Abstract
The paper investigates a system of three equations of strongly elliptic type, supplemented by a boundary condition for one variable and a periodicity condition for the second. This system corresponds to the problem of determining the displacements of a momentless lattice cylindrical shell from an initial state prestressed by internal pressure. The paper provides sufficient conditions on the dimensions of the domain under which the boundary value problem under consideration is well-posed. These conditions are refined for the case where the solution to the original problem is assumed to be symmetric a priori. 2 illustrations. 7 references.
Full Text
Preamble
This work, published in 1967 (Vol. III, No. 8), addresses problems in mathematical physics related to boundary value problems for partial differential equations, specifically focusing on the methods developed by E. G. D'yakonov and I. K. Larin \cite{1, 2}. The study considers systems of equations arising in the theory of elasticity and shell theory (including plates and shallow shells), which are characterized by specific boundary conditions. The approach builds upon the foundational work in \cite{1}, which utilized iterative methods for solving elliptic systems, including those related to the Lamé equations and various problems in the bending of plates.
The present investigation extends the results of E. G. D'yakonov and I. K. Larin regarding the convergence of iterative processes for these systems. In particular, we examine the spectral properties and the choice of optimal parameters for the iterative schemes, building on the estimates provided in \cite{2, 3}. The analysis involves the use of functional spaces and energy estimates to establish the stability and convergence of the numerical solutions.
Problem Formulation and Operator Estimates
We consider the operator equation of the form:
$$Lu(x) + Ru(x) + Qu(x) = f(x)$$
where $u(x) = (u, v, w)^T$ is the vector of unknown functions representing displacements in the $x_1, x_2$ directions and the normal deflection. The operators $L, R,$ and $Q$ are defined based on the geometric and physical properties of the shell, involving differential operators $D_1$ and $D_2$ with respect to the spatial coordinates. The parameter $p$ characterizes the geometry, with $0 < p < \pi/2$.
The domain of interest is defined as $\Omega = { -\infty < x_1 < +\infty, -l_2 < x_2 < l_2 }$, assuming periodicity in the $x_1$ direction with period $2l_1$. The boundary conditions at $x_2 = \pm l_2$ are taken to be homogeneous, $u|{x_2 = \pm l_2} = 0$. We work within the Sobolev space $W_2^1$, equipped with the norm:
$$||u||$$} = \left( \sum_{|\alpha| \le 1} ||D^\alpha u||^2 \right)^{1/2
where $(\cdot, \cdot)$ denotes the standard $L_2$ inner product.
Energy Estimates and Convergence
To establish the convergence of the iterative method, we derive lower bounds for the operator $L$. Specifically, we seek a constant $\gamma > 0$ such that for any $u \in W_2^1$:
$$[Lu, u] \ge \gamma ||u||_{W_2^1}^2$$
The derivation involves several lemmas. Lemma 1 and Lemma 2 provide estimates for the bilinear forms $[Ru, u]$ and $[Qu, u]$ in terms of the integrals $J_1$ and $J_2$, which involve the derivatives of the displacement components. By combining these results, Lemma 3 establishes the positive definiteness of the total operator under certain conditions on the parameters $p, q,$ and $r$.
Specifically, we analyze the expression:
$$[Lu, u] = (1 - s^2)(J_1 + J_2) + J$$
where $s = \sin p$. By introducing auxiliary parameters and applying Cauchy-Schwarz inequalities, we can bound the terms involving cross-derivatives. The goal is to find the supremum of the lower bound $\gamma(t, p)$ over the admissible range of parameters.
Numerical Results and Parameter Optimization
The effectiveness of the iterative scheme depends heavily on the spectral radius of the transition operator. We investigate the behavior of the convergence factor $\gamma$ as a function of the geometric parameter $t = \tan p$. Our analysis shows that for $0 < p < 1$, the operator $L$ remains strongly elliptic, and the iterative process converges at a rate determined by the ratio of the minimum and maximum eigenvalues of the associated operator.
As shown in the figures and tables provided in the full text, the optimal parameters $q$ and $r$ can be determined to maximize the convergence rate. For the case $p=0$, our results coincide with the classical estimates for plates provided in \cite{1}. The extension to $p > 0$ allows for the treatment of curved shells, where the coupling between the membrane and bending states becomes significant.
References
- D'yakonov, E. G. On some iterative methods for solving systems of difference equations. Computational Methods and Programming, Moscow State University, 1959.
- Larin, I. K. On the solution of equations for shallow shells. Journal of Computational Mathematics and Mathematical Physics, No. 1, 81–89, 1966.
- Mikhlin, S. G. Variational Methods in Mathematical Physics, Gostekhizdat, 1952.
- Sobolev, S. L. Applications of Functional Analysis in Mathematical Physics, Leningrad State University, 1950.
- D'yakonov, E. G. Difference Methods for Solving Boundary Value Problems, Moscow State University, 1965.
- Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity, 1957.
- Vlasov, V. Z. General Theory of Shells and its Applications in Engineering, 1951.