An extended stochastic pseudo-spectral Galerkin finite element method (XS-PS-GFEM) for elliptic equations with hybrid uncertainties

Nouy and Clement introduced the stochastic extended finite element method to solve linear elasticity problem defined on random domain. The material properties and boundary conditions were assumed to be deterministic. In this work, we extend this framework to account for multiple independent input uncertainties, namely, material, geometry, and external force uncertainties. The stochastic field is represented using the polynomial chaos expansion. The challenge in numerical integration over multidimensional probabilistic space is addressed using the pseudo-spectral Galerkin method. Thereafter, a sensitivity analysis based on Sobol indices using the derived stochastic extended Finite Element Method solution is presented. The efficiency and accuracy of the proposed novel framework against conventional Monte Carlo methods is elucidated in detail for a few one and two dimensional problems.     

Mesh independent modelling of cracks by using higher order shape functions

A mesh independent crack modelling approach based on displacement approximation with higher order shape functions is proposed. The Heaviside step function based local enrichment method, known as eXtended Finite Element Method, is modified by replacing the step function with a higher order shape functions approximation. Polynomial B‐spline approximation functions are used in the present paper. An advantage of the proposed method is that its implementation only involves integration of the products of original shape functions and their derivatives and does not require modification of the integration domains. A volume integral based expression is proposed to calculate the effective surface area of the crack modelled by using an approximate step function. It is shown to give the actual crack surface area in the limit of the approximate step function approaching the Heaviside function. The convergence and accuracy of the method is illustrated in examples of transverse and oblique (with respect to loading direction) crack problems in rectangular plates. Uniaxial tension of a unidirectional composite with an open hole is considered. Hoop stress relaxation due to longitudinal splitting is successfully modelled by the method proposed and compared to direct modelling by using ANSYS software. Published in 2002 by John Wiley & Sons, Ltd.     

Extended stochastic FEM for diffusion problems with uncertain material interfaces

This paper is concerned with the prediction of heat transfer in composite materials with uncertain inclusion geometry. To numerically solve the governing equation, which is defined on a random domain, an approach based on the combination of the Extended finite element method (X-FEM) and the spectral stochastic finite element method is studied. Two challenges of the extended stochastic finite element method (X-SFEM) are choosing an enrichment function and numerical integration over the probability domain. An enrichment function, which is based on knowledge of the interface location, captures the C ⁰-continuous solution in the spatial and probability domains without a conforming mesh. Standard enrichment functions and enrichment functions tailored to X-SFEM are analyzed and compared, and the basic elements of a successful enrichment function are identified. We introduce a partition approach for accurate integration over the probability domain. The X-FEM solution is studied as a function of the parameters describing the inclusion geometry and the different enrichment functions. The efficiency and accuracy of a spectral polynomial chaos expansion and a finite element approximation in the probability domain are compared. Numerical examples of a two-dimensional heat conduction problem with a random inclusion show the spectral PC approximation with a suitable choice of enrichment function is as accurate and more efficient than the finite element approach. Though focused on heat transfer in composite materials, the techniques and observations in this paper are also applicable to other types of problems with uncertain geometry.     

Embedded interfaces by polytope FEM

In this paper, the Polytope Finite Element Method is employed to model an embedded interface through the body, independent of the background FEM mesh. The elements that are crossed by the embedded interface are decomposed into new polytope elements which have some nodes on the interface line. The interface introduces discontinuity into the primary variable (strong) or into its derivatives (weak). Both strong and weak discontinuities are studied by the proposed method through different numerical examples including fracture problems with traction‐free and cohesive cracks, and heat conduction problems with Dirichlet and Dirichlet–Neumann types of boundary conditions on the embedded interface. For traction‐free cracks which have tip singularity, the nodes near the crack tip are enriched with the singular functions through the eXtended Finite Element Method. The concept of Natural Element Coordinates (NECs) is invoked to drive shape functions for the produced polytopes. A simple treatment is proposed for concave polytopes produced by a kinked interface and also for locating crack tip inside an element prior to using the singularity enrichment. The proposed method pursues some implementational details of eXtended/Generalized Finite Element Methods for interfaces. But here the additional DOFs are constructed on the interface lines in contrast to X/G‐FEM, which attach enriched DOFs to the previously existed nodes. Copyright © 2011 John Wiley & Sons, Ltd.     

Strong and weak arbitrary discontinuities in spectral finite elements

Methods for constructing arbitrary discontinuities within spectral finite elements are described and studied. We use the concept of the eXtended Finite Element Method (XFEM), which introduces the discontinuity through a local partition of unity, so there is no requirement for the mesh to be aligned with the discontinuities. A key aspect of the implementation of this method is the treatment of the blending elements adjacent to the local partition of unity. We found that a partition constructed from spectral functions one order lower than the continuous approximation is optimal and no special treatment is needed for higher order elements. For the quadrature of the Galerkin weak form, since the integrand is discontinuous, we use a strategy of subdividing the discontinuous elements into 6‐ and 10‐node triangles; the order of the element depends on the order of the spectral method for curved discontinuities. Several numerical examples are solved to examine the accuracy of the methods. For straight discontinuities, we achieved the optimal convergence rate of the spectral element. For the curved discontinuity, the convergence rate in the energy norm error is suboptimal. We attribute the suboptimality to the approximations in the quadrature scheme. We also found that modification of the adjacent elements is only needed for lower order spectral elements. Copyright © 2005 John Wiley & Sons, Ltd.     

The Discontinuity‐Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity‐Enriched Finite Element Method (DE‐FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction‐free and cohesive crack examples. We show that DE‐FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.     

Generalized boundary element method for galerkin boundary integrals

A meshless approach to the Boundary Element Method in which only a scattered set of points is used to approximate the solution is presented. Moving Least Square approximations are used to build a Partition of Unity on the boundary and then used to construct, at low cost, trial and test functions for Galerkin approximations. A particular case in which the Partition of Unity is described by linear boundary element meshes, as in the Generalized Finite Element Method, is then presented. This approximation technique is then applied to Galerkin boundary element formulations. Finally, some numerical accuracy and convergence solutions for potential problems are presented for the singular, hypersingular and symmetric approaches.     

A high‐order hybridizable discontinuous Galerkin method for elliptic interface problems

We present a high‐order hybridizable discontinuous Galerkin method for solving elliptic interface problems in which the solution and gradient are nonsmooth because of jump conditions across the interface. The hybridizable discontinuous Galerkin method is endowed with several distinct characteristics. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the global degrees of freedom. Second, they provide, for elliptic problems with polygonal interfaces, approximations of all the variables that converge with the optimal order of k + 1 in the L²(Ω)‐norm where k denotes the polynomial order of the approximation spaces. Third, they possess some superconvergence properties that allow the use of an inexpensive element‐by‐element postprocessing to compute a new approximate solution that converges with order k + 2. However, for elliptic problems with finite jumps in the solution across the curvilinear interface, the approximate solution and gradient do not converge optimally if the elements at the interface are isoparametric. The discrepancy between the exact geometry and the approximate triangulation near the curved interfaces results in lower order convergence. To recover the optimal convergence for the approximate solution and gradient, we propose to use superparametric elements at the interface. Copyright © 2012 John Wiley & Sons, Ltd.     

基于ABAQUS平台的扩展有限元法

以ABAQUS为平台,提出了一种预设虚节点法,首次在通用有限元程序上嵌入了扩展有限元法的功能。推导了扩展有限元法中的子域积分同Heaviside函数的关系,并改进了一种三角形子域积分算法。对三点弯梁的开裂过程进行了模拟。计算结果表明,扩展有限元法对非连续位移场的表达不依赖于单元边界,是一种模拟裂纹扩展过程等涉及移动非连续问题的有效方法。与通用有限元软件的结合则为应用该方法解决实际复杂问题提供了方便的途径。     

Crack tip enrichment in the XFEM using a cutoff function

We consider a variant of the eXtended Finite Element Method (XFEM) in which a cutoff function is used to localize the singular enrichment surface. The goal of this variant is to obtain numerically an optimal convergence rate while reducing the computational cost of the classical XFEM with a fixed enrichment area. We give a mathematical result of quasi‐optimal error estimate. One of the key points of this paper is to prove the optimality of the coupling between the singular and the discontinuous enrichments. Finally, we present some numerical computations validating the theoretical result. These computations are compared with those of the classical XFEM and a non‐enriched method. Copyright © 2008 John Wiley & Sons, Ltd.     

A computationally efficient method for the buckling analysis of shells with stochastic imperfections

A computationally efficient method is presented for the buckling analysis of shells with random imperfections, based on a linearized buckling approximation of the limit load of the shell. A Stochastic Finite Element Method approach is used for the analysis of the “imperfect” shell structure involving random geometric deviations from its perfect geometry, as well as spatial variability of the modulus of elasticity and thickness of the shell, modeled as random fields. A corresponding eigenproblem for the prediction of the buckling load is solved at each MCS using a Rayleigh quotient-based formulation of the Preconditioned Conjugate Gradient method. It is shown that the use of the proposed method reduces drastically the computational effort involved in each MCS, making the implementation of such stochastic analyses in real-world structures affordable.     

Uncertainty analysis of elastostatic problems incorporating a new hybrid stochastic-spectral finite element method

This article aims to present a combination of stochastic finite element and spectral finite element methods as a new numerical tool for uncertainty quantification. One of the well-established numerical methods for reliability analysis of engineering systems is the stochastic finite element method. In this article, a commonly used version of the stochastic finite element method is combined with the spectral finite element method. Furthermore, the spectral finite element method is a numerical method employing special orthogonal polynomials (e.g., Lobatto) and quadrature schemes (e.g., Gauss-Lobatto-Legendre), leading to suitable accuracy, and much less domain discretization with excellent convergence as well. The proposed method of this article is a hybrid method utilizing efficiencies of both methods for analysis of stochastically linear elastostatic problems. Moreover, a spectral finite element method is proposed for numerical solution of a Fredholm integral equation followed by the present method, to provide further efficiencies to accelerate stochastic computations. Numerical examples indicate the efficiency and accuracy of the proposed method.     

Stress intensity factors computation for bending plates with extended finite element method

The modelization of bending plates with through‐the‐thickness cracks is investigated. We consider the Kirchhoff–Love plate model, which is valid for very thin plates. Reduced Hsieh–Clough–Tocher triangles and reduced Fraejis de Veubeke–Sanders quadrilaterals are used for the numerical discretization. We apply the eXtended Finite Element Method strategy: enrichment of the finite element space with the asymptotic bending singularities and with the discontinuity across the crack. The main point, addressed in this paper, is the numerical computation of stress intensity factors. For this, two strategies, direct estimate and J‐integral, are described and tested. Some practical rules, dealing with the choice of some numerical parameters, are underlined. Copyright © 2012 John Wiley & Sons, Ltd.     

An isogeometric extension of Trefftz method for elastostatics in two dimensions

In this paper, an approach to blend the Hybrid‐Trefftz Finite Element Method (HTFEM) and the Isogeometric Analysis (IGA) called the Isogeometric Trefftz (IGAT) method is presented. The structure of the isogeometric extension of the Trefftz method is formally the same as for its conventional counterpart, except the approximation of the boundary displacements and geometry that are carried out using the Non‐Uniform Rational B‐Splines (NURBS) instead of polynomials. In other words, only the element boundaries are approximated using NURBS basis while the Trefftz approximation is used in the interior of the elements. For that reason, IGAT can be ranked alongside recently developed Isogeometric Boundary Element Method (IGABEM), the NURBS‐Enhanced Finite Element Method (NEFEM), the Isogeometric Local Maximum Entropy (IGA‐LME) method, and the Isogeometrically enhanced Scaled‐Boundary element method (SBFEM), which all use NURBS approximation at the domain boundary only. Theoretical conjectures made in this paper are accompanied by three examples that show that IGAT leads to excellent results using only a few elements.     

eXtended hybridizable discontinuous Galerkin for incompressible flow problems with unfitted meshes and interfaces

The eXtended hybridizable discontinuous Galerkin (X-HDG) method is developed for the solution of Stokes problems with void or material interfaces. X-HDG is a novel method that combines the hybridizable discontinuous Galerkin (HDG) method with an eXtended finite element strategy, resulting in a high-order, unfitted, superconvergent method, with an explicit definition of the interface geometry by means of a level-set function. For elements not cut by the interface, the standard HDG formulation is applied, whereas a modified weak form for the local problem is proposed for cut elements. Heaviside enrichment is considered on cut faces and in cut elements in the case of bimaterial problems. Two-dimensional numerical examples demonstrate that the applicability, accuracy, and superconvergence properties of HDG are inherited in X-HDG, with the freedom of computational meshes that do not fit the interfaces     

基于水平集算法的扩展有限元方法研究

扩展有限元是一种以单位分解思想为基础,在常规有限元位移中加入跳跃函数和渐近位移场函数,以处理不连续问题的数值方法。将水平集算法应用到裂纹界面的描述及加强单元类型的判别,并与扩展有限元相结合,用于分析材料断裂问题。相比传统有限元,有限元网格与裂纹面位置相互独立,不需满足裂纹为单元边、裂尖为单元节点和在裂纹附近进行高密度的...     

Local enrichment of the finite cell method for problems with material interfaces

This paper proposes an efficient, hierarchical high-order enrichment approach for the finite cell method applied to problems of solid mechanics involving discontinuities and singularities. In contrast to the standard extended finite element method, where new degrees of freedom are introduced for all finite elements located in the enrichment zone, we define the enrichment on a so-called overlay mesh which is superimposed over the base mesh. The approximation on the base mesh is obtained by means of the finite cell method where the hp-d method is employed to introduce the hierarchical extension on the overlay mesh. We present two different strategies for defining the enrichment on the superimposed overlay mesh. In the first approach, the enrichment is based on a local h-, p- or hp-refinement utilizing the finite element method on the overlay mesh. Alternatively, the enrichment is constructed by means of the partition of unity method introducing carefully selected enrichment functions suitable for the problem at hand. Our results reveal that the proposed method improves the accuracy of the finite cell method significantly with only a minimum number of additional degrees of freedom. In this paper we will focus on examples with material interfaces although the method can also be applied to problems involving strong discontinuities and singularities. Accurate stress distribution and an exponential rate of convergence are the two striking characteristics of the proposed method. Due to the hierarchical approach it paves the way to using different approaches for the approximation on the base and the overlay mesh and accordingly allows multiscale problems to be addressed as well.     

A hybrid extended finite element/level set method for modeling phase transformations

A hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented. We focus attention on two‐dimensional solidification problems, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front. The enrichment strategies of the eXtended Finite Element Method (X‐FEM) are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary. A new approach with the X‐FEM is suggested for this class of problems whereby the partition of unity is constructed with C¹(Ω) polynomials and enriched with a C⁰(Ω) function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to significantly improve estimates for the front velocity. Temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a finer finite difference grid, the Level Set method is used to represent the evolution of the phase interface. An iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary. The robustness and utility of the method is demonstrated with several benchmark problems of phase transformation. Copyright © 2002 John Wiley & Sons, Ltd.     

基于统计模型的结构损伤识别

提出了一种基于递推随机有限元方法(RSFEM)的随机结构损伤识别方法。在定义了随机损伤指数概念的基础上,考虑模型误差的不确定性和测量噪声的影响,建立了关于随机损伤指数的控制方程。然后,利用RSFEM得到了结构随机损伤指数的统计特性。数值算例的结果显示,新的方法能在考虑模型误差和测量噪声的情况下对结构损伤进行有效识别,且在结构随机参数有较大涨落情况下,该方法仍能有效识别出结构损伤,识别结果与蒙特卡洛模拟解非常吻合。