Reproducing Kernel Element Method for Galerkin Solution of Elastostatic Problems
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Keywords
Elasticity, convergence, RKEM, continuity, Galerkin methods.
Abstract
The Reproducing Kernel Element Method (RKEM) is a relatively new technique developed to have two distinguished features: arbitrary high order smoothness and arbitrary interpolation order of the shape functions. This paper provides a tutorial on the derivation and computational implementation of RKEM for Galerkin discretizations of linear elastostatic problems for one and two dimensional space. A key characteristic of RKEM is that it do not require mid-side nodes in the elements to increase the interpolatory power of its shape functions, and contrary to meshless methods, the same mesh used to construct the shape functions is used for integration of the stiffness matrix. Furthermore, some issues about the quadrature used for integration arediscussed in this paper. Its hopes that this may attracts the attention of mathematicians.
MSC: 65N30
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References
[2] D. C. S. Jr, N. Collier, M. Juha, L. Whitenack, “A Framework For Studying The RKEM Representation of Discrete Point Sets”, in Meshfree Methods for Partial
Differential Equations IV, M. Griebel y M. A. Schweitzer, Eds. Springer Berlin Heidelberg, 2008, pp. 301-314. Referenced in 70
[3] T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, 2000. Referenced in 70, 73, 77, 82
[4] L. Piegl, W. Tiller, The NURBS book. New York, NY, USA: Springer-Verlag New York, Inc., 1997. Referenced in 71
[5] G. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, 1.a ed. CRC Press, 2002. Referenced in 71
[6] W. K. Liu, W. Han, H. Lu, S. Li, y J. Cao, “Reproducing kernel element method. Part I: Theoretical formulation”, Computer Methods in Applied Mechanics and Engineering, vol. 193, n.o 12-14, pp. 933-951, mar. 2004. Referenced in 72
[7] S. Li, H. Lu, W. Han, W. Liu, D. Simkins, “Reproducing kernel element method Part II: Globally conforming Im/Cn hierarchies”, Computer Methods in Applied Mechanics and Engineering, vol. 193, n.o 12-14, pp. 953-987, mar. 2004.
Referenced in 72
[8] H. Lu, S. Li, D. C. Simkins Jr., W. Kam Liu, J. Cao, “Reproducing kernel element method Part III: Generalized enrichment and applications”, Computer Methods in Applied Mechanics and Engineering, vol. 193, n.o 12-14, pp. 989-1011, mar. 2004. Referenced in 73
[9] K.-J. Bathe, Finite element procedures. Prentice Hall, 1995. Referenced in 78
[10] N. Collier y D. C. Simkins, “The quasi-uniformity condition for reproducing kernel element method meshes”, Computational Mechanics, vol. 44, n.o 3, pp. 333-342, mar. 2009. Referenced in 79, 85
[11] A. H. Stroud, D. Secrest, Gaussian quadrature formulas. Prentice-Hall, 1966. Referenced in 80
[12] S. Timoshenko y J. N. Goodier, Theory of elasticity. McGraw-Hill, 1969. Referenced in 86
[13] S. Fernández-Méndez y A. Huerta, “Imposing essential boundary conditions in mesh-free methods”, Computer Methods in Applied Mechanics and Engineering, vol. 193, n.o 12-14, pp. 1257-1275, mar. 2004. Referenced in 90