FEM elements enriched with meshfree functions: overview and application

Bekhta M. I., Savula Ya. G.
PDF icon 2014_1_001_016.pdf640.2 KB
A specific method of coupling FEM and meshless/meshfree methods is presented. This method is based on placing meshfree nodes inside the finite element and as a result improving the overall approximation on that element. Advantages and disadvantages of such approach are explained. It is shown that such approach is a version of a more general one. Numerical experiments are presented and analyzed.
  1. Huerta A., Belytschko T., Fernández-Méndez S., Rabczuk T., Meshfree methods, Encyclopedia of Computational Mechanics, John Wiley & Sons, Ltd. 1 (10), 279 (2004).
  2. Huerta A., Fernández-Méndez S., Enrichment and coupling of the finite element and meshless methods. Int. J. Numer. Methods Eng. 48 (11), 1615 (2000).
  3. Fernández-Méndez S., Diez P., Huerta A. Convergence of finite elements enriched with mesh-less methods. Numer. Math. 96 (1), 4 (2003).
  4. Fernández-Méndez S., Huerta A. Imposing essential boundary conditions in mesh-free methods Comput. Methods Appl. Mech. Eng. 193 (12–14), 1257 (2004).
  5. Fernández-Méndez S. Mesh-free methods and finite elements: friend or foe? Universitat Politécnica de Catalunya, 162 p. (2002).
  6. Fries T. P. A stabilized and coupled meshfree/meshbased method for fluid-structure-interaction problems. Dissertation, In Braunschweiger Schriften zur Mechanik (H. Antes, Ed.), Nr. 59-2005, Braunschweig, 165 p. (2005).
  7. Fries T. P., Matthies H. G. A stabilized and coupled meshfree/meshbased method for the incompressible Navier-Stokes equations – part II: Coupling Comp. Methods in Appl. Mech. Engrg. 195, 6191 (2006).
  8. Liu G. R. Meshfree Methods: Moving Beyond the Finite Element Method, Second Edition CRC Press, 792 p. (2009).
  9. Li S., Liu W. K., Meshfree and particle methods and their applications Applied Mechanics Review 55, 1 (2002).
  10. Dolbow J., Belytschko T. An Introduction to Programming the Meshless Element Free Galerkin Method. Archives of Computational Methods in Engineering, 5 (3), 207 (1998).
  11. Belytschko T., Organ D., Krongauz Y. A Coupled Finite Element–Element-free Galerkin Method Comput. Mech. 17, 186 (1995).
  12. Li S., Liu W. K., Belytschko T. Moving least-squares reproducing kernel method Part I: Methodology and convergence. Comp. Methods Appl. Mech. Engrg. 143 (1–2), 113 (1997).
  13. Lancaster P., Salkauskas K. Surfaces Generated by Moving Least Squares Methods Math. Comput. 37, 141 (1981).
  14. Savula Ya. Numerical analysis of problems of mathematical physics by variational methods. Lviv, 221 p. (2004).
  15. Bekhta M., Savula Ya. Problem of optimal parameter selection for combination of meshfree and finite elements methods. Volyn Mathematical Visnyk. 8 (17), 15 (2011).
Math. Model. Comput. Vol.1, No.1, pp.1-16 (2014)