Abstract
Modelling and simulation of macrosegregation has experienced explosive growth since the pioneering studies of Flemings and co-workers in the mid-1960s. This paper presents a review of recent macrosegregation models, with particular emphasis on their application to selected industrially relevant casting processes. The successes and shortfalls of the models in predicting measured macrosegregation patterns are noted. Although it is still early in their development, advanced macrosegregation models that include a detailed consideration of the solidification microstructure are also reviewed. Recent studies involving direct numerical simulation of macrosegregation related phenomena on a microscopic scale are highlighted. Important issues deserving future research attention are identified.
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References
1.
BECKERMANN
C.
WANG
C. Y.
: in 'Annual review of heat transfer VI', Vol. 6 , (ed.
Tien
C. L.
), 115 –198 ; 1995 , New York , Begell House .
2.
PRESCOTT
R J.
INCROPERA
F. P.
: in 'Advances in heat transfer', (ed.
Poulikakos
D.
231 –338 ; 1996 , San Diego , CA, Academic Press .
3.
COMBEAU
H.
APPOLAIRE
B.
LESOULT
G.
: in 'Modeling of casting, welding and advanced solidification processes VIII', (ed.
Thomas
B. G.
Beckermann
C.
), 245 –256 ; 1998 , Warrendale , PA, TMS .
4.
FLEMINGS
M. C.
: in 'Modeling of casting, welding and advanced solidification processes VIII', (ed.
Thomas
B. G.
Beckermann
C.
), 1 –13 ; 1998 , Warrendale , PA, TMS .
5.
FLEMINGS
M. C.
NEREO:
G. E.
Trans. AIME, 1967 , 239, 1449 –1461 .
6.
FLEMINGS
M. C.
MEHRABIAN
R.
NEREO:
G. E.
Trans. AIME, 1968 , 242, 41 –49 .
7.
FLEMINGS
M. C.
NEREO:
G. E.
Trans. AIME, 1968 , 242, 50 –55 .
8.
MEHRABIAN
R.
KEANE
M.
FLEMINGS:
AND M. C.
Metall. Trans., 1970 , 1, 1209 –1220 .
9.
PIWONKA
T. S.
FLEMINGS
M. C.
: Trans. AIME , 1966 , 236, 1157 –1165 .
10.
APELIAN
D.
FLEMINGS
M. C.
MEHRABIAN:
R.
Metall. Trans., 1974 , 5, 2533 –2537 .
11.
MEHRABIAN
R.
KEANE
M. A.
FLEMINGS:
AND M. C.
Metall. Trans., 1970 , 1, 3238 –3241 .
12.
KOU
S.
POIRIER
D. R.
FLEMINGS:
AND M. C.
Metall. Trans. B, 1978 , 9B, 711 –719 .
13.
MEHRABIAN
R.
FLEMINGS:
M. C.
Metall. Trans ., 1970 , 1, 455 –464 .
14.
FUJII
T.
POIRIER
D. R.
FLEMINGS:
AND M. C.
Metall. Trans. B, 1979 , 10B, 331 –339 .
15.
RIDDER
S. D.
KOU
S.
MEHRABIAN:
R.
Metall. Trans. B, 1981 , 12B, 435 –447 .
16.
PRANTIL
V. C.
DAWSON
P. R.
: in HTD-Vol. 29, 47 –54 ; 1983 , New York , ASME .
17.
HILLS
R. N.
LOPER
D. E.
ROBERTS:
P. H.
Q. J. Mech. Appl. Math., 1983 , 36, 505 –539 .
18.
DREW:
D. A.
Ann. Rev. Fluid Mech., 1983 , 15, 261 –291 .
19.
BENNON
W. D.
INCROPERA:
F. P.
Int. J. Heat Mass Transf, 1987 , 30, 2161 –2170 .
20.
BECKERMANN
C.
VISKANTA:
R.
PhysicoChem. Hydrodyn., 1988 , 10, 195 –213 .
21.
VOLLER
V. R.
BRENT
A. D.
PRAKASH:
C.
Int. J. Heat Mass Transf, 1989 , 32, 1718 –1731 .
22.
GANESAN
S.
POIRIER:
D. R.
Metall. Trans. B, 1990 , 21B, 173 –181 .
23.
POIRIER
D. R.
NANDAPURKAR
P. J.
GANESAN:
S.
Metall. Trans. B, 1991 , 22B, 889 –900 .
24.
BENNON
W. D.
INCROPERA:
F. P.
Metall. Trans. B, 1987 , 18B, 611 –616 .
25.
FELICELLI
S. D.
HEINRICH
J. C.
POIRIER:
D. R.
Metall. Trans. B, 1991 , 22B, 847 –859 .
26.
BECKERMANN
C.
VISKANTA:
R.
AppL Mech. Rev., 1993 , 46, 1 –27 .
27.
WORSTER:
M. G.
Ann. Rev. Fluid Mech., 1997 , 29, 91 –122 .
28.
VOLLER
V. R.
: in 'Advances in numerical heat transfer', Vol. 1 , (ed.
Minkowycz
W. J.
Sparrow
E. M.
), 341 –375 ; 1996 , Washington , DC, Taylor & Francis .
29.
VOLLER
V. R.
SWAMINATHAN
C. R.
THOMAS:
B. G.
Int. J. Numer. Methods Eng., 1990 , 30, 875 –898 .
30.
PRAKASH
C.
VOLLER:
V. R.
Numer. Heat Transf. B, 1989 , 15, 171 –189 .
31.
FLEMINGS
M. C.
: 'Solidification processing'; 1974 , New York , McGraw-Hill .
32.
GU
J. P.
BECKERIVIANN:
C.
Metall. Mater. Trans. A, 1999 , 30A, 1357 –1366 .
33.
VANNIER
I.
COMBEAU
H.
LESOULT:
G.
Mater. Sci. Eng., 1993 , A173 , 317 –321 .
34.
SCHNEIDER
M. C.
BECKERMANN:
C.
Metall. Mater. Trans. A, 1995 , 26A, 2373 –2388 .
35.
SUNDARRAJ
S.
MILLER
G. A.
Proc. Merton C. Flemings Symp. on 'Solidification and materials processing', (ed.
Abbaschian
R.
et al.), 311 –316 ; 2001 , Warrendale , PA, TMS .
36.
OLSSON
A.
WEST
R.
FREDRIKSSON:
H.
Scand. J. Metall., 1986 , 15, 104 –112 .
37.
FLEMINGS:
M. C.
ISIJ Int., 2000 , 40, 833 –841 .
38.
KAJITANI
T.
DREZET
J.-M.
RAPPAZ:
M.
Metall. Mater. Trans. A, 2001 , 32A, 1479 –1491 .
39.
LESOULT
G.
SELLA:
S.
Solid State Phenom., 1988 , 3 –4 , 167 –178 .
40.
FREDRIKSSON:
H.
Can. Metall. Q., 1991 , 30, 235 –244 .
41.
ABOUTALEBI
M. R.
HASAN
M.
GUTHRIE:
R. I. L.
Metall. Mater. Trans. B, 1995 , 26B, 731 –744 .
42.
YANG
H. L.
ZHAO
L. G.
ZHANG
X. Z.
DENG
K. W.
LI
W. C.
GAN:
Y.
Metall. Mater. Trans. B, 1998 , 29B, 1345 –1356 .
43.
LEE
S. Y.
CHUNG
S. I.
HONG
C. P.
: in 'Modeling of casting, welding and advanced solidification processes IX', (ed.
Sahm
P. R.
et al.), 648 –655 ; 2000 , Aachen , Shaker Verlag .
44.
OLDENBURG
C. M.
SPERA:
F. J.
Numer. Heat Transf. B, 1992 , 21, 217 –229 .
45.
CHOUDHARY
S. K.
GOSH:
A.
ISIJ Int., 1994 , 34, 338 –345 .
46.
RAIHLE
C. M.
FREDRIKSSON:
H.
Metall. Mater. Trans. B, 1994 , 25B, 123 –133 .
47.
EL-BEALY
M.
FREDRIKSSON:
H.
Scand. J. Metall., 1994 , 23, 140 –150 .
48.
EL-BEAIY:
M.
Scand. J. Metall., 1995 , 24, 106 –120 .
49.
JANSSEN
R. J. A.
BART
G. C. J.
CORNELISSEN
M. C. M.
RABENBERG:
J. M.
AppL Sci. Res., 1994 , 52, 21 –35 .
50.
MIYAZAWA
K.
SCHWERDTFEGER:
K.
Arch. Eisenhattenwes., 1981 , 52, 415 –422 .
51.
OHNAKA
I.
SHIMAZU
T.
Proc. 6th Int. Iron and Steel Congress, Nagoya, Japan, 1990 , 681 –688 ; 1990 , Tokyo , Iron and Steel Institute of Japan .
52.
FINN
T. L.
CHU
M. G.
BENNON
W. D.
: in 'Micro/macro scale phenomena in solidification', (ed.
Beckermann
C.
et al.), 17 –26 ; 1992 , New York , ASME .
53.
REDDY
A. V.
BECKERMANN:
C.
Metall. Mater. Trans. B, 1997 , 28B, 479 –489 .
54.
MO:
A.
Int. J. Heat Mass Transf, 1993 , 36, 4335 –4340 .
55.
HAUG
E.
MO
A.
THEVIK:
H. J.
Int. J. Heat Mass Transf., 1995 , 38, 1553 –1563 .
56.
THEVIK
H. J.
MO:
A.
Int. J. Heat Mass Transf., 1997 , 40, 2055 –2065 .
57.
THEVIK
H. J.
MO:
A.
Metall. Mater. Trans. B, 1997 , 28B, 665 –669 .
58.
THEVIK
H. J.
MO
A.
RUSTEN:
T.
Metall. Mater. Trans. B, 1999 , 30B, 135 –142 .
59.
CHU
M. G.
JACOBY
J. E.
: in 'Light metals 1990', (ed.
Bickert
C. M.
), 925 –930 ; 1990 , Warrendale , PA, TMS .
60.
REDDY
A. V.
BECKERMANN
C.
: in 'Materials processing in the computer age 11', (ed.
Voller
V. R.
et al.), 89 –102 ; 1995 , Warrendale , PA, TMS .
61.
VREEMAN
C. J.
KRANE
M. J. M.
INCROPERA:
F. P.
Int. J. Heat Mass Transf, 2000 , 43, 677 –686 .
62.
VREEMAN
C. J.
INCROPERA:
F. P.
Int. J. Heat Mass Transf., 2000 , 43, 687 –704 .
63.
MO
A.
THEVIK:
H. J.
Metall. Mater. Trans. A, 1998 , 29A, 2189 –2194 .
64.
JOLY
A.
GRUN
G. U.
DALOZ
D.
COMBEAU
H.
LESOULT:
G.
Mater. Sci. Forum, 2000 , 329, 111 –119 .
65.
SCHNEIDER
M. C.
GU
J. P.
BECKERMANN
C.
BOETTINGER
W. J.
KATTNER:
U. R.
Metall. Mater. Trans. A, 1997 , 28A, 1517 –1531 .
66.
GU
J. P.
BECKERMANN
C.
GIAMEI:
A. F.
Metall. Mater. Trans. A, 1997 , 28A, 1533 –1542 .
67.
FELICELLI
S. D.
POIRIER
D. R.
HEINRICH:
J. C.
J. Cryst. Growth, 1997 , 177, 145 –161 .
68.
FELICELLI
S. D.
POIRIER
D. R.
GIAMEI
A. F.
HEINRICH:
J. C.
JOM, 1997 , 49, (3) , 21 –25 .
69.
FELICELLI
S. D.
HEINRICH
J. C.
POIRIER:
D. R.
Int. J. Numer. Methods Fluids, 1998 , 27, 207 –227 .
70.
NEILSON
D. G.
INCROPERA:
F. P.
Numer. Heat Transf. A, 1993 , 23, 1 –20 .
71.
FELICELLI
S. D.
HEINRICH
J. C.
POIRIER:
D. R.
J. Clyst. Growth, 1998 , 191, 879 –888 .
72.
FELICELLI
S. D.
POIRIER
D. R.
HEINRICH:
J. C.
Metall. Mater. Trans. B, 1998 , 29B, 847 –855 .
73.
BECKERMANN
C.
GU
J. P.
BOETTINGER:
W. J.
Metall. Mater. Trans. A, 2000 , 31A, 2545 –2557 .
74.
POLLOCK
T. M.
MURPHY:
AK H.
Metall. Mater. Trans. A, 1996 , 27A, 1081 –1094 .
75.
AUBURTIN
P.
WANG
T.
COCKCROFT
S. L.
MITCHELL:
A.
Metall. Mater. Trans. A, 2000 , 31A, 801 –811 .
76.
MORTENSEN
A.
MICHAUD:
V. J.
Metall. Trans. A, 1990 , 21A, 2059 –2072 .
77.
MICHAUD
V. J.
MORTENSEN:
A.
Metall. Trans. A, 1992 , 23A, 2263 –2280 .
78.
JARRY
P.
MICHAUD
V. J.
MORTENSEN
A.
DUBUS
A.
TIRARD-COLLET:
R.
Metall. Trans. A, 1992 , 23A, 2281 –2289 .
79.
MORTENSEN
A.
JIM:
I.
Int. Mater. Rev., 1992 , 37, 101 –128 .
80.
FELLER
R. J.
BECKERMANN:
C.
Metall. Mater. Trans. B, 1997 , 28B, 1165 –1183 .
81.
HAN
Q.
HUNT:
J. D.
ISIJ Int., 1995 , 35, 693 –699 .
82.
GIULIANI
J.
VAFAI:
K.
J. Fluid. Eng., 1999 , 121, 155 –162 .
83.
SEKHAR
J. A.
TRIVEDI
R.
: m 'Solidification of metal matrix composites', (ed.
Rohatgi
P.
39 –50 ; 1990 , Warrendale , PA, TMS .
84.
MOON
H. K.
CORNIE
J. A.
FLEMINGS:
AND M. C.
Mater. Sci. Eng., 1991 , A144 , 253 –265 .
85.
SCHNEIDER
M. C.
BECKERMANN
C.
LIPINSKI
D. M.
SCHAEFER
W.
: in 'Modeling of casting, welding and advanced solidification processes VIII', (ed.
Thomas
B. G.
Beckermann
C.
), 257 –264 ; 1998 , Warrendale , PA, TMS .
86.
POIRIER:
D. R.
Metall. Trans. B, 1987 , 18B, 245 –255 .
87.
GANESAN
S.
CHAN
C. L.
POIRIER:
D. R.
Mater. Sci. Eng., 1992 , A151 , 97 –105 .
88.
BHAT
M. S.
POIRIER
D. R.
HEINRICH
J. C.
NAGELHOUT:
D.
Scr. Metall. Mater., 1994 , 31, 339 –344 .
89.
BHAT
M. S.
POIRIER
D. R.
HEINRICH:
J. C.
Metall. Mater. Trans. B, 1995 , 26B, 1091 –1092 .
90.
BHAT
M. S.
POIRIER
D. R.
HEINRICH:
J. C.
Metall. Mater. Trans. B, 1995 , 26B, 1049 –1056 .
91.
GOYEAU
B.
BENIHADDADENE
T.
GOBIN
D.
QUINTARD:
M.
Metall. Mater. Trans. B, 1999 , 30B, 613 –622 .
92.
WANG
C. Y.
AHUJA
S.
BECKERMANN
C.
C
H. C.
Metall. Trans. B, 1995 , 26B, 111 –119 .
93.
NIELSEN
O.
ARNBERG
L.
MO
A.
THEVIK:
H.
Metall. Mater. Trans. A, 1999 , 30A, 2455 –2462 .
94.
NIELSEN
O.
ARNBERG:
L.
Metall. Mater. Trans. A, 2000 , 31A, 3149 –3153 .
95.
POIRIER
D. R.
GANESAN:
S.
Mater. Sci. Eng., 1992 , A157 , 113 –123 .
96.
OCANSEY
P.
POIRIER:
D. R.
Mater. Sci. Eng., 1993 , A171 , 231 –240 .
97.
DUNCAN
A. J.
IAN
Q.
VISWANATHAN:
S.
Metall. Mater. Trans. A, 1999 , 30A, 745 –750 .
98.
III
H. C. DE GROH
WEIDMAN
P. D.
ZAKHEM
D.
AHUJA
S.
BECKERMANN:
C.
Metall. Trans. B, 1993 , 24B, 749 –753 .
99.
ARNBERG
L.
CHAI
G.
BACKERUD:
L.
Mater. Sci. Eng., 1993 , A173 , 101 –103 .
100.
RAPPAZ
M.
VOLLER:
V. R.
Metall. Mater. Trans. A, 1990 , 21A, 749 –753 .
101.
SUNDAKRAJ
S.
VOLLER:
V. R.
Int. Commun. Heat Mass Transf, 1993 , 21, 189 –198 .
102.
SUNDAKRAJ
S.
VOLLER:
V. R.
Int. J. Numer. Methods Heat Fluid Flow, 1997 , 7, 181 –199 .
103.
MO:
A.
Metall. Trans. B, 1994 , 25B, 597 –605 .
104.
COMBEAU
H.
DREZET
J.-M.
MO
A.
RAPPAZ:
M.
Metall. Mater. Trans. A, 1996 , 27A, 2305 –2313 .
105.
SCHNEIDER
M. C.
BECKERMANN:
C.
Int. J. Heat Mass Transf., 1995 , 38, 3455 –3473 .
106.
COMBEAU
H.
MO:
A.
Metall. Mater. Trans. A, 1997 , 28A, 2705 –2714 .
107.
BECKERMANN
C.
SCHNEIDER
M. C.
: ISIJ Int ., 1995 , 35, 1300 –1307 .
108.
KRANE
M. J. M.
INCROPERA:
F. P.
Int. J. Heat Mass Transf, 1997 , 40, 3827 –3835 .
109.
KRANE
M. J. M.
INCROPERA:
F. P.
Int. J. Heat Mass Transf, 1997 , 40, 3837 –3847 .
110.
KRANE
M. J. M.
INCROPERA
F. P.
GASKELL:
D. R.
Metall. Mater. Trans. A, 1998 , 29A, 843 –853 .
111.
EL-BEALY
M.
FREDRIKSSON:
H.
Metall. Mater. Trans. B, 1996 , 27B, 999 –1014 .
112.
M&HAMDI,
M.
BOBADILLA
M.
COMBEAU
H.
LESOULT
G.
RN 'Modeling of casting, welding and advanced solidification processes VIII', (ed.
Thomas
B. G.
Beckermann
C.
), 375 –382 ; 1998 , Warrendale , PA, TMS ..
113.
M&apos
M.
;HAMDI, H. COMBEAU, and G . LESOULT. Int. J. Numer. Methods Heat Fluid Flow , 1999 ,9 , 296 –317 .
114.
WANG
C. Y.
BECKERMANN:
C.
Int. J. M ultiphase Flow, 1993 , 19, 397 –407 .
115.
WANG
C. Y.
BECKERMANN:
C.
Metall. Trans. A, 1993 , 24A, 2787 –2802 .
116.
WANG
C. Y.
BECKERMANN:
C.
Mater. Sci. Eng., 1993 , A171 , 199 –211 .
117.
WANG
C. Y.
BECKERMANN:
C.
Metall. Trans. A, 1994 , 25A, 1081 –1093 .
118.
DUSTIN
I.
KURZ:
W.
Z. Metallkd., 1986 , 77, 265 –273 .
119.
RAPPAZ
M.
THEVOZ
PH.
: Acta Metall ., 1987 , 35, 1487 –1497 .
120.
TONG
X.
BECKERMANN:
C.
J. Ctyst. Growth, 1998 , 187, 289 –302 .
121.
RAMANI
A.
BECKERMANN:
C.
Scr. Mater., 1997 , 36, 633 –638 .
122.
APPOLAIRE
B.
ALBERT
V.
COMBEAU
H.
LESOULT:
G.
Acta Mater., 1998 , 46, 5851 –5862 .
123.
APPOLAIRE
B.
ALBERT
V.
COMBEAU
H.
LESOULT:
G.
ISIJ Int., 1999 , 39, 263 –270 .
124.
GERARDIN
S.
COMBEAU
H.
LESOULT:
G.
J. Phys. IV , 2001 , 11, 143 –150 .
125.
RAPPAZ:
M.
Int. Mater. Rev., 1989 , 34, 93 –123 .
126.
GANDIN
CH.-A.
JALANTI
T.
RAPPAZ
M.
: in 'Modeling of casting, welding and advanced solidification processes VIII', (ed.
Thomas
B. G.
Beckermann
C.
), 363 –374 ; 1998 , Warrendale , PA, TMS .
127.
RAPPAZ
M.
DESBIOLLES
J. L.
GANDIN:
C. A.
Mater. Sci. Forum, 2000 , 329, 389 –396 .
128.
NI
J.
BECKERMANN:
C.
Metall. Trans. B, 1991 , 22B, 349 –361 .
129.
WANG
C. Y.
BECKERMANN:
C.
Metall. Mater. Trans. A, 1996 , 27A, 2754 –2764 .
130.
WANG
C. Y.
BECKERMANN:
C.
Metall. Mater. Trans. A, 1996 , 27A, 2765 –2783 .
131.
BECKERMANN
C.
WANG:
C. Y.
Metall. Mater. Trans. A, 1996 , 27A, 2784 –2795 .
132.
NI
J.
INCROPERA:
F. P.
Int. J. Heat Mass Transf., 1995 , 38, 1271 –1284 .
133.
NI
J.
INCROPERA:
F. P.
Int. J. Heat Mass Transf., 1995 , 38, 1285 –1296 .
134.
PRESCOTT
P. J.
INCROPERA
F. P.
BENNON:
W. D.
Int. J. Heat Mass Transf, 1991 , 34, 2351 –2359 .
135.
FLOOD
s. C.
HUNT
J. D.
: AppL Sci. Res ., 1987 , 44, 27 –42 .
136.
BARRERE
J.
GIPOULOUX
O.
WHITAKER:
S.
Transp. Porous Media, 1992 , 7, 209 –222 .
137.
QUINTARD
M.
WHITAKER:
S.
Transp. Porous Media, 1994 , 15, 31 –49 .
138.
BOUSQUET-MELOU
P.
GOYEAU
B.
QUINTARD
M.
FICHOT
F.
GOBIN
D.
: in 'Modeling of casting, welding and advanced solidification processes IX', (ed.
Sahm
P. R.
et al.), 616 –623 ; 2000 , Aachen , Shaker Verlag .
139.
PARADIES
C. J.
KIM
G. T.
GLICKSMAN
M. E.
SMITH
R. N.
: in 'Modeling of casting, welding and advanced solidification processes VI', (ed.
Piwonka
T. S.
et al), 309 –316 ; 1993 , Warrendale , PA, TMS .
140.
WHEELER
A. A.
BOETTINGER
W. J.
MCFADDEN:
G. B.
Phys. Rev. A, 1992 , 45, 7424 .
141.
KOBAYASHI:
R.
Physica D , 1993 , 63 , 410.
142.
KARMA
A.
RAPPEL:
W. J.
Phys. Rev. E, 1998 , 57, 4323 .
143.
BECKERMANN
C.
DIEPERS
H. J.
STEINBACH
I.
KARMA
A.
TONG:
X.
J. Comput. Phys., 1999 , 154, 468 –496 .
144.
TONG
X.
BECKERMANN
C.
KARMA:
A.
Phys. Rev. E, 2000 , 61 , R49—R52 .
145.
TONG
X.
BECKERMANN
C.
KARMA
A.
LI:
Q.
Phys. Rev. E, 2001 , 63 , 061601.
146.
JEONG
J. H.
GOLDENFELD
N.
DANTZIG:
J. A.
Phys. Rev. E, 2001 , 64 , 041602.
147.
DIEPERS
H. J.
BECKERMANN
C.
STEINBACH:
I.
Ada Mater., 1999 , 47, 3663 –3678 .
148.
MCCARTHY:
J. F.
Acta Metall. Mater., 1994 , 42, 1573 –1581 .
149.
PRESCOTT
P. J.
INCROPERA:
F. P.
J. Heat Transf, 1995 , 117, 716 –724 .
150.
HUANG
H.-W.
HEINRICH
J. C.
POIRIER:
D. R.
Numer. Heat Transf. A, 1996 , 29, 639 –644 .
151.
SUNG
P. K.
POIRIER
D. R.
FELICELLE
S. D.
Int. J. Numer. Methods Fluids, 2001 , 35, 357 –370 .
152.
FRUEH
C.
POIRIER
D. R.
FELICELLE
S. D.
Metall. Mater. Trans. A, 2000 , 31A, 3129 –3135 .
153.
SUNG
P. K.
POIRIER
D. R.
FELICELLE
S. D.
Metall. Mater. Trans. A, 2001 , 32A, 202 –207 .
154.
AHMAD
N.
COMBEAU
H.
DESBIOLLES
J. L.
JALANTI
T.
LESOULT
G.
RAPPAZ
J.
RAPPAZ
M.
STOMP:
C.
Metall. Mat. Trans. A, 1998 , 29A, 617 –630 .
155.
KAEMPFER
Th. U.
RAPPAZ
M.
in 'Modeling of casting, welding and advanced solidification processes IX', (ed.
Sahm
P. R.
et al), 640 –647 ; 2000 , Aachen , Shaker Verlag .
156.
VOLLER
V. R.
Proc. 2001 TMS Fall Extraction and Process Metallurgy Meeting: 'Computational modeling of materials, minerals, and metals', (ed.
Cross
M.
41 –61 ; 2001 , Warrendale, PA, TMS (in press) .
157.
VOLLER
V. R.
PORTE-AGEL:
F.
J. Comput. Phys., 2002 , 179, 698 –703 .