Quading triangular meshes with certain topological constraints

In computer graphics and geometric modeling, shapes are often represented by triangular meshes (also called 3D meshes or manifold triangulations). The quadrangulation of a triangular mesh has wide applications. In this paper, we present a novel method of quading a closed orientable triangular mesh into a quasi-regular quadrangulation, i.e., a quadrangulation that only contains vertices of degree four or five. The quasi-regular quadrangulation produced by our method also has the property that the number of quads of the quadrangulation is the smallest among all the quasi-regular quadrangulations. In addition, by constructing the so-called orthogonal system of cycles our method is more effective to control the quality of the quadrangulation.     

有限元网格的超收敛测度及其应用

给出线性有限元求解二阶椭圆问题的有限元网格超收敛测度及其应用.有限元超收敛经常是在具有一定结构的特殊网格条件下讨论的,而本文从一般网格出发,导出一种网格的范数用来描述超收敛所需要的网格条件以及超收敛的程度.并且通过对这种网格范数性质的考察,可以证明对于通常考虑的一些特殊网格的超收敛的存在性.更进一步,我们可以通过正则细分的方式在一般区域上也可以自动获得超收敛网格.最后给出相关的数值结果来验证本文的理论分析.     

On using meshes of mixed element types in adaptive finite element analysis

Four different automatic mesh generators capable of generating either triangular meshes or hybrid meshes of mixed element types have been used in the mesh generation process. The performance of these mesh generators were tested by applying them to the adaptive finite element refinement procedure. It is found that by carefully controlling the quality and grading of the quadrilateral elements, an increase in efficiency over pure triangular meshes can be achieved. Furthermore, if linear elements are employed, an optimal hybrid mesh can be obtained most economically by a combined use of the mesh coring technique suggested by Lo and Lau and a selective removal of diagonals from the triangular element mesh. On the other hand, if quadratic elements are used, it is preferable to generate a pure triangular mesh first, and then obtain a hybrid mesh by merging of triangles.     

Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been successful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.     

Planar mesh refinement cannot be both local and regular

We show that two desirable properties for planar mesh refinement techniques are incompatible. Mesh refinement is a common technique for adaptive error control in generating unstructured planar triangular meshes for piecewise polynomial representations of data. Local refinements are modifications of the mesh that involve a fixed maximum amount of computation, independent of the number of triangles in the mesh. Regular meshes are meshes for which every interior vertex has degree 6. At least for some simple model meshing problems, optimal meshes are known to be regular, hence it would be desirable to have a refinement technique that, if applied to a regular mesh, produced a larger regular mesh. We call such a technique a regular refinement. In this paper, we prove that no refinement technique can be both local and regular. Our results also have implications for non-local refinement techniques such as Delaunay insertion or Rivara's refinement. Received August 1, 1996 / Revised version received February 28, 1997     

A variational approach to optimal meshes

Summary. A variational approach for the optimization of triangular or tetrahedral meshes is presented. Starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity. It will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions. In addition the injectivity of the deformed mesh is discussed. Thereby it is possible to construct suitable meshes for various numerical applications. Received March 14, 1994 / Revised version received August 8, 1994     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

On the adjacencies of triangular meshes based on skeleton-regular partitions

For any 2D triangulation τ, the 1-skeleton mesh of τ is the wireframe mesh defined by the edges of τ, while that for any 3D triangulation τ, the 1-skeleton and the 2-skeleton meshes, respectively, correspond to the wireframe mesh formed by the edges of τ and the “surface” mesh defined by the triangular faces of τ. A skeleton-regular partition of a triangle or a tetrahedra, is a partition that globally applied over each element of a conforming mesh (where the intersection of adjacent elements is a vertex or a common face, or a common edge) produce both a refined conforming mesh and refined and conforming skeleton meshes. Such a partition divides all the edges (and all the faces) of an individual element in the same number of edges (faces). We prove that sequences of meshes constructed by applying a skeleton-regular partition over each element of the preceding mesh have an associated set of difference equations which relate the number of elements, faces, edges and vertices of the nth and (n−1)th meshes. By using these constitutive difference equations we prove that asymptotically the average number of adjacencies over these meshes (number of triangles by node and number of tetrahedra by vertex) is constant when n goes to infinity. We relate these results with the non-degeneracy properties of longest-edge based partitions in 2D and include empirical results which support the conjecture that analogous results hold in 3D.     

Robust intersection of structured hexahedral meshes and degenerate triangle meshes with volume fraction applications

Two methods for calculating the volume and surface area of the intersection between a triangle mesh and a rectangular hexahedron are presented. The main result is an exact method that calculates the polyhedron of intersection and thereafter the volume and surface area of the fraction of the hexahedral cell inside the mesh. The second method is approximate, and estimates the intersection by a least squares plane. While most previous publications focus on non-degenerate triangle meshes, we here extend the methods to handle geometric degeneracies. In particular, we focus on large-scale triangle overlaps, or double surfaces. It is a geometric degeneracy that can be hard to solve with existing mesh repair algorithms. There could also be situations in which it is desirable to keep the original triangle mesh unmodified. Alternative methods that solve the problem without altering the mesh are therefore presented. This is a step towards a method that calculates the solid area and volume fractions of a degenerate triangle mesh including overlapping triangles, overlapping meshes, hanging nodes, and gaps. Such triangle meshes are common in industrial applications. The methods are validated against three industrial test cases. The validation shows that the exact method handles all addressed geometric degeneracies, including double surfaces, small self-intersections, and split hexahedra.     

Four-triangles adaptive algorithms for RTIN terrain meshes

We present a refinement and coarsening algorithm for the adaptive representation of Right-Triangulated Irregular Network (RTIN) meshes. The refinement algorithm is very simple and proceeds uniformly or locally in triangle meshes. The coarsening algorithm decreases mesh complexity by reducing unnecessary data points in the mesh after a given error criterion is applied. We describe the most important features of the algorithms and give a brief numerical study on the propagation associated with the adaptive scheme used for the refinement algorithm. We also present a comparison with a commercial tool for mesh simplification, Rational Reducer, showing that our coarsening algorithm offers better results in terms of accuracy of the generated meshes.     

Parallel anisotropic mesh adaptivity with dynamic load balancing for cardiac electrophysiology

Simulations in cardiac electrophysiology generally use very fine meshes and small time steps to resolve highly localized wavefronts. This expense motivates the use of mesh adaptivity, which has been demonstrated to reduce the overall computational load. However, even with mesh adaptivity performing such simulations on a single processor is infeasible. Therefore, the adaptivity algorithm must be parallelised. Rather than modifying the sequential adaptive algorithm, the parallel mesh adaptivity method introduced in this paper focuses on dynamic load balancing in response to the local refinement and coarsening of the mesh. In essence, the mesh partition boundary is perturbed away from mesh regions of high relative error, while also balancing the computational load across processes. The parallel scaling of the method when applied to physiologically realistic heart meshes is shown to be good as long as there are enough mesh nodes to distribute over the available parallel processes. It is shown that the new method is dominated by the cost of the sequential adaptive mesh procedure and that the parallel overhead of inter-process data migration represents only a small fraction of the overall cost.     

An inequality on the edge lengths of triangular meshes

We give a short proof of the following geometric inequality: for any two triangular meshes A and B of the same polygon C, if the number of vertices in A is at most the number of vertices in B, then the maximum length of an edge in A is at least the minimum distance between two vertices in B. Here the vertices in each triangular mesh include the vertices of the polygon and possibly additional Steiner points. The polygon must not be self-intersecting but may be non-convex and may even have holes. This inequality is useful for many purposes, especially in proving performance guarantees of mesh generation algorithms. For example, a weaker corollary of the inequality confirms a conjecture of Aurenhammer et al. [Theoretical Computer Science 289 (2002) 879-895] concerning triangular meshes of convex polygons, and improves the approximation ratios of their mesh generation algorithm for minimizing the maximum edge length and the maximum triangle perimeter of a triangular mesh.     

Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation

In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.     

Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's,Poisson's,and the elasticity equations

In this article we address the problem of the existence of superconvergence points for finite element solutions of systems of linear elliptic equations. Our approach is quite different from all other studies of superconvergence. We prove that the existence of superconvergence points can be guaranteed by a numerical algorithm, which employs a finite number of operations (provided that there is no roundoff-error). By employing this approach, we can reproduce all known results on superconvergence of finite element solutions for linear elliptic problems and we can obtain many new results. Here, in particular, we address the problem of the superconvergence points for the gradient of finite element solutions of Laplace's and Poisson's equations and we show that the sets of superconvergence points are very different for these two cases. We also study the superconvergence of the components of the gradient of the displacement, the strain and stress for finite element solutions of the equations of elasticity. For Laplace's and Poisson's equations (resp. the equations of elasticity), we consider meshes of triangular as well as square elements of degree p, 1 ? p ? 7 (resp. 1 ? p ? 4). For the meshes of triangular elements we investigate the effect of the geometry of the mesh by considering four mesh patterns that typically occur in practical meshes, while in the case of square elements, we study the effect of the element type (tensor-product, serendipity, or other). © 1996 John Wiley & Sons, Inc.     

Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

Robust a posteriori error estimates for conforming discretizations of a singularly perturbed reaction-diffusion problem on anisotropic meshes

In this paper, we derive robust a posteriori error estimates for conforming approximations to a singularly perturbed reaction-diffusion problem on anisotropic meshes, since the solution in general exhibits anisotropic features, e.g., strong boundary or interior layers. Based on the anisotropy of the mesh elements, we improve the a posteriori error estimates developed by Cheddadi et al., which are reliable and efficient on isotropic meshes but fail on anisotropic ones. Without the assumption that the mesh is shape-regular, the resulting mesh-dependent error estimator is shown to be reliable, efficient and robust with respect to the reaction coefficient, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming one, like the piecewise linear finite element one. Our estimates are based on the usual H(div)-conforming, locally conservative flux reconstruction in the lowest-order Raviart-Thomas space on a dual mesh associated with the original anisotropic simplex one. Numerical experiments in 2D confirm that our estimates are reliable, efficient and robust on anisotropic meshes.     

Development of a P2 element with optimal L2 convergence for biharmonic equation

It is well known that convergence rate of finite element approximation is suboptimal in the L² norm for solving biharmonic equations when P₂ or Q₂ element is used. The goal of this paper is to derive a weak Galerkin (WG) P₂ element with the L² optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle.     

An efficient linear second order unconditionally stable direct discretization method for the phase-field crystal equation on surfaces

We develop an unconditionally stable direct discretization scheme for solving the phase-field crystal equation on surfaces. The surface is discretized by using an unstructured triangular mesh. Gradient, divergence, and Laplacian operators are defined on triangular meshes. The proposed numerical method is second-order accurate in space and time. At each time step, the proposed computational scheme results in linear elliptic equations to be solved, thus it is easy to implement the algorithm. It is proved that the proposed scheme satisfies a discrete energy-dissipation law. Therefore, it is unconditionally stable. A fast and efficient biconjugate gradients stabilized solver is used to solve the resulting discrete system. Numerical experiments are conducted to demonstrate the performance of the proposed algorithm.     

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.     

Supercloseness of the SDFEM on Shishkin triangular meshes for problems with exponential layers

In this paper, we analyze the supercloseness property of the streamline diffusion finite element method (SDFEM) on Shishkin triangular meshes, which is different from one in the case of rectangular meshes. The analysis depends on integral inequalities for the parts related to the diffusion in the bilinear form. Moreover, our result allows the construction of a simple postprocessing that yields a more accurate solution. Finally, numerical experiments support these theoretical results.