Abstract
A method is proposed for arbitrary discontinuities, without the need for a mesh that aligns with the interfaces, and without introducing additional unknowns as in the extended finite element method. The approximation space is built by special shape functions that are able to represent the discontinuity, which is described by the level-set method. The shape functions are constructed by means of the moving least-squares technique. This technique employs special mesh-based weight functions such that the resulting shape functions are discontinuous along the interface. The new shape functions are used only near the interface, and are coupled with standard finite elements, which are employed in the rest of the domain for efficiency. The coupled set of shape functions builds a linear partition of unity that represents the discontinuity. The method is illustrated for linear elastic examples involving strong and weak discontinuities.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P.M.A. Areias and T. Belytschko, Letter to the editor, Comp. Methods Appl. Mech. Engrg. 195 (2004), 1275–1276.
T. Belytschko, L. Gu, and Y. Y. Lu, Fracture and crack growth by element free galerkin methods, Modelling Simul. Material Science Eng. 2 (1994), 519–534.
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: An overview and recent developments, Comp. Methods Appl. Mech. Engrg. 139 (1996), 3–47.
T. Belytschko, W.K. Liu, and B. Moran, Nonlinear finite elements for continua and structures, John Wiley & Sons, Chichester, 2000.
T. Belytschko, Y.Y. Lu, and L. Gu, Element-free Galerkin methods, Internat. J. Numer. Methods Engrg. 37 (1994), 229–256.
T. Belytschko, N. Moës, S. Usui, and C. Parimi, Arbitrary discontinuities in finite elements, Internat. J. Numer. Methods Engrg. 50 (2001), 993–1013.
T. Belytschko, D. Organ, and Y. Krongauz, A coupled finite element-elementfree Galerkin method, Comput. Mech. 17 (1995), 186–195.
J. Chessa, H.Wang, and T. Belytschko, On the construction of blending elements for local partition of unity enriched finite elements, Internat. J. Numer. Methods Engrg. 57 (2003), 1015–1038.
H. Ewalds and R. Wanhill, Fracture mechanics, Edward Arnold, New York, 1989.
T.P. Fries and H.G. Matthies, Classification and overview of meshfree methods, Informatikbericht-Nr. 2003–03, Technical University Braunschweig, (http://opus.tu-bs.de/opus/volltexte/2003/418/), Brunswick, 2003.
A. Hansbo and P. Hansbo, A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Comp. Methods Appl. Mech. Engrg. 193 (2004), 3523–3540.
A. Huerta and S. Fernández-Méndez, Enrichment and coupling of the finite element and meshless methods, Internat. J. Numer. Methods Engrg. 48 (2000), 1615–1636.
P. Krysl and T. Belytschko, Element-free Galerkin method: Convergence of the continuous and discontinuous shape functions, Comp. Methods Appl. Mech. Engrg. 148 (1997), 257–277.
P. Lancaster and K. Salkauskas, Surfaces generated by moving least squares methods, Math. Comput. 37 (1981), 141–158.
W.K. Liu, S. Li, and T. Belytschko, Moving least square reproducing kernel methods (i) methodology and convergence, Comp. Methods Appl. Mech. Engrg. 143 (1997), 113–154.
N. Moës, J. Dolbow, and T. Belytschko, A finite element method for crack growth without remeshing, Internat. J. Numer. Methods Engrg. 46 (1999), 131–150.
D. Organ, M. Fleming, T. Terry, and T. Belytschko, Continous meshless approximations for nonconvex bodies by diffraction and transparency, Comput. Mech. 18 (1996), 225–235.
S. Osher and R.P. Fedkiw, Level set methods and dynamic implicit surfaces, Springer Verlag, Berlin, 2003.
T. Rabczuk, T. Belytschko, S. Fernández-Méndez, and A. Huerta, Meshfree methods, Encyclopedia of Computational Mechanics (E. Stein, R. De Burst, T.J.R. Hughes, ed.), vol. 1, John Wiley & Sons, Chichester, 2004.
G. Strang and G. Fix, An analysis of the finite element method, Prentice-Hall, Englewood Cliffs, NJ, 1973.
N. Sukumar, D.L. Chopp, N. Moës, and T. Belytschko, Modeling holes and inclusions by level sets in the extended finite-element method, Comp. Methods Appl. Mech. Engrg. 190 (2001), 6183–6200.
O.C. Zienkiewicz and R.L. Taylor, The finite element method, vol. 1 -3, Butterworth-Heinemann, Oxford, 2000.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer
About this chapter
Cite this chapter
Fries, TP., Belytschko, T. (2007). New Shape Functions for Arbitrary Discontinuities without Additional Unknowns. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations III. Lecture Notes in Computational Science and Engineering, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-46222-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-46222-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-46214-9
Online ISBN: 978-3-540-46222-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)