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Abstract
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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
package optnodes.
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Keywords
finite elements, optimal nodal distribution, Lebesgue
constant
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Mathematical Subject Classification
Primary: 65N30
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Milestones
Received: 3 October 2024
Revised: 7 February 2025
Accepted: 3 March 2025
Published: 10 April 2025
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© 2025 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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