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Evaluation of the integral terms in reproducing kernel methods

机译：再现核方法中积分项的评估

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摘要

Reproducing kernel method (RKM) has its origins in wavelets and it is based on convolution theory. Being their continuous version, RKM is often referred as the general framework for meshless methods. In fact, since in real computation discretization is inevitable, these integrals need to be evaluated numerically, leading to the creation of reproducing kernel particle method RKPM and moving least squares MLS approximation. Nevertheless, in this paper the integrals in RKM are explicitly evaluated for polynomials basis function and simple geometries in one, two and three dimensions even with conforming holes. Moreover, a general formula is provided for complicated shapes also for multiple connected domains. This is possible through a boundary formulation where domain integrals involved in RKM are transformed by Gauss theorem in circular or flux integrals. Parallelization is readily enabled since no preliminary arrangements of nodes is needed for the moments matrix. Furthermore, using symbolic inversion, computation of shape functions in RKM is considerably speeded up.
机译：再生核方法（RKM）起源于小波，它基于卷积理论。作为其连续版本，RKM通常被称为无网格方法的通用框架。实际上，由于在实际计算中不可避免要进行离散化，因此需要对这些积分进行数值评估，从而导致创建了再生核粒子方法RKPM和移动最小二乘MLS近似。尽管如此，本文仍然明确评估了RKM中的积分的多项式基函数和一维，二维和三维的简单几何形状，甚至具有一致的孔。此外，还提供了用于多个连接域的复杂形状的通用公式。这可以通过边界公式来实现，其中RKM中涉及的域积分通过高斯定理以圆形或通量积分形式变换。并行化很容易实现，因为矩矩阵不需要节点的任何初步安排。此外，使用符号反演，可大大加快RKM中形状函数的计算速度。

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