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

INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2010年第3期84卷 253-304页

作者： Fries, Thomas-Peter Belytschko, Ted Rhein Westfal TH Aachen D-52062 Aachen Germany Northwestern Univ Dept Mech Engn Evanston IL 60208 USA

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 interlaces: Among them are the simulation of cracks, shear hands, dislocations, solidification, and multi-field problems. Copyright (C) 2010 John Wiley & Sons, Ltd.

关键词： review extended finite element method XFEM generalized finite element method GFEM partition of unity method PUM

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Bridging multiple structural scales with a generalized finite element method

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2015年第3-4期102卷 180-201页

作者： Plews, J. A. Duarte, C. A. Univ Illinois Dept Civil & Environm Engn Urbana IL 61801 USA

This paper presents a generalized FEM based on the solution of interdependent coarse-scale (global) and fine-scale (local) problems in order to resolve multiscale effects due to fine-scale heterogeneities. Overall structural behavior is captured by the global problem, while local problems focus on the resolution of fine-scale solution features in regions where material heterogeneities may govern the structural response. Fine-scale problems are accurately solved in parallel, and, to address the intrinsic coupling of scales, these solutions are embedded into the global solution space using a partition of unity approach. This method is demonstrated on representative heat transfer examples in order to examine its accuracy, efficiency, and flexibility. Copyright (C) 2014 John Wiley & Sons, Ltd.

关键词： generalized finite element method extended finite element method multiscale method heat transfer

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Approximation properties of the generalized finite element method

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2011年第4期34卷 369-390页

作者： Anitescu, C. Banerjee, U. Syracuse Univ Dept Math Syracuse NY 13244 USA

In this paper, we have obtained an approximation result in the generalized finite element method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h (l + k) ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.

关键词： generalized finite element method Partition of unity Approximation Quasi-interpolation Error estimates

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WAVENUMBER EXPLICIT CONVERGENCE OF A MULTISCALE generalized finite element method FOR HETEROGENEOUS HELMHOLTZ PROBLEMS*

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2023年第3期61卷 1546-1584页

作者： Ma, Chupeng Alber, Christian Scheichl, Robert Great Bay Univ Sch Sci Dept Math Dongguan 523000 Peoples R China Heidelberg Univ Inst Appl Math Neuenheimer Feld 205 D-69120 Heidelberg Germany Heidelberg Univ Interdisciplinary Ctr Sci Comp Neuenheimer Feld 205 D-69120 Heidelberg Germany

In this paper, a generalized finite element (FE) method with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, (i) wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains and (ii) a quasi-optimal convergence of the method is established by assuming that the size of subdomains is O(1/k) (k is the wavenumber). A novel resonance effect between the wavenumber and the dimension of local spaces on the decay of error with respect to the oversampling size is implied by the analysis. Furthermore, for fixed dimensions of local spaces, the discrete local errors are proved to converge as h \rightarrow 0 (h denoting the mesh size) toward the continuous local errors. The method at the continuous level extends the plane wave partipp. 727--758] to the heterogeneous-coefficients case, and at the discrete level, it delivers an efficient noniterative domain decomposition method for solving discrete Helmholtz problems resulting from standard FE discretizations. Numerical results are provided to confirm the theoretical analysis and to validate the proposed method.

关键词： generalized finite element method Helmholtz equation multiscale method Trefftz methods local spectral spaces

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Approximate and low regularity Dirichlet boundary conditions in the generalized finite element method

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MATHEMATICAL MODELS & methodS IN APPLIED SCIENCES 2007年第12期17卷 2115-2142页

作者： Babuska, Ivo Nistor, Victor Tarfulea, Nicolae Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA Penn State Univ Dept Math University Pk PA 16802 USA Purdue Univ Calumet Dept Math Hammond IN 46323 USA

We propose a method for treating Dirichlet boundary conditions for the Laplacian in the framework of the generalized finite element method (GFEM). A particular interest is taken in boundary data with low regularity (possibly a distribution). Our method is based on using approximate Dirichlet boundary conditions and polynomial approximations of the boundary. The sequence of GFEM-spaces consists of nonzero boundary value functions, and hence it does not conform to one of the basic finite element method (FEM) conditions. We obtain quasi-optimal rates of convergence for the sequence of GFEM approximations of the exact solution. We also extend our results to the inhomogeneous Dirichlet boundary value problem, including the case when the boundary data has low regularity (i.e. is a distribution). Finally, we indicate an effective technique for constructing sequences of GFEM-spaces satisfying our assumptions by using polynomial approximations of the boundary.

关键词： generalized finite element method Dirichlet boundary conditions distributional data

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Improvement of System Quality in a generalized finite element method Using Discrete Curvelet Transform 17

Improvement of System Quality in a Generalized Finite Elemen...

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17th IEEE Biennial Conference on Electromagnetic Field Computation (IEEE CEFC)

作者： Mandinejad, Naier Mota, Hilton O. Silva, Elson J. Adriano, Ricardo Univ Fed Minas Gerais Grad Program Elect Engn BR-31270901 Belo Horizonte MG Brazil Univ Fed Minas Gerais Dept Elect Engn BR-31270901 Belo Horizonte MG Brazil

ISBN: (纸本)9781509010325

The generalized finite element method (GFEM) enriched by Plane Waves has been proved suitable to solve the two-dimensional Helmholtz equation. However, when the numbers of unknowns increase, some difficulties arise, such as bad condition number. To reduce the total number of degrees of freedom, this paper presents a method to identify the most significant plane waves in different locations in the problem. It is based on the analysis of the traditional finite element method solution with a poor mesh resolution using the Fast Discrete Curvelet Transform (FDCT) through wedge wrapping.

关键词： Fast Discrete Curvelet Transform generalized finite element method Plane wave enrichment

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Bridging Scales with a generalized finite element method

Bridging Scales with a Generalized Finite Element Method

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IUTAM Symposium on Linking Scales in Computation - From Microstructure to Macroscopic Properties

作者： Garzon, J. Gupta, V. Simone, A. Duarte, C. A. Univ Illinois Dept Civil & Environm Engn Newmark Lab 2122 MC 250205 N Mathews Av Urbana IL 61801 USA Delft Univ Technol Fac Civil Engn & Geosci NL-2600 GA Delft Netherlands

The generalized FEM (GFEM) has been successfully applied to the simulation of dynamic propagating fractures, polycrystalline and fiber-reinforced microstructures, porous materials, etc. A-priori knowledge about the solution of these problems are used in the definition of their GFEM approximation spaces. This leads to more accurate and robust simulations than available finite element methods while relaxing some meshing requirements. This is demonstrated in a simulation of intergranular crack propagation in a brittle polycrystal using simple background meshes. For many classes of problems - like those with material non-linearities or involving multiscale phenomena - a-priori knowledge of the solution behavior is limited. In this paper, we present a GFEM based on the solution of interdependent global (structural) and fine-scale or local problems. The local problems focus on the resolution of fine-scale features of the solution in the vicinity of, e. g., evolving fracture process zones while the global problem addresses the macroscale structural behavior. Fine-scale solutions are accurately solved using an hp-adaptive GFEM and thus the proposed method does not rely on analytical solutions. These solutions are embedded into the global solution space using the partition of unity method. This GFEM enables accurate modeling of problems involving multiple scales of interest using meshes with elements that are orders of magnitude larger than those required by the FEM. Numerical examples illustrating the application of this class of GFEM to high-cycle fatigue crack growth of small cracks and to problems exhibiting localized non-linear material responses are presented. (C) 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of Dr. Oana Cazacu.

关键词： generalized finite element method Extended finite element method Multiscale Polycrystals Fracture Crack Growth Global-Local Analysis

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The generalized finite element method: Numerical Treatment of Singularities, Interfaces, and Boundary Conditions

The Generalized Finite Element Method: Numerical Treatment o...

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作者： Qu, Qingqin PennState University Libraries

学位级别：Doctor of Philosophy

This dissertation is devoted to numerical approximation of partial differential equations by generalized finite element method (GFEM), which is closely related to some other methods, such as the $hp$ cloud method and the extended finite element method. As an extension of the standard finite element method (FEM), the GFEM has more flexibility in dealing with complicated domain geometry, corner singularities, transmission problems and mixed boundary conditions. It differs from the standard FEM in the construction of the finite-dimensional space in which the approximate solution is sought. Instead of using piecewise polynomials on each element of a triangulation of the domain, the global GFEM space is defined by using partition of unity in combination with local approximation spaces defined in each patch of the partition. In this sense, the GFEM is an example of so called \emph{meshless methods}, since the partition of unity need not be subordinated to a particular triangulation of the domain as the standard FEM does. The GFEM allows one to incorporate a priori knowledge of the local behavior of the solution in the construction of the approximation space, and gives the option of constructing trial spaces of any desired regularity. For transmission (interface) problems on domains with smooth, curved boundaries, we establish quasi-optimal rate of convergence of the numerical solution to the true solution by using anon-conforming GFEM. To achieve this goal, we construct a sequence of approximation spaces $S_n$, satisfying the following two conditions: (1) {\em nearly zero boundary and interface matching}, (2) {\em approximability}. We then seek the numerical solution in this spaces as the Galerkin approximation to the true solution, and show that the approximation error of order $O(h_{n}^{m})$, where $h_{n}$ is the typical size of elements in the GFEM space $S_{n}$, and $m$ is the degree of polynomials used for the local approximation of the solution. Numerical experimen

关键词： Interface problems Partition of unity generalized finite element method Weighted Sobolev spaces Optimal rate of Convergence

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MULTILEVEL methodS FOR THE generalized finite element method DISCRETIZATIONS

MULTILEVEL METHODS FOR THE GENERALIZED FINITE ELEMENT METHOD...

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作者： Cho, Durkbin PennState University Libraries

学位级别：Doctor of Philosophy

This dissertation is focused on multilevel methods for generalized finite element method (GFEM) discretizations. Our first result is an estimate for the rate of convergence of the method of successive subspace corrections (MSSC) in terms of the method of parallel subspace corrections (MPSC). This is motivated by a new representation for the convergence rate of the MSSC. We also provide some estimates for the convergence rate of multilevel preconditioned *** on the multilevel preconditioners and the auxiliary space lemma, we study the efficient solution of linear systems arising in discretizations of second order elliptic partial differential equations (PDEs) via the generalized finite element method (GFEM). Our results apply for GFEM equations on unstructured simplicial grids in 2 and 3 spatial dimensions. We then propose a uniform preconditioner for GFEM linear systems. Moreover, we show that this result has an application in the design of a multilevel preconditioner for the linear elasticity *** have also included our work on applying the multilevel methods to the exact controllability methods for the time-periodic wave equation, which is equivalent to the Helmholtz equation.

关键词： generalized finite element method Subspace correction methods Auxiliary space methods Multilevel methods Exact controllability technique

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Numerical experiments with stable versions of the generalized finite element method

Numerical experiments with stable versions of the Generalize...

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作者： Fernando Massami Sato UNIVERSIDADE DE SAO PAULO

学位级别：硕士

The generalized finite element method (GFEM) is essentially a partition of unity based method (PUM) that explores the Partition of Unity (PoU) concept to match a set of functions chosen to efficiently approximate the solution locally. Despite its well-known advantages, the method may present some drawbacks. For instance, increasing the approximation space through enrichment functions may introduce linear dependences in the solving system of equations, as well as the appearance of blending elements. To address the drawbacks pointed out above, some improved versions of the GFEM were developed. The Stable GFEM (SGFEM) is a first version hereby considered in which the GFEM enrichment functions are modified. The Higher Order SGFEM proposes an additional modification for generating the shape functions attached to the enriched patch. This research aims to present and numerically test these new versions recently proposed for the GFEM. In addition to highlighting its main features, some aspects about the numerical integration when using the higher order SGFEM, in particular are also addressed. Hence, a splitting rule of the quadrilateral element area, guided by the PoU definition itself is described in detail. The examples chosen for the numerical experiments consist of 2-D panels that present favorable geometries to explore the advantages of each method. Essentially, singular functions with good properties to approximate the solution near corner points and polynomial functions for approximating smooth solutions are examined. Moreover, a comparison among the conventional FEM and the methods herein described is made taking into consideration the scaled condition number and rates of convergence of the relative errors on displacements. Finally, the numerical experiments show that the Higher Order SGFEM is the more robust and reliable among the versions of the GFEM tested.

关键词： generalized finite element method Stable generalized finite element method Higher Order Stable generalized finite element method Rate of convergence Scaled condition number

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