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.

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.

The coupling-matrix-free iterative s-version finite element method is extended to multiple local meshes in order to model multiple local features such as holes, inclusions, and cracks. The formulation with multiple local meshes is presented, along with the stress transfer method between the local meshes. The present method does not require the generation of coupling stiffness matrices with a very sophisticated numerical integration method between the global and local meshes or between the local meshes. Instead, stress transfers between pairs of overlapping meshes are performed. Several numerical examples, such as a patch test, a two-hole problem, and a structure with many holes, are presented. The examples demonstrate that the present method is capable of representing a uniform stress distribution as well as capturing the stress concentration of multiple holes that are located in the vicinity of each other. Moreover, numerical integration methods of the global mesh and local meshes are found to have significant influences on the convergence of the iteration. In these numerical examples, the straightforward Gaussian quadrature could not achieve convergence, whereas the element subdivision technique could. Thus, sufficient element subdivisions in the global mesh and the local meshes are necessary in order to produce a converged solution.

From the mechanical point of view, welding processes can be modeled as a coupling problem between heat conduction and thermal elastic–plastic problems. Such a welding mechanics problem generally requires a large amount of computational time due to its nonlinearity as well as a lot of time steps with a moving heat source. To overcome this difficulty, we are developing a large-scale welding simulator based on the domain decomposition method, which is one of parallel finite element methods. In the present paper, the methodology of the domain decomposition method that is applied to welding analysis is presented, followed by the algorithm. Then, a bead-on-plate problem, which is a popular benchmark problem in the field of computational welding mechanics, was analyzed by the simulator. The bead-on-plate problem was successfully analyzed within very small numbers of iteration steps of the Newton–Raphson and conjugate gradient methods.

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