On the optimum support size in meshfree methods: A variational adaptivity approach with maximum-entropy approximants

International Journal for Numerical Methods in Engineering

Volumn 82, Issue 7, 2010, Pages 868-895

  Rosolen, Adrian ^a   Millán, Daniel ^a   Arroyo, Marino ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Local maximum entropy approximants; Meshfree methods; Support size; Variational adaptivity

Indexed keywords

ADAPTIVITY; APPROXIMANTS; LOCAL MAXIMUM; LOCAL MAXIMUM-ENTROPY APPROXIMANTS; MESH-FREE METHOD; VARIATIONAL ADAPTIVITY;

FUNCTION EVALUATION; VARIATIONAL TECHNIQUES;

ENTROPY;

EID: 77951279810     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2793     Document Type: Article

References (29)

1. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 1996; 139:3-47.
2. Li S, Liu W. Meshfree and particle methods and their applications. Applied Mechanics Reviews 2002; 55(1):1-34.
3. Huerta A, Belytscko T, Fernández-Méndez S, Rabczuk T. Meshfree methods, vol. 1. Encyclopedia of Computational Mechanics. Wiley: Chichester, 2004; 279-309.
4. Sukumar N, Wright R. Overview and construction of meshfree basis functions: from moving least squares to entropy approximants. International Journal for Numerical Methods in Engineering 2007; 70(2):181-205.
5. Lancaster P, Salkauskas K. Surfaces generated by Moving Least-Squares methods. Mathematics of Computation 1981; 37(155):141-158.
6. Arroyo M, Ortiz M. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. International Journal for Numerical Methods in Engineering 2006; 65:2167-2202.
7. Cyron CJ, Arroyo M, Ortiz M. Smooth, second order, non-negative meshfree approximants selected by maximum entropy. International Journal for Numerical Methods in Engineering 2009; 79(13):1605-1632.
8. Sukumar N. Construction of polygonal interpolants: a maximum entropy approach. International Journal for Numerical Methods in Engineering 2004; 61(12):2159-2181.
9. Ma Z, Chen H, Zhou C. A study of point moving adaptivity in gridless method. Computer Methods in Applied Mechanics and Engineering 2008; 197:1926-1937.
10. Vidal Y, Parés N, Díez P, Huerta A. Bounds for quantities of interest and adaptivity in the element free Galerkin method. International Journal for Numerical Methods in Engineering 2008; 76(77):1782-1818.
11. Liu W, Li S, Belytschko T. Moving least square reproducing kernel methods. Part i: methodology and convergence. Computer Methods in Applied Mechanics and Engineering 1997; 143(1-2):113-154.
12. Fries T, Matthies H. Classification and overview of meshfree methods. Technical Report, Institute of Scientific Computing, Technical University Braunschweig, Brunswick, Germany, July 2004.
13. Kuhl E, Askes H, Steinmann P. An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: generic hyperelastic formulation. Computer Methods in Applied Mechanics and Engineering 2004; 193(39-41):4207-4222.
14. Thoutireddy P, Ortiz M. A variational r-adaption and shape-optimization method for finite-deformation elasticity. International Journal for Numerical Methods in Engineering 2004; 61:1-21.
15. Mosler J, Ortiz M. On the numerical implementation of variational arbitrary Lagrangian-Eulerian (VALE) formulations. International Journal for Numerical Methods in Engineering 2006; 67:1272-1289.
16. Sukumar N, Moran B, Belytschko T. The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering 1998; 43(5):839-887.
17. Cirak F, Ortiz M, Schröder P. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. International Journal for Numerical Methods in Engineering 2000; 47(12):2039-2072.
18. Hughes T, Cottrell J, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 2005; 194:4135-4195.
19. d. Discrete and Computational Geometry 1994; 12(2): 189-202.
20. Parés N, Díez P, Huerta A. Exact bounds for linear outputs of the advection-diffusion-reaction equation using flux-free error estimates. SIAM Journal on Scientific Computing 2009; 31(4):3064-3089.
21. Thoutireddy P. Variational arbitrary Lagrangian-Eulerian method. Ph.D. Thesis, California Institute of Technology 2003.
22. Askes H, Kuhl E, Steinmann P. An ALE formulation based on spatial and material settings of continuum mechanics. Part 2: classification and applications. Computer Methods in Applied Mechanics and Engineering 2004; 193(39-41):4223-4245.
23. Askes H, Bargmann S, Kuhl E, Steinmann P. Structural optimization by simultaneous equilibration of spatial and material forces. Communications in Numerical Methods in Engineering 2005; 21:433-442.
24. Zielonka M, Ortiz M, Marsden J. Variational R-adaption in elastodynamics. International Journal for Numerical Methods in Engineering 2008; 74:1162-1197.
25. Nocedal J. Updating quasi-Newton matrices with limited storage. Mathematics of Computation 1980; 35:773-782.
26. Byrd R, Lu P, Nocedal J. A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific and Statistical Computing 1995; 16(5):1190-1208.
27. Timoshenko S, Goodier J. Theory of Elasticity. McGraw-Hill: New York, 1951.
28. Krysl P, Belytschko T. Element-free Galerkin method: convergence of the continuous and discontinuous shape functions. Computer Methods in Applied Mechanics and Engineering 1997; 148:257-277.
29. Mosler J, Ortiz M. An error-estimate-free and remapping-free variational mesh refinement and coarsening method for dissipative solids at finite strains. International Journal for Numerical Methods in Engineering 2009; 77(3):437-450.