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http://dx.doi.org/10.12989/sem.2021.78.6.651

A meshfree method based on weak-strong form for structural analysis  

El Kadmiri, Redouane (Hassan II University of Casablanca, National Higher School of Arts and Crafts (ENSAM CASABLANCA))
Belaasilia, Youssef (Hassan II University of Casablanca, National Higher School of Arts and Crafts (ENSAM CASABLANCA))
Timesli, Abdelaziz (Hassan II University of Casablanca, National Higher School of Arts and Crafts (ENSAM CASABLANCA))
Kadiri, M. Saddik (Sultan Moulay Slimane University, National School of Applied Sciences of Khouribga, LIPIM Laboratory)
Publication Information
Structural Engineering and Mechanics / v.78, no.6, 2021 , pp. 651-664 More about this Journal
Abstract
In this work, we propose a novel method associating a weak form Moving Least Square (MLS) method, also called Element Free Galerkin (EFG) method, and a strong form MLS method to solve the structural problems in two-dimensional elasticity. Therefore we use the displacement compatibility and the force equilibrium conditions on the interface to ensure the coupling between meshfree weak form method and meshfree strong form method. The strong form MLS method is easy to implement and computationally efficient, but it can be unstable and less precise for problems with Neumann boundary conditions. On the other hand, the weak form MLS method ensures very good stability and excellent precision, but it requires the numerical integration which makes this method not "truly" meshless and computationally expensive. Among of the advantages of the proposed method are the following: (i) numerical integrations are avoided for all nodes in the domain of the strong form approximation, (ii) the weak form can be used for nodes on the Neumann boundary, (iii) the strong form can be used in the region of large deformation. Comparative studies with analytical solutions and weak form methods are presented to show the effectiveness and performance of the proposed method.
Keywords
meshfree weak-strong form method; Moving Least Square (MLS); coupling weak-strong forms;
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1 Liu, G.R. (2002), Mesh Free Methods: Moving Beyond the Finite Element Method, CRC press, Boca Raton, USA.
2 Liu, G.R. (2003), Mesh Free Methods, C.R.C Press.
3 Liu, W.K., Jun, S. and Zhang, Y.F. (1995), "Reproducing Kernel Particle Methods", Int. J. Numer. Meth. Fluid., 20, 1081-1106. https://doi.org/10.1002/d.1650200824.   DOI
4 Lucy, L.B. (1977), "A numerical approach to the testing of the fission hypothesis", Astronom. J., 82, 1013-1024.   DOI
5 Nayroles, B., Touzot, G. and Villon, P. (1991), "The diffuse approximation", C. R. Acad. Sci., 313, 293-296.
6 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.   DOI
7 Nguyen, V.P., Rabczuk, T., Bordas, S. and Duot, M. (2008), "Meshless methods: A review and computer implementation aspects", Math. Comput. Simul., 79, 763-813. https://doi.org/10.1016/j.matcom.2008.01.003.   DOI
8 Niroumand, H., Mehrizi, M.E.M. and Saaly, M. (2016), "Application of mesh-free smoothed particle hydrodynamics (SPH) for study of soil behavior", Geomech. Eng., 11, 1-39. http://doi.org/10.12989/gae.2016.11.1.001.   DOI
9 Orkisz, J. (1998), Handbook of Computational Solid Mechanics. Finite Difference Method (Part III), Ed. I.M. Kleiber, Springer-Verlag, Berlin.
10 Liu, G.R. and Gu, Y.T. (2001), "A point interpolation method for two-dimensional solids", Int. J. Numer. Meth. Eng., 50, 937-951. https://doi.org/10.1002/1097-0207(20010210)50:4<937::AIDNME62>3.0.CO;2-X.   DOI
11 Saffah, Z., Timesli, A., Lahmam, H., Azouani, A. and Amdi, M. (2021), "New collocation path-following approach for the optimal shape parameter using Kernel method", SN Appl. Sci., 3, 249. https://doi.org/10.1007/s42452-021-04231-1.   DOI
12 Lancaster, P. and Salkauskas, K. (1981), "Surfaces generated by moving least squares methods", Math. Comput., 37, 141-158. https://doi.org/10.1090/S0025-5718-1981-0616367-1.   DOI
13 Timesli, A. (2020b), "Prediction of the critical buckling load of SWCNT reinforced concrete cylindrical shell embedded in an elastic foundation", Comput. Concrete, 26(1), 53-62. http://doi.org/10.12989/cac.2020.26.1.053.   DOI
14 Attaway, S.W., Heinstein, M.W. and Swegle, J.W. (1994), "Coupling of smooth particle hydrodynamics with the finite element method", Nucl. Eng. Des., 150, 199-205. https://doi.org/10.1016/0029-5493(94)90136-8.   DOI
15 Orkisz, J. (1998), "Meshless finite difference method II. Adaptive approach", Comput. Mech., Eds. Idelson, Onate, Duorkin, iacm, CINME.
16 Orkisz, J. (1998), "Meshless finite difference method I. Basic approach", Comput. Mech., Eds, Idelson, Onate, Duorkin, iacm, CINME.
17 Chen, M. and Ling, L. (2019), "Kernel-based meshless collocation methods for solving coupled bulk-surface partial differential equations", J. Sci. Comput., 81, 375-391. https://doi.org/10.1007/s10915-019-01020-2.   DOI
18 Kanok-Nukulchai, W., Barry, W.J. and Saran-Yasoontorn, K. (2001), "Meshless formulation for shear-locking free bending elements", Struct. Eng. Mech., 11, 123-132. https://doi.org/10.12989/sem.2001.11.2.123.   DOI
19 Hegen, D. (1996), "Element-free Galerkin methods in combination with finite element approaches", Comput. Meth. Appl. Mech. Eng., 135, 143-166. https://doi.org/10.1016/0045-7825(96)00994-2.   DOI
20 Babuska, I. and Melenk, J.M. (1997), "The partition of unity method", Int. J. Numer. Meth. Eng., 40, 727-758. https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N.   DOI
21 Peyroteo, M.M.A.P., Belinha, J., Dinis, L.M.J.S. and Jorge, R.M.N. (2019), "A new biological bone remodeling in silico model combined with advanced discretization methods", Int. J. Numer. Meth. Biomed. Eng., 35, e3196. https://doi.org/10.1002/cnm.3196.   DOI
22 Rad, M.H.G., Shahabian, F. and Hosseini, S.M. (2019), "Nonlocal geometrically nonlinear dynamic analysis of nanobeam using a meshless method", Steel Compos. Struct., 32, 293-304. http://doi.org/10.12989/scs.2019.32.3.293.   DOI
23 Rad, M.H.G., Shahabian, F., Mishra, B.K. and Hosseini, S.M. (2020), "Geometrically nonlinear dynamic analysis of FG graphene platelets-reinforced nanocomposite cylinder: MLPG method based on a modified nonlinear micromechanical model", Steel Compos. Struct., 35, 77-92. https://doi.org/10.12989/scs.2020.35.1.077.   DOI
24 Rohit, G.R., Prajapati, J.M. and Patel, V.B. (2020), "Coupling of Finite Element and Meshfree Method for Structure Mechanics Application: A Review", Int. J. Comput. Meth., 17, 1850151. https://doi.org/10.1142/S0219876218501517.   DOI
25 Shedbale, A.S., Singh, I.V. and Mishra, B.K. (2016), "A coupled FE-EFG approach for modeling crack growth in ductile materials", Fatig. Fract. Eng. Mater. Struct., 39, 1204-1225. https://doi.org/10.1111/_e.12423.   DOI
26 Shedbale, A.S., Singh, I.V., Mishra, B.K. and Sharma, K. (2017), "Ductile failure modeling and simulations using coupled FE-EFG approach", Int. J. Fract., 203, 183-209. https://doi.org/10.1111/_e.12423.   DOI
27 Strouboulis, T., Babus, I. and Copps, K. (2000), "The design and analysis of the generalized finite element method", Comput. Meth. Appl. Mech. Eng., 181, 43-69. https://doi.org/10.1002/d.1650200824.   DOI
28 Timesli, A. (2020a), "Buckling analysis of double walled carbon nanotubes embedded in Kerr elastic medium under axial compression using the nonlocal Donnell shell theory", Adv. Nano Res., 9, 69-82. http://dx.doi.org/10.12989/anr.2020.9.2.069.   DOI
29 Belytschko, T., Lu, Y.Y. and Gu, L. (1994), "Element-free Galerkin methods", Int. J. Numer. Meth. Eng., 37, 229-256. https://doi.org/10.1002/nme.1620370205.   DOI
30 Belaasilia, Y., Timesli, A., Braikat, B. and Jamal, M. (2017), "A numerical mesh-free model for elasto-plastic contact problems", Eng. Anal. Bound. Elem., 82, 168-78. https://doi.org/10.1016/j.enganabound.2017.05.010.   DOI
31 Belytschko, T., Organ, D. and Krongauz, Y. (1995), "A coupled finite element-element free Galerkin method", Comput. Mech., 17, 186-195. https://doi.org/10.1007/BF00364080.   DOI
32 Chen, T. and Raju, I.S. (2003), "A coupled finite element and Meshless local Petrov-Galerkin method for twodimensional potential problems", Comput. Meth. Appl. Mech. Eng., 192, 4533-4550. https://doi.org/10.1016/S0045-7825(03)00421-3.   DOI
33 Li, S. and Liu, W.K. (1999), "Reproducing kernel hierarchical partition of unity, Part I-Formulation and theory", Int. J. Numer. Meth. Eng., 45, 251-288. https://doi.org/10.1002/(SICI)1097-0207(19990530)45:3<251::AID-NME583>3.0.CO;2-I.   DOI
34 Huerta, A. and Fernandez-Mendez, S. (2000), "Enrichment and coupling of the finite element and Meshless methods", Int. J. Numer. Methods Eng., 48, 1615-1636. https://doi.org/10.1002/1097-0207(20000820)48:11<1615::AID-NME883>3.0.CO;2-S.   DOI
35 Jaskowiec, J. and Milewski, S. (2016), "Coupling _nite element method with meshless finite difference method in thermomechanical problems", Compt. Math. Appl., 72, 2259-2279. https://doi.org/10.1016/j.camwa.2016.08.020.   DOI
36 Krongauz, Y. and Belytschko, T. (1996), "Enforcement of essential boundary conditions in meshless approximations using finite elements", Comput. Meth. Appl. Mech. Eng., 131, 133-145. https://doi.org/10.1016/0045-7825(95)00954-X.   DOI
37 Liska, T. (1984), "An interpolation method for an irregular net of nodes", Int. J. Numer. Meth. Eng., 192, 1599-1612. https://doi.org/10.1002/nme.1620200905.   DOI
38 Dilts, G.A. (2000), "Moving-least-square-particle hydrodynamics II: Conservation and boundaries", Int. J. Numer. Meth. Eng., 48, 1503-1524. https://doi.org/10.1002/1097-0207(20000810)48:10<1503::AID-NME832>3.0.CO;2-D.   DOI
39 Rao, B.N. (2011), "Coupled meshfree and fractal finite element method for unbounded problems", Comput. Geotech., 38, 697-708. https://doi.org/10.1016/j.compgeo.2011.02.009.   DOI
40 Daux, C., Moes, N., Dolbow, J., Sukumar, N. and Belytschko, T. (1992), "Arbitrary branched and intersecting cracks with the extended finite element method", Int. J. Numer. Meth. Eng., 48, 1741-1760. https://doi.org/10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L.   DOI
41 Dolbow, J. and Belytschko, T. (1998), "An introduction to programming the meshless element free Galerkin method", Arch. Comput. Meth. Eng., 5, 207-241. https://doi.org/10.1007/BF02897874.   DOI
42 Duarte, C.A.M. and Oden, J.T. (1996), "hp clouds - A meshless method to solve boundary value problems", Numer. Meth. Partial. Differ. Equ., 12, 673-705.   DOI
43 Belytschko, T. and Organ, D. (1995), "A coupled _nite elementelement-free Galerkin method", Comput. Mech., 17, 186-195. https://doi.org/10.1007/BF00364080.   DOI
44 Ferezghi, Y.S., Sohrabi, M. and Nezhad, S.M.M. (2020), "Meshless Local Petrov-Galerkin (MLPG) method for dynamic analysis of non-symmetric nanocomposite cylindrical shell", Struct. Eng. Mech., 74, 679-698. http://doi.org/10.12989/sem.2020.74.5.679.   DOI
45 Wu, C.P. and Liu, Y.C. (2016), "A state space meshless method for the 3D analysis of FGM axisymmetric circular plates", Steel Compos. Struct., 22, 161-182. http://doi.org/10.12989/scs.2016.22.1.161.   DOI
46 Shepard, D. (1968), "A two-dimensional interpolation function for irregularly spaced data", Proceedings of the 1968 ACM National Conference, 517-524. https://doi.org/10.1145/800186.810616.   DOI
47 Fernandez-Mendez, S. and Huerta, A. (2004), "Imposing essential boundary conditions in mesh-free methods", Comput. Meth. Appl. Mech. Eng., 193, 1257-1275. https://doi.org/10.1016/j.cma.2003.12.019.   DOI
48 Dilts, G.A. (1999), "Moving-least-square-particle hydrodynamics I: Consistency and stability", Int. J. Numer. Meth. Eng., 44, 1115-1155. https://doi.org/10.1002/(SICI)1097-0207(19990320)44:8<1115::AID-NME547>3.0.CO;2-L.   DOI
49 Gingold, R.A. and Monaghan, J.J. (1977), "Smoothed particle hydrodynamics theory and application to non-spherical stars", Mon. Notices Royal Astron. Soc., 181, 375-389. https://doi.org/10.1093/mnras/181.3.375.   DOI
50 Liska, T. and Orkisz, J. (1980), "The finite difference method at arbitrary irregular grids and its application in applied mechanics", Comput. Struct., 2, 83-95. https://doi.org/10.1016/0045-7949(80)90149-2.   DOI
51 Wu, Y., Choi, H.J., Li, H. and Crawford, J.E. (2013), "Concrete fragmentation modeling using coupled finite element-meshfree formulations", Interact. Multisc. Mech., 6, 173-195. http://doi.org/10.12989/imm.2013.6.2.173.   DOI
52 Xiao, Y. and Wu, H. (2020), "An explicit coupled method of FEM and meshless particle method for simulating transient heat transfer process of friction stir welding", Math. Prob. Eng., 2020, ID 2574127, 16. https://doi.org/10.1155/2020/2574127.   DOI
53 Zhang, X., Liu, X.H., Song, K.Z. and Lu, M.W. (2001), "Leastsquare collocation meshless method", Int. J. Numer. Meth. Eng., 51, 1089-1100. https://doi.org/10.1016/j.enganabound.2014.09.011.   DOI
54 Zhang, Y., Ge, W., Tong, X. and Ye, M. (2018), "Topology optimization of structures with coupled finite element - element free galerkin method", Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci., 232, 1089-1100. https://doi.org/10.1177/0954406216688716.   DOI
55 Timesli, A. (2021b), "Analytical modeling of buckling behavior of porous FGM cylindrical shell embedded within an elastic foundation", Gazi Univ. J. Sci., 1-1. https://doi.org/10.35378/gujs.860783.   DOI
56 Cheng, J.Q., Lee, H.P. and Li, H. (2004), "Development of a meshless finite mixture (MFM) method", Struct. Eng. Mech., 17, 671-690. http://doi.org/10.12989/sem.2004.17.5.671.   DOI
57 Gu, Y.T. and Zhang, L.C. (2008), "Coupling of the meshfree and _nite element methods for determination of the crack tip fields", Eng. Fract. Mech., 75, 986-1004. https://doi.org/10.1016/j.engfracmech.2007.05.003.   DOI
58 Timesli, A. (2021a), "Optimized radius of influence domain in meshless approach for modeling of large deformation problems", Iran. J. Sci. Technol. Trans. Mech. Eng., 1-11. https://doi.org/10.1007/s40997-021-00427-3.   DOI
59 Timesli, A., Braikat, B., Lahmam, H. and Zahrouni, H. (2015), "A new algorithm based on moving least square method to simulate material mixing in friction stir welding", Eng. Anal. Bound. Elem., 50, 372-380. https://doi.org/10.1016/j.enganabound.2014.09.011.   DOI
60 Wang, J.G. and Liu, G.R. (2002), "On the optimal shape parameters of radial basis functions used for 2-D meshlesss methods", Comput. Meth. Appl. Mech. Eng., 191, 2611-2630. https://doi.org/10.1016/S0045-7825(01)00419-4.   DOI