DOĞRUSAL OLMAYAN SINIR KOŞULLARINA SAHİP ISI TRANSFERİ PROBLEMLERİNİN SINIR ELEMAN YÖNTEMİ İLE ANALİZİNE YÖNELİK YENİ BİR FORMÜLASYON

Year 2019, Volume: 39 Issue: 2, 229 - 236, 31.10.2019

Besim Baranoğlu

Abstract

Bu çalışmada Fourier denklemi ile ifade edilen ısı transferi problemlerinin bir ya da daha fazla bölgesinde tanımlı doğrusal olmayan sınır koşulları altında çözümüne yönelik sınır eleman yöntemi tabanlı etkili bir sayısal çözüm sunulmaktadır. Çözüm, sınır eleman yöntemi sistem matrislerinin üzerinde yapılan matematiksel işlemler ile bilinmeyenleri sadece doğrusal olmayan sınır bölgesindeki sıcaklık farkı olan indirgenmiş matris denklemleri elde etmektedir. Bu sayede doğrusal olmayan sınır koşullarına dayalı iterasyonlar daha hızlı gerçekleştirilebilmektedir. Doğrusal olmayan sınır koşullarının tanımlı olduğu bölgelerde çözüm elde edildikten sonra tüm sınır çözümü tanımlı bir son-işlem ile gerçekleştirilebilmektedir. Gerçek çözümü elde edilmiş bir örnek kullanılarak elde edilen sonuçlar değerlendirilmiştir.

Keywords

Sınır eleman yöntemi, ısı transferi, lineer (doğrusal) olmayan sınır koşulları

A NEW FORMULATION FOR THE BOUNDARY ELEMENT ANALYSIS OF HEAT CONDUCTION PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS

Abstract

An effective numerical method based on the boundary element formulation is presented to solve heat conduction equations which are governed by the Fourier equation, with nonlinear boundary conditions on one or more sections of the prescribed boundary. The solution involves the manipulation of the system matrices of the boundary element method and obtaining a smaller ranked matrix equation in which the unknown is only the temperature difference over the nonlinear boundary condition region. This way, the iterations to deal with the nonlinear conditions are performed faster. After finding the solution over the nonlinear boundary condition region, the solution over the entire boundary is obtained as a post-process through a prescribed relation. An example with a proven exact solution is employed to assess the results.

Keywords

Boundary element method, heat conduction, nonlinear boundary conditions

References

• Becker, A. A., 1992, The Boundary Element Method in Engineering, A complete course. McGraw-Hill.
• Beer, G., Smith, I., and Duenser, C., 2008, The Boundary Element Method with Programming. Springer-Verlag/Wien.
• Bialecki, R. and Nowak, A., 1981, Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions. Applied Mathematical Modelling, 5(6):417–421.
• Chan, C. L., 1993, A local iteration scheme for nonlinear two-dimensional steady-state heat conduction: a bem approach. Applied Mathematical Modeling, 17:650–657.
• Dehghani, A., Moradi, A., Dehghani, M., and Ahani, A., 2011, Nonlinear solution for radiation boundary condition of heat transfer process in human eye. 33rd Annual International Conference of the IEEE EMBS Boston, Massachusetts USA, August 30 - September 3, 2011.
• Karakaya, Z., Baranoglu, B., Cetin, B., and Yazici, A., 2015, A Parallel Boundary Element Formulation for Tracking Multiple Particle Trajectories in Stoke’s Flow for Microfluidic Applications. CMES-Computer Modeling Engineering & Sciences, 104(3):227–249.
• Lesnic, D., Onyago, T. T. M., and Ingham, D. B., 2009), The boundary element method for the determination of nonlinear boundary conditions in heat conduction. WIT Transactions on Modelling and Simulation, 48:45–55.
• Mengi, Y. and Argeso, H., 2006, A unified approach for the formulation of interaction problems by the boundary element method. International Journal for Numerical Methods in Engineering, 66(5):816–842.
• Mosayebidorcheh, S., Ganji, D., and Farzinpoor, M., 2014, Approximate solution of the nonlinear heat transfer equation of a fin with the power-law temperature-dependent thermal conductivity and heat transfer coefficient. Propulsion and Power Research, 3(1):41 – 47.
• Mukherjee, S. and Morjaria, M., 1984 On the efficiency and accuracy of the boundary element method and the finite element method. International Journal for Numerical Methods in Engineering, 20:515–522.
• Nowacki, W.,1967, Thermoelasticity. Addison-Wesley.
• Onyago, T. T. M., Ingham, D. B., and Lesnic, D., 2009, Reconstruction of boundary condition laws in heat conduction using the boundary element method. Computers and Mathematics with Applications, 57:153–168.
• Santos, W., Brasil, S., Antonio Fontes Santiago, J., and Telles, J. C., 2018, A new solution technique for cathodic protection systems with homogeneous region by the boundary element method. Revue europeenne de mecanique numerique, 27(5-6 Advances in Boundary Element Techniques):368–382.
• Slodicka, M., Lesnic, D., and Onyago, T. T. M., 2010, Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem. Inverse Problems in Science and Engineering, 18(1):65–81.
• Wrobel, L. and Brebbia, C. A., editors, 1992, Boundary Element Methods in Heat Transfer. International Series on Computational Engineering, Springer.
• Xu, S. Q. and Kamiya, N., 1997, An adaptive boundary element mesh for the problem with nonlinear robin-type boundary condition. Advances in Engineering Software, 28:533–538.
• Yalçın, O. F. and Mengi, Y., 2013, A new boundary element formulation for wave load analysis. Computational Mechanics, 52(4):815–826.