Smoothed particle hydrodynamics (SPH) and its applications in geomechanics: from solid fracture to granular behaviour and multiphase flows in porous media

We present the fundamental concepts of SPH with particular emphasis on its state-of-the-art applications in geomechanics and geotechnical engineering. In the first part of the paper, we focus on establishing fundamental SPH equations and discussing how they are used to solve partial differential equations (PDEs) in geomechanics. Through this process, we expect to provide readers with a better understanding of SPH formulations to avoid misuse or misinterpretation of its capacity and limitation. Discussions on several outstanding issues and recommendations for further developments are also be presented. Of particular interest through this revisit of the key SPH concepts is a new and robust SPH approximation formulation for the Laplacian, which involves the second-order derivatives of a field quantity. This new formulation is proven to outperform existing SPH formulations and achieve high accuracy. The second part of the paper focuses on demonstrating the applications of SPH in the fields of geomechanics and geotechnical engineering through various examples, ranging from the most fundamental to more complex applications involving multi-phase flows. We hope that this paper will become a useful resource to provide readers with a better understanding of SPH and its potential in solving complex problems in geomechanics and geotechnical engineering.