[Paper Review] Low-order continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes

Maxence Reberol, Bruno Lévy|arXiv (Cornell University)|May 9, 2016

Computational Geometry and Mesh Generation19 references3 citations

TL;DR

This paper proposes a method to construct $C^0$-continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes by using quadratic mappings for tetrahedra to match non-planar hexahedral faces and enforcing continuity constraints via quadratic Lagrange basis functions. The approach achieves accuracy comparable to pure hexahedral meshes and significantly better than linear tetrahedral meshes, offering a robust alternative for complex geometries where full hexahedral meshing is infeasible.

ABSTRACT

This article deals with solving partial differential equations with the finite element method on hybrid non-conforming hexahedral-tetrahedral meshes. By non-conforming, we mean that a quadrangular face of a hexahedron can be connected to two triangular faces of tetrahedra. We introduce a set of low-order continuous (C0) finite element spaces defined on these meshes. They are built from standard tri-linear and quadratic Lagrange finite elements with an extra set of constraints at non-conforming hexahedra-tetrahedra junctions to recover continuity. We consider both the continuity of the geometry and the continuity of the function basis as follows: the continuity of the geometry is achieved by using quadratic mappings for tetrahedra connected to tri-affine hexahedra and the continuity of interpolating functions is enforced in a similar manner by using quadratic Lagrange basis on tetrahedra with constraints at non-conforming junctions to match tri-linear hexahedra. The so-defined function spaces are validated numerically on simple Poisson and linear elasticity problems for which an analytical solution is known. We observe that using a hybrid mesh with the proposed function spaces results in an accuracy significantly better than when using linear tetrahedra and slightly worse than when solely using tri-linear hexahedra. As a consequence, the proposed function spaces may be a promising alternative for complex geometries that are out of reach of existing full hexahedral meshing methods.

Motivation & Objective

• To address the challenge of constructing continuous finite element spaces on hybrid non-conforming hexahedral-tetrahedral meshes where hexahedra and tetrahedra meet with non-planar faces.
• To overcome geometric and functional discontinuities arising from non-conforming interfaces between tri-linear hexahedra and linear tetrahedra.
• To enable high-accuracy finite element solutions on hex-dominant meshes that are otherwise difficult to generate with pure hexahedral elements.
• To provide a formal, mathematically grounded framework for continuity enforcement at non-conforming hexahedron-tetrahedron interfaces, avoiding reliance on pyramidal elements.

Proposed method

• Uses quadratic isoparametric mappings for tetrahedra to exactly fit the non-planar quadrilateral faces of adjacent tri-linear hexahedra, ensuring geometric continuity.
• Employs quadratic Lagrange finite elements ($\mathbb{P}_2$) on tetrahedra to match the function space of tri-linear hexahedra ($\mathbb{Q}_1$) at interfaces.
• Imposes constraints at non-conforming junctions to enforce $C^0$ continuity of the global finite element function space.
• Defines the function basis on tetrahedra using barycentric coordinates and edge midpoints, ensuring interpolation at vertices and mid-edges.
• Applies a reference element approach: maps all elements to reference tetrahedra ($\hat{T}$) and reference hexahedra ($\hat{Q}$) to standardize basis functions and constraints.
• Derives continuity conditions by matching function values and gradients at shared interface points, particularly at vertices and edge midpoints across the interface.

Experimental results

Research questions

• RQ1 Can a continuous $C^0$ finite element space be constructed on hybrid non-conforming hexahedral-tetrahedral meshes with geometric and functional continuity?
• RQ2 How can geometric continuity be achieved when a hexahedron’s quadrilateral face is non-planar and connected to two tetrahedra?
• RQ3 What constraints are required to ensure that the finite element functions remain continuous across non-conforming hexahedron-tetrahedron interfaces?
• RQ4 Can the resulting function space achieve accuracy comparable to pure hexahedral meshes while remaining more flexible than pure tetrahedral meshes?
• RQ5 Is it possible to avoid pyramidal elements in hybrid meshes by directly coupling hexahedra and tetrahedra with appropriate continuity constraints?

Key findings

• The proposed method achieves geometric continuity by using quadratic mappings for tetrahedra to exactly match the non-planar faces of tri-linear hexahedra.
• Function continuity is enforced through quadratic Lagrange basis functions on tetrahedra with constraints at vertices and edge midpoints to match the tri-linear hexahedral basis.
• Numerical experiments on Poisson and linear elasticity problems show accuracy significantly better than linear tetrahedral elements and slightly worse than pure tri-linear hexahedral meshes.
• The method enables the use of hex-dominant meshes with scattered non-conforming junctions, which is more suitable for complex geometries than localized transition layers.
• The approach avoids the use of pyramidal elements and provides a formal, mathematically consistent framework for $C^0$ continuity on hybrid non-conforming meshes.