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\def\author{\it Bulletin of TICMI}
\def\thema{\it Volume 2, 1998}
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\newcommand{\nc}{\newcommand}
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%\nc{\ov}{\overline}
\def\Qalph{\mathop{Q_\alpha}}
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\def\Fk{\mathop{F_{ij,1}}}
\def\Fkn{\mathop{F_{ij,2}}}

\begin{center}
{TO DESIGN OF BENDING PROBLEMS WITH BOUNDARY LAYERS FOR ELASTIC PLATES}

\vspace*{0.3cm}
{\it A. Muradova}

\vspace*{0.3cm}
{\it I. Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.3cm}
\par Let an elastic plate be isotropic and homogenuous, then equations with
respect to a deflection and shearing forces are represented 
as a split system of a biharmonic equations 
and two Helmholtz' equations (see e.g. [1]).
Consider a case, when boundary conditions of the
problem for the deflection and the shearing forces are also split.
\par
Thus, assume that for the deflection we have the following problem
\begin{equation}
D\Delta^2w(x,y)=f(x,y),
\end{equation}
\begin{equation}
w(x,y)\big|_{\partial G}=\frac{\partial}{\partial n}w(x,y)\bigg|_{\partial G}=0,
\end{equation}
where 
$(x,y)\in G=(-1,1)$x$(-1,1),$ $\big\{\partial G:\;|x|=1,\;|y|=1\big\},$
$D$ is cylindrical rigidity, and
if we change the independent variables $x,\;y,$ for the shearing forces 
we have the problems:
$$\eps\Delta Q_\alpha(\xi,\eta)-Q_\alpha(\xi,\eta)=F_\alpha(\xi,\eta),\;\;\;
0<\xi,\eta<1,\;\;\;\alpha=1,2,\eqno(3_\alpha)$$
$$\begin{array}{c}
\displaystyle
\partial_1Q_1(\xi,\eta)\big\vert_{\xi=0}=\mu_{01}(\eta),\;\;\;
\partial_1Q_1(\xi,\eta)\big\vert_{\xi=1}=\mu_{11}(\eta),\\
\\ [-0.3cm]
Q_1(\xi,0)=\nu_{01}(\xi),\;\;\;Q_1(\xi,1)=\nu_{11}(\xi),
\end{array}\eqno(4_1)$$
$$\begin{array}{c}
\displaystyle
Q_2(0,\eta)=\mu_{02}(\eta),\;\;\;
Q_2(1,\eta)=\mu_{12}(\eta),\\
\\ [-0.3cm]
\partial_2Q_2(\xi,\eta)\big\vert_{\eta=0}=\nu_{02}(\xi),
\;\;\;\partial_2Q_2(\xi,\eta)\big\vert_{\eta=1}=\nu_{12}(\xi),
\end{array}\eqno(4_2)$$
where $\displaystyle \eps=\frac{h^2}{12}(1+2\gamma),$
$2h$ is a thickness of the plate, $\gamma$ is an arbitrary control parameter, 
$\nu$ is a coefficient of rigidity,
$\partial_1=\partial/\partial\xi,\;
\partial_2=\partial/\partial\eta.$
\par
For solving (1), (2) we use the projective method by [2].
Then the solution of (1), (2) has the following form:
$$w(x,y)=\sum\limits_{i,j=1}^{m}w^{ij}\chi^2P_i(x)\chi^2P_j(y),$$
where
$\chi^2P_i(x),\;\chi^2P_j(y)$ are second order divided differences with respect to
indeces from the Legendre polynomials:
\par
For solving (3$_\alpha$), (4$_\alpha$) an alternating direction iterative scheme is used,
which is a modification of the continual analogue of the alternating
direction method [3] for discrete version. This scheme has the following form: 
$$\begin{array}{c}
\displaystyle
\hspace*{-3.7cm}
(r+1)\Qalph\limits^{(k+\frac{1}{2})}(\xi,\eta_j)+\eps A_1\Qalph\limits^{(k+\frac{1}{2})}(\xi,\eta)\bigg|_{\eta=\eta_j}=\\
\\ [-0.3cm]
\displaystyle
\hspace*{3cm}=r\Qalph\limits^{(k)}(\xi,\eta_j)-\eps A_2\Qalph\limits^{(k)}(\xi,\eta)\bigg|_{\eta=\eta_j}-
F_\alpha(\xi,\eta_j),
\end{array}\eqno(5_\alpha)$$
$$\partial_1\Qm\limits^{(k+\frac{1}{2})}(\xi,\eta)\big|_{\xi=0,\;\eta=\eta_j}=\mu_{01}(\eta_j),\;\;\;
\partial_1\Qm\limits^{(k+\frac{1}{2})}(\xi,\eta)\big|_{\xi=1,\;\eta=\eta_j}=\mu_{11}(\eta_j),
\eqno(6_1)$$
$$\Qn\limits^{(k+\frac{1}{2})}(0,\eta_j)=\mu_{02}(\eta_j),\;\;\;
\Qn\limits^{(k+\frac{1}{2})}(1,\eta_j)=\mu_{12}(\eta_j),\;\;\;
j=\ov{2,n},\eqno(6_2)$$
$$\begin{array}{c}
\displaystyle
\hspace*{-3.7cm}
(r+1)\Qalph\limits^{(k+1)}(\xi_i,\eta)+\eps A_2\Qalph\limits^{(k+1)}(\xi,\eta)\bigg|_{\xi=\xi_i}=\\
\\ [-0.3cm]
\displaystyle
\hspace*{3cm}=r\Qalph\limits^{(k+\frac{1}{2})}(\xi_i,\eta)-
\eps A_1\Qalph\limits^{(k+\frac{1}{2})}(\xi,\eta)\bigg|_{\xi=\xi_i}-
F_\alpha(\xi_i,\eta),
\end{array}\eqno(7_\alpha)$$
$$\Qm\limits^{(k+1)}(\xi_i,0)=\nu_{01}(\xi_i),\;\;\;
\Qm\limits^{(k+1)}(\xi_i,1)=\nu_{11}(\xi_i),\eqno(8_1)$$
$$\partial_2\Qn\limits^{(k+\frac{1}{2})}(\xi,\eta)\big|_{\xi=\xi_i,\eta=0}=\nu_{02}(\xi_i),\;\;\;
\partial_2\Qn\limits^{(k+\frac{1}{2})}(\xi,\eta)\big|_{\xi=\xi_i,\eta=1}=\nu_{12}(\xi_i),
\;\;\;i=\ov{2,n},\;(8_2)$$
where $A_1=-\partial^2/\partial\xi^2,\;A_2=-\partial^2/\partial\eta^2,\;r$
is an iterative parameter. 
\par
The problems (5$_\alpha$), (6$_\alpha$) and (7$_\alpha$), (8$_\alpha$) are the problems with boundary layer for second 
order ordinary differential equations. 
For solving these problems we apply a generalized finite difference method  
and (Q)-formulas [3]. The iterative schemes have the form:
$$
\hspace*{-3.7cm}
(r+1)(B_{1,h}\Qalph\limits^{(k+\frac{1}{2})})_{ij}+
\eps(A_{\alpha,01,h}\Qalph\limits^{(k+\frac{1}{2})})_{ij}=
r(B_{1,h}\Qalph\limits^{(k)})_{ij}-\eqno(9_\alpha)
$$
$$\begin{array}{c}
\displaystyle
-\eps(B_{1,h}B_{2,h}^{-1}A_{\alpha,02,h}\Qalph\limits^{(k)})_{ij}
-\eps(B_{1,h}B_{2,h}^{-1}F_{\alpha}^2)_{ij}-(B_{1,h}F_\alpha)_{ij}+F_{\alpha,ij}^1,
\;i=\ov{n_1,n},
\end{array}$$
$$\partial_1\mathop{Q_{1,1j}}\limits^{(k+\frac{1}{2})\ \ \ }=\mu_{01}(\eta_j),\;\;\;
\partial_1\mathop{Q_{1,n+1,j}}\limits^{(k+\frac{1}{2})\ \ \ \ \ \ }=\mu_{11}(\eta_j),
\eqno(10_1)$$
$$\mathop{Q_{2,1j}}\limits^{(k+1)\ \ \ }=\mu_{02}(\eta_j),\;\;\;
\mathop{Q_{2,n+1,j}}\limits^{(k+1)\ \ \ \ \ \ }=\mu_{12}(\eta_j),
\;\; j=\ov{2,n},
\eqno(10_2)$$
$$
\displaystyle
\hspace*{-3.7cm}
(r+1)(B_{2,h}\Qalph\limits^{(k+1)})_{ij}+
\eps(A_{\alpha,02,h}\Qalph\limits^{(k+1)})_{ij}=
r(B_{2,h}\Qalph\limits^{(k+\frac{1}{2})})_{ij}-
\eqno(11_\alpha)$$
$$\begin{array}{c}
\displaystyle
-\eps(B_{2,h}B_{1,h}^{-1}A_{\alpha,01,h}\Qalph\limits^{(k+\frac{1}{2})})_{ij}
-\eps(B_{2,h}B_{1,h}^{-1}F_{\alpha}^1)_{ij}-(B_{2,h}F_\alpha)_{ij}+F_{\alpha,ij}^2,
\;\;j=\ov{n_2,n},
\end{array}$$
$$\mathop{Q_{1,i1}}\limits^{(k+1)\ \ \ }=\nu_{01}(\xi_i),\;\;\;
\mathop{Q_{1,i,n+1}}\limits^{(k+1)\ \ \ \ \ \ }=\nu_{11}(\xi_i),\eqno(12_1)$$
$$\partial_2\mathop{Q_{2,i1}}\limits^{(k+\frac{1}{2})\ \ \ }=\nu_{02}(\xi_i),\;\;\;
\partial_2\mathop{Q_{2,i,n+1}}\limits^{(k+\frac{1}{2})\ \ \ \ \ \ }=\nu_{12}(\xi_i),
\eqno(12_2)$$
\hspace*{10cm}$i=\ov{2,n}.$\\
here $A_{\alpha,01,h},\;A_{\alpha,02,h},$ $B_{1,h},\;B_{2,h},$ 
$F_{\alpha,ij}^1,\;F_{\alpha,ij}^2$ ($F_{1,ij}^1=0,\;F_{2,ij}^2=0$) 
are defined by the formulas of the generalized finite difference method 
and (Q) formulas; $n_1=1,$ $n_2=2$ for $\alpha=1$ and $n_1=2$ $n_2=1$ for $\alpha=2.$
\par
{\bf Theorem.} {\it If the operators $B_{1,h}^{-1}A_{\alpha,01,h},\;
B_{2,h}^{-1}A_{\alpha,02,h}$ are positive definite, 
then the iterative processes {\rm(9$_\alpha$), (10$_\alpha$)} and 
{\rm(11$_\alpha$)}, {\rm(12$_\alpha$)} converge for $r\ge 0$ and any $\Qalph\limits^{(0)},$ 
$A_2\Qalph\limits^{(0)}.$} 

\vskip 0.3cm
Acknowledgment. The work is written under the supervision of Professor T. Vashakmadze,
the author is grateful to him.

\vspace*{0.4cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.2cm}
\end{center}
\par[1] Vashakmadze T.S., Some Problems of Mathematical Theory of Anysotropic
Elastic Plates. Tbilisi, 1986. (in Russian).
\par[2] Vashakmadze  T.S., On the Application of One Numerical Process in the Dirichlet
Problem of Theory of Shells. Weimar, Wissensch. Z. Hochsch Arch.
Baumwesen, N2, 1972, pp. 228-231 (In Russian).
\par[3] Vashakmadze T.S., Method of Generalized Factorization  and its 
Application to Numerical Solution Some Problems  Mathem. Physics.,
Proceedings of Symp. Continuum Mechanics and Related Problems of
Analysis. Tbilisi: Metsniereba, v. 1, 1972, pp. 36-45. (in Russian).


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