Silling S, Epton M, Weckner O, Xu J, Askari E (2007) Peridynamic states and constitutive modeling. J Elast 88(2):151. doi:10.1007/s10659-007-9125-1
Silling S, Lehoucq R (2008) Convergence of peridynamics to classical elasticity theory. J Elast 93(1):13. doi:10.1007/s10659-008-9163-3
Silling S (2000) Reformulation of elasticity theory for discontinuities and long-range. J Mech Phys Solids 48(1):175. doi:10.1016/S0022-5096(99)00029-0
Warren TL, Silling SA, Askari A, Weckner O, Epton MA, Xu J (2009) A non-ordinary state-based peridynamic method to model solid material deformation and fracture. Int J Solids Struct 46(5):1186. doi:10.1016/j.ijsolstr.2008.10.029
Foster JT, Silling SA, Chen WW (2010) Viscoplasticity using peridynamics. Int J Numer Methods Eng 81(10):1242. doi:10.1002/nme.2725
Tupek M, Rimoli J, Radovitzky R (2013) An approach for incorporating classical continuum damage models in state-based peridynamics. Comput Methods Appl Mech Eng 263:20. doi:10.1016/j.cma.2013.04.012
Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9):1081. doi:10.1002/fld.1650200824
Li S, Liu WK (2002) Meshfree and particle methods and their applications. Appl Mech Rev 55(1):1
Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139(1–4):3. doi:10.1016/S0045-7825(96)01078-X
Li S, Liu WK (2004) Meshfree particle methods. Springer, Berlin
Nguyen VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79(3):763. doi:10.1016/j.matcom.2008.01.003
Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10(5):307. doi:10.1007/BF00364252
Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229. doi:10.1002/nme.1620370205
Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38(10):1655. doi:10.1002/nme.1620381005
Liu WK, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods (I) Methodology and convergence. Comput Methods Appl Mech Eng 143(1–2):113. doi:10.1016/S0045-7825(96)01132-2
Liu WK, Chen Y, Uras R, Chang CT (1996) Generalized multiple scale reproducing kernel particle methods. Comput Methods Appl Mech Eng 139(1–4):91. doi:10.1016/S0045-7825(96)01081-X
Liu W, Chen Y, Chang C, Belytschko T (1996) Advances in multiple scale kernel particle methods. Comput Mech 18(2):73. doi:10.1007/BF00350529
Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics-theory and application to non-spherical stars. Mon Notices R Astron Soc 181:375
Oñate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) Finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Eng 39(22):3839
Kim DW, Liu WK (2006) Maximum principle and convergence analysis for the meshfree point collocation method. SIAM J Numer Anal 44(2):515
Hu HY, Chen JS, Hu W (2011) Error analysis of collocation method based on reproducing kernel approximation. Numer Methods Partial Differ Equ 27(3):554. doi:10.1002/num.20539
Li S, Liu WK (1998) Synchronized reproducing kernel interpolant via multiple wavelet expansion. Comput Mech 21(1):28. doi:10.1007/s004660050281
Kim DW, Yoon YC, Liu WK, Belytschko T (2007) Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative. J Comput Phys 221(1):370. doi:10.1016/j.jcp.2006.06.023
Yoon YC, Song JH (2013) Extended particle difference method for weak and strong discontinuity problems: part II. Formulations and applications for various interfacial singularity problems. Comput Mech 13:1–24. doi:10.1007/s00466-013-0951-7
Yoon YC, Song YH (2013) Extended particle difference method for weak and strong discontinuity problems: part I. Derivation of the extended particle derivative approximation for the representation of weak and strong discontinuities. Comput Mech 3:1–17. doi:10.1007/s00466-013-0950-8
Fornberg B (1988) Generation of finite difference formulas on arbitrarily spaced grids. Math Comput 51(184):699
Lanczos C (1956) Applied analysis. Prentice-Hall, Old Tappan
Savitzky A, Golay MJE (1964) Smoothing and differentiation of data by simplified least squares procedures. Anal Chem 36(8):1627. doi:10.1021/ac60214a047
Liu GR, Gu Y (2001) A point interpolation method for two-dimensional solids. Int J Numer Methods Eng 50(4):937
Liu GR (2010) Meshfree methods: moving beyond the finite element method. CRC Press, Boca Raton
Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Methods Eng 54(11):1623. doi:10.1002/nme.489
Liu G, Kee BB, Chun L (2006) A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis. Comput Methods Appl Mech Eng 195(37):4843
Hu HY, Chen JS, Hu W (2007) Weighted radial basis collocation method for boundary value problems. Int J Numer Methods Eng 69(13):2736. doi:10.1002/nme.1877
Wang L, Chen JS, Hu HY (2010) Subdomain radial basis collocation method for fracture mechanics. Int J Numer Methods Eng 83(7):851. doi:10.1002/nme.2860
Auricchio F, Veiga LBD, Hughes TJR, Reali A, Sangalli G (2010) Isogeometric collocation methods. Math Models Methods Appl Sci 20(11):2075. doi:10.1142/S0218202510004878
Auricchio F, da Veiga LB, Hughes T, Reali A, Sangalli G (2012) Isogeometric collocation for elastostatics and explicit dynamics. Comput Methods Appl Mech Eng 2:249–252. doi:10.1016/j.cma.2012.03.026
Lehoucq R, Silling S (2008) Force flux and the peridynamic stress tensor. J Mech Phys Solids 56(4):1566. doi:10.1016/j.jmps.2007.08.004
Zhou K, Du Q (2010) Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J Numer Anal 48(5):1759. doi:10.1137/090781267
Silling S, Lehoucq R (2010) Peridynamic theory of solid mechanics. Adv Appl Mech 44:73. doi:10.1016/S0065-2156(10)44002-8
Seleson P, Parks M (2011) On the role of the influence function in the peridynamic theory. Int J Multiscale Comput Eng 9(6):689
Diekema E, Koornwinder TH (2012) Differentiation by integration using orthogonal polynomials, a survey. J Approx Theory 164(5):637. doi:10.1016/j.jat.2012.01.003
Foster J, Silling SA, Chen W (2011) An energy based failure criterion for use with peridynamic states. Int J Multiscale Comput Eng 9(6):675
Zhu T, Atluri S (1998) A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Comput Mech 21(3):211
Fernández-Méndez S, Huerta A (2004) Imposing essential boundary conditions in mesh-free methods. Comput Methods Appl Mech Eng 193(12):1257
Tian R (2013) Extra-dof-free and linearly independent enrichments in GFEM. Comput Meth Appl Mech Eng 266:1. doi:10.1016/j.cma.2013.07.005
Chen JS, Hu W, Hu HY (2008) Reproducing kernel enhanced local radial basis collocation method. Int J Numer Methods Eng 75(5):600. doi:10.1002/nme.2269
Chi SW, Chen JS, Luo H, Hu HY, Wang L (2012) Dispersion and stability properties of radial basis collocation method for elastodynamics. Numer Methods Partial Diff Equ 29:818–842. doi:10.1002/num.21732
Hamming RW (1989) Digital filters. Courier Dover Publications, Cambridge