A finite element analysis of the mixed mode bi-material fracture mechanics problem

Two aspects of the mixed mode bi-material fracture mechanics problem are investigated using finite elements. The stress intensity factors for an inclined crack at various distances from a bi-material interface are established as a function of inclination for two material pair combinations. The probable angle of crack extension is established for this problem using the maximum hoop stress criterion. The inclined terminal crack problem is studied using variable power singular elements at the interface. Crack tip stress distributions and probable angle of crack extension are presented as functions of crack inclination and material pair combinations. Crack tip stress distributions assuming an interfacial debonding criterion are also presented as functions of crack inclination and material pair combinations.     

Stability and convergence proofs for a discontinuous-Galerkin-based extended finite element method for fracture mechanics

We prove the optimal convergence of a discontinuous-Galerkin-based extended finite element method for two-dimensional linear elastostatic problems over cracked domains. The method, which we proposed earlier [1], has two distinctive traits: a) it enriches the finite element space with the modes I and II singular asymptotic crack tip fields over a neighborhood of the crack tip termed the enrichment region, and b) it allows functions in the finite element space to be discontinuous across the boundary between the enrichment region and the rest of the domain. The treatment for this discontinuity, generally a non-polynomial function, is facilitated by a specially designed discontinuous Galerkin method based on the Bassi–Rebay numerical flux. The stability of the method is contingent upon an inf–sup condition, which we have proved to hold for any quasiuniform mesh family with sufficiently fine meshes. We have also shown the optimal convergence of the displacement and stress fields, and the convergence of the stress intensity factors extracted as the coefficients of the enrichment functions.     

Elastic-plastic nonlinearities considering fracture mechanics

The solution of fracture mechanics type problems including material and geometric non-linearities along with time-independent and time-dependent constitutive relations is discussed. The effect of solution tolerances on a fracture type problem are discussed and the use of Green's strain tensor and the Piola-Kirchhoff stress tensor is examined in a 1-D analysis and a 2-D fracture problem. A comparison of large and small displacement analysis for a center-cracked panel with elastic-plastic material is made. Small displacement, viscoplastic analysis results also are presented for a center-cracked panel.     

Improved equilibrium finite elements

A decisive improvement in the finite element equilibrium model is presented. A technique for construction of high precision, yet simple, single-sheet elements is demonstrated. Operating upon displacements, the proposed elements are directly suitable for automated computer handling. Application is made to two-dimensional problems. Computed results exhibits good agreement with known solutions and verify the destined rate of convergence as well. Application to mixed models, being essentially perturbed equilibrium models [1],is straightforward.     

Stochastic perturbation finite elements

This paper extends the stochastic perturbation method to vector-valued and matrix-valued functions. The numerical method for the response and reliability of uncertain structures is formulated using matrix calculus, Kronecker algebra and perturbation theory. Random variables and system derivatives are conveniently arranged into two-dimensional matrices and generalized mathematical formulae are obtained. The results derived are easily amenable to computational procedures.     

Self-adaptive boundary elements with h-hierarchical shape functions

A self-adaptive boundary element method as implemented in the new computer program is presented. Error indicators are obtained by comparing two BEM solutions obtained from exactly the same discretization but with (partly) different sets of collocation points. The code has some advanced features, including fully automatic mesh refinement and user-friendly interfaces. h-heirarchical quadratic shape functions are employed to improve the computational efficiency. Numerical results for elastic problems are presented. Pointwise as well as global convergence is always reached in a few adaptive steps.     

分析结构力学与有限元

分析力学历来是在动力学范围内论述的,结构力学与最优控制模拟关系的共同基础就是分析力学．这表明在结构力学与最优控制理论的架构内也应有分析力学的整套理论．本文就结构力学讲述分析力学,称分析结构力学．保守体系可用Hamilton体系的方法描述,其特点是保辛．保辛给出保守体系结构最重要的特性．有限元法是从结构力学发展的,有限元的单元刚度阵应保持对称性,其实这就是保辛．根据区段单元变形能只与其两端位移有关,就可通过数学分析得到Lagrange括号与Poisson括号,展示了其辛对偶体系、正则方程、正则变换等的内容．     

自适应学习的多特征元素协同表示分类算法

针对基于传统的稀疏表示分类算法的单特征鉴别性较弱这一不足,提出一种基于自适应学习的多特征元素协同表示分类算法SLMCE_CRC。该算法结合多特征子字典的思想,对样本提出特征元素的双重分解,并分别从特征和元素角度分别进行相应的协同表示,自适应地学习出各个特征的稀疏权重和元素的残差权重,并进行线性加权,从而实现目标的分类。实验结果表明,使用该方法能显著提高识别率,尤其对含有较多特征细节的图像数据,具有一定的实用价值。     

Neural classification of finite elements

The work deals with a comparative performance of finite elements, making use of their formulation as vectors (or patterns) in a multi-dimensional space of proper attributes. Since the attributes control the performance, elements defined by similar patterns and related to the same class should show similar behavior. The pattern classification may be carried out with the help a self-organizing feature map of Kohonen with the patterns corresponding to the input space. These networks learn both the distribution and topology of a set of input space. At the end of the learning process, the neurons become selectively tuned to classes of input patterns, thus specifying “family relationships” among the elements. The work makes use of the four attributes: the element dimensionality, its number of nodes, maximum degree of interpolation polynomials and number of degrees of freedom per node, though a more general characterization is also possible.     

Functional programming for finite elements

The problems of software reliability can, to some extent, be attributed to the nature of conventional imperative programming languages (FORTRAN, Pascal, etc.). Functional programming languages, on the other hand, adopt a radically different approach where the only control structure is the recursive application of a pre-defined function. This gives the advantage that programs are mathematical expressions and can be treated formally as such for the purpose of correctness proofs. The suitability of the functional approach is examined by writing a standard finite element program in SASL, which is a purely functional language. The clarity and conciseness of the solution is noted, and although certain problems of efficiency and optimal algorithm design require further research, it is clear that the overall top-down design imposed by a functional language is beneficial to the computational engineer.