DOI QR코드

DOI QR Code

A boundary radial point interpolation method (BRPIM) for 2-D structural analyses

  • Gu, Y.T. (Centre for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore) ;
  • Liu, G.R. (Centre for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore)
  • 투고 : 2002.05.08
  • 심사 : 2003.03.13
  • 발행 : 2003.05.25

초록

In this paper, a boundary-type meshfree method, the boundary radial point interpolation method (BRPIM), is presented for solving boundary value problems of two-dimensional solid mechanics. In the BRPIM, the boundary of a problem domain is represented by a set of properly scattered nodes. A technique is proposed to construct shape functions using radial functions as basis functions. The shape functions so formulated are proven to possess both delta function property and partitions of unity property. Boundary conditions can be easily implemented as in the conventional Boundary Element Method (BEM). The Boundary Integral Equation (BIE) for 2-D elastostatics is discretized using the radial basis point interpolation. Some important parameters on the performance of the BRPIM are investigated thoroughly. Validity and efficiency of the present BRPIM are demonstrated through a number of numerical examples.

키워드

참고문헌

  1. Atluri, S.N. and Zhu, T. (1998). "A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics", Comput. Mech., 22, 117-127. https://doi.org/10.1007/s004660050346
  2. Belytschko, T., Lu, Y.Y. and Gu, L. (1994). "Element-Free Galerkin methods", Int. J. Numer. Methods Engrg., 37, 229-256. https://doi.org/10.1002/nme.1620370205
  3. Belytschko, T. and Organ, D. (1995). "Coupled finite element-element-free Galerkin method", Comput. Mech., 17, 186-195. https://doi.org/10.1007/BF00364080
  4. Brebbia, C.A. (1978). The Boundary Element Method for Engineers. Pentech Press, London, Halstead Press, New York.
  5. Brebbia, C.A., Telles, J.C. and Wrobel, L.C. (1984). Boundary Element Techniques. Springer Verlag, Berlin.
  6. Chati, M.K., Mukherjee, S. and Mukherjee, Y.X. (1999). "The boundary node method for three-dimensional linear elasticity", Int. J. Numer. Methods Engrg., 46, 1163-1184. https://doi.org/10.1002/(SICI)1097-0207(19991120)46:8<1163::AID-NME742>3.0.CO;2-Y
  7. Chati, M.K. and Mukherjee, S. (2000). "The boundary node method for three-dimensional problems in potential theory", Int. J. Numer. Methods Engrg., 47, 1523-1547. https://doi.org/10.1002/(SICI)1097-0207(20000330)47:9<1523::AID-NME836>3.0.CO;2-T
  8. De, S. and Bathe, K.J. (2000). "The method of finite spheres", Comput. Mech., 25, 329-345. https://doi.org/10.1007/s004660050481
  9. Franke, C. and Schaback, R. (1997). "Solving partial differential equations by collocation using radial basis functions", Appl. Math. Comput., 93, 73-82.. https://doi.org/10.1016/S0096-3003(97)10104-7
  10. Gu, Y.T. and Liu, G.R. (2001). "A coupled Element Free Galerkin/Boundary Element method for stress analysis of two-dimension solid", Comput. Methods Appl. Mech. Engrg., 190, 4405-4419. https://doi.org/10.1016/S0045-7825(00)00324-8
  11. Hegen, D. (1996). "Element-free Galerkin methods in combination with finite element approaches", Comput. Methods Appl. Mech. Engrg., 135, 143-166. https://doi.org/10.1016/0045-7825(96)00994-2
  12. Kansa, E.J. (1990). "Multiquadrics-a scattered data approximation scheme with applications to computational fluid dynamics", Computers Math. Applic., 19(8/9), 127-145.
  13. Kothnur, V.S., Mukherjee, S. and Mukherjee, Y.X. (1999). "Two-dimensional linear elasticity by the boundary node method", Int. J. Solids Struct., 36, 1129-1147. https://doi.org/10.1016/S0020-7683(97)00363-6
  14. Liu, G.R. (2002), Mesh Free Methods: Moving Beyond the Finite Element Method. CRC press, Boca Raton, USA.
  15. Liu, G.R. and Gu, Y.T. (2000). "Coupling element free Galerkin and hybrid boundary element methods using modified variational formulation", Comput. Mech., 26, 166-173. https://doi.org/10.1007/s004660000164
  16. Liu, G.R. and Gu, Y.T. (2001a). "A point interpolation method for two-dimensional solid", Int. J. Numer. Methods Engrg., 50, 937-951. https://doi.org/10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X
  17. Liu, G.R. and Gu, Y.T. (2001b). "A local point interpolation method for stress analysis of two-dimensional solids", Struct. Eng. Mech., 11(2), 221-236. https://doi.org/10.12989/sem.2001.11.2.221
  18. Liu, G.R. and Gu, Y.T. (2001c), "A Local Radial Point Interpolation Method (LRPIM) for free vibration analyses of 2-D solids", J. Sound Vib., 246(1), 29-46. https://doi.org/10.1006/jsvi.2000.3626
  19. Liu, G.R., Yan, L., Wang, J.G. and Gu, Y.T. (2002). "Point interpolation method based on local residual formulation using radial basis functions", Struct. Eng. Mech., 14(6), 713-732. https://doi.org/10.12989/sem.2002.14.6.713
  20. Mukherjee, Y.X. and Mukherjee, S. (1997). "Boundary node method for potential problems", Int. J. Numer. Methods Engrg., 40, 797-815. https://doi.org/10.1002/(SICI)1097-0207(19970315)40:5<797::AID-NME89>3.0.CO;2-#
  21. Nayroles, B., Touzot, G. and Villon, P. (1992). "Generalizing the finite element method: diffuse approximation and diffuse elements", Comput. Mech., 10, 307-318. https://doi.org/10.1007/BF00364252
  22. Roark, R.J. and Young, W.C. (1975). Formulas for Stress and Strain. McGraw-hill, London.
  23. Sharan, M., Kansa, E.J. and Gupta, S. (1997). "Application of the multiquadric method for numerical solution of elliptic partial differential equations", Applied Mathematics and Computation, 84, 275-302. https://doi.org/10.1016/S0096-3003(96)00109-9
  24. Wang, J.G. and Liu, G.R. (2002), "On the optimal shape parameters of radial basis functions used for 2-D meshlesss methods", Comput. Methods Appl. Mech. Eng., 191, 2611-2630. https://doi.org/10.1016/S0045-7825(01)00419-4
  25. Timoshenko, S.P. and Goodier, J.N. (1970). Theory of Elasticity. 3rd Edition. McGraw-hill, New York.

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  1. Meshfree weak-strong (MWS) form method and its application to incompressible flow problems vol.46, pp.10, 2004, https://doi.org/10.1002/fld.785
  2. An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields vol.78, pp.1, 2011, https://doi.org/10.1016/j.engfracmech.2010.10.014
  3. THE COMPLEX VARIABLE ELEMENT-FREE GALERKIN (CVEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS vol.01, pp.02, 2009, https://doi.org/10.1142/S1758825109000162
  4. Smoothed Point Interpolation Method for Elastoplastic Analysis vol.12, pp.04, 2015, https://doi.org/10.1142/S0219876215400137
  5. A scaled boundary radial point interpolation method for 2-D elasticity problems vol.112, pp.7, 2017, https://doi.org/10.1002/nme.5534
  6. Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation vol.34, pp.4, 2010, https://doi.org/10.1016/j.enganabound.2009.10.010
  7. Inverse Cauchy problem of annulus domains in the framework of spectral meshless radial point interpolation vol.33, pp.3, 2017, https://doi.org/10.1007/s00366-016-0482-x
  8. Meshless method of dual reciprocity hybrid radial boundary node method for elasticity vol.23, pp.5, 2010, https://doi.org/10.1016/S0894-9166(10)60047-X
  9. Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem vol.7, pp.3, 2016, https://doi.org/10.1016/j.asej.2015.07.009
  10. A pseudo-elastic local meshless method for analysis of material nonlinear problems in solids vol.31, pp.9, 2007, https://doi.org/10.1016/j.enganabound.2006.12.008
  11. Virtual boundary meshless least square collocation method for calculation of 2D multi-domain elastic problems vol.36, pp.5, 2012, https://doi.org/10.1016/j.enganabound.2011.12.008
  12. Advanced Implicit Meshless Approaches for the Rayleigh–Stokes Problem for a Heated Generalized Second Grade Fluid with Fractional Derivative 2017, https://doi.org/10.1142/S0219876218500329
  13. Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and multiquadrics discretizations vol.29, pp.12, 2005, https://doi.org/10.1016/j.enganabound.2005.07.004
  14. The Interpolating Element-Free Galerkin Method for 2D Transient Heat Conduction Problems vol.2014, 2014, https://doi.org/10.1155/2014/712834
  15. An extended boundary node method for modeling normal derivative discontinuities in potential theory across edges and corners vol.28, pp.9, 2004, https://doi.org/10.1016/j.enganabound.2004.01.007
  16. THE MESHLESS GALERKIN BOUNDARY NODE METHOD FOR TWO-DIMENSIONAL SOLIDS vol.10, pp.04, 2013, https://doi.org/10.1142/S0219876213500138
  17. MESHFREE METHODS AND THEIR COMPARISONS vol.02, pp.04, 2005, https://doi.org/10.1142/S0219876205000673
  18. Development of a meshless Galerkin boundary node method for viscous fluid flows vol.82, pp.2, 2011, https://doi.org/10.1016/j.matcom.2011.07.004
  19. Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter vol.86, pp.4, 2008, https://doi.org/10.1016/j.compstruct.2008.07.025
  20. Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation vol.130, pp.2, 2015, https://doi.org/10.1140/epjp/i2015-15033-5
  21. A hybrid radial boundary node method based on radial basis point interpolation vol.33, pp.11, 2009, https://doi.org/10.1016/j.enganabound.2009.06.003
  22. Error analysis for moving least squares approximation in 2D space vol.238, 2014, https://doi.org/10.1016/j.amc.2014.04.037
  23. A new spectral meshless radial point interpolation (SMRPI) method: A well-behaved alternative to the meshless weak forms vol.54, 2015, https://doi.org/10.1016/j.enganabound.2015.01.004
  24. Local Heaviside-weighted LRPIM meshless method and its application to two-dimensional potential flows vol.59, pp.5, 2009, https://doi.org/10.1002/fld.1810
  25. Solving high order ordinary differential equations with radial basis function networks vol.62, pp.6, 2005, https://doi.org/10.1002/nme.1220
  26. Virtual boundary meshless least square integral method with moving least squares approximation for 2D elastic problem vol.37, pp.3, 2013, https://doi.org/10.1016/j.enganabound.2013.01.006
  27. A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems vol.35, pp.2, 2011, https://doi.org/10.1016/j.apm.2010.07.030
  28. Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics vol.37, pp.12, 2013, https://doi.org/10.1016/j.enganabound.2013.10.002
  29. Pricing European and American options by radial basis point interpolation vol.251, 2015, https://doi.org/10.1016/j.amc.2014.11.016
  30. THE INTERPOLATING ELEMENT-FREE GALERKIN (IEFG) METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEMS vol.03, pp.04, 2011, https://doi.org/10.1142/S1758825111001214
  31. Adaptive meshless Galerkin boundary node methods for hypersingular integral equations vol.36, pp.10, 2012, https://doi.org/10.1016/j.apm.2011.12.033
  32. Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation vol.39, pp.7, 2016, https://doi.org/10.1002/mma.3604
  33. Complex variable boundary element-free method for two-dimensional elastodynamic problems vol.198, pp.49-52, 2009, https://doi.org/10.1016/j.cma.2009.08.020
  34. Local integration of 2-D fractional telegraph equation via moving least squares approximation vol.56, 2015, https://doi.org/10.1016/j.enganabound.2015.02.012
  35. A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation vol.340, 2017, https://doi.org/10.1016/j.jcp.2017.03.061
  36. Meshless local Petrov–Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation vol.50, 2015, https://doi.org/10.1016/j.enganabound.2014.08.014
  37. A boundary element-free method (BEFM) for three-dimensional elasticity problems vol.36, pp.1, 2005, https://doi.org/10.1007/s00466-004-0638-1
  38. Analysis of Composite Plates Using a Layerwise Theory and Multiquadrics Discretization vol.12, pp.2, 2005, https://doi.org/10.1080/15376490490493952
  39. Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform vol.64, pp.12, 2005, https://doi.org/10.1002/nme.1417
  40. Meshless boundary node methods for Stokes problems vol.39, pp.7, 2015, https://doi.org/10.1016/j.apm.2014.10.009
  41. Moving Least Squares (MLS) Method for the Nonlinear Hyperbolic Telegraph Equation with Variable Coefficients vol.14, pp.03, 2017, https://doi.org/10.1142/S0219876217500268
  42. The Interpolating Boundary Element-Free Method for Unilateral Problems Arising in Variational Inequalities vol.2014, 2014, https://doi.org/10.1155/2014/518727
  43. Meshless simulation of equilibrium swelling/deswelling of pH-sensitive hydrogels vol.44, pp.2, 2006, https://doi.org/10.1002/polb.20698
  44. MODIFIED CHOLESKY DECOMPOSITION FOR SOLVING THE MOMENT MATRIX IN THE RADIAL POINT INTERPOLATION METHOD vol.11, pp.06, 2014, https://doi.org/10.1142/S0219876213500886
  45. Analysis of the Time Fractional 2-D Diffusion-Wave Equation via Moving Least Square (MLS) Approximation vol.3, pp.3, 2017, https://doi.org/10.1007/s40819-016-0247-7
  46. A Kriging interpolation-based boundary face method for 3D potential problems vol.37, pp.5, 2013, https://doi.org/10.1016/j.enganabound.2013.02.006
  47. Simulation of an extruded quadrupolar dielectrophoretic trap using meshfree approach vol.30, pp.11, 2006, https://doi.org/10.1016/j.enganabound.2006.03.014
  48. The complex variable reproducing kernel particle method for two-dimensional elastodynamics vol.19, pp.9, 2010, https://doi.org/10.1088/1674-1056/19/9/090204
  49. Time-dependent fractional advection-diffusion equations by an implicit MLS meshless method vol.88, pp.13, 2011, https://doi.org/10.1002/nme.3223
  50. Meshless local radial point interpolation (MLRPI) on the telegraph equation with purely integral conditions vol.129, pp.11, 2014, https://doi.org/10.1140/epjp/i2014-14241-9
  51. The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem vol.33, pp.3, 2017, https://doi.org/10.1007/s00366-016-0489-3
  52. An Enriched Radial Point Interpolation Method Based on Weak-Form and Strong-Form vol.18, pp.8, 2011, https://doi.org/10.1080/15376494.2011.621832
  53. Application of meshfree methods for solving the inverse one-dimensional Stefan problem vol.40, 2014, https://doi.org/10.1016/j.enganabound.2013.10.013
  54. AN INTERPOLATING BOUNDARY ELEMENT-FREE METHOD WITH NONSINGULAR WEIGHT FUNCTION FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS vol.10, pp.06, 2013, https://doi.org/10.1142/S0219876213500436
  55. Inverse analysis of heat transfer across a multilayer composite wall with Cauchy boundary conditions vol.79, 2014, https://doi.org/10.1016/j.ijheatmasstransfer.2014.08.041
  56. Meshless analysis and applications of a symmetric improved Galerkin boundary node method using the improved moving least-square approximation vol.40, pp.4, 2016, https://doi.org/10.1016/j.apm.2015.09.080
  57. Local integration of population dynamics via moving least squares approximation vol.32, pp.2, 2016, https://doi.org/10.1007/s00366-015-0424-z
  58. Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM) vol.181, pp.4, 2010, https://doi.org/10.1016/j.cpc.2009.12.010
  59. The meshless virtual boundary method and its applications to 2D elasticity problems vol.20, pp.1, 2007, https://doi.org/10.1007/s10338-007-0704-2
  60. A new hybrid boundary node method based on Taylor expansion and the Shepard interpolation method vol.102, pp.8, 2015, https://doi.org/10.1002/nme.4861
  61. An implicit RBF meshless approach for time fractional diffusion equations vol.48, pp.1, 2011, https://doi.org/10.1007/s00466-011-0573-x
  62. Stability and convergence of spectral radial point interpolation method locally applied on two-dimensional pseudoparabolic equation vol.33, pp.3, 2017, https://doi.org/10.1002/num.22119
  63. Analysis of meshless local and spectral meshless radial point interpolation (MLRPI and SMRPI) on 3-D nonlinear wave equations vol.89, 2014, https://doi.org/10.1016/j.oceaneng.2014.08.007
  64. On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three-dimensional wave equations vol.105, pp.2, 2016, https://doi.org/10.1002/nme.4960
  65. AN ADVANCED MESHLESS METHOD FOR TIME FRACTIONAL DIFFUSION EQUATION vol.08, pp.04, 2011, https://doi.org/10.1142/S0219876211002745
  66. An improved pseudospectral meshless radial point interpolation (PSMRPI) method for 3D wave equation with variable coefficients pp.1435-5663, 2018, https://doi.org/10.1007/s00366-018-0656-9
  67. A novel hybrid meshless method for seepage flows in non-homogeneous and anisotropic soils vol.35, pp.2, 2018, https://doi.org/10.1108/EC-07-2017-0245
  68. Pseudospectral Meshless Radial Point Hermit Interpolation Versus Pseudospectral Meshless Radial Point Interpolation vol.17, pp.7, 2003, https://doi.org/10.1142/s0219876219500233