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.

中文翻译：

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无边界缺陷的归一化 SPH 及其在瞬态固体力学问题中的应用

光滑粒子流体动力学 (SPH) 方法是建模固体力学问题的有力工具，尤其是大变形问题。但是，它存在边界不足和边界条件处理困难的问题。在这项工作中，提出了一种归一化的 SPH 方法来克服这些问题。该方法基于新开发的归一化粒子近似。为了导出这个粒子近似，构造了一个对问题域中任何地方的线性函数的导数都是准确的归一化核近似，并且归一化核近似的所有积分项包括边界项都通过粒子总和离散化。归一化粒子近似没有矩阵求逆，因此与其他校正粒子近似相比，在计算稳定性和简单性方面具有吸引力。它的近似精度通过计算测试函数的导数来证明。基于这种粒子近似，推导出了瞬态固体力学问题的归一化 SPH 方法的公式。此外，通过使用归一化粒子近似的边界项，提出了一种处理牵引边界条件的直接方法。通过计算固体中的弹性波传播并与常用的 SPH 方法进行比较，验证了归一化 SPH 方法的准确性和能力。推导了瞬态固体力学问题的归一化 SPH 方法的公式。此外，通过使用归一化粒子近似的边界项，提出了一种处理牵引边界条件的直接方法。通过计算固体中的弹性波传播并与常用的 SPH 方法进行比较，验证了归一化 SPH 方法的准确性和能力。推导了瞬态固体力学问题的归一化 SPH 方法的公式。此外，通过使用归一化粒子近似的边界项，提出了一种处理牵引边界条件的直接方法。通过计算固体中的弹性波传播并与常用的 SPH 方法进行比较，验证了归一化 SPH 方法的准确性和能力。