We present an open-source software RKPM2D for solving PDEs under the reproducing kernel particle method (RKPM)-based meshfree computational framework. Compared to conventional mesh-based methods, RKPM provides many attractive features, such as arbitrary order of continuity and discontinuity, relaxed tie between the quality of the discretization and the quality of approximation, simple h-adaptive refinement, and ability to embed physics-based enrichment functions, among others, which make RKPM promising for solving challenging engineering problems. The aim of the present software package is to support reproducible research and serve as an efficient test platform for further development of meshfree methods. The RKPM2D software consists of a set of data structures and subroutines for discretizing two-dimensional domains, nodal representative domain creation by Voronoi diagram partitioning, boundary condition specification, reproducing kernel shape function generation, domain integrations with stabilization, a complete meshfree solver, and visualization tools for post-processing. In this paper, a brief overview that covers the key theoretical aspects of RKPM is given, such as the reproducing kernel approximation, weak form using Nitsche’s method for boundary condition enforcement, various domain integration schemes (Gauss quadrature and stabilized nodal integration methods), as well as the fully discrete equations. In addition, the computer implementation aspects employed in RKPM2D are discussed in detail. Benchmark problems solved by RKPM2D are presented to demonstrate the convergence, efficiency, and robustness of the RKPM implementation.

中文翻译：

---

RKPM2D：求解偏微分方程的节点积分复制核粒子方法的开源实现

我们提出了一个开源软件RKPM2D，用于在基于再生核粒子法（RKPM）的无网格计算框架下求解PDE。与传统的基于网格的方法相比，RKPM提供了许多吸引人的功能，例如连续性和不连续性的任意顺序，离散化质量和近似质量之间的轻松联系，简单的h自适应的改进以及嵌入基于物理的扩展功能的能力，这使得RKPM在解决具有挑战性的工程问题方面很有前途。本软件包的目的是支持可重现的研究，并作为进一步开发无网格方法的有效测试平台。RKPM2D软件由一组用于离散化二维域的数据结构和子例程，通过Voronoi图分区创建节点代表域，边界条件指定，再现内核形状函数生成，具有稳定作用的域集成，完整的无网格求解器以及可视化组成后处理工具。本文对RKPM的关键理论方面进行了简要概述，例如再现核逼近，弱形式，使用Nitsche方法执行边界条件，采用各种域积分方案（高斯积分和稳定节点积分方法），以及完全离散的方程。此外，还将详细讨论RKPM2D中采用的计算机实现方面。提出了RKPM2D解决的基准问题，以证明RKPM实现的收敛性，效率和健壮性。