Element subdivision technique for coupling-matrix-free iterative s-version FEM and investigation of sufficient element subdivision

Abstract

In s-version finite element method (s-FEM), a local mesh is superposed on a global mesh, and they are solved monolithically. The local mesh represents the local feature such as a hole, whereas the global mesh does the shape of a structure. Since the two meshes are generated independently, mesh generation becomes very tractable. However, s-FEM has a difficulty that the generation of coupling stiffness matrices takes a lot of computational efforts. To overcome this difficulty, the authors proposed coupling-matrix-free iterative s-FEM. In this method, the coupling stiffness matrices are computed implicitly by stress integration on one mesh and stress transfer from one mesh to the other mesh. Converged solution is obtained by iteration. However, in practical cases, unnatural stress oscillations can occur with conventional Gaussian quadrature. In this paper, in order to smooth unnatural stress oscillations, element subdivision technique is applied. Element subdivision technique can deal with the discontinuity of discretized stress along element interfaces. The stress transfer scheme in coupling-matrix-free iterative s-FEM is designed to work harmoniously with element subdivision. Moreover, element subdivision strategy to obtain smooth stress distribution is investigated in the numerical experiments of a circular hole problem. We propose the element subdivision strategy as the following two items. First, at least 4×4 element subdivision should be adopted to obtain smooth stress distribution. Second, the local mesh should be fine enough to evaluate stress concentration accurately. To confirm this strategy, global and local meshes for elliptical hole problems are designed by following this strategy. Even in severe stress concentration problems, unnatural stress oscillations, which occur with conventional quadrature, is smoothed obviously with 4×4 element subdivision. In addition, stress concentration is evaluated accurately due to the second item of the strategy.

Analysis of a many-hole problem using coupling-matrix-free iterative s-version FEM with multiple local meshes

“…In particular, the present method with multiple local meshes requires stress transfers between pairs of overlapping local meshes. Finally, the element subdivision technique (Yumoto et al, 2016b) is described. This technique is the key to achieving convergence of coupling-matrix-free iterative s-FEM in multiple local mesh problems.…”

“…Recently, the coupling-matrix-free iterative s-FEM (Yumoto et al, 2016a(Yumoto et al, , 2016b was proposed as an extension of the original s-FEM. The principal advantage of this method as compared to the original s-FEM is the elimination of coupling stiffness matrices, which requires a very sophisticated volume integration algorithm, as mentioned above.…”

“…Such a method can be viewed as an iterative solver such as the Gauss-Seidel method . However, in our previous studies (Yumoto et al, 2016a(Yumoto et al, , 2016b, the stress transfers were designed for a single overlapping local mesh. This method is applicable to multiple local meshes when local meshes do not overlap each other .…”

“…It is shown that many holes can be introduced at arbitrary positions in a structure. Moreover, the element subdivision technique (Yumoto et al, 2016b) of the global mesh and the local meshes is found to play a significant role in achieving convergence. In this technique, a finite element is divided into multiple element subdivisions, by which numerical integrals based on the Gaussian quadrature are performed.…”

“…In this technique, a finite element is divided into multiple element subdivisions, by which numerical integrals based on the Gaussian quadrature are performed. In our previous study (Yumoto et al, 2016b), the element subdivision technique was used only in the global mesh to improve the accuracy of the analysis, whereas it is used in both the global and local meshes for the convergence in the present study.…”