Numerical approximation of the solution of an obstacle problem modelling the displacement of elliptic membrane shells via the penalty method. (English) Zbl 07822050
Summary: In this paper we establish the convergence of a numerical scheme based, on the Finite Element Method, for a time-independent problem modelling the deformation of a linearly elastic elliptic membrane shell subjected to remaining confined in a half space. Instead of approximating the original variational inequalities governing this obstacle problem, we approximate the penalized version of the problem under consideration. A suitable coupling between the penalty parameter and the mesh size will then lead us to establish the convergence of the solution of the discrete penalized problem to the solution of the original variational inequalities. We also establish the convergence of the Brezis-Sibony scheme for the problem under consideration. Thanks to this iterative method, we can approximate the solution of the discrete penalized problem without having to resort to nonlinear optimization tools. Finally, we present numerical simulations validating our new theoretical results.
MSC:
| 74M15 | Contact in solid mechanics |
| 74K20 | Plates |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 49J40 | Variational inequalities |
Keywords:
obstacle problem; variational inequalities; elasticity theory; finite difference quotient; penalty method; finite element methodSoftware:
ParaViewReferences:
| [1] | Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12, 623-727, 1959 · Zbl 0093.10401 · doi:10.1002/cpa.3160120405 |
| [2] | Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Commun. Pure Appl. Math., 17, 35-92, 1964 · Zbl 0123.28706 · doi:10.1002/cpa.3160170104 |
| [3] | Ahrens, J.; Geveci, B.; Law, C., ParaView: An End-User Tool for Large Data Visualization, 2005, Berlin: Elsevier, Berlin |
| [4] | Alexandrescu, O., Théorème d’existence pour le modèle bidimensionnel de coque non linéaire de W. T. Koiter, C. R. Acad. Sci. Paris Sér. I Math., 319, 899-902, 1994 · Zbl 0809.73040 |
| [5] | Brenner, S.; Scott, LR, The Mathematical Theory of Finite Element Methods, 2008, New York: Springer, New York · Zbl 1135.65042 · doi:10.1007/978-0-387-75934-0 |
| [6] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011, New York: Springer, New York · Zbl 1220.46002 · doi:10.1007/978-0-387-70914-7 |
| [7] | Brezis, H., Sibony, M.: Méthodes d’approximation et d’itération pour les opérateurs monotones. Arch. Ration. Mech. Anal. 28, 59-82 (1967/1968) · Zbl 0157.22501 |
| [8] | Brezis, H.; Stampacchia, G., Sur la régularité de la solution d’inéquations elliptiques, Bull. Soc. Math. France, 96, 153-180, 1968 · Zbl 0165.45601 · doi:10.24033/bsmf.1663 |
| [9] | Caffarelli, LA; Friedman, A., The obstacle problem for the biharmonic operator, Ann. Scuola Norm. Sup. Pisa CI Sci., 6, 151-184, 1979 · Zbl 0405.31007 |
| [10] | Caffarelli, LA; Friedman, A.; Torelli, A., The two-obstacle problem for the biharmonic operator, Pac. J. Math., 103, 325-335, 1982 · Zbl 0511.49002 · doi:10.2140/pjm.1982.103.325 |
| [11] | Chapelle, D.; Bathe, K-J, The Finite Element Analysis of Shells—Fundamentals, 2011, Berlin: Springer, Berlin · Zbl 1211.74002 · doi:10.1007/978-3-642-16408-8 |
| [12] | Chen, Z., Glowinski, R., Li, K.: Current Trends in Scientific Computing: ICM 2002 Beijing Satellite Conference on Scientific Computing, August 15-18, 2002, Xi’an Jiaotong University, Xi’an, China. American Mathematical Society, Providence (2003) · Zbl 1171.86303 |
| [13] | Ciarlet, PG, The Finite Element Method for Elliptic Problems, 1978, Amsterdam: North-Holland, Amsterdam · Zbl 0383.65058 |
| [14] | Ciarlet, PG, Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity, 1988, Amsterdam: North-Holland, Amsterdam · Zbl 0648.73014 |
| [15] | Ciarlet, PG, Mathematical Elasticity. Vol. III: Theory of Shells, 2000, Amsterdam: North-Holland, Amsterdam · Zbl 0953.74004 |
| [16] | Ciarlet, PG, An Introduction to Differential Geometry with Applications to Elasticity, 2005, Dordrecht: Springer, Dordrecht · Zbl 1100.53004 |
| [17] | Ciarlet, PG, Linear and Nonlinear Functional Analysis with Applications, 2013, Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1293.46001 · doi:10.1137/1.9781611972597 |
| [18] | Ciarlet, PG; Destuynder, P., A justification of the two-dimensional linear plate model, J. Mécanique, 18, 315-344, 1979 · Zbl 0415.73072 |
| [19] | Ciarlet, PG; Lods, V., On the ellipticity of linear membrane shell equations, J. Math. Pures Appl., 75, 107-124, 1996 · Zbl 0870.73037 |
| [20] | Ciarlet, PG; Lods, V., Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations, Arch. Ration. Mech. Anal., 136, 2, 119-161, 1996 · Zbl 0887.73038 · doi:10.1007/BF02316975 |
| [21] | Ciarlet, PG; Piersanti, P., Obstacle problems for Koiter’s shells, Math. Mech. Solids, 24, 3061-3079, 2019 · Zbl 07273353 · doi:10.1177/1081286519825979 |
| [22] | Ciarlet, PG; Piersanti, P., A confinement problem for a linearly elastic Koiter’s shell, C.R. Acad. Sci. Paris Sér. I, 357, 221-230, 2019 · Zbl 1409.74029 · doi:10.1016/j.crma.2019.01.004 |
| [23] | Ciarlet, PG; Sanchez-Palencia, E., An existence and uniqueness theorem for the two-dimensional linear membrane shell equations, J. Math. Pures Appl., 75, 51-67, 1996 · Zbl 0856.73038 |
| [24] | Ciarlet, PG; Mardare, C.; Piersanti, P., Un problème de confinement pour une coque membranaire linéairement élastique de type elliptique, C. R. Math. Acad. Sci. Paris, 356, 10, 1040-1051, 2018 · Zbl 1402.74067 · doi:10.1016/j.crma.2018.08.002 |
| [25] | Ciarlet, PG; Mardare, C.; Piersanti, P., An obstacle problem for elliptic membrane shells, Math. Mech. Solids, 24, 5, 1503-1529, 2019 · Zbl 1440.74205 · doi:10.1177/1081286518800164 |
| [26] | Duan, W., Piersanti, P., Shen, X., Yang, Q.: Numerical corroboration of Koiter’s model for all the main types of linearly elastic shells in the static case. Math. Mech. Solids |
| [27] | Eggleston, HG, Convexity. Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, 1958, New York: Cambridge University Press, New York · Zbl 0086.15302 |
| [28] | Evans, LC, Part. Differ. Equ., 2010, Providence: American Mathematical Society, Providence · Zbl 1194.35001 |
| [29] | Falk, RS, Error estimates for the approximation of a class of variational inequalities, Math. Comput., 28, 963-971, 1974 · Zbl 0297.65061 · doi:10.1090/S0025-5718-1974-0391502-8 |
| [30] | Frehse, J., Zum Differenzierbarkeitsproblem bei Variationsungleichungen höherer Ordnung, Abh. Math. Sem. Univ. Hamburg, 36, 140-149, 1971 · Zbl 0219.35029 · doi:10.1007/BF02995917 |
| [31] | Frehse, J., On the regularity of the solution of the biharmonic variational inequality, Manuscr. Math., 9, 91-103, 1973 · Zbl 0252.35031 · doi:10.1007/BF01320669 |
| [32] | Ganesan, S.; Tobiska, L., Finite Elements: Theory and Algorithms, 2017, Cambridge: Cambridge University Press, Cambridge · Zbl 1382.65396 · doi:10.1017/9781108235013 |
| [33] | Genevey, K., A regularity result for a linear membrane shell problem, Math. Modell. Numer., 30, 467-488, 1996 · Zbl 0853.73038 · doi:10.1051/m2an/1996300404671 |
| [34] | Geymonat, G.: Sui problemi ai limiti per i sistemi lineari ellittici. In: Atti del Convegno su le Equazioni alle Derivate Parziali (Nervi, 1965), pp. 60-65. Edizioni Cremonese, Rome (1966) · Zbl 0151.15301 |
| [35] | Grisvard, P.: Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original [MR0775683], With a foreword by Susanne C. Brenner · Zbl 1231.35002 |
| [36] | Hörmander, L.: The analysis of Linear Partial Differential Operators. I, Volume 256 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Distribution Theory and Fourier Analysis, 2edn. Springer, Berlin (1990) |
| [37] | Langtangen, H.P., Logg, A.: Solving PDEs in Python, Volume 3 of Simula SpringerBriefs on Computing. Springer, Cham (2016) · Zbl 1376.65144 |
| [38] | Léger, A.; Miara, B., Mathematical justification of the obstacle problem in the case of a shallow shell, J. Elast., 90, 241-257, 2008 · Zbl 1133.74033 · doi:10.1007/s10659-007-9141-1 |
| [39] | Léger, A.; Miara, B., Erratum to: Mathematical justification of the obstacle problem in the case of a shallow shell, J. Elast., 98, 115-116, 2010 · Zbl 1308.74117 · doi:10.1007/s10659-009-9230-4 |
| [40] | Léger, A.; Miara, B., A linearly elastic shell over an obstacle: the flexural case, J. Elast., 131, 19-38, 2018 · Zbl 1387.74021 · doi:10.1007/s10659-017-9643-4 |
| [41] | Li, K.; Huang, A.; Huang, Q., Finite Element Method and Its Applications, 2015, Beijing: Science Press, Beijing |
| [42] | Lions, J-L, Quelques méthodes de résolution des problèmes aux limites non linéaires, 1969, Paris: Dunod; Gauthier-Villars, Paris · Zbl 0189.40603 |
| [43] | Mezabia, ME; Chacha, DA; Bensayah, A., Modelling of frictionless Signorini problem for a linear elastic membrane shell, Appl. Anal., 101, 6, 2295-2315, 2022 · Zbl 1491.74077 · doi:10.1080/00036811.2020.1807008 |
| [44] | Nečas, J., Direct Methods in the Theory of Elliptic Equations, 2012, Heidelberg: Springer, Heidelberg · Zbl 1246.35005 · doi:10.1007/978-3-642-10455-8 |
| [45] | Piersanti, P., On the improved interior regularity of the solution of a second order elliptic boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle, Discrete Contin. Dyn. Syst., 42, 2, 1011-1037, 2022 · Zbl 1481.74516 · doi:10.3934/dcds.2021145 |
| [46] | Piersanti, P., Asymptotic analysis of linearly elastic elliptic membrane shells subjected to an obstacle, J. Differ. Equ., 320, 114-142, 2022 · Zbl 1486.35222 · doi:10.1016/j.jde.2022.02.053 |
| [47] | Piersanti, P., On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell subject to an obstacle, Asymptot. Anal., 127, 1-2, 35-55, 2022 · Zbl 1509.35309 |
| [48] | Piersanti, P., Asymptotic analysis of linearly elastic flexural shells subjected to an obstacle in absence of friction, J. Nonlinear Sci., 33, 4, 39, 2023 · Zbl 1519.35167 · doi:10.1007/s00332-023-09916-y |
| [49] | Piersanti, P.; Shen, X., Numerical methods for static shallow shells lying over an obstacle, Numer. Algorithms, 1, 623-652, 2020 · Zbl 1465.65144 · doi:10.1007/s11075-019-00830-7 |
| [50] | Piersanti, P.; Temam, R., On the dynamics of grounded shallow ice sheets. Modelling and analysis, Adv. Nonlinear Anal., 12, 1, 40, 2023 · Zbl 1518.35465 |
| [51] | Piersanti, R.; Africa, PC; Fedele, M.; Vergara, C.; Dedè, L.; Corno, AF; Quarteroni, A., Modeling cardiac muscle fibers in ventricular and atrial electrophysiology simulations, Comput. Methods Appl. Mech. Eng., 373, 113468, 2021 · Zbl 1506.74211 · doi:10.1016/j.cma.2020.113468 |
| [52] | Piersanti, P.; White, K.; Dragnea, B.; Temam, R., Modelling virus contact mechanics under atomic force imaging conditions, Appl. Anal., 101, 11, 3947-3957, 2022 · Zbl 1495.74046 · doi:10.1080/00036811.2022.2044027 |
| [53] | Piersanti, P.; White, K.; Dragnea, B.; Temam, R., A three-dimensional discrete model for approximating the deformation of a viral capsid subjected to lying over a flat surface, Anal. Appl., 20, 6, 1159-1191, 2022 · Zbl 1501.35224 · doi:10.1142/S0219530522400024 |
| [54] | Regazzoni, F.; Dedè, L.; Quarteroni, A., Active force generation in cardiac muscle cells: mathematical modeling and numerical simulation of the actin-myosin interaction, Vietnam J. Math., 49, 1, 87-118, 2021 · Zbl 1472.65114 · doi:10.1007/s10013-020-00433-z |
| [55] | Rodríguez-Arós, A., Mathematical justification of the obstacle problem for elastic elliptic membrane shells, Appl. Anal., 97, 1261-1280, 2018 · Zbl 1390.74132 · doi:10.1080/00036811.2017.1337894 |
| [56] | Scholz, R., Numerical solution of the obstacle problem by the penalty method, Computing, 32, 4, 297-306, 1984 · Zbl 0528.65057 · doi:10.1007/BF02243774 |
| [57] | Stampacchia, G.: Èquations elliptiques du second ordre à coefficients discontinus, volume 1965 of Séminaire de Mathématiques Supérieures, No. 16 (Été. Les Presses de l’Université de Montréal, Montreal (1966) · Zbl 0151.15501 |
| [58] | Sun, W.; Yuan, Y-X, Optimization Theory and Methods, Volume 1 of Springer Optimization and Its Applications, 2006, New York: Springer, New York · Zbl 1129.90002 |
| [59] | Zingaro, A.; Dedè, L.; Menghini, F.; Quarteroni, A., Hemodynamics of the heart’s left atrium based on a variational multiscale-LES numerical method, Eur. J. Mech. B Fluids, 89, 380-400, 2021 · Zbl 1492.76159 · doi:10.1016/j.euromechflu.2021.06.014 |
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