For some "nite element analyses of stresses in engineering components, low-order elements can be preferred. This choice, however, results in slow convergence, especially at key stress concentrations. To overcome this di$culty, submodelling of stress concentrators can be employed. With submodelling, a subregion within the original global con"guration and centred on the stress concentrator of interest is analysed by itself, with a consequent reduction in computation. The more aggressive the submodelling, the smaller the subregion and the greater the computational savings. To realize such savings in actuality, it is necessary that appropriate boundary conditions be applied to the subregion. Some of these boundary conditions must be drawn from a global analysis of the original con"guration: then it is essential to ensure that such boundary conditions are determined su$ciently accurately. This paper describes a procedure for being reasonably certain that such is the case. The procedure is evaluated on a series of test problems and demonstrated on a contact application. Results show that good engineering estimates of peak stresses can be obtained even in regions of unusually high stress gradients. Furthermore, these estimates can be obtained in return for quite moderate levels of computational e!ort.R Domain decomposition is a related technique where the mesh is broken apart and sent to separate processors: see e.g. Farhat et al. [7] S There is also an approach where the region of interest is not broken out of the global grid but remeshed with a "ner mesh. This is sometimes referred to as submodelling, but really is grid gradation application. Similar experience with non-linear spring elements has lead ANSYS to make the same recommendation for problems including these elements.There are several approaches in the literature that can increase the e$ciency of "nite element analysis so that such challenges can be met. We brie#y review some of these approaches.One is to use multigrid techniques. With these techniques, a coarse grid is solved and then used as an initial guess for an iterative procedure for reducing the sti!ness matrices. An explanation of the method can be found in [2]. Some solid mechanics applications are given in [3,4]. This technique naturally combines with adaptive re"nement. In adaptive re"nement, the "nite element mesh transitions from a coarse mesh in the far "eld to a "ner mesh around the region of interest. A recent example of this combination is given in [5].Another approach is to use substructuring. In substructuring, symmetries of the mesh are exploited to reduce the size of sti!ness matrices. An explanation of the method is presented in [6, Chapter 10].R Reviews of the approach are provided in [8,9]. An example of a recent application is given in [10].A further approach, and the one of principal concern here, is submodelling. In this technique, a global grid is run and the region of greatest interest is broken out as a submodel and analysed separately with a "ner mesh. &Boundary' conditions are...
