The paper presents a non-element method of solving boundary problems defined on polygonal domains modeled by corner points. To solve these problems a parametric integral equation system (PIES) is used. The system is characterized by a separation of the approximation of boundary geometry from the approximation of boundary functions. This feature makes it possible to effectively investigate the convergence of the obtained solutions with no need of performing the approximation of boundary geometry. The testing examples included confirm high accuracy of the solutions.
This paper presents a new boundary shape representation for 3D boundary value problems based on parametric triangular Bézier surface patches. Formed by the surface patches, the graphical representation of the boundary is directly incorporated into the formula of parametric integral equation system (PIES). This allows us to eliminate the need for both boundary and domain discretizations. The possibility of eliminating the discretization of the boundary and the domain in PIES significantly reduces the number of input data necessary to define the boundary. In this case, the boundary is described by a small set of control points of surface patches. Three numerical examples were used to validate the solutions of PIES with analytical and numerical results available in the literature.
This paper presents a modification of the classical boundary integral equation method (BIEM) for two-dimensional potential boundary-value problem. The proposed modification consists in describing the boundary geometry by means of Hermite curves. As a result of this analytical modification of the boundary integral equation (BIE), a new parametric integral equation system (PIES) is obtained. The kernels of these equations include the geometry of the boundary. This new PIES is no longer defined on the boundary, as in the case of the BIE, but on the straight line for any given domain. The solution of the new PIES does not require boundary discretization as it can be reduced merely to an approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained compared with exact solutions.
The paper uses analytical modification of the classical boundary integral equations (BIEs) for the Helmholtz equation to facilitate the process of practical definition of the boundary geometry. Instead of defining the boundary by means of a boundary integral, the modification makes use of Bézier curves exclusively. As a result, a new parametric integral equation system (PIES) is obtained in which boundary geometry is taken into account in original fundamental boundary solutions. Such boundary definition makes it easy to approximate boundary functions. The proposed method to obtain numerical solution of the PIES for the Helmholtz equation is characterized by high effectiveness.
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