Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

A hybrid finite element analysis of interface cracks

A versatile hybrid finite element scheme consisting of special crack-tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack-tip element incorporates higher-order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack-tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.     

Stress‐based finite element analysis of plane plasticity problems

A stress‐based model of the finite element method is evolved for two‐dimensional quasi‐static plasticity problems. The self‐equilibrating fields of stresses are constructed by means of the Airy stress function, which is approximated by three types of elements: the Bogner–Fox–Schmit rectangle, the Hsieh–Clough–Tocher triangle and its reduced variant. Traction boundary conditions are imposed by the use of the Lagrange multiplier method which gives the possibility of calculation of displacements for boundary points. The concept of multi‐point‐constraints elements is applied in order to facilitate the application of this technique. The iterative algorithm, analogous to the closest‐point‐projection method commonly used in the displacement‐based finite element model, is proposed for solving non‐linear equations for each load increment. Two numerical examples with stress‐ and displacement‐controlled load are considered. The results are compared with those obtained by the displacement model of FEM. Bounds for limit loads are obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element analysis of three dimensional crack growth by the use of a boundary element sub model

A new automated method to model non-planar three dimensional crack growth is proposed which combines the advantages of both the boundary element method and the finite element method. The proposed method links the two methods by a submodelling strategy in which the solution of a global finite element model containing an approximation of the crack is interpolated to a much smaller boundary element model containing a fine discretization of the real crack. The method is validated through several numerical comparisons and by comparison to crack growth measured in a test specimen for an engineering structure.     

Boundary element crack closure calculation of three-dimensional stress intensity factors

A general method for boundary element-crack closure integral calculation of three-dimensional stress intensity factors is presented. An equation for the strain energy release rate in terms of products of nodal values of tractions and displacements is obtained. Embedded and surface cracks of modes I, II, and III are analyzed using the proposed method. The multidomain boundary element technique is introduced so that the crack surface geometry is correctly modeled and the unsymmetrical boundary conditions for mode's II and III crack analysis are handled conveniently. Conventional quadrilateral elements are sufficient for this method and the selection of the size of the crack front elements is independent of the crack mode and geometry. For all of the examples demonstrated in this paper, 54 boundary elements are used, and the most suitable ratio of the width of the crack front elements to the crack depth is 1/10 and the calculation error is kept within ±1.5 percent. Compared to existing analytical and finite element solutions the boundary element-crack closure integral method is very efficient and accurate and it can be easily applied to general three-dimensional crack problems.     

A hybrid finite element method for heterogeneous materials with randomly dispersed rigid inclusions

A hybrid finite element approach is proposed for the mechanical response of two-dimensional heterogeneous materials with linearly elastic matrix and randomly dispersed rigid circular inclusions of arbitrary sizes. In conventional finite element methods, many elements must be used to represent one inclusion. In this work, each inclusion is embedded inside a polygonal element and only one element is required to represent one inclusion. In numerically approximating stress and displacement distributions around the inclusion, classical elasticity solutions for a multiply-connected region are employed. A modified hybrid functional is used as the basis of the element formulation where the displacement boundary conditions of the element are automatically considered in a variational sense. The accuracy and efficiency of the proposed method are demonstrated by two boundary value problems. In one example, the results based on the proposed method with only 64 hybrid elements (450 degrees of freedom) are shown to be almost identical to those based on the traditional method with 2928 conventional elements (5526 degrees of freedom).     

The treatment of natural boundary conditions in the finite element and finite difference methods

To isolate the relative accuracy in the numerical approximation of the natural boundary conditions, a plane- stress example is presented for which the field equations using finite element and finite difference methods are identical. For this example, and by implication a wider class of problem, it is demonstrated that by using conventional practice in both methods the implicit representation of the natural boundary conditions in the finite element gives rise to a lower numerical accuracy than that in the less convenient explicit satisfaction of these boundary conditions in finite difference method.     

基于Voxel有限元网格对球形夹杂复合材料的应力分析

通过Voxel有限元网格对球形夹杂复合材料进行应力分析时,由于Voxel网格在两相界面呈现阶梯状,所以在两相界面附近的单元会表现出明显的应力集中现象。提出采用局部应力平均方法来处理由于Voxel有限元网格而引起的应力集中,并且考虑应力平均区域、应力平均加权函数以及网格密度的影响。结果表明:该局部应力平均方法能够有效地去除两相界面附近单元的应力集中,但应力平均区域不能过大也不能过小。通过计算发现采用2个Voxel网格深度的平均区域为最优,并且具有网格不依赖性。该方法也可以进一步用于球形夹杂复合材料的积累损伤演化分析。     

Stress analysis of a front bumper fascia using the boundary element method

Stress concentration is one of the most common problems related to automotive components and numerical analysis can be of great interest to deal with such problems. The boundary element method (BEM) is a technique which can be used in stress analysis and it is specifically applied to the design of a car component here. This work presents the efficacy of the application of a procedure based on BEM sub-model for stress analysis in a proposed design change of a front bumper fascia. The results confirm the consistency of the proposed procedure compared to the finite element method (FEM), a consolidated method for stress analysis in the automotive industry.     

Effective stress-based finite element error estimation for composite bodies

This paper presents a discretization error estimator for displacement-based finite element analysis applicable to multi-material bodies such as composites. The proposed method applies a specific stress continuity requirement across the intermaterial boundary consistent with physical principles. This approach estimates the discretization error by comparing the discontinuous finite element effective stress function with a smoothed (C⁰ continuous) effective stress function for non-intermaterial boundary elements with a smoothed pseudo-effective stress function for elements which lie on the intermaterial boundary. Examples are presented which illustrate the effectiveness of the multi-material error estimator. The pointwise pseudo-effective stress and the L₂ norm of the estimated stress error are seen to converge with mesh refinement, while Zienkiewicz and Zhu's error estimator failed to converge for elements on the intermaterial boundary due to the physically admissible stress discontinuities that exist on the intermaterial boundary.     

Generalized Kelvin solution based boundary element method for crack problems in multilayered solids

This paper presents a new boundary element method (BEM) for linear elastic fracture mechanics in three-dimensional multilayered solids. The BEM is based on a generalized Kelvin solution. The generalized Kelvin solution is the fundamental singular solution for a multilayered elastic solid subject to point concentrated body-forces. For solving three-dimensional elastic crack problems in a finite region, a multi-region method is also employed in the present BEM. For crack problems in an infinite space, a large finite body is used to approximate the infinite body. In addition, eight-node traction-singular boundary elements are used in representing the displacements and tractions in the vicinity of a crack front. The incorporation of the generalized Kelvin solution into the boundary integral formulation has the advantages in elimination of the element discretization at the interfaces of different elastic layers. Three numerical examples are presented to illustrate the proposed method for the calculation of stress intensity factors for cracks in layered solids. The results obtained using the proposed method are well compared with the existing results available in the relevant literature.     

双层介质声传播问题的标准解及有限元解

文章通过双层介质中的声传播问题,研究了有限元方法在水下声场计算中的应用。基于传统的Galerkin方法推导出水下声场的有限元方程,采用四节点四边形单元离散求解物理域,可选择辐射边界条件、DtN (Dirichletto Neumann)非局部算子、完美匹配层来处理出射声场,得到有限元解。为了验证该有限元模型,需要高精度的参考解。水平不变均匀介质中的声传播问题存在解析解,但双层介质问题不存在解析解。因此,对于双层介质声传播问题,使用波数积分法推导出标准解。分别考虑了有限深度和无限深度双层介质两种情况,并进行了数值模拟。数值结果表明,文章所提的有限元模型与参考解非常吻合。此外,还发现当某号简正波的本征值非常接近割线时,简正波模型KRAKEN难以准确计算该号简正波的本征值,从而声场计算结果存在明显误差;但是有限元方法不需要计算本征值,所以当KRAKEN模型出现此类问题时,有限元方法仍能给出准确的声场计算结果,表明有限元方法在普适性方面优于简正波方法。     

Analysis of cracked anisotropic plates using the hybrid finite element method

This paper presents a convenient and efficient method to obtain accurate stress intensity factors for cracked anisotropic plates. In this method, a complex variable formulation in conjunction with a hybrid displacement finite element scheme is used to carry out the stiffness and stress calculations of finite cracked plates subjected to general boundary and loading conditions. Unlike other numerical methods used for local analysis such as the boundary element method, the present method results in a symmetric stiffness matrix, which can be directly incorporated into the stiffness matrix representing other structural parts modeled by conventional finite elements. Therefore, the present method is ideally suited for modeling cracked plates in a large complex structure.     

The stochastic Galerkin scaled boundary finite element method on random domain

The research work extends the scaled boundary finite element method to non‐deterministic framework defined on random domain wherein random behaviour is exhibited in the presence of the random‐field uncertainties. The aim is to blend the scaled boundary finite element method into the Galerkin spectral stochastic methods, which leads to a proficient procedure for handling the stress singularity problems and crack analysis. The Young's modulus of structures is considered to have random‐field uncertainty resulting in the stochastic behaviour of responses. Mathematical expressions and the solution procedure are derived to evaluate the statistical characteristics of responses (displacement, strain, and stress) and stress intensity factors of cracked structures. The feasibility and effectiveness of the presented method are demonstrated by particularly chosen numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

A finite element model for plane elasticity problems using the complementary energy theorem

A finite element stress analysis capability for plane elasticity problems, employing the principle of stationary complementary energy, is developed. Two models are investigated. The first is a 24 d.o.f. rectangular finite element. The second model consists of an 18 d.o.f. triangular element. In order to allow for self-equilibrating stresses which are continuous within the element, the well-known Airy stress function ø is used. The function ø is represented by means of quintic Hermitian polynomials within the finite element. The values of the ø function and its derivatives up to order two are used as nodal parameters. For matching the stress function with the prescribed boundary tractions, additional equations are developed considering the force and moment equilibrium equations on the boundary consistent with the assumed stress function. These additional boundary equations are incorporated into the system equations using the Lagrangian multiplier technique. Excellent results are obtained for linear elastic problems even with coarse finite element discretization. Some examples of plane elasticity problems are solved and results compared.     

Coupling of element free Galerkin and hybrid boundary element methods using modified variational formulation

 A novel method is proposed by coupling the element free Galerkin (EFG) and the hybrid boundary element (HBE) methods to achieve solution efficiency and accuracy for stress analysis in solids. A modified variational formulation is derived for the present coupled EFG/HBE method so that the continuity and compatibility can be preserved on the interface between the domains of EFG and HBE. The coupled EFG/HBE method has been coded in FORTRAN. The validity and efficiency of the proposed method are demonstrated through a number of example problems. It is found that the present method can take advantages of both EFG and HBE methods. The present method is very easy to implement, and very flexible for obtaining displacements and stresses of desired accuracy in solids, as the efforts for meshing the problem domain have been significantly reduced due to the use of boundary element method (BEM).     

Modeling of concrete cracking—A hybrid technique of using displacement discontinuity element method and direct boundary element method

This paper presents a new approach by making use of a hybrid method of using the displacement discontinuity element method and direct boundary element method to model concrete cracking by incorporating fictitious crack model. Fracture mechanics approach is followed using the Hillerborg's fictitious crack model. A boundary element based substructure method and a hybrid technique of using displacement discontinuity element method and direct boundary element method are compared in this paper. In order to represent the process zone ahead of the crack, closing forces are assumed to act in such a way that they obey a linear normal stress-crack opening displacement law. Plain concrete beams with and without initial crack under three-point loading were analyzed by both the methods. The numerical results obtained were shown to agree well with the results from existing finite element method. The model is capable of reproducing the whole range of load–deflection response including strain-softening and snap-back behavior as illustrated in the numerical examples.     

无限元边界条件下的浅海海底地震波场建模

针对模拟无限大的海底地震波传播问题,应用无限元理论,提出了一种基于无限元边界条件的地震波建模方法,仿真了海底应力场,与高阶交错网格有限差分法仿真结果比对,验证建模方法的正确性。进一步分析无限元边界对海底地震波的吸收效果,结果表明无限元边界对纵波、横波的吸收效果好,可应用于海底地震波场建模中。     

圆管刚性法兰加劲肋角焊缝受力特性及强度计算理论

该文针对圆钢管刚性法兰角焊缝在轴拉荷载作用下的受力性能开展了非线性有限元分析,研究了法兰竖向、横向和环向角焊缝在不同荷载水平下的应力发展规律,指出了法兰在轴拉荷载下三道焊缝的传力分配,进行了不同加劲肋高宽比对竖向焊缝受力影响的研究,提出了更加科学合理的角焊缝计算公式,并通过刚性法兰节点轴拉静力试验证实了所提出的计算公式的合理性。结合有限元分析及试验研究结果得到如下结论:圆钢管刚性法兰竖向角焊缝在轴拉荷载下的焊喉截面正应力呈上大下小的非对称分布,竖向焊缝的拉压分界线位于焊缝长度 约1/5处;角焊缝落弧端存在应力集中;在允许焊缝部分进入塑性的情况下,结构依然是安全的;修正后的角焊缝强度计算公式可应用于圆钢管结构刚性法兰的连接设计。     

伴流声场计算的有限元方法及其应用

推导了势流中的声波方程,并运用伽辽金加权余量法建立了相应的有限元(Finite Element Method,FEM)弱形式。对于管道声学问题的计算,给出了所需边界条件的处理方法,通过离散和装配得到有限元矩阵方程。使用自行编写的有限元程序计算分析了Herschel-Quincke(H-Q)管的消声特性。结果表明,在中低频段有限元计算结果与一维理论计算结果吻合良好,从而验证了该方法的正确性;频率较高时,两者出现明显差异,主要是由于主管和支管交接处的三维波和三维流效应所致。介质流动影响H-Q管的消声特性,特别是共振频率和通过频率,马赫数越大,影响越显著。结构形式的改变可以大大改善特定频率范围的消声性能。