新型节点到边有限元与传统节点单元在电磁分析中的性能比较
Comparative Performance of Novel Nodal-to-Edge finite elements over Conventional Nodal element for Electromagnetic Analysis
摘要：
在基于节点的有限元法（FEM）中，自由度与单元的节点相关联，而对于边缘FEM，自由度被分配给单元的边缘。基于Whitney空间构造了边元。节点元素在界面边界上施加矢量或标量场的切向和法向连续性。但在棱边单元中，只有切向连续性被施加在界面边界上，这与电磁场问题是一致的。因此，在电磁分析中所需的连续性是直接获得与边缘元素，而在节点元素，他们通过潜在的配方。因此，当使用边缘单元时，场变量被直接计算，但是使用节点单元，需要后处理以从电势获得场变量。这里，我们提出了有限元列式，包括棱边元和节点元。此后，我们已经证明了不同的节点和边缘元素的相对性能，通过一系列的例子。所有可能的复杂性，如曲线边界，非凸区域，尖角，非均匀域已在这些例子中解决。在分析中，边缘元素在预测具有尖锐棱角的域的奇异特征值方面的鲁棒性是显而易见的。一个更好的粗网格精度已被观察到的所有边缘元素相比，各自的节点元素。边元素也不易受网格变形的影响。
英文摘要：
In nodal based finite element method (FEM), degrees of freedom are associated with the nodes of the element whereas, for edge FEM, degrees of freedom are assigned to the edges of the element. Edge element is constructed based on Whitney spaces. Nodal elements impose both tangential and normal continuity of vector or scalar fields across interface boundaries. But in edge elements only tangential continuity is imposed across interface boundaries, which is consistent with electromagnetic field problems. Therefore, the required continuities in the electromagnetic analysis are directly obtained with edge elements, whereas in nodal elements they are attained through potential formulations. Hence, while using edge elements, field variables are directly calculated, but with nodal elements, post-processing is required to obtain the field variables from the potentials. Here, we present the finite element formulations with the edge element as well as with nodal elements. Thereafter, we have demonstrated the relative performances of different nodal and edge elements through a series of examples. All possible complexities like curved boundaries, non-convex domains, sharp corners, non-homogeneous domains have been addressed in those examples. The robustness of edge elements in predicting the singular eigen values for the domains with sharp edges and corners is evident in the analysis. A better coarse mesh accuracy has been observed for all the edge elements as compared to the respective nodal elements. Edge elements are also not susceptible to mesh distortion.
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