Abstract
This contribution presents two advances in the formulation of discontinuous approximations in finite elements. The first method relies on Schwarz-Christoffel mapping for integration on arbitrary polygonal domains [1]. When an element is split into two subdomains by a piecewise continuous discontinuity, each of these polygonal domains is mapped onto a unit disk on which cubature rules are utilized. This suppresses the need for the usual two-level isoparametric mapping. The second method relies on strain smoothing applied to discontinuous finite element approximations. By writing the strain field as a non-local weighted average of the compatible strain field, integration on the surface of the finite elements is transformed into boundary integration, so that the usual subdivision into integration cells is not required, an isoparametric mapping is not needed and the derivatives of the shape (enrichment) functions do not need to be computed. Results in fracture mechanics and composite materials are presented and both methods are compared in terms of accuracy and simplicity. The interested reader is referred to [1,6,13] for more details and should contact the authors to receive a version of the MATLAB codes used to obtain the results herein.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S Natarajan, S Bordas, DR Mahapatra, International Journal of Numerical Methods in Engineering, 80(1), 103 (2009), DOI:10.1002/nme.2589
T Belytschko, T Black, International Journal of Numerical Methods in Engineering, 45, 601 (1999)
J Dolbow, N Moes, T Belytschko, Computer Methods in Applied Mechanics and Engineering, 19, 6825 (2001)
N Moes, J Dolbow, T Belytschko, International Journal of Numerical Methods in Engineering, 46, 131 (1999)
GR Liu, TT Nguyen, KY Dai, KY Lam, International Journal of Numerical Methods in Engineering, 71(8), 902 (2007)
SPA Bordas, T Rabczuk, NX Hung, VP Nguyen, S Natarajan, T Bog, DM Quan, NV Hiep, Computers and Structures (2009), DOI: 10.1016/j.compstruc.2008.07.006
AT Driscoll, NL Trefethen, Cambridge Monographs on Applied and Computational Mathematics 8(1) (2002)
AT Driscoll, NL Trefethen, ACM Transactions on Mathematical Software, 22 (2), 168 (1996)
J Chen, C Wu, S Yoon, Y You, International Journal of Numerical Methods in Engineering, 50, 435 (2001)
S Bordas, V Nguyen, C Dunant, ND Hung, A Guidoum, International Journal of Numerical Methods in Engineering (2007), DOI:10.1002/nme.1966
Y Krongauz, T Belytschko, International Journal of Numerical Methods in Engineering, 41(1), 1215 (1998)
N Sukumar, DL Chopp, N Moes, T Belytschko, Computer Methods in Applied Mechanics and Engineering, 190 (46–47), 6183 (2001), DOI: 10.1016/S0045-7825(01)00215-8
S Natarajan, DR Mahapatra, S Bordas, International Journal of Numerical Methods in Engineering (2009), DOI: 10.1002/nme.2798
JS Chen, HP Wang, Computational Mechanics, 25, 137 (2000)
H Nguyen-Xuan, S Bordas, H Nguyen-Dang, International Journal of Numerical Methods in Engineering, 74, 175 (2008). DOI: 10.1002/nme.2146
S Bordas, S Natarajan, International Journal of Numerical Methods in Engineering, (2009), DOI: 10.1002/nme.2713
V Nguyen Phu, T Rabczuk, S Bordas, M Duflot, Mechanics and Computer in Simulation, 79, 763 (2008), DOI: 10.1016/j.matcom.2008.01.003
Acknowledgements
The first author acknowledges the financial support of (1) Overseas Research Students Awards Scheme; (2) the Faculty of Engineering, for period Jan. 2008 – Sept. 2009 and of (3) School of Engineering (Cardiff University) for the period Sept. 2009 onwards. The second author gratefully acknowledge the financial support of the Royal Academy of Engineering and of the Leverhulme Trust Senior Research Fellowship (2009-2010) [Towards the Next Generation Surgical Simulators] http://www.raeng.org.uk/research/researcher/leverhulme/current.htm
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this paper
Cite this paper
Natarajan, S., Bordas, S.P.A., Mahapatra, D.R. (2010). On Numerical Integration of Discontinuous Approximations in Partition of Unity Finite Elements. In: Dattaguru, B., Gopalakrishnan, S., Aatre, V. (eds) IUTAM Symposium on Multi-Functional Material Structures and Systems. IUTAM Bookseries (closed), vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3771-8_30
Download citation
DOI: https://doi.org/10.1007/978-90-481-3771-8_30
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3770-1
Online ISBN: 978-90-481-3771-8
eBook Packages: EngineeringEngineering (R0)