×

\(hp\)-adaptive least squares spectral element method for population balance equations. (English) Zbl 1138.65110

Summary: Population balance equations (PBE) are encountered in numerous scientific and engineering disciplines. This equation describes complex processes where the accurate prediction of the dispersed phase plays a major role for the overall behavior of the system. The PBE is a nonlinear partial integro-differential equation which is computationally intensive.
This paper discusses the application of an \(hp\)-adaptive refinement technique applied to a least squares spectral element formulation for solving population balance equations. The refinement is based on an estimate of the local Sobolev regularity index of the underlying solution by monitoring the decay rate of its Legendre expansion coefficients. The performance of the method is demonstrated numerically by using analytical test cases.

MSC:

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
45G10 Other nonlinear integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Bochev, Finite element methods based on least-squares and modified variational principles, Technical report, POSTECH, 2001; P. Bochev, Finite element methods based on least-squares and modified variational principles, Technical report, POSTECH, 2001
[2] Chen, M.; Hwang, C.; Shih, Y., A wavelet-Galerkin method for solving population balance equations, Comput. Chem. Eng., 20, 2, 131-145 (1996)
[3] Deville, M. O.; Fischer, P. F.; Mund, E. H., High-Order Methods for Incompressible Fluid Flow (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1007.76001
[4] Dorao, C. A.; Jakobsen, H. A., A least squares method for the solution of population balance problems, Comput. Chem. Eng., 30, 15, 535-547 (2005)
[5] C.A. Dorao, H.A. Jakobsen, An evaluation of selected numerical methods for solving the population balance equation, in: Fourth International Conference on CFD in the Oil and Gas, Metallurgical & Process Industries, SINTEF/NTNU Trondheim, Norway, 6-8 June 2005; C.A. Dorao, H.A. Jakobsen, An evaluation of selected numerical methods for solving the population balance equation, in: Fourth International Conference on CFD in the Oil and Gas, Metallurgical & Process Industries, SINTEF/NTNU Trondheim, Norway, 6-8 June 2005
[6] Dorao, C. A.; Jakobsen, H. A., Time-space least squares spectral element method for the population balance problems, (Simos, T.; Maroulis, G., Advances in Computational Methods in Science and Engineering 2005, ICCMSE, 21-26 October 2005, Greece. Advances in Computational Methods in Science and Engineering 2005, ICCMSE, 21-26 October 2005, Greece, Lect. Ser. Comput. Comput. Sci., vol. 4A (2005)), 171-174 · Zbl 1193.92096
[7] Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, Math. Sci. Eng., vol. 87 (1972), Academic Press: Academic Press San Diego · Zbl 0319.49020
[8] A. Galvaõ, M. Gerritsma, B. De Maerschalck, hp; A. Galvaõ, M. Gerritsma, B. De Maerschalck, hp
[9] P. Houston, B. Senior, E. Süli, Sobolev regularity estimation for hp; P. Houston, B. Senior, E. Süli, Sobolev regularity estimation for hp · Zbl 1043.65114
[10] Houston, P.; Süli, E., A note on the design of hp-adaptive finite element methods for elliptic differential equations, Comput. Methods Appl. Mech. Eng., 194, 229-243 (2005) · Zbl 1074.65131
[11] Jakobsen, H. A.; Lindborg, H.; Dorao, C. A., Modeling of bubble column reactors: Progress and limitations, Ind. Eng. Chem. Res., 44, 14, 5107-5151 (2005)
[12] Jiang, B., The Least-Square Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics (1998), Springer: Springer Berlin
[13] Lasheras, J. C.; Eastwood, C.; Matínez-Bazán, C.; Montañés, J. L., A review of statistical models for the break-up of an immiscible fluid immersed into a fully developed turbulent flow, Int. J. Multiphase Flow, 28, 247-278 (2001) · Zbl 1136.76556
[14] Liu, J., Exact a posteriori error analysis of the least squares finite element method, Appl. Math. Comput., 116, 297-305 (2000) · Zbl 1023.65117
[15] Liu, Y.; Cameron, T., A new wavelet-based method for the solution of the population balance equation, Chem. Eng. Sci., 56, 5283-5294 (2001)
[16] Maday, Y.; Rønquist, E. M., Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries, Comput. Methods Appl. Mech. Eng., 9, 91-115 (1990) · Zbl 0728.65078
[17] De Maerschalck, B.; Gerritsma, M. I., The use of the Chebyshev approximation in the space-time least-squares spectral element method, Numer. Algorithms, 38, 13, 173-196 (2005) · Zbl 1066.65110
[18] De Maerschalck, B.; Gerritsma, M. I., High-order Gauss-Lobatto integration for non-linear hyperbolic equations, J. Sci. Comput., 27, 13, 201-214 (2006) · Zbl 1115.65103
[19] Mantzaris, N. V.; Daoutidis, P.; Srienc, F., Numerical solution of multi-variable cell population balance models. II. Spectral methods, Comput. Chem. Eng., 25, 1441-1462 (2001)
[20] Patil, D. P.; Andrews, J. R.G., An analytical solution to continuous population balance model describing floc coalescence and breakage—A special case, Chem. Eng. Sci., 53, 3, 599-601 (1998)
[21] Pontaza, J. P.; Reddy, J. N., Spectral/hp least-squares finite element formulation for the incompressible Navier-Stokes equation, J. Comput. Phys., 190, 2, 523-549 (2003) · Zbl 1077.76054
[22] Pontaza, J. P.; Reddy, J. N., Space-time coupled spectral/hp least squares finite element formulation for the incompressible Navier-Stokes equation, J. Comput. Phys., 190, 2, 418-459 (2004) · Zbl 1106.76403
[23] Proot, M. M.J.; Gerritsma, M. I., A least-squares spectral element formulation for Stokes problem, J. Sci. Comput., 17, 1-4, 285-296 (2002) · Zbl 1036.76046
[24] Ramkrishna, D., Population Balances, Theory and Applications to Particulate Systems in Engineering (2000), Academic Press: Academic Press San Diego
[25] Rønquist, E. M.; Patera, A. T., Spectral element multigrid. I. Formulation and numerical results, J. Sci. Comput., 4, 389-402 (1987) · Zbl 0666.65055
[26] Subramain, G.; Ramkrishna, D., On the solution of statistical models of cell populations, Math. Biosci., 10, 1-23 (1971)
[27] Wulkow, M.; Gerstlauer, A.; Nieken, U., Modeling and simulation of crystallization processes using parsival, Chem. Eng. Sci., 56, 2575-2588 (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.