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.     

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.     

A remark on the optimality of adaptive finite element methods

We show that the standard assumption on the smallness of the marking parameter θ in adaptive finite element methods can be avoided for the proof of the optimality of the algorithm. To this end we propose a new technique based on comparison of the solutions of different finite element spaces obtained by different refinements of a given mesh. We consider conforming and nonconforming low-order finite elements on triangular and tetrahedral meshes.     

A multi-mesh finite element method for Lagrange elements of arbitrary degree

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviors of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid–solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier–Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy way to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.     

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.     

The finite element method for parabolic equations

Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.The work was partially supported by ONR Contract N00014-77-C-0623     

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.     

Toward anisotropic mesh construction and error estimation in the finite element method

Directional, anisotropic features like layers in the solution of partial differential equations can be resolved favorably by using anisotropic finite element meshes. An adaptive algorithm for such meshes includes the ingredients Error estimation and Information extraction/Mesh refinement. Related articles on a posteriori error estimation on anisotropic meshes revealed that reliable error estimation requires an anisotropic mesh that is aligned with the anisotropic solution. To obtain anisotropic meshes the so‐called Hessian strategy is used, which provides information such as the stretching direction and stretching ratio of the anisotropic elements. This article combines the analysis of anisotropic information extraction/mesh refinement and error estimation (for several estimators). It shows that the Hessian strategy leads to well‐aligned anisotropic meshes and, consequently, reliable error estimation. The underlying heuristic assumptions are given in a stringent yet general form. Numerical examples strengthen the exposition. Hence the analysis provides further insight into a particular aspect of anisotropic error estimation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 625–648, 2002; DOI 10.1002/num.10023     

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.     

A wirebasket preconditioner for the mortar boundary element method

We present and analyze a preconditioner of the additive Schwarz type for the mortar boundary element method. As a basic splitting, on each subdomain we separate the degrees of freedom related to its boundary from the inner degrees of freedom. The corresponding wirebasket-type space decomposition is stable up to logarithmic terms. For the blocks that correspond to the inner degrees of freedom standard preconditioners for the hypersingular integral operator on open boundaries can be used. For the boundary and interface parts as well as the Lagrangian multiplier space, simple diagonal preconditioners are optimal. Our technique applies to quasi-uniform and non-uniform meshes of shape-regular elements. Numerical experiments on triangular and quadrilateral meshes confirm theoretical bounds for condition and MINRES iteration numbers.     

Mesh and solver co‐adaptation in finite element methods for anisotropic problems

Mesh generation and algebraic solver are two important aspects of the finite element methodology. In this article, we are concerned with the joint adaptation of the anisotropic triangular mesh and the iterative algebraic solver. Using generic numerical examples pertaining to the accurate and efficient finite element solution of some anisotropic problems, we hereby demonstrate that the processes of geometric mesh adaptation and the algebraic solver construction should be adapted simultaneously. We also propose some techniques applicable to the co‐adaptation of both anisotropic meshes and linear solvers. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates

This paper presents an alternative alpha finite element method using triangular meshes (AαFEM) for static, free vibration and buckling analyses of laminated composite plates. In the AαFEM, an assumed strain field is carefully constructed by combining compatible strains and additional strains with an adjustable parameter α which can produce an effectively softer stiffness formulation compared to the linear triangular element. The stiffness matrices are obtained based on the strain smoothing technique over the smoothing domains and the constant strains on triangular sub-domains associated with the nodes of the elements. The discrete shear gap (DSG) method is incorporated into the AαFEM to eliminate transverse shear locking and an improved triangular element termed as AαDSG3 is proposed. Several numerical examples are then given to demonstrate the effectiveness of the AαDSG3.     

Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Natural superconvergence of the least-squares finite element method is surveyed for the one-and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method. The second author was supported in part by the US National Science Foundation under Grant DMS-0612908.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

Finite element and boundary element contact stress analysis with remeshing technique

The fundamental part of the contact stress problem solution using a finite element method is to locate possible contact areas reliably and efficiently. In this research, a remeshing technique is introduced to determine the contact region in a given accuracy. In the proposed iterative method, the meshes near the contact surface are modified so that the edge of the contact region is also an element’s edge. This approach overcomes the problem of surface representation at the transition point from contact to non-contact region. The remeshing technique is efficiently employed to adapt the mesh for more precise representation of the contact region. The method is applied to both finite element and boundary element methods. Overlapping of the meshes in the contact region is prevented by the inclusion of displacement and force constraints using the Lagrange multipliers technique. Since the method modifies the mesh only on the contacting and neighbouring region, the solution to the matrix system is very close to the previous one in each iteration. Both direct and iterative solver performances on BEM and FEM analyses are also investigated for the proposed incremental technique. The biconjugate gradient method and LU with Cholesky decomposition are used for solving the equation systems. Two numerical examples whose analytical solutions exist are used to illustrate the advantages of the proposed method. They show a significant improvement in accuracy compared to the solutions with fixed meshes.     

Polynomial preserving recovery for quadratic elements on anisotropic meshes

Polynomial preserving gradient recovery technique under anisotropic meshes is further studied for quadratic elements. The analysis is performed for highly anisotropic meshes where the aspect ratios of element sides are unbounded. When the mesh is adapted to the solution that has significant changes in one direction but very little, if any, in another direction, the recovered gradient can be superconvergent. The results further explain why recovery type error estimator is robust even under nonstandard and highly distorted meshes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

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     

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.     

A non-local damage approach for the boundary element method

The conventional (local) constitutive modelling of materials exhibiting strain softening behaviour is susceptive to a spurious mesh dependence caused by numerically induced strain localization. Also, for refined meshes, numerical instabilities may be verified, mainly if the simulations are performed by the boundary element method. An alternative to overcome such difficulties is the adoption of the so called non-local constitutive models. In these approaches, some internal variables of the constitutive model in a single point are averaged considering its values of the neighbouring points. In this paper, the implicit formulation of the boundary element method for physically non-linear problems in solid mechanics is used with a non-local isotropic damage model and a very simple averaging scheme, over internal cells, is introduced. It is shown that the analysis become more stable in comparison to the case of a local application of the same model and that the results recover the desired objectiveness to mesh refinement.     

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.