Locking Phenomenon in Computational Methods of the Shell Theory
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International Applied Mechanics, Vol. 56, No. 3, May, 2020
LOCKING PHENOMENON IN COMPUTATIONAL METHODS OF THE SHELL THEORY
V. À. Maksymyuk
The causes of computational locking in the shell theory are analyzed. The general cause of the phenomenon is shown to be rooted in variations calculus and related to the relationship between variable functions. As exemplified by a numerical case, the convergence may depend on the type of load. Keywords: elliptical cylinder, stress-strain state, locking, finite-difference method Introduction. Cylindrical shells having a noncircular cross-section are of practical interest in engineering [1, 3, 5, 18]. From a theoretical point of view, the calculation of the stress–strain state (SSS) for such shells under certain loads can be a test for numerical methods. One of the simplest tests is the one-dimensional problem [1, 3, 19, 20] of the strain of a long cylindrical shell with an elliptical cross-section (ring) under a constant internal pressure, which has turned one hundred and fifty years. This problem has made it possible to formulate some general conclusions about the accuracy of numerical methods of the shell theory. At a certain stage, the grid computational methods of shell mechanics faced the so-called problem [2, 5, 12, 13, and 15] of locking, which manifested itself in their slow but, importantly, stable convergence. Mathematicians and mechanicians explain the causes of locking in different ways. The former refer to “ill-posed problem” [6, 14], “ill-conditioned matrix” [4, 7, 16], and “small coefficients of higher derivatives” [17], i.e., to the causes stemming from algebra or differential equations. The latter talk about “rigid-body displacements” [10], “share of various strains in the energy: [9], i.e., consider mechanical causes. However, some researchers [2, 15] of the computational locking phenomenon have already expressed doubts about mathematical formulations, while others have changed their minds in favor of mechanical formulations. 1. Analysis of Locking Causes. Apparently, there is a common cause of locking that lies in the field of variations calculus. Following discretization, the grid methods lead to a system of linear algebraic equations (SLAE). Usually, obtaining SLAE is preceded by varying a certain functional. Only independent functions are variable. Because of an incorrectly chosen coordinate system, system of variable functions, or some features of the deformation of the structure, some relationships may arise between the variable functions, which lead to the computational locking. For example, in the test problem [13] of strain-free displacement of a ring (Fig. 1), in the polar coordinate, the displacements are linked by the Pythagorean theorem, whereas, this is not the case in the Cartesian one. The displacement is provided by an additional condition on the line j = 0°: u r = h. The solution of this plane problem is written as u j = -h sin j , u r = h cos j . Obviously, the displacement components in the polar system are not independent. In addition to
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