RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations. 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.

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  1. Huang, Tsung-Hui: A variational multiscale stabilized and locking-free meshfree formulation for Reissner-Mindlin plate problems (2022)
  2. Huang, Tsung-Hui; Chao, Chia-Lien: A stabilized one-point integrated mixed formulation for finite element and meshfree methods in modeling nearly incompressible materials (2022)
  3. Huang, Tsung-Hui; Chen, Jiun-Shyan; Tupek, Michael R.; Beckwith, Frank N.; Fang, H. Eliot: A variational multiscale immersed meshfree method for fluid structure interactive systems involving shock waves (2022)
  4. Huang, Tsung-Hui; Chen, Jiun-Shyan; Tupek, Michael R.; Beckwith, Frank N.; Koester, Jacob J.; Fang, H. Eliot: A variational multiscale immersed meshfree method for heterogeneous materials (2021)
  5. Li, Chen: A partial differential equation-based image restoration method in environmental art design (2021)
  6. Neofytou, Andreas; Huang, Tsung-Hui; Kambampati, Sandilya; Picelli, Renato; Chen, Jiun-Shyan; Kim, H. Alicia: Level set topology optimization with nodally integrated reproducing kernel particle method (2021)
  7. Pasetto, Marco; Baek, Jonghyuk; Chen, Jiun-Shyan; Wei, Haoyan; Sherburn, Jesse A.; Roth, Michael J.: A Lagrangian/semi-Lagrangian coupling approach for accelerated meshfree modelling of extreme deformation problems (2021)
  8. Qian, Zhihao; Wang, Lihua; Gu, Yan; Zhang, Chuanzeng: An efficient meshfree gradient smoothing collocation method (GSCM) using reproducing kernel approximation (2021)
  9. Wang, Lihua; Liu, Yijia; Zhou, Yueting; Yang, Fan: A gradient reproducing kernel based stabilized collocation method for the static and dynamic problems of thin elastic beams and plates (2021)