Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor

Authors:
Hung V. Ly and Hien T. Tran

Journal:
Quart. Appl. Math. **60** (2002), 631-656

MSC:
Primary 76M25; Secondary 65K10, 76M35, 76N25

DOI:
https://doi.org/10.1090/qam/1939004

MathSciNet review:
MR1939004

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Proper orthogonal decomposition (which is also known as the Karhunen-Loève decomposition) is a reduction method that is used to obtain low-dimensional dynamic models of distributed parameter systems. Roughly speaking, proper orthogonal decomposition (POD) is an optimal technique of finding a basis that spans an ensemble of data, collected from an experiment or a numerical simulation of a dynamical system, in the sense that when these basis functions are used in a Galerkin procedure, they will yield a finite-dimensional system with the smallest possible degrees of freedom. Thus, the technique is well suited to treat optimal control and parameter estimation of distributed parameter systems. In this paper, the method is applied to analyze the complex flow phenomenon in a horizontal chemical vapor deposition (CVD) reactor. In particular, we show that POD can be used to efficiently approximate solutions to the compressible viscous flows coupled with the energy and the species equations. In addition, we also examine the feasibility and efficiency of the POD method in the optimal control of the source vapors to obtain the most uniform deposition profile at the maximum growth rate. Finally, issues concerning the implementation of the method and numerical calculations are discussed.

**[1]**Nadine Aubry, Philip Holmes, John L. Lumley, and Emily Stone,*The dynamics of coherent structures in the wall region of a turbulent boundary layer*, J. Fluid Mech.**192**(1988), 115–173. MR**984943**, https://doi.org/10.1017/S0022112088001818**[2]**Nadine Aubry, Wen Yu Lian, and Edriss S. Titi,*Preserving symmetries in the proper orthogonal decomposition*, SIAM J. Sci. Comput.**14**(1993), no. 2, 483–505. MR**1204243**, https://doi.org/10.1137/0914030**[3]**K. J. Bachmann, N. Sukidi, C. Hopfner, C. Harris, N. Dietz, H. T. Tran, S. Beeler, K. Ito, and H. T. Banks,*Real-time monitoring of steady-state pulsed chemical beam epitaxy by p-polarized reflectance*, J. of Crystal Growth**183**, 323-337 (1998)**[4]**K. S. Ball, L. Sirovich, and L. R. Keefe,*Dynamical eigenfunction decomposition of turbulent channel flow*, International Journal for Numerical Methods in Fluids**12**, 585-604 (1991)**[5]**H. T. Banks, K. Ito, J. S. Scroggs, H. T. Tran, N. Dietz, and K. J. Bachmann,*Modeling and control of advanced chemical vapor deposition processes*, in*Mathematics of Microstructure Evolution*(eds.: L. Q. Chen, et al.), SIAM/TMS Publications, 327-344 (1996)**[6]**G. Berkooz,*Observations on the proper orthogonal decomposition*, in*Studies in Turbulence*(eds.: T. B. Gatski, S. Sarkar, and C. G. Speziale), Springer-Verlag, New York, 1992, pp. 229-247**[7]**Gal Berkooz, Philip Holmes, and John L. Lumley,*The proper orthogonal decomposition in the analysis of turbulent flows*, Annual review of fluid mechanics, Vol. 25, Annual Reviews, Palo Alto, CA, 1993, pp. 539–575. MR**1204279****[8]**Philip J. Holmes, John L. Lumley, Gal Berkooz, Jonathan C. Mattingly, and Ralf W. Wittenberg,*Low-dimensional models of coherent structures in turbulence*, Phys. Rep.**287**(1997), no. 4, 337–384. MR**1471174**, https://doi.org/10.1016/S0370-1573(97)00017-3**[9]**D. H. Chambers, R. J. Adrian, P. Moin, D. S. Stewart, and H. J. Sung,*Karhunen-Loève expansion of Burgers' model of turbulence*, Phys. Fluids**31**, 2573-2582 (1988)**[10]**M. E. Coltrin, R. J. Kee, and J. A. Miller,*A mathematical model of Silicon chemical vapor deposition*, J. Electrochem. Soc.**133**, 1206-1213 (1986)**[11]**D. I. Fotiadis,*Two- and three-dimensional finite element simulations of reacting flows in chemical vapor deposition of compound semiconductors*, Ph.D. thesis, Univ. Minn., Minneapolis, 1990**[12]**E. Fujii, H. Nakamura, K. Haruna, and Y. Koga,*A quantitative calculation of the growth rate of epitaxial silicon from SiCl**in a barrel reactor*, J. Electrochem. Soc.**119**, 1106-1113 (1972)**[13]**S. A. Gokoglu, M. Kuczmarski, P. Tsui, and A. Chait,*Convection and chemistry effects in CVD--A*3 -*D analysis for silicon deposition*, Journal de Physique**50**, N5-17 (1989)**[14]**M. Graham and I. G. Kevrekidis,*Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis*, Computers Chem. Engineering**20**, N5:495-506 (1996)**[15]**R. Hilai and J. Rubinstein,*Recognition of rotated images by invariant Karhunen-Loève expansion*, Journal of the Optical Society of America A--Optics Image Science and Vision**11**, N5:1610-1618 (1994)**[16]**Kazufumi Ito, Hien T. Tran, and Jeffery S. Scroggs,*Mathematical issues in optimal design of a vapor transport reactor*, Flow control (Minneapolis, MN, 1992) IMA Vol. Math. Appl., vol. 68, Springer, New York, 1995, pp. 197–218. MR**1348648**, https://doi.org/10.1007/978-1-4612-2526-3_9**[17]**Kazufumi Ito, Jeffrey S. Scroggs, and Hien T. Tran,*Optimal control of thermally coupled Navier-Stokes equations*, Optimal design and control (Blacksburg, VA, 1994) Progr. Systems Control Theory, vol. 19, Birkhäuser Boston, Boston, MA, 1995, pp. 199–214. MR**1352258**, https://doi.org/10.1109/TCST.2010.2041930**[18]**K. F. Jensen,*Chemical vapor deposition*, in Microelectronics Processing: Chemical Engineering Aspects, D. W. Hess and K. F. Jensen, eds., 1989, pp. 199-263**[19]**K. F. Jensen, E. O. Einset, and D. I. Fotiadis,*Flow phenomena in chemical vapor deposition of thin films*, Annual Rev. Fluid Mech.**23**, 197-232 (1991)**[20]**Ram P. Kanwal,*Linear integral equations*, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 1997. MR**1427946****[21]**Kari Karhunen,*Zur Spektraltheorie stochastischer Prozesse*, Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys.**1946**(1946), no. 34, 7 (German). MR**0023012****[22]**M. Kirby and L. Sirovich,*Application of the Karhunen-Loève procedure for the characterization of human faces*, IEEE Transactions on Pattern Analysis and Machine Intelligence**12**, N1:103-108 (1990)**[23]**M. Kirby, J. P. Boris, and L. Sirovich,*A proper orthogonal decomposition of a simulated supersonic shear layer*, International Journal for Numerical Methods in Fluids**10**, 411-428 (1990)**[24]**M. Loève,*Quelques propriétés des fonctions aléatoires de second ordre*, Comptes Rendus Acad. Sci. Paris**222**, 469-470 (1946)**[25]**J. L. Lumley,*The structure of inhomogeneous turbulent flows*, in Atmospheric turbulence and radio wave propagation, A. M. Yaglom and V. I. Tatarski, eds., Moscow: Nauka, 1967, pp. 166-178**[26]**John L. Lumley,*Stochastic tools in turbulence*, Academic Press, New York-London, 1970. Applied Mathematics and Mechanics, Vol. 12. MR**0451408****[27]**H. V. Ly and H. T. Tran,*Applications of proper orthogonal decomposition in simulations and optimal control of the Rayleigh-Bénard Convection*, 1998, to be submitted**[28]**H. K. Moffat and K. F. Jensen,*Complex flow phenomena in MOCVD reactors*, Journal of Crystal Growth**77**, 108-119 (1986)**[29]**H. K. Moffat and K. F. Jensen,*Three-dimensional flow effects in Silicon CVD in horizontal reactors*, J. Electrochem. Soc.**135**, 459-471 (1988)**[30]**Parviz Moin and Robert D. Moser,*Characteristic-eddy decomposition of turbulence in a channel*, J. Fluid Mech.**200**(1989), 471–509. MR**990171**, https://doi.org/10.1017/S0022112089000741**[31]**A. M. Obuhov,*The statistical description of continuous fields*, Trudy Geofiz. Inst.**1954**(1954), no. 24(151), 3–42. MR**0067404****[32]**S. Ostrach,*Low-gravity fluid flows*, Ann. Rev. Fluid Mech.**14**, 313-345 (1982)**[33]**Klaus Oswatitsch,*Gas Dynamics*, Academic Press Inc., New York, 1956. English version by Gustav Kuerti. MR**0081069****[34]**J. Ouazzani, Kuan-Cheng Chiu, and F. Rosenberger,*On the*2*D modeling of horizontal CVD reactors and its limitations*, Journal of Crystal Growth**91**, 497-508 (1988)**[35]**J. Ouazzani and F. Rosenberger,*Three-dimensional modeling of horizontal chemical vapor deposition*I.*MOCVD at atmospheric pressure*, Journal of Crystal Growth**100**, 545-576 (1990)**[36]**R. Pollard and J. Newman,*Silicon deposition on a rotating disk*, J. Electrochem. Soc.**127**, 744-752 (1980)**[37]**V. S. Pougachev,*General theory of the correlations of random functions*, Izv. Akad. Nauk USSR, Ser. Mat.,**17**, 1401-1402 (1953)**[38]**M. Rajaee, S. K. F. Karlson, and L. Sirovich,*Low-dimensional description of free-shear-flow coherent structures and their dynamical behavior*, Journal of Fluid Mechanics**258**, 1-29 (1994)**[39]**R. S. Reichert, F. F. Hatay, S. Biringer, and A. Husser,*Proper orthogonal decomposition applied to turbulent flows in a square duct*, Phys. Fluids Mechanics**6**, N9:3086-3092 (1994)**[40]**M. A. Saad,*Compressible Fluid Flow*, Prentice-Hall, New Jersey, 1985**[41]**L. Sirovich,*Chaotic dynamics of coherent structures*, Phys. D**37**(1989), no. 1-3, 126–145. Advances in fluid turbulence (Los Alamos, NM, 1988). MR**1024387**, https://doi.org/10.1016/0167-2789(89)90123-1**[42]**L. Sirovich,*Analysis of turbulent flows by means of the empirical eigenfunctions*, Fluid Dynamics Research**8**, 85-100 (1991)**[43]**J. S. Scroggs, H. T. Banks, K. Ito, S. Ravindran, H. T. Tran, K. J. Bachmann, H. Castleberry, and N. Dietz,*High pressure vapor transport of ZnGeP*: II,*three-dimensional simulation of gas dynamics under microgravity conditions*, in Proceedings of the 1995 TMS Annual Meeting, Las Vegas, Nevada, 1995**[44]**T. Theodorsen,*Mechanism of Turbulence*, in Proc. 2nd Midwestern Conference on Fluid Mechanics, Ohio State University, Columbus, OH, 1952**[45]**A. A. Townsend,*The structure of turbulent shear flow*, Cambridge University Press, New York, 1956. MR**0078813****[46]**G. W. Young, S. I. Hariharan, and R. Carnahan,*Flow effects in a vertical CVD reactor*, SIAM J. Appl. Math.**52**(1992), no. 6, 1509–1532. MR**1191348**, https://doi.org/10.1137/0152088

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
76M25,
65K10,
76M35,
76N25

Retrieve articles in all journals with MSC: 76M25, 65K10, 76M35, 76N25

Additional Information

DOI:
https://doi.org/10.1090/qam/1939004

Article copyright:
© Copyright 2002
American Mathematical Society