A finite element alternative to infinite elements

In this paper, a simple idea based on midpoint integration rule is utilized to solve a particular class of mechanics problems; namely static problems defined on unbounded domains where the solution is required to be accurate only in an interior (and not in the far field). By developing a finite element mesh that approximates the stiffness of an unbounded domain directly (without approximating the far-field displacement profile first), the current formulation provides a superior alternative to infinite elements (IEs) that have long been used to incorporate unbounded domains into the finite element method (FEM). In contrast to most IEs, the present formulation (a) requires no new shape functions or special integration rules, (b) is proved to be both accurate and efficient, and (c) is versatile enough to handle a large variety of domains including those with anisotropic, stratified media and convex polygonal corners. In addition to this, the proposed model leads to the derivation of a simple error expression that provides an explicit correlation between the mesh parameters and the accuracy achieved. This error expression can be used to calculate the accuracy of a given mesh a-priori. This in-turn, allows one to generate the most efficient mesh capable of achieving a desired accuracy by solving a mesh optimization problem. We formulate such an optimization problem, solve it and use the results to develop a practical mesh generation methodology. This methodology does not require any additional computation on the part of the user, and can hence be used in practical situations to quickly generate an efficient and near optimal finite element mesh that models an unbounded domain to the required accuracy. Numerical examples involving practical problems are presented at the end to illustrate the effectiveness of this method.