Extended Finite Element Method

Theory and Applications

Amir R. Khoei

• Publisher: Wiley
• Year: 2014
• ISBN:
  9781118869680
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mechanics

Introduces the theory and applications of the extended finite element method (XFEM) in the linear and nonlinear problems of continua, structures and geomechanics

• Explores the concept of partition of unity, various enrichment functions, and fundamentals of XFEM formulation.
• Covers numerous applications of XFEM including fracture mechanics, large deformation, plasticity, multiphase flow, hydraulic fracturing and contact problems
• Accompanied by a website hosting source code and examples

1
Introduction

1.1 Introduction

The finite element method (FEM) is one of the most common numerical tools for obtaining the approximate solutions of partial differential equations. It has been applied successfully in many areas of engineering sciences to study, model, and predict the behavior of structures. The area ranges across aeronautical and aerospace engineering, the automobile industry, mechanical engineering, civil engineering, biomechanics, geomechanics, material sciences, and many more. The FEM does not operate on differential equations; instead, continuous boundary and initial value problems are reformulated into equivalent variational forms. The FEM requires the domain to be subdivided into non-overlapping regions, called the elements. In the FEM, individual elements are connected together by a topological map, called a mesh, and local polynomial representation is used for the fields within the element. The solution obtained is a function of the quality of mesh and the fundamental requirement is that the mesh has to conform to the geometry. The main advantage of the FEM is that it can handle complex boundaries without much difficulty. Despite its popularity, the FEM suffers from certain drawbacks. There are number of instances where the FEM poses restrictions to an efficient application of the method. The FEM relies on the approximation properties of polynomials; hence, they often require smooth solutions in order to obtain optimal accuracy. However, if the solution contains a non-smooth behavior, like high gradients or singularities in the stress and strain fields, or strong discontinuities in the displacement field as in the case of cracked bodies, then the FEM becomes computationally expensive to get optimal convergence.

One of the most significant interests in solid mechanics problems is the simulation of fracture and damage phenomena (Figure 1.1). Engineering structures, when subjected to high loading, may result in stresses in the body exceeding the material strength and thus, in progressive failure. These material failure processes manifest themselves in various failure mechanisms such as the fracture process zone (FPZ) in rocks and concrete, the shear band localization in ductile metals, or the discrete crack discontinuity in brittle materials. The accurate modeling and evolution of smeared and discrete discontinuities have been a topic of growing interest over the past few decades, with quite a few notable developments in computational techniques over the past few years. Early numerical techniques for modeling discontinuities in finite elements can be seen in the work of Ortiz, Leroy, and Needleman (1987) and Belytschko, Fish, and Englemann (1988). They modeled the shear band localization as a “weak” (strain) discontinuity that could pass through the finite element mesh using a multi-field variational principle. Dvorkin, Cuitiño, and Gioia (1990) considered a “strong” (displacement) discontinuity by modifying the principle of virtual work statement. A unified framework for modeling the strong discontinuity by taking into account the softening constitutive law and the interface traction–displacement relation was proposed by Simo, Oliver, and Armero (1993). In the strong discontinuity approach, the displacement consists of regular and enhanced components, where the enhanced component yields a jump across the discontinuity surface. An assumed enhanced strain variational formulation is used, and the enriched degrees of freedom (DOF) are statically condensed on an element level to obtain the tangent stiffness matrix for the element. An alternative approach for modeling fracture phenomena was introduced by Xu and Needleman (1994) based on the cohesive surface formulation, which was used later by Camacho and Ortiz (1996) to model the damage in brittle materials. The cohesive surface formulation is a phenomenological framework in which the fracture characteristics of the material are embedded in a cohesive surface traction–displacement relation. Based on this approach, an inherent length scale is introduced into the model, and in addition, no fracture criterion is required so the crack growth and the crack path are outcomes of the analysis.

Figure 1.1 Building destroyed by a 8.8 magnitude earthquake on Saturday, February 27, 2010, with intense shaking lasting for about 3 minutes, which occurred off the coast of central Chile.

(Source: Vladimir Platonow (Agência Brasil) [CC-BY-3.0-br (http://creativecommons.org/licenses/by/3.0/br/deed.en)], via Wikimedia Commons; http://commons.wikimedia.org/wiki/File:Terremoto_no_Chile_2010.webp)

In the FEM, the non-smooth displacement near the crack tip is basically captured by refining the mesh locally. The number of DOF may drastically increase, especially in three-dimensional applications. Moreover, the incremental computation of a crack growth needs frequent remeshings. Reprojecting the solution on the updated mesh is not only a costly operation but also it may have a troublesome impact on the quality of results. The classical FEM has achieved its limited ability for solving fracture mechanics problems. To avoid these computational difficulties, a new approach to the problem consists in taking into account the a priori knowledge of the exact solution. Applying the asymptotic crack tip displacement solution to the finite element basis seems to have been a somewhat early idea. A significant improvement in crack modeling was presented with the development of a partition of unity (PU) based enrichment method for discontinuous fields in the PhD dissertation by Dolbow (1999), which was referred to as the extended FEM (X-FEM). In the X-FEM, special functions are added to the finite element approximation using the framework of PU. For crack modeling, a discontinuous function such as the Heaviside step function and the two-dimensional linear elastic asymptotic crack tip displacement fields, are used to account for the crack. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces. The location of the crack discontinuity can be arbitrary with respect to the underlying finite element mesh, and the crack propagation simulation can be performed without the need to remesh as the crack advances. A particularly appealing feature is that the finite element framework and its properties, such as the sparsity and symmetry, are retained and a single-field (displacement) variational principle is used to obtain the discrete equations. This technique provides an accurate and robust numerical method to model strong (displacement) discontinuities.

The original research articles on the X-FEM were presented by Belytschko and Black (1999) and Moës, Dolbow, and Belytschko (1999) for elastic fracture propagation on the topic of “A FEM for crack growth without remeshing”. They presented a minimal remeshing FEM for crack growth by including the discontinuous enrichment functions to the finite element approximation in order to account for the presence of the crack. The essential idea was based on adding enrichment functions to the approximation space that contains a discontinuous displacement field. Hence, the method allows the crack to be arbitrarily aligned within the mesh. The same span of functions was earlier developed by Fleming et al. (1997) for the enrichment of the element-free Galerkin method. The method exploits the PU property of finite elements that was noted by Melenk and Babuška (1996), namely that the sum of the shape functions must be unity. This property has long been known, since it corresponds to the ability of the shape functions to reproduce a constant that represents translation, which is crucial for convergence.

The X-FEM provides a powerful tool for enriching solution spaces with information from asymptotic solutions and other knowledge of the physics of the problem. This has proved very useful for cracks and dislocations where near-field solutions can be embedded by the PU method to tremendously increase the accuracy of relatively coarse meshes. The technique offers possibilities in treating phenomena such as surface effects in nano-mechanics, void growth, subscale models of interface behavior, and so on. Thus, the X-FEM method has greatly enhanced the power of the FEM for many of the problems of interest in mechanics of materials. The aim of this chapter is to provide an overview of the X-FEM with an emphasis on various applications of the technique to materials modeling problems, including linear elastic fracture mechanics ( LEFM); cohesive fracture mechanics; composite materials and material inhomogeneities; plasticity, damage and fatigue problems; shear band localization; fluid–structure interaction; fluid flow in fractured porous media; fluid flow and fluid mechanics problems; phase transition and solidification; thermal and thermo-mechanical problems; plates and shells; contact problems; topology optimization; piezoelectric and magneto-electroelastic problems; and multi-scale modeling.

1.2 An Enriched Finite Element Method

The FEM is widely used in industrial design applications, and many different software packages based on FEM techniques have been developed. It has undoubtedly become the most popular and powerful analytical tool for studying the behavior of a wide range of engineering and physical problems. Its applications have been developed from basic mechanical problems to fracture mechanics, fluid dynamics, nano-structures, electricity, chemistry, civil engineering, and material science (Figure 1.2). The FEM has proved to be very well suited to the study of fracture mechanics. However, modeling the propagation of a crack through a finite element mesh turns out to be difficult because of the modification of mesh topology. To accurately model discontinuities with FEMs, it is necessary to conform the discretization to the discontinuity. This becomes a major difficulty when treating problems with evolving discontinuities where the mesh must be regenerated at each step. Reprojecting the solution on the updated mesh is not only a costly operation but also it may have a troublesome impact on the quality of results.

Figure 1.2 Bridge damage in Shaharah, Yemen, August 1986. The failure of bridges is of special concern to structural engineers in trying to learn lessons vital to bridge design, construction, and maintenance.

(Source: Bernard Gagnon [CC-BY-3.0-br (http://creativecommons.org/licenses/by/3.0.en)], via Wikimedia Commons; http://en.wikipedia.org/wiki/File:Shehara_02.webp)

Modeling moving discontinuities within the classical finite element is quite cumbersome due to the necessity of the mesh to conform to discontinuity surfaces. Mesh generation of complex geometries can be very time consuming with a classical finite element analysis. The main difficulty arises from the necessity of the mesh to conform to physical surfaces. Discontinuities such as holes, cracks, and material interfaces may not cross mesh elements. Moreover, local refinements close to discontinuities and mesh modification to track the geometrical and topological changes in crack propagation problems for example, can be difficult. Also, when geometries evolve and history dependent models are used, robust methods to transfer the solution to the new mesh are needed. This issue is particularly significant, since computed fields defined on these discontinuities are often the most important ones. In order to overcome these mesh-dependent difficulties, the generalized finite element method (G-FEM) and the X-FEM have been developed to facilitate the modeling of arbitrary moving discontinuities through the partition of unity enrichment of finite elements (PUM), in which the main idea is to extend a classical approximate solution basis by a set of locally supported enrichment functions that carry information about the character of the solution, for example, singularity, discontinuity, and boundary layer. As it permits arbitrary functions to be locally incorporated in the FEM or the meshfree approximation, the PUM gives flexibility in modeling moving discontinuities without changing the underlying mesh, while the set of enrichment functions evolve (and/or their supports) with the interface geometry. In addition to facilitating the modeling of moving discontinuities, enrichment also increases the local approximation power of the solution space by allowing the introduction of arbitrary functions within the basis. This is particularly useful for problems with singularities or boundary layers.

Basically, the G-FEM and the X-FEM are versatile tools for the analysis of problems characterized by discontinuities, singularities, localized deformations, and complex geometries. These methods can dramatically simplify the solution of many problems in material modeling, such as the propagation of cracks, the evolution of dislocations, the modeling of grain boundaries, and the evolution of phase boundaries. The advantage of these methods is that the finite element mesh can be completely independent of the morphology of these entities. The G-FEM and the X-FEM incorporate the analytically known or numerically computed handbook functions within some range of their applicability into the traditional FE (finite element) approximation with the PU (partition of unity) method to enhance the local and global accuracy of the computed solution. Both the X-FEM and G-FEM meshes need not conform to the boundaries of the problem. The FEM is used as the building block in the X-FEM and the G-FEM; hence, much of the theoretical and numerical developments in FEs can be readily extended and applied. Moreover, the X-FEM and G-FEM make possible an accurate solution of engineering problems in complex domains that may be practically impossible to solve using the FE method. The X-FEM and G-FEM are basically identical methods; the X-FEM was originally developed for discontinuities, such as cracks, and used local enrichments while the G-FEM was first involved with global enrichments of the approximation space. The X-FEM and G-FEM can be used with both structured and unstructured meshes. The structured meshes are appealing for many studies in materials science, where the objective is to determine the properties of a unit cell of the material. However, the unstructured meshes tend to be widely used for the analysis of engineering structures and components since it is often desirable to conform the mesh to the external boundaries of the component, although some methods under development today are able to treat even complicated geometries with structured meshes (Belytschko, Gracie, and Ventura 2009). The G-FEM allows for p–adaptivity and provides accurate numerical solutions with coarse or practically ac...

Table of contents

• Cover
• Wiley Series in Computational Mechanics
• Title page
• Copyright page
• Dedication page
• Series Preface
• Preface
• 1 Introduction
• 2 Extended Finite Element Formulation
• 3 Enrichment Elements
• 4 Blending Elements
• 5 Large X-FEM Deformation
• 6 Contact Friction Modeling with X-FEM
• 7 Linear Fracture Mechanics with the X-FEM Technique
• 8 Cohesive Crack Growth with the X-FEM Technique
• 9 Ductile Fracture Mechanics with a Damage-Plasticity Model in X-FEM
• 10 X-FEM Modeling of Saturated/Semi-Saturated Porous Media
• 11 Hydraulic Fracturing in Multi-Phase Porous Media with X-FEM
• 12 Thermo-Hydro-Mechanical Modeling of Porous Media with X-FEM
• References
• Index
• End User License Agreement