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A stabilized conforming nodal integration for Galerkin mesh-free methods. (English) Zbl 1011.74081
From the summary: Domain integration by Gauss quadrature in Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under-integration and vanishing derivatives of shape functions at the nodes. Here we propose a strain smoothing stabilization for nodal integration to eliminate spatial instability in nodal integration. For convergence, an integration constraint is introduced as a necessary condition for linear exactness in mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy the integration constraint by using a divergence theorem. The numerical results show that the accuracy and convergent rates in the mesh-free method with direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Nayroles, Computational Mechanics 10 pp 307– (1992) · Zbl 0764.65068 · doi:10.1007/BF00364252
[2] Belytschko, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205
[3] Liu, International Journal for Numerical Methods in Fluids 20 pp 1081– (1995) · Zbl 0881.76072 · doi:10.1002/fld.1650200824
[4] Chen, Computer Methods in Applied Mechanics and Engineering 139 pp 195– (1996) · Zbl 0918.73330 · doi:10.1016/S0045-7825(96)01083-3
[5] Duarte, Computer Methods in Applied Mechanics and Engineering 139 pp 237– (1996) · Zbl 0918.73328 · doi:10.1016/S0045-7825(96)01085-7
[6] Melenk, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0
[7] Sulsky, Computer Methods in Applied Mechanics and Engineering 118 pp 179– (1994) · doi:10.1016/0045-7825(94)90112-0
[8] Atluri, Computational Mechanics 22 pp 117– (1998) · Zbl 0932.76067 · doi:10.1007/s004660050346
[9] Monaghan, Computer Physics Communications 48 pp 89– (1988) · Zbl 0673.76089 · doi:10.1016/0010-4655(88)90026-4
[10] Randles, Computer Methods in Applied Mechanics and Engineering 139 pp 375– (1996) · Zbl 0896.73075 · doi:10.1016/S0045-7825(96)01090-0
[11] Beissel, Computer Methods in Applied Mechanics and Engineering 139 pp 49– (1996) · Zbl 0918.73329 · doi:10.1016/S0045-7825(96)01079-1
[12] Dolbow, Computational Mechanics 23 pp 219– (1999) · Zbl 0963.74076 · doi:10.1007/s004660050403
[13] On neighbors, derivatives, and viscosity in particle codes. Proceeding of ECCM Conference, Munich, Germany, 31 August-3 September 1999.
[14] Bonet, International Journal for Numerical Methods in Engineering 47 pp 1189– (1999) · Zbl 0964.76071 · doi:10.1002/(SICI)1097-0207(20000228)47:6<1189::AID-NME830>3.0.CO;2-I
[15] Belytschko, Journal of Computational Applied Mathematics 74 pp 111– (1996) · Zbl 0862.73058 · doi:10.1016/0377-0427(96)00020-9
[16] Breitkopf, Computational Mechanics 25 pp 199– (2000) · Zbl 0979.74077 · doi:10.1007/s004660050469
[17] Liszka, Computer and Structures 11 pp 83– (1980) · Zbl 0427.73077 · doi:10.1016/0045-7949(80)90149-2
[18] Krongauz, Computer Methods in Applied Mechanics and Engineering 131 pp 133– (1996) · Zbl 0881.65098 · doi:10.1016/0045-7825(95)00954-X
[19] Chen, Computer Methods in Applied Mechanics and Engineering 187 pp 441– (2000) · Zbl 0980.74077 · doi:10.1016/S0045-7825(00)80004-3
[20] Krongauz, International Journal for Numerical Methods in Engineering 146 pp 371– (1997)
[21] Johnson, International Journal for Numerical Methods in Engineering 39 pp 2725– (1996) · Zbl 0880.73076 · doi:10.1002/(SICI)1097-0207(19960830)39:16<2725::AID-NME973>3.0.CO;2-9
[22] Generalized nonlocal meshfree method in strain localization. Proceeding of International Conference on Computational Engineering Science, Atlanta, Georgia, 6-9 October 1998.
[23] Chen, International Journal for Numerical Methods in Engineering 47 pp 1303– (2000) · Zbl 0987.74079 · doi:10.1002/(SICI)1097-0207(20000310)47:7<1303::AID-NME826>3.0.CO;2-5
[24] Simo, Journal of Applied Mechanics 53 pp 51– (1986) · Zbl 0592.73019 · doi:10.1115/1.3171737
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