Abstract The nonlocal operator method (NOM) is initially proposed as a particle-based method, which has difficulties in imposing accurately the boundary conditions of various orders. In this paper, we converted the particle-based NOM into a scheme with approximation property. The new scheme describes partial derivatives of various orders at a point by the nodes in the support and takes advantage of the background mesh for numerical integration. The boundary conditions are enforced via the modified variational principle. The particle-based NOM can be viewed as a special case of NOM with approximation property when nodal integration is used. The scheme based on numerical integration greatly improves the stability of the method. As a consequence, the requirement of the operator energy functional in particle-based NOM is avoided. We demonstrate the capabilities of the proposed method by solving gradient elasticity problems and comparing the numerical results with exact solutions.

中文翻译：

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梯度实体数值积分的非局部算子方法

摘要 非局部算子法（NOM）最初是作为一种基于粒子的方法而提出的，它难以准确施加各种阶次的边界条件。在本文中，我们将基于粒子的 NOM 转换为具有近似属性的方案。新方案描述了支撑中节点在一个点上的各种阶次的偏导数，并利用背景网格进行数值积分。边界条件通过修改后的变分原理强制执行。当使用节点积分时，基于粒子的 NOM 可以看作是具有近似属性的 NOM 的特例。基于数值积分的方案大大提高了方法的稳定性。因此，避免了在基于粒子的 NOM 中对算子能量泛函的要求。