Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T09:04:42.164Z Has data issue: false hasContentIssue false
  • English
  • Français

A unified boundary integral equation method for a class of second order elliptic boundary value problems

Published online by Cambridge University Press:  17 February 2009

M. Rezayat ,
F. J. Rizzo  and
D. J. Shippy
D. J. Shippy
Affiliation:
Department of Engineering Mechanics, University of Kentucky, Lexington, Kentucky 40506, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A generalized integral equation formulation and a systematic numerical solution procedure are presented for a class of boundary value problems governed by a general second-order differential equation of elliptic type. Diverse numerical examples include problems of plane-wave scattering, three-dimensional fluid flow, and plane heat transfer for a body with a moving flame boundary. The last example employs certain representation functions useful to increase solution effectiveness in problems with an isolated integrable singularity.

Type
Research Article
Information
The ANZIAM Journal , Volume 25 , Issue 4 , April 1984 , pp. 501 - 517
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Altenkirch, R. A., Rezayat, M., Eichhorn, R. and Rizzo, F. J., “A study of heat conduction forward of flame spread in solids; by the boundary integral equation method”, Trans. ASME Ser. C J. Heat Transfer 104 (1982), 734–740.CrossRefGoogle Scholar
[2]Banaugh, R. P. and Goldsmith, W., “Diffraction of steady acoustic waves by surfaces of arbitrary shape”, J. Acoust. Soc. Amer. 35 (1963), 1590–1601.CrossRefGoogle Scholar
[3]Carslaw, H. S. and Jaeger, J. C., Conduction of heat in solids (Oxford University Press, London, 1967).Google Scholar
[4]Chen, L. H. and Schweikert, D. G., “Sound radiation from an arbitrary body”, J. Acoust. Soc. Amer. 35 (1963), 1626–1632.CrossRefGoogle Scholar
[5]Copley, L. G., “Fundamental results concerning integral representations in acoustic radiation”, J. Acoust. Soc. Amer. 44 (1968), 28–32.CrossRefGoogle Scholar
[6]DeWiest, Roger J. M., “Flow to an eccentric well in a leaky circular aquifer with varied lateral replenishment”, Geofis. Pura Appl. 54 (1963), 90–102.Google Scholar
[7]Frey, A. E. and T'ien, J. S., “A theory of flame spread over a solid fuel including finite-rate chemical kinetics”, Combustion and Flame 36 (1979), 263–289.CrossRefGoogle Scholar
[8]John, F., Plane waves and spherical means (Interscience, New York, 1st edition, 1955).Google Scholar
[9]Kellogg, O. D., Foundatins of potential theory (Dover, New York, 1st edition, 1953).Google Scholar
[10]Lafe, O. E., “Boundary integral solutions to nearly horizontal flows in multiply zoned aquifers”, Ph.D. Thesis, Cornell University, 1981.Google Scholar
[11]Martin, P. A., “On the null-field equations for the exterior problems of acoustics”, Quart. J. Mech. Appl. Math. 33 (1980), 385–396.CrossRefGoogle Scholar
[12]Meyer, W. L., Bell, W. A., Zinn, B. T. and Stallybrass, M. P., “Boundary integral solutions of three dimensional acoustic radiation problems”, J. Sound and Vibration 59 (1978), 245–262.CrossRefGoogle Scholar
[13]Pao, Y. H. and Mow, Ch. Ch., Diffraction of elastic waves and dynamic stress concentrations (Crane, Russak and Company Inc., New York, 1973).CrossRefGoogle Scholar
[14]Rice, J. R., “On the degree of convergence of nonlinear spline approximation”, in Approximation with special emphasis on spline functions (ed. Schoenberg, I. J.), (Academic Press, New York, 1969).Google Scholar
[15]Rizzo, F. J. and Shippy, D. J., “A method of solution for certain problems of transient heat conduction”, AIAA J. 8 (1970), 2004–2009.CrossRefGoogle Scholar
[16]Rizzo, F. J. and Shippy, D. J., “The boundary integral equation method with application to certain stress concentration problems in elasticity”, J. Austral. Math. Soc. Ser. B 22 (1981), 381–393.CrossRefGoogle Scholar
[17]Rizzo, F. J. and Shippy, D. J., “An advanced boundary integral equation method for three-dimensional thermoelasticity”, Internat. J. Numer. Methods Engrg. 11 (1977), 1753–1768.CrossRefGoogle Scholar
[18]Rouse, H. (ed.), Advanced mechanics of fluids (John Wiley and Sons, Inc., New York, 1969).Google Scholar
[19]Schenck, H. A., “Improved integral formulation for acoustic radiation problems”, J. Acoust. Soc. Amer. 44 (1968), 41–58.CrossRefGoogle Scholar
[20]Shaw, R. P., “Boundary integral equation methods applied to wave problems”, in Developments in boundary element methods-1, Chapter 6, 121–154.Google Scholar
[21]Sobolov, S. L., Partial differential equations of mathematical physics (Pergamon Press, London, 1964).Google Scholar
[22]Stern, M., “On calculating thermally induced stress singularities”, Conference on thermal stresses in severe environment (Plenum Press, 1980), 123–134.CrossRefGoogle Scholar
[23]Williams, M. L., “Stress singularities resulting from various boundary conditions in angular corners of plates in extension”, J. Appl. Mech. 74 (1952), 526–528.CrossRefGoogle Scholar
[24]Wu, Y. S., “The boundary integral equation method using various approximation techniques for problems governed by Laplace's equation”, Master's thesis, University of Kentucky, 1976.CrossRefGoogle Scholar
[25]Zienkiewicz, O. C., The finite element method (McGraw-Hill Book Company (UK) Limited, London, 3rd edition, 1977).Google Scholar