A multiscale extended finite element method for crack propagation

In this paper, we propose a multiscale strategy for crack propagation which enables one to use a refined mesh only in the crack’s vicinity where it is required. Two techniques are used in synergy: a multiscale strategy based on a domain decomposition method to account for the crack’s global and local effects efficiently, and a local enrichment technique (the X-FEM) to describe the geometry of the crack independently of the mesh. The focus of this study is the avoidance of meshing difficulties and the choice of an appropriate scale separation to make the strategy efficient. We show that the introduction of the crack’s discontinuity both on the microscale and on the macroscale is essential for the numerical scalability of the domain decomposition method to remain unaffected by the presence of a crack. Thus, the convergence rate of the iterative solver is the same throughout the crack’s propagation.     

A hybrid method for finite element ordering

Nodal ordering for the formation of suitable sparsity patterns for stiffness matrices of finite element meshes are often performed using graph theory and algebraic graph theory. In this paper a hybrid method is presented employing the main features of each theory. In this method, vectors containing certain properties of graphs are taken as Ritz vectors, and using methods for constructing a complementary Laplacian, a reduced eigenproblem is formed. The solution of this problem results in coefficients of the Ritz vectors, indicating the significance of each considered vector.The present method uses the global properties of graphs in ordering, and the local properties are incorporated using algebraic graph theory. The main feature of this method is its capability of transforming a general eigenproblem into an efficient approach incorporating graph theory. Examples are included to illustrate the efficiency of the presented method.     

A two-scale extended finite element method for modelling 3D crack growth with interfacial contact

3D fatigue crack growth problems are nowadays handled using X-FEM coupled with level set techniques. It is also well established that such an approach allows mesh-independent crack modelling and no remeshing during crack propagation. However, when contact and friction occur along the crack faces, a discretization of the internal variables linked to the interface law is necessary. The interface discretization is generally constructed from the finite elements cut by the crack. As a consequence, a mesh dependency between the bulk discretization and the interface discretization is introduced. However, the dimension of the possible non-linearities arising at the crack interface (like confined plasticity or unilateral contact with friction) may be several orders of magnitude finer than the crack size. A finer discretization is thus required to accurately capture these non-linearities. The aim of the present paper is to develop a method considering the 3D cracked structure and the crack interface as two independent global and local problems characterized by different length scales and different behaviors. Here, the interface is seen as an autonomous entity with its own discretization, variables and constitutive law. A formulation involving three-fields is used. The interface is linked to the global problem in a weak sense in order to avoid instabilities in the contact solution. Two iterative strategies are considered to solve the contact problem. Two-dimensional and three-dimensional numerical examples are presented to demonstrate the ability of the model to solve the contact at the crack interface with or without propagation at a given level of accuracy.     

A mixed finite element method for boundary flux computation

Most finite element schemes for thermal problems estimate boundary heat flux directly from the derivative of the finite element solution. The boundary flux calculated by this approach is typically inaccurate and does not guarantee a global heat balance.In this paper we present a mixed finite element method for calculating the boundary flux and show the superiority of this method through numerical examples of both diffusion and advection-diffusion problems.     

A novel finite element method for two-point boundary value problems

A finite element method is presented for solving boundary value problems for ordinary differential equations in which the general solution of the differential equation is computed first, followed by a selection procedure for the particular solution of the boundary value problem from the general solution. In this method, the discrete representation of the differential equation is a singular matrix equation, which is solved by using generalized matrix inversion. The technique is applied to both linear and nonlinear boundary value problems and to boundary value problems requiring eigenvalue evaluation. The solution of several examples involving different types of two-point boundary value problems is presented.     

A finite element immersed boundary method for fluid flow around moving objects

This paper presents the extension of a recently proposed immersed boundary method to the solution of the flow around moving objects. Solving the flow around objects with complex shapes may involve extensive meshing work that has to be repeated each time a change in the geometry is needed. Mesh generation and solution interpolation between successive grids may be costly and introduce errors if the geometry changes significantly during the course of the computation. These drawbacks are avoided when the solution algorithm can tackle grids that do not fit the shape of immersed objects. This work presents an extension of our recently developed finite element Immersed Boundary (IB) method to transient applications involving the movement of immersed fluid/solid interfaces. As for the fixed solid boundary case, the method produces solutions of the flow satisfying accurately boundary conditions imposed on the surface of immersed bodies. The proposed algorithm enriches the finite element discretization of interface elements with additional degrees of freedom, the latter being eliminated at element level. The boundary of immersed objects is defined using a time dependent level-set function. Solutions are shown for various flow problems and the accuracy of the present approach is measured with respect to solutions on body-conforming meshes.     

A nonconforming finite element method for a singularly perturbed boundary value problem

We analyze a new nonconforming Petrov-Galerkin finite element method for solving linear singularly perturbed two-point boundary value problems without turning points. The method is shown to be convergent, uniformly in the perturbation parameter, of orderh ^1/2 in a norm slightly stronger than the energy norm. Our proof uses a new abstract convergence theorem for Petrov-Galerkin finite element methods.     

Structural scale modelling by finite element method approach

The structural scale modelling by the finite element method approach is explained. This approach not only has the advantages of the finite element method but also gives insight and ready utility to the modeller. According to the scaling laws obtained by this work, a theoretical example is illustrated to show the workability of the laws. Some practical considerations/limitations of this work are also mentioned. Further extensions to this work, under progress, are also stated.     

An h-hierarchical adaptive scaled boundary finite element method for elastodynamics

A posteriori h-hierarchical adaptive scaled boundary finite element method (ASBFEM) for transient elastodynamic problems is developed. In a time step, the fields of displacement, stress, velocity and acceleration are all semi-analytical and the kinetic energy, strain energy and energy error are all semi-analytically integrated in subdomains. This makes mesh mapping very simple but accurate. Adaptive mesh refinement is also very simple because only subdomain boundaries are discretised. Two 2D examples with stress wave propagation were modelled. It is found that the degrees of freedom needed by the ASBFEM are only 5%–15% as needed by adaptive FEM for the examples.     

在裂纹尖端引入奇异单元的偶应力有限元法

针对在微观状态下结构力学行为会受尺度效应影响的问题,在偶应力理论中考虑微观结构的旋转梯度可以较好解释结构的尺度效应.建立基于一般偶应力理论的有限元法的基本方程,并在裂纹尖端引入奇异单元,计算受单向拉伸的中心斜裂纹板裂纹尖端场的应力强度因子(Stress Intensity Factor,SIF),分析特征长度变化对SIF的影响,对比偶应力理论下的结果与经典理论下的结果.结果表明：在裂纹尖端引入奇异单元可以提高计算精度和稳定性;偶应力使得裂纹尖端SIF比经典理论下的值小,并且SIF随着特征长度增大而减小.     

A finite element creep crack growth model for Waspaloy

Plastic and creep deformations lead to reduced stress levels ahead of the crack tip in a creep crack growth test. However, they can also cause microcracks, cavities and other defects forcing fracture. Numerous damage models are reported in the literature to describe the behavior. In this article, a damage model will be developed from different theories and will be used to describe the creep crack growth behavior of Waspaloy at 973 K. Material parameters for this model are adjusted to uniaxial creep and tensile tests. The calculated creep crack growth curves match very well with the experimental ones supporting the model.     

Recent developments in the Trefftz method for finite element and boundary element applications

In 1926 E. Trefftz published a paper about a variational formulation which utilizes boundary integrals. Almost half a century later researchers became interested again in the ideas of Trefftz when the potential advantage of the Trefftz-method for an efficient use in numerical application on a computer was recognized. The concept of Trefftz can be used both for finite element and boundary element applications. A crucial ingredient of the Trefftz- method is a set of linearly independent trial functions which a priori satisfy the governing differential equations under consideration. In this paper an overview of some recent developments to construct trial functions for the Trefftz-method in a systematic manner is given. Using different types of approximation functions (singular or non-singular) we can obtain very accurate finite element and boundary element algorithms.     

A finite element model for delamination propagation in composites

In this paper the theory and application of a modelling technique for three-dimensional planar delamination growth in laminated composites has been presented. The method is based on linear elastic fracture mechanics assumptions for delamination cracks and uses a strain energy release rate criterion. For a given component, strain energy release rate is considered to be a non-linear function of the location of the delamination front. Hence, satisfaction of the growth criterion reduces to the solution of a non-linear system of equations. A generalized secant method, by the Broyden's update method, is used to solve the system of non-linear equations. Several examples of the application of the technique are presented.     

On the CFL condition for the finite element immersed boundary method

The finite element version of the immersed boundary method proved to be a robust alternative to the original one which was based on finite differences. In this paper we highlight the advantages of the new method and discuss a stability analysis for its space–time discretization. Numerical experiments confirm the theoretical results.     

A new method for the coupling of finite element and boundary element discretized subdomains of elastic bodies

A new and efficient approach for the coupling of subregions of elastic solids discretized by means of finite elements (FE) and boundary elements (BE), respectively, is presented. The method is characterized by so-called ‘bi-condensation’ of nodal degrees of freedom followed by the transformation of the resulting BEM-related traction-displacement equations for the interface(s) of the BE subregion(s) and the FE subdomain(s) to ‘FEM-like’ force-displacement relations which are assembled with the FEM-related force-displacement equations for the interface(s). The presented ‘local FE coupling approach’ is computationally more economic than a global coupling approach since it only requires the inversion of BEM-related coefficient matrices referred to the interfaces of BE subregions and FE subdomains. Depending on whether the principle of virtual displacements or the principle of minimum of potential energy is used for the generation of force-displacement equations for the coupling interface(s), unsymmetric or symmetric coefficient matrices are obtained. Since the two principles are mechanically equivalent, identical results would be achieved in the limit of finite discretizations.The numerical investigation has shown that, depending on the problem and the discretization, the results obtained on the basis of symmetric coefficient matrices may be poor. This applies to ‘edge problems’ characterized by discontinuous tractions along the edges. On the basis of unsymmetric coefficient matrices, however, satisfactory results are obtained even for relatively coarse discretizations.     

A Dorodnitsyn finite element formulation for turbulent boundary layers

The Dorodnitsyn boundary layer formulation is combined with a modified Galerkin finite element formulation and an implicit, non-iterative marching scheme to generate a computational algorithm that is both accurate and very economical. For four representative pressure gradient cases taken from the 1968 Stanford Turbulent Boundary Layer Conference the Dorodnitsyn finite element formulation is compared with a Dorodnitsyn spectral formulation and a representative finite difference package. All methods produce solutions of high accuracy but the Dorodnitsyn finite element formulation is about ten times more economical than the other methods.     

An extended finite element method for modeling crack growth with frictional contact

A new technique for the finite element modeling of crack growth with frictional contact on the crack faces is presented. The eXtended Finite Element Method (X-FEM) is used to discretize the equations, allowing for the modeling of cracks whose geometry are independent of the finite element mesh. This method greatly facilitates the simulation of a growing crack, as no remeshing of the domain is required. The conditions which describe frictional contact are formulated as a non-smooth constitutive law on the interface formed by the crack faces, and the iterative scheme implemented in the LATIN method [Nonlinear Computational Structural Mechanics, Springer, New York, 1998] is applied to resolve the nonlinear boundary value problem. The essential features of the iterative strategy and the X-FEM are reviewed, and the modifications necessary to integrate the constitutive law on the interface are presented. Several benchmark problems are solved to illustrate the robustness of the method and to examine convergence. The method is then applied to simulate crack growth when there is frictional contact on the crack faces, and the results are compared to both analytical and experimental results.     

基于二次插值重构有限元法的动态裂纹扩展模拟

基于二次插值重构有限元法（Twice interpolation Finite Element Method, TFEM）分析动态断裂力学问题并进行数值实验,考察TFEM在裂纹动态扩展模拟中的准确性和可靠性.由于TFEM保证节点处梯度场的连续性,因此裂尖附近的应力场可以得到较好的逼近.把该算法成功移植到自主开发的三维裂纹扩展仿真软件（ZonCrack）中.利用ZonCrack进行的裂纹扩展,分析结果表明：TFEM得到裂尖应力强度因子（Stress Intensity Factor, SIF)与解析解基本一致;裂纹扩展的模拟结果与实验值吻合良好.     

A finite element method for flow separation

The finite element model has been developed in order to solve separation pattern of the flow past an obstruction in a two-dimensional flow field. The Helmholtz-Poisson form of the Reynolds equations are solved alternately until a stable flow separation in the neighbourhood of the obstruction is obtained. In order to check the results of the finite element model, an experimental separation pattern using Pitot-tube measurements has been conducted. The computed and the experimental flow separation patterns show a good agreement.