In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-tr...In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.展开更多
The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and ...The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.展开更多
文摘In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.
文摘The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.
