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Failure of heterogeneous materials using non-linear continuum laws
Published in Georgios A. Drosopoulos, Georgios E. Stavroulakis, Nonlinear Mechanics for Composite Heterogeneous Structures, 2022
Georgios A. Drosopoulos, Georgios E. Stavroulakis
When the size of the zone in front of the crack tip, which is called fracture process zone and shown in figure 3.5 of section 3.3, is sufficiently small in comparison with the structural dimensions, then principles from linear fracture mechanics can be adopted to simulate cracks in composite structures. In this case, potential inelastic phenomena are restricted to very small regions and can be neglected. Representative examples of materials depicting this behaviour, are metals or brittle materials. One of the methods which can be adopted to capture the response of these materials, is the Extended Finite Element Method (XFEM) as presented in section 5.2 of chapter 5. This method relies on the partition of unity property of finite element shape functions stating that the sum of the shape functions must be unity [22, 147]. Using this method, arbitrarily aligned cracks within the finite element mesh can be simulated, by introducing proper enrichment functions.
Other Meshless Methods
Published in Jichun Li, Yi-Tung Chen, Computational Partial Differential Equations Using MATLAB®, 2019
Functions that are identically 1 on a subset of their support can form a partition of unity. Prove that ϕ(x − xi), i = 0,…, n, form a partition of unity subordinate to patches Ωi, where ϕ(x)={32+2xh,ifx∈(−34h,−h4],1,ifx∈(−h4,h4],32−2xh,ifx∈(h4,34h),0elsewhere.
Representing a Surface Using Nonuniform Rationalized B-Spline
Published in Buntara S. Gan, Condensed Isogeometric Analysis for Plate and Shell Structures, 2019
These basis functions will give values less than 1 along the ξ-axis. Moreover, if we sum all the basis functions’ values at a particular location of the parameter ξ, we will get a constant value of 1. All types of basis functions with this kind of characteristic are said to have a partition of unity. The partition of unity is an important property that is required in selecting a shape function to be used in the finite element formulations.
Inverse problem techniques for multiple crack detection in 2D elastic continua based on extended finite element concepts
Published in Inverse Problems in Science and Engineering, 2021
The solution to this problem is straightforward by conventional numerical methods. The X-FEM which was originally proposed by Belytschko et al. [6] is one of the best candidates. As Figure 2 shows, the main idea of X-FEM is to separate the interfaces from the background mesh so that they can pass through the elements. This removes the remeshing step for moving interface problems like crack propagation. The method is based on the concept of the partition of unity (PUM), proposed by Babuska et al. [46] that allows the inclusion of prior knowledge of the problem by enhancement of the finite element space with enrichment functions. As Equation (3) shows, the displacement field for LEFM problems, has three main parts; the first part is the standard term which is similar to the conventional finite element displacement field and it models the continuous part of the field; the next two parts are the enriched terms which simulate the crack body discontinuity and crack tip singularities, respectively. In this equation, is the finite element shape function matrix of node I, H is the Heaviside enrichment function that is used to model the crack body discontinuity and it is defined as As Figure 3 illustrates, ϕ is the level-set function which is used to describe the interface geometry and it is usually defined as the signed distance function: where is the closest projection of a point on the interface and is the normal vector to the interface at .
Simulation of second order singularly-perturbed boundary-value problems using improved meshless method
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2018
Shape functions for meshless techniques need to satisfy certain conditions such as adherence to partition of unity , compact domain of influence, adapt to randomness of nodes, to name a few. Computational efficiency is significantly affected by choice of shape function.