Bézier curves in the modification of boundary integral equations (BIE) for potential boundary-values problems

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

“…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. The expression resulting from the modification of the traditional BIE for the Laplace's and Helmholtz equations is called the parametric integral equation system (PIES) as the boundary geometry in the PIES kernels has been analytically defined in a parametric way (Zieniuk, 2003).…”

“…The solution of the PIES in a direct way is independent of the shape of the boundary geometry as the boundary is defined analytically in the PIES mathematical formalism (Zieniuk, 2003). In other words, in PIES the approximation of boundary geometry is independent of the approximation of boundary functions.…”

“…The number of these points is considerably smaller than the number of nodes in the case of the traditional FEM. For curvilinear boundary shapes we use Bézier curves (Zieniuk, 2003) to obtain their continuous definition.…”

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.…”

“…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. The shape of the boundary could be defined by such curves as Bézier [15], Hermite [16] and B-spline [17], and its definition is practically reduced to giving a small set of control points. The complexity of modeling the shape of the boundary in PIES depends on the complexity of the shape of the concerned boundary problem.…”