Advances in Enriched Finite Element Methods

Core Legato-team member: S Natarajan, Indian Institute of Technology-Madras, India.
Collaborators: C Song, The University of New South Wales, ET Ooi, Federation University, Australia.

The finite element method’s efficiency to model cracks or discontinuities or large gradients has always been considered an area for improvement since early 1970. Though the idea of augmenting (or enriching) the finite element space appropriately dates back to 1970’s, the pioneering work of Melenk & Babuska, Belytschko & Others, Duarte & Simone to name a few, has led to some of the robust methods available to model cracks or discontinuities or large gradients independent of the underlying FE mesh.

Representation of discontinuity within XFEM framework

Representation of discontinuity within XFEM framework

 

The flexibility provided by the XFEM also leads to some difficulties: (a) singular and discontinuous integrands; (b) blending different partitions of unity; (c) poor convergence rate and (d) ill-conditioning and additional unknowns. We have proposed two novel techniques to address the difficulty in the numerical integration. The first method relies on conformal mapping, where the regions intersected by the discontinuity surface are mapped onto a unit disk. The second method relies on strain smoothing applied to discontinuous field approximations. By writing the strain field as a non-local weighted average of the compatible strain field, integration on the interior of the finite elements is transformed into boundary integration, so that no-subdivision into integration cells is required.

 

Integration over an element with discontinuity using conformal mapping.

Integration over an element with discontinuity using conformal mapping.

Integration over an element with discontinuity using strain smoothing technique.

Integration over an element with discontinuity using strain smoothing technique.

 

Combined XFEM – SBFEM

The success of the XFEM when applied to moving boundary problems, esp, crack growth, relies on the a priori knowledge of the set of functions that span the asymptotic fields ahead of the crack tip. This requirement for a priori knowledge of the asymptotic fields hinders the application of the XFEM directly to heterogeneous materials for which the asymptotic fields do not exist in closed form or are very complex.

In order to alleviate the aforementioned shortcoming, we proposed the extended scaled boundary finite element method, which combines the best features of the XFEM and the scaled boundary FEM (SBFEM). The SBFEM is a fundamental solution-less method and has emerged as an attractive alternate to model problems with singularities. It by itself combines the best features of the FEM and the boundary element method. Numerical solutions are sought on the boundary, whilst the solution along the radial lines emanating from the ‘scaling center’ is represented analytically. Moreover, by utilizing the special features of the scaling center, the method allows the computation of the stress intensity factor directly.

Another striking feature of the combined method is that, the method can be directly applied to problems with cracks at interfaces or cracks terminating at the interfaces. The method does not require special numerical integration scheme to compute the terms in the stiffness matrix. This numerical technique can also be used to derive new enrichment functions for materials with complex microstructure or heterogeneous medium.

Order of singularity at a triple junction

Reference

S Natarajan, D Roy Mahapatra, S Bordas, Integrating strong and weak discontinuities without integration subcells and example applications in XFEM/GFEM framework, International Journal for Numerical Methods in Engineering, v83, 269-294, 2010. Preprint

S Bordas, S Natarajan, P Kerfriden, CE Augarde, DR Mahapatra, S Dal Pont, On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM), International Journal for Numerical Methods in Engineering, v86, 637-666, 2011. Preprint

S Bordas, et al., Strain smoothing in FEM and XFEM, Computers and Structures, v88, 1419-1443, 2010. Preprint

PM Baiz, S Natarajan, S Bordas, P Kerfriden, T Rabczuk, Linear buckling analysis of cracked plates by SFEM and XFEM, Journal of Mechanics and Structures, v6, 1213-1238, 2011. Preprint

S Natarajan, P Kerfriden, D Roy Mahapatra, S Bordas, Numerical analysis of the inclusion-crack interaction by the extended finite element method, International Journal for Computational Methods in Engineering, v15, 26-32, 2014. Preprint

S Natarajan, C Song, Representation of singular fields without asymptotic enrichment in the extended finite element method, International Journal for Numerical Methods in Engineering, v96, 813-841, 2013

S Natarajan, C Song, S Belouettar, Numerical evaluation of stress intensity factors and T-stress for interfacial cracks and cracks terminating at the interface without asymptotic enrichment, Computer Methods in Applied Mechanics and Engineering, v279, 86-112, 2014