Adaptive hpq finite element methods for the analysis of 3D-based models of complex structures. I: Hierarchical modeling and approximations. (English) Zbl 1231.74449
Summary: This is a paper out of a series of three papers devoted to model- and \(h\)pq-adaptive finite element methods assigned for modeling and analysis of complex structures. In this paper we focus our attention on hierarchical models and hierarchical approximations, while the issues of error estimation and adaptivity control will be presented in the next papers of the series. In all the papers we consider the elastic structures of complex mechanical description. We apply 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within our finite element formulation. In the proposed approach, the mechanical model and discretization parameters, such as the size \(h\) of the element, and the longitudinal and transverse approximation orders, \(p\) and \(q\), can vary locally, i.e. they can be different in each finite element. The a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors. The error-controlled adaptive procedures are derived from a four-step strategy, with possible iterations on \(h\)- and \(p\)-steps.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74K99 | Thin bodies, structures |
Keywords:
hierarchical modeling; hierarchical approximations; finite elements; adaptivity; complex structuresReferences:
| [1] | Actis, R. L.; Szabó, B. A., Hierarchic models for bidirectional composites, Finite Element Anal. Design, 13, 149-168 (1993) · Zbl 0780.73044 |
| [2] | Actis, R. L.; Szabó, B. A.; Schwab, C., Hierarchic models for laminated plates and shells, Comput. Methods Appl. Mech. Engrg., 172, 79-107 (1999) · Zbl 0959.74061 |
| [3] | 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 |
| [4] | Ainsworth, M.; Senior, B., Aspects of an adaptive hp-finite element method: adaptive strategy, conforming approximations, and efficient solvers, Comput. Methods Appl. Mech. Engrg., 150, 65-87 (1997) · Zbl 0906.73057 |
| [5] | Arnold, D. N.; Falk, R. S., A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 21, 1276-1290 (2001) · Zbl 0696.73040 |
| [6] | Babuška, I.; Guo, B. Q., Approximation properties of the hp version of the finite element method, Comput. Methods Appl. Mech. Engrg., 133, 319-346 (1996) · Zbl 0882.65096 |
| [7] | Babuška, I.; Li, L., Hierarchic modelling of plates, Comput. Struct., 40, 419-430 (1991) |
| [8] | Babuška, I.; Li, L., The hp version of the finite element method in plate modelling problem, Commun. Appl. Numer. Methods, 8, 17-26 (1992) · Zbl 0751.73052 |
| [9] | Babuška, I.; Li, L., The problem of plate modelling: theoretical and computational results, Comput. Methods Appl. Mech. Engrg., 100, 249-273 (1992) · Zbl 0764.73040 |
| [10] | 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 |
| [11] | Babuška, I.; Suri, M., The \(h-p\) version of the finite element method with quasi-uniform meshes, SIAM J. Numer. Anal., 21, 199-238 (1987) · Zbl 0623.65113 |
| [12] | Betsch, P.; Stein, E., An assumed strain approach avoiding artificial thickness straining for a 4-node shell element, Commun. Numer. Methods Engrg., 11, 899-909 (1995) · Zbl 0833.73051 |
| [14] | Chinosi, C.; Della Croce, L.; Scapolla, T., Hierarchic finite elements for thin plates and shells, Comput. Assist. Mech. Engrg. Sci., 5, 151-160 (1998) · Zbl 0948.74058 |
| [15] | Ciarlet, P. G., Mathematical Elasticity. Mathematical Elasticity, Three-Dimensional Elasticity, vol. 1 (1988), North-Holland: North-Holland Amsterdam · Zbl 0648.73014 |
| [16] | Ciarlet, P. G., Mathematical Elasticity. Mathematical Elasticity, Theory of Shells, vol. 3 (2000), North-Holland: North-Holland Amsterdam · Zbl 0953.74004 |
| [17] | Chang, Y. C.; Demkowicz, L., Solution of viscoelastic scattering problems in linear acoustics using hp boundary/finite element method, Int. J. Numer. Methods Engrg., 44, 1885-1907 (1999) · Zbl 0948.74027 |
| [18] | 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 |
| [19] | 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 |
| [20] | Dauge, M.; Gruais, I.; Rossle, A., The influence of lateral boundary conditions on the asymptotics in the thin elastic plates, SIAM J. Math. Anal., 31, 305-345 (2000) · Zbl 0958.74034 |
| [21] | Dauge, M.; Yosibash, Z., Boundary layer realization in thin elastic three-dimensional domains and two-dimensional hierarchic plate models, Int. J. Solids Struct., 37, 2443-2471 (2000) · Zbl 0973.74043 |
| [22] | Della Croche, L.; Scapolla, T., High order finite elements for thin to moderately thick plates, Comput. Mech., 10, 263-279 (1992) · Zbl 0825.73676 |
| [24] | Demkowicz, L., Computing with hp-Adaptive Finite Elements. Computing with hp-Adaptive Finite Elements, One- and Two-Dimensional Elliptic and Maxwell Problems, vol. 1 (2007), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 1111.65103 |
| [26] | Demkowicz, L.; Kurtz, J.; Pardo, D.; Paszyski, M.; Rachowicz, W.; Zdunek, A., Computing with hp-Adaptive Finite Elements. Computing with hp-Adaptive Finite Elements, Three-Dimensional Elliptic and Maxwell Problems with Applications, vol. 2 (2008), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL · Zbl 1148.65001 |
| [27] | Demkowicz, L.; Oden, J. T., Application of the hp-adaptive BE/FE methods to elastic scattering, Comput. Methods Appl. Mech. Engrg., 133, 287-317 (1996) · Zbl 0918.73289 |
| [28] | Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O., Towards a universal hp adaptive finite element strategy. Part 1. A constrained approximation and data structure, Comput. Methods Appl. Mech. Engrg., 77, 79-112 (1989) · Zbl 0723.73074 |
| [29] | Düster, A.; Nigll, A.; Rank, E., Applying the \(hp -d\) version of the FEM to locally enhance dimensionally reduced models, Comput. Methods Appl. Mech. Engrg., 196, 3768-3782 (2007) · Zbl 1173.74413 |
| [30] | Düster, A.; Parvanian, J.; Yang, Z.; Rank, E., The finite cell method for three-dimensional problems of solid mechanics, Comput. Methods Appl. Mech. Engrg., 197, 3524-3533 (2008) · Zbl 1173.74413 |
| [31] | Fish, J.; Guttal, R., The \(p\)-version of the finite element method for shell analysis, Comput. Mech., 16, 328-340 (1995) · Zbl 0836.73068 |
| [32] | Fish, J.; Guttal, R., The \(s\)-version of finite element method for laminated composites, Int. J. Numer. Methods Engrg., 39, 3641-3662 (1996) · Zbl 0888.73060 |
| [33] | Fish, J.; Guttal, R., Hierarchical assumed strain-vorticity shell element with drilling degrees of freedom, Finite Elements Anal. Design, 33, 61-70 (1999) · Zbl 0952.74068 |
| [34] | Gordon, W. J.; Hall, C. A., Transfinite element methods: blending function interpolation over arbitrary curved element domain, Numer. Math., 21, 109-129 (1973) · Zbl 0254.65072 |
| [35] | Guo, B.; Babuška, I., The \(h-p\) version of finite element method. Part 2. General results and applications, Comput. Mech., 1, 203-220 (1986) · Zbl 0634.73059 |
| [36] | Gui, W.; Babuška, I., The \(h, p\) and \(h-p\) versions of the finite element methods in one dimension. Part 2. The error analysis of the \(h\) and \(h-p\) versions, Numer. Math., 49, 613-657 (1986) · Zbl 0614.65089 |
| [37] | Hakula, H.; Leino, Y.; Pitkaranta, J., Scale resolution, locking and high-order finite element modelling of shells, Comput. Methods Appl. Mech. Engrg., 133, 157-182 (1996) · Zbl 0918.73111 |
| [38] | Hauptmann, M.; Schweizerhof, K., A systematic development of solid-shell element formulation for linear and nonlinear analyses employing only displacement degrees of freedom, Int. J. Numer. Methods Engrg., 42, 49-70 (1998) · Zbl 0917.73067 |
| [39] | Karamian, P.; Sanchez-Hubert, J.; Palencia, E., A model problem for boundary layers of thin elastic shells, ESAIM Math. Model. Numer. Anal., 34, 1-30 (2000) · Zbl 1004.74050 |
| [40] | Legat, V.; Oden, J. T., An adaptive hp finite element methods for incompressible free surface flows of generalized Newtonian fluids, (Theoretical, Experimental and Numerical Contributions to the Mechanics of Fluids and Solids (1995), Birkhauser), 643-678 · Zbl 0830.76050 |
| [41] | Leino, Y.; Pitkaranta, J., On membrane locking of hp finite elements in a cylindrical shell problem, Int. J. Numer. Methods Engrg., 37, 1053-1070 (1994) · Zbl 0799.73071 |
| [42] | Pitkaranta, J., The problem of membrane locking in the finite element analysis of cylindrical shells, Numer. Math., 61, 523-542 (1992) · Zbl 0768.73079 |
| [44] | Oden, J. T.; Demkowicz, L.; Rachowicz, W.; Westermann, T. A., Towards 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 |
| [45] | Oden, J. T.; Cho, J. R., Adaptive hpq finite element methods of hierarchical models for plate- and shell-like structures, Comput. Methods Appl. Mech. Engrg., 136, 317-345 (1996) · Zbl 0892.73068 |
| [46] | Oden, J. T.; Lee, C. Y., An adaptive hp-finite element method in frictional contact problems, (Proceedings of the Second Asian-Pacific Conference on Comput. Mech. August 1993, Sydney (Australia) (1993), A.A. Balkema: A.A. Balkema Rotterdam) · Zbl 0847.73062 |
| [47] | Oden, J. T.; Wu, W.; Legat, V., An hp adaptive strategy for finite element approximations of the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 20, 831-852 (1995) · Zbl 0836.76050 |
| [48] | Ovaskainen, O.; Pitkaranta, J., An \(h-p-n\) adaptive finite element scheme for shell problems, Adv. Engrg. Software, 26, 201-207 (1996) |
| [49] | Rachowicz, W.; Demkowicz, L., An hp-adaptive finite element method for electromagnetics. Part 2: A 3D implementation, Int. J. Numer. Methods Engrg., 53, 147-180 (2002) · Zbl 0994.78012 |
| [50] | Rachowicz, W.; Oden, J. T.; Demkowicz, L., Towards a universal hp adaptive finite element strategy. Part 3. Design of \(h-p\) meshes, Comput. Methods Appl. Mech. Engrg., 77, 181-212 (1989) · Zbl 0723.73076 |
| [51] | Rachowicz, W.; Pardo, D.; Demkowicz, L., Fully automatic hp-adaptivity in three dimensions, Comput. Methods Appl. Mech. Engrg., 195, 4816-4842 (2006) · Zbl 1193.65203 |
| [52] | Rank, E., A zooming technique using hierarchical hp-version of the finite element method, (Whitemann, J., The Mathematics of Finite Elements and Applications - Highlights (1993), Elsevier) · Zbl 0828.65126 |
| [53] | Rank, E.; Bröker, H.; Düster, A.; Nübel, V., High order solid elements for thin-walled structures: no tricks? - no crimes!, (Trends Comput. Struct. Mech. (2001), CIMNE: CIMNE Barcelona), 368-378 |
| [54] | Rank, E.; Düster, A.; Alam, Z., Adaptive computation of thin-walled structures based on high-order solid elements, (Proceedings of WCCM VI with APCOM’04 (2004), Tsinghua Univ. Press and Springer-Verlag: Tsinghua Univ. Press and Springer-Verlag Beijing) |
| [55] | Reddy, J. N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis (2003), CRC Press · Zbl 0899.73002 |
| [56] | Reddy, M. P.; Wastermann, E. G.; Schaub, T. A.; Oden, J. T., Numerical simulation of the three-dimensional extrusion process using hp-adaptive finite element model, (Proceedings of NAFEMS World Congress 97, April 9-11 (1997), Universitat Stuttgart: Universitat Stuttgart Stuttgart) |
| [57] | Safjan, A.; Demkowicz, L.; Oden, J. T., Adaptive finite element methods for hyperbolic systems with application to transient acoustics, Int. J. Numer. Methods Engrg., 32, 677-707 (1991) · Zbl 0755.76055 |
| [58] | Schwab, C., A posteriori modelling error estimation for hierarchic plate models, Numer. Math., 74, 221-259 (1996) · Zbl 0856.73031 |
| [59] | Schwab, C., \(p\)- and hp FEM (1998), Oxford University Press: Oxford University Press Oxford · Zbl 1298.74237 |
| [60] | Schwab, C.; Suri, M., Locking and boundary layer effects in the finite element approximation of the Reissner-Mindlin plate model, Proc. Symp. Appl. Math., 48, 367-371 (1994) · Zbl 0815.73061 |
| [61] | Schwab, C.; Surim, M., The \(p\) and hp versions of the finite element method for problems with boundary layers, Math. Comput., 65, 1403-1429 (1996) · Zbl 0853.65115 |
| [62] | Schwab, C.; Suri, M.; Xenophontos, C., The hp finite element method for problems in mechanics with boundary layers, Comput. Methods Appl. Mech. Engrg., 157, 311-333 (1998) · Zbl 0959.74073 |
| [63] | Schwab, C.; Wright, S., Boundary layers in hierarchical beam and plate models, report, J. Elasticity, 38, 1-40 (1995) · Zbl 0834.73040 |
| [64] | Solin, P.; Demkowicz, L., Goal-oriented hp-adaptivity for elliptic problems, Comput. Methods Appl. Mech. Engrg., 193, 444-468 (2004) · Zbl 1044.65082 |
| [65] | Surana, K. S.; Sorem, R. M., \(p\)-version hierarchical three-dimensional curved shell element for elastostatics, Int. J. Numer. Methods Engrg., 31, 649-676 (1991) · Zbl 0825.73798 |
| [66] | Suri, M.; Babuška, I.; Schwab, C., Locking effects in the finite element approximation of plate model, Math. Comput., 64, 210, 461-482 (1995) · Zbl 0827.73063 |
| [67] | 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 (1997) · Zbl 0954.74072 |
| [68] | Stein, E.; Rust, W.; Ohnimus, S., \(h\)- and \(d\)-Adaptive FE methods for two-dimensional structural problems including post-buckling of shells, Comput. Methods Appl. Mech. Engrg., 101, 315-354 (1992) · Zbl 0779.73070 |
| [69] | Stein, E.; Rutter, M.; Ohnimus, S., Adaptive finite element analysis and modeling of solids and structures. Findings, problems and trends, Int. J. Numer. Methods Engrg., 60, 103-138 (2004) · Zbl 1060.74642 |
| [70] | Stein, E.; Rutter, M.; Ohnimus, S., Error-controlled adaptive goal oriented modelling and finite element approximations in elasticity, Comput. Methods Appl. Mech. Engrg., 196, 3598-3613 (2007) · Zbl 1173.74443 |
| [71] | Szabó, B. A.; Babuška, I., Finite Element Analysis (1991), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York · Zbl 0792.73003 |
| [72] | Szabó, B. A.; Sahrmann, G. J., Hierarchic plate and shell models based on \(p\)-extension, Int. J. Numer. Methods Engrg., 26, 1855-1881 (1988) · Zbl 0661.73048 |
| [73] | Tews, R.; Rachowicz, W., Application of an automatic hp adaptive finite element method for thin walled structures, Comput. Methods Appl. Mech. Engrg., 198, 1967-1984 (2009) · Zbl 1227.74099 |
| [74] | 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 |
| [76] | Wempner, G., Mechanics of Solids, with Applications to Thin Bodies (2007), Springer-Verlag |
| [77] | Yosibash, Z., Numerical analysis of edge sigularities in three-dimensional elasticity, Int. J. Numer. Methods Engrg., 40, 4611-4632 (1997) · Zbl 0910.73066 |
| [78] | Yosibash, Z.; Padan, R.; Joskowicz, L.; Milgrom, C., A CT-based high-order finite element analysis of the human proximal femur compared to in vitro experiments, J. Biomech. Engrg. Trans. ASME, 129, 297-309 (2007) |
| [79] | Yosibash, Z.; Trabelsi, N.; Hellmich, C., Subject-specific \(p\)-FE analysis of the proximal femur utilizing micromechanics-based material properties, Int. J. Multiscale Comput. Engrg., 6, 483-498 (2008) |
| [83] | Zboiński, G., Unresolved problems of adaptive hierarchical modelling and hp-adaptive analysis within computational solid mechanics, (Kuczma, M.; Wilmański, K., Computational Methods in Mechanics (2010), Springer: Springer Berlin), 111-145 |
| [84] | 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 |
| [85] | Zboiński, G., 3D-based hp-adaptive first order shell finite element for modelling and analysis of complex structures - Part 2. Application to structural analysis, Int. J. Numer. Methods Engrg., 70, 1546-1580 (2007) · Zbl 1194.74486 |
| [88] | Zboiński, G.; Jasiński, M., 3D-based hp-adaptive first order shell finite element for modelling and analysis of complex structures - Part 1. The model and the approximation, Int. J. Numer. Methods Engrg., 70, 1513-1545 (2007) · Zbl 1194.74487 |
| [89] | Zboiński, G.; Ostachowicz, W., An algorithm of a family of 3D-based solid-to-shell transition hpq/hp-adaptive finite elements, J. Theor. Appl. Mech., 38, 791-806 (2000) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
