Simultaneous analysis of continuously embedded Reissner-Mindlin shells in 3D bulk domains. (English) Zbl 07898476
Summary: A mechanical model and numerical method for the analysis of Reissner-Mindlin shells with geometries implied by a continuous set of level sets (isosurfaces) over some three-dimensional bulk domain is presented. A three-dimensional mesh in the bulk domain is used in a tailored FEM formulation where the elements are by no means conforming to the level sets representing the shape of the individual shells. However, the shell geometries are bounded by the intersection curves of the level sets with the boundary of the bulk domain so that the boundaries are meshed conformingly. This results in a method which was coined Bulk Trace FEM before. The simultaneously considered, continuously embedded shells may be useful in the structural design process or for the continuous reinforcement of bulk domains. Numerical results confirm higher-order convergence rates.
© 2024 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.
© 2024 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 74K25 | Shells |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S22 | Isogeometric methods applied to problems in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
bulk trace FEM; higher-order convergence studies; level-set method; PDEs on manifolds; Reissner-Mindlin shellReferences:
| [1] | CalladineCR. Theory of Shell Structures. Cambridge University Press; 1983. · Zbl 0507.73079 |
| [2] | ZingoniA. Shell Structures in Civil and Mechanical Engineering: Theory and Analysis. ICE Publishing; 2018. |
| [3] | BaşarY, KrätzigW. Mechanik der Flächentragwerke. Vieweg [( + \]\) Teubner Verlag; 1985. |
| [4] | BischoffM, RammE, IrslingerJ. Models and finite elements for thin‐walled structures. In: SteinE (ed.), BorstR (ed.), HughesT (ed.), eds. Encyclopedia of Computational Mechanics. Vol 2. 2nd ed.John Wiley & Sons; 2017. |
| [5] | ChapelleD, BatheK. The Finite Element Analysis of Shells - Fundamentals. Springer; 2011. · Zbl 1211.74002 |
| [6] | ZienkiewiczO, TaylorR, FoxD. The Finite Element Method for Solid and Structural Mechanics. Butterworth‐Heinemann; 2014. · Zbl 1307.74003 |
| [7] | LoveAEH. The small free vibrations and deformation of a thin elastic shell. Philos Trans R Soc. 1888;179:491‐546. · JFM 20.1075.01 |
| [8] | EchterR, OesterleB, BischoffM. A hierarchic family of isogeometric shell finite elements. Comp Methods Appl Mech Engrg. 2013;254:170‐180. doi:10.1016/j.cma.2012.10.018 · Zbl 1297.74071 |
| [9] | SchöllhammerD, FriesT. Kirchhoff-Love shell theory based on tangential differential calculus. Comput Mech. 2019;64:113‐131. doi:10.1007/s00466‐018‐1659‐5 · Zbl 1465.74117 |
| [10] | KiendlJ, BletzingerK, LinhardJ, WüchnerR. Isogeometric shell analysis with Kirchhoff-Love elements. Comp Methods Appl Mech Engrg. 2009;198:3902‐3914. doi:10.1016/j.cma.2009.08.013 · Zbl 1231.74422 |
| [11] | ReissnerE. The effect of transverse shear deformation on the bending of elastic plates. J Appl Mech. 1945;12:A69‐A77. doi:10.1115/1.4009435 · Zbl 0063.06470 |
| [12] | SchöllhammerD, FriesT. Reissner-Mindlin shell theory based on tangential differential calculus. Comp Methods Appl Mech Engrg. 2019;352:172‐188. doi:10.1016/j.cma.2019.04.018 · Zbl 1441.74117 |
| [13] | ZouZ, ScottM, MiaoD, BischoffM, OesterleB, DornischW. An isogeometric Reissner-Mindlin shell element based on Bézier dual basis functions: Overcoming locking and improved coarse mesh accuracy. Comp Methods Appl Mech Engrg. 2020;370:113283. doi:10.1016/j.cma.2020.113283 · Zbl 1506.74195 |
| [14] | Nguyen‐ThanhN, ValizadehN, NguyenM, et al. An extended isogeometric thin shell analysis based on Kirchhoff-Love theory. Comp Methods Appl Mech Engrg. 2015;284:265‐291. doi:10.1016/j.cma.2014.08.025 · Zbl 1423.74811 |
| [15] | DornischW, KlinkelS, SimeonB. Isogeometric Reissner-Mindlin shell analysis with exactly calculated director vectors. Comp Methods Appl Mech Engrg. 2013;253:491‐504. doi:10.1016/j.cma.2012.09.010 · Zbl 1297.74070 |
| [16] | DornischW, KlinkelS. Treatment of Reissner-Mindlin shells with kinks without the need for drilling rotation stabilization in an isogeometric framework. Comp Methods Appl Mech Engrg. 2014;276:35‐66. doi:10.1016/j.cma.2014.03.017 · Zbl 1423.74571 |
| [17] | BensonD, BazilevsY, HsuM, HughesT. Isogeometric shell analysis: The Reissner-Mindlin shell. Comp Methods Appl Mech Engrg. 2010;199:276‐289. doi:10.1016/j.cma.2009.05.011 · Zbl 1227.74107 |
| [18] | OsherS, FedkiwR. Level Set Methods and Dynamic Implicit Surfaces. Springer; 2006. |
| [19] | FriesT, OmerovićS, SchöllhammerD, SteidlJ. Higher‐order meshing of implicit geometries - Part I: Integration and interpolation in cut elements. Comp Methods Appl Mech Engrg. 2017;313:759‐784. doi:10.1016/j.cma.2016.10.019 · Zbl 1439.65212 |
| [20] | FriesT, SchöllhammerD. Higher‐order meshing of implicit geometries, Part II: Approximations on manifolds. Comp Methods Appl Mech Engrg. 2017;326:270‐297. doi:10.1016/j.cma.2017.07.037 · Zbl 1439.65213 |
| [21] | DelfourM, ZolesioJ. A boundary differential equation for thin shells. J Differ Equ. 1995;119:426‐449. doi:10.1006/jdeq.1995.1097 · Zbl 0827.73038 |
| [22] | DelfourM, ZolésioJ. Tangential Differential Equations for Dynamical Thin/Shallow Shells. J Differ Equ. 1996;128:125‐167. doi:10.1006/jdeq.1996.0092 · Zbl 0852.73035 |
| [23] | DelfourM, ZolésioJ. Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. SIAM; 2011. · Zbl 1251.49001 |
| [24] | FriesT, SchöllhammerD. A unified finite strain theory for membranes and ropes. Comp Methods Appl Mech Engrg. 2020;365:113031. doi:10.1016/j.cma.2020.113031 · Zbl 1442.74112 |
| [25] | SchöllhammerD, FriesT. A higher‐order Trace finite element method for shells. Int J Numer Methods Eng. 2021;122:1217‐1238. doi:10.1002/nme.6558 · Zbl 07863109 |
| [26] | HansboP, LarsonM. Finite element modeling of a linear membrane shell problem using tangential differential calculus. Comp Methods Appl Mech Engrg. 2014;270:1‐14. doi:10.1016/j.cma.2013.11.016 · Zbl 1296.74054 |
| [27] | KaiserM, FriesT. Curved, Linear Kirchhoff Beams Formulated Using Tangential Differential Calculus and Lagrange Multipliers. GAMM. Vol 22. John Wiley & Sons; 2023; Chichester:e202200042. |
| [28] | HansboP, LarsonM, LarssonK. Variational formulation of curved beams in global coordinates. Comput Mech. 2014;53:611‐623. doi:10.1007/s00466‐013‐0921‐0 · Zbl 1320.74069 |
| [29] | DziukG, ElliottC. Finite element methods for surface PDEs. Acta Numer. 2013;22:289‐396. doi:10.1017/s0962492913000056 · Zbl 1296.65156 |
| [30] | FriesT, KaiserM. On the simultaneous solution of structural membranes on all level sets within a bulk domain. Comp Methods Appl Mech Engrg. 2023;415:116223. doi:10.1016/j.cma.2023.116223 · Zbl 1539.74079 |
| [31] | DziukG, ElliottC. Eulerian finite element method for parabolic PDEs on implicit surfaces. Interface Free Bound. 2008;10:119‐138. · Zbl 1145.65073 |
| [32] | DziukG, ElliottC. An Eulerian approach to transport and diffusion on evolving implicit surfaces. Comput Vis Sci. 2010;13:17‐28. doi:10.1007/s00791‐008‐0122‐0 · Zbl 1220.65132 |
| [33] | BurgerM. Finite element approximation of elliptic partial differential equations on implicit surfaces. Comput Vis Sci. 2009;12:87‐100. doi:10.1007/s00791‐007‐0081‐x · Zbl 1173.65353 |
| [34] | DeckelnickK, DziukG, ElliottC, HeineC. An [( h \]\)-narrow band finite-element method for elliptic equations on implicit surfaces. IMA J Numer Anal. 2010;30:351‐376. doi:10.1093/imanum/drn049 · Zbl 1191.65151 |
| [35] | DeckelnickK, ElliottC, RannerT. Unfitted finite element methods using bulk meshes for surface partial differential equations. SIAM J Numer Anal. 2014;52:2137‐2162. doi:10.1137/130948641 · Zbl 1307.65164 |
| [36] | BertalmioM, ChengL, OsherS, SapiroG. Variational problems and partial differential equations on implicit surfaces. J Comput Phys. 2001;174:759‐780. doi:10.1006/jcph.2001.6937 · Zbl 0991.65055 |
| [37] | GreerJ. An improvement of a recent Eulerian method for solving PDEs on general geometries. J Sci Comput. 2006;29:321‐352. doi:10.1007/s10915‐005‐9012‐5 · Zbl 1122.65073 |
| [38] | GreerJ, BertozziA, SapiroG. Fourth order partial differential equations on general geometries. J Comput Phys. 2006;216:216‐246. doi:10.1016/j.jcp.2005.11.031 · Zbl 1097.65087 |
| [39] | OlshanskiiM, ReuskenA, XuX. A stabilized finite element method for advection‐diffusion equations on surfaces. IMA J Numer Anal. 2014;34:732‐758. doi:10.1093/imanum/drt016 · Zbl 1293.65159 |
| [40] | OlshanskiiM, ReuskenA. Trace finite element methods for PDEs on surfaces. In: BordasS (ed.), BurmanE (ed.), LarsonM (ed.), OlshanskiiM (ed.), eds. Geometrically Unfitted Finite Element Methods and Applications. Lecture Notes in Computational Science and Engineering. Vol 121. Springer Nature; 2017:211‐258. · Zbl 1448.65245 |
| [41] | ChernyshenkoA, OlshanskiiM. An adaptive octree finite element method for PDEs posed on surfaces. Comp Methods Appl Mech Engrg. 2015;291:146‐172. doi:10.1016/j.cma.2015.03.025 · Zbl 1425.65155 |
| [42] | GrandeJ, ReuskenA. A higher order finite element method for partial differential equations on surfaces. SIAM J Numer Anal. 2016;54:388‐414. doi:10.1137/14097820x · Zbl 1382.65398 |
| [43] | BurmanE, HansboP, LarsonM. A stabilized cut finite element method for partial differential equations on surfaces: The Laplace-Beltrami operator. Comp Methods Appl Mech Engrg. 2015;285:188‐207. doi:10.1016/j.cma.2014.10.044 · Zbl 1425.65152 |
| [44] | BurmanE, HansboP, LarsonM, MassingA. Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions. ESAIM: Math Model Numer Anal. 2018;52:2247‐2282. doi:10.1051/m2an/2018038 · Zbl 1417.65199 |
| [45] | CenanovicM, HansboP, LarsonM. Cut finite element modeling of linear membranes. Comp Methods Appl Mech Engrg. 2016;310:98‐111. doi:10.1016/j.cma.2016.05.018 · Zbl 1439.74396 |
| [46] | GfrererM. A [( {C}^1 \]\)‐continuous Trace‐Finite‐Cell‐Method for linear thin shell analysis on implicitly defined surfaces. Comput Mech. 2020;67:679‐697. doi:10.1007/s00466‐020‐01956‐5 · Zbl 07360524 |
| [47] | JankuhnT, OlshanskiiM, ReuskenA. Incompressible fluid problems on embedded surfaces: Modeling and variational formulations. Interface Free Bound. 2018;20:353‐377. doi:10.4171/ifb/405 · Zbl 1406.35224 |
| [48] | FriesT. Higher‐order surface FEM for incompressible Navier‐Stokes flows on manifolds. Int J Numer Methods Fluids. 2018;88:55‐78. doi:10.1002/fld.4510 |
| [49] | FedererH. Geometric Measure Theory. Springer; 1969. · Zbl 0176.00801 |
| [50] | MorganF. Geometric Measure Theory: A Beginner’s Guide. Academic press; 1988. · Zbl 0671.49043 |
| [51] | KiendlJ, MarinoE, LorenzisLD. Isogeometric collocation for the Reissner-Mindlin shell problem. Comp Methods Appl Mech Engrg. 2017;325:645‐665. doi:10.1016/j.cma.2017.07.023 · Zbl 1439.74433 |
| [52] | SimoJC, FoxDD, RifaiMS. On a stress resultant geometrically exact shell model. Part II: The linear theory; Computational aspects. Comp Methods Appl Mech Engrg. 1989;73:53‐92. doi:10.1016/0045‐7825(89)90098‐4 · Zbl 0724.73138 |
| [53] | OlshanskiiM, QuainiA, ReuskenA, YushutinV. A finite element method for the surface Stokes problem. SIAM J Sci Comput. 2018;40:A2492‐A2518. doi:10.1137/18M1166183 · Zbl 1397.65287 |
| [54] | BurmanE, HansboP. Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl Numer Math. 2012;62:328‐341. doi:10.1016/j.apnum.2011.01.008 · Zbl 1316.65099 |
| [55] | BurmanE. A penalty‐free nonsymmetric Nitsche‐type method for the weak imposition of boundary conditions. SIAM J Numer Anal. 2012;50:1959‐1981. doi:10.1137/10081784X · Zbl 1262.65165 |
| [56] | SchillingerD, HarariI, HsuM, et al. The non‐symmetric Nitsche method for the parameter‐free imposition of weak boundary and coupling conditions in immersed finite elements. Comp Methods Appl Mech Engrg. 2016;309:625‐652. doi:10.1016/j.cma.2016.06.026 · Zbl 1439.65192 |
| [57] | BabuškaI. Error‐bounds for finite element method. Numer Math. 1971;16:322‐333. doi:10.1007/BF02165003 · Zbl 0214.42001 |
| [58] | BrezziF. On the existence, uniqueness and approximation of saddle‐point problems arising from Lagrange multipliers. RAIRO Anal Numér. 1974;R‐2:129‐151. doi:10.1051/m2an/197408R201291 · Zbl 0338.90047 |
| [59] | ZienkiewiczO, TaylorR, ZhuJ. The Finite Element Method: Its Basis and Fundamentals. Butterworth‐Heinemann; 2013. · Zbl 1307.74005 |
| [60] | AltenbachH, AltenbachJ, NaumenkoK. Ebene Flächentragwerke. Springer; 2016. |
| [61] | BelytschkoT, StolarskiH, LiuW, CarpenterN, OngJ. Stress projection for membrane and shear locking in shell finite elements. Comp Methods Appl Mech Engrg. 1985;51:221‐258. doi:10.1016/0045‐7825(85)90035‐0 · Zbl 0581.73091 |
| [62] | ChapelleD, BatheK. Fundamental considerations for the finite element analysis of shell structures. Comput Struct. 1998;66:19‐36. doi:10.1016/s0045‐7949(97)00078‐3 · Zbl 0934.74073 |
| [63] | BatheK, IosilevichA, ChapelleD. An evaluation of the MITC shell elements. Comput Struct. 2000;75:1‐30. doi:10.1016/s0045‐7949(99)00214‐x |
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.
