The partition of unity finite element method: Basic theory and applications

Melenk, Jens Markus; Babuška, Ivo

doi:10.1016/s0045-7825(96)01087-0

Cited by 2,987 publications

(1,963 citation statements)

References 12 publications

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“…The partition of unity finite element method [19] is a generalization of the standard Galerkin finite element method. In the literature, numerical techniques such as the extended finite element method (X-FEM) [17,18], generalized finite element method [23], or the element partition method [24] are all particular instances of the partition of unity method.…”

Section: Extended Finite Element Methodsmentioning

confidence: 99%

“…SROLOVITZ 365 tions. In the X-FEM, the framework of partition of unity [19] is used to enrich the classical displacement-based finite element approximation with a discontinuous function (Heaviside step function) and the two-dimensional asymptotic crack-tip fields. This provides a means to model the crack independently of the underlying finite element mesh.…”

Section: Introductionmentioning

confidence: 99%

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Finite element-based model for crack propagation in polycrystalline materials

Sukumar

Srolovitz

2004

Mat. apl. comput.

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Abstract.In this paper, we use an extended form of the finite element method to study failure in polycrystalline microstructures. Quasi-static crack propagation is conducted using the extended finite element method (X-FEM) and microstructures are simulated using a kinetic Monte Carlo Potts algorithm. In the X-FEM, the framework of partition of unity is used to enrich the classical finite element approximation with a discontinuous function and the two-dimensional asymptotic crack-tip fields. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces, and hence crack growth simulations can be carried out without the need for remeshing. First, the convergence of the method for crack problems is studied and its rate of convergence is established. Microstructural calculations are carried out on a regular lattice and a constrained Delaunay triangulation algorithm is used to mesh the microstructure. Fracture properties of the grain boundaries are assumed to be distinct from that of the grain interior, and the maximum energy release rate criterion is invoked to study the competition between intergranular and transgranular modes of crack growth.Mathematical subject classification: 74R10.

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Section: Extended Finite Element Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finite element-based model for crack propagation in polycrystalline materials

Sukumar

Srolovitz

2004

Mat. apl. comput.

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“…GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in [4,5,24]. In particular, we show that the superconvergence points for the gradient of the approximate are zeros of certain systems of non-linear equations that do not depend on the solution of the boundary value problem.…”

Section: Introductionmentioning

confidence: 92%

Superconvergence in the Generalized Finite Element Method

Babuška¹,

Banerjee²,

Osborn³

2005

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In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in [4,5,24]. In particular, we show that the superconvergence points for the gradient of the approximate are zeros of certain systems of non-linear equations that do not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of certain systems of non-linear equations. We note that it is easy to construct smooth generalized finite element approximation.

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“…Such approach brings need for regularization [3,2]. Several other approaches incorporating discontinuities into FEM models has been presented in literature [10].…”

Section: Introductionmentioning

confidence: 99%

Simulations of Concrete Fracture at Various Strain Rates: Parametric Study

Květoň¹,

Eliáš²

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. The contribution presents numerical simulations of a recently published experimental series on concrete specimens. The numerical model mimics the meso-structure of the concrete by representing material via system of interconnected rigid polyhedral particles. These particles correspond to mineral aggregates and surrounding matrix. The geometry of particles is based on Voronoi tessellation. The particles are rigid, but they may move and rotate. The displacement jump at their contacts gives rise interaction forces defined by the constitutive relation in a vectorial form. The constitutive equations are independent of the strain rate. Simulations are dynamical, the time dependent response is calculated using Newmark implicit scheme. Full inertia tensor of every particle is taken into account when assembling the mass matrix.To demonstrate the ability of the model to capture the strain rate dependence of the experimental results, series of specimens loaded under various displacement rates are simulated. For quasi-static loading, the initial microcracks usually localize into one highly damaged zone where main crack appears. With increasing strain rate, the amount of energy accumulated in the specimen body is not consumed by one crack only and crack branching occurs. The peak loading force increases with increasing rate as well.Both the strain rate dependence of the loading force and crack branching phenomena are captured by the presented numerical model. The obtained results are compared with the experimental response of the simulated specimens. Parametric study of the model is performed showing different effect of various parameters under different strain rates.

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The partition of unity finite element method: Basic theory and applications

Cited by 2,987 publications

References 12 publications

Finite element-based model for crack propagation in polycrystalline materials

Finite element-based model for crack propagation in polycrystalline materials

Superconvergence in the Generalized Finite Element Method

Simulations of Concrete Fracture at Various Strain Rates: Parametric Study

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