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A fast object-oriented MATLAB implementation of the reproducing kernel particle method. (English) Zbl 1398.74450
Summary: Novel numerical methods, known as Meshless Methods or Meshfree Methods and, in a wider perspective, Partition of Unity Methods, promise to overcome most of disadvantages of the traditional finite element techniques. The absence of a mesh makes meshfree methods very attractive for those problems involving large deformations, moving boundaries and crack propagation. However, meshfree methods still have significant limitations that prevent their acceptance among researchers and engineers, namely the computational costs. This paper presents an in-depth analysis of computational techniques to speed-up the computation of the shape functions in the Reproducing Kernel Particle Method and Moving Least Squares, with particular focus on their bottlenecks, like the neighbour search, the inversion of the moment matrix and the assembly of the stiffness matrix. The paper presents numerous computational solutions aimed at a considerable reduction of the computational times: the use of \(kd-\)trees for the neighbour search, sparse indexing of the nodes-points connectivity and, most importantly, the explicit and vectorized inversion of the moment matrix without using loops and numerical routines.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
Software:
ESFLIB; Matlab; Mfree2D
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