Skip to main content
SpringerLink
Log in

Computational efficiency of meshfree methods with local-coordinates algorithm

  • Published:
International Journal of Precision Engineering and Manufacturing Aims and scope Submit manuscript

Abstract

In this study, meshfree methods with uniform nodal distribution and local-coordinates shape functions are investigated. The proposed meshfree method can be used with various shape functions and is tested on a test patch with a Laplace governing equation and both essential and Neumann boundary conditions. It is shown to reduce the computational time dependency on the number of nodes by 36%. This reduction of dependency is significant as it reduces the computational time by several orders for analysis of problems requiring very fine nodal distribution. The meshfree method with uniform nodal distribution and local coordinates shape function reduces and converges to the perfectly uniform nodal distribution with finer nodal distributions. The technique is illustrated on a rectangular Kirchhoff-Love plate to show the practical use of the technique for allowing higher order shape function and finer nodal distribution to be used with multiple overlapping boundary conditions as well as on an electromagnetic problem to explain the uses of this technique on solving multiple similar problem cases with slight changes in geometry of boundary nodes. Overall, the present methodology provides a simple way to increase the computational efficiency of meshfree methods in range of an order while retaining many of its benefits.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

φ(x):

shape function

W(x):

weight function

n:

number of supporting nodes

d:

nodal value

Ω:

domain

Γ:

boundary

References

  1. Monaghan, J. J., “An Introduction to SPH,” Computer Physics Communications, Vol. 48, No. 1, pp. 89–96, 1988.

    Article  MATH  Google Scholar 

  2. Nayroles, B., Touzot, G., and Villon, P., “Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements,” Computational Mechanics, Vol. 10, No. 5, pp. 307–318, 1992.

    Article  MATH  Google Scholar 

  3. Belytschko, T., Lu, Y. Y., and Gu, L., “ElementFree Galerkin Methods,” International Journal for Numerical Methods in Engineering, Vol. 37, No. 2, pp. 229–256, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  4. Melenk, J. M. and Babuška, I., “The Partition of Unity Finite Element Method: Basic Theory and Applications,” Computer Methods in Applied Mechanics and Engineering, Vol. 139, No. 1, pp. 289–314, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  5. Babuska, I. and Melenk, J. M., “The Partition of Unity Method,” International Journal of Numerical Methods in Engineering, Vol. 40, No. 4, pp. 727–758, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  6. Sibson, R., “A Vector Identity for the Dirichlet Tessellation,” Proc. of the Cambridge Philosophical Society, Vol. 87, No. 1, pp. 151–155, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  7. Balachandran, G., Rajagopal, A., and Sivakumar, S., “Mesh Free Galerkin Method based on Natural Neighbors and Conformal Mapping,” Computational Mechanics, Vol. 42, No. 6, pp. 885–905, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu, G. R. and Gu, Y. T., “An Introduction to Meshfree Methods and their Programming,” Springer Science & Business Media, pp. 13, 54–144, 2005.

    Google Scholar 

  9. Liu, G. R., “Meshfree Methods: Moving Beyond the Finite Element Method,” CRC Press, pp. 14–18, 37–40, 42–49, 60–68, 2009.

    Book  Google Scholar 

  10. Strouboulis, T., Copps, K., and Babuška, I., “The Generalized Finite Element Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 32, pp. 4081–4193, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  11. Belytschko, T., Moës, N., Usui, S., and Parimi, C., “Arbitrary Discontinuities in Finite Elements,” International Journal for Numerical Methods in Engineering, Vol. 50, No. 4, pp. 993–1013, 2001.

    Article  MATH  Google Scholar 

  12. Duarte, C. A. and Oden, J. T., “Hp Clouds-an Hp Meshless Method,” Numerical Methods for Partial Differential Equations, Vol. 12, No. 6, pp. 673–706, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  13. Aluru, N. R., and Li, G., “Finite Cloud Method: A True Meshless Technique based on a Fixed Reproducing Kernel Approximation,” International Journal for Numerical Methods in Engineering, Vol. 50, No. 10, pp. 2373–2410, 2001.

    Article  MATH  Google Scholar 

  14. Fries, T. P. and Matthies, H. G., “Classification and Overview of Meshfree Methods,” Department of Mathematics and Computer Science, Technical University of Braunschweig, 2004.

    Google Scholar 

  15. Kaufmann, T., Yu, Y., Engström, C., Chen, Z., and Fumeaux, C., “Recent Developments of the Meshless Radial Point Interpolation Method for TimeDomain Electromagnetics,” International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, Vol. 25, No. 5–6, pp. 468–489, 2012.

    Article  Google Scholar 

  16. Li, S. and Liu, W. K., “Moving Least-Square Reproducing Kernel Method Part II: Fourier Analysis,” Computer Methods in Applied Mechanics and Engineering, Vol. 139, No. 1–4, pp. 159–193, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  17. Angulo, A., Pozo, L. P., and Perazzo, F., “A Posteriori Error Estimator and an Adaptive Technique in Meshless Finite Points Method,” Engineering Analysis with Boundary Elements, Vol. 33, No. 11, pp. 1322–1338, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  18. Wang, H. P. and Wang, D., “Efficient Meshfree Computation with Fast Treatment of Essential Boundary Conditions for Industrial Applications,” Journal of Engineering Mechanics, Vol. 135, No. 10, pp. 1147–1154, 2009.

    Article  Google Scholar 

  19. Nakata, S., “Parallel Meshfree Computation for Parabolic Equations on Graphics Hardware,” International Journal of Computer Mathematics, Vol. 88, No. 9, pp. 1909–1919, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  20. Barbieri, E. and Meo, M., “A Fast Object-Oriented Matlab Implementation of the Reproducing Kernel Particle Method,” Computational Mechanics, Vol. 49, No. 5, pp. 581–602, 2012.

    Article  MATH  MathSciNet  Google Scholar 

  21. Cavoretto, R. and De Rossi, A., “A meshless Interpolation Algorithm using a Cell-based Searching Procedure,” Computers & Mathematics with Applications, Vol. 67, No. 5, pp. 1024–1038, 2014.

    Article  MathSciNet  Google Scholar 

  22. De Vuyst, T., Vignjevic, R., and Campbell, J., “Coupling between Meshless and Finite Element Methods,” International Journal of Impact Engineering, Vol. 31, No. 8, pp. 1054–1064, 2005.

    Article  Google Scholar 

  23. Belytschko, T., Krongauz, Y., Fleming, M., Organ, D., and Liu, W. K. S., “Smoothing and Accelerated Computations in the Element Free Galerkin Method,” Journal of Computational and Applied Mathematics, Vol. 74, No. 1, pp. 111–126, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  24. Breitkopf, P., Rassineux, A., Touzot, G., and Villon, P., “Explicit Form and Efficient Computation of MLS Shape Functions and their Derivatives,” International Journal for Numerical Methods in Engineering, Vol. 48, No. 3, pp. 451–466, 2000.

    Article  MATH  Google Scholar 

  25. Li, G., Sidibe, K., and Liu, G., “Meshfree Method for 3D Bulk Forming Analysis with Lower Order Integration Scheme,” Engineering Analysis with Boundary Elements, Vol. 28, No. 10, pp. 1283–1292, 2004.

    Article  MATH  Google Scholar 

  26. Rabczuk, T. and Belytschko, T., “Adaptivity for Structured Meshfree Particle Methods in 2D and 3D,” International Journal for Numerical Methods in Engineering, Vol. 63, No. 11, pp. 1559–1582, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  27. Rabczuk, T. and Belytschko, T., “A Three-Dimensional Large Deformation Meshfree Method for Arbitrary Evolving Cracks,” Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 29, pp. 2777–2799, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  28. Liu, W. K. and Chen, Y., “Wavelet and Multiple Scale Reproducing Kernel Methods,” International Journal for Numerical Methods in Fluids, Vol. 21, No. 10, pp. 901–931, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu, W. K., Jun, S., and Zhang, Y. F., “Reproducing Kernel Particle Methods,” International Journal for Numerical Methods in Fluids, Vol. 20, No. 89, pp. 1081–1106, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu, W. K., Chen, Y., Uras, R. A., and Chang, C. T., “Generalized Multiple Scale Reproducing Kernel Particle Methods,” Computer Methods in Applied Mechanics and Engineering, Vol. 139, No. 1, pp. 91–157, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  31. Lancaster, P. and Salkauskas, K., “Surfaces Generated by Moving Least Squares Methods,” Mathematics of Computation, Vol. 37, No. 155, pp. 141–158, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  32. Onate, E., Idelsohn, S., Zienkiewicz, O. C., and Taylor, R., “A Finite Point Method in Computational Mechanics. Applications to Convective Transport and fluid flow,” International Journal for Numerical Methods in Engineering, Vol. 39, No. 22, pp. 3839–3866, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  33. Lima, N. Z., Mesquita, R. C., Facco, W. G., Moura, A. S., and Silva, E. J., “The Nonconforming Point Interpolation Method Applied to Electromagnetic Problems,” IEEE Transactions on Magnetics, Vol. 48, No. 2, pp. 619–622, 2012.

    Article  Google Scholar 

  34. Liu, G. R., Huynh, D. B. P., and Gu, Y. T., “An Adaptive Meshfree Collocation Method for Static and Dynamic Nonlinear Problems,” Computational Methods, pp. 1459–1464, 2006.

    Chapter  Google Scholar 

  35. Ling, L., “An AdaptiveHybrid Meshfree Approximation Method,” International Journal for Numerical Methods in Engineering, Vol. 89, No. 5, pp. 637–657, 2012.

    Article  MATH  MathSciNet  Google Scholar 

  36. Lee, S. C., Kim, C. K., Song, H. E., and Kim, Y. S., “Finite Element Analysis of Crystalline Silicon Solar Cell in Screen Printing Process by using Taguchi Method,” Int. J. Precis. Eng. Manuf., Vol. 14, No. 4, pp. 635–642, 2013.

    Article  Google Scholar 

  37. Choi, Y. J., Park, K. H., Hong, Y. H., Kim, K. T., Lee, S. W., and Choi, H. Z., “Effect of Ultrasonic Vibration in Grinding; Horn Design and Experiment,” Int. J. Precis. Eng. Manuf., Vol. 14, No. 11, pp. 1873–1879, 2013.

    Article  Google Scholar 

  38. Kim, N., Kim, H., and Jin, K., “Minimizing The Axial Force and The Material Build-up in the Tube Flow Forming Process,” Int. J. Precis. Eng. Manuf., Vol. 14, No. 2, pp. 259–266, 2013.

    Article  Google Scholar 

  39. Kim, H., Lee, Y., and Lim, E., “A Quartz-Bar Megasonic System for Nano-Pattern Cleaning,” Int. J. Precis. Eng. Manuf. Vol. 14, No. 10, pp. 1713–1718, 2013.

    Article  Google Scholar 

  40. Reddy, J. N., “Theory and Analysis of Elastic Plates and Shells,” Taylor and Francis, 2nd Ed., 1945.

    Google Scholar 

  41. Ng, T. Y., Li, H., Cheng, J. Q., Lam, K. Y., and Yew, Y. K., “A Novel True Meshless Numerical Technique (hM-DOR Method) for the Deformation Control of Circular Plates Integrated with Piezoelectric Sensors/Actuators,” Smart Materials and Structures, Vol. 12, No. 6, pp. 955–961, 2003.

    Article  Google Scholar 

  42. Yang, E., Wang, X., Mo, J., Cao, J., and Wang, S., “Model of Streamline Upwinding and Meshless Method for Electromagnetic Field Involving Moving Conductor,” International Journal of Applied Electromagnetics and Mechanics, Vol. 31, No. 3, pp. 147–160, 2009.

    Google Scholar 

  43. Paranjape, A. A., Guan, J., Chung, S.-J., and Krstic, M., “Pde Boundary Control for Flexible Articulated Wings on a Robotic Aircraft,” IEEE Transactions on Robotics, Vol. 29, No. 3, pp. 625–640, 2013.

    Article  Google Scholar 

  44. García-Martín, J., Gómez-Gil, J., and Vázquez-Sánchez, E., “Non-Destructive Techniques based on Eddy Current Testing,” Sensors, Vol. 11, No. 3, pp. 2525–2565, 2011.

    Article  Google Scholar 

  45. Cacciola, M., Calcagno, S., Megali, G., Pellicano, D., Versaci, M., and Morabito, F. C., “Eddy Current Modeling in Composite Materials,” Progress in Electromagnetics Research Symposium, Vol. 5, No. 6, pp. 591–595, 2009.

    Google Scholar 

  46. Agarwal, P. K. and Sharir, M., “Efficient Algorithms for Geometric Optimization,” ACM Computing Surveys (CSUR), Vol. 30, No. 4, pp. 412–458, 1998.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Mechanics, Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore, Singapore

    Wei Xuan Chan & Yong-Jin Yoon

  2. Energy Research Institute @ NTU (ERI@N), 1 CleanTech Loop, 06-04 CleanTech One, 637141, Singapore, Singapore

    Wei Xuan Chan

  3. School of Mechanical and Nuclear Engineering, Ulsan National Institute of Science and Technology, UNIST-gil 50, Ulsan, 689-798, South Korea

    HungSun Son

Authors
  1. Wei Xuan Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. HungSun Son
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong-Jin Yoon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong-Jin Yoon.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chan, W.X., Son, H. & Yoon, YJ. Computational efficiency of meshfree methods with local-coordinates algorithm. Int. J. Precis. Eng. Manuf. 16, 547–556 (2015). https://doi.org/10.1007/s12541-015-0074-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12541-015-0074-5

Keywords

Search

Navigation