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**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 |

### Keywords:

potential flow; method of fundamental solutions; desingularisation; two-dimensional Cartesian coordinates; single layer fundamental solution; double layer fundamental solution
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\textit{B. Šarler}, Eng. Anal. Bound. Elem. 33, No. 12, 1374--1382 (2009; Zbl 1244.76084)

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