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Consistent generalized finite element method: An accurate and robust mesh-based method even in distorted meshes

ENGINEERING ANALYSIS WITH BOUNDARY elementS 2025年 171卷

作者： Ma, Jinwei Duan, Qinglin Tian, Rong Shu, Siqi Dalian Univ Technol State Key Lab Struct Anal Optimizat & CAE Software Dalian 116024 Liaoning Peoples R China CAEP Software Ctr High Performance Numer Simulat Beijing 100088 Peoples R China Inst Appl Phys & Computat Math Beijing 100088 Peoples R China Chinese Acad Geol Sci Inst Geomech Beijing 100081 Peoples R China

A consistent generalized finite element method (C-GFEM) is proposed, showing excellent accuracy and convergence in distorted quadrilateral and hexahedral meshes. Both displacement approximation and domain integration are taken into consideration regarding the declining performance of the finite element method (FEM) in distorted meshes. In the displacement approximation, extra-degrees of freedom-free and linearly independent enrichments developed in GFEM are employed, which restores the reproducibility of the approximation in distorted meshes. In the domain integration, the idea of correcting nodal derivatives in the framework of the Hu-Washizu three-field variational principle is introduced into GFEM, based on which consistent integration schemes using quadrilateral and hexahedral elements are developed in this work. Furthermore, to consistently enforce the essential boundary condition, additional terms of boundary integral are introduced into the weak form. As a result, the proposed C-GFEM can pass patch tests and keep high accuracy even though the computational mesh is distorted. Its perfect performance in distorted meshes is sufficiently demonstrated by the numerical investigation of several benchmark examples.

关键词： generalized finite element method FEM Mesh distortion Domain integration Approximation

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generalized finite element method for second-order elliptic operators with Dirichlet boundary conditions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2008年第1期218卷 175-183页

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

We introduce a method for approximating essential boundary conditions-conditions of Dirichlet type-within the generalized finite element method (GFEM) framework. Our results apply to general elliptic boundary value problems of the form -Sigma(n)(i,j=1) (a(ij) u(xi))x(j) + Sigma(n)(i=1) b(i)u(xi) + cu = f in Omega, u=0 on partial derivative Omega, where Omega is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, {S-mu}(mu >= 1), which are nonconforming (that is S-mu not subset of H-0(1)(Omega)). We assume that parallel to nu parallel to(L2(partial derivative Omega)) <= Ch(mu)(m)parallel to nu parallel to(H1(Omega)), for all v is an element of S-mu, and there exists u(1) is an element of S-mu, such that parallel to u-u(1)parallel to(H1(Omega)) <= Ch(mu)(j)parallel to u parallel to(Hj+1(Omega)), 0 <= j <= m, where u is an element of Hm+1(Omega) is the exact solution, m is the expected order of approximation, and hp is the typical size of the elements defining S, Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence up E S, of u. Next, we extend our analysis to the inhomogeneous boundary value problem -Sigma(n)(i,j=1)(a(ij)u(xi))(xj) + Sigma(n)(i=1) b(i)u(xi) + cu = f in Omega, u =g on partial derivative Omega. Finally, we outline the construction of a sequence of GFEM spaces S-mu subset of (S) over tilde (mu), mu = 1, 2,..., that satisfies out assumptions. (C) 2007 Published by Elsevier B.V.

关键词： generalized finite element method Dirichlet boundary conditions

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generalized finite element method in linear and nonlinear structural dynamic analyses

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ENGINEERING COMPUTATIONS 2016年第3期33卷 806-830页

作者： Neto, Dorival Piedade Baroncini Proenca, Sergio Persival Univ Sao Paulo Dept Struct Engn Sao Carlos SP Brazil

Purpose - The purpose of this paper is to evaluate the accuracy and numerical stability of the generalized finite element method (GFEM) for solving structural dynamic problems. Design/methodology/approach - The GFEM is a numerical method based on the partition of unity concept. The method can be understood as an extension of the conventional finite element method for which the local approximation provided by the shape functions can be improved by means of enrichment functions. Polynomial enrichment functions are hereby used combined with an implicit time-stepping integration technique for improving the dynamical response of the models. Both consistent and lumped mass matrices techniques are tested. The method accuracy and stability are investigated through linear and nonlinear elastic problems. Findings - The results indicate that the adopted strategies can provide stable and accurate responses for GFEM in dynamic analyses. Furthermore, the mass lumping technique provided remarkable reductions of the system of equation condition number, therefore leading to more stable numerical models. Originality/value - The evaluated features of GFEM models for implicit time-stepping integration schemes represent new information of great deal of interest regarding linear and nonlinear dynamic analyses using such a method.

关键词： generalized finite element method Implicit time integration Nonlinear dynamics Structural dynamics

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generalized finite element method FOR NONLINEAR THREE-DIMENSIONAL ANALYSIS OF SOLIDS

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2008年第1期5卷 37-62页

作者： Baroncini Proenca, Sergio Persival Ruiz Torres, Ivan Francisco Univ Sao Paulo Sch Engn Sao Carlos Dept Struct Engn BR-13566590 Sao Paulo Brazil

The generalized finite element method (GFEM) is employed in this paper for the numerical analysis of three-dimensional solids tinder nonlinear behavior. A brief summary of the GFEM as well as a description of the formulation of the hexahedral element based oil the proposed enrichment strategy are initially presented. Next, in order to introduce the nonlinear analysis of solids, two constitutive models are briefly reviewed: Lemaitre's model, in which damage and plasticity are coupled, and Mazars's damage model suitable for concrete tinder increased loading. Both models are employed in the framework of a nonlocal approach to ensure solution objectivity. In the numerical analyses carried out, a selective enrichment of approximation at regions of concern in the domain (mainly those with high strain and damage gradients) is exploited. Such a possibility makes the three-dimensional analysis less expensive and practicable since re-meshing resources, characteristic of h-adaptivity, can be minimized. Moreover, a combination of three-dimensional analysis and the selective enrichment presents a valuable good tool for a better description of both damage and plastic strain scatterings.

关键词： generalized finite element method nonlinear analysis of solids damage mechanics

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generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2008年第51-52期197卷 4882-4893页

作者： Srinivasan, K. R. Matous, K. Geubelle, P. H. Univ Illinois Dept Aerosp Engn Urbana IL 61801 USA

An extension of the generalized finite element method to the class of mixed finite element methods is presented to tackle heterogeneous systems with nearly incompressible non-linear hyperelastic material behavior. In particular, heterogeneous systems with large modulus mismatch across the material interface undergoing large strains are investigated using two formulations, one based on a continuous deformation map, the other on a discontinuous one. A bimaterial patch test is formulated to assess the ability of the two formulations to reproduce constant stress fields, while a mesh convergence study is used to examine the consistency of the formulations. Finally, compression of a model heterogeneous propellant pack is simulated to demonstrate the robustness of the proposed discontinuous deformation map formulation. (C) 2008 Elsevier B.V. All rights reserved.

关键词： generalized finite element method Mixed finite element method Lagrange multipliers Bimaterial solid Incompressible hyperelasticity

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The generalized finite element method: an example of its implementation and illustration of its performance

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2000年第8期47卷 1401-1417页

作者： Strouboulis, T Copps, K Babuska, I Texas A&M Univ Dept Aerosp Engn College Stn TX 77843 USA Univ Texas Texas Inst Computat & Appl Math Austin TX 78712 USA

The generalized finite element method (GFEM) was introduced in Reference [1] as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright (C) 2000 John Wiley & Sons, Ltd.

关键词： generalized finite element method hybrid meshless meshfree numerical integration

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generalized finite element method using proper orthogonal decomposition

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2009年第7期79卷 887-906页

作者： Aquino, W. Brigham, J. C. Earls, C. J. Sukumar, N. Cornell Univ Sch Civil & Environm Engn Ithaca NY 14853 USA Univ Pittsburgh Dept Civil & Environm Engn Pittsburgh PA 15260 USA Univ Calif Davis Dept Civil & Environm Engn Davis CA 95616 USA

A methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or Simulated data. The approach is based oil the proper orthogonal decomposition (POD) technique, which is used to generate low-order representations of data that contain general information about the solution of partial differential equations. One of the main challenges ill Such enriched finite element methods is knowing, how to choose, a priori, enrichment functions that capture the nature of the Solution of the governing equations. POD produces low-order subspaces, that are optimal in some norm, for approximating a given data set. For most problems, since the Solution error in Galerkin methods is bounded by the error in the best approximation, it is expected that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. We demonstrate the potential of this approach through three numerical examples. Best-approximation Studies are conducted that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis. Copyright (C) 2009 John Wiley & Sons, Ltd.

关键词： POD partition of unity enrichment enriched finite elements generalized finite element method

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Analysis and applications of a generalized finite element method with global-local enrichment functions

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2008年第6-8期197卷 487-504页

作者： Duarte, C. A. Kim, D. -J. Univ Illinois Dept Civil & Environm Engn Newmark Lab 2122 Urbana IL 61801 USA

This paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method. The procedure combines classical global-local finite element method concepts with the partition of unity approach. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. The computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently. The effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper. We also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution. Key aspects of the computational implementation, in particular, the numerical integration of generalized FEM approximations built with global-local enrichment functions, are presented. The method is applied to fracture mechanics problems with multiple cracks in the domain. Our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems. Our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global-local enrichment functions. (c) 2007 Elsevier B.V. All rights reserved.

关键词： generalized finite element method extended finite element method meshfree global-local fracture mechanics multiple cracks

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Accurate assessment of natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by adaptive generalized finite element method

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ENGINEERING COMPUTATIONS 2016年第5期33卷 1586-1609页

作者： Arndt, Marcos Machado, Roberto Dalledone Scremin, Adriano Univ Fed Parana Dept Civil Construct Curitiba Parana Brazil Univ Fed Parana Dept Mech Engn Curitiba Parana Brazil

Purpose - The purpose of this paper is devoted to present an accurate assessment for determine natural frequencies for uniform and non-uniform Euler-Bernoulli beams and frames by an adaptive generalized finite element method (GFEM). The present paper concentrates on developing the C-1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. Design/methodology/approach - The variational problem of free vibration is formulated and the main aspects of the adaptive GFEM are presented and discussed. The efficiency and convergence of the proposed method in vibration analysis of uniform and non-uniform Euler-Bernoulli beams are checked. The application of this technique in a frame is also presented. Findings - The present paper concentrates on developing the C-1 element of the adaptive GFEM for vibration analysis of Euler-Bernoulli beams and frames. The GFEM, which was conceived on the basis of the partition of unity method, allows the inclusion of enrichment functions that contain a priori knowledge about the fundamental solution of the governing differential equation. The proposed enrichment functions are dependent on the geometric and mechanical properties of the element. This approach converges very fast and is able to approximate the frequency related to any vibration mode. Originality/value - The main contribution of the present study consisted in proposing an adaptive GFEM for vibration analysis of Euler-Bernoulli uniform and non-uniform beams and frames. The GFEM results were compared with those obtained by the h and p-versions of FEM and the c-version of the CEM. The adaptive GFEM has shown to be efficient in the vibration analysis of beams and has indicated that it can be applied even for a coarse discretization scheme in complex practical problems.

关键词： Dynamic analysis Free vibration Partition of unity generalized finite element method

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An Object-Oriented class design for the generalized finite element method programming

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LATIN AMERICAN JOURNAL OF SOLIDS AND STRUCTURES 2013年第6期10卷 1267-1291页

作者： Piedade Neto, Dorival Costa Ferreira, Manoel Denis Baroncini Proenca, Sergio Persival Univ Sao Paulo Dept Struct Engn Sch Engn Sao Carlos Sao Carlos SP Brazil

The generalized finite element method (GFEM) is a numerical method based on the finite element method (FEM), presenting as its main feature the possibility of improving the solution by means of local enrichment functions. In spite of its advantages, the method demands a complex data structure, which can be especially benefited by the Object-Oriented Programming (OOP). Even though the OOP for the traditional FEM has been extensively described in the technical literature, specific design issues related to the GFEM are yet little discussed and not clearly defined. In the present article it is described an Object-Oriented (OO) class design for the GFEM, aiming to achieve a computational code that presents a flexible class structure, circumventing the difficulties associated to the method characteristics. The proposed design is evaluated by means of some numerical examples, computed using a code implemented in Python programming language.

关键词： Object-Oriented programming generalized finite element method Python programming language

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