New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation and adaptive refinement technique for fracture analysis of 2-D/3-D crack problems is the state-of-the-art. The objective of the present paper is to propose a new a posteriori error estimator based on strain energy release rate (SERR) or stress intensity factor (SIF) at the crack tip region and to use this along with the stress based error estimator available in the literature for the region away from the crack tip. The proposed a posteriori error estimator is called the K-S error estimator. Further, an adaptive mesh refinement (h-) strategy which can be used with K-S error estimator has been proposed for fracture analysis of 2-D crack problems. The performance of the proposed a posteriori error estimator and the h-adaptive refinement strategy have been demonstrated by employing the 4-noded, 8-noded and 9-noded plane stress finite elements. The proposed error estimator together with the h-adaptive refinement strategy will facilitate automation of fracture analysis process to provide reliable solutions.     

A posteriori error estimates and adaptivity for finite element solutions in finite elasticity

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.     

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.     

A posteriori error estimates for numerical solutions to inverse problems of elastography

The article deals with one of inverse problems of elastography: knowing displacement of compressed tissue finds the distribution of Young’s modulus in the investigated specimen. The direct problem is approximated and solved by the finite element method. The inverse problem can be stated in different ways depending on whether the solution to be found is smooth or discontinuous. Tikhonov regularization with appropriate regularizing functionals is applied to solve these problems. In particular, discontinuous Young’s modulus distribution can be found on the class of 2D functions with bounded variation of Hardy–Krause type. It is shown in the paper that a variant of Tikhonov regularization provides for such discontinuous distributions the so-called piecewise uniform convergence of approximate solutions as the error levels of the data vanish. The problem of practical a posteriori estimation of the accuracy for obtained approximate solutions is under consideration as well. A method of such estimation is presented. As illustrations, model inverse problems with smooth and discontinuous solutions are solved along with a posteriori estimations of the accuracy.     

A posteriori pointwise upper bound error estimates in the finite element method

An error estimate for the finite element method is presented in this paper. The error is identified as the response to a set of residual forces, and a complementary analysis provides an upper bound estimate of the global energy of the error. The inequality proposed by Babu?ka and Miller¹ is then employed to bound the error in stress and displacement at a point. The formula is derived for two-dimensional elasticity, but the procedure is general; and can be applied to three-dimensional and other problems. Numerical experiments using the procedure are carried out and the results are given for the four-node bilinear compatible element and plane stress.     

A posteriori error estimation for finite element solutions of Helmholtz' equation—Part II: estimation of the pollution error

In part I of this investigation, we proved that the standard a posteriori estimates, based only on local computations, may severely underestimate the exact error for the classes of wave-numbers and the types of meshes employed in engineering analyses. We showed that this is due to the fact that the local estimators do not measure the pollution effect inherent to the FE-solutions of Helmholtz' equation with large wavenumber. Here, we construct a posteriori estimates of the pollution error. We demonstrate that these estimates are reliable and can be used to correct the standard a posteriori error estimates in any patch of elements of interest. © 1997 John Wiley & Sons, Ltd.     

The post-processing approach in the finite element method—Part 3: A posteriori error estimates and adaptive mesh selection

This paper is the final in a series of three in which we have discussed a finite element post-processing technique. Here we shall deal with the questions of adaptive mesh selection and a posteriori error estimation. Some numerical examples computed by the FEARS program will be used to illustrate the approaches taken.     

A simple a-posteriori error estimation for adaptive BEM in elasticity

In this paper, the properties of various boundary integral operators are investigated for error estimation in adaptive BEM. It is found that the residual of the hyper-singular boundary integral equation (BIE) can be used for a-posteriori error estimation for different kinds of problems. Based on this result, a new a-posteriori error indicator is proposed which is a measure of the difference of two solutions for boundary stresses in elastic BEM. The first solution is obtained by the conventional boundary stress calculation method, and the second one by use of the regularized hyper-singular BIE for displacement derivative. The latter solution has recently been found to be of high accuracy and can be easily obtained under the most commonly used C ⁰ continuous elements. This new error indicator is defined by a L ₁ norm of the difference between the two solutions under Mises stress sense. Two typical numerical examples have been performed for two-dimensional (2D) elasticity problems and the results show that the proposed error indicator successfully tracks the real numerical errors and effectively leads a h-type mesh refinement procedure.     

A posteriori error estimation for finite element solutions of Helmholtz’ equation. part I: the quality of local indicators and estimators

This paper contains a first systematic analysis of a posteriori estimation for finite element solutions of the Helmholtz equation. In this first part, it is shown that the standard a posteriori estimates, based only on local computations, severely underestimate the exact error for the classes of wave numbers and the types of meshes employed in engineering analysis. This underestimation can be explained by observing that the standard error estimators cannot detect one component of the error, the pollution error, which is very significant at high wave numbers. Here, a rigorous analysis is carried out on a one-dimensional model problem. The analytical results for the residual estimator are illustrated and further investigated by numerical evaluation both for a residual estimator and for the ZZ-estimator based on smoothening. In the second part, reliable a posteriori estimators of the pollution error will be constructed. © 1997 by John Wiley & Sons, Ltd.     

An <Emphasis Type="Italic">hp</Emphasis>-adaptive finite element/boundary element coupling method for electromagnetic problems

We present an hp-version of the finite element / boundary element coupling method to solve the eddy current problem for the time-harmonic Maxwell’s equations. We use H(curl, Ω -conforming vector-valued polynomials to approximate the electric field in the conductor Ω and surface curls of continuous piecewise polynomials on the boundary Γ of Ω to approximate the twisted tangential trace of the magnetic field on Γ. We present both a priori and a posteriori error estimates together with a three-fold hp-adaptive algorithm to compute the fem/bem coupling solution with appropriate distributions of polynomial degrees on suitably refined meshes.     

Prediction of material coupling effect on structural damping of composite beams and blades

The present paper investigates the effect of material coupling on static and modal characteristics of composite structures. Incorporation of stiffness and damping coupling terms into a beam formulation yields equivalent section stiffness and damping properties. Building upon the damping mechanics, an extended beam finite element is developed capable of providing the stiffness and damping matrices of the structure. Validation cases on beams and blades demonstrate the importance of all stiffness and damping terms. Numerical results validate the predicted effect of material coupling on static characteristics of composite box-section beams. The effect of the full coupling damping matrices on modal frequencies and structural modal damping of composite beams is investigated. Box-section beams and small blade models with various ply angle laminations at the girder segments are considered. Finally, the developed finite element is applied to the prediction of the modal characteristics of a 19 m realistic wind-turbine model blade.     

A priori and a posteriori analysis of the meshless Galerkin boundary node method for three-dimensional elasticity

The meshless Galerkin boundary node method is presented in this paper for boundary-only analysis of three-dimensional elasticity problems. In this method, boundary conditions can be implemented directly and easily despite the employed moving least-squares shape functions lack the delta function property, and the resulting system matrices are symmetric and positive definite. A priori error estimates and the consequent rate of convergence are presented. A posteriori error estimates are also provided. Reliable and efficient error estimators and an efficient and convergent adaptive meshless algorithm are then derived. Numerical examples showing the efficiency of the method, confirming the theoretical properties of the error estimates, and illustrating the capability of the adaptive algorithm, are reported.     

A posteriori estimates of the solution error caused by discretization in the finite element,finite difference and boundary element methods

A theory is described which guarantees an upper and lower bound estimate of the discretization error in numerical solutions of elliptic boundary value problems. This method gives bounded global estimates of the error in the energy norm. Pointwise estimates of the error in the solution variable or its derivatives can then be obtained if the numerical solution is exhibiting pointwise monotonic convergence. The versatility of this method is illustrated by its application to numerical solutions from finite element, finite difference and boundary element methods.     

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part I: Stress recovery and a posteriori error estimation

In this study, an adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D elastostatic problems is suggested. This adaptive refinement procedure is based on the Zienkiewicz and Zhu (ZZ) error estimator for the a posteriori error estimation and an adaptive finite point mesh generator for new point mesh generation. The presentation of the work is divided into two parts. In Part I, concentration will be paid on the stress recovery and the a posteriori error estimation processes for the RKPM. The proposed error estimator is different from most recovery type error estimators suggested previously in such a way that, rather than using the least-squares fitting approach, the recovery stress field is constructed by an extraction function approach. Numerical studies using 2D benchmark boundary value problems indicated that the recovered stress field obtained is more accurate and converges at a higher rate than the RKPM stress field. In Part II of the study, concentration will be shifted to the development of an adaptive refinement algorithm for the RKPM.     

Stress intensity factor error index for finite element analysis with singular elements

An error index for the stress intensity factor (SIF) obtained from the finite element analysis results using singular elements is proposed. The index was developed by considering the facts that the analytical function shape of the crack tip displacement is known and that the SIF can be evaluated from the displacements only. The advantage of the error index is that it has the dimension of the SIF and converges to zero when the actual error of the SIF by displacement correlation technique converges to zero. Numerical examples for some typical crack problems, including a mixed mode crack, whose analytical solutions are known, indicated the validity of the index. The degree of actual SIF error seems to be approximated by the value of the proposed index.     

A model for calculating geometry factors for a mixed-mode I–II single edge notched tension specimen

In the present study, a novel approach is presented to obtain closed-form solutions for the geometry factors, which are used to determine the stress intensity factors for various configurations. A single edge notched tension specimen with an angled-crack is used as an example to demonstrate the applicability, simplicity and flexibility of the new approach. Several values for crack inclination angles, plate widths and crack lengths, including micro-cracks, are considered in the analysis. The new approach is validated through comparison with existing analytical and numerical solutions as well as experimental results.     

A modification of the Petrov–Galerkin method for the transient convection–diffusion equation

A variation of the Petrov–Galerkin method of solution of a partial differential equation is presented in which the weight function applied to the time derivative term of the transient convection–diffusion equation is different from the weight function applied to the special derivatives. This allows for the formulation of fourth-order explicit and centred difference schemes. Comparison with analytic solutions show that these methods are able to capture steep wave fronts. The ability of the explicit method to capture wave fronts increases as the amount of convective transport increases.     

Mismatch limit loads and approximate <Emphasis Type="Italic">J</Emphasis> estimates for tensile plates with constant-depth surface cracks in the center of welds

The present work provides mismatch limit loads and approximate J estimates for tensile plates with constant-depth, part-through surface cracks in the center of the weld metal. Based on systematic three-dimensional FE limit analyses, effects of strength mismatch related variables on limit loads are firstly quantified by the strength mismatch ratio and one geometry-related parameter. Mismatch limit loads for part-through surface cracks are then correlated to those for two-dimensional, through-wall crack problems. Based on the proposed limit load solutions, the applicability of the reference stress based J estimates is also investigated. When the reference stress is defined by the mismatch limit load, predicted J values agree overall well with FE results.     

Random response of a rotating composite blade with flexure–torsion coupling effect by the finite element method

A finite element model is employed to investigate the mean square response of a damped rotating composite blade with flexure–torsion interaction under stationary or non-stationary random excitation. The effects of transverse shear deformation and rotary inertia are considered. The finite element model can satisfy all the geometric and natural boundary conditions of a thick blade. The blade is considered to be subjected to white noise, band-limited white noise or filtered white noise excitation. The numerical results indicate that the increment of rotational speed will reduce the mean square response. It is also found that the mean square response decreases when the low natural frequency of base decreases. Inversely, the mean square response increases when the high natural frequency of base decreases. It is also shown that the fiber orientations have a significant effect on the mean square response of an orthotropic blade under random excitations. Moreover, the flexure–torsion coupling effect on the mean square response is changed by different fiber orientations.