A new integral identity for potential polygonal domain problems described by parametric linear functions

Non-element method of solving 2D boundary problems defined on polygonal domains modeled by Navier equation

“…In the research (Zieniuk, 2001(Zieniuk, , 2002 carried out by the author, a new approach to boundary problems has been proposed: an approach that would no longer require traditional boundary discretization or at least, eliminate its use to minimum. To make it possible, it is necessary to analytically modify the traditional boundary integral equation (BIE), which, in general, consists in getting rid of the necessity of defining the shape of the boundary by means of boundary integral.…”

“…Potentially, it creates greater possibilities of choosing more effective ways of both geometry modeling and choosing more effective methods of the approximation of the boundary functions. Thus, to practically define, for example, a polygonal domain, only corner points are posed (Zieniuk, 2002). The number of these points is considerably smaller than the number of nodes in the case of the traditional FEM.…”

Triangular Bézier surface patches in modeling shape of boundary geometry for potential problems in 3D

“…Presented briefly, the modification of traditional BIE is considered as a generalization of the modification applied to 2D problems as in [10][11][12][13][14][15][16][17]. In general, it consists in analytically defining curvilinear boundary geometry in traditional BIE with the help of triangular Bézier surface patches.…”

“…To find a solution in the domain, we need to obtain an integral identity known for BIE that makes use of the solution on the boundary obtained by PIES. After using similar modifications as in the case of 2D problems [10,11], we have had an integral identity which used solutions (23) and (24) at the boundary, previously obtained by the PIES solution. The modified identity takes the following form: …”

“…So far, however, PIES has been mainly used to solve 2D potential boundary value problems modeled by partial differential equations such as: Laplace [10,11], Poisson [12], Helmholtz [13] and Navier-Lame [14]. These equations have been written in the alternative form, with the help of PIES, which takes into account in its mathematical formalism the shape of the boundary modeled by curves known from computer graphics.…”