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Adaptive \(hpq\) finite element methods for the analysis of 3D-based models of complex structures. Part II: A posteriori error estimation. (English) Zbl 1286.74112

Summary: This is the second paper out of a series of papers devoted to model- and \(hpq\)-adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description. In our previous publication [G. Zboinski, Comput. Methods Appl. Mech. Eng. 199, No. 45–48, 2913–2940 (2010; Zbl 1231.74449)] we investigated the issue of hierarchical models and approximations of such structures. We applied 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within the proposed finite element formulation. Furthermore, we assumed that the mechanical model and discretization parameters (such as: the size h of the element, and the longitudinal and transverse approximation orders, \(p\) and \(q\)) could vary locally, i.e. they could be different in each finite element. The a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints. The generalized method is applied to the assessment of the total and approximation errors, while the modeling error is calculated as the difference between the former two errors. The corresponding error-controlled adaptive procedures are based on a three-step strategy, with possible iterations of \(h\)- and \( p\)-steps.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
74K25 Shells
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 1231.74449

Software:

3DmhpqAP
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Full Text: DOI

References:

[1] Ainsworth, M., A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains, Numer. Math., 80, 325-362 (1998) · Zbl 0912.65087
[2] Ainsworth, M., A synthesis of a posteriori error estimation techniques for conforming, non-conforming and discontinuous Galerkin finite element methods, (Shi, Z.-C.; Chen, Z.; Tang, T.; Yu, D., Recent Advances in Adaptive Computation. Recent Advances in Adaptive Computation, Contemporary Mathematics, vol. 383 (2005), AMS: AMS Providence), 1-14 · Zbl 1098.65106
[3] Ainsworth, M.; Babušhka, I., Reliable and robust a posteriori error estimation for singularly perturbed reaction diffusion problems, SIAM J. Numer. Anal., 36, 331-353 (1999)
[4] Ainsworth, M.; Demkowicz, L.; Kim, C. W., Analysis of the equilibrated residual method for a posteriori error estimation on meshes with hanging nodes, Comput. Methods Appl. Mech. Engrg., 196, 3493-3507 (2007) · Zbl 1173.76424
[5] Ainsworth, M.; Oden, J. T., A unified approach to a posteriori error estimation using element residual methods, Numer. Math., 65, 23-50 (1993) · Zbl 0797.65080
[6] Ainsworth, M.; Oden, J. T., A procedure for a posteriori error estimation for \(h-p\) finite element methods, Comput. Methods Appl. Mech. Engrg., 101, 73-96 (1992) · Zbl 0778.73060
[7] Ainsworth, M.; Oden, J. T., A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg., 142, 1-88 (1997) · Zbl 0895.76040
[8] Ainsworth, M.; Oden, J. T., A posteriori error estimators for second order elliptic systems: Part 1. Theoretical foundations and a posteriori error analysis, Comput. Math. Appl., 25, 101-113 (1993) · Zbl 0764.65060
[9] Ainsworth, M.; Oden, J. T., A posteriori error estimators for second order elliptic systems: Part 2. An optimal order process for calculating self-equilibrating fluxes, Comput. Math. Appl., 26, 75-87 (1993) · Zbl 0789.65083
[10] Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis (2000), Wiley & Sons, Inc.: Wiley & Sons, Inc. New York · Zbl 1008.65076
[11] Ainsworth, M.; Oden, J. T.; Wu, W., A posteriori error estimation for hp approximation in elastostatics, Appl. Numer. Math., 14, 23-55 (1994) · Zbl 0802.73072
[12] Babušhka, I.; Lee, I.; Schwab, C., On the a posteriori error estimation of the modeling error for the heat conduction in a plate and its use for adaptive hierarchical modeling, Appl. Numer. Math., 14, 5-21 (1994) · Zbl 0809.65095
[13] Babuška, I.; Rheinboldt, W. C., A posteriori error estimates for finite element method, Int. J. Numer. Methods Engrg., 12, 1597-1615 (1978) · Zbl 0396.65068
[14] Babuška, I.; Rheinboldt, W. C., Reliable error estimation and mesh adaptation for the finite element method, (Oden, J. T., Computational Method in Nonlinear Mechanics (1980), North Holland: North Holland Amsterdam), 67-108
[15] Babuška, I.; Schwab, C., A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains, SIAM J. Numer. Anal., 33, 221-246 (1996) · Zbl 0846.65056
[17] Bank, R. E.; Weiser, A., Some a posteriori error estimators for elliptic partial differential equations, Math. Comput., 44, 283-301 (1985) · Zbl 0569.65079
[18] Billade, N.; Vemaganti, K., Goal-oriented error estimation for hierarchical models of thin structures, (Yao, Z. H.; Yuan, M. W.; Zhong, W. X., Conf. Proc. WCCM VI & APCOM’04 (2004), Tsinghua Univ. Press, Springer: Tsinghua Univ. Press, Springer Beijing), 1-10
[20] Cho, J. R.; Oden, J. T., A priori modeling error estimates of hierarchical models for elasticity problems for plate- and shell-like structures, Math. Comput. Model., 23, 10, 117-133 (1996) · Zbl 0854.73082
[21] Cho, J. R.; Oden, J. T., A priori error estimations of hp-finite element approximations for hierarchical models of plate- and shell-like structures, Comput. Methods Appl. Mech. Engrg., 132, 135-177 (1996) · Zbl 0884.73063
[22] Cho, J. R.; Oden, J. T., Locking and boundary layer in hierarchical models for thin elastic structures, Comput. Methods Appl. Mech. Engrg., 149, 33-48 (1997) · Zbl 0918.73130
[24] Demkowicz, L.; Oden, J. T.; Strouboulis, T., Adaptive finite elements for flow problems with moving boundaries. Part 1: Variational principles and a posteriori error estimates, Comput. Methods Appl. Mech. Engrg., 46, 217-251 (1984) · Zbl 0583.76025
[25] Demkowicz, L.; Oden, J. T.; Strouboulis, T., An adaptive \(p\)-version finite element method for transient flow problems with moving boundaries, (Gallagher, R. H., Finite Elements in Fluids VI (1985), Wiley: Wiley New York), 291-305
[26] Gago, J.; Kelly, D. W.; Zienkiewicz, O. C.; Babuška, I., A posteriori error analysis and adaptive process in the finite element method. Part II: Adaptive mesh refinement, Int. J. Numer. Methods Engrg., 19, 1621-1656 (1983) · Zbl 0534.65069
[27] Kelly, D. W., The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method, Int. J. Numer. Methods Engrg., 20, 1491-1506 (1984) · Zbl 0575.65100
[28] Kelly, D. W.; Gago, J.; Zienkiewicz, O. C.; Babuška, I., A posteriori error analysis and adaptive process in the finite element method. Part I: Error analysis, Int. J. Numer. Methods Engrg., 19, 1593-1619 (1983) · Zbl 0534.65068
[29] Ladeveze, P.; Maunder, E. A.W., A general method for recovering equilibrating element tractions, Comput. Methods Appl. Mech. Engrg., 137, 111-151 (1996) · Zbl 0886.73065
[30] Ladeveze, P.; Maunder, E. A.W., A general methodology for recovering equilibrating finite element tractions and stress fields for plate and solid elements, Comput. Assist. Mech. Engrg. Sci., 20, 533-548 (1997) · Zbl 0974.74060
[31] (Ladevéze, P.; Oden, J. T., Advances in Adaptive Computational Methods in Mechanics (1998), Elsevier: Elsevier Amsterdam) · Zbl 0915.76001
[32] Ladevéze, P.; Leguillon, L., Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal., 20, 485-509 (1983), (for nonlinear finite element analysis, in: P. Ladevéze, J.T. Oden (Eds.), Advances in Adaptive Computational Methods in Mechanics, Elsevier, Amsterdam, 1998, pp. 231-256.) · Zbl 0582.65078
[33] Nochetto, R. H., Pointwise a posteriori error estimates of elliptic problems on highly graded meshes, Mat. Comput., 64, 1-22 (1995) · Zbl 0920.65063
[34] Oden, J. T., Error estimation and control in computational fluid dynamics. The O.C. Zienkiewicz lecture, (Proc. Math. of Finite Elements - MAFELAP VIII (1993), Brunnel Univ.: Brunnel Univ. Uxbridge), 1-36
[35] Oden, J. T.; Cho, J. R., Adaptive finite element methods of hierarchical shell models for plate- and shell-like structures, Comput. Methods Appl. Mech. Engrg., 136, 317-345 (1996) · Zbl 0892.73068
[37] Oden, J. T.; Prudhomme, S., Estimation of modeling error in computational mechanics, J. Comput. Phys., 182, 496-515 (2002) · Zbl 1053.74049
[38] Oden, J. T.; Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl., 41, 735-756 (2001) · Zbl 0987.65110
[39] Oden, J. T.; Prudhomme, S.; Bauman, P. T.; Chamoin, L., Estimation and control of modeling error: a general approach to multi-scale modeling, (Fish, J., Bridging the Scales in Science and Engineering (2008), Oxford University Press: Oxford University Press Oxford), 285-304 · Zbl 1221.82012
[40] Oden, J. T.; Prudhomme, S.; Hammerand, D. C.; Kuczma, S., Modeling error and adaptivity in nonlinear continuum mechanics, Comput. Methods Appl. Mech. Engrg., 191, 6663-6684 (2001) · Zbl 1012.74081
[41] Oden, J. T.; Vemaganti, K., Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part I: Error estimates and adaptive algorithms, J. Comput. Phys., 164, 22-47 (2000) · Zbl 0992.74072
[42] Prudhomme, S.; Oden, J. T., Modeling error estimation and adaptivity for multi-scale problems, (Yao, Z. H.; Yuan, M. W.; Zhong, W. X., Conf. Proc. WCCM VI & APCOM’04 (2004), Tsinghua Univ. Press, Springer: Tsinghua Univ. Press, Springer Beijing), 553-557
[43] Prudhomme, S.; Oden, J. T., On goal-oriented error estimation for elliptic problems: application to control of pointwise errors, Comput. Methods Appl. Mech. Engrg., 176, 313-331 (1999) · Zbl 0945.65123
[44] Prudhomme, S.; Oden, J. T.; Westermann, T.; Bass, J.; Botkin, M. E., Practical methods for a posteriori error estimation in engineering applications, Int. J. Numer. Methods Engrg., 56, 1193-1224 (2003) · Zbl 1038.74045
[45] Schwab, C., A-posteriori modeling error estimation for hierarchic plate models, Numer. Math., 74, 221-259 (1996) · Zbl 0856.73031
[46] Stein, E.; Ohnimus, S., Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems, Comput. Methods Appl. Mech. Engrg., 176, 363-385 (1999) · Zbl 0954.74072
[47] Stein, E.; Ohnimus, S., Coupled model- and solution adaptivity in the finite element method, Comput. Methods Appl. Mech. Engrg., 150, 327-350 (1997) · Zbl 0926.74127
[48] Stein, E.; Ohnimus, S., Equilibrium method for postprocessing and error estimation in the finite element method, Comput. Assist. Mech. Engrg. Sci., 4, 645-666 (1997) · Zbl 0974.74066
[49] Verfürth, R., A posteriori error estimation and adaptive mesh refinement techniques, J. Comput. Appl. Math., 50, 67-83 (1994) · Zbl 0811.65089
[50] Verfürth, R., A review of a posteriori error estimation techniques for elasticity problems, Comput. Methods Appl. Mech. Engrg., 176, 419-440 (1999) · Zbl 0935.74072
[51] Vogelius, M.; Babuška, I., On a dimensional reduction method. I. The optimal selection of basis functions, Math. Comput., 37, 155, 31-46 (1981) · Zbl 0495.65049
[53] Zboiński, G., A posteriori error estimation for hp-approximation of the 3D-based first order shell model. Part 1. Theoretical aspects, Appl. Math. Inf. Mech., 8, 1, 104-125 (2003) · Zbl 1078.74056
[54] Zboiński, G., A posteriori error estimation for hp-approximation of the 3D-based first order shell model. Part 2. Implementation aspects, Appl. Math. Inf. Mech., 8, 2, 58-83 (2003) · Zbl 1079.74058
[55] Zboiński, G., 3D-based hp-adaptive first order shell finite element for modeling and analysis of complex structures. Part 2. Application to structural analysis, Int. J. Numer. Methods Engrg., 70, 1546-1580 (2007) · Zbl 1194.74486
[56] Zboiński, G., Application of the three-dimensional triangular-prism hpq adaptive finite element to plate and shell analysis, Comput. Struct., 65, 497-514 (1997) · Zbl 0922.73071
[57] Zboiński, G., Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. Part 1. Hierarchical modeling and approximation, Comput. Methods Appl. Mech. Engrg., 199, 2913-2940 (2010) · Zbl 1231.74449
[59] Zboiński, G., Unresolved problems of adaptive hierarchical modeling and hp-adaptive analysis within computational solid mechanics, (Kuczma, M.; Wilmański, K., Computational Methods in Mechanics (2010), Springer: Springer Berlin), 111-145
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