# zbMATH — the first resource for mathematics

Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. (English) Zbl 1244.76084
Summary: This paper describes an application of the recently proposed modified method of fundamental solutions (MMFS) to potential flow problems. The solution in two-dimensional Cartesian coordinates is represented in terms of the single layer and the double layer fundamental solutions. Collocation is used for the determination of the expansion coefficients. This novel method does not require a fictitious boundary as the conventional method of fundamental solutions (MFS). The source and the collocation points thus coincide on the physical boundary of the system. The desingularised values, consistent with the fundamental solutions used, are deduced from the direct boundary element method (BEM) integral equations by assuming a linear shape of the boundary between the collocation points. The respective values of the derivatives of the fundamental solution in the coordinate directions, as required in potential flow calculations, are calculated indirectly from the considerations of the constant potential field. The normal on the boundary is calculated by parametrisation of its length and the use of the cubic radial basis functions with the second-order polynomial augmentation. The components of the normal are calculated in an analytical way. A numerical example of potential flow around a two-dimensional circular region is presented. The results with the new MMFS are compared with the results of the classical MFS and the analytical solution. It is shown that the MMFS gives better accuracy for the potential, velocity components (partial derivatives of the potential), and absolute value of the velocity as compared with the classical MFS. The results with the single layer fundamental solution are more accurate than the results with the double layer fundamental solution.

##### MSC:
 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing 65N80 Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Golberg, M.A.; Chen, C.S., Discrete projection methods for integral equations, (1997), Computational Mechanics Pubications Southampton [2] Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130 [3] () [4] Wrobel, L.C., The boundary element method, volume I: applications in termo-fluids and acoustics, (2002), Willey London [5] Aliabadi, M.H., The boundary element method, volume II: applications in solids and structures, (2002), Willey London [6] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Advances in computational mathematics, 9, 69-95, (1998) · Zbl 0922.65074 [7] Balakrishnan, K.; Ramachadran, P.A., The method of fundamental solutions for linear diffusion-reaction equations, Mathematical and computer modelling, 31, 221-237, (2000) · Zbl 1042.35569 [8] Hou, T.Y.; Lowengrub, J.S.; Shelley, M.J., Boundary integral methods for multicomponent fluids and multiphase materials, Journal of computational physics, 169, 302-326, (2001) · Zbl 1046.76029 [9] Johnston, R.L.; Fairweather, G., The method of fundamental solutions for problems in potential flow, Applied mathematical modelling, 8, 265-270, (1984) · Zbl 0546.76021 [10] Fenner, R.T., Source field superposition analysis of two dimensional potential problems, International journal of numerical methods in engineering, 32, 1079-1091, (1991) · Zbl 0755.76060 [11] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for the solution of nonlinear plane potential problems, IMA journal of numerical analysis, 9, 231-242, (1989) · Zbl 0676.65110 [12] Karageorghis, A., The method of fundamental solutions for the solution of steady state free boundary problems, Journal of computational physics, 98, 119-128, (1992) · Zbl 0745.65075 [13] Šarler, B., Solution of a two-dimensional bubble shape in potential flow by the method of fundamental solutions, Engineering analysis with boundary elements, 30, 227-235, (2006) · Zbl 1195.76323 [14] Leitao, V.M.A., On the implementation of a multi-region Trefftz-collocation formulation for 2-D potential problems, Engineering analysis with boundary elements, 20, 51-61, (1997) [15] Berger, J.R.; Karageorghis, A., The method of fundamental solutions for heat conduction in layered materials, International journal for numerical methods in engineering, 45, 1618-1694, (1999) · Zbl 0972.80014 [16] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for axisymmetric potential problems, International journal of numerical methods in engineering, 44, 1653-1669, (1999) · Zbl 0932.74078 [17] Šarler, B.; Vertnik, R.; Savić, M.; Chen, C.S., A meshless approach to hollow-brick temperature field, (), 6 [18] Johnston, R.L.; Fairweather, G., The method of fundamental solutions for problems in potential flow, Applied mathematical modelling, 8, 265-270, (1984) · Zbl 0546.76021 [19] Young, D.L.; Chen, K.H.; Lee, C.W., Novel meshless method for solving the potential problems with arbitrary domain, Journal of computational physics, 209, 290-322, (2005) · Zbl 1073.65139 [20] Young, D.L.; Chen, J.T.; Kao, J.H., A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains, CMES: computer modelling in engineering and sciences, 19, 197-221, (2007) · Zbl 1184.65116 [21] Banerjee, P.K.; Butterfield, R., Boundary element methods in engineering, (1981), McGraw Hill New York · Zbl 0499.73070 [22] Šarler, B., Desingularised method of double layer fundamental solutions for potential flow problems, (), 159-168 [23] Rek, Z.; Šarler, B., Analytical integration of elliptic 2D fundamental solution and its derivatives for straight-line elements with constant interpolation, Engineering analysis with boundary elements, 5-6, 515-525, (1999) · Zbl 0940.65130 [24] Šarler, B., Towards a mesh-free solution of transport phenomena, Engineering analysis with boundary elements, 26, 731-738, (2002) · Zbl 1032.76628 [25] Šarler, B., Axisymmetric augmented thin plate splines, Engineering analysis with boundary elements, 21, 81-85, (1998) · Zbl 0973.74647 [26] Šarler, B.; Jelić, N.; Kovačević, I.; Lakner, M.; Perko, J., Axisymmetric multiquadrics, Engineering analysis with boundary elements, 30, 137-142, (2006) · Zbl 1195.65190
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.