Journal ArticleDOI
The Partition of Unity Method
Ivo Babuška,Jens Markus Melenk +1 more
TLDR
In this article, a new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved, which can therefore be more efficient than the usual finite element methods.Abstract:
A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved This new method can therefore be more efficient than the usual finite element methods An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers The basic estimates for a posteriori error estimation for this new method are also proved © 1997 by John Wiley & Sons, Ltdread more
Citations
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Journal ArticleDOI
The partition of unity finite element method: Basic theory and applications
Jens Markus Melenk,Ivo Babuška +1 more
TL;DR: In this article, the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM) are presented and a detailed and illustrative analysis is given for a one-dimensional model problem.
Journal ArticleDOI
A phase-field description of dynamic brittle fracture
TL;DR: It is shown that the combination of the phase-field model and local adaptive refinement provides an effective method for simulating fracture in three dimensions.
Journal ArticleDOI
The extended/generalized finite element method: An overview of the method and its applications
TL;DR: An overview of the extended/generalized finite element method (GEFM/XFEM) with emphasis on methodological issues is presented in this article, which enables accurate approximation of solutions that involve jumps, kinks, singularities, and other locally non-smooth features within elements.
Journal ArticleDOI
Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments
TL;DR: In this paper, a modified regularized formulation of the Ambrosio-Tortorelli type was proposed to avoid crack interpenetration and predicts asymmetric results in traction and in compression.
Journal ArticleDOI
A new method for modelling cohesive cracks using finite elements
TL;DR: In this paper, a model which allows the introduction of displacements jumps to conventional finite elements is developed, where the path of the discontinuity is completely independent of the mesh structure.
References
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Book
Table of Integrals, Series, and Products
TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Journal ArticleDOI
Element‐free Galerkin methods
Ted Belytschko,Y. Y. Lu,L. Gu +2 more
TL;DR: In this article, an element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems, where moving least-squares interpolants are used to construct the trial and test functions for the variational principle.
Proceedings ArticleDOI
A two-dimensional interpolation function for irregularly-spaced data
TL;DR: In many fields using empirical areal data there arises a need for interpolating from irregularly-spaced data to produce a continuous surface as discussed by the authors, and it is assumed that a unique number (such as rainfall in meteorology, or altitude in geography) is associated with each data point.
Journal ArticleDOI
Reproducing kernel particle methods
TL;DR: A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed and is called the reproducingkernel particle method (RKPM).
Related Papers (5)
The partition of unity finite element method: Basic theory and applications
Jens Markus Melenk,Ivo Babuška +1 more