迈向守恒SPH近似的高阶一致性与收敛性
Towards high-order consistency and convergence of conservative SPH approximations
AI总结：
本文提出反向核梯度修正（RKGC）公式，解决守恒SPH近似的高阶一致性问题，显著提升拉格朗日和欧拉SPH方法的精度，并缓解自由表面流模拟的高耗散问题。
摘要：
光滑粒子流体动力学（SPH）在模拟许多工程问题方面具有独特优势，但在其守恒公式中实现高阶一致性仍有待解决。尽管对于SPH梯度近似，可以通过粒子对差分和核梯度修正（KGC）方法分别获得零阶和高阶一致性，但由于其非守恒特性，它们在离散实际模拟中的守恒律时的适用性受到限制。虽然标准反对称SPH近似能够通过粒子松弛实现守恒的零阶一致性，但其与KGC的直接扩展既无法满足零阶也无法满足高阶一致性。本文提出了反向KGC（RKGC）公式，该公式是守恒的，并且当粒子基于KGC矩阵松弛时，能够精确满足零阶和一阶一致性。大量数值算例表明，新公式显著提高了拉格朗日SPH方法的精度。特别是，它能够解决模拟自由表面流中长期存在的高耗散问题。此外，在完全松弛的粒子下，即使光滑长度与粒子间距之比大幅减小，它也能提高欧拉SPH方法的精度。事实上，反向KGC公式具有扩展到更高阶一致性的潜力。然而，解决相应的粒子松弛问题仍是一个悬而未决的挑战。
英文摘要：
Smoothed particle hydrodynamics (SPH) offers distinct advantages for modeling many engineering problems, yet achieving high-order consistency in its conservative formulation remains to be addressed. While zero- and higher-order consistencies can be obtained using particle-pair differences and the kernel gradient correction (KGC) approaches, respectively, for SPH gradient approximations, their applicability for discretizing conservation laws in practical simulations is limited due to their non-conservative feature. Although the standard anti-symmetric SPH approximation is able to achieve conservative zero-order consistency through particle relaxation, its straightforward extensions with the KGC fail to satisfy either zero- or higher-order consistency. In this paper, we propose the reverse KGC (RKGC) formulation, which is conservative and able to precisely satisfy both zero- and first-order consistencies when particles are relaxed based on the KGC matrix. Extensive numerical examples show that the new formulation considerably improves the accuracy of the Lagrangian SPH method. In particular, it is able to resolve the long-standing high-dissipation issue for simulating free-surface flows. Furthermore, with fully relaxed particles, it enhances the accuracy of the Eulerian SPH method even when the ratio between the smoothing length and the particle spacing is considerably reduced. Indeed, the reverse KGC formulation holds the potential for the extension to even higher-order consistencies. However, addressing the corresponding particle relaxation problem remains a pending challenge.
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