An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented. This method enables the accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements. This is achieved by enriching the polynomial approximation space of the classical finite element method. The GEFM/XFEM has shown its potential in a variety of applications that involve non-smooth solutions near interfaces: Among them are the simulation of cracks, shear bands, dislocations, solidification, and multi-field problems. Copyright © 2010 John Wiley & Sons, Ltd.

The extended/generalized finite element method: An overview of the method and its applications

▪ Abstract We provide an overview of level set methods, introduced by Osher and Sethian, for computing the solution to fluid-interface problems. These are computational techniques that rely on an implicit formulation of the interface, represented through a time-dependent initial-value partial-differential equation. We discuss the essential ideas behind the techniques, the coupling of these techniques to finite-difference methods for incompressible and compressible flow, and a collection of applications including two-phase flow, ship hydrodynamics, and ink-jet-printhead design.

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Level set methods for fluid interfaces

The extended and generalized finite element methods are reviewed with an emphasis on their applications to problems in material science: (1) fracture, (2) dislocations, (3) grain boundaries and (4) phases interfaces. These methods facilitate the modeling of complicated geometries and the evolution of such geometries, particularly when combined with level set methods, as for example in the simulation growing cracks or moving phase interfaces. The state of the art for these problems is described along with the history of developments.

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A review of extended/generalized finite element methods for material modeling

The extended finite element method (XFEM) enables local enrichments of approximation spaces. Standard finite elements are used in the major part of the domain and enriched elements are employed where special solution properties such as discontinuities and singularities shall be captured. In elements that blend the enriched areas with the rest of the domain problems arise in general. These blending elements often require a special treatment in order to avoid a decrease in the overall convergence rate. A modification of the XFEM approximation is proposed in this work. The enrichment functions are modified such that they are zero in the standard elements, unchanged in the elements with all their nodes being enriched, and varying continuously in the blending elements. All nodes in the blending elements are enriched. The modified enrichment function can be reproduced exactly everywhere in the domain and no problems arise in the blending elements. The corrected XFEM is applied to problems in linear elasticity and optimal convergence rates are achieved. Copyright © 2007 John Wiley & Sons, Ltd.

A corrected XFEM approximation without problems in blending elements

http://dilbert.engr.ucdavis.edu/~suku/xfem/papers/xfem_partI.pdf

Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation

We consider a problem stemming from recent models of phase transitions in stimulus-responsive hydrogels, wherein a sharp interface separates swelled and collapsed phases. Extended finite element methods that approximate the local solution with an enriched basis such that the mesh need not explicitly 'fit' the interface geometry are emphasized. Attention is focused on the weak enforcement of the normal configurational force balance and various options for evaluating the jump in the normal component of the solute flux at the interface. We show that as the reciprocal interfacial mobility vanishes, it plays the role of a penalty parameter enforcing a pure Dirichlet constraint, eventually triggering oscillations in the interfacial velocity. We also examine alternative formulations employing a Lagrange multiplier to enforce this constraint. It is shown that the most convenient choice of basis for the Lagrange multiplier results in oscillations in the multiplier field and a decrease in accuracy and rate of convergence in local error norms, suggesting a lack of stability in the discrete formulation. Under such conditions, neither the direct evaluation of the gradient of the approximation at the phase interface nor the interpretation of the Lagrange multiplier field provide a robust means to obtain the jump in the normal component of solute flux. Fortunately, the adaptation and use of local, domain-integral methodologies considerably improves the flux evaluations. Several example problems are presented to compare and contrast the various techniques and methods.

On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method

A hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented. We focus attention on two-dimensional solidification problems, where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front. The enrichment strategies of the eXtended Finite Element Method (X-FEM) are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary. A new approach with the X-FEM is suggested for this class of problems whereby the partition of unity is constructed with C1(Ω) polynomials and enriched with a C0(Ω) function. This approach leads to jumps in temperature gradient occurring only at the phase boundary, and is shown to significantly improve estimates for the front velocity. Temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment. In conjunction with a finer finite difference grid, the Level Set method is used to represent the evolution of the phase interface. An iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary. The robustness and utility of the method is demonstrated with several benchmark problems of phase transformation. Copyright © 2002 John Wiley & Sons, Ltd.

H. Ji

Papers

On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method

A hybrid extended finite element/level set method for modeling phase transformations