Development of High-Order Infinite Element Method for Stress Analysis of Elastic Bodies with Singularities

Abstract

Structural analysis problems are traditionally solved using the Finite Element Method (FEM). However, when the structure of interest contains discontinuities such as cracks or re-entrant corners, a large number of elements are required to accurately reproduce the stress characteristics in the region of the discontinuities. As a result, the FEM method requires a large storage space and has a slow convergence speed. It has been shown that the Infinite Element Method (IEM) overcomes these limitations and provides a feasible means of solving various types of elasticity and singularity problems. Previous studies have generally focused on the use of IEM formulations based on low-order elements (2×2). By contrast, this study develops a high-order (3×3) IEM formulation. The solutions obtained using the proposed IEM method for various 2D elasto-static problems are compared with the results obtained using the traditional low-order IEM method and the analytical solutions presented in the literature. It is shown that the results obtained using the proposed method are more accurate than those obtained using the low-order IEM method and are in excellent agreement with the analytical solutions. 1132heterogeneous properties, those with inhomogeneous behaviors pose a far greater challenge. Various researchers have suggested that the limitations of the FEM and BEM methods can be overcome by the Infinite Element Method (IEM) (3) . The basic principle of IEM involves discretizing the problem domain in to a large numbers of elements using a suitable partitioning technique and nodal sequence such that each element layer is similar to the other layers. Ying (4) , Han and Ying (5) , and Ying and Pan (6) developed the mathematical foundations of the IEM formulation for the Laplace equation, exterior Stokes problem, corner problem, and planar elasticity problem in fracture mechanics. Ying (7) proved the existence of a transformation matrix relating the nodal displacement vectors of the inner and outer layers, respectively, and showed that the stiffness matrices associated with each layer could therefore be combined to form an overall stiffness matrix defined in terms of the boundary nodes and tractions only. In addition, it was shown that the combined stiffness matrix converges to a certain constant quantity as the number of element layers approaches infinity. Guo (8) proved that the concept of similar elements could be extended to arbitrary isoparametric elements and also presented a transformation procedure for constructing the overall stiffness matrix of the body of interest by combining the local stiffness matrices of each layer. Liu and Chiou (9)-(12) recently developed both 2D (2×2) and 3D (2×2×2) IEM formulations to solve a variety of elasticity and singularity problems. In addition, the authors proposed a convergence criterion for the IEM formulation which extended the imaginary "infinity layer" concept to practical applications and ensured the convergence of the overall IE stiffness matrix.Previous researchers have generally focu...

Development of High-Order Infinite Element Method for Bending Analysis of Mindlin–Reissner Plates

“…Liu et al [18,19] have combined the IEM with the FEM to solve static linear problems and have continuously extended equations from 2D to 3D. Liu et al [20] further derived a high-order IEM equation for analyzing various 2D elastic static problems; they compared their results with those of the traditional loworder IEM and with analytical solutions provided in the literature. eir findings revealed that the results obtained using their method were more accurate than those obtained using the low-order IEM and were in good agreement with the analytical solutions.…”