Maximum Angle Condition Research Articles

Similarity Classes in the Eight-Tetrahedron Longest-Edge Partition of a Regular Tetrahedron

A tetrahedron is called regular if its six edges are of equal length. It is clear that, for an initial regular tetrahedron R0, the iterative eight-tetrahedron longest-edge partition (8T-LE) of R0 produces an infinity sequence of tetrahedral meshes, τ0={R0}, τ1={Ri1}, τ2={Ri2},…, τn={Rin},…. In this paper, it is proven that, in the iterative process just mentioned, only two distinct similarity classes are generated. Therefore, the stability and the non-degeneracy of the generated meshes, as well as the minimum and maximum angle condition straightforwardly follow. Additionally, for a standard-shape tetrahedron quality measure (η) and any tetrahedron Rin∈τn, n>0, then ηRin≥23ηR0. The non-degeneracy constant is c=23 in the case of the iterative 8T-LE partition of a regular tetrahedron.

On degenerating finite element tetrahedral partitions

Degenerating tetrahedral partitions show up quite often in modern finite element analysis. Actually the commonly used maximum angle condition allows some types of element degeneracies. Also, mesh generators and various adaptive procedures may easily produce degenerating mesh elements. Finally, complicated forms of computational domains (e.g. along with a priori known solution layers, etc) may demand the usage of elements of various degenerating shapes. In this paper, we show that the maximum angle condition presents a threshold property in interpolation theory, as the interpolation error may grow (or at least does not decay) if this condition is violated (which does not necessarily imply that FEM error grows). We also demonstrate that the popular red refinements, if done inappropriately, may lead to degenerating partitions which break the maximum angle condition. Finally, we prove that not all tetrahedral elements from a family of tetrahedral partitions are badly shaped when the discretization parameter tends to zero.

Similarity classes generated by the 8T-LE partition applied to trirectangular tetrahedra

When the 8T-LE partition is recursively applied to any initial trirectangular tetrahedron T, only a finite number of dissimilar tetrahedra are generated. It implies the stability of the meshes. At each step of refinement the number of right-type or path tetrahedra grows, so the quality of the obtained meshes improves. The minimum angle condition and the maximum angle condition are trivially satisfied, since the number of similarity classes is finite.

A virtual finite element method for two-dimensional Maxwell interface problems with a background unfitted mesh

A virtual element method (VEM) with the first-order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual [Formula: see text]-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.

Improved Maximum Angle Estimate for Longest-Edge Bisection

In this note we construct an upper estimate on the maximum angles of triangles generated by the longest-edge bisection algorithms applied to a triangle. This upper bound considerably improves upon one inherited from the well-known minimum angle estimation, thus providing tighter two-sided estimation of the angles generated by bisections. This estimate also guarantees the validity of the maximum angle condition, widely used in finite element analysis and computer graphics.

A new geometric condition equivalent to the maximum angle condition for tetrahedrons

For a tetrahedron, suppose that all internal angles of faces and all dihedral angles are less than a fixed constant C that is smaller than π. Then, it is said to satisfy the maximum angle condition with the constant C. The maximum angle condition is important in the error analysis of Lagrange interpolation on tetrahedrons. This condition ensures that we can obtain an error estimation, even on certain kinds of anisotropic tetrahedrons. In this paper, using two quantities that represent the geometry of tetrahedrons, we present an equivalent geometric condition to the maximum angle condition for tetrahedrons.

Crouzeix–Raviart and Raviart–Thomas finite-element error analysis on anisotropic meshes violating the maximum-angle condition

We investigate the piecewise linear nonconforming Crouzeix–Raviart and the lowest order Raviart–Thomas finite-element methods for the Poisson problem on three-dimensional anisotropic meshes. We first give error estimates of the Crouzeix–Raviart and the Raviart–Thomas finite-element approximate problems. We next present the equivalence between the Raviart–Thomas finite-element method and the enriched Crouzeix–Raviart finite-element method. We emphasize that we do not impose either shape-regular or maximum-angle condition during mesh partitioning. Numerical results confirm the results that we obtained.

Condition numbers of finite element methods on a class of anisotropic meshes

We study the behavior of the finite element condition numbers on a class of anisotropic meshes. These newly-developed mesh algorithms can produce numerical approximations with optimal convergence to isotropic and anisotropic singular solutions of elliptic boundary value problems in two- and three-dimensions. Despite the simplicity and fewer geometric constraints in implementation, these meshes can be highly anisotropic and do not maintain the maximum angle condition. We formulate a unified refinement principle and establish sharp estimates on the growth rate of the condition numbers of the stiffness matrix from these meshes. These results are important for effective applications of these meshes and for the design of fast numerical solvers. Numerical tests validate the theoretical analysis.

General theory of interpolation error estimates on anisotropic meshes

We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.

A Priori Analysis of an Anisotropic Finite Element Method for Elliptic Equations in Polyhedral Domains

Abstract Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces H σ {H^{\sigma}} with σ ∈ [ 0 , 1 ) {\sigma\in[0,1)} . We propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with fewer geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires the pure Dirichlet boundary condition. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations with the price to have “smoother” interior data. Numerical tests validate the theoretical analysis.

Error analysis of Lagrange interpolation on tetrahedrons

This paper describes the analysis of Lagrange interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle condition. In this paper, we present a new estimation in which the error is bounded in terms of the diameter and projected circumradius of the tetrahedron. Because we do not impose any geometric restrictions on the tetrahedron itself, our error estimation may be applied to any tetrahedralizations of domains including very thin tetrahedrons.

Anisotropic interpolation error estimate for arbitrary quadrilateral isoparametric elements

The aim of this paper is to show that, for any $$p \in [1,\infty )$$ , the $$W^{1,p}$$ -anisotropic interpolation error estimate holds on quadrilateral isoparametric elements verifying the maximum angle condition (MAC) and the property of comparable lengths for opposite sides (clos), i.e., on all those quadrilaterals with interior angles uniformly bounded away from $$\pi $$ and with both pairs of opposite sides having comparable lengths. For rectangular elements our interpolation error estimate agrees with the usual one whereas for perturbations of rectangles (the most general quadrilateral elements previously considered as far as we know) our result has some advantages with respect to the pre-existing ones: the interpolation error estimate that we proved is written by using two neighboring sides of the element instead of the sides of the unknown perturbed rectangle and, on the other hand, conditions MAC and clos are much simpler requirements to verify and with a clearer geometrical sense than those involved in the definition of perturbations of a rectangle.

On the generalization of the Synge–Křížek maximum angle condition for d-simplices

In this note we present a generalization of the maximum angle condition, proposed by J.L. Synge in 1957 and M. Křížek in 1992 for triangular and tetrahedral elements, respectively, for the case of higher-dimensional simplicial finite elements. Its relations to the other angle-type conditions commonly used in interpolation theory and in finite element analysis are discussed.

English

The non-conforming linear (P1) triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuska-Aziz maximum angle condition is required just as in the case of the conforming P1 triangle. Some applications and numerical results are also included to see the validity and effectiveness of our analysis.

English

We propose an analogue of the maximum angle condition (commonly used in finite element analysis for triangular and tetrahedral meshes) for the case of prismatic elements. Under this condition, prisms in the meshes may degenerate in certain ways, violating the so-called inscribed ball condition presented by P.G.Ciarlet (1978), but the interpolation error remains of the order O(h) in the H1-norm for sufficiently smooth functions.

Sharp geometric requirements in the Wachspress interpolation error estimate

AbstractGeometric conditions on general polygons are given in [9] in order to guarantee the error estimate for interpolants built from generalized barycentric coordinates, and the question about identifying sharp geometric restrictions in this setting is proposed. In this work, we address the question when the construction is made by using Wachspress coordinates. We basically show that the imposed conditionsbounded aspect ratio property(barp),maximum angle condition(MAC) andminimum edge length property(melp) are actually equivalent to (MAC, melp), and if any of these conditions is not satisfied, then there is no guarantee that the error estimate is valid. In this sense, (MAC) and (melp) can be regarded as sharp geometric requirements in the Wachspress interpolation error estimate.

Regularity and a priori error analysis on anisotropic meshes of a Dirichlet problem in polyhedral domains

Consider the Poisson equation on a polyhedral domain with the given data in a weighted $$L^2$$ space. We establish new regularity results for the solution with possible vertex and edge singularities and propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with less geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires smoother given data. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations by allowing almost- $$L^2$$ data. Numerical tests validate the theoretical analysis.

An anisotropic finite element method on polyhedral domains: Interpolation error analysis

On a polyhedral domain Ω ⊂ R 3 \Omega \subset \mathbb R^3 , consider the Poisson equation with the Dirichlet boundary condition. For singular solutions from the non-smoothness of the domain boundary, we propose new anisotropic mesh refinement algorithms to improve the convergence of finite element approximation. The proposed algorithm is simple, explicit, and requires less geometric conditions on the mesh and on the domain. Then, we develop interpolation error estimates in suitable weighted spaces for the anisotropic mesh, especially for the tetrahedra violating the maximum angle condition. These estimates can be used to design optimal finite element methods approximating singular solutions. We report numerical test results to validate the method.

Approximating surface areas by interpolations on triangulations

We consider surface area approximations by Lagrange and Crouzeix--Raviart interpolations on triangulations. For Lagrange interpolation, we give an alternative proof for Young's classical result that claims the areas of inscribed polygonal surfaces converge to the area of the original surface under the maximum angle condition on the triangulation. For Crouzeix--Raviart interpolation we show that the approximated surface areas converge to the area of the original surface without any geometric conditions on the triangulation.

English

The maximum angle condition of J. L. Synge was originally introduced in interpolation theory and further used in finite element analysis and applications for triangular and later also for tetrahedral finite element meshes. In this paper we present some of its generalizations to higher-dimensional simplicial elements. In particular, we prove optimal interpolation properties of linear simplicial elements in ℝd that degenerate in some way.