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Adaptive FEM analysis of selected elastic-visco-plastic problems. (English) Zbl 1173.74402
From the abstract: Elastic-visco-plastic deformations modeled by the Bodner-Partom constitutive equations have been evaluated by the adaptive finite element method. We use the explicit residual error estimate supplemented with a special term accounting for the fact that the right hand side of the equations we use is only approximated since it depends on the inelastic strains which are computed only at the Gauss integration points. We also address the problems of adaptation strategy and the method of solution transfer to a new mesh. Results of recent tests confirm efficiency and robustness of the approach.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
Software:
2Dhp90
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References:
[1] Ainsworth, M.; Oden, J.T., A posteriori error estimation in finite element analysis, (2000), J. Wiley & Sons · Zbl 1008.65076
[2] Akin, J.E., Finite elements for analysis and design, (1995), Academic Press · Zbl 0877.73059
[3] Anderson, H., An implicit formulation of the bodner – partom constitutive equations, Comput. struct., 81, 1405-1414, (2003)
[4] Babuška, I., The finite element method with Lagrangian multipliers, Numer. math., 20, 179-192, (1973) · Zbl 0258.65108
[5] Babuška, I.; Melenk, J., The partition of unity method, Int. J. numer. methods engrg., 40, 727-758, (1997) · Zbl 0949.65117
[6] Babuška, I.; Miller, A., A feedback finite element method with a posteriori error estimation. part 1, Comput. methods appl. mech. engrg., 61, 1-40, (1987) · Zbl 0593.65064
[7] Babuška, I.; Rheinboldt, W.C., Error estimates for adaptive finite element computations, Int. J. numer. methods engrg., 12, 1597-1615, (1978) · Zbl 0396.65068
[8] Babuška, I.; Strouboulis, T.; Upadhyay, C.S., A model study of the quality of a posteriori error estimators for linear elliptic problems. error estimation in the interior of patchwise uniform grids of triangles, Comput. methods appl. mech. engrg., 114, 307-378, (1994)
[9] Babuška, I.; Suri, M., The p and hp versions of the finite element methods, basic principles and properties, SIAM rev., 36, 578-632, (1994) · Zbl 0813.65118
[10] Barlow, J., Optimal stress locations in finite element models, Int. J. numer. methods engrg., 10, 243-251, (1976) · Zbl 0322.73049
[11] Bass, J.M.; Oden, J.T., Adaptive finite element methods for a class of evolution problems in viscoplasticity, Int. J. engrg. sci., 25, 623-653, (1987) · Zbl 0612.73072
[12] Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, Int. J. numer. methods engrg., 45, 601-620, (1999) · Zbl 0943.74061
[13] Belytschko, T.; Tabbana, M., h-adaptive finite element method for dynamic problems with emphasis on localization, Int. J. numer. methods engrg., 36, 4245-4265, (1993) · Zbl 0794.73071
[14] Bodner, S.R., Review of a unified elasto – viscoplastic theory, () · Zbl 0800.73108
[15] Bodner, S.R., Unified plasticity for engineering applications, (2002), Kluwer Academic
[16] Bodner, S.R.; Partom, Y., Constitutive equations elastic viscoplastic strain-hardening materials, J. appl. mech., 42, 385-389, (1975)
[17] Boroomand, B.; Zienkiewicz, O., Recovery by equilibrium in patches (REP), Int. J. numer. methods engrg., 40, 137-164, (1997)
[18] Braess, D., Finite elements, theory, fast solvers, and applications in solid mechanics, (1997), Cambridge University Press · Zbl 0894.65054
[19] Brezzi, F.; Fortin, M., Mixed and hybrid finite element method, (1991), Springer Verlag · Zbl 0788.73002
[20] Brezzi, F.; Marini, L.D., A survey on mixed finite element methods, IEEE trans. magn., 30, 5, 3547-3551, (1994)
[21] Carstensen, C.; Alberty, J., Averaging techniques for reliable a posteriori fe-error control in elastoplasticity with hardening, Comput. methods appl. mech. engrg., 192, 1435-1450, (2003) · Zbl 1026.74067
[22] Cecot, W., Adaptive integration of bodner – partom’s constitutive model, ()
[23] Cecot, W., Analysis of selected in-elastic problems by h-adaptive finite element method, Mechanics, vol. 323, (2005), Cracow University of Technology Press
[24] Cecot, W., On application of the boundary element methods for analysis of the shakedown problems, J. theor. appl. mech., 36, 387-401, (1998) · Zbl 0929.74117
[25] Cecot, W., Application of h-adaptive FEM and zarka’s approach to analysis of shakedown problems, Int. J. numer. methods engrg., 61, 2139-2158, (2004) · Zbl 1075.74639
[26] Cecot, W.; Rachowicz, W., Adaptive solution of problems modeled by unified state variable constitutive equations, Comput. assist. mech. engrg. sci., 7, 479-492, (2000) · Zbl 0964.74508
[27] Demkowicz, L., Selected topics from mathematical theory of finite elements, (), 47-84, (Chapter 2)
[28] Demkowicz, L.; Oden, J.T.; Rachowicz, W.; Hardy, O., Toward a universal hp-adaptive finite element strategy. part 1: constrained approximation and data structure, Comput. methods appl. mech. engrg., 77, 79-112, (1989) · Zbl 0723.73074
[29] L. Demkowicz, W. Rachowicz, K. Banas, J. Kucwaj, 2-D hp adaptive package, Technical Report 4/92, Cracow University of Technology, 1992.
[30] L. Demkowicz, W. Rachowicz, K. Gerdes, A. Bajer, 2D hp-adaptive finite element package, Fortran 90 implementation (2Dhp90), Technical Report 98-14, TICAM, http://www.ticam.utexas.edu, 1998.
[31] Dyduch, M.; Habraken, A.M.; Cescotto, S., Automatic adaptive remeshing for numerical simulations of metalforming, Comput. methods appl. mech. engrg., 101, 283-298, (1992) · Zbl 0777.73064
[32] Fotiu, P., A modified generalized midpoint rule for the integration of rate-dependent thermo-elastic—plastic constitutive equations, Came, 122, 105-129, (1995) · Zbl 0854.73078
[33] Gallimard, L.; Ladeveze, P.; Pelle, J.P., An enhanced error estimator on the constitutive relation for plasticity problems, Comput. struct., 78, 801-810, (2000)
[34] Gratsch, T.; Bathe, K.J., A posteriori error estimation techniques in practical finite element analysis, Comput. struct., 83, 235-265, (2005)
[35] Hinton, E.; Campbell, J., Local and global smoothing of discontinued finite element functions using a least square method, Int. J. numer. methods engrg., 8, 461-480, (1974) · Zbl 0286.73066
[36] Huerta, A.; Diez, P., Error estimation including pollution assessment for nonlinear finite element analysis, Comput. methods appl. mech. engrg., 181, 21-41, (2000) · Zbl 0964.74067
[37] Ionescu, I.R.; Sofonea, M., Functional and numerical methods in viscoplasticity, (1993), Oxford Univ. Press · Zbl 0787.73005
[38] Johnson, C.; Hansbo, P., Adaptive finite element methods in computational mechanics, Comput. methods appl. mech. engrg., 101, 143181, (1992)
[39] Kelly, D.W.; Gago, J.; Zienkiewicz, O.C.; Babuška, I., A posteriori error analysis and adaptive process in finite element method. part 1: error analysis, Int. J. numer. methods engrg., 19, 1593-1619, (1983) · Zbl 0534.65068
[40] J. Krok, J. Orkisz, Application of the generalized FD approach to stress evaluation in the FE solution, in: Proceedings of the Int. Conf. on Comp. Mech., Tokyo, 1986, pp. 31-36.
[41] Kumar, V.; Morjaria, M.; Mukherjee, S., Numerical integration of some stiff constitutive models of inelastic deformation, Trans. ASME, 102, 92-96, (1980)
[42] Lackner, R.; Mang, H., Adaptive FE analysis of RC shells I: theory, J. engrg. mech., 127, 1203-1212, (2001)
[43] Lackner, R.; Mang, H., Adaptive FE analysis of RC shells II: applications, J. engrg. mech., 127, 1213-1222, (2001)
[44] Ladeveze, P.; Moes, N.; Douchin, B., Constitutive relation error estimators for (visco)plastic analysis with softening, Comput. methods appl. mech. engrg., 176, 247-264, (1999) · Zbl 0948.74062
[45] Ladeveze, P.; Pelle, J.P.; Rougeot, P., Error estimate procedure in the finite element method and application, SIAM J. numer. anal., 20, 485-509, (1983) · Zbl 0582.65078
[46] Lancaster, P.; Salkauskas, K., Surface generated by moving least squares methods, Math. comput., 155, 37, 141-158, (1981) · Zbl 0469.41005
[47] Lee, N.S.; Bathe, K.J., Error indicators and adaptive remeshing in large deformation finite element analysis, Finite element anal. des., 16, 99-139, (1994) · Zbl 0804.73064
[48] T. Liszka, J. Orkisz, Finite difference method of arbitrary irregular meshes in non-linear problems of applied mechanics, in: Proceedings of the XIV Polish Conference on Computer Methods, 1977. · Zbl 0453.73086
[49] Martshuk, G.I., Numerical analysis of the problems of mathematical physics, (1983), PWN, (in Polish)
[50] Min, J.B.; Tworzydlo, W.W.; Xiques, K.E., Adaptive finite element methods for continuum damage modeling, Comput. struct., 58, 887-900, (1995) · Zbl 0900.73811
[51] Oden, J.T.; Brauchli, H., On the calculation of consistent stress distribution in the finite element approximations, Int. J. numer. methods engrg., 3, 317-325, (1971) · Zbl 0251.73056
[52] Oden, J.T.; Carey, G.F., Finite elements. mathematical aspects, vol. IV, (1983), Prentice-Hall · Zbl 0496.65055
[53] Oden, J.T.; Demkowicz, L.; Rachowicz, W.; Westermann, T.A., Toward a universal hp-adaptive finite element strategy. part 2: A posteriori error estimation, Comput. methods appl. mech. engrg., 77, 113-180, (1989) · Zbl 0723.73075
[54] Orlando, A.; Peric, D., Analysis of transfer procedures in elastoplasticity based on the error in the constitutive equations: theory and numerical illustration, Int. J. numer. methods engrg., 60, 1595-1631, (2004) · Zbl 1060.74519
[55] Peric, D.; Hochard, Ch.; Dutko, M.; Owen, D., Transfer operators for evolving meshes in small strain elasto-plasticity, Comput. methods appl. mech. engrg., 137, 331-344, (1996) · Zbl 0884.73071
[56] Peric, D.; Yu, J.; Owen, D., On error estimates and adaptivity in elastoplastic solids, Int. J. numer. methods engrg., 37, 1351-1379, (1994) · Zbl 0805.73066
[57] Press, W.H., Numerical recipes, (1992), Cambridge University Press · Zbl 0778.65003
[58] Rachowicz, W., Evaluation and comparison of residual error estimates with selfequilibrated fluxes in three dimensions, ()
[59] Rachowicz, W.; Oden, J.T.; Demkowicz, L., Toward a universal hp-adaptive finite element strategy. part 3: design of hp meshes, Comput. methods appl. mech. engrg., 77, 181-212, (1989) · Zbl 0723.73076
[60] Repin, S.; Xanthis, L., A posteriori error estimation for elasto-plastic problems based on duality theory, Comp. methods appl. mech. engrg., 138, 317-339, (1996) · Zbl 0886.73082
[61] Sandhu, J.S.; Liebowitz, H., Examples of adaptive FEA in plasticity, Engrg. fract. mech., 50, 947-956, (1995)
[62] Schatzman, M., Numerical analysis. A mathematical introduction, (2002), Oxford University Press
[63] Shillor, M.; Sofonea, M.; Telega, J., Models and analysis of quasistatic contact, (2004), Springer-Verlag · Zbl 1180.74046
[64] Simo, J.C.; Hughes, T.J.R., Computational inelasticity, (1998), Springer-Verlag · Zbl 0934.74003
[65] ()
[66] Stein, E.; Schmidt, M., Adaptive FEM for elasto-plastic deformations, (), 53-108, (Chapter 3)
[67] Steward, J.R.; Hughes, T.J., A tutorial in elementary finite element error analysis: a systematic presentation of a priori and a posteriori error estimation, Comput. methods appl. mech. engrg., 158, 1-22, (1998) · Zbl 0945.65121
[68] Suli, E.; Mayers, D., An introduction to numerical analysis, (2003), Cambridge University Press · Zbl 1033.65001
[69] Szefer, G., Variational principles in computational mechanics, (), 65-84, (in Polish)
[70] Tworzydło, W.; Cecot, W.; Oden, J.T.; Yew, C.H., Computational micro- and macroscopic models of contact and friction: formulation, approach and applications, Wear, 220, 113-140, (1998)
[71] Verfurth, R., A posteriori error estimates for nonlinear problems. finite element discretizations of elliptic equations, Math. comput., 62, 445-475, (1994) · Zbl 0799.65112
[72] Verfurth, R., A review of a posteriori error estimation techniques for elasticity problems, Comput. methods appl. mech. engrg., 176, 419-440, (1999) · Zbl 0935.74072
[73] Wheeler, M.F.; Whiteman, J.R., Superconvergent recovery of gradients on subdomains from piecewise linear approximations, Numer. methods part. diff. equations, 3, 65-82, (1987) · Zbl 0706.65107
[74] Wiberg, N.E.; Abdulwhab, F., A posteriori error estimation based on superconvergent derivatives and equilibrium, Int. J. numer. methods engrg., 36, 2703-2724, (1993) · Zbl 0800.73463
[75] Wiberg, N.E.; Abdulwhab, F.; Ziukas, S., Enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions, Int. J. numer. methods engrg., 37, 3417-3440, (1994) · Zbl 0825.73777
[76] Wunderlich, W.; Cramer, H.; Steinl, G., An adaptive finite element approach in associated and non-associated plasticity considering localization phenomena, (), 293-332 · Zbl 1011.74067
[77] Zienkiewicz, O.C.; Zhu, J.Z., A simple error estimate and adaptive procedure for practical engineering analysis, Int. J. numer. methods engrg., 24, 337-357, (1987) · Zbl 0602.73063
[78] Zienkiewicz, O.C.; Zhu, J.Z., The superconvergent patch recovery and a posteriori error estimates. parts 1 and 2, Int. J. numer. methods engrg., 33, 1331-1382, (1992) · Zbl 0769.73085
[79] Zienkiewicz, O.C.; Zhu, J.Z., Superconvergence and the superconvergent patch recovery, Finite elem. anal. des., 19, 11-23, (1995) · Zbl 0875.73292
[80] Zlamal, M., Superconvergence and reduced integration in the finite elements, Math. comput., 32, 663-685, (1977) · Zbl 0448.65068
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