×

zbMATH — the first resource for mathematics

Avoiding membrane locking with Regge interpolation. (English) Zbl 07337803
Summary: In this paper we present a method to overcome membrane locking of thin shells. An interpolation operator into the so-called Regge finite element space is inserted in the membrane energy term to weaken the implicitly given kernel constraints. The number of constraints is decreased on triangular meshes compared to reduced integration techniques, in the lowest order case asymptotically by a factor two. A tying point procedure to accomplish the interpolation is presented revealing a strong connection to MITC shell elements. Provided the interpolant, this approach can be incorporated easily into existing shell elements. The performance of the proposed method is demonstrated by means of several benchmark examples.
MSC:
74-XX Mechanics of deformable solids
41-XX Approximations and expansions
Software:
Netgen; NGSolve
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Babuška, I.; Suri, M., On locking and robustness in the finite element method, SIAM J. Numer. Anal., 29, 5, 1261-1293 (1992) · Zbl 0763.65085
[2] Chenais, D.; Paumier, J.-C., On the locking phenomenon for a class of elliptic problems, Numer. Math., 67, 4, 427-440 (1994) · Zbl 0798.73054
[3] Chapelle, D.; Bathe, K. J., Fundamental considerations for the finite element analysis of shell structures, Comput. Struct., 66, 1, 19-36 (1998) · Zbl 0934.74073
[4] Arnold, D. N., Discretization by finite elements of a model parameter dependent problem, Numer. Math., 37, 3, 405-421 (1981) · Zbl 0446.73066
[5] Bathe, K.-J.; Brezzi, F., A simplified analysis of two plate bending elements – the MITC4 and MITC9 elements, (Pande, G. N.; Middleton, J., Numerical Techniques for Engineering Analysis and Design (1987), Springer Netherlands: Springer Netherlands Dordrecht), 407-417
[6] Brezzi, F.; Fortin, M.; Stenberg, R., Error analysis of mixed-interpolated elements for Reissner-Mindlin plates, Math. Models Methods Appl. Sci., 1, 2, 125-151 (1991) · Zbl 0751.73053
[7] Pechstein, A.; Schöberl, J., The TDNNS method for Reissner-Mindlin plates, Numer. Math., 137, 3, 713-740 (2017) · Zbl 1457.65211
[8] Zienkiewicz, O. C.; Taylor, R. L.; Too, J. M., Reduced integration technique in general analysis of plates and shells, Internat. J. Numer. Methods Engrg., 3, 2, 275-290 (1971) · Zbl 0253.73048
[9] Park, K. C.; Stanley, G. M., A curved C0 shell element based on assumed natural-coordinate strains, J. Appl. Mech., 53, 2, 278-290 (1986) · Zbl 0588.73137
[10] Hughes, T. J.R.; Tezduyar, T. E., Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element, J. Appl. Mech., 48, 3, 587-596 (1981) · Zbl 0459.73069
[11] Macneal, R. H., Derivation of element stiffness matrices by assumed strain distributions, Nucl. Eng. Des., 70, 1, 3-12 (1982)
[12] Huang, H. C.; Hinton, E., A new nine node degenerated shell element with enhanced membrane and shear interpolation, Internat. J. Numer. Methods Engrg., 22, 1, 73-92 (1986) · Zbl 0593.73076
[13] Bathe, K.-J.; Dvorkin, E. N., A formulation of general shell elements-the use of mixed interpolation of tensorial components, Internat. J. Numer. Methods Engrg., 22, 3, 697-722 (1986) · Zbl 0585.73123
[14] Jang, J.; Pinsky, P. M., An assumed covariant strain based 9-node shell element, Internat. J. Numer. Methods Engrg., 24, 12, 2389-2411 (1987) · Zbl 0623.73090
[15] Tessler, A.; Hughes, T. J.R., A three-node Mindlin plate element with improved transverse shear, Comput. Methods Appl. Mech. Engrg., 50, 1, 71-101 (1985) · Zbl 0562.73069
[16] Simo, J. C.; Rifai, M. S., A class of mixed assumed strain methods and the method of incompatible modes, Internat. J. Numer. Methods Engrg., 29, 8, 1595-1638 (1990) · Zbl 0724.73222
[17] Stolarski, H.; Belytschko, T., Membrane locking and reduced integration for curved elements, J. Appl. Mech., 49, 1, 172-176 (1982) · Zbl 0482.73060
[18] Stolarski, H.; Belytschko, T., Shear and membrane locking in curved C0 elements, Comput. Methods Appl. Mech. Engrg., 41, 3, 279-296 (1983) · Zbl 0509.73072
[19] Kim, D.-N.; Bathe, K.-J., A triangular six-node shell element, Comput. Struct., 87, 23, 1451-1460 (2009)
[20] Chapelle, D.; Bathe, K.-J., The Finite Element Analysis of Shells - Fundamentals (2011), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 1103.74003
[21] Arnold, D.; Brezzi, F., Locking-free finite element methods for shells, Math. Comput. Am. Math. Soc., 66, 217, 1-14 (1997) · Zbl 0854.65095
[22] Chapelle, D.; Stenberg, R., Stabilized finite element formulations for shells in a bending dominated state, SIAM J. Numer. Anal., 36, 1, 32-73 (1998) · Zbl 0940.74059
[23] Echter, R.; Oesterle, B.; Bischoff, M., A hierarchic family of isogeometric shell finite elements, Comput. Methods Appl. Mech. Engrg., 254, 170-180 (2013) · Zbl 1297.74071
[24] Koschnick, F.; Bischoff, M.; Camprubí, N.; Bletzinger, K.-U., The discrete strain gap method and membrane locking, Comput. Methods Appl. Mech. Engrg., 194, 21, 2444-2463 (2005) · Zbl 1082.74053
[25] Quaglino, A., A framework for creating low-order shell elements free of membrane locking, Internat. J. Numer. Methods Engrg., 108, 1, 55-75 (2016)
[26] Pitkäranta, J., The problem of membrane locking in finite element analysis of cylindrical shells, Numer. Math., 61, 1, 523-542 (1992) · Zbl 0768.73079
[27] Gerdes, K.; Matache, A. M.; Schwab, C., Analysis of membrane locking in hp FEM for a cylindrical shell, ZAMM Z. Angew. Math. Mech., 78, 10, 663-686 (1998) · Zbl 0908.73074
[28] Choi, D.; Palma, F. J.; Sanchez-Palencia, E.; Vilarino, M. A., Membrane locking in the finite element computation of very thin elastic shells, ESAIM Math. Model. Numer. Anal., 32, 2, 131-152 (1998) · Zbl 0905.73066
[29] Suri, M., Analytical and computational assessment of locking in the hp finite element method, Comput. Methods Appl. Mech. Engrg., 133, 3, 347-371 (1996) · Zbl 0893.73070
[30] Hakula, H.; Leino, Y.; Pitkäranta, J., Scale resolution, locking, and high-order finite element modelling of shells, Comput. Methods Appl. Mech. Engrg., 133, 3, 157-182 (1996) · Zbl 0918.73111
[31] Regge, T., General relativity without coordinates, Il Nuovo Cimento (1955-1965), 19, 3, 558-571 (1961)
[32] Williams, R. M.; Tuckey, P. A., Regge calculus: a brief review and bibliography, Classical Quantum Gravity, 9, 5, 1409-1422 (1992) · Zbl 0991.83532
[33] Cheeger, J.; Müller, W.; Schrader, R., Kinematic and tube formulas for piecewise linear spaces, Indiana Univ. Math. J., 35, 4, 737-754 (1986) · Zbl 0615.53058
[34] Cheeger, J.; Müller, W.; Schrader, R., On the curvature of piecewise flat spaces, Comm. Math. Phys., 92, 3, 405-454 (1984) · Zbl 0559.53028
[35] Whitney, H., Geometric Integration Theory (1957), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0083.28204
[36] Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15, 1-155 (2006) · Zbl 1185.65204
[37] Arnold, D.; Falk, R.; Winther, R., Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47, 2, 281-354 (2010) · Zbl 1207.65134
[38] Christiansen, S. H., A characterization of second-order differential operators on finite element spaces, Math. Models Methods Appl. Sci., 14, 12, 1881-1892 (2004) · Zbl 1076.83008
[39] Christiansen, S. H., On the linearization of Regge calculus, Numer. Math., 119, 4, 613-640 (2011) · Zbl 1269.83022
[40] Li, L., Regge Finite Elements with Applications in Solid Mechanics and Relativity (2018), University of Minnesota, URL http://hdl.handle.net/11299/199048
[41] Hiptmair, R., Canonical construction of finite elements, Math. Comput. Am. Math. Soc., 68, 228, 1325-1346 (1999) · Zbl 0938.65132
[42] Nédélec, J. C., Mixed finite elements in R3, Numer. Math., 35, 3, 315-341 (1980)
[43] Nédélec, J. C., A new family of mixed finite elements in R3, Numer. Math., 50, 1, 57-81 (1986) · Zbl 0625.65107
[44] Abramowitz, M., (Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables (1974), Dover Publications, Inc.: Dover Publications, Inc. New York, NY, USA)
[45] Andrews, G. E.; Askey, R.; Roy, R., Special functions, (Encyclopedia of Mathematics and its Applications (1999), Cambridge University Press)
[46] Beuchler, S.; Schöberl, J., New shape functions for triangular p-FEM using integrated Jacobi polynomials, Numer. Math., 103, 3, 339-366 (2006) · Zbl 1095.65101
[47] Dziuk, G.; Elliott, C. M., Finite element methods for surface PDEs, Acta Numer., 22, 289-396 (2013) · Zbl 1296.65156
[48] Ciarlet, P. G., An introduction to differential geometry with applications to elasticity, J. Elasticity, 78-79, 1, 1-215 (2005)
[49] Bischoff, M.; Ramm, E.; Irslinger, J., Models and finite elements for thin-walled structures, (Encyclopedia of Computational Mechanics Second Edition (2017), American Cancer Society), 1-86
[50] Braess, D., (Finite Elemente - Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie (2013), Springer-Verlag: Springer-Verlag Berlin Heidelberg) · Zbl 1264.65189
[51] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method. Vol. 1: The Basis (2000), Butterworth-Heinemann: Butterworth-Heinemann Oxford · Zbl 0991.74002
[52] Zaglmayr, S., High Order Finite Element Methods for Electromagnetic Field Computation (2006), Johannes Kepler Universität Linz, URL https://www.numerik.math.tugraz.at/ zaglmayr/pub/szthesis.pdf
[53] Ko, Y.; Lee, P.-S.; Bathe, K.-J., A new MITC4+ shell element, Comput. Struct., 182, 404-418 (2017)
[54] Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (2000), Dover Publications Inc.: Dover Publications Inc. Mineola, New York. · Zbl 1191.74002
[55] Hauret, P.; Hecht, F., A discrete differential sequence for elasticity based upon continuous displacements, SIAM J. Sci. Comput., 35, 1, B291-B314 (2013) · Zbl 1372.74086
[56] Neunteufel, M.; Schöberl, J., The Hellan-Herrmann-Johnson method for nonlinear shells, Comput. Struct., 225, Article 106109 pp. (2019)
[57] Schöberl, J., NETGEN an advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci., 1, 1, 41-52 (1997) · Zbl 0883.68130
[58] Schöberl, J., C++ 11 implementation of finite elements in NGSolve, (Institute for Analysis and Scientific Computing (2014), Vienna University of Technology), URL https://www.asc.tuwien.ac.at/ schoeberl/wiki/publications/ngs-cpp11.pdf
[59] Pitkäranta, J.; Leino, Y.; Ovaskainen, O.; Piila, J., Shell deformation states and the finite element method: A benchmark study of cylindrical shells, Comput. Methods Appl. Mech. Engrg., 128, 1, 81-121 (1995) · Zbl 0861.73046
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.