Abstract

We present a general framework to construct symmetric, well-conditioned, cross-element compatible nodal distributions that can be used for high-order and high-dimensional finite elements. Starting from the inherent symmetries of an element geometry, we construct node groups in a systematic and efficient manner utilizing the natural coordinates of each element, while ensuring nodes stay within the elements. Proper constraints on the symmetry group lead to nodal distributions that ensure cross-element compatibility (i.e., nodes of adjacent elements are co-located) on both homogeneous and mixed meshes. The final nodal distribution is defined as a minimizer of an optimization problem over symmetry group parameters with linear constraints that ensure nodes remain with an element and enforce other properties (e.g., cross-element compatibility). We demonstrate the merit of this framework by comparing the proposed optimization-based nodal distributions with other popular distributions available in the literature, and its robustness by generating optimized nodal distributions for otherwise difficult elements (such as simplex and pyramid elements). All nodal distributions are tabulated in the $optnodes$ package optnodes.

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