This paper is an extended version of the ICCS conference paper (Kapturczak et al., 2020, [1]) and presents a way to improve the boundary shape modeling process in solving boundary value problems in elasticity. The inclusion of NURBS curves into the mathematical formalism of the parametric integral equations system method (PIES) is proposed. The advantages of such an application are widely discussed. Recently, the Bezier curves, mainly the cubic curves (of third-degree), were used. The segments of the boundary shape were modeled by such curves (with ensuring continuity at the connection points). Using NURBS curves, the boundary shape can be modeled with only one curve. So, continuity is automatically ensured. Additionally, the second degree NURBS curve is enough to obtain the shape with high accuracy (better than cubic Bezier curves). The NURBS curve is defined by points, their weights, and the knots vector. Such parameters significantly improve the shape modification process, which can directly improve e.g. the shape identification process. The examples of shape modification using such parameters are presented. The boundary shapes of the examples (even defined by both linear and curvilinear segments) can be defined using only one closed NURBS curve. The impact of modeling accuracy on the final PIES solutions is examined on examples described by the Navier–Lamé equations. To improve calculations, the PIES method using NURBS curves was implemented as a computer program. Then, it was decided to verify the accuracy of the obtained solutions. For comparison, the solutions were also obtained using analytical solutions, boundary element method, and PIES method (with the Bezier curves). An improvement in the boundary shape modeling was noticed. It significantly affects the accuracy of solutions. As a result, the consumption of computer resources was reduced, while the process of boundary shape modeling and the accuracy of the obtained results were improved.

本文是ICCS会议论文的扩展版本（Kapturczak等，2020，[1]），提出了一种改进边界形状建模过程以解决弹性边界值问题的方法。提出将NURBS曲线包含在参数积分方程系统方法（PIES）的数学形式中。广泛讨论了这种应用程序的优点。最近，使用了贝塞尔曲线，主要是三次曲线（三次）。边界形状的线段通过此类曲线建模（并确保连接点的连续性）。使用NURBS曲线，边界形状只能用一条曲线建模。因此，自动确保连续性。此外，二阶NURBS曲线足以获得高精度的形状（优于三次贝塞尔曲线）。NURBS曲线由点，其权重和结矢量定义。这样的参数显着改善了形状修改过程，其可以直接改善例如形状识别过程。给出了使用此类参数进行形状修改的示例。可以仅使用一条闭合的NURBS曲线来定义示例的边界形状（甚至由线性和曲线段定义）。在Navier–Lamé方程描述的示例中检查了建模精度对最终PIES解决方案的影响。为了改进计算，使用NURBS曲线的PIES方法被实现为计算机程序。然后，决定验证所获得解决方案的准确性。为了进行比较，还使用解析解，边界元法，和PIES方法（使用Bezier曲线）。注意到边界形状建模方面的改进。它会严重影响解决方案的准确性。结果，减少了计算机资源的消耗，同时提高了边界形状建模的过程和所获得结果的准确性。