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Low-ram algorithm for solving 3-d natural convection problems using orthogonal collocation

 

Algoritmo de bajo consumo de ram para resolver problemas de convección natural 3-d, usando colocación ortogonal

 

H. Jiménez-Islas¹*, M. Calderón-Ramírez³, F.I. Molina-Herrera¹, G.M. Martínez-González², J.L. Navarrete-Bolaños¹, E.O. Castrejón-González²

 

¹ Departamento de Ingeniería Bioquímica. *Corresponding author. E-mail: hugo.jimenez@itcelaya.edu.mx. Tel. +52 (461) 611-75-75.

² Departamento de Ingeniería Química.

³ Departamento de Ciencias Básicas. Instituto Tecnológico de Celaya. Av. Tecnológico y Antonio García Cubas sn. Celaya, Guanajuato. C.P.38010. México.

 

Received May 13, 2013.
Accepted January 21, 2014.

 

Abstract

Computational code IMPLI-C3 is a low-RAM consumption program designed to solve three-dimensional parabolic partial differential nonlinear equations. The spatial coordinates are discretized using orthogonal collocation with Legendre polynomials while time was discretized via backward finite differences, generating an implicit method that originates a set of algebraic equations, which are solved by nonlinear relaxation for each step of time integration. Nonlinear relaxation is an iterative method that only uses the Jacobian diagonal and voids the RAM storage of the entire Jacobian matrix. This allows the simulation of physical systems that require greater number of nodes that otherwise would use too much RAM when trying to solve by Newton-Raphson. The code was successfully evaluated using several problems related to natural convection previously reported in literature, observing that nonlinear relaxation only requires 0.3%-1.5% of the memory required by Newton-Raphson for the same problems. Furthermore, one can be conclude that, in problems with many nodes, the use of multivariate Newton-Raphson is unfeasible due to high consumption of RAM that can even cause it to overflow.

Keywords: nonlinear relaxation, natural convection, orthogonal collocation, parabolic partial differential equations.

 

Resumen

IMPLI-C3 es un código computacional que utiliza bajos recursos de memoria RAM, diseñado para resolver ecuaciones diferenciales parciales parabólicas no lineales en tres dimensiones. Los ejes coordenados en el espacio se discretizaron usando colocación ortogonal con polinomios de Legendre mientras que el tiempo se discretizó mediante diferencias finitas hacia atrás, generando un esquema implícito que origina un conjunto de ecuaciones algebraicas, las cuales se resolvieron mediante relajación no lineal para cada etapa de integración en el tiempo. La relajación no lineal, es un método iterativo que emplea solamente la diagonal del Jacobiano para evitar que se almacene toda la matriz Jacobiana en la memoria RAM de la computadora. Lo anterior permite la simulación de sistemas físicos que requieren mayor cantidad de nodos que, de otra manera emplearían demasiada RAM al intentar resolverlos mediante Newton-Raphson. El código se evaluó satisfactoriamente usando varios problemas relacionados con el fenómeno de convección natural previamente reportados en la literatura, observando que el método de relajación no lineal solamente utiliza entre 0.3% a 1.5% de la memoria con respecto al método de Newton-Raphson. Además se pudo corroborar que, en problemas con muchos nodos, el uso de Newton-Raphson multivariable no es factible debido al consumo elevado de memoria RAM que, inclusive puede provocar su desbordamiento.

Palabras clave: relajación no lineal, convección natural, colocación ortogonal, ecuaciones diferenciales parciales parabólicas.

 

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Acknowledgments

We gratefully acknowledge the financial support of CONACYT, Mexico via grants Basic Science SEP-2004-CO1-46230 and CB-2011/167095, and CONCYTEG (Guanajuato State Government) via grants 07-09-k662-050, 07-09-k119-107, and 09-09k662-065.

 

References

Abalone, R., Gastón, A., Cassinera, A. and Lara, M. A. (2006). Modelización de la Distribución de Temperatura y Humedad de Granos Almacenados en Silos. Asociación Argentina de Mecánica Computacional. Mecánica Computacional; 233-247.         [ Links ]

Balzi, U. R., Gastón, A., and Abalone, R. (2008). Efecto de la Convección Natural en la Distribución de Temperatura y Migración de Humedad en Granos Almacenados en Silos. Asociación Argentina de Mecánica Computacional. Mecánica Computacional, 1471-1485.         [ Links ]

Barrozo, M. A. S., Henrique, H. M., Sartori, D. J. M. and Freire, J. T. (2006). The use of the orthogonal collocation method on the study of the drying kinetics of soybean seeds. Journal of Stored Products Research 42, 348-356.         [ Links ]

Bessonov, O. A., Brailovskaya, V.A., Nikitin, S.A., and Polezhaev, V. I. (1998). Three-Dimensional Natural Convection in a Cubical Enclosure: A Benchmark Numerical Solution. Advances in Computational Heat Transfer Procedures of International Symposium Cesme, 1.         [ Links ]

Brahim, B. B. and Taieb, L. (2009). Transient natural convection in 3D tilted enclosure heated from two opposite sides. International Communications in Heat and Mass Transfer 36, 604-613.         [ Links ]

Brooker, D. B., Bakker-Arkema, F. W. and Hall C. W. (1974). Drying Cereal Grains. AVI Publishing Company: Westport, Connecticut. USA.         [ Links ]

Carrera-Rodríguez, M., Martínez-González, G. M., Navarrete-Bolaños, J. L., Botello-Álvarez, J. E., Rico-Martínez, R. and Jiménez-Islas, H. (2011). Transient numerical study of the effect of ambient temperature on 2-D cereal grain storage in cylindrical silos. Journal of Stored Products Research 47, 106-122.         [ Links ]

De Vahl, Davis, G. (1983). Natural Convection of Air in a Square Cavity: A Benchmark Numerical Solution. International Journal for Numerical Methods in Fluids 3, 249-264.         [ Links ]

Ebrahimi, A. A., Ebrahim, H. A. and Jamshidi, E. (2008). Solving partial differential equations of gas-solid reactions by orthogonal collocation. Computers & Chemical Engineering 32, 1746-1759.         [ Links ]

Finlayson, B. A. (1972). The Method of Weighted Residuals and Variational Principles, Academic Press. USA.         [ Links ]

Finlayson, B. A. (1980). Nonlinear Analysis in Chemical Engineering. McGraw-Hill College. USA.         [ Links ]

Jiménez-Islas, H. and López-Isunza, F. (1994). ELI-COL Programa para resolver sistemas de ecuaciones diferenciales parciales elípticas, por doble colocación ortogonal. Avances en Ingeniería Química 2, 82-86.         [ Links ]

Jiménez-Islas H. and López-Isunza F. (1996). PAR-COL2, Programa para resolver EDP parabólicas bidimensionales lineales, no por doble colocación ortogonal. Avances en Ingeniería Química 4, 168-173.         [ Links ]

Jiménez-Islas, H. (1999). Modelamiento Matemático de los Procesos de Transferencia de Momentum, Calor y Masa en Medios Porosos. PhD Thesis (in Spanish), Universidad Autónoma Metropolitana Unidad Iztapalapa, México, D.F.         [ Links ]

Jiménez-Islas, H., Navarrete-Bolaños, J. L. and Botello-Álvarez, E. (2004). Numerical Study of the Natural Convection of Heat and 2-D Mass of Grain Stored in Cylindrical Silos. Agrociencia 38, 325-342.         [ Links ]

Jiménez-Islas, H. (2001). Natural Convection in a Cubical Porous Cavity: Solution by Orthogonal Collocation. Computational Fluid Dynamics; Proceedings of the Fourth UNAM Supercomputing Conference. World Scientific Publishing Co. Singapore, 173-180.         [ Links ]

Leonardi, E. A. (1984). Numerical Study of the Effects of Fluid Properties on Natural Convection. Ph.D. Thesis. University of New South Wales, Australia.         [ Links ]

Lo, D. C., Young, D. L., Murugesan, K., Tsai, C. C., and Gou, M. H. (2007). Velocity-Vorticity formulation for 3D natural convection in an inclined cavity by DQ method. International Journal of Heat and Mass Transfer 50, 479-491.         [ Links ]

Mohsenin, N. N. (1980). Thermal Properties of Foods and Agricultural Materials (2nd ed. pp 407). Gordon and Breach Science Publishers: NY. USA.         [ Links ]

Nield, D. A. and Bejan, A. (1992). Convection in Porous Media. Springer-Verlag, USA.         [ Links ]

Ostrach, S. (1988). Natural Convection in Enclosures. Journal of Heat Transfer 110, 1175-1190.         [ Links ]

Ozoe, H. and Toh, K. A. (1998). Technique to circumvent a singularity at a radial center with application for a three-dimensional cylindrical system. Numerical Heat Transfer, Part B. Fundamentals. An International Journal of Computation and Methodology 33, 355-365.         [ Links ]

Ravnik, J., Skerget, L. and Zunic, Z. (2008). Velocity-Vorticity formulation for 3-D natural convection in an inclined enclosure by FEM. International Journal of Heat and Mass Transfer 51, 4517-4527.         [ Links ]

Roache, P.J. (1972). Computational Fluid Dynamics. Hermosa Publisher: Albuquerque, N.M. USA.         [ Links ]

Tric, E., Labrose, G. and Betrouni, M. (2000). A first incursion into the 3D structure of natural convection of air in a differentially heated cubic cavity from accurate numerical solutions. International Journal of Heat and Mass Transfer 43, 4043-4056.         [ Links ]

Vemuri, V. R. and Karplus, W. J. (1981). Digital Computer Treatment of Partial Differential Equations. Prentice-Hall Co. USA.         [ Links ]

Villadsen, J. V. and Stewart, W. E. (1967). Solution of Boundary-Value Problems by Orthogonal Collocation. Chemical Engineering Science 22, 1483-1501.         [ Links ]

Wakashima, S. and Saitoh, T. S. (2004). Benchmark solutions for natural convection in a cubic cavity using the high-order time-space method. International Journal of Heat and Mass Transfer 47, 853-864.         [ Links ]

Wilson, M. T. (1999). A Model for Predicting Mould Growth and Subsequent heat Generation in Bulk Stored Grain. Journal of Stored Products Research 35, 1-13.         [ Links ]

Woot-Tsuen, W. and Flores, M. (1970). Tabla de Composición de Alimentos para uso en América Latina (pp 150). Editorial Interamericana, S.A., México.         [ Links ]