A meshfree unification: reproducing kernel peridynamics
Tóm tắt
This paper is the first investigation establishing the link between the meshfree state-based peridynamics method and other meshfree methods, in particular with the moving least squares reproducing kernel particle method (RKPM). It is concluded that the discretization of state-based peridynamics leads directly to an approximation of the derivatives that can be obtained from RKPM. However, state-based peridynamics obtains the same result at a significantly lower computational cost which motivates its use in large-scale computations. In light of the findings of this study, an update to the method is proposed such that the limitations regarding application of boundary conditions and the use of non-uniform grids are corrected by using the reproducing kernel approximation.
Tài liệu tham khảo
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