
1 %& & & C%< + & 9B
6 % > 7 8 %& %> D 1 % >% %& D # D %& > & % D , %> D & & C%< -% C < %%% # %+ %%= & & & D % # '9 + )(4( # % D D, & .9 & A > %& D 7 & , 4'.8 % & 7%% & , 4'.8 & D & 7 , 4'.8 # =% % % , 4'.% C<> C < " & C< &
# >.. 7 * 4! /
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>< '# '
6 % %& % & > # & > % $ ! 3 & & =% & ' 1 , '. 1 ! & L % %& & , 4'; % L & + 2 & % K & > & &&
& % % %
# >.0 - 4! ! * ! 94! 2 8 ' 8
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>= ";' '-'
> K K % % #% % % % # > % K %=9' & % % K %& ! ( & % 6 % K & 3 %
% K 3
B ' . 4.)
% + % K %%% L N 6 K>% !$ & & L
% 2 7 L % K 8 % % # K>% %> K % % # %% !$ % K >
& & % # % # % % & % ; B ; '# %% 3H1 % +
= 7 8A
% ;! % .
./
! 4.4
P P' ∆η
n
A
A
P
P
Boundary at time tn
Boundary at time tn + ∆tn
# >. @ *
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A
% &#&
&#
&
/
&#
&
!
&#
&
&#&
&
6 % % % %%
Time = 9 hours Time = 18 hours
Time = 36 hours Time = 54 hours
# >.8 A 0 " 4
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# %%= 99 1 L /B ' # % 7 > % 8 ;( # B' L =% # % ; B9 # % L & % B' ;( = L ;.Z ;( %% 1 %% /B % # % %%> & % K & % >% %& & )B % K +. L , 4'/ & =% %
& L %
!'
' $ 6&& " .A , # ,* : '(4(
. H@ 5# - ?1 '(*B
; "! 0 12 5 : H 78 <%
! '(4*/ @ 6 + D > '(*)
9 @ "! 0 12 %&A =%
2 2 2 2 2 33 9/434/ '(*)) "! 0 12 @! 6 + 31 K , % %%
"2 2 2 2 2 27?2: )4;3*( '(4(4 " + > > > D
" 7 @# " 8 %% .)'3*4 >
6 '(*B* 6 + K
" < .4;34* $1 5 '(**
( @@ ! !6 $&& ,! ( , ,* >$> : $ '(4)
'B @@ "<$ & 6 % 2 2<2 3 '/3.) '(4.
'' !% ! 52 ! L % 2 2 <2 26 ;)93*. '(*/
'. HH , 6 >% >
2 > K 2 "2 2 82 /9/3*; '(*;'; 5 H 5 : , > > K
& 2 1 "2 ,! 7 + 7 !6
$&& 8 %% ..;3./. '(4*
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'/ # @# >% + > %% > K 2 52 +2 :9 /B.(3/B '(*'
'9 H@ ,= , > > %& # 2 2 2 3;;4'3*4 '(49
') ! # @ # > %% + > %%
" , 5 '.93/* '(49'4 @ ! 6 % + %& 2 2 2
2 22 '93;' '(4;
'* % @ @ % > % K H1 %%= & 2 7' +2 2,, 4;*3/) '(4.
'( , > > K
2 2 5 ;;43/* '(4(.B @ ! # + D
F . % 9 %% ;;B3* "OH65 & '4 >
" H 2 '(4(.' @ ! ! , + >
92 2 +2 2 #2 2,3 94'3(. '(4(
.. 5# 5 H1 + > %% K & ,! ,*! 7 # 8 #1 #1 '(*.
.; 6 3H1 K %% + > >K % 2 "2 2 83 ;';3;( '(*;
./ 5 F H#6, K> 6 ,"5#56 F
% > K : " 52 23 '9'34; '(*)
.9 6 6 %% + >
D ,! ,*! 7 # 8 %% *;93/. #1 #1 '(*.
.) 6 # %% + >
3 2 2 2 2 , 6 '>.. '(*/.4 5# "! 0 12 % + >
K 2 2 2 2 , 2 ('34 '(*'.* HH , 6 > %
> K 2 "2 2 89 .*43;.; '(*/.( ,GA > > + > ,"5#56 %
> K " 52 25 ..93*9 '(*4
;B $ !6 , @! "! 0 12 6 6 % 2 & 8: " "2 & H "& '(4*
;' #1 # ? # % =% + %% 2 2 2 2 2 23 ;;'39' '(4*
;. @ : @ "! 0 12 # H1
%& 3 "2 2 2 2 2 92;9(3)( '(*4
;; 5 : H 6 K + % " , 7 .B43.* '(4(
;/ " & @ 6 @ % %& & D 2 2 2 2 2 37 ;3.B'(*(
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;9 H :& $ := 5 R K >F % R <R R + G , 5% O/' '(*.
;) # # 1 , K % & 2 2 2 2 , 25 (;(39; '(('
;4 $ H ! 6 % & %%
K 2 2 2 2 , 35 .(3/) '(();* $ # #2 #>% =% + %>
D % % 2 2 2 2 ,
32 **93(BB '((9;( !%% ! 2 , , %%=
K ! " + D2 / '((9
/B !%% ! 2 , , %%= K " + D2 / '(()
/' "! 0 12 @ @ 6 >% =% K 2 2 #2 "2 2 ;'3/( '((;
/. "! 0 12 "2 6 %> & +
K 2 2 2 2 , 3, 'B)'3*B '((9/; "! 0 12 "2 # & %
> D 1 2 # .
'((*// "2 "! 0 12 # & %% & D
, ,2 = 2 " '((9/9 5 :Q "! 0 12 # > %& K>
& + 2 2 2 2 , 6 'B/;3); '(*//) 12 "! 0 12 ! 6
=% + %%= K 2
2 "2 , ,* $ L F . %% '34 '(*B/4 #& 1 % +
D 2 2 2 2 , 3 *(3''. '(*.
/* 5 : H H 6 % D 2#" 2 .2 -2 2,67'B8 '/B(3.* '(4*
/( 5 % + H=>4 '(*'9B 6 $ : & 2
2 2 2 , 3 .);34) '(*.9' ! 0 :1 # % +
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