# Comparative study of conjugate gradient algorithms performance on the example of steady-state axisymmetric heat transfer problem

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## Abstract

The finite element method (FEM) is one of the most frequently used numerical methods for finding the approximate discrete point solution of partial differential equations (PDE). In this method, linear or nonlinear systems of equations, comprised after numerical discretization, are solved to obtain the numerical solution of PDE. The conjugate gradient algorithms are efficient iterative solvers for the large sparse linear systems. In this paper the performance of different conjugate gradient algorithms: conjugate gradient algorithm (CG), biconjugate gradient algorithm (BICG), biconjugate gradient stabilized algorithm (BICGSTAB), conjugate gradient squared algorithm (CGS) and biconjugate gradient stabilized algorithm with l GMRES restarts (BICGSTAB(l)) is compared when solving the steady-state axisymmetric heat conduction problem. Different values of l parameter are studied. The engineering problem for which this comparison is made is the two-dimensional, axisymmetric heat conduction in a finned circular tube.

[1] Zienkiewicz O.C., Taylor R.L: Finite Element Method. Elsevier, 2006.

[2] Kwon Y.W., Bang H.: The Finite Element Method using MATLAB. CRC Press, 2000.

[3] Lewis R.W., Nithiarasu P., Seetharamu K..N.: Fundamentals of the FiniteElement Method for Heat and Fluid Flow. Wiley, West Sussex 2004.

[4] Ocłoń P., Taler J.: Mixed finite element and finite volume formulation - linearquadrilateral elements. In: Encyclopedia of Thermal Stresses (R. Hetnarski, Ed.) Springer (accepted for print).

[5] Taler J., Duda P.: Solving Direct and Inverse Heat Conduction Problems. Springer, Berlin 2006.

[6] Taler J., Ocłoń P.: Finite element method in steady state and transient heatconduction. In: Encyclopedia of Thermal Stresses (R. Hetnarski, Ed.). Springer (accepted for print).

[7] Anderson J.D.: Computational Fluid Dynamics. McGraw-Hill Science, Boca Raton 1995.

[8] Chung T.J.: Computational Fluid Dynamics. Cambridge University Press, 2010.

[9] Barrett R., Berry M., Chan T.F. et al.: Templates for the solution of linearsystems: building blocks for iterative methods. SIAM, Philadelphia 1999.

[10] Krizek M., Neittaanmäki P., Glowinski P.R., Korotov S.: Conjugate GradientAlgorithms and Finite Element Methods. Springer 2004.

[11] Łopata S., Ocłoń P.: The analysis of gradient algorithm effectiveness - twodimensional heat transfer problem. Arch. Thermodyn. 31(2010), 4, 37-50.

[12] Meurant G.: The Lanczos and Conjugate Gradient Alghoritms. SIAM, Philadelphia 2006.

[13] Saad Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia 1999.

[14] Incopera F.P., DeWitt D.P.: Introduction to Heat Transfer. Wiley, Hoboken 2006.

[15] Sonneveld P.: CGS: A fast Lanczos-type solver for nonsymmetric linear systems. SIAM J. Sci. Comput. 10(1989), 1, 36-52.

[16] Parihar S.K., Wright N.T.: Thermal contact resistance at elastomer to metalinterfaces. Int. Commun. Heat Mass Transf. 24(1997), 8, 1083-1092.

[17] Wolff E.G., Schneider D.A.: Prediction of thermal contact resistance betweenpolished surfaces. Int. Commun. Heat Mass Transf. 41(1988), 22, 3469-3482.

[18] Massé H., Arquis E.E., Delaunay D., Quilliet S., Le Bot P.H.: Heat transferwith mechanically driven thermal contact resistance at the polymer-mold interfacein injection molding of polymers. Int. J. Heat and Mass Transf. 47(2004), 8-9, 2015-2027.

[19] Łopata S., Ocłoń P.: Modelling and optimizing operating conditions of heat exchangerwith finned elliptical tubes. In: Computational Modeling and Applications Fluid Dynamics (L.H. Juarez, Ed.), InTech ISBN: 978-953-51-0052-2, 327-356.

[20] Łopata S., Ocłoń P.: Analysis of operating conditions for high performance heatexchanger with the finned elliptical tube. Rynek Energii 5 (102)(2012), 112-124.

[21] Xiangqiao Y.: Finite element formulation of a heat transfer problem for an axisymmetriccomposite structure. Comput. Mech. 36(2005), 76-82.

[22] Sleijpen G.L., Van der Vorst H.A., Fokkema D.R.: BICGstab(l) and otherhybrid Bi-CG methods. Numer. Algorithms (1994)7: 75-109.

[23] Van der Vorst H.A.: Bi-CGSTAB A fast and smoothly converging variant ofBi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(1992), 2, 631-644.

[24] Gerub W.: Linear Algebra. Springer, Berlin, 1975.

[25] Taler D.: Effect of thermal contact resistance on the heat transfer in plate finnedtube heat exchangers. In: Proc. of the Conf. Heat Exchanger Fouling and Cleaning, 1-6 Jul., Tomar, Eng. Conf. Int., New York 2007, 255-364.

[26] Taler D: Determining thermal resistance between tube and fins in tube and platefin heat exchangers. In: Wspołczesne technologie i urządzenia energetyczne. Cracow University of Technology, Cracow 2007, 649-668.

[27] Cebula A., Taler D.: Determining thermal contact resistance of the fin-to-tubeattachment in plate fin-and tube heat exchangers. EngOpt 2010, The 2nd Int. Conf. on Engineering Optimization, Lisbon, 6-9 Sep., 2010, Book of Abstracts, 310-320, CD-ROM proc. 1-10.

[28] Taler D., Cebula A.: Analysis of thermal contact resistance on the heat transferin plate fin-and-tube heat exchangers. HTRSE Heat Transfer and Renewable Sources of Energy, Szczecin 2010, 331-340.

[29] Taler D., Cebula A.: A new method for determination of thermal contact resistanceof a fin-to-tube attachment in plate fin-and-tube heat exchangers. Chem. Process Eng. J. 31(2010), 839-855.

[30] Ocłoń P.: Gradient algorithms in solving steady state and transient heat conductionproblems. In: Proc, of the ECCOMAS Special lnterest Conference, Numerical Heat Transfer 2012, CD-ROM, Wroclaw, 2012, 583-595.

[31] MATLAB online documentation. http://www.mathworks.com/help/matlab

[32] ANSYS v.121, Theory reference for Mechanical ADPL and Mechanical Applications, www1.ansys.com/customer/content/documentation/121/ansthry.pdf

[33] TEWI Project, http://www.tewi.ics.p.lodz.pl

# Archives of Thermodynamics

## The Journal of Committee on Thermodynamics and Combustion of Polish Academy of Sciences

### Journal Information

CiteScore 2016: 0.54

SCImago Journal Rank (SJR) 2016: 0.319
Source Normalized Impact per Paper (SNIP) 2016: 0.598

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