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Diffusion Problems

  • Chapter
Boundary Element Techniques

Abstract

In this chapter, we study boundary integral solutions to the diffusion equation

$${{\nabla }^{2}}u\left( x,t \right)-\frac{1}{k}\frac{\partial u\left( x,1 \right)}{\partial t}=0x\in \Omega $$
(4.1)

with boundary conditions of the following types:

$$\begin{array}{*{20}{c}} {u\left( {x,t} \right) = \bar{u}\left( {x,t} \right),} & {x \in {{\Gamma }_{1}},} \\ {q\left( {x,t} \right) = \frac{{\partial u\left( {x,t} \right)}}{{\partial n\left( x \right)}} = \bar{q}\left( {x,t} \right)} & {x \in {{\Gamma }_{2}}} \\ \end{array}$$
(4.2)

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References

  1. Rizzo, F. J., and Shippy, D. J., A method of solution for certain problems of transient heat conduction, AIAA J. 8, 2004–2009 (1970).

    Article  MATH  Google Scholar 

  2. Butterfield, R., and Tomlin, G. R., Integral techniques for solving zoned anisotropic continuum problems, in Proc. Int. Conf. on Variational Methods in Engineering, Vol. 2 ( C. A. Brebbia and H. Tottenham, Eds.), Southampton University Press, Southampton 1972.

    Google Scholar 

  3. Tomlin, G. R., Numerical analysis of continuum problems in zoned anisotropic media, Ph.D. Thesis, Southampton University, 1972.

    Google Scholar 

  4. Chang, Y. P., Kang, C. S., and Chen, D. J., The use of fundamental Green’s functions for the solution of problems of heat conduction in anisotropic media, Int. J. Heat Mass Transfer 16, 1905–1918 (1973).

    Article  MATH  Google Scholar 

  5. Shaw, R. P., An integral equation approach to diffusion, Int. J. Heat Mass Transfer 17, 693–699 (1974).

    Article  Google Scholar 

  6. Wrobel, L. C., and Brebbia, C. A., The boundary element method for steady-state and transient heat conduction, in Numerical Methods in Thermal Problems ( R. W. Lewis and K. Morgan, Eds.), Pineridge Press, Swansea, Wales, 1979.

    Google Scholar 

  7. Wrobel, L. C., and Brebbia, C. A., A formulation of the boundary element method for axisymmetric transient heat conduction, Int. J. Heat Mass Transfer 24, 843–850 (1981).

    Article  MATH  Google Scholar 

  8. Brebbia, C. A., and Walker, S., Boundary Element Techniques in Engineering, NewnesButterworths, London, 1980.

    MATH  Google Scholar 

  9. Widder, V. D., The Laplace Transform, Princeton University Press, Princeton, 1946.

    Google Scholar 

  10. Abramowitz, M., and Stegun, I. A. (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.

    Google Scholar 

  11. Lachat, J. C., and Combescure, A., Laplace transform and boundary integral equation: application to transient heat conduction problems, in Innovative Numerical Analysis in Applied Engineering Science (T. A. Cruse et al,Eds.), CETIM, Versailles, 1977.

    Google Scholar 

  12. Schapery, R. A., Approximate methods of transform inversion for visco-elastic stress analysis, in Proc. Fourth U.S. National Congress on Applied Mechanics, Vol. 2, 1962.

    Google Scholar 

  13. Liggett, J. A., and Liu, P. L. F., Unsteady flow in confined aquifers: A comparison of two boundary integral methods, Water Resources Res. 15, 861–866 (1979).

    Article  Google Scholar 

  14. Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford, 1959.

    Google Scholar 

  15. Curran, D. A. S., Cross, M., and Lewis, B. A., Solution of parabolic differential equations by the boundary element method using discretization in time, Appl. Math. Modelling 4, 398–400 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  16. Morse, P. M., and Feshbach, H., Methods of Theoretical Physics, Mc-Graw-Hill, New York, 1953.

    MATH  Google Scholar 

  17. Roach, G. F., Green’s Functions: Introductory Theory with Applications, Van Nostrand Reinhold, London, 1970.

    Google Scholar 

  18. Gradshteyn, I. S., and Ryzhik, I. M., Table of Integrals, Series and Products, Academic Press, London, 1965.

    Google Scholar 

  19. Wrobel, L. C., and Brebbia, C. A., Time-dependent potential problems, in Progress in Boundary Element Methods, Vol. 1 ( C. A. Brebbia, Ed.), Pentech Press, London, Halstead Press, NY, 1981.

    Google Scholar 

  20. Dubois, M., and Buysse, M., Transient heat transfer analysis by the boundary integral equation method, in New Developments in Boundary Element Methods ( C. A. Brebbia, Ed.), CML Publications, Southampton, 1980.

    Google Scholar 

  21. Curran, D., Cross, M., and Lewis, B. A., A preliminary analysis of boundary element methods applied to parabolic partial differential equations, in New Developments in Boundary Element Methods ( C. A. Brebbia, Ed.), CML Publications, Southampton, 1980.

    Google Scholar 

  22. Thaler, R. H., and Mueller, W. K., A new computational method for transient heat conduction in arbitrarily shaped regions, in Fourth Int. Heat Transfer Conference ( U. Grigull and E. Hahne, Eds.), Elsevier Publishing Co., Amsterdam, 1970.

    Google Scholar 

  23. Onishi, K, and Kuroki, T., Boundary element method in transient heat transfer problems, Bull. Inst. Advan. Res. Fukuoka Univ. 52 (1981).

    Google Scholar 

  24. Hammer, P. C., Marlowe, O. J., and Stroud, A. H., Numerical integration over simplexes and cones, Math. Comput. 10, 130–137 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  25. Wrobel, L. C., and Brebbia, C. A., Boundary elements in thermal problems, in Numerical Methods in Heat Transfer ( R. Lewis, K. Morgan, and O. C. Zienkiewicz, Eds.), Wiley, Chichester, 1981.

    Google Scholar 

  26. Wrobel, L. C., Potential and viscous flow problems using the boundary element method, Ph.D. Thesis, Southampton University, 1981.

    Google Scholar 

  27. Onishi, K, Convergence in the boundary element method for heat equation, TRU Math. 17, 213–225 (1981).

    MathSciNet  MATH  Google Scholar 

  28. Bruch, J. C., Jr., and Zyvoloski, G., Transient two-dimensional heat conduction solved by the finite element method, Int. J. Numerical Methods Engng. 8, 481–494 (1974).

    Article  MATH  Google Scholar 

  29. Zienkiewicz, O. C., and Parekh, C. J., Transient field problems: Two-dimensional and three-dimensional analysis by isoparametric finite elements, Int. J. Numerical Methods Engng. 2, 61–71 (1970).

    Article  MATH  Google Scholar 

  30. Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1944.

    MATH  Google Scholar 

  31. Haji-Sheik, A., and Sparrow, E. M., Transient heat conduction in a prolate spheroidal solid, J. Heat Transfer Trans. ASME 88 C, 331–333 (1966).

    Google Scholar 

  32. Anderson, C. A., and Zienkiewicz, O. C., Spontaneous ignition: Finite element solutions for steady and transient conditions, J. Heat Transfer Trans. ASME 96 C, 398–404 (1977).

    Google Scholar 

  33. Rubinstein, L. I., The Stefan Problem, AMS Translations of Mathematical Monographs, Vol. 27, Amer. Math. Soc., Providence, R. I., 1971.

    Google Scholar 

  34. Chuang, Y. K., The melting and dissolution of a solid in a liquid with a strong exothermic heat of solution, Ph.D. Thesis, State University of New York at Buffalo, 1971.

    Google Scholar 

  35. Chuang, Y. K, and Szekely, J., On the use of Green’s functions for solving melting or solidification problems, Int. J. Heat Mass Transfer 14, 1285–1294 (1971).

    Article  Google Scholar 

  36. Chuang, Y. K., and Szekely, J., The use of Green’s functions for solving melting or solidification problems in the cylindrical coordinate system, Int. J. Heat Mass Transfer 15, 1171–1174 (1972).

    Article  Google Scholar 

  37. Chuang, Y. K., and Ehrich, O., On the integral technique for spherical growth problems, Int. J. Heat Mass Transfer 17, 945 – 953 (1974).

    Google Scholar 

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Authors and Affiliations

  1. Dept. of Civil Engineering, University of Southampton, SO9 5NH, Southampton, UK

    C. A. Brebbia

  2. Programa de Engenharia Civil, COPPE — Univ. Federal do Rio de Janeiro, Caixa Postal 68506, 21944, Rio de Janeiro, Brazil

    J. C. F. Telles & L. C. Wrobel

Authors
  1. C. A. Brebbia
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  2. J. C. F. Telles
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  3. L. C. Wrobel
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© 1984 Springer-Verlag Berlin, Heidelberg

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Brebbia, C.A., Telles, J.C.F., Wrobel, L.C. (1984). Diffusion Problems. In: Boundary Element Techniques. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48860-3_4

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  • DOI: https://doi.org/10.1007/978-3-642-48860-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-48862-7

  • Online ISBN: 978-3-642-48860-3

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