An accuracy analysis of Galerkin meshfree methods accounting for numerical integration

Junchao Wu1,2, Dongdong Wang2
1College of Civil Engineering, Huaqiao University, Xiamen, Fujian 361021, China
2Department of Civil Engineering, Xiamen University, Xiamen, Fujian, 361005, China

Tài liệu tham khảo

Belytschko, 1996, Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Engrg., 139, 3, 10.1016/S0045-7825(96)01078-X Atluri, 2002 Babuška, 2003, Survey of meshless and generalized finite element methods: a unified approach, Acta Numer., 12, 1, 10.1017/S0962492902000090 Li, 2004 Zhang, 2004 Nguyen, 2008, Meshless methods: A review and computer implementation aspects, Math. Comput. Simulation, 79, 763, 10.1016/j.matcom.2008.01.003 Liu, 2009 Chen, 2017, Meshfree methods: progress made after 20 years, J. Eng. Mech. ASCE, 143, 10.1061/(ASCE)EM.1943-7889.0001176 Nayroles, 1992, Generalizing the finite element method: Diffuse approximation and diffuse elements, Comput. Mech., 10, 307, 10.1007/BF00364252 Belytschko, 1994, Element-free Galerkin methods, Internat. J. Numer. Methods Engrg., 37, 229, 10.1002/nme.1620370205 Belytschko, 1995, Element-free Galerkin methods for static and dynamic fracture, Int. J. Solids Struct., 32, 2547, 10.1016/0020-7683(94)00282-2 Liu, 1995, Reproducing kernel particle methods, Internat. J. Numer. Methods Fluids, 20, 1081, 10.1002/fld.1650200824 Chen, 1996, Reproducing kernel particle methods for large deformation analysis of nonlinear structures, Comput. Methods Appl. Mech. Engrg., 139, 195, 10.1016/S0045-7825(96)01083-3 Duarte, 1996, An hp adaptive method using clouds, Comput. Methods Appl. Mech. Engrg., 139, 237, 10.1016/S0045-7825(96)01085-7 Babuška, 1997, The partition of unity method, Internat. J. Numer. Methods Engrg., 40, 727, 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N Sukumar, 1998, The natural element method in solid mechanics, Internat. J. Numer. Methods Engrg., 43, 839, 10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R Atluri, 1998, A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics, Comput. Mech., 22, 117, 10.1007/s004660050346 Wang, 2002, A point interpolation meshless method based on radial basis functions, Internat. J. Numer. Methods Engrg., 54, 1623, 10.1002/nme.489 Liu, 2004, Reproducing kernel element method. Part I: Theoretical formulation, Comput. Methods Appl. Mech. Engrg., 193, 933, 10.1016/j.cma.2003.12.001 Sukumar, 2004, Construction of polygonal interpolants: a maximum entropy approach, Internat. J. Numer. Methods Engrg., 61, 2159, 10.1002/nme.1193 Arroyo, 2006, Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods, Internat. J. Numer. Methods Engrg., 65, 2167, 10.1002/nme.1534 Liu, 2007, A smoothed finite element method for mechanics problems, Comput. Mech., 39, 859, 10.1007/s00466-006-0075-4 Guan, 2011, Semi-Lagrangian reproducing kernel particle method for fragment-impact problems, Int. J. Impact Eng., 38, 1033, 10.1016/j.ijimpeng.2011.08.001 Wang, 2014, Quasi-convex reproducing kernel meshfree method, Comput. Mech., 54, 689, 10.1007/s00466-014-1022-4 Metsis, 2015, A new hierarchical partition of unity formulation of EFG meshless methods, Comput. Methods Appl. Mech. Engrg., 283, 782, 10.1016/j.cma.2014.10.016 Yreux, 2017, A quasi-linear reproducing kernel particle method, Internat. J. Numer. Methods Engrg., 109, 1045, 10.1002/nme.5319 Wu, 2018, Simulation of impact penetration and perforation of metal targets using the smoothed particle Galerkin method, J. Eng. Mech. ASCE, 144, 10.1061/(ASCE)EM.1943-7889.0001470 Trask, 2020, A conservative, consistent, and scalable meshfree mimetic method, J. Comput. Phys., 409, 10.1016/j.jcp.2019.109187 Lancaster, 1981, Surfaces generated by moving least squares methods, Math. Comp., 37, 141, 10.1090/S0025-5718-1981-0616367-1 Dolbow, 1999, Numerical integration of the Galerkin weak form in meshfree methods, Comput. Mech., 23, 219, 10.1007/s004660050403 Beissel, 1996, Nodal integration of the element-free Galerkin method, Comput. Methods Appl. Mech. Engrg., 139, 49, 10.1016/S0045-7825(96)01079-1 Dyka, 1997, Stress points for tension instability in SPH, Internat. J. Numer. Methods Engrg., 40, 2325, 10.1002/(SICI)1097-0207(19970715)40:13<2325::AID-NME161>3.0.CO;2-8 Rabczuk, 2004, Stable particle methods based on Lagrangian kernels, Comput. Methods Appl. Mech. Engrg., 193, 1035, 10.1016/j.cma.2003.12.005 Wu, 2018, A stable and convergent Lagrangian particle method with multiple local stress points for large strain and material failure analysis, Finite Elem. Anal. Des., 146, 96, 10.1016/j.finel.2018.04.007 Hillman, 2016, An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics, Internat. J. Numer. Methods Engrg., 107, 603, 10.1002/nme.5183 Silva-Valenzuela, 2020, A nodal integration scheme for meshfree Galerkin methods using the virtual element decomposition, Internat. J. Numer. Methods Engrg., 121, 2174, 10.1002/nme.6304 De, 2001, The method of finite spheres with improved numerical integration, Comput. Struct., 79, 2183, 10.1016/S0045-7949(01)00124-9 Carpinteri, 2002, The partition of unity quadrature in meshless methods, Internat. J. Numer. Methods Engrg., 54, 987, 10.1002/nme.455 Liu, 2010, A new support integration scheme for the weakform in mesh-free methods, Internat. J. Numer. Methods Engrg., 82, 699, 10.1002/nme.2780 Chen, 2001, A stabilized conforming nodal integration for Galerkin mesh-free methods, Internat. J. Numer. Methods Engrg., 50, 435, 10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A Chen, 2002, Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods, Internat. J. Numer. Methods Engrg., 53, 2587, 10.1002/nme.338 Irons, 1972, Experience with the patch test for convergence of finite element methods, 557 Zienkiewicz, 1997, The finite element patch test revisited a computer test for convergence, validation and error estimates, Comput. Methods Appl. Mech. Eng., 149, 223, 10.1016/S0045-7825(97)00085-6 Wang, 2004, Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation, Comput. Methods Appl. Mech. Eng., 193, 1065, 10.1016/j.cma.2003.12.006 Chen, 2006, A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates, Internat. J. Numer. Methods Engrg., 68, 151, 10.1002/nme.1701 Wang, 2008, A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration, Internat. J. Numer. Methods Engrg., 74, 368, 10.1002/nme.2175 Wang, 2008, An efficient Galerkin meshfree analysis of shear deformable cylindrical panels, Interact. Multiscale Mech., 1, 339, 10.12989/imm.2008.1.3.339 Wang, 2011, Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures, Comput. Mech., 48, 47, 10.1007/s00466-011-0580-y Wang, 2013, A Hermite reproducing kernel Galerkin meshfree approach for buckling analysis of thin plates, Comput. Mech., 51, 1013, 10.1007/s00466-012-0784-9 Wang, 2015, A circumferentially enhanced Hermite reproducing kernel meshfree method for buckling analysis of Kirchhoff–Love cylindrical shells, Int. J. Struct. Stab. Dyn., 15, 10.1142/S0219455414500904 Wang, 2011, Three dimensional efficient meshfree simulation of large deformation failure evolution in soil medium, Sci. China Tech. Sci., 54, 573, 10.1007/s11431-010-4287-7 Wu, 2014, Three dimensional fragmentation simulation of concrete structures with a nodally regularized meshfree method, Theor. Appl. Fract. Mech., 72, 89, 10.1016/j.tafmec.2014.04.006 Wu, 2016, A direct displacement smoothing meshfree particle formulation for impact failure modeling, Int. J. Impact Eng., 87, 169, 10.1016/j.ijimpeng.2015.03.013 Wang, 2019, A three dimensional two-level gradient smoothing meshfree method for rainfall induced landslide simulations, Front. Struct. Civ. Eng., 13, 337, 10.1007/s11709-018-0467-5 Pan, 2019, A momentum-consistent stabilization algorithm for Lagrangian particle methods in the thermo-mechanical friction drilling analysis, Comput. Mech., 64, 625, 10.1007/s00466-019-01673-8 Huang, 2020, A MUSCL-SCNI approach for meshfree modeling of shock waves in fluids, Comput. Part. Mech., 7, 329, 10.1007/s40571-019-00248-x Duan, 2012, Second-order accurate derivatives and integration schemes for meshfree methods, Internat. J. Numer. Methods Engrg., 92, 399, 10.1002/nme.4359 Duan, 2014, Consistent element-free Galerkin method, Internat. J. Numer. Methods Engrg., 99, 79, 10.1002/nme.4661 Wang, 2019, Consistent integration schemes for meshfree analysis of strain gradient elasticity, Comput. Methods Appl. Mech. Engrg., 357, 10.1016/j.cma.2019.112601 Chen, 2013, An arbitrary order variationally consistent integration for Galerkin meshfree methods, Internat. J. Numer. Methods Engrg., 95, 387, 10.1002/nme.4512 Hillman, 2014, Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems, Comput. Part. Mech., 1, 245, 10.1007/s40571-014-0024-5 Wang, 2016, An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods, Comput. Methods Appl. Mech. Engrg., 298, 485, 10.1016/j.cma.2015.10.008 Wang, 2019, An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature, Comput. Methods Appl. Mech. Engrg., 349, 628, 10.1016/j.cma.2019.02.029 Wu, 2020, An efficient gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture, Comput. Part. Mech., 7, 193, 10.1007/s40571-019-00240-5 Strang, 1972, Variational crimes in finite element method, 689 Strang, 2018 Liu, 1996, Advances in multiple scale kernel particle methods, Comput. Mech., 18, 73, 10.1007/BF00350529 Liu, 1997, Moving least-square reproducing kernel methods (I) methodology and convergence, Comput. Methods Appl. Mech. Engrg., 143, 113, 10.1016/S0045-7825(96)01132-2 Chen, 2003, A reproducing kernel method with nodal interpolation property, Internat. J. Numer. Methods Engrg., 56, 935, 10.1002/nme.592 Han, 2001, Error analysis of the reproducing kernel particle method, Comput. Methods Appl. Mech. Engrg., 190, 6157, 10.1016/S0045-7825(01)00214-6 Babuška, 2008, Quadrature for meshless methods, Internat. J. Numer. Methods Engrg., 76, 1434, 10.1002/nme.2367 Rüter, 2017, An enhanced-strain error estimator for Galerkin meshfree methods based on stabilized conforming nodal integration, Comput. Math. Appl., 74, 2144, 10.1016/j.camwa.2017.06.052 Fernández-Méndez, 2004, Imposing essential boundary conditions in mesh-free methods, Comput. Methods Appl. Mech. Engrg., 193, 1257, 10.1016/j.cma.2003.12.019 Quarteroni, 1994 Burden, 2010 Davis, 2007 Philippe, 2013 Hughes, 2000 Zienkiewicz, 2015 Timoshenko, 1970