This paper presents a new Smooth Particle Hydrodynamics computational framework for the solution of inviscid free surface flow problems. The formulation is based on the Total Lagrangian description of a system of first-order conservation laws written in terms of the linear momentum and the Jacobian of the deformation. One of the aims of this paper is to explore the use of Total Lagrangian description in the case of large deformations but without topological changes. In this case, the evaluation of spatial integrals is carried out with respect to the initial undeformed configuration, yielding an extremely efficient formulation where the need for continuous particle neighbouring search is completely circumvented. To guarantee stability from the SPH discretisation point of view, consistently derived Riemann-based numerical dissipation is suitably introduced where global numerical entropy production is demonstrated via a novel technique in terms of the time rate of the Hamiltonian of the system. Since the kernel derivatives presented in this work are fixed in the reference configuration, the non-physical clumping mechanism is completely removed. To fulfil conservation of the global angular momentum, a posteriori (least-squares) projection procedure is introduced. Finally, a wide spectrum of dedicated prototype problems is thoroughly examined. Through these tests, the SPH methodology overcomes by construction a number of persistent numerical drawbacks (e.g. hour-glassing, pressure instability, global conservation and/or completeness issues) commonly found in SPH literature, without resorting to the use of any ad-hoc user-defined artificial stabilisation parameters. Crucially, the overall SPH algorithm yields equal second order of convergence for both velocities and pressure.

本文提出了一种新的光滑粒子流体动力学计算框架，用于解决无粘性自由表面流问题。该公式基于一阶守恒律系统的总拉格朗日描述，该定律以线性动量和变形的雅可比公式表示。本文的目的之一是探索在变形较大但没有拓扑变化的情况下使用总拉格朗日描述的方法。在这种情况下，将对初始未变形构型进行空间积分评估，从而获得一种极为有效的公式，其中完全避免了对连续粒子相邻搜索的需求。为了从SPH离散化角度保证稳定性，适当地引入基于一致的派生的基于Riemann的数值耗散，其中通过新技术根据系统的哈密顿量的时间率来证明全局数值熵的产生。由于本文中介绍的内核导数已固定在参考配置中，因此非物理成簇机制已被完全消除。为了实现全局角动量的守恒，引入了后验（最小二乘）投影程序。最后，彻底检查了各种专用原型问题。通过这些测试，SPH方法通过构造克服了SPH文献中常见的许多持久的数值缺陷（例如，沙漏，压力不稳定，整体守恒和/或完整性问题），无需使用任何临时的用户定义的人工稳定参数。至关重要的是，总体SPH算法在速度和压力方面都产生相等的二阶收敛性。