Improved implementation and robustness study of the X‐FEM for stress analysis around cracks

Numerical crack propagation schemes were augmented in an elegant manner by the X‐FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X‐FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a ‘topological’ enrichment (only the elements touching the front are enriched). We suggest a ‘geometrical’ enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X‐FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that ‘off the shelf’ iterative solver packages can be used and perform as well on X‐FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X‐FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness. Copyright © 2005 John Wiley & Sons, Ltd.     

On the numerical stability and mass‐lumping schemes for explicit enriched meshfree methods

Meshfree methods (MMs) such as the element free Galerkin (EFG)method have gained popularity because of some advantages over other numerical methods such as the finite element method (FEM). A group of problems that have attracted a great deal of attention from the EFG method community includes the treatment of large deformations and dealing with strong discontinuities such as cracks. One efficient solution to model cracks is adding special enrichment functions to the standard shape functions such as extended FEM, within the FEM context, and the cracking particles method, based on EFG method. It is well known that explicit time integration in dynamic applications is conditionally stable. Furthermore, in enriched methods, the critical time step may tend to very small values leading to computationally expensive simulations. In this work, we study the stability of enriched MMs and propose two mass‐lumping strategies. Then we show that the critical time step for enriched MMs based on lumped mass matrices is of the same order as the critical time step of MMs without enrichment. Moreover, we show that, in contrast to extended FEM, even with a consistent mass matrix, the critical time step does not vanish even when the crack directly crosses a node. Copyright © 2011 John Wiley & Sons, Ltd.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

The Discontinuity‐Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity‐Enriched Finite Element Method (DE‐FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction‐free and cohesive crack examples. We show that DE‐FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.     

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 John Wiley & Sons, Ltd.     

Implicit boundary method for analysis using uniform B‐spline basis and structured grid

Several analysis techniques such as extended finite element method (X‐FEM) have been developed recently, which use structured grid for the analysis. Implicit boundary method uses implicit equations of the boundary to apply boundary conditions in X‐FEM framework using structured grids. Solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary. In this paper, this method is applied for analysis using uniform B‐spline basis defined over a structured grid. Solution structures that are C¹ or C² continuous throughout the analysis domain can be constructed using B‐spline basis functions. As a structured grid does not conform to the geometry of the analysis domain, the boundaries of the analysis domain are defined independently using equations of the boundary curves/surfaces. Compared with conforming mesh, it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted. Numerical examples are presented to demonstrate the performance of these B‐spline elements. The results are compared with analytical solutions as well as with traditional finite element solutions. Convergence studies for several examples show that B‐spline elements provide accurate solutions with fewer elements and nodes compared with traditional FEM. They also provide continuous stress and strain in the analysis domain, thus eliminating the need for smoothing stress/strain results. Copyright © 2008 John Wiley & Sons, Ltd.     

An edge‐based smoothed finite element method for 3D analysis of solid mechanics problems

The edge‐based smoothed finite element method (ES‐FEM) was proposed recently in Liu, Nguyen‐Thoi, and Lam to improve the accuracy of the FEM for 2D problems. This method belongs to the wider family of the smoothed FEM for which smoothing cells are defined to perform the numerical integration over the domain. Later, the face‐based smoothed FEM (FS‐FEM) was proposed to generalize the ES‐FEM to 3D problems. According to this method, the smoothing cells are centered along the faces of the tetrahedrons of the mesh. In the present paper, an alternative method for the extension of the ES‐FEM to 3D is investigated. This method is based on an underlying mesh composed of tetrahedrons, and the approximation of the field variables is associated with the tetrahedral elements; however, in contrast to the FS‐FEM, the smoothing cells of the proposed ES‐FEM are centered along the edges of the tetrahedrons of the mesh. From selected numerical benchmark problems, it is observed that the ES‐FEM is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the FS‐FEM for a given number of degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

Electrostatic simulation using XFEM for conductor and dielectric interfaces

Many Micro‐Electro‐Mechanical Systems (e.g. RF‐switches, micro‐resonators and micro‐rotors) involve mechanical structures moving in an electrostatic field. For this type of problems, it is required to evaluate accurately the electrostatic forces acting on the devices. Extended Finite Element (X‐FEM) approaches can easily handle moving boundaries and interfaces in the electrostatic domain and seem therefore very suitable to model Micro‐Electro‐Mechanical Systems. In this study we investigate different X‐FEM techniques to solve the electrostatic problem when the electrostatic domain is bounded by a conducting material. Preliminary studies in one‐dimension have shown that one can obtain good results in the computation of electrostatic potential using X‐FEM. In this paper the extension of these preliminary studies to 2D problem is presented. In particular, a new type of enrichment functions is proposed in order to treat accurately Dirichlet boundary conditions on the interface. Copyright © 2010 John Wiley & Sons, Ltd.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

The effects of element shape on the critical time step in explicit time integrators for elasto‐dynamics

In this paper, the effects of element shape on the critical time step are investigated. The common rule‐of‐thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four‐noded bilinear quadrilaterals and three‐noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements. Copyright © 2014 John Wiley & Sons, Ltd.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.     

A cell‐based smoothed discrete shear gap method (CS‐FEM‐DSG3) based on the C0‐type higher‐order shear deformation theory for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle

A cell‐based smoothed discrete shear gap method (CS‐FEM‐DSG3) based on the first‐order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the CS‐FEM‐DSG3 is extended to the C⁰‐type higher‐order shear deformation plate theory (C⁰‐HSDT) and is incorporated with damping–spring systems for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle. At each time step of dynamic analysis, one four‐step procedure is performed including the following: (1) transformation of the weight of a four‐wheel vehicle into the sprung masses at wheels; (2) dynamic analysis of the sprung mass of wheels to determine the contact forces; (3) transformation of the contact forces into loads at nodes of plate elements; and (4) dynamic analysis of the plate elements on viscoelastic foundations. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Selective smoothed finite element methods for extremely large deformation of anisotropic incompressible bio‐tissues

Two dynamic selective smoothed FEM (selective S‐FEM) are proposed for analysis of extremely large deformation of anisotropic incompressible bio‐tissues using the simplest four‐node tetrahedron elements. In the present two Selective S‐FEMs, the method that consists of the face‐based smoothed FEM (FS‐FEM) used for the deviatoric part of deformation and the node‐based smoothed FEM (NS‐FEM) used for the volumetric part is called FS/NS‐FEM; another method that replaces the deviatoric part of deformation in the first one by the edge‐based smoothed FEM (3D‐ES‐FEM) is call 3D‐ES/NS‐FEM. Both selective S‐FEMs can achieve outstanding accuracy, and stability of volumetric locking free. This is because the NS‐FEM offers an ‘overly‐soft’ feature (in contrast to the standard FEM ‘overly‐stiff’ model), which can be used to effectively mitigate the volumetric locking, and on the other hand, the 3D‐ES‐FEM and FS‐FEM produce close to exact stiffness for the discretized model leading to accurate solution. Numerical examples are presented to examine the performance of the selective S‐FEM methods, including soft bio‐tissues that may be isotropic, transversely isotropic, and anisotropic arterial layered materials. The present methods are found having good accuracy and performance. The examples also demonstrate that the proposed methods are very robust and possess remarkable capabilities of handling element distortion, which is very useful for simulating soft materials including bio‐tissues. Copyright © 2014 John Wiley & Sons, Ltd.     

A new coupling technique for the combination of wavelet‐Galerkin method with finite element method in solids and structures

In this paper, a coupling technique is developed for the combination of the wavelet‐Galerkin method (WGM) with the finite element method (FEM). In this coupled method, the WGM and FEM are respectively used in different sub‐domains. The WGM sub‐domain and the FEM sub‐domain are connected by a transition region that is described by B‐spline basis functions. The basis functions of WGM and FEM are modified in the transition region to ensure the basic polynomial reconstruction condition and the compatibility of displacements and their gradients. In addition, the elements of FEM and WGM are not necessary to conform to the transition region. This newly developed coupled method is applied to the stress analysis of 2D and 3D elasticity problems in order to investigate its performance and study parameters. Numerical results show that the present coupled method is accurate and stable. The new method has a promising potential for practical applications and can be easily extended for coupling of other numerical methods. Copyright © 2017 John Wiley & Sons, Ltd.     

Level set X‐FEM non‐matching meshes: application to dynamic crack propagation in elastic–plastic media

This paper develops two aspects improving crack propagation modelling with the X‐FEM method. On the one hand, it explains how one can use at the same time a regular structured mesh for a precise and efficient level set update and an unstructured irregular one for the mechanical model. On the other hand, a new numerical scheme based on the X‐FEM method is proposed for dynamic elastic–plastic situations. The simulation results are compared with two experiments on PMMA for which crack speed and crack path are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Overcoming element quality dependence of finite elements with adaptive extended stencil FEM (AES‐FEM)

The finite element methods (FEMs) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the Adaptive Extended Stencil Finite Element Method (AES‐FEM) as a means for overcoming this dependence on element shape quality. Our method replaces the traditional basis functions with a set of generalized Lagrange polynomial basis functions, which we construct using local weighted least‐squares approximations. The method preserves the theoretical framework of FEM and allows imposing essential boundary conditions and integrating the stiffness matrix in the same way as the classical FEM. In addition, AES‐FEM can use higher‐degree polynomial basis functions than the classical FEM, while virtually preserving the sparsity pattern of the stiffness matrix. We describe the formulation and implementation of AES‐FEM and analyze its consistency and stability. We present numerical experiments in both 2D and 3D for the Poisson equation and a time‐independent convection–diffusion equation. The numerical results demonstrate that AES‐FEM is more accurate than linear FEM, is also more efficient than linear FEM in terms of error versus runtime, and enables much better stability and faster convergence of iterative solvers than linear FEM over poor‐quality meshes. Copyright © 2016 John Wiley & Sons, Ltd.     

Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods

We show both theoretically and numerically a connection between the smoothed finite element method (SFEM) and the virtual element method and use this approach to derive stable, cheap and optimally convergent polyhedral FEM. We show that the stiffness matrix computed with one subcell SFEM is identical to the consistency term of the virtual element method, irrespective of the topology of the element, as long as the shape functions vary linearly on the boundary. Using this connection, we propose a new stable approach to strain smoothing for polygonal/polyhedral elements where, instead of using sub‐triangulations, we are able to use one single polygonal/polyhedral subcell for each element while maintaining stability. For a similar number of degrees of freedom, the proposed approach is more accurate than the conventional SFEM with triangular subcells. The time to compute the stiffness matrix scales with the in case of the conventional polygonal FEM, while it scales as in the proposed approach. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Copyright © 2015 John Wiley & Sons, Ltd.     

Three‐dimensional non‐planar crack growth by a coupled extended finite element and fast marching method

A numerical technique for non‐planar three‐dimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method (X‐FEM) and the fast marching method (FMM). In crack modeling using X‐FEM, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields. The initial crack geometry is represented by two level set functions, and subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated without remeshing. Crack growth is conducted using FMM; unlike a level set formulation for interface capturing, no iterations nor any time step restrictions are imposed in the FMM. Planar and non‐planar quasi‐static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique. Copyright © 2008 John Wiley & Sons, Ltd.     

A novel numerical method of high‐order accuracy for flow in unsaturated porous media

We construct finite volume schemes of very high order of accuracy in space and time for solving the nonlinear Richards equation (RE). The general scheme is based on a three‐stage predictor–corrector procedure. First, a high‐order weighted essentially non‐oscillatory (WENO) reconstruction procedure is applied to the cell averages at the current time level to guarantee monotonicity in the presence of steep gradients. Second, the temporal evolution of the WENO reconstruction polynomials is computed in a predictor stage by using a global weak form of the governing equations. A global space–time DG FEM is used to obtain a scheme without the parabolic time‐step restriction caused by the presence of the diffusion term in the RE. The resulting nonlinear algebraic system is solved by a Newton–Krylov method, where the generalized minimal residual method algorithm of Saad and Schulz is used to solve the linear subsystems. Finally, as a third step, the cell averages of the finite volume method are updated using a one‐step scheme, on the basis of the solution calculated previously in the space–time predictor stage. Our scheme is validated against analytical, experimental, and other numerical reference solutions in four test cases. A numerical convergence study performed allows us to show that the proposed novel scheme is high order accurate in space and time. Copyright © 2011 John Wiley & Sons, Ltd.