An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence

Sashikumaar Ganesan

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 6, page 1447-1465
  • ISSN: 0764-583X

Abstract

top
We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.

How to cite

top

Ganesan, Sashikumaar. "An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1447-1465. <http://eudml.org/doc/222160>.

@article{Ganesan2012,
abstract = {We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.},
author = {Ganesan, Sashikumaar},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Population balance equations; operator-splitting method; error analysis; streamline Upwind Petrov Galerkin finite element methods; backward Euler scheme; convergence; population balance equation; streamline-upwind Petrov Galerkin finite element method; stability; accuracy; diffusion-convection equation; consistency; error estimate; numerical results},
language = {eng},
month = {5},
number = {6},
pages = {1447-1465},
publisher = {EDP Sciences},
title = {An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence},
url = {http://eudml.org/doc/222160},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Ganesan, Sashikumaar
TI - An operator-splitting Galerkin/SUPG finite element method for population balance equations : stability and convergence
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/5//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1447
EP - 1465
AB - We present a heterogeneous finite element method for the solution of a high-dimensional population balance equation, which depends both the physical and the internal property coordinates. The proposed scheme tackles the two main difficulties in the finite element solution of population balance equation: (i) spatial discretization with the standard finite elements, when the dimension of the equation is more than three, (ii) spurious oscillations in the solution induced by standard Galerkin approximation due to pure advection in the internal property coordinates. The key idea is to split the high-dimensional population balance equation into two low-dimensional equations, and discretize the low-dimensional equations separately. In the proposed splitting scheme, the shape of the physical domain can be arbitrary, and different discretizations can be applied to the low-dimensional equations. In particular, we discretize the physical and internal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG) finite elements, respectively. The stability and error estimates of the Galerkin/SUPG finite element discretization of the population balance equation are derived. It is shown that a slightly more regularity, i.e. the mixed partial derivatives of the solution has to be bounded, is necessary for the optimal order of convergence. Numerical results are presented to support the analysis.
LA - eng
KW - Population balance equations; operator-splitting method; error analysis; streamline Upwind Petrov Galerkin finite element methods; backward Euler scheme; convergence; population balance equation; streamline-upwind Petrov Galerkin finite element method; stability; accuracy; diffusion-convection equation; consistency; error estimate; numerical results
UR - http://eudml.org/doc/222160
ER -

References

top
  1. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3th edition. Springer (2008).  Zbl1135.65042
  2. E. Burman, Consistent SUPG-method for transient transport problems : Stability and convergence. Comput. Methods Appl. Mech. Eng.199 (2010) 1114–1123.  Zbl1227.76047
  3. I.T. Cameron, F.Y. Wang, C.D. Immanuel and F. Stepanek, Process systems modelling and applications in granulation : a review. Chem. Eng. Sci.60 (2005) 3723–375.  
  4. F.B. Campos and P.L.C. Lage, A numerical method for solving the transient multidimensional population balance equation using an Euler-Lagrange formulation. Chem. Eng. Sci.58 (2003) 2725–2744.  
  5. P. Chen, J. Sanyal and M.P. Dudukovic, CFD modeling of bubble columns flows : implementation of population balance. Chem. Eng. Sci.59 (2004) 5201–5207.  
  6. K. Eriksson and C. Johnson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput.60 (1993) 167–188.  Zbl0795.65074
  7. S. Ganesan and L. Tobiska, Implementation of an operator-splitting finite element method for high-dimensional parabolic problems. Faculty of Mathematics, University of Magdeburg, Preprint No. 11-04 (2011).  
  8. S. Ganesan and L. Tobiska, An operator-splitting finite element method for the efficient parallel solution of multidimensional population balance systems. Chem. Eng. Sci.69 (2012) 59–68.  
  9. R. Glowinski, E.J. Dean, G. Guidoboni, D.H. Peaceman and H.H. Rachford, Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation. Japan J. Ind. Appl. Math.25 (2008) 1–63.  
  10. R. Gunawan, I. Fusman and R.D. Braatz, High resolution algorithms for multidimensional population balance equations. AIChE J.50 (2004) 2738–2749.  
  11. R. Gunawan, I. Fusman and R.D. Braatz, Parallel high-resolution finite volume simulation of particulate processes. AIChE J.54 (2008) 1449–1458.  
  12. T.J.R. Hughes and A.N. Brooks, A multi-dimensional upwind scheme with no cross-wind diffusion, in Finite element methods for convection dominated flows, edited by T.J.R. Hughes. ASME, New York (1979) 19–35.  
  13. H.M. Hulburt and S. Katz, Some problems in particle technology : A statistical mechanical formulation. Chem. Eng. Sci.19 (1964) 555–574.  
  14. V. John and J. Novo, Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations. SIAM J. Numer. Anal.49 (2011) 1149–1176.  Zbl1233.65065
  15. V. John, M. Roland, T. Mitkova, K. Sundmacher, L. Tobiska and A. Voigt, Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Eng. Sci.64 (2009) 733–741.  
  16. V. Kulikov, H. Briesen, R. Grosch, A. Yang, L. von Wedel and W. Marquardt, Modular dynamic simulation for integrated particulate processes by means of tool integration. Chem. Eng. Sci.60 (2005) 2069–2083.  
  17. V. Kulikov, H. Briesen and W. Marquardt, A framework for the simulation of mass crystallization considering the effect of fluid dynamics. Chem. Eng. Sci.45 (2006) 886–899.  
  18. G. Lian, S. Moore and L. Heeney, Population balance and computational fluid dynamics modelling of ice crystallisation in a scraped surface freezer. Chem. Eng. Sci.61 (2006) 7819–7826.  
  19. D.L. Ma, D.K. Tafti and R.D. Braatz, High-resolution simulation of multidimensional crystal growth. Ind. Eng. Chem. Res.41 (2002) 6217–6223.  
  20. D.L. Ma, D.K. Tafti and R.D. Braatz, Optimal control and simulation of multidimensional crystallization processes. Comput. Chem. Eng.26 (2002) 1103–1116.  
  21. A. Majumder, V. Kariwala, S. Ansumali and A. Rajendran, Fast high-resolution method for solving multidimensional population balances in crystallization. Ind. Eng. Chem. Res.49 (2010) 3862–3872.  
  22. D. Marchisio and R. Fox, Solution of population balance equations using the direct quadrature method of moments. J. Aero. Sci.36 (2005) 43–73.  
  23. D. L. Marchisio and R.O. Fox, Solution of population balance equations using the direct quadrature method of moments. J. Aero. Sci.36 (2005) 43–73.  
  24. M. N. Nandanwara and S. Kumar, A new discretization of space for the solution of multi-dimensional population balance equations : Simultaneous breakup and aggregation of particles. Chem. Eng. Sci.63 (2008) 3988–3997.  
  25. D. Ramkrishna, Population Balances, Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego (2000).  
  26. D. Ramkrishna and A.W. Mahoney, Population balance modeling : Promise for the future. Chem. Eng. Sci.57 (2002) 595–606.  
  27. V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, 3th edition. Springer (2008).  Zbl1105.65102

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.