论文标题
无网状近似和helmholtz-hodge矢量场的分解
Meshless Approximation and Helmholtz-Hodge Decomposition of Vector Fields
论文作者
论文摘要
对矢量场的分析对于理解几种物理现象，例如自然事件（例如，波浪分析），扩散过程，电气和电磁场至关重要。虽然先前的工作主要集中在体积或表面上的2D或3D向量场的分析上，但我们解决了对任意域上定义的向量场的无网格分析，而没有对其维度和离散化的假设。通过表达其成分作为径向基函数的线性组合并计算相应的保守，无旋转和谐波成分作为对A最小量或微分问题的解决方案，可以实现其成分作为线性组合的潜力，从而实现其旋转场的无网近似值。为此，我们确定了径向基础函数内核上的条件，以保证其衍生物的存在。最后，我们在通过传感器测量或通过模拟生成的2D和3D矢量场上演示了我们的方法。
The analysis of vector fields is crucial for the understanding of several physical phenomena, such as natural events (e.g., analysis of waves), diffusive processes, electric and electromagnetic fields. While previous work has been focused mainly on the analysis of 2D or 3D vector fields on volumes or surfaces, we address the meshless analysis of a vector field defined on an arbitrary domain, without assumptions on its dimension and discretisation. The meshless approximation of the Helmholtz-Hodge decomposition of a vector field is achieved by expressing the potential of its components as a linear combination of radial basis functions and by computing the corresponding conservative, irrotational, and harmonic components as solution to a least-squares or to a differential problem. To this end, we identify the conditions on the kernel of the radial basis functions that guarantee the existence of their derivatives. Finally, we demonstrate our approach on 2D and 3D vector fields measured by sensors or generated through simulation.
