Abstract
Meshfree methods such as the Reproducing Kernel Particle Method and the Element Free Galerkin method have proven to be excellent choices for problems involving complex geometry, evolving topology, and large deformation, owing to their ability to model the problem domain without the constraints imposed on the Finite Element Method (FEM) meshes. However, meshfree methods have an added computational cost over FEM that come from at least two sources: increased cost of shape function evaluation and the determination of adjacency or connectivity. The focus of this paper is to formally address the types of adjacency information that arises in various uses of meshfree methods; a discussion of available techniques for computing the various adjacency graphs; propose a new search algorithm and data structure; and finally compare the memory and run time performance of the methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Available at https://graphics.stanford.edu/data/3Dscanrep/.
References
Atluri SN, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22(2):117–127
Barbieri E, Meo M (2012) A fast object-oriented matlab implementation of the reproducing kernel particle method. Comput Mech 49(5):581–602
Belytschko T, Liu Y, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256
Belytschko T, Krongauz Y, Fleming M, Organ D, Liu W (1996) Smoothing and accelerated computations in the element free Galerkin method. J Comput Appl Math 74:111–126
Bentley JL (1975a) Multidimensional binary search trees used for associative searching. Commun ACM 18(9):509–517
Bentley JL (1975b) A survey of techniques for fixed radius near neighbor searching. Technical report, Stanford Linear Accelerator Center, Stanford, CA, USA
Berg Md, Cheong O, Mv Kreveld, Overmars M (2008) Computational geometry. Springer, Berlin
Cartwright C, Oliveira S, Stewart DE (2006) Parallel support set searches for meshfree methods. SIAM J Sci Comput 28(4):1318–1334
Cavoretto R, De Rossi A, Perracchione E (2016) Efficient computation of partition of unity interpolants through a block-based searching technique. Comput Math Appl 71(12):2568–2584
Chartrand G (1985) Introductory graph theory. Dover, New York
Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50:435–466
Chen JS, Hillman M, Rüter M (2013) An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int J Numer Methods Eng 95(5):387–418
Dolbow J, Belytschko T (1999) Numerical integration of the Galerkin weak from in meshfree methods. Comput Mech 23:219–30
Duarte C, Oden J (1996) An h-p adaptive method using clouds. Comput Methods Appl Mech Eng 139(1–4):237–262
Fuchs H, Kedem ZM, Naylor BF (1980) On visible surface generation by a priori tree structures. SIGGRAPH Comput Graph 14(3):124–133
Fujimoto A, Iwata K (1985) Accelerated ray tracing. Springer, Tokyo, pp 41–65
Han X, Oliveira S, Stewart D (2000) Finding sets covering a point with application to mesh-free Galerkin methods. SIAM J Comput 30(4):1368–1383
Karatarakis A, Metsis P, Papadrakakis M (2013) GPU-acceleration of stiffness matrix calculation and efficient initialization of EFG meshless methods. Comput Methods Appl Mech Eng 258:63–80
Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55(1):1–34
Li S, Liu WK (2004) Meshfree particle methods. Springer, Berlin
Li S, Qian D, Wk Liu, Belytschko T (2001) A meshfree contact-detection algorithm. Comput Methods Appl Mech Eng 190:3271–3292
Liu G (2009) Meshfree methods moving beyond the finite element method. CRC Press, Boca Raton
Liu G, Tu Z (2002) An adaptive procedure based on background cells for meshless methods. Comput Methods Appl Mech Eng 191:1923–1943
Liu W, Jun S, Li S, Adee J, Belytschko T (1995a) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38:1655–1679
Liu W, Jun S, Zhang Y (1995b) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106
Onate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Eng 39(22):3839–3866
Rubin SM, Whitted T (1980) A 3-dimensional representation for fast rendering of complex scenes. SIGGRAPH Comput Graph 14(3):110–116
Tapia-Fernández S, Romero I, García-Beltrán A (2017) A new approach for the solution of the neighborhood problem in meshfree methods. Engineering with Computers 33:239–247
Wald I, Ize T, Kensler A, Knoll A, Parker SG (2006) Ray tracing animated scenes using coherent grid traversal. ACM Trans Graph 25(3):485–493
Wang D, Zhou Y, Shao S (2016) Efficient implementation of smoothed particle hydrodynamics (SPH) with plane sweep algorithm. Commun Comput Phys 19(3):770–800
Wendland H (2004) Scattered data approximation, vol 17. Cambridge University Press, Cambridge
Acknowledgements
The authors would like to thank and acknowledge Dr. Jen Bright for the data provided for the Scarlet Macaw bird skull. The author’s would also like to thank Formerics, LLC for the use of their software.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Olliff, J., Alford, B. & Simkins, D.C. Efficient searching in meshfree methods. Comput Mech 62, 1461–1483 (2018). https://doi.org/10.1007/s00466-018-1574-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-018-1574-9