SSPH basis functions for meshless methods, and comparison of solutions with strong and weak formulations

We propose a new and simple technique called the Symmetric Smoothed Particle Hydrodynamics (SSPH) method to construct basis functions for meshless methods that use only locations of particles. These basis functions are found to be similar to those in the Finite Element Method (FEM) except that the basis for the derivatives of a function need not be obtained by differentiating those for the function. Of course, the basis for the derivatives of a function can be obtained by differentiating the basis for the function as in the FEM and meshless methods. These basis functions are used to numerically solve two plane stress/strain elasto-static problems by using either the collocation method or a weak formulation of the problem defined over a subregion of the region occupied by the body; the latter is usually called the Meshless Local Petrov–Galerkin (MLPG) method. For the two boundary-value problems studied, it is found that the weak formulation in which the basis for the first order derivatives of the trial solution are derived directly in the SSPH method (i.e., not obtained by differentiating the basis function for the trial solution) gives the least value of the L²-error norm in the displacements while the collocation method employing the strong formulation of the boundary-value problem has the largest value of the L²-error norm. The numerical solution using the weak formulation requires more CPU time than the solution with the strong formulation of the problem. We have also computed the L²-error norm of displacements by varying the number of particles, the number of Gauss points used to numerically evaluate domain integrals appearing in the weak formulation of the problem, the radius of the compact support of the kernel function used to generate the SSPH basis, the order of complete monomials employed for constructing the SSPH basis, and boundary conditions used at a point on a corner of the rectangular problem domain. It is recommended that for solving two-dimensional elasto-static problems, the MLPG formulation in which derivatives of the trial solution are found without differentiating the SSPH basis function be adopted.