This manuscript provides a practical overview of meshless methods for solid mechanics, focusing on global weak forms. It includes a well-structured MATLAB code to illustrate the concepts, which is available for download. The code covers intrinsic and extrinsic enrichment, point collocation methods, boundary condition enforcement schemes, and test cases for one and two-dimensional elastostatic problems, including weak and strong discontinuities.
The paper is organized into several sections, covering basic approximations, kernel functions, completeness, partition of unity, intrinsic and extrinsic meshless methods, weighted residual methods, discrete equations for elastostatics, and error estimation and adaptivity. It discusses various methods such as smooth particle hydrodynamics (SPH), reproducing kernel particle method (RKPM), moving least squares (MLS) approximation, partition of unity finite element method (PUFEM), and hp-clouds. The implementation aspects, including efficient shape function computation, Gauss point generation, assembly procedures, and integration on essential boundaries, are also detailed.
The manuscript concludes with numerical examples and a discussion on the advantages and disadvantages of meshless methods, emphasizing their simplicity in incorporating $h$-adaptivity, handling moving discontinuities, and large deformations. It also addresses the challenges of essential boundary conditions and shape function derivatives, which are not straightforward in meshless methods compared to finite element methods.This manuscript provides a practical overview of meshless methods for solid mechanics, focusing on global weak forms. It includes a well-structured MATLAB code to illustrate the concepts, which is available for download. The code covers intrinsic and extrinsic enrichment, point collocation methods, boundary condition enforcement schemes, and test cases for one and two-dimensional elastostatic problems, including weak and strong discontinuities.
The paper is organized into several sections, covering basic approximations, kernel functions, completeness, partition of unity, intrinsic and extrinsic meshless methods, weighted residual methods, discrete equations for elastostatics, and error estimation and adaptivity. It discusses various methods such as smooth particle hydrodynamics (SPH), reproducing kernel particle method (RKPM), moving least squares (MLS) approximation, partition of unity finite element method (PUFEM), and hp-clouds. The implementation aspects, including efficient shape function computation, Gauss point generation, assembly procedures, and integration on essential boundaries, are also detailed.
The manuscript concludes with numerical examples and a discussion on the advantages and disadvantages of meshless methods, emphasizing their simplicity in incorporating $h$-adaptivity, handling moving discontinuities, and large deformations. It also addresses the challenges of essential boundary conditions and shape function derivatives, which are not straightforward in meshless methods compared to finite element methods.