Summary
Application of the boundary integral equation (BIE) or boundary element method to two-dimensional transient heat flow problems using higher-order spatial shape functions is presented. Many different formulations have been proposed for the treatment of heat conduction (diffusion) problems by the BIE method, the most efficient of which is the one which employs a time dependent fundamental solution. The formulation adopted for this analysis employs space and time dependent fundamental solutions to derive the boundary integral equation in the time domain. It is an implicit time-domain formulation and is valid for both regular and unbounded domains. A time stepping scheme (time integral method) is then used to solve the boundary initial value problem by marching forward in time. Constant and linear temporal interpolation and quadratic shape functions are used to approximate field quantities in the time and space domains, respectively. Temporal and spatial integrations are carried out to form a system of linear equations. At the end of each time step, these equations are solved to obtain unknown values at that time. Validity of the BIE formulation is demonstrated by solving some test cases whose analytical solutions are known. Two-dimensional heat flow in a cooled turbine rotor blade is carried out as a practical application.
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References
Rizzo, F. J.; Shippy, D. H.: A method of solution for certain problems of transient heat conduction. AIAAJ 8 (1970) 2004–2009.
Chang, Y. P.; Kang, C. S.; Chen, D. J.: The use of fundamental Green’s function for the solution of problems of heat conduction in anisotropic media. Int. J. Heat Mass Transfer 16 (1973) 1905–1918.
Shaw, R. P.: An integral approach to diffusion. Int. J. Heat Mass Transfer 17 (1974) 693–699.
Wrobel, L. C.; Brebbia, C. A.: Time dependent potential problems: in Progress in boundary element methods: vol. 1 London, Pentech Press (1981).
Qamar, M. A.: A Boundary Integral Equation Method for Two-dimensional Thermoelastic Analysis. Ph.D. Thesis, University of London, (1990) (in preparation).
Banerjee, P.K.; and Butterfield, R.: Boundary Element Methods in Engineering Science. London: McGraw-Hill, 1981.
Sharp, S.: A condition for simplifying the forcing term in boundary element solutions of the diffusion equation. Communications in Applied Numerical Methods 4 (1985) 67–70.
Abramowitz, M.; Stegun, I.: Handbook of mathematical functions. New York, Dover Publications 1972.
Curran, D. A. S; Cross, M.; Lewis, B. A.: Solution of parabolic differential equations by the boundary element method using discretisation in time. Applied Mathematical Modelling 4 (1980) 398–400.
Beskos, D. E.: Boundary element methods in mechanics. New York, Elsevier Science Publishers 1987.
Bruch Jr., J. C.; Zyvoloski, G.: Transient two-dimensional heat conduction problems solved by the finite element method. Int. J. for Num. Methods in Engn 8 (1974) 481–494.
Carslaw, H. S.; Jaeger, J. C.: Conduction of heat in solids. London, Oxford University Press 1959.
Zienkiewwicz, O. C.; Parekh, C. J.: Transient field problems: Two-dimensional and Three-dimensional analysis by isoparametric finite elements. Int. J. Num. Methods Engng 2 (1970) 61–71.
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© 1991 Springer-Verlag Berlin, Heidelberg
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Qamar, M.A., Fenner, R.T., Becker, A.A. (1991). Application of the Boundary Integral Equation (Boundary Element) Method to Time Domain Transient Heat Conduction Problems. In: Morino, L., Piva, R. (eds) Boundary Integral Methods. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85463-7_43
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DOI: https://doi.org/10.1007/978-3-642-85463-7_43
Publisher Name: Springer, Berlin, Heidelberg
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