Local Partition Of Unity Research Articles

Two‐scale method for shear bands: thermal effects and variable bandwidth

AbstractA method for the analysis of shear bands using local partition of unity is developed in the framework of the extended finite element method (XFEM). Enrichments are introduced for both the displacement field and the thermal field. The shear band width is determined by minimizing the plastic work. A coupled finite strain thermo‐elastoplastic constitutive law is used. The enrichment is injected into the mesh when the material law becomes unstable. The criterion based on a complete stability analysis for materials in the finite strain regime including heat conduction, strain hardening, strain rate hardening and thermal softening is presented. A mixed continuous quadrilateral element is employed. The method is applied to the Nesterenko experiments, which exhibit multiple propagating shear bands and other problems. Copyright © 2007 John Wiley & Sons, Ltd.

A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics

This paper proposes a three-dimensional meshfree method for arbitrary crack initiation and propagation that ensures crack path continuity for non-linear material models and cohesive laws. The method is based on a local partition of unity. An extrinsic enrichment of the meshfree shape functions is used with discontinuous and near-front branch functions to close the crack front and improve accuracy. The crack is hereby modeled as a jump in the displacement field. The initiation and propagation of a crack is determined by the loss of hyperbolicity or the loss of material stability criterion. The method is applied to several static, quasi-static and dynamic crack problems. The numerical results very precisely replicate available experimental and analytical results.

Two-scale shear band evolution by local partition of unity

We introduce a methodology to model shear band evolution in the quasi-static regime using the extended finite element method. We enrich the finite element polynomial displacement field with a fine scale function, which models the high displacement gradient in the shear band. For this purpose we use a local partition of unity and a parameterized displacement enrichment based on closed form solutions for one-dimensional shear bands. A stabilized consistent penalty method is used to circumvent locking in the regularized elasto-viscoplastic plane-strain regime and to guarantee element stability. The loss of stability of the boundary value problem is used as an indicator of shear band initiation point and direction. Shear band development examples are shown, illustrating the capabilities of the method to track shear band evolution and strains as high as 1000%. Copyright © 2005 John Wiley & Sons, Ltd.

The Extended Finite Element Method for Dynamic Fractures

A method for modelling arbitrary growth of dynamic cracks without remeshing is presented. The method is based on a local partition of unity. It is combined with level sets, so that the discontinuities can be represented entirely in terms of nodal data. This leads to a simple method with clean data structures that can easily be incorporated in general purpose software. Results for a mixed-mode dynamic fracture problem are given to demonstrate the method.

An enriched finite element method and level sets for axisymmetric two‐phase flow with surface tension

AbstractA finite element method for axisymmetric two‐phase flow problems is presented. The method uses an enriched finite element formulation, in which the interface can move arbitrarily through the mesh without remeshing. The enrichment is implemented by the extended finite element method (X‐FEM) which models the discontinuity in the velocity gradient at the interface by a local partition of unity. It provides an accurate representation of the velocity field at interfaces on an Eulerian grid that is not conformal to the weak discontinuity. The interface is represented by a level set which is also used in the construction of the element enrichment. Surface tension effects are considered and the interface curvature is computed from the level set field. The method is demonstrated by several examples. Copyright © 2003 John Wiley & Sons, Ltd.

Tensor sequences and inductive limits with local partition of unity

In this note we characterize those locally convex hulls\(F = \sum\limits_{i = 1}^\infty {A_i \left( {F_i } \right)} \) for which the e-tensor product\(E \otimes _\varepsilon (\sum\limits_{i = 1}^\infty {A_i (F_i )} )\) is topologically isomorphic to the locally convex hull\(\sum\limits_{i = 1}^\infty { Id \otimes A_i (E \otimes _\varepsilon F_i )} \) for all normed spaces E. By using ⊗-sequences in the sense of W.Kaballo and D.Vogt [18] it is shown that\(F = \sum\limits_{i = 1}^\infty {A_i \left( {F_i } \right)} \) is commutable with e-tensor products if and only if F admits a local partition of unity. In this paper we give examples of locally convex hulls with local partition of unity. From this one obtains conditions under which the e-tensor product of two bornological (DF)-spaces is again a bornological (DF)-space. Applications to sequence spaces are given.