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.

我们介绍 SPH 的基本概念，特别强调其在地质力学和岩土工程中的最新应用。在本文的第一部分，我们重点建立基本的 SPH 方程并讨论如何使用它们来求解地质力学中的偏微分方程 (PDE)。通过这个过程，我们希望让读者更好地了解 SPH 配方，以避免对其容量和局限性的误用或误解。还介绍了对几个悬而未决的问题的讨论和进一步发展的建议。通过对关键 SPH 概念的重新审视，特别令人感兴趣的是拉普拉斯算子的一​​种新的、稳健的 SPH 近似公式，它涉及场量的二阶导数。事实证明，这种新配方优于现有的 SPH 配方并实现了高精度。论文的第二部分侧重于通过各种示例展示 SPH 在地质力学和岩土工程领域的应用，从最基本的应用到涉及多相流的更复杂的应用。我们希望本文将成为有用的资源，让读者更好地了解 SPH 及其在解决地质力学和岩土工程中的复杂问题方面的潜力。