finite element method - fluid dynamics - zienkwicz and taylor
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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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INLET(Free-stream)
EXIT
SIDE(Free-stream)
SLIP or NO-SLIP
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sity
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0 500 1000 1500 2000 2500Iterations
Single step
(d) (e)M = 1.2
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Den
sity
at L
.E.
0 250 500 750 1000 1250 1500No. of iterations
Multi-stepSingle step
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(a) (b)
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0.250–50.00 –40.00 –30.00 –20.00 –10.00 0
Distance X
Den
sity
T–G scheme (Cs = 0.0)T–G scheme (Cs = 0.5)Present schemes (Cs = 0.0)
(a) (b)
(c) (d)M = 0.5
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s
R0
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u1 = u∞, u2 = 0
u1 = u2 = 0 u1 = u2 = 0
u1 = u2 = 0p = 0
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& K 5 & 5 & 'B BBB , /(
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1.0
0.8
0.6
0.4
0.2
0
x2
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'CBS'
(a) Stokes flow, viscosity 1
1.0
0.8
0.6
0.4
0.2
0
x2
–0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0u1
'Ghia''CBS'
(b) Re = 400
1.0
0.8
0.6
0.4
0.2
0
x2
–0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0u1
'Ghia''CBS'
(c) Re = 1000
1.0
0.8
0.6
0.4
0.2
0
x2
–0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0u1
'Ghia''CBS'
(d) Re = 5000
# 88 6 * ' * " B* # $
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6 !% ; % =% %> 1 % N > & & =% % & & & %& L = = # & 'B =% % L %
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0.5
0
–0.5
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x1
p'CBS'
(a) Stokes flow, viscosity 1
–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
'CBS'
(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
–0.100 0.2 0.4 0.6 0.8 1.0
x1
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 % & =% !% ; ;)
8< 1 -) '4
% + !% '/ '9 F ' & & K # % & = D % %& % + % % # % & & % %& D 5 ''3)( & + % % D
)+ .
L %%
1.5
1.0
0.5
0
–0.5
–1.0
–1.50 0.2 0.4 0.6 0.8 1.0
x1
p
'10x_uniform''40x_unifom'
40x_non-uniform'
# 8> , - 4! * # $ *
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+> 7 8 > + % & 2 # + % =% %& & + & + # & & " ;.;;9'9. & & + %
x
u
u
# 85 6 * C B* 8
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% D %& =% > %%# & >
> # 1 > ".
x
u
u
# 8? 6 * C B*
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& 7 .. 8
..
. .
.
./.'
+
2 % %& .. # 1 & =
h
h
hh
u
u
u
u
tn
u
# 8.@ 2! ! " + * B
# 8.. 2! ! " - ! " B
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% 7 , /'.8
= '
*..
./..
5 %
5 ''.B
..
./.;
=% 7/..8 7/.;8 = 6 F ' % % K & & > %& K & # 1 &
..
. /./
# %% & =% 7/./8 % %& % %
..
. /.9
X >%+ % &
N K # 2 +> > % & % $ %
x1
x2
n1 n2
emax
Exact φ
Linear φ
hn1, n2, nodesh, element size
# 8.0 9 !
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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 %%= &
&# &
.
.
B /.(
& % .
%'
&#
.
%'
&#
&
/;B
% = &
% &#& /;'
& C%<
K C > <% 1 & # + %% % '(*4 & .. & D %&4B4; $ %& & % & 6 1 & % # & = & ;(9;9) 0 12 9B )9 %& D = & % & "& % % D % = & + + = 2 & & # %% 1 . K = , /'; ,' ,. = % %
# K & %
.
.,..
.=
.,.'
/;.
#
# =
.,..
.,.'
#$$$$$$% /;;
% & % % & =
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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
# 8. + " ,
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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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# 8.8 6 * B 8 " "
1 -) '4
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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 : - '
: ' %$ !
%& D % = %& /. 7 8
" 3 9 : - '
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#8.=
0
4
!
*
B
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*&
" 3 9 : - '
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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 &
* &#&
&#
&#
//.
% & % # 1 (B(.
'B4'B*# & & % % >
%& D # %%& D D & & %% % = & & & % & # !% '. F ' & % , /'* &
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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
*
# 8.5 - * * "* "
" 3 9 : - '
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% (; D (/() & % % & K B %
% D % % % % % %
1 %& % # & = %& 1
D & + + D %%
8? %; 3 9 -' '#
; 5% % ! !
D % & % & % #% & % =% %
B !' //;
B B 3 % ! % & + K7;;/8 7&8 % # & & & % &
K 7;;;8 %+ & %%% % , /'( #& =% K 7//;8 1 " , /'(7&8 & % %
% & %& 1 , , /'(78 % $ D & = & & # >% D > & # &
!
///
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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•
# 8.? - *
%; 3 9 -' '#
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# % & % 7 F .8 # K 7//'8
1
#
,
//*
1 1 #
#
' 1'
,
1 1 # //(
& % & % % 7 % %
2 + 8 % + + K & L ' ' % 7 8 # + > D %% %
'(4B(4(( 6%% % &K & %*('BB'.4
%% N 2 & D & & % 'B) % = 7 % + 8 -+& # & & % % & )* % %> =% # % & C>%<'B9'B) %= %# % %
% %% & # ''9 '.* '.( % +
; 5 !
# %& #! )* + D , /.B78 %% & & += % % % , /.B7&8 % % > %& % + % 1 + & % &
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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
# 80@ 2" **
%; 3 9 -' '#
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Moving mesh
Extrusion
Rolling
Forging
Cutting
Sheet forming(deep drawing)
(b) Transient
# 80@
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& &
/9B&
& K 7'(8
/9B
62 !% % = , /.' % => %& 7 % 8(;
' 78 . 7+ 8
% = , Z !78 = , Z !78
#)O' .* (B'B '.B. )4*' .9 ((BB B4; 94(4'#)$'O; ;' B/;B .B;. 494) .) .9*B '4* 4*B';#)$'O; .( B;'B '.9. 4;B* .) ..(B ')) )';(.#)O;! .4 (B.9 *'9 *4). .9 (49B B)4 *99;*
= .9 *BBB BBB [ .9 *BBB BBB [
CL
MESH I
CL
MESH I
v = 0u = 1specified
Slip boundary
Free boundary
# 80. # '$ ! *
%; 3 9 -' '#
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& K 7'.8
./9B
% L % K 7/9B8 & & & %&
7 K /)8
( B /9'
# % %& & &1 & % 1 D + &1 % 1 K % % !% . % + '(4;
'(4*'B.'B; & % 'B(''B , /.. % % > %& 'B; %& & % %
& & 1 N % % '') # % % = N %
; - ! %$ $
# % %&& %& #%=% % & , /.; /./ % %% # & %
+ % % & 1 # + & & & & & =%
/9. % % % &
K 7/9'8 %+ %%
( ( B /9;
% 1 %> % =% % % !% ;
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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
#800
-
*
"
!
"
>/
%; 3 9 -' '#
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& % 1 % & % %& =% , /.;*( %
& 1% 1 & 6% & %& & % 3 & 1% % =% , /./ %&';'';.
% % % %& & 2 %+ & % %& % %&
!% '9 F ' % =% & % ';;';/ , /./ %
U
(a) t = 0
u∆t
U
(b) t = 15∆t
30%
U
(c) t = 30∆t
60%
U
(d) t = 45∆t
90%
# 80 #* $<> ; != 9* ) #$ #$ #$ #$ ! # " $
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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
%
t = 0
t = 1
.1s
t = 7
.7s
#808
#$
"
!
#
"
"*
$
%; 3 9 -' '#
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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
# 808 #$ ! */ 0?D3 2" =<
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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
#80<
3
!
"*
4
=
%; 3 9 -' '#
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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
Y
(b) Mesh for establishing die geometry
# 80= 2 " * 0 " ! 0 " 4 ! 8 0 0 " ! 1 ! 8/) ?> ! ! <8:!B # ( $=/
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; !
/(' L %& % & -+ & # & # & %& % " %& C% >&1< % %=
t = 900 s
t = 2400 s
t = 4280 s
(c) Deformed shapes of various times
# 80=
%; 3 9 -' '#
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% % %& L & % '') '(*/ & D D & %& %& % D && # % %& K %= & &- = 1 ! # ,&'9/'); & "& &- & % % &1 & &
8.@ ' - --' ' '#
=% K = & > % % % # %% + % ?6. ?6; % : : :&')/')9 , %&> % N % & %& & % & %&> & %> & % =% & # &1 % & F . &1 &1 1 1 % K = % % + 1 > % % 78 1 %& %& !$ 1 7 8 % & %'))
%& % % & % 1 %& & % % # % % & & %%% % & N & 0 12 ')) , /.4 % & % & % = + % 7, /.4&8 % % K> 7, /.48 % # + K !$ + 7, /.4 8
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(a)
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(a) (b)
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(a)
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50
(a) Da = 10–6, Ra = 108, ε = 0.8
50
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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
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(a)
(d)
(b)
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% # # , )* %= =% )* &
> & A
(c) The corresponding pressure contours
(b) The corresponding density
(a) Sequence of meshes employed
Analysisdomain
22° Flowvelocity
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(c) (d)
(a) (b)
10
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u1 = 1u2 = 0
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1.80
1.60
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MUSCLCBS: Anisotropic Shock CaptureCBS: Second Derivative Shock Capture
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Analysis domain
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& 1 % % , )'B N % & & > % & !:). % 1 % 6 % = % % 7>&8 , )'' % =% % 1
& 1 6 = + & % =% '4 % %
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#> %& D > %& & % %%=> %& > !% ; = > %& %% + , & % > %& =% & > & & % 6 % & & 1 % %& & 1 % L % % %
% = %% %
& & & % % & .) & % 2 6%% = !
% & % & % K + 6 & + % + + &K + %%= + % +
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% % &K # % & % % 6%% = K )9)( % & & & C <
+ # & %% % L
$ # % %2 % % % % %& &-;);*
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%% % 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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,: -496- $ ! 3:3
6 %& % & & & # % % & &
# =. - 0AB;-0 --E #$ #$ #9" #$ * -- " 6 " F* B 3*$
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% # % '9 "& '((4 1 %& & % & % 7 & 8 %& >= # % D % & % 6 K & % = 49B , %&# % &
% % & >% % &
(a)
(b)
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6
4
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CFDresults
KeyM = 0.71M = 1.08M = 0.96M = 1.05
Computerresults
Pendinetest results
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# % % % %+% & B4' B() 'B9 'B*
,: <$ (!
# > =% & K& > 1') .# , )')
(a) Mesh on analysis surface
(b) Mesh on analysis surface (c) Pressure contours
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=? ' '; -'/
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% =% #+ =% % 4; > , )'4 % & > % % 1 6 % , )'*
> %&4; " > % &
NE = 7870NP = 4130
NE = 7377NP = 3867
NE = 6847NP = 3580
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(a) (b)
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10–15 ‘body’ layers of quadrilateralelements with length l, correspondingadaptive layer thickness d and number of layers decided by user
(a) A two-dimensional sublayer of structured quadrilaterals
‘Body’ layer subdivision in three dimensions joining a tetrahedral mesh
(b) A three-dimensional sublayer of prismatic elements
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(a)
(b)
(c)
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X/L
P/P
inf
0.80
0.60
0.40
0.20
00 0.25 0.50 0.75 1.00 1.25 1.50
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MUSCLCBSCarter
MUSCLCBSCarter
(a)
(b)
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Anal
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& & D % = % 1 % 4( %& %% !% / # %+ # %& & %% !$ >% D # D 6!6BB'. # %& = & *B*' , )./ + # % & & & 1*.
%& & % + & 1 + =% =
++ ++
+
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1.50
1.30
1.10
0.90
0.70
0.50
+ + + + ++
++
++ +
+
+ + +
++
First mesh
Third mesh
Experiment
Surface pressure
X/Xshk
+ + + + + + +
+ ++
++++
++
++
+++ ++
+ First mesh
Second mesh
Experiment
Skin friction
0.30
0.20
0.10
–0.00
–0.10
–0.20
–0.50 –0.10 0.30 0.70 1.10 1.50
×10–2
X/Xshk
CF
+ +
++
(d) Surface pressure and skin friction
# =0
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(a)
Density contours(b)
Wake region, density contours(c)
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Initial mesh, 1804 points, 3487 elements Mach number
(a)
Mach number
Mach number
adaptive mesh, 916 points, 1830 elements
(b)
1162 points, 2258 elements(c)
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(b)
(c) (d)
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% & 1 % 1 % 1 ! & %& & L K & % & 31 K
K %2 & % & % 1 6 K &
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% %%& %& & % & % D(/() 3 % K 6%% =
1.5
1.0
0.5
0
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0X/C
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(c)
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h
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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
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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 , 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
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chet
Barry
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ton-
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r-Mar
eAv
onm
outh
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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
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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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& & 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
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>= ";' '-'
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% K 3
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= 7 8A
% ;! % .
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! 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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K 2 2 2 2 , 35 .(3/) '(();* $ # #2 #>% =% + %>
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e tra
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0.1 0.5 1.0 2.0 3.0 4.0 5.0 6.0 6.0 7.0 8.0Wavelength λ/B
Incidentwave
Infiniteelements Special three-
dimensionalfinite elements Three-dimensional
finite elementsBed
8 m
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Unit: m
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Inputs
Preliminaryroutines
Datacheck
Step 1Intermediatemomentum
Step 2Density/pressure
Step 3Momentumcorrection
Energycoupling
Boundaryconditions
Steadystate
Output
End
Make changes
Failed
Passed
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Yes
No
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