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The singular Zienkiewicz tetrahedron: Definition and Integration
Johannes Storn
TL;DR
Three-dimensional H2-conforming finite elements on tetrahedral meshes remain difficult because existing constructions rely on high polynomial degree, refinements, or non-polynomial functions. The paper extends the singular Zienkiewicz triangle with rational shape functions and supplies an exact iterative integration procedure. It constructs and analyzes the resulting element, while numerical experiments compare exact integration with quadrature and show quadrature errors depend on the number of Gauss points.
Problem
Three-dimensional H2-conforming finite element constructions are constrained by high polynomial degree, mesh refinements, or the unresolved extension of non-polynomial approaches.
Method
The paper extends the singular Zienkiewicz triangle to tetrahedra using rational shape functions and develops an exact iterative integration procedure for the associated rational functions.
Results
The construction yields a novel H2-conforming tetrahedral finite element, and experiments show quadrature-induced eigenvalue errors depend on the number of one-dimensional Gauss points.
Takeaways & Limitations
Exact integration enables exact treatment of the rational basis functions, derivatives, and products used in finite element assembly.
Takeaways & Limitations
A reduction step cannot be applied when the critical conditions cannot be satisfied by a single index, requiring a decomposition.
Abstract
from arXiv · showhide
We extend the two-dimensional singular Zienkiewicz element to three dimensions, leading to a novel $H^2$-conforming finite element on tetrahedral meshes based on rational shape functions. Besides the finite element construction and its conformity analysis, we develop an exact iterative integration procedure for the associated class of rational functions. The resulting formulae allow for the exact integration of the basis functions, their derivatives, and the products occurring in finite element assembly.
1. Introduction
The paper addresses the difficulty of constructing globally H2-conforming finite elements in three dimensions, where existing approaches require high polynomial degree, mesh refinements, or non-polynomial functions. It extends the singular Zienkiewicz triangle using rational functions and develops exact integration to support implementation.
- H2-conforming discretizations of fourth-order problems require continuity of functions and first derivatives across element interfaces.
- Three-dimensional constructions include the degree-nine Zienkiewicz element and lower-degree alternatives based on macroelement refinements.
- The paper closes the open non-polynomial 3D gap by extending the singular Zienkiewicz triangle to tetrahedral meshes.
- Rational enrichment preserves suitably low-dimensional traces while avoiding cascading increases in polynomial degree and local dimension.
- The resulting rational shape functions have second derivatives generally only in L∞, so standard polynomial quadrature is inaccurate.
- An exact integration routine is developed for rational functions appearing in the finite element construction and assembly.
2. The polynomial Zienkiewicz tetrahedron
The polynomial Zienkiewicz tetrahedron enriches quadratic polynomials with cubic edge functions and uses vertex Hermite data as its degrees of freedom. Its facet traces support continuity across tetrahedra, but the normal derivative is not determined by shared vertex data, so the global element is H1-conforming but not H2-conforming.
- The space is formed by enriching quadratic polynomials with cubic edge functions on tetrahedra.
- The element has 16 degrees of freedom consisting of function values and gradients at the four vertices.These are the vertex Hermite data z(vi) and ∇z(vi).
- The vertex Hermite degrees of freedom are unisolvent: a function vanishing in all of them must be identically zero.The proof restricts the function to each edge, where cubic Hermite uniqueness eliminates the edge coefficients and the remaining quadratic polynomial.
- On each facet, the scalar trace is the two-dimensional polynomial Zienkiewicz space, while the normal trace consists of quadratic polynomials.
- Matching vertex Hermite data across a shared facet guarantees matching scalar traces but does not generally guarantee matching normal derivatives.An explicit construction gives identical shared-facet vertex data with different normal derivatives on the two tetrahedra.
- Because the global element is not H2-conforming, its piecewise-Laplacian biharmonic method can diverge on certain meshes; polynomial enrichment instead leads to much larger spaces.The cited polynomial alternative has 220 local degrees of freedom.
3. Enrichment of the polynomial Zienkiewicz space
The construction enriches the polynomial Zienkiewicz tetrahedron with rational edge bubbles, then corrects their normal traces to obtain globally H2-conforming spaces with explicit degrees of freedom and dimensions.
- Motivation: Rational bubbles enrich the polynomial Zienkiewicz space because its original degrees of freedom are insufficient for H2-conformity.The enrichment adds degrees of freedom while retaining suitably controlled traces.
- Rational functions: The rational bubble class is differentiated within itself, and barycentric derivatives determine Cartesian derivatives through the chain rule.This provides an explicit symbolic procedure for evaluating derivatives needed in finite element implementations.
- Regularity and integrability: For every singular boundary subsimplex, Lp-integrability requires −|S|/p < AS(α) − BS(β), while derivatives through order m require m − |S|/p < AS(α) − BS(β).These conditions are necessary and sufficient for the stated Sobolev regularity, including the corresponding non-integrability cases when they fail.
- Singular edge bubbles: Edge bubbles supply two additional linearly independent quadratic gradient-trace components on every edge and are added to the polynomial space.The enriched space is defined from these edge-bubble generators.
- Normal trace correction: Rational edge bubbles initially create non-polynomial normal traces on facets, so a correction removes these traces before establishing H2 conformity.The corrected construction yields a finite element with vertex Hermite and edge-gradient degrees of freedom.
4. Integration formula for rational functions
The section develops exact integration for rational monomials on tetrahedra by reducing moments iteratively to one-face, vertex-star, and polynomial cases. The procedure uses admissible reductions that preserve finiteness and terminates when the denominator disappears or the central numerator exponent reaches zero.
- 4. Integration formula for rational functions: Exact integration formulae are developed for rational monomials Rαβ and support implementation of the three-dimensional singular Zienkiewicz element.The resulting routine targets the rational functions used in the finite element construction.
- 4. Integration formula for rational functions: Affine invariance makes the normalized tetrahedral moments independent of the particular tetrahedron, so the analysis uses barycentric coordinates on a reference tetrahedron.The moments are introduced on a tetrahedron with vertices and associated barycentric coordinates.
- 4.1. Reduction to triangular and star moments: Opposite-edge reduction rewrites moments containing both opposite edges as sums of moments with smaller denominator exponents, while preserving finiteness.Iterating this identity yields reduced multi-indices whose supports contain no pair of opposite edges.
- 4.1. Reduction to triangular and star moments: Every reduced support falls into either the one-face case, supported on three edges of a face, or the vertex-star case, supported on three edges meeting at a vertex.The two configurations are defined as “one face” and “vertex star.”
- 4.1.2. Vertex-star: Vertex-star moments are handled by reductions based on critical single and pair inequalities, with admissibility ensured when a single index covers the critical conditions or all pair conditions are critical.The reduction identities lower the central numerator power while maintaining the relevant integrability conditions.
- 4.1.2. Vertex-star: When critical conditions cannot be hit by one index, a separate decomposition handles the case in which all pair conditions are critical.The proof tracks how denominator and leaf numerator exponents change so the conditions remain valid.
5. Numerical scheme
The paper combines exact rational-moment integration with geometry-efficient stiffness assembly, then compares exact integration against tensor-product Gauß quadrature in numerical experiments.
- Exact integration: The recursive rational-moment routine terminates through successive reductions of decreasing multi-index or exponent quantities.The reductions decrease |β|, α0, |δ|, a + b + c, or a + b.
- Implementation: The implementation precomputes geometry-independent rational-function products and integrates them offline for assembly.The singular Zienkiewicz element uses 64 geometry-independent rational functions for 28 local basis functions.
- Efficient assembly: 2,097,152 ≈ 2 · 10^6 floating-point operations are required per tetrahedron by the direct tensor-based assembly strategy.Each local Frobenius inner product costs 512 operations, applied across 64^2 tensor entries.
- Efficient assembly: 21 geometry coefficients reduce online assembly to geometry-dependent linear combinations of precomputed reference matrices.The six independent entries of the symmetric geometry matrix contribute six linear and 15 quadratic coefficients.
- Numerical quadrature: The tensor-product quadrature routine uses n^3 points and is exact for polynomial integrands in P_(2n−3)(bT).The one-dimensional Gauß rules are applied successively on the corresponding intervals and extend affinely to arbitrary tetrahedra.
- Quadrature error: Quadrature performs poorly for the first rational integrand, whereas about n = 30 points per dimension achieve relative error 10^-5 for the bounded integrand.The latter requires 30^3 = 27,000 point evaluations, with slower apparent convergence as n increases.