Acta Numerica 2003: Survey of meshless and generalized finite element methods: A unified approach

By Ivo Babuska, Uday Banerjee, John E. Osborn. 2003

399 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Mechanics of Materials, Safety, Risk, Reliability and Quality

View Paper: https://www.semanticscholar.org/paper/73c0034c4782e8eb076244932c5649032fe2f3c7

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In Civil and Structural Engineering

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Citations in the corpus

Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery (2006, methodology): “[5] surveyed the mathematical theory of meshless methods and XFEM for linear elliptic equations.”
A discontinuous‐Galerkin‐based immersed boundary method (2008, methodology): “This difficulty, which is sometimes called “boundary locking” [48] has been mathematically analyzed by Babuška et al [4] and will be further discussed in the next paragraphs.”
Conforming polygonal finite elements (2004, methodology): “A unified mathematical analysis of meshfree methods was recently proposed by Babuška and co-workers [21].”
On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method (2004, methodology): “Expression (33) is quite similar to domain forms of the J -integral and interaction integral designed to approximate the energy release rate at points along a three-dimensional crack front (see References [23, 32]).”
hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks (2009, methodology): “Theoretical analysis of the penalty method for the GFEM is presented in [2].”
GENERALIZED FINITE ELEMENT METHODS — MAIN IDEAS, RESULTS AND PERSPECTIVE (2004, methodology): “This is an issue with meshfree method; see [Babuška et al. (2003)] for a discussion of techniques to overcome it.”
Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures (2011, background): “The reader may see [8–13] for excellent summary about various types of meshfreemethods and their recent developments.”
Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks (2019, methodology): “Optimization-based methods have been popular for some time in the form of leastsquares finite element methods (LSFEM) [1, 2, 3], and element-free Galerkin methods [4, 5], where even for mesh-free formulations, theoretical convergence criteria have been examined [6].”
Recent progresses in boundary layer theory (2015, methodology): “See [3, 4, 106] for the XFEM method for cracks; see [104, 105] for GFEM; and see e.”
Smooth, second order, non‐negative meshfree approximants selected by maximum entropy (2009, neutral): “As reviewed in [4, 19], a number of methods have been proposed to impose essential boundary conditions in the numerical approximation of boundary value problems with noninterpolatory approximants.”
A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties (2013, methodology): “We here propose a reformulation based on the use of a characteristic function method [33].”
Three-dimensional crack growth with hp-generalized finite element and face offsetting methods (2010, methodology): “It combines the systematic way of building discretizations from the standard FEM and the approximation function flexibility enjoyed by meshfree methods [ 3 ,6,27,35,36].”
Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM (2009, methodology): “The issues arising in numerical integration are probably similar to the similar issues arising in the implementation of the Generalized Finite Element method [10] and in the determination of eigenvalues [14].”