Abstract
This paper is concerned with automatic enrichment in the particle-partition of unity method (PPUM). The goal of our automatic enrichment is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an automatically determined enrichment zone hierarchically near the singularities of the solution. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner. The results of our numerical experiments clearly show that the hierarchically enriched PPUM recovers the optimal convergence rate globally and even shows a kind of superconvergence within the enrichment zone. The condition number of the stiffness matrix is independent of the employed enrichment and the relative size of the enrichment zone.
This is a preview of subscription content, log in via an institution to check access.
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
I. Babuška, U. Banerjee, and J. E. Osborn, Survey of Meshless and Generalized Finite Element Methods: A Unified Approach, Acta Numerica, (2003), pp. 1–125.
I. Babuška and J. M. Melenk, The Partition of Unity Method, Int. J. Numer. Meth. Engrg., 40 (1997), pp. 727–758.
S. Beissel and T. Belytschko, Nodal Integration of the Element-Free Galerkin Method, Comput. Meth. Appl. Mech. Engrg., 139 (1996), pp. 49–74.
T. Belytschko and T. Black, Elastic crack growth in finite elements with minimal remeshing, Int. J. Numer. Meth. Engrg., 45 (1999), pp. 601–620.
T. Belytschko, Y. Y. Lu, and L. Gu, Crack propagation by element-free galerkin methods, Engrg. Frac. Mech., 51 (1995), pp. 295–315.
T. Belytschko, N. Moës, S. Usui, and C. Parimi, Arbitrary discontinuities in finite elements, Int. J. Numer. Meth. Engrg., 50 (2001), pp. 993–1013.
E. Chahine, P. Laborde, J. Pommier, Y. Renard, and M. Salaün, Study of some optimal xfem type methods, in Advances in Meshfree Techniques, V. M. A. Leitao, C. J. S. Alves, and C. A. M. Duarte, eds., vol. 5 of Computational Methods in Applied Sciences, Springer, 2007.
J. S. Chen, C. T. Wu, S. Yoon, and Y. You, A Stabilized Conforming Nodal Integration for Galerkin Mesh-free Methods, Int. J. Numer. Meth. Engrg., 50 (2001), pp. 435–466.
J. Dolbow and T. Belytschko, Numerical Integration of the Galerkin Weak Form in Meshfree Methods, Comput. Mech., 23 (1999), pp. 219–230.
C. A. Duarte, L. G. Reno, and A. Simone, A higher order generalized fem for through-the-thickness branched cracks, Int. J. Numer. Meth. Engrg., 72 (2007), pp. 325–351.
C. A. M. Duarte, I. Babuška, and J. T. Oden, Generalized. Finite Element Methods for Three Dimensional Structural Mechanics Problems, Comput. Struc., 77 (2000), pp. 215–232.
C. A. M. Duarte, O. N. H. T. J. Liszka, and W. W. Tworzydlo, A generalized finite element method for the simulation of three-dimensional dynamic crack propagation, Int. J. Numer. Meth. Engrg., 190 (2001), pp. 2227–2262.
M. Griebel, P. Oswald, and M. A. Schweitzer, A Particle-Partition of Unity Method—Part VI: A p-robust Multilevel Preconditioner, in Meshfree Methods for Partial Differential Equations II, M. Griebel and M. A. Schweitzer, eds., vol. 43 of Lecture Notes in Computational Science and Engineering, Springer, 2005, pp. 71–92.
M. Griebel and M. A. Schweitzer, A Particle-Partition of Unity Method— Part II: Efficient Cover Construction and Reliable Integration, SIAM J. Sci. Comput., 23 (2002), pp. 1655–1682.
-, A Particle-Partition of Unity Method—Part III: A Multilevel Solver, SIAM J. Sci. Comput., 24 (2002), pp. 377–409.
-, A Particle-Partition of Unity Method—Part V: Boundary Conditions, in Geometric Analysis and Nonlinear Partial Differential Equations, S. Hildebrandt and H. Karcher, eds., Springer, 2002, pp. 517–540.
-, A Particle-Partition of Unity Method—Part VII: Adaptivity, in Meshfree Methods for Partial Differential Equations III, M. Griebel and M. A. Schweitzer, eds., vol. 57 of Lecture Notes in Computational Science and Engineering, Springer, 2006, pp. 121–148.
N. Moës, J. Dolbow, and T. Belytschko, A finite element method for crack growth without remeshing, Int. J. Numer. Meth. Engrg., 46 (1999), pp. 131–150.
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1970–1971), pp. 9–15.
J. T. Oden and C. A. Duarte, Clouds, Cracks and FEM’s, Recent Developments in Computational and Applied Mechanics, 1997, pp. 302–321.
M. A. Schweitzer, A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations, vol. 29 of Lecture Notes in Computational Science and Engineering, Springer, 2003.
-, An adaptive hp-version of the multilevel particle-partition of unity method, Comput. Meth. Appl. Mech. Engrg., (2008), accepted.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schweitzer, M.A. (2008). A Particle-Partition of Unity Method Part VIII: Hierarchical Enrichment. In: Griebel, M., Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations IV. Lecture Notes in Computational Science and Engineering, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79994-8_16
Download citation
DOI: https://doi.org/10.1007/978-3-540-79994-8_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79993-1
Online ISBN: 978-3-540-79994-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)