Automatic adaptive refinement for shell analysis using nine-node assumed strain element

An automatic adaptive refinement procedure for the analysis of shell structures using the nine-node degenerated solid shell element is suggested. The basic adaptive refinement principle and the effects of singularities and boundary layers on the convergence rate of the nine-node element used are discussed. A new stress recovery procedure based on the patch convective co-ordinate system concept is developed for the construction of a continuous smoothed stress field over the shell domains. The stress recovery procedure is easy to implement, requires a modest computational effort and needs only local patch information. It can be applied to shells with non-uniform thickness as well as to multi-layered shell structures. The smoothed recovered stress obtained is then used with the Zienkiewicz and Zhu error estimator for a posteriori error estimation during the adaptive refinement analysis. Numerical results which are in good agreement with theoretical predictions are obtained and they indicate that the current adaptive refinement procedure can eliminate the effect of singularities inside the problem domains so that a near-optimal convergence rate is achieved in all the numerical examples. This also indicates that the stress recovery procedure can produce an accurate stress field and as a result the error estimator can reflect the error distribution of the finite element solution. Even though in the current study only one type of element is used in the analysis, the whole adaptive refinement scheme can be readily applied to any other types of degenerated solid element. © 1997 John Wiley & Sons, Ltd.     

A general methodology for structural shape optimization problems using automatic adaptive remeshing

The proposed methodology is based on the use of the adaptive mesh refinement (AMR ) techniques in the context of 2D shape optimization problems analysed by the finite element method. A suitable and very general technique for the parametrization of the optimization problem, using B-splines to define the boundary, is first presented. Then mesh generation, using the advancing frontal method, the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems In particular, the analytical sensitivity analysis of the different items ruling the problem (B-splines. finite element mesh, structural behaviour and error estimator) is studied in detail. The sensitivities of the finite element mesh and error estimator permit their projection from one design to the next one leading to an a priori knowledge of the finite element error distribution on the new design without the necessity of any additional structural analysis. With this information the mesh refinement criterion permits one to build up a finite element mesh on the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked by means of several examples.     

An automatic adaptive refinement finite element procedure for 2D elastostatic analysis

An automatic adaptive refinement procedure for finite element analysis is presented. The procedure is applied to two-dimensional elastostatic problems to obtain solutions of prescribed accuracy. Through the combined use of new mesh generator using contour developed by Lo¹ and the concept of strain energy concentration, high-quality graded finite element meshes are generated. The whole process is fully automatic and no user intervention is required during the successive cycles of the mesh refinements. The Zienkiewicz and Zhu² error estimator is found to be effective and has been adopted for the present implementation. In the numerical examples tested, the error estimator gives an accurate error norm estimation and the effectivity index of the estimator converges to a value close to unity.     

On using degenerated solid shell elements in adaptive refinement analysis

Three different degenerated shell elements are studied in an adaptive refinement procedure for the solution of shell problems. The stress recovery procedure expressed in a convective patch co‐ordinate system is used for the construction of continuous smoothed stress fields for the a posteriori error estimation. The performance of the stress recovery procedure, the error estimator and the adaptive refinement strategy are tested by solving three benchmark shell problems. It is found that when adaptive refinement is used, the adverse effects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates. It is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations. In addition, if complete Lagrangian polynomial terms are used, the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed. Copyright © 1999 John Wiley & Sons, Ltd.     

Automatic adaptive refinement for plate bending problems using Reissner–Mindlin plate bending elements

The influence of the presence of singular points and boundary layers associated with the edge effects in a Reissner–Mindlin (RM) plate in the design of an optimal mesh for a finite element solution is studied, and methods for controlling the discretization error of the solution are suggested. An effective adaptive refinement strategy for the solution of plate bending problems based on the RM plate bending model is developed. This two-stage adaptive strategy is designed to control both the total and the shear error norms of a plate in which both singular points and boundary layers are present. A series of three different order assumed strain RM plate bending elements has been used in the adaptive refinement procedure. The locations of optimal sampling points and the effect of element shape distortions on the theoretical convergence rate of these elements are given and discussed. Numerical experiments show that the suggested refinement procedure is effective and that optimally refined meshes can be generated. It is also found that all the plate bending elements used can attain their full convergence rates regardless of the presence of singular points and boundary layers inside the problem domain. Boundary layer effects are well captured in all the examples tested and the use of a second stage of refinement to control the shear error is justified. In addition, tests on the Zienkiewicz–Zhu error estimator show that their performances are satisfactory. Finally, tests of the relative effectiveness of the plate bending elements used have also been made and it is found that while the higher order cubic element is the most accurate element tested, the quadratic element tested is the most efficient one in terms of CPU time used. © 1998 John Wiley & Sons, Ltd.     

Adaptive mesh refinement of the boundary element method for potential problems by using mesh sensitivities as error indicators

This paper presents a novel method for error estimation and h-version adaptive mesh refinement for potential problems which are solved by the boundary element method (BEM). Special sensitivities, denoted as mesh sensitivities, are used to evaluate a posteriori error indicators for each element, and a global error estimator. A mesh sensitivity is the sensitivity of a physical quantity at a boundary node with respect to perturbation of the mesh. The element error indicators for all the elements can be evaluated from these mesh sensitivities. Mesh refinement can then be performed by using these element error indicators as guides.The method presented here is suitable for both potential and elastostatics problems, and can be applied for adaptive mesh refinement with either linear or quadratic boundary elements. For potential problems, the physical quantities are potential and/or flux; for elastostatics problems, the physical quantities are tractions/displacements (or tangential derivatives of displacements). In this paper, the focus is on potential problems with linear elements, and the proposed method is validated with two illustrative examples. However, it is easy to extend these ideas to elastostatics problems and to quadratic elements.The computing for this research has been supported by the Cornell National Supercomputer Facility.     

An automatic adaptive refinement procedure using triangular and quadrilateral meshes

An automatic adaptive refinement procedure for finite element analysis for two-dimensional stress analysis problems is presented. Through the combined use of the new mesh generator developed by the authors (to appear) for adaptive mesh generation and the Zienkiewicz-Zhu [Int. J. numer. Meth. Engng31, 1331–1382 (1992)] error estimator based on the superconvergent patch recovery technique, an adaptive refinement procedure can be formulated which can achieve the aimed accuracy very economically in one or two refinement steps. A simple method is also proposed to locate the existence and the position of singularities in the problem domain. Hence, little or no a priori knowledge about the location and strength of the singularities is required. The entire adaptive refinement procedure has been made fully automatic and no user intervention during successive cycles of mesh refinements is needed. The robustness and reliability of the refinement procedure have been tested by solving difficult practical problems involving complex domain geometry with many singularities. We found that in all the examples studied, regardless of the types of meshes employed, triangular and quadrilateral meshes, nearly optimal overall convergence rate is always achieved.     

Automatic adaptive FE analysis of thin‐walled structures using 3D solid elements

In this study, a new automatic adaptive refinement procedure for thin‐walled structures using 3D solid elements is suggested. This procedure employs a specially designed superconvergent patch recovery (SPR) procedure for stress recovery, the Zienkiewicz and Zhu (Z–Z) error estimator for the a posteriori error estimation, a new refinement strategy for new element size prediction and a special mesh generator for adaptive mesh generation. The proposed procedure is different from other schemes in such a way that the problem domain is separated into two distinct parts: the shell part and the junction part. For stress recovery and error estimation in the shell part, special nodal coordinate systems are used and the stress field is separated into two components. For the refinement strategy, different procedures are employed for the estimation of new element sizes in the shell and the junction parts. Numerical examples are given to validate the effectiveness of the suggested procedure. It is found that by using the suggested refinement procedure, when comparing with uniform refinement, higher convergence rates were achieved and more accurate final solutions were obtained by using fewer degrees of freedoms and less amount of computational time. Copyright © 2008 John Wiley & Sons, Ltd.     

EXTENSION OF THE ZIENKIEWICZ–ZHU ERROR ESTIMATOR TO SHAPE SENSITIVITY ANALYSIS

The application of the Zienkiewicz–Zhu estimator was extended to the estimation of the discretization error arising from shape sensitivity analysis using the finite element method. The sensitivity error was quantified from the sensitivity of the energy norm by using an estimator specially developed for this purpose. Sensitivity analyses were carried out using the discrete analytical approach, which introduced no additional errors other than the discretization error. In this work, direct nodal averaging was used for linear triangular elements and the SPR technique for quadratic elements in order to obtain the smoothed stress and sensitivities fields. Two examples with an exact solution are used to analyse the effectivity of the proposed estimator and its convergence with the h-adaptive refinement. © 1997 by John Wiley & Sons, Ltd.     

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.     

An automatic adaptive refinement procedure for the reproducing kernel particle method. Part II: Adaptive refinement

In Part II of this study, an automatic adaptive refinement procedure using the reproducing kernel particle method (RKPM) for the solution of 2D linear boundary value problems is suggested. Based in the theoretical development and the numerical experiments done in Part I of this study, the Zienkiewicz and Zhu (Z–Z) error estimation scheme is combined with a new stress recovery procedure for the a posteriori error estimation of the adaptive refinement procedure. By considering the a priori convergence rate of the RKPM and the estimated error norm, an adaptive refinement strategy for the determination of optimal point distribution is proposed. In the suggested adaptive refinement scheme, the local refinement indicators used are computed by considering the partition of unity property of the RKPM shape functions. In addition, a simple but effective variable support size definition scheme is suggested to ensure the robustness of the adaptive RKPM procedure. The performance of the suggested adaptive procedure is tested by using it to solve several benchmark problems. Numerical results indicated that the suggested refinement scheme can lead to the generation of nearly optimal meshes for both smooth and singular problems. The optimal convergence rate of the RKPM is restored and thus the effectivity indices of the Z–Z error estimator are converging to the ideal value of unity as the meshes are refined.     

CALCULATION OF THE STRESS INTENSITY FACTOR AND ESTIMATION OF ITS ERROR BY A SHAPE SENSITIVITY ANALYSIS

Abstract— The precision with which the stress intensity factor (SIF) can be calculated from a finite element solution depends essentially on the extraction method and on the discretization error. In this paper, the influence of the discretization error in the SIF calculation was studied and a method for estimating the resulting error was developed. The SIF calculation method used is based on a shape design sensitivity analysis; this assures that the resulting error in the extracted SIF depends solely on the global discretization error present in the finite element solution. Moreover, this method allows us to extend the Zienkiewicz-Zhu discretization error estimator to the SIF calculation. The reliability of the proposed method was analysed solving a two-dimensional problem using an h -adaptive process. Also the convergence of the error with the h -adaptive refinement was studied.     

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.     

弹性力学问题自适应有限元及其局部多重网格法

针对平面弹性力学问题,利用最新顶点二分法,设计了一种不需要标记振荡项和加密单元不需要满足“内节点”性质的自适应有限元法;利用自适应加密过程中每层网格上只有局部单元需要加密这一特性,设计了一种基于局部松弛的多重网格法.数值实验结果表明:该文设计的自适应有限元法具有一致收敛性和拟最优计算复杂度,基于局部松弛的多重网格法对求解弹性力学问题自适应网格下的有限元方程具有很好的计算效率和鲁棒性.     

Smart computational fracture of materials and structures

The use of the finite element method for complex engineering problems is now common. To ease the burden on the engineer the development of smart or adaptive computational methods is now required to model complex problems. In this paper we investigate the development of an adaptive finite element method for fracture-related problems. The adaptive method involves various stages which include the finite element analysis, error estimation/indication, mesh refinement and fracture/failure analysis in a loop. Some simple error estimators, based on stress projection, are used to investigate the adaptive finite element process. Element refinement is based on three schemes; the first and second are a simple and hierarchical refinement scheme with transitioning which avoids the need for constraint equations between element boundaries. Another scheme based on constraint equations between elements is also examined. The energy norm is used to estimate the element error. The software has the ability to introduce a discrete fracture in the structure according to standard fracture analysis practice. Crack tip parameters are calculated using a least-squares fit of the displacements into the asymptotic crack tip displacement field. Some simple examples are used to investigate the adaptive process, its behavior and some of the practical problems encountered. The convergence and equilibrium of the adaptive process, in terms of global error in the energy norm, are investigated. In the example the same problem is analyzed using both a fine computational grid and a coarse one. The coarse mesh is then adapted using the three different procedures available. The estimated error in the solution and the stress intensity are shown against the number of elements and number of iterations. Some further areas of research in adaptive finite element analysis are discussed.     

Adaptive finite element analysis of mixed‐mode crack problems with automatic mesh generator

Fully automatic advancing front type mesh generator to take care of crack and fracture problems has been presented. It is coupled with Zienkiewicz and Zhu error estimator, the refinement methodology depends on the concept of strain energy concentration for adaptive analysis of mixed‐mode crack problems. No investigation is reported in this direction so far. It has been found that the above combination proved to be very powerful for adaptive finite element analysis of mixed‐mode crack problems in two‐dimensional isotropic solids. Very accurate stress intensity factors have been obtained for a target error of 10 per cent with a minimum number of steps. Copyright © 2000 John Wiley & Sons, Ltd.     

Some benchmark problems and basic ideas on the accuracy of boundary element analysis

Taken the linear elasticity problems as examples, some benchmark problems are presented to investigate the influence of calculation error and discretization error on the accuracy of boundary element analysis. For the conventional boundary element analysis based on singular kernel function of fundamental solution and using Gaussian elimination method, the main calculation error comes from the integration of kernel and shape function product on each element. Based on some benchmark problems of “simple problem” without discretization error, it can be observed that sometimes a large number of integration points in Gaussian quadrature should be adopted. To guarantee the integration accuracy reliably, an improved adaptive Gaussian quadrature approach is presented and verified. The main error of boundary element analysis is the discretization error, provided the calculation error has been reduced effectively. Based on some benchmark problems, it can be observed that for the bending problems both the constant and linear element are not efficient, the quadratic element with a reasonable refined mesh is required to guarantee the accuracy of boundary element analysis. An error indicator to guide the mesh refinement in the convergence test towards the converged accurate results based on the distribution of boundary effective stress is presented and verified.     

An adaptive finite element approach for frictionless contact problems

The derivation of an a posteriori error estimator for frictionless contact problems under the hypotheses of linear elastic behaviour and infinitesimal deformation is presented. The approximated solution of this problem is obtained by using the finite element method. A penalization or augmented‐Lagrangian technique is used to deal with the unilateral boundary condition over the contact boundary. An a posteriori error estimator suitable for adaptive mesh refinement in this problem is proposed, together with its mathematical justification. Up to the present time, this mathematical proof is restricted to the penalization approach. Several numerical results are reported in order to corroborate the applicability of this estimator and to compare it with other a posteriori error estimators. Copyright © 2001 John Wiley & Sons, Ltd.     

Voxel‐based meshing and unit‐cell analysis of textile composites

Unit‐cell homogenization techniques are frequently used together with the finite element method to compute effective mechanical properties for a wide range of different composites and heterogeneous materials systems. For systems with very complicated material arrangements, mesh generation can be a considerable obstacle to usage of these techniques. In this work, pixel‐based (2D) and voxel‐based (3D) meshing concepts borrowed from image processing are thus developed and employed to construct the finite element models used in computing the micro‐scale stress and strain fields in the composite. The potential advantage of these techniques is that generation of unit‐cell models can be automated, thus requiring far less human time than traditional finite element models. Essential ideas and algorithms for implementation of proposed techniques are presented. In addition, a new error estimator based on sensitivity of virtual strain energy to mesh refinement is presented and applied. The computational costs and rate of convergence for the proposed methods are presented for three different mesh‐refinement algorithms: uniform refinement; selective refinement based on material boundary resolution; and adaptive refinement based on error estimation. Copyright © 2003 John Wiley & Sons, Ltd.     

An adaptive IgA-BEM with hierarchical B-splines based on quasi-interpolation quadrature schemes

The isogeometric formulation of the boundary element method (IgA-BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The proposed quadrature schemes are based on a spline quasi-interpolation (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual-based error estimator. Numerical examples show that the optimal convergence rate of the Galerkin solution is recovered by the proposed adaptive method.