An Improved Finite Cloud Method With Uniformly Distributed Clouds and Enhanced Boundary Conditions

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Miew Leng Oh, See Pheng Hang

Abstract

Finite cloud method (FCM) employs the fixed kernel reproducing technique to construct the interpolation function and point collocation approach is adopted for the discretization. In this study, an improved FCM is proposed such that a node of interest is approximated with its nearest cloud. This feature enables a set of uniformly distributed clouds of various densities such that all the information in the problem domain is captured and stored in the clouds. Additionally, the instability of FCM near the boundaries is treated by having the boundary nodes also satisfy the governing differential equation. Besides, a splitting mechanism is suggested for the node refinement to improve the accuracy of solution. Parameters are introduced to control the density of clouds and the singularity of the moment matrices associated with the clouds. Thus, a more controllable numerical simulation is developed. Numerical examples are presented and the results have shown that the improved FCM produces a stable and better accuracy of solution.

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References

  1. J.S. Chen, M. Hillman, S.W. Chi, Meshfree Methods: Progress Made After 20 Years, J. Eng. Mech. 143 (2017), 04017001. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001176.
  2. S.D. Daxini, J.M. Prajapati, A Review on Recent Contribution of Meshfree Methods to Structure and Fracture Mechanics Applications, Sci. World J. 2014 (2014), 247172. https://doi.org/10.1155/2014/247172.
  3. S. Garg, M. Pant, Meshfree Methods: A Comprehensive Review of Applications, Int. J. Comput. Methods. 15 (2018), 1830001. https://doi.org/10.1142/S0219876218300015.
  4. Q. Duan, X. Gao, B. Wang, X. Li, H. Zhang, T. Belytschko, Y. Shao, Consistent Element‐Free Galerkin Method, Int. J. Numer. Methods Eng. 99 (2014), 79-101. https://doi.org/10.1002/nme.4661.
  5. Q. Duan, T. Belytschko, Gradient and Dilatational Stabilizations for Stress‐Point Integration in the Element‐Free Galerkin Method, Int. J. Numer. Methods Eng. 77 (2009), 776-798. https://doi.org/10.1002/nme.2432.
  6. M. Najafi, V. Enjilela, Natural Convection Heat Transfer at High Rayleigh Numbers–Extended Meshless Local Petrov–Galerkin (MLPG) Primitive Variable Method, Eng. Anal. Bound. Elements. 44 (2014), 170-184. https://doi.org/10.1016/j.enganabound.2014.01.022.
  7. R. Singh, K.M. Singh, Interpolating Meshless Local Petrov-Galerkin Method for Steady State Heat Conduction Problem, Eng. Anal. Bound. Elements. 101 (2019), 56-66. https://doi.org/10.1016/j.enganabound.2018.12.012.
  8. R. Divya, V. Sriram, K. Murali, Wave-Vegetation Interaction Using Improved Meshless Local Petrov Galerkin Method, Appl. Ocean Res. 101 (2020), 102116. https://doi.org/10.1016/j.apor.2020.102116.
  9. S.S. Mulay, H. Li, S. See, On the Development of Adaptive Random Differential Quadrature Method with an Error Recovery Technique and its Application in the Locally High Gradient Problems, Comput. Mech. 45 (2010), 467-493. https://doi.org/10.1007/s00466-010-0468-2.
  10. C.W. Bert, M. Malik, Differential Quadrature Method in Computational Mechanics: A Review, Appl. Mech. Rev. 49 (1996), 1–28. https://doi.org/10.1115/1.3101882.
  11. X. Liang, T. Wang, D. Huang, Z. Liu, R. Zhu, C. Wang, An Improved RBF Based Differential Quadrature Method, Eng. Anal. Bound. Elements. 135 (2022), 299–314. https://doi.org/10.1016/j.enganabound.2021.11.023.
  12. T. Liu, J. Yu Ding, X. Yu Xu, Differential Quadrature Method for Partial Differential Dynamic Equations of Beam–Ring Structure, AIAA J. 60 (2022), 2542-2554. https://doi.org/10.2514/1.J061113.
  13. F.X. Sun, J.F. Wang, Y.M. Cheng, An Improved Interpolating Element-Free Galerkin Method for Elasticity, Chin. Phys. B. 22 (2013), 43–50. https://doi.org/10.1088/1674-1056/22/12/120203.
  14. S. Wu, Y. Xiang, B. Liu, G. Li, A Weak-Form Interpolation Meshfree Method for Computing Underwater Acoustic Radiation, Ocean Eng. 233 (2021), 109105. https://doi.org/10.1016/j.oceaneng.2021.109105.
  15. S. R. Idelsohn, E. Onate, N. Calvo, F. Del Pin, The Meshless Finite Element Method, Int. J. Numer. Methods Eng. 58 (2003), 893–912. https://doi.org/10.1002/nme.798.
  16. Y. Chai, C. Cheng, W. Li, Y. Huang, A Hybrid Finite Element-Meshfree Method Based on Partition of Unity for Transient Wave Propagation Problems in Homogeneous and Inhomogeneous Media, Appl. Math. Model. 85 (2020), 192–209. https://doi.org/10.1016/j.apm.2020.03.026.
  17. H. Li, Q. Zhang, A Meshfree Finite Volume Method With Optimal Numerical Integration and Direct Imposition of Essential Boundary Conditions, Appl. Numer. Math. 153 (2020), 98-113. https://doi.org/10.1016/j.apnum.2020.02.005.
  18. I. Jaworska and S. Milewski, On Two-Scale Analysis of Heterogeneous Materials by Means of the Meshless Finite Difference Method, Int. J. Multiscale Comput. Eng. 14 (2016), 113–134. https://doi.org/10.1615/IntJMultCompEng.2016014435.
  19. Y. Jiang, Algebraic-Volume Meshfree Method for Application in Finite Volume Solver, Comput. Methods Appl. Mech. Eng. 355 (2019), 44–66. https://doi.org/10.1016/j.cma.2019.05.048.
  20. Y. Jiang, General Mesh Method: A Unified Numerical Scheme, Comput. Methods Appl. Mech. Eng. 369 (2020), 1–28. https://doi.org/10.1016/j.cma.2020.113049.
  21. Y. Gu, L.C. Zhang, Coupling of the Meshfree and Finite Element Methods for Determination of the Crack Tip Fields, Eng. Fracture Mech. 75 (2008), 986-1004. https://doi.org/10.1016/j.engfracmech.2007.05.003.
  22. M. L. Oh and S. H. Yeak, A Hybrid Multiscale Finite Cloud Method and Finite Volume Method in Solving High Gradient Problem, Int. J. Comput. Methods, 19 (2022), 2250002. https://doi.org/10.1142/S0219876222500025.
  23. D.R. Burke, T.J. Smy, Optical Mode Solving for Complex Waveguides Using a Finite Cloud Method, Optics Express. 20 (2012), 17783-17796. https://doi.org/10.1364/OE.20.017783.
  24. D.R. Burke, T.J. Smy, Thermal Models for Optical Circuit Simulation Using a Finite Cloud Method and Model Reduction Techniques, IEEE Trans. Computer-Aided Design Integrated Circuits Syst. 32 (2013), 1177-1186. https://doi.org/10.1109/TCAD.2013.2253835.
  25. D.R. Burke, A Meshless Approach to Solving Partial Differential Equations Using the Finite Cloud Method for the Purposes of Computer Aided Design, Doctoral Dissertation, Carleton University, 2013.
  26. S.K. De, N.R. Aluru, A Chemo-Electro-Mechanical Mathematical Model for Simulation of pH Sensitive Hydrogels, Mech. Mater. 36 (2004), 395–410. https://doi.org/10.1016/S0167-6636(03)00067-X.
  27. X.Z. Jin, G. Li, N.R. Aluru, On the Equivalence between Least-Squares and Kernel Approximations in Meshless Methods, Computer Model. Eng. Sci. 2 (2001), 447-462.
  28. X.Z. Jin, G. Li, N.R. Aluru, Positivity Conditions in Meshless Collocation Methods, Computer Methods Appl. Mech. Eng. 193 (2004), 1171-1202. https://doi.org/10.1016/j.cma.2003.12.013.
  29. X.Z. Jin, G. Li, N.R. Aluru, New Approximations and Collocation Schemes in the Finite Cloud Method, Comput. Struct. 83 (2005), 1366-85. https://doi.org/10.1016/j.compstruc.2004.08.030.
  30. W.X. Chan, H. Son, Y.J. Yoon, Computational Efficiency of Meshfree Methods With Local-Coordinates Algorithm, Int. J. Precis. Eng. Manuf. 16 (2015), 547–556. https://doi.org/10.1007/s12541-015-0074-5.
  31. G.R. Liu, Y.T. Gu, A Point Interpolation Method, In: Proceedings of the Fourth Asia-Pacific Conference on Computational Mechanics, Singapore. (1999), 1009-1014.
  32. N.R. Aluru, G. Li, Finite Cloud Method: A True Meshless Technique Based on a Fixed Reproducing Kernel Approximation, Int. J. Numer. Meth. Eng. 50 (2001), 2373-2410. https://doi.org/10.1002/nme.124.