Navier-Stokes方程的集中质量非协调有限元法

本文通过所谓的速度-压力型公式讨论了Navier-Stokes方程的集中质量非协调有限元法(半离散情形)。首先给出了所讨论方程的集中质量非协调有限元逼近格式,其次对所讨论方程的真解与逼近格式的解之间的误差进行了分析,最后利用Navier-Stokes投影算子及其性质,得到了在确定模意义下的速度、压力误差估计,且某些误差估计能达到最优。     

On error estimates and adaptivity in elastoplastic solids: Applications to the numerical simulation of strain localization in classical and Cosserat continua

The a posteriori error estimates based on the post-processing approach are introduced for elastoplastic solids. The standard energy norm error estimate established for linear elliptic problems is generalized here to account for the presence of internal variables through the norm associated with the complementary free energy. This is known to represent a natural metric for the class of elastoplastic problems of evolution. In addition, the intrinsic dissipation functional is utilized as a basis for a complementary a posteriori error estimates. A posteriori error estimates and adaptive refinement techniques are applied to the finite element analysis of a strain localization problem. As a model problem, the constitutive equations describing a generalization of standard J₂-elastoplasticity within the Cosserat continuum are used to overcome serious limitations exhibited by classical continuum models in the post-instability region. The proposed a posteriori error estimates are appropriately modified to account for the Cosserat continuum model and linked with adaptive techniques in order to simulate strain localization problems. Superior behaviour of the Cosserat continuum model in comparison to the classical continuum model is demonstrated through the finite element simulation of the localization in a plane strain tensile test for an elastopiastic softening material, resulting in convergent solutions with an h-refinement and almost uniform error distribution in all considered error norms.     

Finite element analysis for a class of nonlinear variational inequalities

In this paper, we prove that the error estimates for the finite element approximation of a class of variational inequalities arising in elastostatics is of order h in the energy norm. In fact, our estimates improve all of the previously known error estimates for elliptic variational inequalities.     

Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models

In this paper, stabilized Crank-Nicolson/Adams-Bashforth schemes are presented for the Allen-Cahn and Cahn-Hilliard equations. It is shown that the proposed time discretization schemes are either unconditionally energy stable, or conditionally energy stable under some reasonable stability conditions. Optimal error estimates for the semi-discrete schemes and fully-discrete schemes will be derived. Numerical experiments are carried out to demonstrate the theoretical results.     

On computational methods for variational inequalities

In this note, we focus on optimised mesh design for the Finite Element (FE) method for variational inequalities using global norm estimates for local error control. The strategies are based on the so called dual-weighted-residual (DWR) approach to a posteriori error control for FE-schemes (see, e.g., Rannacher et al. [19, 6, 2]), where error control for the primal problem is established by solving an auxiliary (dual) problem. In this context we blamed (cf. e.g., Rannacher and Suttmeier [18, 19]) global norm estimates being not that useful in applications. But having a closer look at the DWR-concept, one observes that in fact global (energy) error bounds can be employed to establish local error control. Our ideas and techniques are illustrated at the socalled obstacle problem.It turns out, that reliable and efficient energy error control is one main ingredient to establish useful a posteriori error bounds for local quantities. Therefore, in addition, we derive an unified approach to a posteriori error control in the energy norm for elliptic variational inequalities of first kind. Eventually, this framework is applied to Signorinis problem.     

A posteriori error control in finite element methods via duality techniques: Application to perfect plasticity

In this paper a new technique for a posteriori error control and adaptive mesh design is presented for finite element models in perfect plasticity. The approach is based on weighted a posteriori error estimates derived by duality arguments as proposed in Becker and Rannacher (1996) and Rannacher and Suttmeier (1997) for linear problems. The conventional strategies for mesh refinement in finite element methods are mostly based on a posteriori error estimates for the global energy norm in terms of local residuals of the computed solution. These estimates reflect the approximation properties of the trial functions by local interpolation constants while the stability property of the continuous model enters through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in computing local quantities as point values or contour integrals and in the case of nonlinear material behavior. More accurate and efficient error estimation can be achieved by using suitable weights which can be obtained numerically in the course of the refinement process from the solutions of linearized dual problems. This feed-back approach is developed here for primal-mixed finite element models in linear-elastic perfect plasticity.     

The post-processing approach in the finite element method—part 1: Calculation of displacements,stresses and other higher derivatives of the displacements

This is the first in a series of three papers in which we discuss a method for ‘post-processing’ a finite element solution to obtain high accuracy approximations for displacements, stresses, stress intensity factors, etc. Rather than take the values of these quantities ‘directly’ from the finite element solution, we evaluate certain weighted averages of the solution over the entire region. These yield approximations are of the same order of accuracy as the strain energy. We obtain error estimates, and also present some numerical examples to illustrate the practical effectiveness of the technique. In the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation.     

A feed-back approach to error control in finite element methods: application to linear elasticity

Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.     

A-posteriori error estimates for the finite element method

Computable a-posteriori error estimates for finite element solutions are derived in an asymptotic form for h → 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one-dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self-adjoint operator of the second order. Generalizations to more general one-dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a-posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.     

Adaptive component mode synthesis in linear elasticity

Component mode synthesis (CMS) is a classical method for the reduction of large‐scale finite element models in linear elasticity. In this paper we develop a methodology for adaptive refinement of CMS models. The methodology is based on a posteriori error estimates that determine to what degree each CMS subspace influence the error in the reduced solution. We consider a static model problem and prove a posteriori error estimates for the error in a linear goal quantity as well as in the energy and L² norms. Automatic control of the error in the reduced solution is accomplished through an adaptive algorithm that determines suitable dimensions of each CMS subspace. The results are demonstrated in numerical examples. Copyright © 2010 John Wiley & Sons, Ltd.     

A priori error estimation of finite element models from first principles

The quality of finite element computational results can be assessed only by providing rational criteria for evaluating errors. Most exercises in this direction are based ona posteriori error estimates, based primarily on experience and intuition. If finite element analysis has to be considered a rational science, it is imperative that procedures to computea priori error estimates from first principles are made available. This paper captures some efforts in this direction.     

Convergence analysis of a hierarchical enrichment of Dirichlet boundary conditions in a mesh‐free method

Implementation of Dirichlet boundary conditions in mesh‐free methods is problematic. In Wagner and Liu (International Journal for Numerical Methods in Engineering 2001; 50 :507), a hierarchical enrichment technique is introduced that allows a simple implementation of the Dirichlet boundary conditions. In this paper, we provide some error analysis for the hierarchical enrichment mesh‐free technique. We derive optimal order error estimates for the hierarchical enrichment mesh‐free interpolants. For one‐dimensional elliptic boundary value problems, we can directly apply the interpolation error estimates to obtain error estimates for the mesh‐free solutions. For higher‐dimensional problems, derivation of error estimates for the mesh‐free solutions depends on the availability of an inverse inequality. Numerical examples in 1D and 2D are included showing the convergence behaviour of mesh‐free interpolants and mesh‐free solutions when the hierarchical enrichment mesh‐free technique is employed. Copyright © 2001 John Wiley & Sons, Ltd.     

Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM

In this paper a new technique aimed to obtain accurate estimates of the error in energy norm using a moving least squares (MLS) recovery-based procedure is presented. In the techniques based on the superconvergent patch recovery (SPR) the continuity of the recovered field is provided by the shape functions of the underlying mesh. We explore the capabilities of a recovery technique based on an MLS fitting, more flexible than SPR techniques as it directly provides continuous interpolated fields without relying on any FE mesh, to obtain estimates of the error in energy norm as an alternative to SPR. In the enhanced MLS proposed in the paper, boundary equilibrium is enforced using a nearest point approach that modifies the MLS functional. Lagrange multipliers are used to impose a nearly exact satisfaction of the internal equilibrium equation. The numerical results indicate the high accuracy of the proposed error.     

An error estimate in the EFG method

In this paper, local and global error estimates for the element-free Galerkin (EFG) method are proposed. The essence of proposed error estimates is to use the difference between the values of the projected stress and these given directly by the EFG solution. The stress projection can be obtained simply by taking product of shape function based on a different domain of influence with the stresses at nodes. In this study, it was found that the effectivity index is optimized if the domain of influence in stress projection procedure is the smallest that retains regularity of the matrices in EFG. Numerical tests are shown for various 1D and 2D examples illustrating the good effectiveness of the proposed error estimator in the global energy norm and in the local error estimates.     

抛物问题的质量集中非协调有限元法

主要讨论了一类抛物问题的质量集中非协调有限元方法。首先,我们给出了所讨论问题的质量集中非协调有限元Crank-Nicolson全离散逼近格式。其次,对讨论问题的解与所给出逼近格式的解之间的误差估计进行了分析研究。最后利用椭圆投影算子,我们得到了关于L2模和能量模方面的一些误差估计式。     

A short note on the effect of sample size on the estimation error in Cp

Process capability indices such as C_p are used extensively in manufacturing industries to assess processes in order to decide about purchasing. In practice, the parameter for calculating C_p is rarely known and is frequently replaced with estimates from an in-control reference sample. This article explores the optimal sample size required to achieve a desired error of estimation using absolute percentage error of different C_p estimates. Moreover, some practical tools are created to allow practitioners to find sample size in different situations.     

Error estimates for frequency responses calculated from time-domainmeasurements

Estimates are derived for the errors in frequency responses calculated from measured step or impulse response data. Three error sources are considered-aliasing errors, errors due to random noise added to the amplitudes of the data, and errors due to random noise added to the time coordinates of the data. Two types of error estimates are derived. Pre-measurement estimates are based on easily determined parameters of the signal and noise; post-measurement estimates are based on the difference between two measurements of the same signal. The post-measurement error estimates apply to additional sources of error. The goal of the paper is to give easily used error estimates for easily implemented methods rather than to present the most sophisticated methods     

Two-Grid Finite Element Methods for the Steady Navier-Stokes/Darcy Model

Two-grid finite element methods for the steady Navier-Stokes/Darcy model are considered. Stability and optimal error estimates in the $H^1$-norm for velocity and piezometric approximations and the $L^2$-norm for pressure are established under mesh sizes satisfying $h=H^2$. A modified decoupled and linearised two-grid algorithm is developed, together with some associated optimal error estimates. Our method and results extend and improve an earlier investigation, and some numerical computations illustrate the efficiency and effectiveness of the new algorithm.     

Practical methods for a posteriori error estimation in engineering applications

This work presents an extension of the goal‐oriented error estimation technique to the engineering analysis of three‐dimensional linear elastic bodies. In the series of examples shown, the errors are estimated with respect to local displacement and stress components. The paper also introduces novel means to compute lower bounds on the error in the energy norm based on a cost‐effective postprocessing of the upper bound error estimates. The numerical results indicate that the method can be used effectively for complex engineering applications. Copyright © 2003 John Wiley & Sons, Ltd.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.