Showing papers by "Sundararajan Natarajan published in 2011"

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

Abstract: By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25: 137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6): 859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright (C) 2011 John Wiley & Sons, Ltd.

Linear buckling analysis of cracked plates by SFEM and XFEM

Abstract: In this paper, the linear buckling problem for isotropic plates is studied using a quadrilateral element with smoothed curvatures and the extended finite element method. First, the curvature at each point is obtained by a nonlocal approximation via a smoothing function. This element is later coupled with partition of unity enrichment to simplify the simulation of cracks. The proposed formulation suppresses locking and yields elements which behave very well, even in the thin plate limit. The buckling coefficient and mode shapes of square and rectangular plates are computed as functions of crack length, crack location, and plate thickness. The effects of different boundary conditions are also studied.

Enriched finite element methods: advances & applications

Abstract: The accuracy and the efficiency of both the methods are studied numerically with problems involving strong and weak discontinuities. The XFEM is applied to study the crack inclusion interaction in a particle reinforced composite material. The influence of the crack length, the number of inclusions and the geometry of the inclusions on the crack tip stress field is numerically studied. Linear natural frequencies of cracked functionally graded material plates are studied within the framework of the XFEM. The effect of the plate aspect ratio, the crack length, the crack orientation, the gradient index and the influence of cracks is numerically studied.

Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework

Abstract: Partition of unity methods, such as the extended finite element method (XFEM) allow discontinuities to be simulated independently of the mesh [1]. This eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re-meshing, as the discontinuity evolves. However, to compute the stiffness matrix of the elements intersected by the discontinuity, a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted. In this paper, we use a simple integration technique, proposed for polygonal domains [2] to suppress the need for element subdivision. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi-material problem, show that the proposed method yields accurate results. Owing to its simplicity, the proposed integration technique can be easily integrated in any existing code.

On the approximation in the smoothed finite element method (SFEM)

Abstract: This letter aims at resolving the issues raised in the recent short communication [1] and answered by [2] by proposing a systematic approximation scheme based on non-mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) [3, 4, 5, 6, 7, 8, 9] and resolve the existence, linearity and positivity deficiencies pointed out in [1]. We show that Wachspress interpolants [10] computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results which are almost identical to those of the SFEM initially proposed in [3]. These results that the proposed approximation scheme forms a strong and rigorous basis for construction of smoothed finite element methods.