An interface‐enriched generalized finite element analysis for electromagnetic problems with non‐conformal discretizations

An interface‐enriched generalized finite element method is presented for analyzing electromagnetic problems involving highly inhomogeneous materials. To avoid creating conformal meshes within a complex computational domain and preparing multiple meshes during optimization, enriched vector basis functions are introduced over the finite elements that intersect the material interfaces to capture the normal derivative discontinuity of the tangential field component. These enrichment functions are directly constructed from a linear combination of the vector basis functions of the sub‐elements. Several numerical examples are presented to verify the method with analytical solutions and demonstrate its h‐refinement convergence rate. The proposed interface‐enriched generalized finite element method is shown to achieve the same level of accuracy as the standard finite element method based on conformal meshes. Two examples, involving multiple microvascular channels and circular inclusions of different radii, are analyzed to illustrate the capability of the proposed approach in handling complicated inhomogeneous geometries. Copyright © 2015 John Wiley & Sons, Ltd.     

求解不可压缩粘性流的GLS单元之比较

比较了用于求解不可压缩粘性流的四边形双线性、双二次单元及三角形二次单元的性能,这些单元采用GLS稳定化有限元格式,而压力和速度采用等阶数插值。对得出的非线性有限元方程,使用Newton-Raphson迭代来求解,推导了计算切线矩阵的所需公式。完成了对雷诺数分别为1000、5000、10000和20000的方腔上板流的数值模拟,并对不同单元的结果的精度和收敛率进行了比较。数值算例显示,较之于另两种单元,三角形的二次单元在精度和收敛性上达到最好的匹配。     

Numerical analysis of dynamic debonding under 2D in-plane and 3D loading

We present a numerical scheme specially developed for 2D and 3D dynamic debonding problems. The method, referred to as spectral scheme, allows for a precise modeling of stationary and/or spontaneously expanding interfacial cracks of arbitrary shapes and subjected to an arbitrary combination of time- and space-dependent loading conditions. It is based on a spectral representation of the elastodynamic relations existing between the displacement components along the interface plane and the corresponding dynamic stresses. A general stress-based cohesive failure model is introduced to model the spontaneous progressive failure of the interface. The numerical scheme also allows for the introduction of a wide range of contact relations to model the possible interactions between the fracture surfaces. Simple 2D problems are used to investigate the accuracy and stability of the proposed scheme. Then, the spectral method is used in various 2D and 3D interfacial fracture problems, with special emphasis on the issue of the limiting speed for a spontaneously propagating debonding crack in the presence of frictional contact. This revised version was published online in July 2006 with corrections to the Cover Date.     

Plane wave propagation in 2D and 3D monodisperse periodic granular media

Plane wave propagation in periodic ordered granular media comprising of elastic spherical particles is investigated. The spheres are under zero precompression and are assumed to interact via the Hertzian contact potential. Various two- and three-dimensional granular structures such as hexagonal packing (2D and 3D), face-centered cubic and body-centered cubic packings are considered in the present study, with the plane impact either normal or oblique to the granular system. For the normal impact case, 1D chains equivalent to the 2D and 3D structures are obtained. A universal relation between the wavefront speed and the force amplitude is derived, valid for all the granular structures studied. In the angular impact case, the shear component of the amplitude of the particle velocity is found to initially decay exponentially and further in a series of linear regimes. By employing simpler models, semi-analytical predictions are obtained for the decay of shearing effect.     

Energy release rate approximation for edge cracks using higher-order topological derivatives

Sub-micron films deposited on a flexible substrate are now commonly used in electronic industry. The main damaging mode of these systems is a multi-cracking of the film under the action of thermal and mechanical stresses. This multi-cracking phenomenon is described using the coupled criterion based on the simultaneous fulfilment of an energy and a stress criteria. The coupled criterion is implemented in a representative volume element and it allows to decide whether the stress or the energy condition governs the cracking mechanism. It is found that the energy conditions predominates for very thin films whereas the stress condition can take place for thicker films. The initial density of cracks is determined and is in good agreement with the experimental measures. Further subdivisions, when increasing the load, are also predicted. Moreover, under some conditions, a master curve can rule the density of cracks function of the applied strain, showing a good agreement between predictions and experiments for a wide range of film thicknesses.     

Impact response of granular layers

Motivated by the wave tailoring potential of granular media, this study aims at evaluating force transmission through granular layers made of spherical particles. 2D simulations based on Hertzian contact law between adjacent particles are performed on two distinct systems: (1) layers consisting of ordered bimaterial lattices, and (2) single material layers with random packing. For the ordered systems, force transmission properties are found to vary with material mismatch and layer thickness. Transmitted force-decay in random configurations is substantially higher than those in the ordered systems.

     

Coupled multi‐scale cohesive modeling of failure in heterogeneous adhesives

A multi‐scale cohesive numerical framework is proposed to simulate the failure of heterogeneous adhesively bonded systems. This multi‐scale scheme is based on Hill's variational principle of energy equivalence between the higher and lower level scales. It provides an easy way to obtain accurate homogenized macroscopic properties while capturing the physics of failure processes at the micro‐scale in sufficient detail. We use an isotropic rate‐dependent damage model to mimic the failure response of the constituents of heterogeneous adhesives. The finite element method is used to solve the equilibrium equation at each scale. A nested iterative scheme inspired by the return mapping algorithm used in computational inelasticity is implemented. We propose a computationally attractive technique to couple the macro‐ and micro‐scales for rate‐dependent constitutive laws. We introduce an adhesive patch test to study the numerical performance, including spatial and temporal convergence of the multi‐scale scheme. We compare the solution of the multi‐scale cohesive scheme with a direct numerical simulation. Finally, we solve mode I and mode II fracture problems to demonstrate failure at the macro‐scale. Copyright © 2010 John Wiley & Sons, Ltd.     

Generalized finite element enrichment functions for discontinuous gradient fields

A general GFEM/XFEM formulation is presented to solve two‐dimensional problems characterized by C⁰ continuity with gradient jumps along discrete lines, such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media. The new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines, such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks. We show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines. The use of the proposed enrichments in both linear and quadratic approximations is investigated, as well as their combination with interface enrichment functions available in the literature. Through a detailed convergence study, we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems. In the linear case, the effectiveness of correction functions proposed in the literature is also investigated. Copyright © 2009 John Wiley & Sons, Ltd.