Adaptive finite element procedures in elasticity and plasticity

The paper discusses error estimation and h-adaptive finite element procedures for elasticity and plasticity problems. For the spatial discretization error, an enhanced Superconvergent Patch Recovery (SPR) technique which improves the error estimation by including fulfillment of equilibrium and boundary conditions in the smoothing procedure is discussed. It is known that an accurate error estimation on an early stage of analysis results in a more rapid and optimal adaptive process. It is shown that node patches and element patches give similar quality of the postprocessed solution. For dynamic problems, a postprocessed type of error estimate and an adaptive procedure for the semidiscrete finite element method are discussed. It is shown that the procedure is able to update the spatial mesh and the time step size so that both spatial and time discretization errors are controlled within specified tolerances. A time-discontinuous Galerkin method for solving the second-order ordinary differential equations in structural dynamics is also presented. Many advantages of the new approach such as high order accuracy, possibility to filter effects of spurious modes and convenience to apply adaptive analysis are observed. For plasticity problems, some recent work that improved plastic strains and plastic localization is discussed.     

Postprocessing techniques and h-adaptive finite element-eigenproblem analysis

The paper presents postprocessing techniques based on locally improved finite element (FE) solutions of the basic field variables. This opens up the possibility to control both “strain energy” terms and “kinetic energy” terms in the governing equations. The proposed postprocessing technique on field variables is essentially a least square fit of the prime variables (displacements) at superconvergent points. Its performance is compared with other well-known techniques, showing a good performance. A h-adaptive FE strategy for acoustic problems is presented where, for adaptive mesh generation and remeshing the commercial software package i-deas has been applied and for the FE analysis the commercial software package sysnoise. The paper also presents an adaptive h-version FE approach to control the discretisation error in free vibration analysis. The postprocessing technique used here is a mix of local and global updating methods. Rapid convergence of the preconditioned conjugate gradient method is enhanced by choosing the initial trial eigenmodes as the superconvergent patch recovery technique for displacements improved FE eigenmodes. Numerical examples show nice properties of the final local and global updated solution as a basis for an error estimator and the error indicator in an adaptive process.     

An improved superconvergent patch recovery technique for the axisymmetrical problems

The enhancement of the standard post processing technique proposed by Zienkiewicz-Zhu[1] is studied for the axisymmetrical problems using 4-node quadrilateral elements. We have implemented and compared the various a posteriori error estimators in this category. The presented numerical results show that the modification applied to the standard superconvergent patch recovery method improves the reliability of this procedure. As an outcome of our study, the reliability of error estimator based on averaging and extrapolation technique seems indubitably questionable.     

Superconvergence of Discontinuous Finite Element Solutions for Transient Convection–diffusion Problems

We present a study of the local discontinuous Galerkin method for transient convection–diffusion problems in one dimension. We show that p-degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δx ^p+2) superconvergent at Radau points. For diffusion- dominated problems, the solution’s derivative is O(Δx ^p+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Object-Oriented Programming in Field Recovery and Error Estimation

This paper considers an object-oriented implementation of a generic program module for field recovery and recovery-based error stimation. The field recovery is based on the superconvergent patch recovery technique by Zienkiewicz and Zhu. The current implementation is problem independent, and is organized as a set of C++ classes based on the software library Diffpack. The program may be run stand-alone as a post-processor, reading finite element data and results from ASCII files. Alternatively, it may be linked into existing simulation codes through a Fortran interface, thereby enabling error estimation within the time-step loop of a transient simulation. The use of the field recovery and error estimation module is demonstrated on an isotropic linear elastic problem with known analytical solution, such that also the exact error, and not only the estimated error, may be computed. The computational efficiency of the object-oriented module is assessed by comparing the time consumption with a similar program implemented in Fortran.     

The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis

In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h ^p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h ^p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.     

An adaptive mesh refinement of quadrilateral finite element meshes based upon a posteriori error estimation of quantities of interest: linear static response

A posteriori error estimation in finite element analysis serves as an important guide to the meshing tool in an adaptive refinement process. However, the traditional posteriori error estimates, which are often defined in the energy or energy-type norms over the entire domain, provide users insufficient information regarding the accuracy of specific quantities in the solution. This paper describes an adaptive quadrilateral refinement process with a goal-oriented error estimation, in which a posteriori error is estimated with respect to the specified quantity of interest. A highlight of this paper is the demonstration of tools described in the paper used in a practical industrial environment. The performance of this process is demonstrated on several practical problems where the comparison is with the adaptive process based on the traditional error estimation.     

H-R自适应边界元的模糊误差估计方法

为克服在利用传统自适应边界元法求解弹性问题时需对不同问题设计不同误差估计公式的缺点,以专家经验为基础,利用模糊逻辑理论,提出一种新的误差分析方法.基于H-R自适应边界元法,用FORTRAN编写求解2个经典平面弹性静力学问题的程序.分析表明该误差分析方法能较好地估计边界元解的误差.     

An improved adaptive formulation for the computation of limit analysis problems of thin axisymmetrical shells

An improved adaptive technique for the finite element formulation of limit analysis problems of symmetrically loaded thin shells of revolution is presented. Bending, membrane behavior and changes of curvature are simulated by using an axisymmetrical thin shell element of three nodes associated with a new piecewise linear (PWL) yield surface. The results obtained show better upper bounds when compared to other numerical and analytical solutions reported in the literature. Finally, an a posteriori error measure is re-written to include the new displacement rate and PWL strain rate fields, resulting in an adaptive scheme for refining the finite element mesh.     

Interactive-adaptive geometrically nonlinear analysis of shell structures

This paper presents an investigation of interactive-adaptive techniques for nonlinear finite element structural analysis. In particular, effective methods leading to reliable automated, finite element solutions of nonlinear shell problems are of primary interest here. This includes automated adaptive nonlinear solution procedures based on error estimation and adaptive step length control, reliable finite elements that account for finite deformations and finite rotations, three-dimensional finite element modeling, and an easy-to-use, easy-to-learn graphical user interface with three-dimensional graphics. A computational environment, which interactively couples a comprehensive geometric modeler, an automatic three-dimensional mesh generator and an advanced nonlinear finite element analysis program with real-time computer graphics and animation tools, is presented. Three examples illustrate the merit and potential of the approaches adopted here and confirm the feasibility of developing fully automated computer aided engineering environments.     

Derivative calculation from finite element solutions

A technique is considered whereby very accurate derivatives of a finite element solution can be calculated efficiently. It is demonstrated here that most of the necessary quantities for this subsidiary computation are available as computed by-products in the preceding finite element solution procedure. The calculation is shown in this note to be a particular form of a procedure for which superconvergent theoretical error estimates have been proven elsewhere. Numerical experiments confirm the superior accuracy in the computed derivative (stress or flux).     

Adaptive space–time finite element methods for the wave equation on unbounded domains

Comprehensive adaptive procedures with efficient solution algorithms for the time-discontinuous Galerkin space–time finite element method (DGFEM) including high-order accurate nonreflecting boundary conditions (NRBC) for unbounded wave problems are developed. Sparse multi-level iterative schemes based on the Gauss–Seidel method are developed to solve the resulting fully-discrete system equations for the interior hyperbolic equations coupled with the first-order temporal equations associated with auxiliary functions in the NRBC. Due to the local nature of wave propagation, the iterative strategy requires only a few iterations per time step to resolve the solution to high accuracy. Further cost savings are obtained by diagonalizing the mass and boundary damping matrices. In this case the algebraic structure decouples the diagonal block matrices giving rise to an explicit multi-corrector method. An h-adaptive space–time strategy is employed based on the Zienkiewicz–Zhu spatial error estimate using the superconvergent patch recovery (SPR) technique, together with a temporal error estimate arising from the discontinuous jump between time steps of both the interior field solutions and auxiliary boundary functions. For accurate data transfer between meshes, a new enhanced interpolation (EI) method is developed and compared to standard interpolation and projection. Numerical studies of transient radiation and scattering demonstrate the accuracy, reliability and efficiency gained from the adaptive strategy.     

An Adaptive SDG Method for the Stokes System

Staggered grid techniques are attractive ideas for flow problems due to their more enhanced conservation properties. Recently, a staggered discontinuous Galerkin method is developed for the Stokes system. This method has several distinctive advantages, namely high order optimal convergence as well as local and global conservation properties. In addition, a local postprocessing technique is developed, and the postprocessed velocity is superconvergent and pointwisely divergence-free. Thus, the staggered discontinuous Galerkin method provides a convincing alternative to existing schemes. For problems with corner singularities and flows in porous media, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, we will derive a computable error indicator for the staggered discontinuous Galerkin method and prove that this indicator is both efficient and reliable. Moreover, we will present some numerical results with corner singularities and flows in porous media to show that the proposed error indicator gives a good performance.     

Adaptive multiple-levelh-refinement in automated finite element analyses

This paper discusses an automatic, adaptive finite element modeling system consisting of mesh generation, finite element analysis, and error estimation. The individual components interact with one another and efficiently reduce the finite element error to within an acceptable value and perform only a minimum number of finite element analyses.One of the necessary components in the automated system is a multiple-level local remeshing algorithm. Givenh-refinement information provided by an a posteriori error estimator, and adjacency information available in the mesh data structures, the local remeshing algorithm grades the refinement toward areas requesting refinement. It is shown that the optimal asymptotic convergence rate is achieved, demonstrating the effectiveness of the intelligent multiple-level localh-refinement.     

The Superconvergent Cluster Recovery Method

A new gradient recovery technique SCR (Superconvergent Cluster Recovery) is proposed and analyzed for finite element methods. A linear polynomial approximation is obtained by a least-squares fitting to the finite element solution at certain sample points, which in turn gives the recovered gradient at recovering points. Compared with similar techniques such as SPR and PPR, our approach is cheaper and efficient, while having same or even better accuracy. In additional, it can be used as an a posteriori error estimator, which is relatively simple to implement, cheap in terms of storage and computational cost for adaptive algorithms. We present some numerical examples illustrating the effectiveness of our recovery procedure.     

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O(h^p+1) and is spanned by two (p+1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of (p+1)-degree Radau polynomial. For a linear model problem, the p-degree discontinuous Galerkin solution flux exhibits a strong O(h^2p+2) local superconvergence on average at the element outflow boundary. We further establish an O(h^2p+1) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.     

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。     

h andh-p version error estimation and adaptive procedures from theory to practice

In this paper we introduce techniques that allow us to define a posteriori error estimators via well-known recovery techniques. These allow us to construct a posteriori error estimators for relatively general problems. Further, we introduce new adaptive procedures that make use of these estimators and, in particular, describe anh-p procedure that is simple to implement and that, as numerical experiments have shown, attains an accelerated rate of convergence expected from theh-p version.