Variable‐node elements for non‐matching meshes by means of MLS (moving least‐square) scheme

Abstract

SUMMARYA new class of finite elements is described for dealing with non-matching meshes, for which the existing finite elements are hardly efficient. The approach is to employ the moving least-square (MLS) scheme to devise a class of elements with an arbitrary number of nodal points on the parental domain. This approach generally leads to elements with the rational shape functions, which significantly extends the function space of the conventional finite element method. With a special choice of the nodal points and the base functions, the method results in useful elements with the polynomial shape functions for which the C 1 continuity breaks down across the boundaries between the subdomains comprising one element. The present scheme possesses an extremely high potential for applications which deal with various problems with discontinuities, such as material inhomogeneity, crack propagation, phase transition and contact mechanics. The effectiveness of the new elements for handling the discontinuities due to non-matching interfaces is demonstrated using appropriate examples.

A new computational approach to contact mechanics using variable‐node finite elements

“…Infact we can show that some of them [14][15][16] are capable of passing the patch test when the stabilized conforming nodal integration [19][20][21] is employed [22]. Here we choose a (4 + k)-noded MLS-based finite element, which is one among the MLS-based finite elements developed recently [11][12][13]. Compared with the other transition elements [14][15][16][17][18], it requires only 2 × 2 Gaussian integration on each of the subdomains comprising one element to pass the patch test as seen in Figure 3.…”

“…The remaining issue is to make sure that the basic requirement for finite elements is met in terms of compatibility and completeness for this special element. There are a few choices in employing special elements such as the polygonal finite element using natural neighbor approximation or moving least-squares approximation and so on [11][12][13][14][15][16][17][18][19]. However, most of them [14][15][16][17][18][19] asymptotically pass the patch test wherein high-order Gaussian integration is employed.…”

“…This insertion leads to increase in element nodes on each element containing nx k or nz k on the master/slave surface; therefore, all these elements are replaced by the aforementioned variable-node elements [11][12][13]. Then, the node-to-surface contact problem is transformed into the node-to-node contact problem straightforwardly.…”

“…These elements satisfy the linear interpolation property between two neighboring nodes and the Kronecker delta property. The detailed framework of these MLS-based finite elements is well reported in References [11][12][13] and is briefly summarized in Appendix A.…”

“…This transformation may be realized a type of special element that is capable of allowing an arbitrary number of nodes on the contact interface. For this purpose, we adopt a novel variable-node element, which belongs to MLS (moving least square)-based elements [11][12][13][14][15][16], on the contact surfaces. The contact scheme utilizing this element will be shown to provide a distinct advantage over the conventional methodologies in contact mechanics, which are sometimes incapable of Figure 2.…”