A meshless local Petrov-Galerkin Shepard and least-squares method based on duo nodal supports. (English) Zbl 1407.74097
Summary: The meshless Shepard and least-squares (MSLS) interpolation is a newly developed partition of unity- (PU-) based method which removes the difficulties with many other meshless methods such as the lack of the Kronecker delta property. The MSLS interpolation is efficient to compute and retain compatibility for any basis function used. In this paper, we extend the MSLS interpolation to the local Petrov-Galerkin weak form and adopt the duo nodal support domain. In the new formulation, there is no need for employing singular weight functions as is required in the original MSLS and also no need for background mesh for integration. Numerical examples demonstrate the effectiveness and robustness of the present method.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
References:
| [1] | Fries, T.; Matthies, H., Classification and Overview of Meshfree Methods (2004), Brunswick, Germany: Technical University Braunschweig, Brunswick, Germany |
| [2] | Hao, S.; Liu, W. K.; Belytschko, T., Moving particle finite element method with global smoothness, International Journal for Numerical Methods in Engineering, 59, 7, 1007-1020 (2004) · Zbl 1065.74608 · doi:10.1002/nme.999 |
| [3] | Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Mathematics and Computers in Simulation, 79, 3, 763-813 (2008) · Zbl 1152.74055 · doi:10.1016/j.matcom.2008.01.003 |
| [4] | Zhuang, X.; Augarde, C. E.; Mathisen, K. M., Fracture modeling using meshless methods and levels sets in 3D: framework and modeling, International Journal for Numerical Methods in Engineering, 92, 11, 969-998 (2012) · Zbl 1352.74312 · doi:10.1002/nme.4365 |
| [5] | Zhuang, X.; Augarde, C.; Bordas, S., Accurate fracture modelling using meshless methods, the visibility criterion and level sets: formulation and 2D modelling, International Journal for Numerical Methods in Engineering, 86, 2, 249-268 (2011) · Zbl 1235.74346 · doi:10.1002/nme.3063 |
| [6] | Rabczuk, T.; Belytschko, T., Cracking particles: a simplified meshfree method for arbitrary evolving cracks, International Journal for Numerical Methods in Engineering, 61, 13, 2316-2343 (2004) · Zbl 1075.74703 · doi:10.1002/nme.1151 |
| [7] | Rabczuk, T.; Zi, G., A meshfree method based on the local partition of unity for cohesive cracks, Computational Mechanics, 39, 6, 743-760 (2007) · Zbl 1161.74055 · doi:10.1007/s00466-006-0067-4 |
| [8] | Rabczuk, T.; Areias, P. M. A.; Belytschko, T., A meshfree thin shell method for non-linear dynamic fracture, International Journal for Numerical Methods in Engineering, 72, 5, 524-548 (2007) · Zbl 1194.74537 · doi:10.1002/nme.2013 |
| [9] | Jarak, T.; Sorić, J., Analysis of rectangular square plates by the mixed meshless local Petrov-Galerkin (MLPG) approach, Computer Modeling in Engineering & Sciences, 38, 3, 231-261 (2008) · Zbl 1357.74030 |
| [10] | Areias, P.; Rabczuk, T., Finite strain fracture of plates and shells with configurational forces and edge rotation, International Journal For Numerical Methods in Engineering, 94, 1099-1122 (2013) · Zbl 1352.74316 |
| [11] | Areias, P.; Rabczuk, T.; Dias-da-Costa, D., Assumed-metric spherically-interpolated quadrilateral shell element, Finite Elements in Analysis and Design, 66, 53-67 (2013) · Zbl 1282.74057 |
| [12] | Thai, C. H.; Nguyen-Xuan, H.; Nguyen-Thanh, N.; Le, T.-H.; Nguyen-Thoi, T.; Rabczuk, T., Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach, International Journal for Numerical Methods in Engineering, 91, 6, 571-603 (2012) · Zbl 1253.74007 · doi:10.1002/nme.4282 |
| [13] | Chau-Dinh, T.; Zi, G.; Lee, P. S.; Song, J. H.; Rabczuk, T., Phantom-node method for shell models with arbitrary cracks, Computers and Structures, 92-93, 242-256 (2012) · doi:10.1016/j.compstruc.2011.10.021 |
| [14] | Nguyen-Thanh, N.; Kiendl, J.; Nguyen-Xuan, H.; Wüchner, R.; Bletzinger, K. U.; Bazilevs, Y., Rotation free isogeometric thin shell analysis using PHT-splines, Computer Methods in Applied Mechanics and Engineering, 200, 47-48, 3410-3424 (2011) · Zbl 1230.74230 · doi:10.1016/j.cma.2011.08.014 |
| [15] | Nguyen-Thanh, N.; Rabczuk, T.; Nguyen-Xuan, H.; Bordas, S., An alternative alpha finite element method with discrete shear gap technique for analysis of Mindlin-Reissner plates, Finite Elements in Analysis and Design, 47, 519-535 (2011) |
| [16] | Rabczuk, T.; Belytschko, T., A three-dimensional large deformation meshfree method for arbitrary evolving cracks, Computer Methods in Applied Mechanics and Engineering, 196, 29-30, 2777-2799 (2007) · Zbl 1128.74051 · doi:10.1016/j.cma.2006.06.020 |
| [17] | Han, Z. D.; Atluri, S. N., Meshless local Petrov-Galerkin (MLPG) approaches for solving 3D problems in elasto-statics, CMES. Computer Modeling in Engineering & Sciences, 6, 2, 169-188 (2004) · Zbl 1087.74654 |
| [18] | Rabczuk, T.; Bordas, S.; Zi, G., On three-dimensional modelling of crack growth using partition of unity methods, Computers and Structures, 88, 23-24, 1391-1411 (2010) · doi:10.1016/j.compstruc.2008.08.010 |
| [19] | Rabczuk, T.; Gracie, R.; Song, J.-H.; Belytschko, T., Immersed particle method for fluid-structure interaction, International Journal for Numerical Methods in Engineering, 81, 1, 48-71 (2010) · Zbl 1183.74367 · doi:10.1002/nme.2670 |
| [20] | Rabczuk, T.; Samaniego, E., Discontinuous modelling of shear bands using adaptive meshfree methods, Computer Methods in Applied Mechanics and Engineering, 197, 6-8, 641-658 (2008) · Zbl 1169.74655 · doi:10.1016/j.cma.2007.08.027 |
| [21] | Rabczuk, T.; Belytschko, T.; Xiao, S. P., Stable particle methods based on Lagrangian kernels, Computer Methods in Applied Mechanics and Engineering, 193, 12-14, 1035-1063 (2004) · Zbl 1060.74672 · doi:10.1016/j.cma.2003.12.005 |
| [22] | Atluri, S. N.; Zhu, T., The Meshless Local Petrov-Galerkin(MLPG) approach for solving problems in elasto-statics, Computational Mechanics, 25, 169-179 (2000) · Zbl 0976.74078 |
| [23] | Zheng, J.; Long, S.; Xiong, Y.; Li, G., A finite volume meshless local Petrov-Galerkin method for topology optimization design of the continuum structures, Computer Modeling in Engineering & Sciences, 42, 1, 19-34 (2009) · Zbl 1357.74046 |
| [24] | Cai, Y.; Zhuang, X.; Augarde, C., A new partition of unity finite element free from the linear dependence problem and possessing the delta property, Computer Methods in Applied Mechanics and Engineering, 199, 17-20, 1036-1043 (2010) · Zbl 1227.74065 · doi:10.1016/j.cma.2009.11.019 |
| [25] | Cai, Y.; Zhuang, X.; Zhu, H., A generalized and efficient method for finite cover generation in the numerical manifold method, International Journal of Computational Methods, 10, 5 (2013) · Zbl 1359.65283 · doi:10.1142/S021987621350028X |
| [26] | Nguyen-Xuan, H.; Rabczuk, T.; Nguyen-Thoi, T.; Tran, T. N.; Nguyen-Thanh, N., Computation of limit and shakedown loads using a node-based smoothed finite element method, International Journal for Numerical Methods in Engineering, 90, 3, 287-310 (2012) · Zbl 1242.74142 · doi:10.1002/nme.3317 |
| [27] | Thai-Hoang, C.; Nguyen-Thanh, N.; Nguyen-Xuan, H.; Rabczuk, T., An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates, Applied Mathematics and Computation, 217, 17, 7324-7348 (2011) · Zbl 1415.74048 · doi:10.1016/j.amc.2011.02.024 |
| [28] | Vu-Bac, N.; Nguyen-Xuan, H.; Chen, L.; Lee, C. K.; Zi, G.; Zhuang, X.; Liu, G. R.; Rabczuk, T., A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics, Journal of Applied Mathematics, 2013 (2013) · Zbl 1271.74418 |
| [29] | Jia, Y.; Zhang, Y.; Xu, G.; Zhuang, X.; Rabczuk, T., Reproducing kernel triangular B-spline-based FEM for solving PDEs, Computer Methods in Applied Mechanics and Engineering, 267, 342-358 (2013) · Zbl 1286.65161 · doi:10.1016/j.cma.2013.08.019 |
| [30] | Lancaster, P.; Salkauskas, K., Surfaces generated by moving least squares methods, Mathematics of Computation, 37, 155, 141-158 (1981) · Zbl 0469.41005 · doi:10.2307/2007507 |
| [31] | Breitkopf, P.; Rassineux, A.; Touzot, G.; Villon, P., Explicit form and efficient computation of MLS shape functions and their derivatives, International Journal for Numerical Methods in Engineering, 48, 3, 451-466 (2000) · Zbl 0965.65015 |
| [32] | Zhuang, X.; Augarde, C., Aspects of the use of orthogonal basis functions in the element-free Galerkin method, International Journal for Numerical Methods in Engineering, 81, 3, 366-380 (2010) · Zbl 1183.74376 · doi:10.1002/nme.2696 |
| [33] | Zhuang, X.; Heaney, C.; Augarde, C., On error control in the element-free Galerkin method, Engineering Analysis with Boundary Elements, 36, 3, 351-360 (2012) · Zbl 1245.65161 · doi:10.1016/j.enganabound.2011.06.011 |
| [34] | Kaljević, I.; Saigal, S., An improved element free Galerkin formulation, International Journal for Numerical Methods in Engineering, 40, 16, 2953-2974 (1997) · Zbl 0895.73079 · doi:10.1002/(SICI)1097-0207(19970830)40:16<2953::AID-NME201>3.3.CO;2-J |
| [35] | Chen, J.-S.; Wang, H.-P., New boundary condition treatments in meshfree computation of contact problems, Computer Methods in Applied Mechanics and Engineering, 187, 3-4, 441-468 (2000) · Zbl 0980.74077 · doi:10.1016/S0045-7825(00)80004-3 |
| [36] | Cai, Y. C.; Zhu, H. H., A local meshless Shepard and least square interpolation method based on local weak form, Computer Modeling in Engineering & Sciences, 34, 2, 179-204 (2008) · Zbl 1232.65169 |
| [37] | Zhuang, X.; Zhu, H.; Augarde, C., An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function, Computational Mechanics (2013) · doi:10.1007/s00466-031-0912-1 |
| [38] | Liu, G. R.; Gu, Y. T.; Dai, K. Y., Assessment and applications of point interpolation methods for computational mechanics, International Journal for Numerical Methods in Engineering, 59, 1373-1397 (2004) · Zbl 1041.74562 |
| [39] | Dolbow, J.; Belytschko, T., Numerical integration of the Galerkin weak form in meshfree methods, Computational Mechanics, 23, 3, 219-230 (1999) · Zbl 0963.74076 · doi:10.1007/s004660050403 |
| [40] | Beissel, S.; Belytschko, T., Nodal integration of the element-free Galerkin method, Computer Methods in Applied Mechanics and Engineering, 139, 1-4, 49-74 (1996) · Zbl 0918.73329 · doi:10.1016/S0045-7825(96)01079-1 |
| [41] | Chen, J. S.; Wu, C. T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin meshfree methods, International Journal for Numerical Methods in Engineering, 50, 435-466 (2001) · Zbl 1011.74081 |
| [42] | Puso, M. A.; Chen, J. S.; Zywicz, E.; Elmer, W., Meshfree and finite element nodal integration methods, International Journal for Numerical Methods in Engineering, 74, 3, 416-446 (2008) · Zbl 1159.74456 · doi:10.1002/nme.2181 |
| [43] | Atluri, S. N., The Meshless Local Petrov-Galerkin (MLPG)Method for Domain & Boundary Discretizations (2004), Tech Science Press · Zbl 1105.65107 |
| [44] | Carpinteri, A.; Ferro, G.; Ventura, G., The partition of unity quadrature in meshless methods, International Journal for Numerical Methods in Engineering, 54, 7, 987-1006 (2002) · Zbl 1028.74047 · doi:10.1002/nme.455 |
| [45] | Kaljević, I.; Saigal, A. S., An improved element free Galerkin formulation, International Journal for Numerical Methods in Engineering, 40, 16, 2953-2974 (1997) · Zbl 0895.73079 · doi:10.1002/(SICI)1097-0207(19970830)40:16<2953::AID-NME201>3.3.CO;2-J |
| [46] | Rajendran, S.; Zhang, B. R., A “FE-meshfree” QUAD4 element based on partition of unity, Computer Methods in Applied Mechanics and Engineering, 197, 1-4, 128-147 (2007) · Zbl 1169.74628 · doi:10.1016/j.cma.2007.07.010 |
| [47] | Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1951), New York, NY, USA: McGraw-Hill, New York, NY, USA · Zbl 0045.26402 |
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