An s-version finite element method without generation of coupling stiffness matrix by using iterative technique

Abstract

In s-version finite element method (s-FEM) proposed by Fish (1992), a local mesh that represents the local feature such as a hole or a crack is superposed on a global mesh that represents the shape of the whole analysis model. The interaction between global and local meshes is represented by coupling stiffness matrices. Since the global and local meshes can be generated independently, mesh generation efforts are reduced remarkably. However, s-FEM has a common issue. The generation of coupling stiffness matrices takes a considerable amount of program development efforts, which include constructing accurate cross-element integration methodology and programming it for various element types. For such an issue, we propose an iterative s-FEM that does not require the generation of coupling stiffness matrices at all. The coupling term is now evaluated by global and local stresses that are computed on the respective mesh and then transferred to the other by interpolation techniques. The global and local stresses are treated as initial stress in the finite element computations. The global and local analyses are performed alternately under assumed initial stress, and converged solution is achieved by iteration with a monitored residual being sufficiently small. In proposed iterative s-FEM, an issue about linear independence of global and local elements, which is known to occur in the original s-FEM, does not occur. In numerical experiments, converged solution was successfully obtained within several hundred iteration counts. Accurate stress distribution for a stress concentration problem and an accurate stress intensity factor for a linear elastic fracture mechanics problem were computed by proposed iterative s-FEM. In addition, several stress interpolation techniques were compared in the numerical experiments. Nearest neighbor interpolation for the global stress and local least squares interpolation for the local stress showed good convergence and accurate solution.

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

“…The left-hand side is referred to as a relative residual norm. This convergence criterion is an extension of our previous studies (Yumoto et al, 2016a(Yumoto et al, , 2016b. Note that reaction forces at constrained degrees of freedom are included in the denominator of Eq.…”

“…Here, k is the iteration step. These equations can be regarded as an extension of our previous studies on the couplingmatrix-free iterative s-FEM for a single local mesh (Yumoto et al, 2016a(Yumoto et al, , 2016b. In the present method, the computations of Eqs.…”

“…In the coupling-matrix-free iterative s-FEM (Yumoto et al, 2016a(Yumoto et al, , 2016b, we rewrite Eq. (8) for the Gauss-Seidel method as…”

“…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 .…”