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.     

Smooth interpolation of a mesh of curves

The interpolation of a mesh of curves by a smooth regularly parametrized surface with one polynomial piece per facet is studied. Not every mesh with a well-defined tangent plane at the mesh points has such an interpolant: the curvature of mesh curves emanating from mesh points with an even number of neighbors must satisfy an additional vertex enclosure constraint. The constraint is weaker than previous analyses in the literature suggest and thus leads to more efficient constructions. This is illustrated by an implemented algorithm for the local interpolation of a cubic curve mesh by a piecewise [bi]quarticC ¹ surface. The scheme is based on an alternative sufficient constraint that forces the mesh curves to interpolate second-order data at the mesh points. Rational patches, singular parametrizations, and the splitting of patches are interpreted as techniques to enforce the vertex enclosure constraint.Communicated by Wolfgang Dahmen.     

Asymptotic behaviour of the containment of certain mesh patterns

We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between 0 and 1.     

A self-adaptive moving mesh method for the Camassa-Holm equation

A self-adaptive moving mesh method is proposed for the numerical simulations of the Camassa-Holm equation. It is an integrable scheme in the sense that it possesses the exact N-soliton solution. It is named a self-adaptive moving mesh method, because the non-uniform mesh is driven and adapted automatically by the solution. Once the non-uniform mesh is evolved, the solution is determined by solving a tridiagonal linear system. Due to these two superior features of the method, several test problems give very satisfactory results even if by using a small number of grid points.     

A new stabilized finite element method for advection‐diffusion‐reaction equations

In this article, a new stabilized finite element method is proposed and analyzed for advection‐diffusion‐reaction equations. The key feature is that both the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number are reasonably incorporated into the newly designed stabilization parameter. The error estimates are established, where, up to the regularity‐norm of the exact solution, the explicit‐dependence of the diffusivity, advection, reaction, and mesh size (or the dependence of the mesh‐dependent Péclet number and the mesh‐dependent Damköhler number) is revealed. Such dependence in the error bounds provides a mathematical justification on the effectiveness of the proposed method for any values of diffusivity, advection, dissipative reaction, and mesh size. Numerical results are presented to illustrate the performance of the method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 616–645, 2016     

The error‐minimization‐based rezone strategy for arbitrary Lagrangian‐Eulerian methods

The objective of the Arbitrary Lagrangian‐Eulerian (ALE) methodology for solving multidimensional fluid flow problems is to move the computational mesh, using the flow as a guide, to improve the robustness, accuracy and efficiency of a simulation. The main elements in the ALE simulation are an explicit Lagrangian phase, a rezone phase in which a new mesh is defined, and a remapping (conservative interpolation) phase, in which the Lagrangian solution is transferred to the new mesh. In most ALE codes, the main goal of the rezone phase is to maintain high quality of the rezoned mesh. In this article, we describe a new rezone strategy which minimizes the L₂ norm of the solution error and maintains smoothness of the mesh. The efficiency of the new method is demonstrated with numerical experiments. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

On stability ofm-variateC 1 interpolation

A simplicial mesh (triangulation) is constructed that generalizes the two-dimensional 4-direction mesh to ℝ ^m . This mesh, with symmetric, shift-invariant values at the vertices, is shown to admit a bounded C¹ interpolant if and only if the alternating sum of the values at the vertices of any 1-cube is zero. This implies that interpolation at the vertices of an m-dimensional, simplicial mesh by a C¹ piecewise polynomial of degree m+1 with one piece per simplex is unstable.     

Recovering Mesh Geometry from a Stiffness Matrix

We introduce the following class of mesh recovery problems: Given a stiffness matrix A and a PDE, construct a mesh M such that the finite-element formulation of the PDE over M is A. We show, under certain assumptions, that it is possible to reconstruct the original mesh for the special case of the Laplace operator discretized on an unstructured mesh of triangular elements with linear basis functions. The reconstruction is achieved through a series of techniques from graph theory and numerical analysis, some of which are new and can find application in other scientific areas. Finally, we discuss extensions to other operators and some open questions related to this class of problems.     

An accurate solution procedure for fluid flow with natural convection

We consider the Boussinesq model of buoyancy driven fluid flows. This nonlinear system is solved numerically using a two level finite element method. On the first level, a nonlinear system is solved on a very coarse mesh. Thereafter, a linear system is solved on a fine mesh.

In a standard approach, one might obtain the numerical solution from a discretization of the original, nonlinear system using the same fine mesh.

Both solutions are of equal order of accuracy if the mesh widths are properly balanced. Therefore the two level method is very efficient.     

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.     

Upwind Based Parameter Uniform Convergence Analysis for Two Parametric Parabolic Convection Diffusion Problems by Moving Mesh Methods

A parabolic two parametric convection-diffusion reaction problem is considered for the moving mesh error analysis. The continuous problem is discretized by the first order upwind scheme on a non uniform mesh. A curvature based error monitor function is proposed to generate the layer adapted mesh. It is proved that the numerical solution converges to the exact solution on the mesh obtained by the equidistribution of the proposed monitor function. The convergence is first order accurate. The present analysis generalizes the results obtained in earlier publications [8,9]. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adapting nodes in three dimensional irregular regions for meshless-type methods

In this paper, a new adaptive nodes technique based on equi-distribution principles and dimension reduction is presented for irregular regions in three dimensional cases. The mesh generation is performed by first producing some adaptive nodes in a cube based on equi-distribution along the coordinate axes and then transforming the generated nodes to the physical domain followed by a refinement process. The mesh points produced are appropriate for meshless-type methods which need only some scattered points rather than a mesh with some smoothness properties. The effectiveness of the generated mesh points is examined by a collocation meshless method using a well known radial basis function, namely ?(r)?=?r ⁵ which is sufficiently smooth for our purpose. Some experimental results will be presented to illustrate the effectiveness of the proposed method.     

Adaptation de maillages non structurés pour des problèmes instationnairesUnstructured mesh adaptation for transient problems

This Note deals with the adaptation of unstructured meshes for transient CFD problems. The proposed approach is based on a new mesh adaptation algorithm and a metric intersection in time procedure suitable to capture such phenomena. More precisely, a new specific loop is inserted in the main adaptation loop to solve a transient fixed point problem. The mesh adaptation stage consists in optimizing the current mesh so as to obtain a unit mesh with respect to this metric. A 2D example is provided to emphasize the efficiency of the proposed method. To cite this article: F. Alauzet et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 773–778.     

A higher order finite element method with modified graded mesh for singularly perturbed two-parameter problems

This paper presents a modified graded mesh for singularly perturbed two-parameter problems. The mesh is generated recursively using Newton's algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on higher order polynomials of degree p≥1. We prove parameter uniform convergence of optimal order in ε-weighted energy norm. A test example is taken to compare the proposed graded mesh with others found in the literature.     

Numerical approximation of the Smagorinsky model by a two-level method

A two-level method for discretizing the Smagorinsky model for the numerical simulation of turbulent flows is proposed. In the two-level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single-step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two-level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide an a priori error estimate for the two-level method, which yields appropriate scalings between the coarse and fine mesh-sizes (H and h, respectively), and the radius of the spatial filter used in the Smagorinsky model (δ). In addition, we provide an algorithm in which a coarse mesh correction is performed to further enhance the accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity.     

Layered elastic solid method for the generation of unstructured dynamic mesh

The layered elastic solid method (LESM), a modified elastic solid method (ESM) was put forward in the present study. In LESM, the computational zone is divided into several layers and material properties of these layers, which stay a certain value in ESM, are changed with mesh deformation. The deformation capability of mesh in LESM is better than ESM and the quality of the mesh generated by LESM is superior to that generated by ESM when they undergo the same deformation. In ESM, the main influence factors on mesh quality and deformation capability are Poisson’s ratio and single-step rotation angle. In LESM, mesh quality and deformation capability reach a largest value with an increase in Young’s modulus. Meanwhile, the mesh can achieve a larger deformation capability when single-step rotation angle is 0.25°. Finally, numerical simulation on a two-dimensional aerofoil using LESM was carried out. It is found that the results of LESM show a better agreement with experiment results.     

A Comparative Numerical Study of Meshing Functionals for Variational Mesh Adaptation

We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalization of Winslow's variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.     

Anisotropic Wilson element with conforming finite element approximation for a coupled continuum pipe‐flow/Darcy model in Karst aquifers

Numerical method for a coupled continuum pipe‐flow/Darcy model describing flow in porous media with an embedded conduit pipe is considered. Wilson element on anisotropic mesh is used to solve the Darcy equation on porous matrix. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in L² and H¹ norms are established independent of the regularity condition on the mesh. Numerical examples show the efficiency of the presented scheme. With the same number of nodal points, the results using Wilson element on anisotropic mesh are much better than those of the same element and Q₁ element on regular mesh. Copyright © 2014 John Wiley & Sons, Ltd.     

Combined unstructured and hanging-node-based remeshing with boundary control for metal forming simulations

During finite element simulation of metal forming process, the mesh which represents the workpiece undergoes extreme large deformation, which could result in highly distorted mesh and numerical failure in simulation. To overcome the problem and improve computation efficiency, advancing front quad meshing technique and non–conforming mesh refinement approach are combined to generate new mesh according to desired mesh size distribution. Application of the combined remeshing strategy to rolling simulation will be presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)