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A boundary cloud method with a cloud-by-cloud polynomial basis

机译：基于逐个云​​多项式的边界云方法

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摘要

We have recently presented a boundary cloud method (BCM) [Comput. Meth. Appl. Mech. Engng 191 (2002) 2337], which combines boundary integral formulations with scattered point interpolation techniques. A generalized least-squares approach, which requires information about the outward normal to the boundary, is employed to construct interpolation functions. Since an outward normal is not well defined for geometries with corners for 2D problems (or for corners and edges for 3D problems), points could not be placed at corners when discretizing the surface of the object. In this paper, we introduce a new implementation of the BCM, which uses a varying base interpolating polynomial to construct interpolation functions. The key idea is to define an appropriate polynomial basis which ensures linear completeness. The polynomial basis can change from cloud to cloud depending on the definition of the cloud at each point. The new implementation can handle points at corners and is much simpler and at least an order of magnitude faster compared to our original implementation. The original implementation can be more accurate and can give higher order convergence rates, but is limited because it cannot handle points at corners. Numerical results comparing the original and the new implementation are shown for several potential and electrostatic problems.
机译：我们最近提出了一种边界云方法（BCM）[Comput。方法应用机甲Engng 191（2002）2337]，它结合了边界积分公式和散点插值技术。采用通用最小二乘法来构造插值函数，该方法需要有关边界的向外法线的信息。由于对于带有2D问题的拐角的几何图形（或对于3D问题的拐角和边缘的几何图形），不能很好地定义向外的法线，因此在离散化对象的表面时无法将点放置在拐角处。在本文中，我们介绍了BCM的新实现，该实现使用可变基数插值多项式构造插值函数。关键思想是定义一个适当的多项式基础，以确保线性完整性。取决于每个点上云的定义，多项式基础可以在云之间变化。与我们的原始实现相比，新的实现可以处理拐角处的点，并且更简单并且至少快一个数量级。原始实现可能更准确，并且可以提供更高的阶收敛速度，但是由于它无法处理拐角处的点而受到限制。数值结果比较了原始方法和新方法，显示了一些潜在的问题和静电问题。

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