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

Julian M. Kaufmann, Matthew J. Zahr

TL;DR

The paper addresses ill-conditioning of high-order finite-element nodal distributions by introducing a general optimization-based, symmetry-driven framework that constructs cross-element–compatible nodal distributions from intrinsic symmetry orbits. It formalizes an abstract formulation linking natural coordinates, symmetry orbits, and linear constraints, and instantiates it for seven reference element geometries. By minimizing a smooth Lebesgue-constant surrogate using IPOPT, the approach yields nodal layouts that often outperform standard distributions (GLL, Isaac, Warburton, etc.) in interpolation stability and mass-matrix conditioning, while ensuring cross-element compatibility on mixed meshes. This framework enables robust high-order finite-element discretizations across diverse geometries and paves the way for future exploration of other objectives and element types.

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 [22].

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

TL;DR

The paper addresses ill-conditioning of high-order finite-element nodal distributions by introducing a general optimization-based, symmetry-driven framework that constructs cross-element–compatible nodal distributions from intrinsic symmetry orbits. It formalizes an abstract formulation linking natural coordinates, symmetry orbits, and linear constraints, and instantiates it for seven reference element geometries. By minimizing a smooth Lebesgue-constant surrogate using IPOPT, the approach yields nodal layouts that often outperform standard distributions (GLL, Isaac, Warburton, etc.) in interpolation stability and mass-matrix conditioning, while ensuring cross-element compatibility on mixed meshes. This framework enables robust high-order finite-element discretizations across diverse geometries and paves the way for future exploration of other objectives and element types.

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 [22].
Paper Structure (28 sections, 40 equations, 24 figures, 6 tables)

This paper contains 28 sections, 40 equations, 24 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Abstract formulation
  3. Natural coordinates
  4. Intrinsic symmetry orbits
  5. Constrained symmetry orbits
  6. Symmetry orbit collections
  7. Reference domains
  8. Line
  9. Triangle
  10. Quadrilateral
  11. Tetrahedron
  12. Hexahedron
  13. Triangular prism
  14. Square Pyramid
  15. Optimization-based nodal distributions
  16. ...and 13 more sections

Figures (24)

  • Figure 1: Symmetry orbits for the triangular element. Legend: Points generated by $\bar{\mathcal{P}}_1$ (\ref{['line:tridemo1:orb1']}), $\bar{\mathcal{P}}_2(0.2)$ (\ref{['line:tridemo1:orb2']}), and $\bar{\mathcal{P}}_3(0.3,0.2)$ (\ref{['line:tridemo1:orb3']}).
  • Figure 2: Nodal distribution for symmetry orbit collection described in Section \ref{['sec:refdom:tri']} with $\alpha_1 = 0.25$, $\alpha_2 = 0.5$, $\alpha_3 = 0.1$, $\alpha_4 = 0.6$, $\alpha_5 = 0.7$, and $\alpha_6 = 0$. Legend: Points generated by $\bar{\mathcal{P}}_1$ (\ref{['line:tridemo2:orb1']}), $\bar{\mathcal{P}}_2(\alpha_1)$ (\ref{['line:tridemo2:orb2a']}), $\bar{\mathcal{P}}_2(\alpha_2)$ (\ref{['line:tridemo2:orb2b']}), $\bar{\mathcal{P}}_3(\alpha_3,\alpha_4)$ (\ref{['line:tridemo2:orb3a']}), and $\bar{\mathcal{P}}_3(\alpha_5,\alpha_6)$ (\ref{['line:tridemo2:orb3b']}).
  • Figure 3: Symmetry orbits for the square pyramid element including isometric (left), top (middle), and side(right) views. Legend: Points generated by $\bar{\mathcal{P}}_1(0.0)$ (\ref{['line:pyrmddemo1:orb1']}), $\bar{\mathcal{P}}_2(0.75, -0.8)$ (\ref{['line:pyrmddemo1:orb2']}), $\bar{\mathcal{P}}_3(0.15, 0.5)$ (\ref{['line:pyrmddemo1:orb3']}), and $\bar{\mathcal{P}}_4(0.2, 0.4, -0.5)$ (\ref{['line:pyrmddemo1:orb4']}).
  • Figure 4: The Lebesgue constant (left) and mass matrix condition number (right) of the optimized nodes (\ref{['line:lebcnst:0']}), GLL nodes (\ref{['line:lebcnst:2']}), and uniform nodes (\ref{['line:lebcnst:1']}) for the line element. The Lebesgue objective (\ref{['eqn:lebobj']}) for each nodal distribution is also included in the left figure for each nodal distribution (dashed).
  • Figure 5: Zoom of Figure \ref{['fig:compare_line']} to emphasize difference between optimized and GLL nodes.
  • ...and 19 more figures

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Remark 3