Abstract
Mesh methods for discretization of the differential vector relations are generalized as applied to problems of shell theory. In the finite-difference method, covariant derivatives are replaced by vector differences, which are then projected on the vectors of a local basis. In the finite-element method, vector functions are approximated by a Taylor series with tensor coefficients. It is shown that such schemes satisfy the condition of rigid displacement for a deformable body, which improves considerably the convergence of the solution. The proposed schemes, which are sensitive to approximation uncertainties, were tested by solving problems on deformation of shells
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
N. P. Abovskii and A. P. Deruga, “The basic results and trends in the development of the variational-difference method in design of complex shell-rod structures,” in: Three-Dimensional Structures in the Krasnoyarsk Territory [in Russian], Izd. KISI, Krasnoyarsk (1989), pp. 13–26.
A. I. Golovanov and M. S. Kornishin, Introduction to the Finite-Element Method of the Statics of Thin Shells [in Russian], Kazanskii Fiz.-Tekhn. Inst., Kazan' (1989).
E. A. Gotsulyak, V. N. Ermishev, and N. T. Zhadrasinov, Application of the Curvilinear-Mesh Method to Shell Design [in Russian], Manuscript No. 2557-81 deposited at UkrNIINTI 01.06.81, Kiev (1980).
E. A. Gotsulyak, V. N. Ermishev, and N. T. Zhadrasinov, “Convergence of the curvilinear-mesh method in problems of shell theory,” Soprot. Mater. Theor. Sooruzh., Issue 39, 80–84 (1981).
E. A. Gotsulyak and E. V. Kostina, “Analysis of the finite-element method for convergence in vector approximation,” Soprot. Mater. Theor. Sooruzh., Issue 63, 38–47 (1997).
E. O. Gotsulyak and O. V. Kostina, “Special applications of the finite-element method to the design of general-type shells,” Dop. AN Ukrainy, No. 11, 72–75 (1998).
G. Cantin, “Rigid body motions in curved finite elements,” AIAA J., 8, No. 7, 1252–1255 (1970).
G. Cantin and R. W. Clough, “A curved, cylindrical-shell, finite element,” AIAA J., 67, No. 6 (1968).
A. S. Sakharov, “The moment scheme of finite elements with regard for rigid displacements,” Soprot. Mater. Theor. Sooruzh., Issue 24, 147–156 (1981).
G. A. Fonder and R. W. Clough, “Explicit addition of rigid-body motions in curved finite elements,” AIAA J., 11, No. 3, 305–312 (1973).
R. H. Gallagher, The Development and Evaluation of Matrix Methods for Thin Shell Structural Analysis, New York (1966), pp. 24–30.
E. A. Gotsulyak and E. V. Kostina, “Vector relations for finite element digitization of the theory of thin shells of arbitrary form,” in: Proc. Int. Congr. ICSS-98, Moscow (1998), pp. 252–259.
E. A. Gotsulyak and D. E. Prusov, “Numerical study of the stability of geometrically imperfect shells of general form,” Int. Appl. Mech., 35, No. 6, 610–613 (1999).
Ya. V. Grigorenko, Ya. G. Savula, and I. S. Mukha, “Linear and nonlinear problems on the elastic deformation of complex shells and methods of their numerical solution,” Int. Appl. Mech., 36, No. 8, 979–1000 (2000).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gotsulyak, E.A. Mesh Discretization of the Vector Relations of Shell Theory in a Curvilinear Coordinate System. International Applied Mechanics 37, 784–789 (2001). https://doi.org/10.1023/A:1012415308469
Issue Date:
DOI: https://doi.org/10.1023/A:1012415308469