Mixed strong form representation particle method for solids and structures

In this paper, a generalized particle system (GPS) method, a general method to describe multiple strong form representation based particle methods is described. Gradient, divergence, and Laplacian operators used in various strong form based particle method such as moving particle semi-implicit (MPS) method, smooth particle hydrodynamics (SPH), and peridynamics, can be described by the GPS method with proper selection of parameters. In addition, the application of mixed formulation representation to the GPS method is described. Based on Hu- Washizu principle and Hellinger-Reissner principle, the mixed form refers to the method solving multiple primary variables such as displacement, strain and stress, simultaneously in the FEM method; however for convenience in employing FEM with particle methods, a simple representation in construction only is shown. It is usually applied to finite element method (FEM) to overcome numerical errors including locking issues. While the locking issues do not arise in strong form based particle methods, the mixed form representation in construction only concept applied to GPS method can be the first step for fostering coupling of multi-domain problems, coupling mixed form FEM and mixed form representation GPS method; however it is to be noted that the standard GPS particle method and the mixed for representation construction GPS particle method are equivalent. Two dimensional simple bar and beam problems are presented and the results from mixed form GPS method is comparable to the mixed form FEM results.