0
Research Papers: Conduction

Modification of the Classical Boundary Integral Equation for Two-Dimensional Transient Heat Conduction With Internal Heat Source, With the Use of NURBS for Boundary Modeling

[+] Author and Article Information
Eugeniusz Zieniuk

Faculty of Mathematics and Computer Science,
University of Bialystok,
Białystok 15-089, Poland
e-mail: ezieniuk@ii.uwb.edu.pl

Dominik Sawicki

Faculty of Mechanical Engineering,
Bialystok University of Technology,
Białystok 15-089, Poland
e-mail: sawicki.dominik1@gmail.com

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 18, 2016; final manuscript received February 27, 2017; published online April 19, 2017. Editor: Portonovo S. Ayyaswamy.

J. Heat Transfer 139(8), 081301 (Apr 19, 2017) (11 pages) Paper No: HT-16-1215; doi: 10.1115/1.4036099 History: Received April 18, 2016; Revised February 27, 2017

The most popular methods used for solving transient heat conduction problems, like finite element method (FEM) and boundary element method (BEM), require discretization of the domain or the boundary. The discretization problem escalates for unsteady issues, because an iterative process is required to solve them. An alternative to avoid the mentioned problem is parametric integral equations systems (PIESs), which do not require classical discretization of the boundary and the domain, while being numerically solved. PIES have been previously used with success to solve steady-state problems. Moreover, they have been recently tested also with success for transient heat conduction problems, without internal heat sources. The purpose of this paper is to generalize PIES based on analytical modification of classical boundary integral equation (BIE) for transient heat conduction with internal heat source and nonuniform rational basis spline (NURBS) for boundary modeling. The obtained generalization of PIES is tested on examples, mostly with defined exact solution.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Wu, J. , Gui, D. , Liu, D. , and Feng, X. , 2015, “ The Characteristic Variational Multiscale Method for Time Dependent Conduction–Convection Problems,” Int. Commun. Heat Mass Transfer, 68, pp. 58–68. [CrossRef]
Zhang, L. , Zhao, J. M. , and Liu, L. H. , 2016, “ A New Stabilized Finite Element Formulation for Solving Radiative Transfer Equation,” ASME J. Heat Transfer, 138(6), p. 064502.
Carslaw, H. S. , and Jaeger, J. C. , 1959, Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford, UK.
Brian, P. L. T. , 1961, “ A Finite-Difference Method of High-Order Accuracy for the Solution of Three-Dimensional Transient Heat Conduction Problems,” Am. Inst. Chem. Eng. J., 7(3), pp. 367–370. [CrossRef]
Wang, Y. , Qin, Y. , and Zhang, J. , 2011, “ Application of New Finite Volume Method (FVM) on Transient Heat Transferring,” Information Computing and Applications (Communications in Computer and Information Science, Vol. 105), Springer, Berlin, pp. 109–116.
Zienkiewicz, O. C. , and Taylor, R. L. , 2000, The Finite Element Method, Vol. 1–3, Butterworth, Oxford, UK.
Bruch, J. C. , and Zyvoloski, G. , 1974, “ Transient Two-Dimensional Heat Conduction Problems Solved by the Finite Element Method,” Int. J. Numer. Methods, 8(3), pp. 481–494. [CrossRef]
Lewis, R. W. , Morgan, K. , Thomas, H. R. , and Seetharamu, K. , 1996, The Finite Element Method in Heat Transfer Analysis, Wiley, Chichester, UK.
Brebbia, C. A. , Telles, J. C. , and Wrobel, L. C. , 1984, Boundary Element Techniques, Theory and Applications in Engineering, Springer, New York.
Wrobel, L. C. , and Brebbia, C. A. , 1979, The Boundary Element Method for Steady-State and Transient Heat Conduction, Pineridge Press, Swansea, UK.
Tanaka, M. , Matsumoto, T. , and Yang, Q. F. , 1994, “ Time-Stepping Boundary Element Method Applied to 2-D Transient Heat Conduction Problems,” Appl. Math. Modell., 18(10), pp. 569–576. [CrossRef]
Majchrzak, E. , 2001, Boundary Element Method in Heat Transfer, Technical University of Czestochowa, Czestochowa, Poland (in Polish).
Sutradhar, A. , Paulino, G. H. , and Gray, L. J. , 2002, “ Transient Heat Conduction in Homogeneous and Non-Homogeneous Materials by the Laplace Transform Galerkin Boundary Element Method,” Eng. Anal. Boundary Elem., 26(2), pp. 119–132. [CrossRef]
Partridge, P. W. , Brebbia, C. A. , and Wrobel, L. C. , 1992, The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, UK.
Nowak, A. J. , 1989, “ The Multiple Reciprocity Method of Solving Transient Heat Conduction Problems,” Boundary Elements XI, Vol. 2, Computational Mechanics Publications, Southampton, UK.
Yu, K. H. , Kadarman, A. H. , and Djojodihardjo, H. , 2010, “ Development and Implementation of Some BEM Variants—A Critical Review,” Eng. Anal. Boundary Elem., 34(10), pp. 884–899. [CrossRef]
Johansson, B. T. , and Lesnic, D. , 2008, “ A Method of Fundamental Solutions for Transient Heat Conduction,” Eng. Anal. Boundary Elem., 32(9), pp. 697–703. [CrossRef]
Kołodziej, J. A. , Mierzwiczak, M. , and Ciałkowski, M. , 2010, “ Application of the Method of Fundamental Solutions and Radial Basis Functions for Inverse Heat Source Problem in Case of Steady-State,” Int. Commun. Heat Mass Transfer, 37(2), pp. 121–124. [CrossRef]
Mierzwiczak, M. , and Kołodziej, J. A. , 2012, “ Application of the Method of Fundamental Solutions With the Laplace Transformation for the Inverse Transient Heat Source Problem,” J. Theor. Appl. Mech., 50(4), pp. 1011–1023.
Zieniuk, E. , 2003, “ Bézier Curves in the Modification of Boundary Integral Equations (BIE) for Potential Boundary-Values Problems,” Int. J. Solids Struct., 40(9), pp. 2301–2320. [CrossRef]
Zieniuk, E. , 2003, “ Hermite Curves in the Modification of Integral Equations for Potential Boundary-Value Problems,” Eng. Comput., 20(2), pp. 112–128. [CrossRef]
Zieniuk, E. , and Boltuc, A. , 2006, “ Bézier Curves in the Modeling of Boundary Geometry for 2D Boundary Problems Defined by Helmholtz Equation,” J. Comput. Acoust., 14(3), pp. 353–367. [CrossRef]
Zieniuk, E. , and Boltuc, A. , 2006, “ Non-Element Method of Solving 2D Boundary Problems Defined on Polygonal Domains Modeled by Navier Equation,” Int. J. Solids Struct., 43(25–26), pp. 7939–7958. [CrossRef]
Zieniuk, E. , 2007, “ Modelling and Effective Modification of Smooth Boundary Geometry in Boundary Problems Using B-Spline Curves,” Eng. Comput., 4(23), pp. 39–48. [CrossRef]
Bołtuć, A. , and Zieniuk, E. , 2011, “ Modeling Domains Using Bézier Surfaces in Plane Boundary Problems Defined by the Navier–Lame Equation With Body Forces,” Eng. Anal. Boundary Elem., 35(10), pp. 1116–1122. [CrossRef]
Zieniuk, E. , 2013, Computational Method PIES for Solving Boundary Value Problems, Polish Scientific Publishers PWN, Warsaw, Poland (in Polish).
Zieniuk, E. , Sawicki, D. , and Bołtuć, A. , 2014, “ Parametric Integral Equations Systems in 2D Transient Heat Conduction Analysis,” Int. J. Heat Mass Transfer, 78, pp. 571–587. [CrossRef]
Piegl, L. , and Tiller, W. , 1997, The NURBS Book, 2nd ed., Springer-Verlag, Berlin.
Foley, J. , van Dam, A. , Feiner, S. , Hughes, J. , and Phillips, R. , 1994, Introduction to Computer Graphics, Addison-Wesley, Boston, MA.
Farin, G. , 1990, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Academic Press, New York.
Mortenson, M. , 1985, Geometric Modelling, Wiley, New York.
Cody, W. J. , and Thacher, H. C., Jr. , 1968, “ Rational Chebyshev Approximations for Exponential Integral E1 (x),” Math. J. Comput., 22(103), pp. 641–649.
Gottlieb, D. , and Orszag, S. A. , 1977, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, PA.

Figures

Grahic Jump Location
Fig. 1

The considered rectangular domain with boundary conditions

Grahic Jump Location
Fig. 2

L2 norm of temperature error in time, for ten points located on one of the boundary segments, compared to FEM

Grahic Jump Location
Fig. 3

L2 norm of temperature error in time, for 50 points located inside of the domain, compared to FEM

Grahic Jump Location
Fig. 4

L2 norm of temperature error in time, for ten points located on one of the boundary segments, compared to the exact solution

Grahic Jump Location
Fig. 5

L2 norm of temperature error in time, for 50 points located inside of the domain, compared to the exact solution

Grahic Jump Location
Fig. 6

L2 norm of temperature error in time, for ten points located on one of the boundary segments, compared to the exact solution

Grahic Jump Location
Fig. 7

L2 norm of temperature error in time, for 50 points located inside of the domain, compared to the exact solution

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In