Normalized SPH without boundary deficiency and its application to transient solid mechanics problems

Smoothed particle hydrodynamics (SPH) method is a powerful tool for modeling solid mechanics problems, especially for large deformation problems. However, it suffers from boundary deficiency and difficulty of boundary condition treatment. In this work, a normalized SPH method is proposed to overcome these problems. The method is based on a newly developed normalized particle approximation. To derive this particle approximation, a normalized kernel approximation which is accurate for derivatives of linear functions everywhere in a problem domain is constructed, and all integral terms of the normalized kernel approximation including boundary terms are discretized by particle summations. The normalized particle approximation is free of matrix inversion, consequently attractive in computational stability and simplicity compared with other corrective particle approximations. Its approximation accuracy is demonstrated by calculating derivatives of test functions. Based on this particle approximation, the formulation of the normalized SPH method for transient solid mechanics problems is derived. Moreover, a direct method of treating traction boundary conditions is presented by making use of the boundary term of the normalized particle approximation. The accuracy and capability of the normalized SPH method are validated by the calculation of elastic wave propagation in solids and compared with commonly used SPH method.