Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials

Abstract

In this article, we present a strategy of using rectangular and triangular Bézier surface patches for nonelement representation of 3D boundary geometries for problems of linear elasticity. The boundary generated in this way is directly incorporated in the parametric integral equation system (PIES), which has been developed by the authors. The boundary values on each surface patch are approximated by Lagrange polynomials. Three illustrative examples are presented to confirm the effectiveness of the proposed boundary representation in connection with PIES and to show good accuracy of numerical results.

On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

“…The proposed method keeps the simplicity of the standard BEM and is able to solve more challenging problems with complex geometry shape. Future developments on this topic include its extension to both purely 3D geometry (see Reference [23]) and multi-physics problems (e.g. elasticity, heat transfer, coupled problems).…”

“…This approach was successfully used before the isogeometric development. The works of Zieniuk [21], Zieniuk and Boltuc [22], and Zieniuk and Szerszeń [23], among others, proposed the concept of parametric integral equation system (PIES) for accurate representation of boundary geometry. In these works, the boundary geometry was described as a parametric function that was included in the PIES kernels, rather than in the boundary integral as is done in standard BEM.…”

An accurate treatment of non-homogeneous boundary conditions for development of the BEM

“…Hence, many authors have conducted improvements related to geometry representation and field approximation. These studies covered the parametric representation of the geometry [13,14], the isogeometric approach [15,16], and the spectral formulations [17][18][19][20]. These publications have led to progress in the formulation of the BEM.…”