Description
The Generalized or Extended FEM (GFEM/XFEM) has been intensively developed in the last 15 years and today is available in mainstream commercial Finite Element software. The GFEM/XFEM is particularly attractive for the solution of problems involving propagating fractures because it alleviates several meshing requirements of the FEM while retaining its attractive features. If proper enrichment functions are used in a fixed region around the crack front, the GFEM delivers optimal convergence rates – the same rate as in problems with smooth solutions. In this so-called geometrical enrichment, the size of the enrichment zone is independent of the mesh size. However, this strategy leads to extremely high condition numbers. In the topological enrichment strategy, the size of the enrichment reduces with mesh refinement and the conditioning and rate of convergence of the GFEM are the same order as in the FEM. Therefore, significant mesh refinement may be required in this enrichment strategy, in particular in 3D problems. A modified GFEM, referred to as Stable GFEM (SGFEM) [1,2], is optimally convergence and has a condition number of the same order as the FEM. The SGFEM is based on simple local modifications of enrichment functions employed in the GFEM and lends to straightforward implementation in existing GFEM software. In this presentation, we report on a SGFEM for 3D fractures. We show that a straightforward extension of ideas from 1D or 2D fracture problems presented in Refs. [1, 2] is not sufficient to achieve optimal convergence rates in the 3D case. We propose a set of additional enrichments on nodes with singular bases to recover optimal convergence in 3D. Furthermore, the proposed SGFEM for 3D fractures delivers the same rate of growth in conditioning as the standard FEM for proper choice of singular enrichment functions and is more accurate than the GFEM for both geometrical and topological enrichment strategies. REFERENCES [1] Babuska, I., Banerjee, U. Stable generalized finite element method (SGFEM). Comp. Methods in Appl. Mech. Engng. 2012, 201–204 (0), 91–111.. DOI: 10.1016/j.cma.2011.09.012. [2] Gupta, V., Duarte, C.A., Babuska, I., Banerjee, U. A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Comp. Meth Appl. Mech. Engng. 2013, 266(0), 23–39. DOI: 10.1016/j.cma.2013.07.
Recommended Citation
Gupta, V., & Duarte, A. (2014). Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics. In A. Bajaj, P. Zavattieri, M. Koslowski, & T. Siegmund (Eds.). Proceedings of the Society of Engineering Science 51st Annual Technical Meeting, October 1-3, 2014 , West Lafayette: Purdue University Libraries Scholarly Publishing Services, 2014. https://docs.lib.purdue.edu/ses2014/mss/cfm/2
Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics
The Generalized or Extended FEM (GFEM/XFEM) has been intensively developed in the last 15 years and today is available in mainstream commercial Finite Element software. The GFEM/XFEM is particularly attractive for the solution of problems involving propagating fractures because it alleviates several meshing requirements of the FEM while retaining its attractive features. If proper enrichment functions are used in a fixed region around the crack front, the GFEM delivers optimal convergence rates – the same rate as in problems with smooth solutions. In this so-called geometrical enrichment, the size of the enrichment zone is independent of the mesh size. However, this strategy leads to extremely high condition numbers. In the topological enrichment strategy, the size of the enrichment reduces with mesh refinement and the conditioning and rate of convergence of the GFEM are the same order as in the FEM. Therefore, significant mesh refinement may be required in this enrichment strategy, in particular in 3D problems. A modified GFEM, referred to as Stable GFEM (SGFEM) [1,2], is optimally convergence and has a condition number of the same order as the FEM. The SGFEM is based on simple local modifications of enrichment functions employed in the GFEM and lends to straightforward implementation in existing GFEM software. In this presentation, we report on a SGFEM for 3D fractures. We show that a straightforward extension of ideas from 1D or 2D fracture problems presented in Refs. [1, 2] is not sufficient to achieve optimal convergence rates in the 3D case. We propose a set of additional enrichments on nodes with singular bases to recover optimal convergence in 3D. Furthermore, the proposed SGFEM for 3D fractures delivers the same rate of growth in conditioning as the standard FEM for proper choice of singular enrichment functions and is more accurate than the GFEM for both geometrical and topological enrichment strategies. REFERENCES [1] Babuska, I., Banerjee, U. Stable generalized finite element method (SGFEM). Comp. Methods in Appl. Mech. Engng. 2012, 201–204 (0), 91–111.. DOI: 10.1016/j.cma.2011.09.012. [2] Gupta, V., Duarte, C.A., Babuska, I., Banerjee, U. A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Comp. Meth Appl. Mech. Engng. 2013, 266(0), 23–39. DOI: 10.1016/j.cma.2013.07.