RKPM-based smoothed GFEM with Kronecker-Delta property for 2D and 3D solid problems

We propose a reproducing kernel particle method-based smoothed generalized finite element method (RKPM-SGFEM) for 2D and 3D structural analysis. As with partition of unity idea, the displacement function in RKPM-SGFEM is discretized as finite element shape function and local approximation, where the local approximation is obtained by Taylor truncation in nodal support domain. The gradient smoothing reproducing kernel (RK) meshfree approximation is utilized for the derivatives of Taylor polynomials. It is well known that RKPM does not possess Kronecker-Delta property. However, this defect of RKPM is suppressed under the proposed RKPM-SGFEM framework, because of the novel combination of element shape function and Taylor expansion. In addition, the final composite shape function also ensures partitions of unity, which is not affected by the meshfree shape function. Subsequently, we performed a series of numerical tests on the proposed RKPM-SGFEM, which not only passes linear independent and zero energy modal tests, but also shows higher accuracy, error convergence speed, and efficiency in numerical analysis of 2D and 3D problems. Besides, numerical verification also indicates that RKPM-SGFEM is insensitive to mesh distortion, temporally stable, and highly consistent with experimental test in practical engineering analysis.