An adaptive enrichment algorithm for advection‐dominated problems

We are interested in developing a numerical framework well suited for advection–diffusion problems when the advection part is dominant. In that case, given Dirichlet type boundary condition, it is well known that a boundary layer develops. To resolve correctly this layer, standard methods consist in increasing the mesh resolution and possibly increasing the formal accuracy of the numerical method. In this paper, we follow another path: we do not seek to increase the formal accuracy of the scheme but, by a careful choice of finite element, to lower the mesh resolution in the layer. Indeed the finite element representation we choose is locally the sum of a standard one plus an enrichment. This paper proposes such a method and with several numerical examples, we show the potential of this approach. In particular, we show that the method is not very sensitive to the choice of the enrichment and develop an adaptive algorithm to automatically choose the enrichment functions.Copyright © 2012 John Wiley & Sons, Ltd.     

The intrinsic XFEM for two‐fluid flows

In two‐fluid flows, jumps and/or kinks along the interfaces are present in the resulting velocity and pressure fields. Standard methods require mesh manipulations with the aim that either element edges align with the interfaces or that the mesh is sufficiently refined near the interfaces. In contrast, enriched methods, such as the extended finite element method (XFEM), enable the representation of arbitrary jumps and kinks inside elements. Thereby, optimal convergence can be achieved for two‐fluid flows with meshes that remain fixed throughout the simulation. In the intrinsic XFEM, in contrast to other enriched methods, no more unknowns are present in the approximation than in a standard finite element approximation. In this work, the intrinsic XFEM is employed for the simulation of incompressible two‐fluid flows. Numerical results are shown for a number of test cases and prove the success of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

A domain-independent interaction integral for magneto-electro-elastic materials

Magneto-electro-elastic (MEE) materials usually consist of piezoelectric (PE) and piezomagnetic (PM) phases. Between different constituent phases, there exist lots of interfaces with discontinuous MEE properties. Complex interface distribution brings a great difficulty to the fracture analysis of MEE materials since the present fracture mechanics methods can hardly solve the fracture parameters efficiently of a crack surrounded by complex interfaces. This paper develops a new domain formulation of the interaction integral for the computation of the fracture parameters including stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF) for linear MEE materials. The formulation derived here does not involve any derivatives of material properties and moreover, it can be proved that an arbitrary interface in the integral domain does not affect the validity and the value of the interaction integral. Namely, the interaction integral is domain-independent for material interfaces and thus, its application does not require material parameters to be continuous. Due to this advantage, the interaction integral becomes an effective approach for extracting the fracture parameters of MEE materials with complex interfaces. Combined with the extended finite element method (XFEM), the interaction integral is employed to solve several representative problems to verify its accuracy and domain-independence. Good results show the effectiveness of the present method in the fracture analysis of MEE materials with continuous and discontinuous properties. Finally, the particulate MEE composites composed of PE and PM phases are considered and four schemes of different property-homogenization level are proposed for comparing their effectiveness.     

Influence of heat transfer and fluid flow on crack growth in multilayered porous/dense materials using XFEM: Application to Solid Oxide Fuel Cell like material design

An advanced numerical model is developed to investigate the influence of heat transfer and fluid flow on crack propagation in multi-layered porous materials. The fluid flow, governed by the Navier–Stokes and Darcy’s law, is discretized with the nonconforming Crouzeix–Raviart (CR) finite element method. A combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods is used to solve the advection–diffusion heat transfer equation in the flow channel and in the fluid phase within the porous material. The crack is assumed to affect only the heat diffusion within the porous layer, therefore a time splitting technique is used to solve the heat transfer in the fluid and the solid phases separately. Thus, within the porous material, the crack induces a discontinuity of the temperature at the crack surfaces and a singularity of the flux at the crack tip. Conduction in the solid phase is solved using the eXtended Finite Element Method (XFEM) to better handle the discontinuities and singularities caused by the cracks. The XFEM is also used to solve the thermo-mechanical problem and to track the crack propagation. The multi-physics model is implemented then validated for the transient regime, this necessitated a post processing treatment in which, the stress intensity factors (SIF) are computed for each time step. The SIFs are then used in the crack propagation criterion and the crack orientation angle. The methodology seems to be robust accurate and the computational cost is reduced thanks to the XFEM.     

Laser cutting silicon-glass double layer wafer with laser induced thermal-crack propagation

This study was aimed at introducing the laser induced thermal-crack propagation (LITP) technology to solve the silicon-glass double layer wafer dicing problems in the packaging procedure of silicon-glass device packaged by WLCSP technology, investigating the feasibility of this idea, and studying the crack propagation process of LITP cutting double layer wafer. In this paper, the physical process of the 1064 nm laser beam interact with the double layer wafer during the cutting process was studied theoretically. A mathematical model consists the volumetric heating source and the surface heating source has been established. The temperature and stress distribution was simulated by using finite element method (FEM) analysis software ABAQUS. The extended finite element method (XFEM) was added to the simulation as the supplementary features to simulate the crack propagation process and the crack propagation profile. The silicon-glass double layer wafer cutting verification experiment under typical parameters was conducted by using the 1064 nm semiconductor laser. The crack propagation profile on the fracture surface was examined by optical microscope and explained from the stress distribution and XFEM status. It was concluded that the quality of the finished fracture surface has been greatly improved, and the experiment results were well supported by the numerical simulation results.     

光滑扩展有限元方法及其特点研究

将光滑有限元法S-FEM（Smoothed Finite Element Method）的子域光滑应变技术和边域光滑应变技术同时引入到扩展有限元XFEM（Extended Finite Element Method）中，提出一种新的光滑扩展有限元法S-XFEM（Smoothed Extended Finite Element Method）。在单元选取及扩充结点选取时采用ES-FEM的光滑域划分方式，在数值积分计算刚度矩阵时采用基于三角形子域的CS-FEM积分思路，并给出了高斯点的积分策略。设计了S-XFEM程序架构并利用Matlab语言编制了S-XFEM计算程序。通过几个经典算例研究对比了XFEM和S-XFEM的特点，验证了S-XFEM的精确性和适用性。结果表明，XFEM和S-XFEM均具有很高的计算精确性和收敛性，XFEM计算精度略高于S-XFEM，而S-XFEM在网格独立性上则明显优于XFEM。     

Wall modeling via function enrichment within a high‐order DG method for RANS simulations of incompressible flow

We present a novel approach to wall modeling for the Reynolds‐averaged Navier‐Stokes equations within the discontinuous Galerkin method. Wall functions are not used to prescribe boundary conditions as usual, but they are built into the function space of the numerical method as a local enrichment, in addition to the standard polynomial component. The Galerkin method then automatically finds the optimal solution among all shape functions available. This idea is fully consistent and gives the wall model vast flexibility in separated boundary layers or high adverse pressure gradients. The wall model is implemented in a high‐order discontinuous Galerkin solver for incompressible flow complemented by the Spalart‐Allmaras closure model. As benchmark examples, we present turbulent channel flow starting from Re_τ=180 and up to Re_τ=100000 as well as flow past periodic hills at Reynolds numbers based on the hill height of Re_H=10595 and Re_H=19000.     

基于扩展有限元法的裂尖场精度研究

扩展有限元方法基于单元分解的基本思想,通过引入位移加强函数来表征裂纹的不连续性和裂尖的奇异性。在裂尖加强单元与常规单元之间有一层混合单元,当对裂尖特定区域进行加强时,混合单元个数相应增加,混合单元个数与计算精度存在一定联系。本文提出一种正方形裂尖加强区域的选择方式,可得到较单个加强和圆形加强精度更高、更稳定的计算结果。对于不同长度的裂纹,表征裂尖场奇异性所需的裂尖加强范围存在较大差异,以正方形裂尖加强方式进行计算,得到了不同裂纹长度下最优的加强尺寸。     

Numerical investigations of the XFEM for solving two-phase incompressible flows

This paper discusses the application of the extended finite element method (XFEM) to solve two-phase incompressible flows. The Navier–Stokes equations are discretised using the Taylor–Hood finite element. To capture the different discontinuities across the interface, kink or jump enrichments are used for the velocity and/or pressure fields. However, these enrichments may lead to an inappropriate combination of interpolations. Different polynomial enrichment orders and different enrichment functions are investigated; only the stable combination will be used afterward.

In cases with a surface tension force, the accuracy mainly relies on the precise computation of the normal and curvature. A novel method for computing normal vectors to the interface is proposed. This method employs successive mesh refinements inside the cut elements. Comparisons with analytical and numerical solutions demonstrate that the method is effective. Moreover, the mesh refinement improves the sub-integration in the XFEM and allows for a precise re-initialisation procedure.     

Fatigue crack growth simulation in coated materials using X-FEM

In the present work, the eXtended Finite Element Method (XFEM) is used to study the effect of bi-material interfaces on fatigue life in galvanised panels. X-FEM and Paris law are implemented in ABAQUS software using Python code. The XFEM method proved to be an adequate method for stress intensity factor computation, and, furthermore, no remeshing is required for crack growth simulations. A study of fatigue crack growth is conducted for several substrate materials, and the influence of the initial crack angle is ascertained. This study also compares the crack growth rate between three types of bi-materials alloys zinc/steel, zinc/aluminium, and zinc/zinc. The interaction between two cracks and fatigue life, in the presence of bi-material interface, is investigated as well.