利用移动最小二乘和径向基函数增强SPH
Enhancing SPH using moving least-squares and radial basis functions
AI总结：
本文通过移动最小二乘近似和径向基函数改进粒子近似的精度，提出变分一致的水动力学方程，并在Sod激波管问题中验证性能。
摘要：
本文考虑了通过提高粒子近似精度来增强无网格拉格朗日粒子方法——光滑粒子流体动力学（SPH）的两种途径。具体而言，我们将考虑使用移动最小二乘近似（MLS）和径向基函数（RBF）构造的形状函数。使用MLS近似具有吸引力，因为可以强制执行粒子近似的多项式一致性。RBF更具吸引力，因为它们允许省去光滑长度——SPH方法中控制形状函数支撑域内粒子数量的参数。目前，只有选择光滑长度的经验方法。我们确保任何增强都保留SPH的保守性和无网格特性。为此，我们推导了一组新的变分一致的水动力学方程。最后，我们在Sod激波管问题上展示了新方程的性能。
英文摘要：
In this paper we consider two sources of enhancement for the meshfree Lagrangian particle method smoothed particle hydrodynamics (SPH) by improving the accuracy of the particle approximation. Namely, we will consider shape functions constructed using: moving least-squares approximation (MLS); radial basis functions (RBF). Using MLS approximation is appealing because polynomial consistency of the particle approximation can be enforced. RBFs further appeal as they allow one to dispense with the smoothing-length -- the parameter in the SPH method which governs the number of particles within the support of the shape function. Currently, only ad hoc methods for choosing the smoothing-length exist. We ensure that any enhancement retains the conservative and meshfree nature of SPH. In doing so, we derive a new set of variationally-consistent hydrodynamic equations. Finally, we demonstrate the performance of the new equations on the Sod shock tube problem.
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