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

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