2D finite element analysis of thermally bonded nonwoven materials: Continuous and discontinuous models

Due to a random structure of nonwoven materials, their non-uniform local material properties and nonlinear properties of single fibres, it is difficult to develop a numerical model that adequately accounts for these features and properly describes their performance. Two different finite element (FE) models – continuous and discontinuous – are developed here to describe the tensile behaviour of nonwoven materials. A macro-level continuum finite element model is developed based on the classic composite theory by treating the fibrous network as orthotropic material. This model is used to analyse the effect of thermally bonding points on the deformational behaviour and deformation mechanisms of thermally bonded nonwoven materials at macro-scale. To describe the effects of discontinuous microstructure of the fabric and implement the properties of polypropylene fibres, a micro-level discontinuous finite element model is developed. Applicability of both models to describe various deformational features observed in experiments with a real thermally bonded nonwoven is discussed.     

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     

一种非连续介质中热传导过程的数值模拟方法

传统热传导的分析基于连续模型,无法刻画热量在两个接触体之间的传递。该文提出了一种非连续介质中热传导过程的数值计算方法,并编制了相应的C++计算程序。该方法首先将计算域离散为一系列的块体,块体内部划分若干连续介质单元,块体边界设定为潜在接触界面,并利用半弹簧-半棱联合接触模型进行接触对的快速检索及标记。每个块体内部的热传导采用传统连续模型进行计算(该文采用有限体积法),每个接触界面采用点面接触型及棱棱接触型热传导模型进行描述。通过调整接触界面热传导系数中的刚度因子,可以实现接触界面对热传导过程不同的抵抗效应。数值算例表明,该文所述方法可以较为准确地模拟热量在非连续介质中的传递过程;接触界面上的刚度因子越大,界面对热传导过程的抵抗效应越小;当刚度因子大于100,界面抵抗效应基本消失,非连续介质的计算结果与连续介质的计算结果完全一致;此外,接触界面上的刚度因子仅影响热传导的瞬态过程,而不影响其稳态解。     

Discontinuous crack model for piezoelectric materials and its application to CED evaluation

In order to study the influence of electric displacement saturation on fracture behavior of a piezoelectric material, the electric displacement strip-saturation model by [Gao H, Zhang TY, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J Mech Phys Solids 1997;45:491-510] has been applied. However, this model is only applicable to problems such as that of a crack in an infinite plate for which it provides a singular solution. In order to overcome this situation, we developed a crack model for a piezoelectric material named discontinuous crack model that is presented in this paper, to evaluate crack energy density (CED) considering electric yielding and we studied its applicability through finite element analyses. The model is defined and methods to establish its constitutive equation are discussed. Moreover, it is shown that the model by Gao et al. and the ordinary crack model in continuum can be regarded as special cases of the discontinuous crack model. Subsequently, the CED and its derivatives for the discontinuous crack model are defined and their path independent expressions are also derived based on conservation laws. Finally, a finite element formulation is devised and the applicability of the model to the evaluations of CED and its derivatives is studied through finite element analyses of an example.     

Interface crack problems in graded orthotropic media: Analytical and computational approaches

Interface crack problems in graded orthotropic media are considered using analytical and computational techniques. In the analytical formulation an interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate is considered. The principal axes of orthotropy are assumed to be parallel and perpendicular to the crack plane. Mechanical properties of the medium are assumed to be continuous with discontinuous derivatives at the interface. The problem is formulated in terms of the averaged constants of plane orthotropic elasticity and reduced to a pair of singular integral equations which are solved numerically to compute the mixed mode stress intensity factors and the energy release rate. In the second part of the study, enriched finite elements are formulated and implemented for graded orthotropic materials. Comparisons of the finite element and analytical results show that enriched finite element technique is capable of producing highly accurate results for crack problems in graded orthotropic media. Finally, periodic interface cracking and the four point bending test for graded orthotropic solids are modeled using enriched finite elements and the results are briefly discussed.     

Characterization of inplane loaded anisotropic interface corners with the boundary finite element method

Local mechanical fields at inplane loaded interface corners with discontinuous transitions of anisotropic material properties are of particular interest. This is due to the fact that the structural situation of a corner geometry with different material properties can cause singular behavior of the mechanical inplane fields. The boundary finite element method is employed in this study for the investigation of such structural problems. The results obtained with the boundary finite element method agree excellently with reference results and in addition give a deeper insight into the singular behavior of the problem considered.     

多体非连续变形系统的计算力学模型及其应用

针对由不同特性物体所组成的多体系统,探讨了能够涵盖各种变形状态和运动形式的广义有限单元模式及其插值函数形式。对于多体接触问题,发展了能够合理描述界面特性的接触力元模型,即采用某种应力插值函数将界面上的相互作用力由接触对上的接触应力来表达,并将接触对上的接触应力当作需满足界面上屈服准则与流动法则等状态控制条件的参变量,将其作为约束条件加入系统控制方程。根据非连续变形系统的分区参变量最小势能变分原理,联立变分驻值条件与参变量的状态控制条件建立了多体系统非连续变形计算力学分析的基本控制方程,将问题最终归结为一个含有自由变量和等式约束条件的线性互补问题,对此发展了数值解法,并进行了多个算例的数值分析。计算结果表明该模型不仅能够对多体系统进行静、动力耦合分析,而且还能够模拟多体系统的变形与应力及接触界面上的接触应力和相对运动等复杂的非线性过程。     

On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field

Heterogeneous incompressible fluid flows with jumps in the viscous properties are solved with the particle finite element method using continuous and discontinuous pressure fields. We show the importance of using discontinuous pressure fields to avoid errors in the incompressibility condition near the interface.     

A semi‐Lagrangean time‐integration approach for extended finite element methods

Many computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi‐Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapolated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one‐dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two‐dimensional and a three‐dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche‐type formulation for interface problems with embedded strong discontinuities. Copyright © 2014 John Wiley & Sons, Ltd.     

Treatment of material discontinuities in finite element computations

We consider finite element analysis of problems with discontinuous material coefficients. For applications in which the material interface crosses an element, we develop special elements with an embedded flux constraint at the interface. This new procedure is compared with the standard finite element method with interface coincident with the element boundary and with an existing method proposed by Steven.¹ Supporting numerical studies are conducted and rates of convergence for the solution and interface flux are examined. Some local superconvergence behaviour is observed.     

A consistent geometrically non‐linear approach for delamination

A new model is presented for the simulation of delamination in laminated composite materials. A key feature is that the material structure and the finite element mesh are uncoupled. The displacement discontinuities that arise during the delamination process are described mathematically using discontinuous functions. This leads naturally to a set of coupled equations for the continuous and the discontinuous parts of the response. Discontinuities can pass through solid finite elements arbitrarily, with the displacement jump continuous across element boundaries. The performance of the model is demonstrated for several problems of delamination and geometric instability. Copyright © 2002 John Wiley & Sons, Ltd.     

T‐stress evaluations for nonhomogeneous materials using an interaction integral method

A new equivalent domain integral of the interaction integral is derived for the computation of the T‐stress in nonhomogeneous materials with continuous or discontinuous properties. It can be found that the derived expression does not involve any derivatives of material properties. Moreover, the formulation can be proved valid even when the integral domain contains material interfaces. Therefore, the present method can be used to extract the T‐stress of nonhomogeneous materials with complex interfaces effectively. The interaction integral method in conjunction with the extended FEM is used to solve several representative examples to show its validity. Finally, using this method, the influences of material properties on the T‐stress are investigated. Numerical results show that the mechanical properties and their first‐order derivatives affect the T‐stress greatly, while the higher‐order derivatives affect the T‐stress slightly. Copyright © 2012 John Wiley & Sons, Ltd.     

A variationally coupled FE–BE method for transient problems

A variationally coupled finite element–boundary element method is developed for transient problems. A single variational statement is obtained for the entire domain and the unknown tractions, which may be discontinuous on the interface and are often a source of difficulties, are eliminated. Moreover, no interface conditions need be taken into consideration at the level of the discretized equation. The discrete equations for the coupled system can be obtained directly without any intermediate steps. The method generalizes a coupling method previously developed by the authors for statics. Numerical examples show that the solutions obtained by the present method agree very well with those obtained by analytical solutions.     

Modelling and process optimization for functionally graded materials

We optimize continuous quench process parameters to produce functionally graded aluminium alloy extrudates. To perform this task, an optimization problem is defined and solved using a standard non‐linear programming algorithm. Ingredients of this algorithm include (1) the process parameters to be optimized, (2) a cost function: the weighted average of the precipitate number density distribution, (3) constraint functions to limit the temperature gradient (and hence distortion and residual stress) and exit temperature, and (4) their sensitivities with respect to the process parameters. The cost and constraint functions are dependent on the temperature and precipitate size which are obtained by balancing energy to determine the temperature distribution and by using a reaction‐rate theory to determine the precipitate particle sizes and their distributions. Both the temperature and the precipitate models are solved via the discontinuous Galerkin finite element method. The energy balance incorporates non‐linear boundary conditions and material properties. The temperature field is then used in the reaction rate model which has as many as 10⁵ degrees‐of‐freedom per finite element node. After computing the temperature and precipitate size distributions we must compute their sensitivities. This seemingly intractable computational task is resolved thanks to the discontinuous Galerkin finite element formulation and the direct differentiation sensitivity method. A three‐dimension example is provided to demonstrate the algorithm. Copyright © 2004 John Wiley & Sons, Ltd.     

An evaluation of the MPM for simulating dynamic failure with damage diffusion

For dynamic brittle failure, conventional mesh-based methods, such as the finite element method and finite difference method, are handicapped when localized large deformations and subsequent transitions from continuous to discontinuous failure modes occur. To evaluate the potential of the material point method (MPM) in simulating dynamic brittle failure involving different failure modes, the essential features of the MPM are explored for wave and impact problems, and combined wave and diffusion problems are then solved by using the MPM. Through the comparison with the experimental, analytical and numerical data available, it appears that the MPM is a robust tool to simulate multi-physics problems such as dynamic failure under impact.     

An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites

In a nanostructured material, the interface-to- volume ratio is so high that the interface energy, which is usually negligible with respect to the bulk energy in solid mechanics, can no longer be neglected. The interfaces in a number of nanomaterials can be appropriately characterized by the coherent interface model. According to the latter, the displacement vector field is continuous across an interface in a medium while the traction vector field across the same interface is discontinuous and must satisfy the Laplace–Young equation. The present work aims to elaborate an efficient numerical approach to dealing with the interface effects described by the coherent interface model and to determining the size-dependent effective elastic moduli of nanocomposites. To achieve this twofold objective, a computational technique combining the level set method and the extended finite element method is developed and implemented. The numerical results obtained by the developed computational technique in the two-dimensional (2D) context are compared and discussed with respect to the relevant exact analytical solutions used as benchmarks. The computational technique elaborated in the present work is expected to be an efficient tool for evaluating the overall size-dependent elastic behaviour of nanomaterials and nano-sized structures.     

Evaluation of energy release rate for a planar crack in heterogeneous media

A modified version of domain integral method is developed for evaluation of energy release rate with finite element solutions for problems with a 2-D crack located in a heterogeneous elastic field. The heterogeneous field considered in this work generally contains various materials, with discontinuous mechanical moduli across the interfaces. The formulation is proved to be patch-independent, in a generalized sense, and valid for problems under both small and large deformations. The results of calculation appear to be very insensitive to the crack tip finite element models when the tip is away from the material interface. However, strong dependency on the local modeling is observed in case the tip is located at the interface. Alternative studies on this particular case are thus required.     

Meshfree point collocation method with intrinsic enrichment for interface problems

A meshfree collocation method with an intrinsic wedge enrichment is presented for solving interface problems. To approximate the class of functions with discontinuous derivatives on the interface, the wedge is asymptotically added to the basis functions. A general class of wedge basis functions with specified orders of asymptotic behavior at the interface is developed for moving least square approximations. These are implemented in diffuse derivative methods where the shape functions are approximately differentiated. The reproducing properties of these approximations for the polynomial part and for the wedge function along straight boundaries of the basis are demonstrated. For curved boundaries, the reproducing properties of the wedge functions are more restricted. Numerical results show the ease of constructing the intrinsic enrichment and the robustness of the numerical scheme in solving interface problems.     

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...     

Finite elements in space and time for generalized viscoelastic maxwell model

 The generalization of a new numerical approach with simultaneous space–time finite element discretization for viscoelastic problems developed in the papers by Buch et al. (1999) and Idesman et al. (2000) is presented for the case of the generalized viscoelastic Maxwell model. New non-symmetric variational and discretized formulations are derived using the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour described by the generalized Maxwell model is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations with total displacements and internal displacements (internal variables) as unknowns, namely to the equilibrium equation and the evolution equations for the internal displacements which are fulfilled in the weak form. Using continuous test functions in space and time, a continuous space–time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into appropriate time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across time interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case. The proposed approach has some very attractive advantages with respect to semidiscretization methods, regarding the possibility of adaptive space–time refinements and efficient parallel processing on MIMD-parallel computers. The considered numerical examples show the effectiveness of simultaneous space–time finite element calculations and a high convergence rate for adaptive refinement. Numerical efficiency is an advantage of DGM in comparison with CGM for discontinuously changing (e.g. piecewise constant) boundary conditions in time and for solutions with high gradients. Received 7 February 2000