hp-Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization

We consider a family of hp-version discontinuous Galerkin finite element methods with least-squares stabilization for symmetric systems of first-order partial differential equations. The family includes the classical discontinuous Galerkin finite element method, with and without streamline-diffusion stabilization, as well as the discontinuous version of the Galerkin least-squares finite element method. An hp-optimal error bound is derived in the associated DG-norm. If the solution of the problem is elementwise analytic, an exponential rate of convergence under p-refinement is proved. We perform numerical experiments both to illustrate the theoretical results and to compare the various methods within the family.     

An Adaptive Finite Element Method for the Wave Scattering with Transparent Boundary Condition

Consider the acoustic wave scattering by an impenetrable obstacle in two dimensions. The model is formulated as a boundary value problem for the Helmholtz equation with a transparent boundary condition. Based on a duality argument technique, an a posteriori error estimate is derived for the finite element method with the truncated Dirichlet-to-Neumann boundary operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. A new adaptive finite element algorithm is proposed for solving the acoustic obstacle scattering problem, where the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization error. Numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method.     

On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

We investigate the influence of the shape parameter in the meshless Gaussian radial basis function finite difference (RBF-FD) method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds a near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided.     

Moving Mesh Finite Element Methods for an Optimal Control Problem for the Advection-Diffusion Equation

An optimal control problem for the advection-diffusion equation is studied using a Lagrangian-moving mesh finite element method. The weak formulation of the model advection–diffusion equation is based on Lagrangian coordinates, and semi–discrete (in space) error estimates are derived under minimal regularity assumptions. In addition, using these estimates and Brezzi-Rappaz-Raviart theory, symmetric error estimates for the optimality system are derived. The results also apply for advection dominated problems     

涡流检测系统仿真分析的自适应算法

该文针对电磁场有限元计算的特点,深入研究了涡流检测系统中电磁场有限元后验误差估计的误差模选择问题,并在分析Zienkiewicz-Zhu方法在电磁场有限元后验误差估计应用中存在局限性的基础上,提出了一种适合于涡流检测系统中电有限元分析的后验误差估计新方法。在此基础上,结合James R.Stewart和Thomas J.R.Hughes所提出的简单实用的有限元算法,提出了一种适合于涡流检测系统中电磁场有限元分析的hp 自适应新算法。     

Adaptive error control for multigrid finite element

We consider the problem of adaptive error control in the finite element method including the error resulting from, inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element with a canomical multigrid algorithm. The proofs are based on a combination of so-called strong stability and, the orthogonality inherent in both the finite element method can the multigrid algorithm.     

机身壁板稳定性显式非线性分析

对6种不同构型的飞机机身下壁板薄壁结构试验件,进行显示有限元非线性屈曲分析,并与工程方法及试验得到的结果进行对比分析,可知有限元法比工程方法更精确,且与试验相比,误差更稳定.     

Optimal error bounds for the MITC4 plate bending element

Abstract One of the most popular finite element method for Reissner-Mindlin plates is the so-called MITC4 method. Unfortunately, until recently, the best error estimate available in the literature for this element needs a non-optimal and generally unrealistic regularity for the solution. Here we derive optimal error estimates for this well-known low-order finite element method, which are valid for a general family of meshes. A similar result has recently been obtained elsewhere, with the same mesh restrictions. The exposition presented here is totally different.     

An error analysis and mesh adaptation method for shape design of structural components

A finite element error analysis and mesh adaptation method that can be used for improving analysis accuracy in carrying out shape design of structural components is presented in this paper. The simple error estimator developed by Zienkiewicz is adopted in this study for finite element error analysis, using only post-processing finite element data. The mesh adaptation algorithm implemented in ANSYS is investigated and the difficulties found are discussed. An improved algorithm that utilizes ANSYS POST1 capabilities is proposed and found to be more efficient than the ANSYS algorithm. An example is given to show the efficiency. An interactive mesh adaptation method that utilizes PATRAN meshing and result-displaying capabilities is proposed. This proposed method displays error distribution and stress contour of analysis results using color plots, to help the designer in identifying the critical regions for mesh refinement. Also, it provides guidance for mesh refinement by computing and displaying the desired element size information, based on error estimate and a mesh refinement criterion defined by the designer. This method is more efficient and effective than the semi-automatic algorithm implemented in ANSYS, and is suitable for structural shape design. This method can be applied not only to set-up a finite element mesh of the structure at initial design but to ensure analysis accuracy in the design process. Examples are given to demonstrate feasibility of the proposed method.     

Solution of mixed boundary value problems with local error bound by the finite element method

A method of analysis using finite element techniques is presented for second order, mixed boundary value problems in the plane. The technique focuses computational effort on specific points in the domain and provides absolute solution error bounds at those points by applying the hypercircle method. Solution error of less than 0.0003% and solution error bounds of ± 0.012% are obtained in sample problems. The solution accuracy is notably superior to what is obtained in the traditional finite element method with equivalent discretization. Two problems are presented to illustrate both the strengths and weaknesses of the method.     

Strict bounds for computed stress intensity factors

This paper addresses the subject of local error estimators for finite element methods involving local enrichment. More precisely, we focus on the extended finite element method, which was developed in the last decade for fracture mechanics. We present a method for the calculation of accurate strict bounds of mixed-mode stress intensity factors associated with an extraction operator and the theory of error in constitutive relation. This method has already been shown to be effective for standard finite elements. The objective of this paper is to show that it can easily be adapted to the extended finite element method.     

A Posteriori Error Estimates for Parabolic Variational Inequalities

We study a posteriori error estimates in the energy norm for some parabolic obstacle problems discretized with a Euler implicit time scheme combined with a finite element spatial approximation. We discuss the reliability and efficiency of the error indicators, as well as their localization properties. Apart from the obstacle resolution, the error indicators vanish in the so-called full contact set. The case when the obstacle is piecewise affine is studied before the general case. Numerical examples are given.     

Characteristic mixed finite element approximation of transient convection diffusion optimal control problems

In this paper, we investigate a characteristic mixed finite element approximation of transient convection diffusion optimal control problems. The state and the adjoint state are approximated by characteristic mixed finite element method, while the control is discretized by standard finite element method. We derive the continuous and discrete first-order optimality conditions and prove a priori error estimates for the state, the adjoint state and the control. Numerical examples are presented to illustrate the theoretical findings.     

Multidimensional exponentially fitted simplicial finite elements for convection-diffusion equations with tensor-valued diffusion

Abstract In this paper we propose and analyze an exponentially fitted simplicial finite element method for the numerical approximation of solutions to diffusion-convection equations with tensor-valued diffusion coefficients. The finite element method is first formulated using exponentially fitted finite element basis functions constructed on simplicial elements in arbitrary dimensions. Stability of the method is then proved by showing that the corresponding bilinear form is coercive. Upper error bounds for the approximate solution and the associated flux are established.     

A variational multiscale a posteriori error estimation method for mixed form of nearly incompressible elasticity

This paper presents an error estimation framework for a mixed displacement–pressure finite element method for nearly incompressible elasticity. The proposed method is based on Variational Multiscale (VMS) concepts, wherein the displacement field is decomposed into coarse scales that can be resolved by a given finite element mesh and fine scales that are beyond the resolution capacity of the mesh. Variational projection of fine scales onto the coarse-scale space via variational embedding of the fine-scale solution into the coarse-scale formulation leads to the stabilized method with two major attributes: first, it is free of volumetric locking and, second, it accommodates arbitrary combinations of interpolation functions for the displacement and pressure fields. This VMS-based stabilized method is equipped with naturally derived error estimators and offers various options for numerical computation of the error. Specifically, two error estimators are explored. The first method employs an element-based strategy and a representation of error via a fine-scale error equation defined over element interiors which is evaluated by a direct post-solution evaluation. This quantity when combined with the global pollution error results in a simple explicit error estimator. The second method involves solving the fine-scale error equation through localization to overlapping patches spread across the domain, thereby leading to an implicit calculation of the local error. This implicit calculation when combined with the global pollution error results in an implicit error estimator. The performance of the stabilized method and the error estimators is investigated through numerical convergence tests conducted for two model problems on uniform and distorted meshes. The sharpness and robustness of the estimators is shown to be consistent across the test cases employed.     

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.     

Adaptive Finite Element Methods for the Identification of Elastic Constants

In this paper, the elastic constants of a material are recovered from measured displacements where the model is the equilibrium equations for the orthotropic case. The finite element method is used for the discretization of the state equation and the Gauss–Newton method is used to solve the nonlinear least squares problem attained from the parameter estimation problem. A posteriori error estimators are derived and used to improve the accuracy by an appropriate mesh refinement. A numerical experiment is presented to show the applicability of the approach.     

An optimal order numerical quadrature approximation of a planar isoparametric eigenvalue problem on triangular finite element meshes

Abstract This paper deals with a finite element numerical quadrature method. It is applied for a class of second-order self-adjoint elliptic operators defined on a bounded domain in the plane. Isoparametric finite element transformations and triangular Lagrange finite elements are used.We establish the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a numerical quadrature finite element approximation. Thus the relationship between possible quadrature formulas and the optimal and almost optimal precision of the method is established. The emphasis of the paper is on the error analysis of the approximate eigenpairs. Numerical results confirming the theory are presented.     

Superconvergence of least-squares methods for a coupled system of elliptic equations

We present a numerical method for solving a coupled system of elliptic partial differential equations (PDEs). Our method is based on the least-squares (LS) approach. We develop ellipticity estimates and error bounds for the method. The main idea of the error estimates is the establishment of supercloseness of the LS solutions, and solutions of the mixed finite element methods and Ritz projections. Using the supercloseness property, we obtain $L_2$-norm error estimates, and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces. Moreover, we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions. Application to optimal control problems is briefly considered.     

The global approximate particular solution meshless method for two-dimensional linear elasticity problems

The two-dimensional linear elasticity equations are solved by the global method of approximate particular solution as a new meshless option to the conventional finite element discretization. The displacement components are approximated by a linear combination of the elasticity particular solutions and the stress tensor is obtained by differentiating the displacement expressions in terms of the particular solutions. The multiquadric radial basis function (RBF) is employed as the non-homogeneous term in the governing equation to compute the particular solutions. The cantilever beam and the infinite plate with a hole problem are solved to verify the implemented meshless method. For each situation, the trend of the root mean square error is assessed in terms of the shape parameter and the number of nodes. Unlike most of the RBF collocation strategies, it is found that numerical results are in good agreement with the analytical solutions for a wide range of shape parameter values.