Numerical evaluation of elastodynamic energy fracture parameters in 2-D heterogeneous media

The elastodynamic energy fracture parameters for a stationary crack in 2-D heterogeneous media are evaluated with a presented generalized Domain Integral Method (DIM). The method, incorporated with the finite element solutions, is demonstrated to be patch-independent in a generalized sense. In the context of dynamic response, the near-tip region is always involved in the calculation. The method is used for determination of the associated Energy Release Rate (ERR) for the cases when the crack tip is away from the material interface, with the formulation valid for both small and large elastic deformations. Numerical results for such problems appear to be very insensitive to the crack-tip finite element models. As to the instances when the tip terminates normally at the material interface, the ERR is not feasible for use as a fracture criterion. The generalized DIM is then applied for calculation of the alternative elastodynamic energy parameter J/R^λ₀. The exponential order λ, with regard to the strength of stress singularity, is also properly evaluated in the calculation. No particular singular finite element is required throughout the study. © 1998 John Wiley & Sons, Ltd.     

Coupled and decoupled stabilized mixed finite element methods for nonstationary dual-porosity-Stokes fluid flow model

In this paper, we propose and analyze two stabilized mixed finite element methods for the dual-porosity-Stokes model, which couples the free flow region and microfracture-matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual-porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods.     

非均匀材料界面裂纹的Cell-Based光滑有限元法

为提高非均匀材料界面裂纹尖端断裂参数的求解精度,基于非均匀材料界面断裂力学、Cell-Based光滑有限元(Cell-SFEM)和非均匀材料的互交作用积分法,提出了求解非均匀材料界面裂纹尖端断裂参数的CellBased光滑有限元法,推导了基于Cell-Based光滑有限元法的非均匀材料的互交作用积分法,对非均匀材料间的界面裂纹尖端处正则应力强度因子进行了求解,并与参考解进行了比较,讨论了互交积分区域大小和光滑子元个数与正则应力强度因子的关系。数值算例结果表明:本方法具有很高的计算精度,对积分区域大小不敏感,可为设计、制造抗破坏非均匀材料提供依据。     

Numerical implementation of imperfect interfaces

One approach to characterizing interfacial stiffness is to introduce imperfect interfaces that allow displacement discontinuities whose magnitudes depend on interfacial traction and on properties of the interface or interphase region. This work implemented such imperfect interfaces into both finite element analysis and the material point method. The finite element approach defined imperfect interface elements that are compatible with static, linear finite element analysis. The material point method interfaces extended prior contact methods to include interfaces with arbitrary traction-displacement laws. The numerical methods were validated by comparison to new or existing stress transfer models for composites with imperfect interfaces. Some possible experiments for measuring the imperfect interface parameters needed for modeling are discussed.     

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.     

Fracture analysis of a nonhomogeneous coating/substrate system with an interface under thermal shock

The work is devoted to establish a model for the interface problem of a nonhomogeneous coating/substrate system. In the model, according to the distribution of material properties, three types of interface problems are considered: (i) The material properties and their derivatives are continuous on the interface; (ii) the material properties are continuous, but their derivatives are discontinuous on the interface; and (iii) the material properties as well as their derivatives are discontinuous on the interface. In order to solve the complex interface problems, a transient interaction energy integral method (IEIM) is developed in this paper. The transient thermal stress intensity factors are evaluated using the IEIM combined with the finite element method and the finite difference method. The influences of the interface discontinuity and the geometric parameters on the transient TSIFs are investigated. Particularly, the crack growth behavior with different interface discontinuities is discussed.     

The global element method for stationary advective problems

We extend the variation principle used in the global element method for self-adjoint elliptic problems¹, to problems containing advective terms. Used with the global element formalism, the extended principle yields a uniform framework for treating advective or boundary layer problems, with the attractive feature that the implicit treatment of interface conditions between elements yields an effective decoupling between ‘boundary layer’ and ‘interior’ parts of the solution. As a numerical example, we solve the one-dimensional model problem of Christie and Mitchell³, obtaining high accuracy for Peclet numbers up to 10⁶ with no sign of instability. These results suggest that, given a suitable choice of global elements, the decoupling is very effective in damping the oscillations found in standard finite difference or finite element treatment.     

An energy method for calculating the stress intensities in orthotropic bimaterial fracture

An energy based numerical method has been developed for extracting stress intensities at the tip of an interface crack bounded by two orthogonal dissimilar materials and subjected to a general state of stress. The method is most suitable for mixed mode delamination fracture studies often observed in brittle matrix composite laminates. After obtaining the near-tip finite element solution for a given laminated geometry, the elastic energy release rate, i.e., J is computed via the stiffness derivative method. The individual orthotropic stress intensities, K _I ^*, K _II ^* are then calculated at a minimum computational expense from further J calculations perturbed by reciprocal stress intensity increments. Results obtained using the Crack Surface Displacement (CSD) method were found to be in good agreement with those obtained using the energy method. Comparisons with theoretical solutions indicate that the energy method can be used accurately even when relatively coarse finite element meshes containing approximately 200 eight noded isoparametric elements are used. The method provides an effective and reliable tool for studying via the method of finite elements delamination phenomena in composite anisotropic laminates.     

Transient heat conduction analysis in a piecewise homogeneous domain by a coupled boundary and finite element method

A coupled finite element–boundary element analysis method for the solution of transient two‐dimensional heat conduction equations involving dissimilar materials and geometric discontinuities is developed. Along the interfaces between different material regions of the domain, temperature continuity and energy balance are enforced directly. Also, a special algorithm is implemented in the boundary element method (BEM) to treat the existence of corners of arbitrary angles along the boundary of the domain. Unknown interface fluxes are expressed in terms of unknown interface temperatures by using the boundary element method for each material region of the domain. Energy balance and temperature continuity are used for the solution of unknown interface temperatures leading to a complete set of boundary conditions in each region, thus allowing the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, fluxes from specific BEM regions are expressed in terms of common boundary (interface) temperatures, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of finite element method (FEM) regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Stress intensity factors of a central interface crack in a bonded finite plate and periodic interface cracks under arbitrary material combinations

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary material combinations. This paper deals with a central interface crack in a bonded finite plate and periodic interface cracks. Then, the effects of material combination and relative crack length on the stress intensity factors are discussed. A useful method to calculate the stress intensity factor of interface crack is presented with focusing on the stress at the crack tip calculated by the finite element method.     

Finite element analysis of sinusoidal small-amplitude vibrations in deformed viscoelastic solids. Part I: Theoretical development

A general method is presented for the isothermal mechanical analysis of incompressible material solids in which a small-amplitude time harmonic oscillation is superposed on a static finite deformation field. Material behaviour is assumed to be of the ‘fading memory’ type describable by the finite linear viscoelasticity theory of Coleman and Noll.⁴ Existing finite element codes, that treat finite elasticity material behaviour, can be extended with minor modifications to treat the subject problem. A specialized form of the finite linear viscoelasticity constitutive equations proposed by Morman⁸ has been implemented in the MARC nonlinear finite element program for predicting the response of statically deformed elastomeric components to small amplitude vibrations. Numerical results obtained with MARC for the static force-deflection behaviour and dynamic (complex) stiffness for a viscoelastic cylinder subject to combined axial and twisting pre-loads, are in excellent agreement with corresponding analytical results.     

Thermoelastic stress field in a piecewise homogeneous domain under non‐uniform temperature using a coupled boundary and finite element method

This study concerns the development of a coupled finite element–boundary element analysis method for the solution of thermoelastic stresses in a domain composed of dissimilar materials with geometric discontinuities. The continuity of displacement and traction components is enforced directly along the interfaces between different material regions of the domain. The presence of material and geometric discontinuities are included in the formulation explicitly. The unknown interface traction components are expressed in terms of unknown interface displacement components by using the boundary element method for each material region of the domain. Enforcing the continuity conditions leads to a final system of equations containing unknown interface displacement components only. With the solution of interface displacement components, each region has a complete set of boundary conditions, thus leading to the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, stresses from specific BEM regions are first expressed in terms of interface displacements, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of FEM regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

A simple and efficient preconditioning scheme for heaviside enriched XFEM

The extended finite element method (XFEM) is an approach for solving problems with non-smooth solutions, which arise from geometric features such as cracks, holes, and material inclusions. In the XFEM, the approximate solution is locally enriched to capture the discontinuities without requiring a mesh which conforms to the geometric features. One drawback of the XFEM is that an ill-conditioned system of equations results when the ratio of volumes on either side of the interface in an element is small. Such interface configurations are often unavoidable, in particular for moving interface problems on fixed meshes. In general, the ill-conditioning reduces the performance of iterative linear solvers and impedes the convergence of solvers for nonlinear problems. This paper studies the XFEM with a Heaviside enrichment strategy for solving problems with stationary and moving material interfaces. A generalized formulation of the XFEM is combined with the level set method to implicitly define the embedded interface geometry. In order to avoid the ill-conditioning, a simple and efficient scheme based on a geometric preconditioner and constraining degrees of freedom to zero for small intersections is proposed. The geometric preconditioner is computed from the nodal basis functions, and therefore may be constructed prior to building the system of equations. This feature and the low-cost of constructing the preconditioning matrix makes it well suited for nonlinear problems with fixed and moving interfaces. It is shown by numerical examples that the proposed preconditioning scheme performs well for discontinuous problems and \(C^0\) -continuous problems with both the stabilized Lagrange and Nitsche methods for enforcing the continuity constraint at the interface. Numerical examples are presented which compare the condition number and solution error with and without the proposed preconditioning scheme. The results suggest that the proposed preconditioning scheme leads to condition numbers similar to that of a body-fitted mesh using the traditional finite element method without loss of solution accuracy.     

Modeling curved interfaces without element-partitioning in the extended finite element method

In this paper, we model holes and material interfaces (weak discontinuities) in two-dimensional linear elastic continua using the extended finite element method on higher-order (spectral) finite element meshes. Arbitrary parametric curves such as rational Bézier curves and cubic Hermite curves are adopted in conjunction with the level set method to represent curved interfaces. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes' theorem to reduce integration to the boundary of the domain. Numerical integration on cut elements requires the evaluation of a one-dimensional integral over a parametric curve, and hence, the need to partition curved elements is eliminated. To improve stiffness matrix conditioning, ghost penalty stabilization and the Jacobi preconditioner are used. For material interface problems, we develop an enrichment function that captures weak discontinuities on spectral meshes. Taken together, we show through numerical experiments that these advances deliver optimal algebraic rates of convergence with h-refinement (p=1,2,…,5) and exponential rates of convergence with p-refinement (p=1,2,…,7) for elastostatic problems with holes and material inclusions on Cartesian pth-order spectral finite element meshes.     

Convergence analysis of the Q1-finite element method for elliptic problems with non-boundary-fitted meshes

The aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain's boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesian-grid methods. These methods use regular structured meshes (most often Cartesian) for non-aligned domains. Although non-boundary-fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates. In this paper, the convergence of a Q₁-non-conforming finite element method is analyzed for second-order elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard Q₁-rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as non-conforming. A stair-step boundary defined from the Cartesian mesh approximates the original domain's boundary. The convergence analysis of the finite element method for such a kind of non-boundary-fitted stair-stepped approximation is not treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary-fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments. The rate of convergence is proved to be in 𝒪(h^1/2) for the H¹-norm for all general boundary conditions, and classical duality arguments allow one to obtain an 𝒪(h) error estimate in the L²-norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, which impose original boundary conditions on a non-boundary-fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence. Copyright © 2008 John Wiley & Sons, Ltd.     

Level set topology optimization of structural problems with interface cohesion

This paper presents a finite element topology optimization framework for the design of two‐phase structural systems considering contact and cohesion phenomena along the interface. The geometry of the material interface is described by an explicit level set method, and the structural response is predicted by the extended finite element method. In this work, the interface condition is described by a bilinear cohesive zone model on the basis of the traction‐separation constitutive relation. The non‐penetration condition in the presence of compressive interface forces is enforced by a stabilized Lagrange multiplier method. The mechanical model assumes a linear elastic isotropic material, infinitesimal strain theory, and a quasi‐static response. The optimization problem is solved by a nonlinear programming method, and the design sensitivities are computed by the adjoint method. The performance of the presented method is evaluated by 2D and 3D numerical examples. The results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered. In addition, 3D examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality. The optimization framework presented is limited to two‐phase structural systems where the material interface is coincident in the undeformed configuration, and to structural responses that remain valid considering small strain kinematics. Copyright © 2017 John Wiley & Sons, Ltd.     

Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique

This paper describes an interface element to calculate the strain energy release rates based on the virtual crack closure technique (VCCT) in conjunction with finite element analysis (FEA). A very stiff spring is placed between the node pair at the crack tip to calculate the nodal forces. Dummy nodes are introduced to extract information for displacement openings behind the crack tip and the virtual crack jump ahead of the crack tip. This interface element leads to a direct calculation of the strain energy release rate (both components G_I and G_II) within a finite element analysis without extra post-processing. Several examples of stationary cracks under impact loading were examined. Dynamic stress intensity factors were converted from the calculated transient strain energy release rate for comparison with the available solutions by the others from numerical and experimental methods. The accuracy of the element is validated by the excellent agreement with these solutions. No convergence difficulty has been encountered for all the cases studied. Neither special singular elements nor the collapsed element technique is used at the crack tip. Therefore, the fracture interface element for VCCT is shown to be simple, efficient and robust in analyzing crack response to the dynamic loading. This element has been implemented into commercial FEA software ABAQUS^® with the user defined element (UEL) and should be very useful in performing fracture analysis at a structural level by engineers using ABAQUS^®.     

Fracture of metal/ceramic interfaces

The present paper examines metal/ceramic interfaces. Energy release rates are calculated with the finite element method for different elastic–plastic material laws of the metal. The local strain field of the metal is measured during a four-point bending test with an optical method and compared with results from the simulations. The aim of the work is to understand the influence of interface strength and material properties on the energy release rate.     

Galerkin and collocation-Galerkin methods with superconvergence and optimal fluxes

Finite element methods are formulated and investigated for the effectiveness factor problem for heat and mass transfer with chemical reactions in catalyst pellet models. A Galerkin finite element method is compared with a previous C¹ collocation method⁷. A scheme that is conceptually intermediate between these two methods and accordingly has been termed collocation-Galerkin is formulated and numerical experiments considered. Of particular interest here are superconvergence results at the Gauss and Jacobi points, respectively. Numerical studies of superconvergence in the presence of a nonlinear reaction-rate term are presented. An integral formula is devised and used to compute the flux at the pellet surface to optimal accuracy. Numerical experiments are conducted to demonstrate the improvement in computed fluxes.     

A continuation method for rigid‐cohesive fracture in a discontinuous Galerkin finite element setting

An energy minimization formulation of initially rigid cohesive fracture is introduced within a discontinuous Galerkin finite element setting with Nitsche flux. The finite element discretization is directly applied to an energy functional, whose term representing the energy stored in the interfaces is nondifferentiable at the origin. Unlike finite element implementations of extrinsic cohesive models that do not operate directly on the energy potential, activation of interfaces happens automatically when a certain level of stress encoded in the interface potential is reached. Thus, numerical issues associated with an external activation criterion observed in the previous literature are effectively avoided. Use of the Nitsche flux avoids the introduction of Lagrange multipliers as additional unknowns. Implicit time stepping is performed using the Newmark scheme, for which a dynamic potential is developed to properly incorporate momentum. A continuation strategy is employed for the treatment of nondifferentiability and the resulting sequence of smooth nonconvex problems is solved using the trust region minimization algorithm. Robustness of the proposed method and its capabilities in modeling quasistatic and dynamic problems are shown through several numerical examples.