Upper bound shakedown analysis with the nodal natural element method. (English) Zbl 1311.74135
Summary: A novel numerical solution procedure is developed for the upper bound shakedown analysis of elastic-perfectly plastic structures. The nodal natural element method (nodal-NEM) combines the advantages of the NEM and the stabilized conforming nodal integration scheme, and is used to discretize the established mathematical programming formulation of upper bound shakedown analysis based on Koiter’s theorem. In this formulation, the displacement field is approximated by using the Sibson interpolation and the difficulty caused by the time integration is solved by König’s technique. Meanwhile, the nonlinear and non-differentiable characteristic of objective function is overcome by distinguishing non-plastic areas from plastic areas and modifying associated constraint conditions and goal function at each iteration step. Finally, the objective function subjected to several equality constraints is linearized and the upper bound shakedown load multiplier is obtained. This direct iterative process can ensure the shakedown load to monotonically converge to the upper bound of true solution. Several typical numerical examples confirm the efficiency and accuracy of the proposed method.
Keywords:
shakedown analysis; kinematic theorem; Sibson interpolation; stabilized conforming nodal integration; nodal natural element methodReferences:
| [1] | Xu BY, Liu XS (1985) Plastic limit analysis of structures. China Architecture and Building Press, Beijing |
| [2] | Chen G, Liu YH (2006) Numerical theories and engineering methods for structural limit and Shakedown analyses. Science Press, Beijing |
| [3] | Melan E (1938) Zur Plastizitä t des räumlichen Kontinuums. Ing-Arch 9:115-126 · JFM 64.0840.01 · doi:10.1007/BF02084409 |
| [4] | Koiter WT (1956) A new general theorem on shakedown of elastic-plasitc structures. Proc Konink Ned Akad Wet B 59:24-34 · Zbl 0074.40801 |
| [5] | Kamenjarzh J, Merzljakov A (1994) On kinematic method in shakedown theory. I: duality of extremum problems. Int J Plast 10(4):363-380 · Zbl 0808.73023 · doi:10.1016/0749-6419(94)90038-8 |
| [6] | Kamenjarzh J, Merzljakov A (1994) On kinematic method in shakedown theory. II: modified kinematic method. Int J Plast 10(4):381-392 · Zbl 0808.73023 · doi:10.1016/0749-6419(94)90039-6 |
| [7] | Khoi VD, Yan AM, Hung ND (2004) A dual form for discretized kinematic formulation in shakedown analysis. Int J Solids Struct 41(1):267-277 · Zbl 1069.74005 · doi:10.1016/j.ijsolstr.2003.08.013 |
| [8] | Khoi VD, Yan AM, Hung ND (2004) A primal-dual algorithm for shakedown analysis of structures. Comput Methods Appl Mech Eng 193(42-44):4663-4674 · Zbl 1112.74546 |
| [9] | Tran TN, Liu GR, Nguyen-Xuan H, Nguyen-Thoi T (2010) An edge-based smoothed finite element method for primal-dual shakedown analysis of structures. Int J Numer Meth Eng 82(7):917-938 · Zbl 1188.74073 |
| [10] | Tran TN (2011) A dual algorithm for shakedown analysis of plate bending. Int J Numer Meth Eng 86(7):862-875 · Zbl 1235.74318 · doi:10.1002/nme.3081 |
| [11] | Stein E, Zhang GB, Konig JA (1992) Shakedown with nonlinear strain-hardening including structural computation using finite-element method. Int J Plast 8(1):1-31 · Zbl 0766.73026 · doi:10.1016/0749-6419(92)90036-C |
| [12] | Zhang YG (1995) An iteration algorithm for kinematic shakedown analysis. Comput Methods Appl Mech Eng 127(1-4):217-226 · Zbl 0860.73014 · doi:10.1016/0045-7825(95)00121-6 |
| [13] | Hachemi A, Weichert D (1998) Numerical shakedown analysis of damaged structures. Comput Methods Appl Mech Eng 160:57-70 · Zbl 0939.74058 · doi:10.1016/S0045-7825(97)00283-1 |
| [14] | Yan AM, Hung ND (2001) Kinematical shakedown analysis with temperature-dependent yield stress. Int J Numer Meth Eng 50(5):1145-1168 · Zbl 1047.74071 · doi:10.1002/1097-0207(20010220)50:5<1145::AID-NME70>3.0.CO;2-C |
| [15] | Maier G, Pan LG, Perego U (1993) Geometric effects on shakedown and ratchetting of axisymmetric cylindrical shells subjected to variable thermal loading. Eng Struct 15:453-465 · Zbl 0861.73045 · doi:10.1016/0141-0296(93)90063-A |
| [16] | Borino G, Polizzotto C (1996) Dynamic shakedown of structures with variable appended masses and subjected to repeated excitations. Int J Plast 12(2):215-228 · Zbl 0858.73044 · doi:10.1016/S0749-6419(96)00004-6 |
| [17] | Tran TN, Kreissig R, Staat M (2009) Probabilistic limit and shakedown analysis of thin plates and shells. Struct Saf 31(1):1-18 · doi:10.1016/j.strusafe.2007.10.003 |
| [18] | Carvelli V (2004) Shakedown analysis of unidirectional fiber reinforced metal matrix composites. Comput Mater Sci 31(1-2):24-32 · doi:10.1016/j.commatsci.2004.01.030 |
| [19] | Li HX, Yu HS (2006) A non-linear programming approach to kinematic shakedown analysis of composite materials. Int J Numer Meth Eng 66(1):117-146 · Zbl 1110.74836 · doi:10.1002/nme.1547 |
| [20] | Li HX, Yu HS (2006) A nonlinear programming approach to kinematic shakedown analysis of frictional materials. Int J Solids Struct 43(21):6594-6614 · Zbl 1120.74347 · doi:10.1016/j.ijsolstr.2006.01.009 |
| [21] | Feng XQ, Liu XS (1997) On shakedown of three-dimensional elastoplastic strain-hardening structures. Int J Plast 12(10):1241-1256 · Zbl 0895.73017 · doi:10.1016/S0749-6419(95)00050-X |
| [22] | Feng XQ, Sun QP (2007) Shakedown analysis of shape memory alloy structures. Int J Plast 23(2):183-206 · Zbl 1127.74316 · doi:10.1016/j.ijplas.2006.04.001 |
| [23] | Carvelli V, Cen ZZ, Liu YH, Maier G (1999) Shakedown analysis of defective pressure vessels by a kinematic approach. Arch Appl Mech 69(9-10):751-764 · Zbl 0969.74582 · doi:10.1007/s004190050254 |
| [24] | Chen HF, Ure J, Li TB, Chen WH, Mackenzie D (2011) Shakedown and limit analysis of \[90^{\circ }90\]∘ pipe bends under internal pressure, cyclic in-plane bending and cyclic thermal loading. Trans ASME J Press Vessel Technol 88(5-7):213-222 |
| [25] | Gross-Wedge J (1997) On the numerical assessment of the safety factor of elastic-plastic structures under variable loading. Int J Mech Sci 39(4):417-433 · Zbl 0891.73051 · doi:10.1016/S0020-7403(96)00039-2 |
| [26] | Shiau SH (2001) Numerical methods for shakedown analysis of pavements under moving surface loads. PhD thesis, University of Newcastle, NSW, Australia · Zbl 1011.74081 |
| [27] | Konig JA, Maier G (1981) Shakedown analysis of elastoplastic structures: a review of recent developments. Nucl Eng Des 66(1):81-95 · doi:10.1016/0029-5493(81)90183-7 |
| [28] | Panzeca T (1992) Shakedown and limit analysis by the boundary integral equation method. Eur J Mech A-Solids 11(5):685-699 · Zbl 0765.73073 |
| [29] | Zhang XF, Liu YH, Cen ZZ (2004) Boundary element methods for lower bound limit and shakedown analysis. Eng Anal Boundary Elem 28(8):905-917 · Zbl 1130.74477 · doi:10.1016/S0955-7997(03)00117-6 |
| [30] | Liu YH, Zhang XF, Cen ZZ (2005) Lower bound shakedown analysis by the symmetric Galerkin boundary element method. Int J Plast 21(1):21-42 · Zbl 1112.74552 · doi:10.1016/j.ijplas.2004.01.003 |
| [31] | Belytschko T, Lu YY, Gu L (1994) Element free Galerkin method. Int J Numer Meth Eng 37(2):229-256 · Zbl 0796.73077 · doi:10.1002/nme.1620370205 |
| [32] | Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle method. Int J Numer Meth Fluids 20(8-9):1081-1106 · Zbl 0881.76072 · doi:10.1002/fld.1650200824 |
| [33] | Liszka TJ, Duarte CAM, Tworzydlo WW (1996) hp-meshless cloud method. Comput Methods Appl Mech Eng 139(1-4):263-288 · Zbl 0893.73077 · doi:10.1016/S0045-7825(96)01086-9 |
| [34] | Atluri SN, Zhu T (1998) A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech 22(2):117-127 · Zbl 0932.76067 · doi:10.1007/s004660050346 |
| [35] | Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Meth Eng 43(5):839-887 · Zbl 0940.74078 · doi:10.1002/(SICI)1097-0207(19981115)43:5<839::AID-NME423>3.0.CO;2-R |
| [36] | Zhang XG, Liu XH, Song KZ, Lu MW (2001) Least-squares collocation meshless method. Int J Numer Meth Eng 51(9):1089- 1100 · Zbl 1056.74064 |
| [37] | Liu GR, Wang JG (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Meth Eng 54(11):1623-1648 · Zbl 1098.74741 · doi:10.1002/nme.489 |
| [38] | Gu L (2003) Moving Kriging interpolation and element free Galerkin method. Int J Numer Meth Eng 56(1):1-11 · Zbl 1062.74652 · doi:10.1002/nme.553 |
| [39] | 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):370-394 · Zbl 1110.65110 · doi:10.1016/j.jcp.2006.06.023 |
| [40] | Bessa MA, Foster JT, Belytschko T, Liu WK (2013) A meshfree unification: reproducing kernel peridynamics. Comput Mech. doi:10.1007/s00466-013-0969-x · Zbl 1398.74452 |
| [41] | Yoon YC, Song JH (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. doi:10.1007/s00466-013-0950-8 · Zbl 1398.74482 |
| [42] | 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. doi:10.1007/s00466-013-0951-7 · Zbl 0893.73077 |
| [43] | Chen SS, Liu YH, Cen ZZ (2008) Lower bound shakedown analysis by using the element free Galerkin method and nonlinear programming. Comput Methods Appl Mech Eng 197:3911-3921 · Zbl 1194.74514 · doi:10.1016/j.cma.2008.03.009 |
| [44] | Chen SS, Liu YH, Li J, Cen ZZ (2010) Performance of the MLPG method for static shakedown analysis for bounded kinematic hardening structures. Eur J Mech A/Solids 30(2):183-194 · Zbl 1261.74034 · doi:10.1016/j.euromechsol.2010.10.005 |
| [45] | Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139(1-4):49-74 · Zbl 0918.73329 · doi:10.1016/S0045-7825(96)01079-1 |
| [46] | Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Meth Eng 50(2):435-466 · Zbl 1011.74081 · doi:10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A |
| [47] | Chen JS, Yoon S, Wu CT (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Meth Eng 53(12):2587-2615 · Zbl 1098.74732 · doi:10.1002/nme.338 |
| [48] | Chen JS, Wu CT, Belytschko T (2000) Regularization of material instabilities by meshfree approximations with intrinsic length scales. Int J Numer Meth Eng 47(7):1303-1322 · Zbl 0987.74079 · doi:10.1002/(SICI)1097-0207(20000310)47:7<1303::AID-NME826>3.0.CO;2-5 |
| [49] | Wang D, Chen JS (2004) Locking-free stabilized conforming nodal integration for Mindlin-Reissner plate. Comput Methods Appl Mech Eng 193(12-14):1065-1083 · Zbl 1060.74675 |
| [50] | Wang D, Chen JS (2006) A locking-free meshfree curved beam formulation with the stabilized conforming nodal integration. Comput Mech 39(1):83-90 · Zbl 1168.74472 · doi:10.1007/s00466-005-0010-0 |
| [51] | Wang D, Chen JS (2008) A Hermite reproducing kernel approximation for thin plate analysis with sub-domain stabilized conforming integration. Int J Numer Meth Eng 74(3):368-390 · Zbl 1159.74460 · doi:10.1002/nme.2175 |
| [52] | Wang D, Lin Z (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(1):47-63 · Zbl 1398.74478 · doi:10.1007/s00466-011-0580-y |
| [53] | Yoo JW, Moran B, Chen JS (2004) Stabilized conforming nodal integration in the natural-element method. Int J Numer Meth Eng 60:861-890 · Zbl 1060.74677 · doi:10.1002/nme.972 |
| [54] | Puso MA, Chen JS, Zywicz E, Elmer W (2008) Meshfree and finite element nodal integration methods. Int J Numer Meth Eng 74:416-446 · Zbl 1159.74456 · doi:10.1002/nme.2181 |
| [55] | Sibson R (1980) Vector identity for the dirichlet tessellation. Math Proc Camb Philos Soc 87:151-155 · Zbl 0466.52010 · doi:10.1017/S0305004100056589 |
| [56] | Braun J, Sambridge M (1995) A numerical method for solving partial differential equations on highly irregular evolving grids. Nature 376:655-660 · doi:10.1038/376655a0 |
| [57] | Sukumar N, Moran B, Semenov AY, Belikov VV (2001) Natural neighbour Galerkin methods. Int J Numer Meth Eng 50(1):1-27 · Zbl 1082.74554 · doi:10.1002/1097-0207(20010110)50:1<1::AID-NME14>3.0.CO;2-P |
| [58] | Cueto E, Sukumar N, Calvo B, Martinez MA, Cegonino J, Doblare M (2003) Overview and recent advances in natural neighbour Galerkin methods. Arch Comput Methods Eng 10(4):307-384 · Zbl 1050.74001 |
| [59] | Gonzalez D, Cueto E, Doblare M (2004) Volumetric locking in natural neighbour Galerkin methods. Int J Numer Meth Eng 61(4):611-632 · Zbl 1124.74328 |
| [60] | Thiessen AH (1911) Precipitation averages for large areas. Mon Weather Rep 39:1082-1084 |
| [61] | Belikov VV, Ivanov VD, Kontorovich VK, Korytnik SA (1997) The non-Sibsonian interpolation: a new method of interpolation of the values of a function on an arbitrary set of points. Comput Math Math Phys 37(1):9-15 · Zbl 0948.65005 |
| [62] | Hiyoshi H, Sugihara K (1999) Two generalizations of an interpolant based on Voronoi diagrams. Int J Shape Model 5(2):219-231 · doi:10.1142/S0218654399000186 |
| [63] | Alfaro I, Yvonnet J, Chinesta F, Cueto E (2007) A study on the performance of natural neighbour-based Galerkin methods. Int J Numer Meth Eng 71(12):1436-1465 · Zbl 1194.74509 · doi:10.1002/nme.1993 |
| [64] | Cueto E, Doblare M, Gracia L (2000) Imposing essential boundary conditions in the natural element method by means of density-scaled a-shapes. Int J Numer Meth Eng 49:519-546 · Zbl 0989.74077 · doi:10.1002/1097-0207(20001010)49:4<519::AID-NME958>3.0.CO;2-0 |
| [65] | Zhou ST, Liu YH (2012) Upper-bound limit analysis based on the natural element method. Acta Mech Sin 28(5):1398-1415 · Zbl 1345.74106 · doi:10.1007/s10409-012-0149-9 |
| [66] | Himmelblau DM (1972) Appl Nonlinear Progr. McGraw-Hill Book Company, New York · Zbl 0241.90051 |
| [67] | Belytschko T (1972) Plane stress shakedown analysis by finite elements. Int J Mech Sci 14(9):619-625 · doi:10.1016/0020-7403(72)90061-6 |
| [68] | Corradi L, Zavelani A (1974) A linear programming approach to shakedown analysis of structures. Comput Methods Appl Mech Eng 3(1):37-53 · doi:10.1016/0045-7825(74)90041-3 |
| [69] | Nguyen DH, Palgen L (1980) Shakedown analysis by displacement method and equilibrium finite elements. Trans Can Soc Mech Eng 61:34-40 |
| [70] | Genna F (1988) A nonlinear inequality, finite element approach to the direct computation of shakedown load safety factors. Int J Mech Sci 30(10):769-789 · Zbl 0669.73027 · doi:10.1016/0020-7403(88)90041-0 |
| [71] | Zouain N, Borges L, Silveira JL (2002) An algorithm for shakedown analysis with nonlinear yield functions. Comput Methods Appl Mech Eng 191(23-24):2463-2481 · Zbl 1054.74064 · doi:10.1016/S0045-7825(01)00374-7 |
| [72] | Garcea G, Armentano G, Petrolo S, Casciaro R (2005) Finite element shakedown analysis of two-dimensional structures. Int J Numer Meth Eng 63(8):1174-1202 · Zbl 1084.74052 · doi:10.1002/nme.1316 |
| [73] | Tin-Loi F, Ngo NS (2007) Performance of a p-adaptive finite element method for shakedown analysis. Int J Mech Sci 49(10):1168-1178 · doi:10.1016/j.ijmecsci.2007.02.004 |
| [74] | Zhang TG, Raad L (2002) An eigen-mode method in kinematic shakedown analysis. Int J Plast 18(1):71-90 · Zbl 1035.74011 · doi:10.1016/S0749-6419(00)00055-3 |
| [75] | Krabbenhoft K, Lyamin AV, Sloan SW (2007) Bounds to shakedown loads for a class of deviatoric plasticity models. Comput Mech 39(6):879-888 · Zbl 1160.74008 · doi:10.1007/s00466-006-0076-3 |
| [76] | Chen SS (2009) Lower-bound limit and Shakedown analysis based on meshless methods. PhD thesis, Tsinghua University, Beijing, People’s Republic of China |
| [77] | Vu DK (2001) Dual limit and shakedown analysis of structures, dissertation. Universite de Liege, Belgium |
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