Skip to main content
Book cover

International Conference on Computational Science

ICCS 2022: Computational Science – ICCS 2022 pp 55–68Cite as

Multipoint Meshless FD Schemes Applied to Nonlinear and Multiscale Analysis

  • 630 Accesses

Part of the Lecture Notes in Computer Science book series (LNCS,volume 13353)

Abstract

The paper presents computational schemes of the multipoint meshless method – the numerical modeling tool that allows accurate and effective solving of boundary value problems. The main advantage of the multipoint general version is its generality – the basic relations of derivatives from the unknown function depend on the domain discretization only and are independent of the type of problem being solved. This feature allows to divide the multipoint computational strategy into two stages and is advantageous from the calculation efficiency point of view. The multipoint method algorithms applied to such engineering problems as numerical homogenization of heterogeneous materials and nonlinear analysis are developed and briefly presented. The paper is illustrated by several examples of the multipoint numerical analysis.

Keywords

  • Meshless FDM
  • Higher order approximation
  • Multipoint method
  • Homogenization
  • Nonlinear analysis
  • Elastic-plastic problem

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-031-08760-8_5
  • Chapter length: 14 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   89.00
Price excludes VAT (USA)
  • ISBN: 978-3-031-08760-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   119.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.

References

  1. Chen, J.-S., Hillman, M., Chi, S.-W.: Meshfree methods: progress made after 20 years. J Eng Mech 143(4), 04017001 (2017)

    CrossRef  Google Scholar 

  2. Jaworska, I.: On the ill-conditioning in the new higher order multipoint method. Comput. Math. Appl. 66(3), 238–249 (2013)

    CrossRef  MathSciNet  Google Scholar 

  3. Jaworska, I., Orkisz, J.: Higher order multipoint method – from Collatz to meshless FDM. Eng. Anal. Bound. Elem. 50, 341–351 (2015)

    CrossRef  MathSciNet  Google Scholar 

  4. Orkisz, J.: Finite Difference Method (part III). In: Handbook of Computational Solid Mechanics, Springer-Verlag, 336–432 (1998)

    Google Scholar 

  5. Collatz, L.: Numerische Behandlung von Differential-gleichungen, Springer-Verlag (1955)

    Google Scholar 

  6. Atluri, S.N.: The Meshless Method (MLPG) for Domain & Bie Discretizations (2004)

    Google Scholar 

  7. Liszka, T.: An automatic grid generation in flat domain. Mech. i Komp 4, 181–186 (1981)

    Google Scholar 

  8. Schöberl, J.: NETGEN An Advancing Front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1, 41–52 (1997)

    CrossRef  Google Scholar 

  9. Preparata, F.P., Shamos, M.I.: Computational geometry. Springer-Verlag (1985)

    Google Scholar 

  10. Jaworska, I., Milewski, S.: On two-scale analysis of heterogeneous materials by means of the meshless finite difference method. Int. J. Multiscale Comput. Eng. 14(2), 113–134 (2016)

    CrossRef  Google Scholar 

  11. Jaworska, I.: Higher order multipoint meshless FDM for two-scale analysis of heterogeneous materials. Int. J. Multiscale Comput. Eng. 17(3), 239–260 (2019)

    CrossRef  Google Scholar 

  12. Jaworska, I., Orkisz, J.: Estimation of a posteriori computational error by the higher order multipoint meshless FDM. Comput. Inform. 36, 1447–1466 (2017)

    CrossRef  MathSciNet  Google Scholar 

  13. Geers, M.G.D., Kouznetsova, V.G., Brekelmans, W.A.M.: Multi-scale first-order and second-order computational homogenizastion of microstructures towards continua. Int. J. Multiscale Comput. Eng. 1(4), 371–385 (2003)

    CrossRef  Google Scholar 

  14. Jaworska, I., Orkisz, J.: On nonlinear analysis by the multipoint meshless FDM. Eng. Anal. Bound. Elem. 92, 231–243 (2018)

    CrossRef  MathSciNet  Google Scholar 

  15. Milewski, S., Orkisz, J.: In search of optimal acceleration approach to iterative solution methods of simultaneous algebraic equations. Comput Math Appl 68(3), 101–117 (2014)

    CrossRef  MathSciNet  Google Scholar 

  16. Demkowicz, L., Rachowicz, W., Devloo, P.: A fully automatic hp-adaptivity. J. Sci. Comput. 17(1–4), 117–142 (2002)

    CrossRef  MathSciNet  Google Scholar 

  17. Oleksy, M., Cecot, W.: Application of HP-adaptive finite element method to two-scale computation. Arch. Comput. Methods Eng. 22(1), 105–134 (2015)

    CrossRef  MathSciNet  Google Scholar 

  18. Nadai, A.: Der Beginn des Fliesvorganges in einem tordierten Stab, ZAMM3, 442-454 (1923)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cracow University of Technology, Krakow, Poland

    Irena Jaworska

Authors
  1. Irena Jaworska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Irena Jaworska .

Editor information

Editors and Affiliations

  1. Brunel University London, London, UK

    Derek Groen

  2. University of Amsterdam, Amsterdam, The Netherlands

    Clélia de Mulatier

  3. AGH University of Science and Technology, Krakow, Poland

    Prof. Maciej Paszynski

  4. University of Amsterdam, Amsterdam, The Netherlands

    Dr. Valeria V. Krzhizhanovskaya

  5. University of Tennessee at Knoxville, Knoxville, TN, USA

    Prof. Jack J. Dongarra

  6. University of Amsterdam, Amsterdam, The Netherlands

    Prof. Dr. Peter M. A. Sloot

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Jaworska, I. (2022). Multipoint Meshless FD Schemes Applied to Nonlinear and Multiscale Analysis. In: Groen, D., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2022. ICCS 2022. Lecture Notes in Computer Science, vol 13353. Springer, Cham. https://doi.org/10.1007/978-3-031-08760-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-08760-8_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-08759-2

  • Online ISBN: 978-3-031-08760-8

  • eBook Packages: Computer ScienceComputer Science (R0)