On the random differential quadrature (RDQ) method: Consistency analysis and application in elasticity problems. (English) Zbl 1236.74304
Summary: The differential quadrature (DQ) method is an efficient derivative approximation technique, but it requires a regular domain with uniformly arranged nodes. This restricts its application in the case of a regular domain discretized by field nodes in a fixed pattern only. In the presented random differential quadrature (RDQ) method, however, this restriction of the DQ method is removed, and its applicability is extended to the cases of a regular domain discretized by randomly distributed field nodes and an irregular domain discretized by uniform or randomly distributed field nodes. A consistency analysis of the locally applied DQ method is carried out. Based on it, approaches are suggested to obtain the fast convergence of the function value by the RDQ method. The convergence studies are carried out by solving 1D and 2D elasticity problems, and it is concluded that the RDQ method can effectively handle regular as well as irregular domains discretized by random or uniformly distributed field nodes.
MSC:
| 74S60 | Stochastic and other probabilistic methods applied to problems in solid mechanics |
| 74B05 | Classical linear elasticity |
References:
| [1] | Lucy LB (1977) A numerical approach to testing the fission hypothesis. Astronom J 8(12): 1013–1024 · doi:10.1086/112164 |
| [2] | Gingold RA, Monaghan JJ et al (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Notices Roy Astronom Soc 181: 375–389 · Zbl 0421.76032 |
| [3] | Nayroles B, Touzot G, Villon P et al (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10: 307–318 · Zbl 0764.65068 · doi:10.1007/BF00364252 |
| [4] | Belytschko T, Gu L, Lu YY et al (1994) Fracture and crack growth by element free Galerkin method. Modell Simul Mater Sci Eng 2: 519–534 · doi:10.1088/0965-0393/2/3A/007 |
| [5] | Lu YY, Belytschko T, Gu L et al (1994) A new implementation of the element free Galerkin method. Comp Methods Appl Mech Eng 113: 397–414 · Zbl 0847.73064 · doi:10.1016/0045-7825(94)90056-6 |
| [6] | Belytschko T, Lu YY, Gu L et al (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37: 229–256 · Zbl 0796.73077 · doi:10.1002/nme.1620370205 |
| [7] | Braun J, Sambridge M et al (1995) A numerical method for solving partial differential equations on highly irregular evolving grids. Nature 376: 655–660 · doi:10.1038/376655a0 |
| [8] | Liu WK, Jun S et al (1998) Multiple scale reproducing kernel particle methods for large deformation problems. Int J Numer Methods Eng 41: 1339–1362 · Zbl 0916.73060 · doi:10.1002/(SICI)1097-0207(19980415)41:7<1339::AID-NME343>3.0.CO;2-9 |
| [9] | Liu WK, Chen Y, Uras RA, Chang CT et al (1996) Generalized multiple scale reproducing kernel particle method. Comp Methods Appl Mech Eng 139: 91–157 · Zbl 0896.76069 · doi:10.1016/S0045-7825(96)01081-X |
| [10] | Liu WK, Jun S, Zhang YF et al (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20: 1081–1106 · Zbl 0881.76072 · doi:10.1002/fld.1650200824 |
| [11] | Melenk JM, babushka I et al (1996) The partition of unity finite element method: basic theory and applications. Comp Methods Appl Mech Eng 139: 289–314 · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0 |
| [12] | Zhu T, Zhang JD, Atluri SN et al (1998) A local boundary integral equation (LBIE) method in computational mechanics and a meshless discretization approach. Comput Mech 21: 223–235 · Zbl 0920.76054 · doi:10.1007/s004660050297 |
| [13] | Atluri SN, Sladek J, Sladek V, Zhu T et al (2000) The local boundary integral equation (LBIE) and its meshless implementation for linear elasticity. Comput Mech 25: 180–189 · Zbl 0976.74078 · doi:10.1007/s004660050467 |
| [14] | Atluri SN, Zhu T et al (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22: 117–127 · Zbl 0932.76067 · doi:10.1007/s004660050346 |
| [15] | Li H, Wang QX, Lam KY et al (2004) Development of a novel meshless Local Kriging (LoKriging) method for structural dynamics analysis. Comput Methods Appl Mech Eng 193: 2599–2619 · Zbl 1067.74598 · doi:10.1016/j.cma.2004.01.010 |
| [16] | Liu GR, Gu YT et al (2001) A point interpolation method for two-dimensional solids. Int J Numer Eng 50: 937–951 · Zbl 1050.74057 · doi:10.1002/1097-0207(20010210)50:4<937::AID-NME62>3.0.CO;2-X |
| [17] | Onate E, Idelsohn EO, Zienkiewicz OC, Taylor RL et al (1996) A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Eng 39: 3839–3866 · Zbl 0884.76068 · doi:10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R |
| [18] | Li H, Ng TY, Cheng JQ, Lam KY et al (2003) Hermite–Cloud: a novel true meshless method. Comput Mech 33: 30–41 · Zbl 1061.65128 · doi:10.1007/s00466-003-0497-1 |
| [19] | Liu GR, Zhang J, Lam KY, Li H, Xu G, Zhong ZH, Li GY, Han X et al (2008) A gradient smoothing method (GSM) with directional correction for solid mechanics problems. Comput Mech 41: 457–472 · Zbl 1162.74502 · doi:10.1007/s00466-007-0192-8 |
| [20] | Sukumar N, Moran B, Semenov AY, Belicov VV et al (2001) Natural neighbour Galerkin methods. Int J Numer Methods Eng 50: 1–27 · Zbl 1082.74554 · doi:10.1002/1097-0207(20010110)50:1<1::AID-NME14>3.0.CO;2-P |
| [21] | Li H, Yew YK, Ng TY, Lam KY et al (2005) Meshless steady-state analysis of chemo-electro-mechanical coupling behaviour of pH-sensitive hydrogel in buffered solution. J Electroanal Chem 580: 161–172 · doi:10.1016/j.jelechem.2005.03.034 |
| [22] | Sukumar N, Moran B, Belytschko T et al (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43: 839–887 · Zbl 0940.74078 · doi:10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R |
| [23] | Li H, Cheng JQ, Ng TY, Chen J, Lam KY et al (2004) A meshless Hermite–cloud method for nonlinear fluid–structure analysis of near-bed submarine pipelines under current. Eng Struct 26: 531–542 · doi:10.1016/j.engstruct.2003.12.005 |
| [24] | Idelsohn SR, Onate E, Calvo N, Pin FD et al (2003) The meshless finite element method. Int J Numer Methods Eng 58: 893–912 · Zbl 1035.65129 · doi:10.1002/nme.798 |
| [25] | Onate E, Idelsohn EO, Zienkiewicz OC, Taylor RL, Sacco C et al (1996) A stabilized finite point method for analysis of fluid mechanics problems. Comp Methods Appl Mech Eng 139: 315–346 · Zbl 0894.76065 · doi:10.1016/S0045-7825(96)01088-2 |
| [26] | Atluri SN, Cho JY, Kim HG et al (1999) Analysis of thin beams, using meshless local Petrov–Galerkin method, with generalized moving least square interpolations. Comput Mech 24: 334–347 · Zbl 0968.74079 · doi:10.1007/s004660050456 |
| [27] | Liu GR, Zhang J, Li H, Lam KY, Bernard BT Kee et al (2006) Radial point interpolation based finite difference method for mechanics problems. Int J Numer Methods Eng 68: 728–754 · Zbl 1129.74049 · doi:10.1002/nme.1733 |
| [28] | Atluri SN, Kim HG, Cho JY et al (1999) A critical assessment of the truly meshless Local Petrov–Galerkin (MLPG), and local boundary integral equation (LBIE) methods. Comput Mech 24: 348–372 · Zbl 0977.74593 · doi:10.1007/s004660050457 |
| [29] | Gilhooley DF, Xiao JR, Batra RC, McCarthy MA, Gillespie JW et al (2008) Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions. Comput Materials Sci 41: 467–481 · doi:10.1016/j.commatsci.2007.05.003 |
| [30] | Aluru NR, Li G (2001) Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation. Int J Numer Methods Eng 50: 2373–2410 · Zbl 0977.74078 · doi:10.1002/nme.124 |
| [31] | Aluru NR (2000) A point collocation method based on Reprod Kernel Approx. Int J Numer Methods Eng 47: 1083–1121 · Zbl 0960.74078 · doi:10.1002/(SICI)1097-0207(20000228)47:6<1083::AID-NME816>3.0.CO;2-N |
| [32] | Li G, Paulino GH, Aluru NR et al (2003) Coupling of the mesh-free finite cloud method with the boundary element method: a collocation approach. Computer Methods Appl Mech Eng 192: 2355–2375 · Zbl 1074.74058 · doi:10.1016/S0045-7825(03)00258-5 |
| [33] | Jin X, Li G, Aluru NR et al (2005) New approximations and collocation schemes in the finite cloud method. Comp Struct 83: 1366–1385 · doi:10.1016/j.compstruc.2004.08.030 |
| [34] | Aluru NR (1999) A reproducing kernel particle method for meshless analysis of microelectromechanical systems. Comput Mech 23: 324–338 · Zbl 0949.74077 · doi:10.1007/s004660050413 |
| [35] | Jin X, Li G, Aluru NR et al (2004) Positivity conditions in meshless collocation methods. Comp Methods Appl Mech Eng 193: 1171–1202 · Zbl 1060.74667 · doi:10.1016/j.cma.2003.12.013 |
| [36] | Bellman RE, Kashef BG, Casti J et al (1972) Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J Comput Phys 10: 40–52 · Zbl 0247.65061 · doi:10.1016/0021-9991(72)90089-7 |
| [37] | Quan JR, Chang CT et al (1989) New insights in solving distributed system equations by the quadrature method–I. Analysis. Comp Chem Eng 13: 779–788 · doi:10.1016/0098-1354(89)85051-3 |
| [38] | Quan JR, Chang CT et al (1989) New insights in solving distributed system equations by the quadrature method–II. Numerical experiments. Comp Chem Eng 13: 1017–1024 · doi:10.1016/0098-1354(89)87043-7 |
| [39] | Shu C, Khoo BC, Yeo KS et al (1994) Numerical solutions of incompressible Navier–Stokes equations by generalized differential quadrature. Finite Elem Analy Des 18: 83–97 · Zbl 0875.76428 · doi:10.1016/0168-874X(94)90093-0 |
| [40] | Ding H, Shu C, Yeo KS, Xu D et al (2006) Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method. Comp Methods Appl Mech Eng 195: 516–533 · Zbl 1222.76072 · doi:10.1016/j.cma.2005.02.006 |
| [41] | Naadimuthu G, Bellman R, Wang KM et al (1984) Differential quadrature and partial differential equations: some numerical results. J Math Anal Appl 98: 220–235 · Zbl 0521.65089 · doi:10.1016/0022-247X(84)90290-7 |
| [42] | Chang S (2000) Differential quadrature and its application in engineering. Springer, London · Zbl 0944.65107 |
| [43] | Nayroles B, Touzot G, Villon P et al (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10: 307–318 · Zbl 0764.65068 · doi:10.1007/BF00364252 |
| [44] | John S (1989) Finite difference schemes and partial differential equations. Wadsworth and Brooks, California · Zbl 0681.65064 |
| [45] | Christoph U (1997) Numerical computation : methods, software, and analysis. Springer, Germany · Zbl 0866.65001 |
| [46] | Sarra SA (2006) Chebyshev interpolation: an interactive tour. J Online Math Appl 6: 1–13 |
| [47] | Mukherjee YX, Mukherjee S (1997) On boundary conditions in the element-free Galerkin method. Comput Mech 19: 264–270 · Zbl 0884.65105 · doi:10.1007/s004660050175 |
| [48] | Edward S (2006) C++ for mathematicians: an introduction for students and professionals. CRC Press, USA · Zbl 1108.68025 |
| [49] | Fish J, Belytschko T (2007) A first course in finite elements. Wiley, Chichester · Zbl 1135.74001 |
| [50] | Liu GR, Quek SS (2003) The finite element method: a practical course. Butterworth-Heinemann, Oxford · Zbl 1027.74001 |
| [51] | Timoshenko SP, Goodier JN (1970) Theory of elasticity. McGraw-Hill, New york |
| [52] | Zhilun Xu (1992) Applied elasticity. New Age International, India |
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