Infinite elements in finite element method

Abstract

It is natural and common to idealize stress or field problems into finite element models with rigid boundaries remote from the area of interest. However, the degree of accuracy of solutions may be significantly increased, if infinite elements extending to infinity are used all along the rigid boundaries. Infinite elements are introduced and also the history and development of these elements are discussed in detail. The classification of the infinite elements is made as, a) Mapped infinite elements, and b) Decay function infinite elements. Firstly, unidimensional infinite elements are described and after the geometric and field variable interpolation of these elements are expressed; the strain matrix and the stiffness matrix are explicitly obtained. In this presentation, a total of 23 different types of 1-D (5), 2-D (13), and 3-D (5) infinite elements have been investigated. Their geometrical configurations, coordinate mapping and field variable mapping functions are presented explicitly in a systematic fashion. In order to emphasize the high performance and accuracy of the infinite elements, four distinct case studies have been presented. Firstly, the deflection and stress analyses of a point load and a circular uniform distributed load acting on a semi-infinite axisymmetrical medium have been presented with and without infinite elements. The results have been compared with the exact solution by Boussinesq. Secondly, a square plate loading on the axi-symmetric half space has been analyzed by using solid finite and 3-D dynamic infinite elements. Thirdly, the calculation of the vertical vibration of a square rigid plate resting on a semi-infinite half-space has been given. Finally, for the Boussinesq problem, a sensitivity analysis is performed using not only various mesh sizes but also springs all along the truncated boundaries and the results are compared. It is amply demonstrated that the use of infinite elements provides unprecedented high degree of accuracy.