Tensile instability, often observed in smoothed particle hydrodynamics (SPH), is a numerical artifact that manifests itself by unphysical clustering or separation of particles. The instability originates in estimating the derivatives of the smoothing functions which, when interact with material constitution may result in negative stiffness in the discretized system. In the present study, a stable formulation of SPH is developed where the kernel function is continuously adapted at every material point depending on its state of stress. Bspline basis function with a variable intermediate knot is used as the kernel function. The shape of the kernel function is then modified by changing the intermediate knot position such that the condition associated with instability does not arise. While implementing the algorithm the simplicity and computational efficiency of SPH are not compromised. One-dimensional dispersion analysis is performed to understand the effect adaptive kernel on the stability. Finally, the efficacy of the algorithm is demonstrated through some benchmark elastic dynamics problems.

中文翻译：

---

具有自适应 B 样条内核的稳定 SPH

通常在平滑粒子流体动力学 (SPH) 中观察到的拉伸不稳定性是一种数值伪影，它通过粒子的非物理聚类或分离表现出来。不稳定性源于估计平滑函数的导数，当与材料构成相互作用时，可能会导致离散系统中的负刚度。在本研究中，开发了 SPH 的稳定公式，其中核函数根据其应力状态在每个材料点连续调整。具有可变中间节点的 Bspline 基函数用作核函数。然后通过改变中间节点位置来修改核函数的形状，从而不会出现与不稳定性相关的条件。在实现算法的同时，SPH 的简单性和计算效率没有受到影响。进行一维色散分析以了解自适应核对稳定性的影响。最后，通过一些基准弹性动力学问题证明了该算法的有效性。