Representations of hypersurfaces and minimal smoothness of the midsurface in the theory of shells

• Autorzy: Delfour M. C.
• Języki publikacji: EN

Abstrakty

EN

Many hypersurfaces ω in R^N can be viewed as a subset of the boundary Γ of an open subset Ω of R^N. In such cases, the gradient and Hessian matrix of the associated oriented distance function ba to the underlying set Ω completely describe the normal and the N fundamental forms of ω, and a fairly complete intrinsic theory of Sobolev spaces on C1'1-hypersurfaces is available in Delfour (2000). In the theory of thin shells, the asymptotic model only depends on the choice of the constitutive law, the midsurface, and the space of solutions that properly handles the loading applied to the shell and the boundary conditions. A central issue is the minimal smoothness of the midsurface to still make sense of asymptotic membrane shell and bending equations without ad hoc mechanical or mathematical assumptions. This is possible for a C1'1-midsurface with or without boundary and without local maps, local bases, and Christoffel symbols via the purely intrinsic methods developed by Delfour and Zolesio (1995a) in 1992. Anicic, LeDret and Raoult (2004) introduced in 2004 a family of surfaces ω that are the image of a connected bounded open Lipschitzian domain in R² by a bi-Lipschitzian mapping with the assumption that the normal field is globally Lipschizian. >From this, they construct a tubular neighborhood of thickness 2h around the surface and show that for sufficiently small h the associated tubular neighborhood mapping is bi-Lipschitzian. We prove that such surfaces are C1'1-surfaces with a bounded measurable second fundamental form. We show that the tubular neighborhood can be completely described by the algebraic distance function to ω and that it is generally not a Lipschitzian domain in R³ by providing the example of a plate around a flat surface ω verifying all their assumptions. Therefore, the G1-join of K-regular patches in the sense of Le Dret (2004) generates a new K-regular patch that is a C1'1-surface and the join is C1'1. Finally, we generalize everything to hypersurfaces generated by a bi-Lipschitzian mapping defined on a domain with facets (e.g. for sphere, torus). We also give conditions for the decomposition of a C1'1-hypersurface into C1'1-patches.

Słowa kluczowe

EN

thin shell; asymptotic shell; midsurface; smoothness; representation of a surface; oriented distance function; bi-Lipschitz mapping; tubular neighborhood

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2008
• Tom: Vol. 37, no 4
• Strony: 879--911
• Opis fizyczny: Bibliogr. 20 poz., rys.

Twórcy

autor

Delfour M. C.

• Centre de recherches mathematiques et Departement de mathematiques et de statistique, Universite de Montreal, C. P. 6128 succ. Centre-ville, Montreal (Qc), H3C 3J7, Canada,

Bibliografia

• ADAMS, R., ARONSZAJN, N. and SMITH, K.T. (1968) Theory of Bessel Potentials. Part II. Ann. Inst. Fourier (Grenoble) 17 (1967) fasc. 2, 1-133.
• ANICIC S., LE DRET, H. and RAOULT, A. (2004) The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity. Math. Meth. Appl. Sci. 27, 1283-1299.
• BERNADOU, M. and DELFOUR, M.C. (2000) Intrinsic models of piezoelectric shells. In: Proceedings of ECCOMAS 2000 (European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, Sept. 11-14).
• CIARLET, PH.D. (2000) Mathematical Elasticity, vol. III: Theory of Shells. North-Holland, Elsevier Science, Amsterdam.
• DELFOUR, M.C. (1998) Intrinsic differential geometric methods in the asymptotic analysis of linear thin shells. In: M.C. Delfour, ed., Boundaries, interfaces and transitions (Banff, AB, 1995), CRM Proc. Lecture Notes 13, Amer. Math. Soc., Providence, RI, 19-90.
• DELFOUR, M.C. (1999a) Intrinsic P(2,1) thin shell model and Naghdi’s models without a priori assumption on the stress tensor. In: K.H. Hoffmann, G. Leugering and F. Tröltzsch, eds., Optimal control of partial differential equations, Internal. Ser. Numer. Math. 133, Birkhäuser, Basel, 99-113.
• DELFOUR, M.C. (1999b) Characterization of the space of the membrane shell equation for arbitrary C1’1 midsurfaces. Control and Cybernetics 28 (3), 481-501.
• DELFOUR, M.C. (2000) Tangential differential calculus and functional analysis on a C1’1 submanifold. In: R. Gulliver, W. Littman and R. Triggiani, eds., Differential-geometric methods in the control of partial differential equations, Contemp. Math. 268, Amer. Math. Soc., Providence, RI, 83-115.
• DELFOUR, M.C. (2002) Modeling and control of asymptotic shells. In: W. Desch, F. Kappel and K. Kunish, eds., Control and Estimation of Distributed Parameter Systems, International Series of Numerical Mathematics 143, Birkhäuser Verlag, 105-120.
• DELFOUR, M.C. and BERNADOU, M. (2002) Intrinsic asymptotic model of piezoelectric shells. In: K.-H. Hoffmann, I. Lasiecka, G. Leugering, J. Sprekels and F. Tröltzsch, eds., Optimal Control of Complex Structures, International Series of Numerical Mathematics 139, Birkhäuser Verlag, 59-72.
• DELFOUR, M.C. and ZOLÉSIO, J.P. (1994) Shape analysis via oriented distance functions. J. Fund. Anal. 123(1), 129-201.
• DELFOUR, M.C. and ZOLÉSIO, J.P. (1995a) On a variational equation for thin shells. In: J. Lagnese, D.L. Russell and L. White, eds., Control and Optimal Design of Distributed Parameter Systems, IMA Vol. Math. Appl. 70, Springer, New York, 25-37.
• DELFOUR, M.C. and ZOLÉSIO, J.P. (1995b) A boundary differential equation for thin shells. J. Differential Equations 119, 426-449.
• DELFOUR, M.C. and ZOLÉSIO, J.P. (1996) Tangential differential equations for dynamical thin/shallow shells. J. Differential Equations 128, 125-167.
• DELFOUR, M.C. and ZOLÉSIO, J.P. (1997) Differential equations for linear shells: comparison between intrinsic and classical models. In: Luc Vinet, ed., Advances in mathematical sciences: CRM’s 25 years, CRM Proc. Lecture Notes 11, Amer. Math. Soc., Providence, RI, 41-124.
• DELFOUR, M.C. and ZOLÉSIO, J.P. (2001) Shapes and geometries. Analysis, differential calculus, and optimization. Advances in Design and Control 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
• FEDERER, H. (1959) Curvature measures. Trans. Amer. Math. Soc. 93, 418-419.
• FREY, P.J. AND GEORGE, P.L. (2008) Mesh Generation: Application to Finite Element, Second Edition. ISTE Ltd, London and John Wiley and Sons, Inc, Hoboken (NJ).
• LE DRET, H. (2004) Well-posedness for Koiter and Naghdi shells with a G1-midsurface. Analysis and Applications 2 (4), 365-388.
• NAGUMO, M. (1942) Über die Loge der Integralkurven gewöhnlicher Differenialgleichungen. Proc. Phys. Math. Soc. Japan 24, 551-559.