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UPSI Digital Repository (UDRep)
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| Abstract : Universiti Pendidikan Sultan Idris |
| The aim of this research is to produce the system of equations for three different mixed Boundary Value Problems (BVPs). The potential problem which involves the Laplace’s equation on a square shape domain was considered, where the boundary is divided into four sets of linear boundary elements. The Boundary Element Method (BEM) was used to approximate the solutions for BVP. The mixed BVPs were reduced to Boundary Integral Equation (BIEs) by using direct method which were related with Green’s second identity representation formula. Then, linear interpolation was used on the discretized elements. The results showed that, there are three system of equations which were obtained. For some cases of mixed BVPs which involves discontinuous fluxes problems yields underdetermined systems. Out of the three problems that being considered, one of three BVPs leads to the underdetermined system of equations. Therefore, the transformation for the underdetermined system to the standard form is necessary for the numerical purposes. The gradient approach method which is widely applies to the Dirichlet problem was considered. This gradient approach method is extended to the underdetermined system of equations obtained from the mixed BVP which subsequently transformed to the standard system. In conclusion, the mixed BVP that involve discontinuous fluxes problem will yield to the underdetermined system of equations that prohibits in solving the system numerically. However, by the gradient approach method, the underdetermined system can be transformed to the standard form and can be solved the system numerically. The study implicates that the procedure used in this studies can be extended to higher dimensional mixed BVPs which involved the discontinuous fluxes problems. |
| References |
Aliabadi, M.H. &Wen, P.H. (2002). The Boundary Element Method: Applications in Solids and Structures (Vol.2). Chichester: Wiley.
Ali, A. & Rajakumar, C. (2005). The Boundary Element Method Applications in Sound and Vibration. A.A.Balkema, Netherlands.
Ang, W.T. (2007). A Beginner’s Course in Boundary Element Methods. Florida : Boca Raton.
Atkinson, K.E & Chandler, G. (1990). Boundary Integral Equation Methods For Solving Laplace's Equation With Nonlinear Boundary Conditions: The Smooth Boundary Case. Mathematics of Computation, 55(192), 451-47
Beer, G, (et al) (2008), The Boundary Element Method with Programming. Springer Wien New York, Germany.
Brebbia, C.A. & Dominguez, J. (1977). Boundary Element Methods for Potential Problems. Appl. Math. Modelling, Vol 1, 372-378.
Cayol, V. & Cornet, F.H. (1997). 3D Mixed Boundary Elements for Elastostatic Deformation Field Analysis. Int. J. Rock Mech. Min. Sci., 34(2), 275-287.
Cheng, A.H.-D. & Cheng, D.T. (2005). Heritage And Early History Of The Boundary Element Method. Engineering Analysis with Boundary Elements, 29, 268–302.
Chunderlik, R. & Mikula, K. (2014). Numerical Solution of the Fixed AltimetryGravimetry BVP Using the Direct BEM Formulation. Slovak University of Technology, https://www.researchgate.net/publication/226154690. Costable, M. & Stephen, E. (1985). A Direct Boundary Integral Equation Method For Transmission Problems. Journal of Mathematical Analysis and Applications, 106, 3677413.
Dijkstra, W. & Mattheij R.M.M. (2006). The condition number of the BEM-matrix arising from Laplace’s equation. Electronic Journal of Boundary Elements. 4(2), 67-81
Forschung, N. & Praxis, L. Boundary Element Solvers for Linear Water Wave Simulation in a Model Basin. Johanne S Kepler University Atlinz.
Gaul, L. (et al) (2008), Boundary Element Methods for Engineers and Scientists. Springer, New York.
Green, G. (1993). An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. The Mathematical Gazette, 77(478), 26-51.
Hall, W.S. (1994). The Boundary Element Method. Kluwer Academic, United Kingdom.
Hunter, P. & Pullan, A. (2003). FEM/BEM Notes. University of Auckland, New Zealand.
Johnson, A.N. (et al) (2015). Modeling Mixed Boundary Problems with the Complex Variable Boundary Element Method (Cvbem) Using Matlab And Mathematica. Int. J. Comp. Meth. and Exp. Meas., 3(3), 269–278.
Makau, B.K., Gathia, P.W. & Manyanga, D.O. (2016). Numerical Simulation Of A Free Surface Flow Due To A Point Source. International Journal of Advanced Scientific and Technical Research, 6(5).
Mazya, V. & Soloviev, A. (2003). A Direct Method for Boundary Integral Equations On A Contour With A Peak. Georgian Mathematical Journal, 10(3), 573–593.
Menin, O.H. & Rolnik, V. (2013). Relation between Accuracy and Computational Time For Boundary Element Method Applied To Laplace Equation. Journal of Computational Interdisciplinary Sciences. 4(2), 1-6.
Nurul Akmal Mohamed. (2013), “Numerical solution and spectrum of boundary-domain integral equations,”, Ph.D. thesis, Brunel University.
Paris, F. & Canas, J. (1997), Boundary Element Method Fundamentals and Applications. Oxford University, United States.
Zeinab Sedaghatjoo., Mehdi Dehghan., & Hossein Hosseinzadeh. (2013). The use of continuous boundary elements in the boundary elements method for domains with non-smooth boundaries via finite difference approach. Computers and Mathematics with Applications. 65, 983-995. |
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