Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) is one variant of SPH where the variables are described using the fixed reference configuration and a Lagrangian smoothing kernel. TLSPH elevates the computational efficiency of the standard SPH when no topological change is involved, and it alleviates the stability of SPH scheme with respect to tensile loading. However, instabilities associated with spurious mode, or hourglass/zero-energy mode, persists and often affects the simulation of solids undergoing extremely large strain. This work proposes an alternative formulation to compute deformation gradient with improved accuracy and therefore minimising the possibility of encountering the zero-energy mode. Specifically, we leverage the local discrete computation of bond-based (or pairwise) deformation gradient smoothed by the kernel. Additionally, the bond of a particle with itself is considered to preserve the polynomial reproducibility imposed by the kernel correction scheme. We showcase the convergence of the approach using a two-dimensional benchmark example. Furthermore, the accuracy, robustness, and stability of the proposed approach are assessed in various two- and three-dimensional examples, highlighting on the stability improvement that allows for solid dynamic simulations with more extreme elongation.

翻译：全拉格朗日光滑粒子流体动力学（TLSPH）是SPH的一种变体，其变量使用固定的参考构型和拉格朗日光滑核函数进行描述。在不涉及拓扑变化时，TLSPH提升了标准SPH的计算效率，并改善了SPH方案在拉伸载荷下的稳定性。然而，与伪模式或沙漏/零能模式相关的不稳定性仍然存在，并常常影响经历极大应变的固体模拟。本研究提出了一种计算变形梯度的替代公式，该公式具有更高的精度，从而最大限度地减少了遭遇零能模式的可能性。具体而言，我们利用基于键（或成对）变形梯度的局部离散计算，并通过核函数进行光滑化。此外，考虑了粒子与其自身的键，以保持由核函数修正方案所施加的多项式可再现性。我们通过一个二维基准算例展示了该方法的收敛性。此外，在多个二维和三维算例中评估了所提方法的精度、鲁棒性和稳定性，重点突出了其稳定性改进，使得能够对具有更极端伸长率的固体动力学进行模拟。