Symmetric, Optimization-based, Cross-element Compatible Nodal Distributions for High-order Finite Elements

In high-order and high-dimensional finite elements, ill-conditioned nodal distributions are often computationally cost-prohibitive. As a result, uniform distributions quickly fall apart. For tensor-product like elements, Gauss-Legendre-Lobatto (GLL) nodal distributions are often used as a substitute. Besides these, other efficient nodal distributions are difficult to create due to a desired symmetry within elements and conformity with neighboring elements. In this paper, we provide a general framework to construct symmetric, well-conditioned, cross-element compatible nodal distributions which can be used for high-order and high-dimensional finite elements. Starting from the inherent symmetries in any potential element, the framework is used to build up nodal groups in a structured and efficient manner utilizing the natural coordinates of each element, while ensuring nodes stay within the elements. By constructing constrained symmetry groups, the vertices, edges, and faces, of all elements are required to conform to their respective lower-dimensional distributions. Optimizing over these groups yields the desired optimized nodal distributions. We demonstrate the strength of this framework by creating and comparing optimized nodal distributions with GLL distributions (in elements such as the line, quadrilateral, and hexahedron), and its robustness by generating optimized nodal distributions for otherwise difficult elements (such as the triangle, tetrahedron, and triangular prism).