Level Set construction with Fast Marching Method

Finally, one can also generate the Level Set function by using the Fast Marching Method. Given an initial set of computed nodal Level Set values on the nodes located closed and on both sides of the interface, the distance function can be obtained very efficiently by solving the Eikonal equation using FMM. As this method can handle only monotonous speed function, this problem is solved once on each side of the interface with a constant unit speed function.

Chapter

3

The eXtended Finite Element Method

This chapter provides an introduction and a review of the eXtended Finite Element method. At first, the different methods that led to the development of enriching techniques are presented. Then, a simple example is used to illustrate the concept, key points and different approaches of the enrichment techniques, and the general aspects of the method are introduced. The X-FEM is then introduced for three main practical approximations used in mechanics to provide a global view of the enrichment problems and strategies.

3.1

Introduction

Since the beginning of the 60’s, the Finite Element Method (FEM) has been very popular for finding approximate solutions to differential equation problems. Initially developed for the analy- sis of mechanical structures, it was identified early on as a versatile method that could be applied successfully to a wide class of physical and engineering problems. Thus, it was rapidly adopted by researchers and scientists because of its ability to deal with a variety of problems ranging from solid mechanics, electromagnetism, heat transfer, fluid dynamics . . . The large enthusiasm encountered by this method has greatly contributed to its tremendous development. Moreover, thanks to the advancements of the computer science and the rapid development of digital com- puter hardware, the FEM has become an effective method for practical engineering problems. Consequently, due to its simplicity of implementation and its robustness, the industry has now widely adopted the FEM for solving daily problems. While reserved at the beginning to a class of applications requiring high performance such as aerospace applications, the FEM is intensively used to design a large area of products used in every day life.

For physical problems that admit smooth solutions, the standard FEM function space made of piecewise continuous polynomial functions is generally sufficient to obtain accurate results and ensure the convergence. However, due to the nature of the Finite Element approximation space, the study of problems with solution involving discontinuities remains challenging. To accurately model this class of problems, the construction of an appropriate and variable mesh is needed because the element topology has to be aligned with the discontinuity. For instance, dealing with the jumps in the response field arising with cracks, the mesh has to follow the discontinuity so that the edges or faces are coincident with the crack whereas the nodes must be placed on each side of the crack allowing material separation along the crack surface. For problems in which kinks or high gradient are likely to appear, a locally significant mesh refinement or a high

order finite element approximation is generally requested to obtain an accurate solution. Hence, one can easily notice that a special attention has to be paid to the mesh generation step. While the mesh generation technology has continuously improved the quality and the robustness of available finite element meshers, a human intervention is still needed to ensure that the mesh exhibits a good quality. This becomes a highly challenging task when the problem involves evolving discontinuities, therefore needing to repeat the mesh generation task at each time step. Following the evolution of a single discontinuity in 2D problems is affordable but this becomes very arduous with multiple discontinuities or with 3D problems. Moreover, the multiple regener- ations of the mesh are generally expensive in terms of computational cost and in general results in a loss of accuracy when the projection of physical quantities between successive meshes is required. In Idelsohn et al. [86], the authors claims that: It is widely acknowledged that the 3D mesh generation remains the highest part of the total man-hours devoted to solve computational mechanics problems. Also, the major problems come from automatization issue. The generation time remains unbounded, even using the most sophisticated mesh-generators. For a given geome- try, an initial mesh can be obtained very quickly, but it may also need several iterations, including manual intervention, to achieve an acceptable mesh.

From this statement, much attention has been devoted to the development of the so-called mesh- less or meshfree methods that try to overcome the difficulties related to the mesh. The idea of these methods is to get rid of the mesh and to define the FE approximation by constructing the approximate solution on a set of sprinkled points on the computational domain. This ap- proximation is constructed on the nodes only, with associated weight functions having compact support with a simple shape, such as a circle in 2D or a sphere in 3D for instance. Over the years, this idea has attracted a lot of researchers and several meshless methods have been proposed in the literature. Among others, we can mention the Element Free Galerkin (EFG) proposed by Belytschko et al. [22], the Reproducing Kernel Particle Method (RKPM) [103] of Liu et al. The main advantage of these methods is that the onerous mesh generation of conventional mesh based method is circumvented. They can easily cope with evolving discontinuities without remeshing and adaptive mesh refinement is easily accomplished by adding or removing nodes. While these methods have been applied successfully to a wide range of applications such as crack propagations [23], they suffer from some difficulties in practice:

- The imposition of the essential boundary conditions is not straightforward because the shape functions are not interpolants (lack of the Kronecker-δ property).

- The computational cost of meshless method is higher than FEM.

- The shape functions are not polynomial but rational functions and requires therefore careful integration techniques demanding high-order integration schemes.

- The shape functions have to be computed depending on the geometrical distribution of the nodes.

- Due to the numerous differences between FEM and meshless methods, the computer im- plementation differs significantly.

To avoid some inherent drawbacks of the meshless methods still benefiting from the advantage of the smoothness, hybrid methods coupling meshless and mesh-based methods have been pro- posed [141]. However, all the aforementioned drawbacks have severely limited the development

and the adoption of meshless methods in the industry. To the author knowledge, no commercial software has actually implemented a meshless solution.

The high degree of continuity in the solution field of the meshless method is in general a great advantage as the field derivative are smooth. Conversely, when the exact solution of the problem exhibits discontinuities, the accuracy of these methods can be rather poor. Thus the meshless methods solve the mesh related problem but face the same difficulties than the FEM with non smooth problems. To circumvent what is seen as a drawback in a non smooth problem such as crack analysis, Fleming and Belytschko [20] proposed to enrich the approximation space of EFG trial functions by including the crack tip asymptotic field into the displacement field. With this enrichment technique, Krysl [95] has been able to model crack propagation in 2D and 3D. In 1996, Melenk and Babuska [110] showed that the classical finite element basis can be extended to represent a specific given function on the computational domain and that some advantages found in the meshless approach can be realized using a partition of unity method (PUM). The idea of PUM is to enrich or to extend the finite element polynomial approximation space by adding special shape functions, which can represent exactly an a priori known behavior of the solution. Initially, the aim of adding enrichment functions was to improve the performance of the classical finite element approximation on the entire domain and the enrichment was thus carried out globally (on all elements). In their seminal work on PUM, Melenk and Babuska introduced a global harmonic polynomial enrichment to deal with a globally non-smooth solution as it can be the case with high frequency solution of the Helmoltz equation. They showed that such technique of embedding additional function yields accurate solutions and that optimal rates of convergence could be obtained. The idea of enriching the space with custom shape functions was already known, for instance in the Global-local method [119]. However, this method did not get a lot of success due to the fact that the enrichment is global, which destroy the banded structure of the stiffness matrix. In the PUM, the central idea is to multiply the enrichment function with functions satisfying the partition of unity (PU) that results in a conforming approximation. From a certain point of view, this enrichment technique can be seen as a new element type with special shape functions. The idea of constructing specific elements to model non smooth physical problem is not really new too. Among others, we can cite Barsoum [14] who proposed an element dedicated to obtain accurate singular solution around crack tip. While these tech- niques obtained effective results for some specific problems, the general scheme developed by Melenk and Babuska [110] is more attractive due to the fact that it consists in a generalization of the field approximation and that it can equivalently be applied to any numerical method in- cluding the FEM. Moreover, it is less demanding in terms of implementation in existing FE codes. Since its introduction, the PU concept has focused a lot of interest and has been the topic of intensive researches and applications. In [163], Strouboulis and co-authors introduced the Gen- eralized Finite Element Method (GFEM) for solving different elliptic problems by enriching the entire domain. The method was referred to GFEM since the classical FEM is a special case of this method. The enrichment technique improves the solution by introducing additional shape functions but the second advantage of this method is that discontinuous, singular or ’exotic’ shape functions can be added allowing to represent non smooth behavior independently of the mesh. Later, as most non-smooth or non differentiable solution properties, such as jumps, kinks, and singularities are generally local phenomena, they introduced the idea of local enrichment by restricting the enrichment only to a subset of the domain.

Inspired by the possibilities of the GFEM, the first implementation of the so-called eXtended Finite Element Method (X-FEM) has been proposed by Belytschko and Black [19] where they enriched the finite element space with asymptotic crack-tip displacement field to treat 2D static fracture problems. In this seminal work, only the nodes close to the crack tips were enriched with the four functions spanning the near tip fields. At this stage, the method was able to model non conforming cracks tips and capturing accurate stress intensity factors with minimal mesh refinement. In Moës et al. [114] and Dolbow [52], the approach has been extended with the introduction of another enrichment function able to treat the discontinuity occurring across the crack lips. Hence, combining a Heaviside enrichment function and the crack tip field enabled modeling a complete crack embedded inside a non conforming mesh.

Following these pioneering works, the X-FEM approach has rapidly focused the interest of re- searchers and the method has been extended to arbitrary branched, intersecting cracks [51] and 3D cracks [166]. In [162], Stolarska et al. proposed to use the Level Set Method to represent the crack and to model the crack growth in 2D. Beside providing a theoretical method to update the position of the cracks, the use of the Level Set Method offered complementary capabilities such as simplifying the selection of the enriched nodes, defining the enrichment functions as well as localizing the interfaces. In [115], the method has been extended to 3D for non planar 3D crack growth. In [164], Sukumar et al. extended the X-FEM to model holes and inclu- sions. The X-FEM has obtained such promising results in fracture mechanics that some authors have immediately foreseen the opportunity of applying X-FEM to many kinds of problems in which discontinuities and moving boundaries arise. As example publications, we can mention Belytschko et al. [25] who applied the X-FEM to the modeling of composite fiber orientation in a micro-structure or Moës et al. [113] who presented a computational approach to study complex micro structures, Chessa et al. [44] who studied the case of two phase fluid problems. Applications to fluid structure interaction can also be found in Legay et al. [97] or Gerstenberger and Wall [74]. In the first implementation of the GFEM, the enrichment was realized on the entire domain and mainly applied for weak discontinuities or for enriching the approximation space with harmonic functions for instance. The X-FEM has been developed in parallel to the GFEM, with local enrichment devoted to crack simulation. Later, Strouboulis used the acronym GFEM in applica- tion using local enrichments with a discontinuity similar to X-FEM. Thus, X-FEM and GFEM are in fact almost identical and both derive from PUM. They mainly differs in the forms of the PU that is used and in the use of different local spaces. Several rather complete reviews on the X-FEM/GFEM have been published, some of them focus only on crack such as [21, 90, 140] while the others [1, 70] are more general and cover different application fields. A text book by Mohammadi has also been published [116].