# V-ellipticity and finite element approximation of the clamped shallow non circular cylindrical shell problem

Pulin Kumar Bhattacharyya

► To cite this version:

Pulin Kumar Bhattacharyya. V-ellipticity and finite element approximation of the clamped shallow non circular cylindrical shell problem. [Research Report] IRIA-RR-264, IRIA. 1977, pp.35. hal-04716433

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V-ELLIPTICITY AND
FINITE ELEMENT
APPROXIMATION OF THE
CLAMPED SHALLOW
NONCIRCULAR CYLINDRICAL
SHELL PROBLEM

Pulin Kumar BHATTACHARYYA

Rapport de Recherche N° 264

Novembre 1977

35p.
# $\vec{V}$-ELLIPTICITY AND FINITE ELEMENT APPROXIMATION OF THE CLAMPED SHALLOW NONCIRCULAR CYLINDRICAL SHELL PROBLEM*

Pulin Kumar Bhattacharyya**

## Résumé :

On établit la $\vec{V}$-ellipticité, l'existence et l'unicité de la solution du problème continu des coques cylindriques en général non circulaires, et peu profondes au sens de W.T. Koiter, les coques étant encastrées au bord. Alors, on donne une estimation de l'erreur pour la solution du problème discretisé à l'aide d'une méthode d'éléments finis conformes.

## Abstract :

The $\vec{V}$-ellipticity, the existence and uniqueness of solution of the (continuous) problem of shallow noncircular cylindrical shells clamped along straight and curvilinear edges have been established, assuming W.T. Koiter's definition of «shallowness». Then an error estimate for the conforming finite element solution of the corresponding discrete problem is given.

<footnote>* Ce travail a pu être réalisé grâce aux échanges culturels franco-indiens.</footnote>

<footnote>** IIT, Math. Department, New Delhi 110029 INDE et IRIA/LABORIA</footnote>
# INTRODUCTION

Of all consistent general theories of thin elastic shells, Koiter's [13], [15] and Naghdi's [19], [20] formulations have found interesting responses. The $\vec{V}$-ellipticity of the Koiter's formulation has been established by BERNADOU and CIARLET [2], while that of Naghdi's formulation has been proved by COUTRIS [8]. ROUGEE [23] has dealt with the $\vec{V}$-ellipticity of elastic supported cylindrical shells. In this connection, one must mention also the MIKHLIN's results on shell operators in [18]. The present report deals with the $\vec{V}$-ellipticity of clamped shallow noncircular cylindrical shells and their conforming finite element analysis. As it is well-known, there exist various formulations of the shallow shell theory [11]–[15], [17], [24], [22]. In this report, Koiter's tensoral approach [13]–[15] will be used, accepting his definition of "shallowness" [15]. For the validity of approximations of shallow shell theory, one may find interesting results in [5], [11], [22], [24], besides the papers of KOITER [13] [15].

In conforming finite element analysis, the basic results of CIARLET in [6], [7] will be utilized.

# 1. - GEOMETRY OF THE MIDDLE SURFACE OF A CYLINDRICAL SHELL

## 1.1. Definition of the middle surface S

Let $\vec{\phi}$ be a mapping from $\varepsilon^2$ into $\varepsilon^3$, i.e.

$$ (1.1.1) \qquad \vec{\phi} : \varepsilon^2 \to \varepsilon^3. $$

Let $\Omega$ be an open rectangle $(-s_o, s_o) \times (-l, l)$, $(s_o > 0, l > 0)$ in $\varepsilon^2$, and $\Gamma$ be its boundary as shown in the Fig. 1.1, i.e.

$$ (1.1.2) \qquad \Omega = \{ (\xi^1, \xi^2) : -s_o < \xi^1 < s_o, -l < \xi^2 < l \} \subset \varepsilon^2. $$

**Remark 1.1.1** : The symmetry of $\Omega$ is assumed only for the sake of simplicity, but *not* necessary, in general. ■

Let $S$ be the middle surface of the cylindrical shell in $\varepsilon^3$ and $\partial S$ be its boundary such that
<figure><img src="image_4.png" /><figcaption>Fig. 1.1</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 1.2</figcaption></figure>
$$
(1.1.3) \quad \left\{
\begin{aligned}
S &= \vec{\phi}(\Omega), \quad \partial S = \vec{\phi}(\Gamma), \quad \text{i.e.} \\
\forall \xi &= (\xi^1, \xi^2) \in \Omega \subset \mathcal{E}^2, \quad \vec{\phi}(\xi) \in S \subset \mathcal{E}^3, \\
\forall \xi &= (\xi^1, \xi^2) \in \Gamma \subset \mathcal{E}^2, \quad \vec{\phi}(\xi) \in \partial S \subset \mathcal{E}^3.
\end{aligned}
\right.
$$

Here, the mapping $\vec{\phi} : \Omega \to S$, $\Gamma \to \partial S$ is not a unique one. For noncircular (circular one is a particular case) cylindrical surface, a convenient definition of the mapping $\vec{\phi}$ in the explicit form is given by :

$$
(1.1.4) \quad \vec{\phi}(\xi) = \vec{\phi}(\xi^1, \xi^2) = \vec{r}_1(\xi^1) + \xi^2 \vec{e}_3 \quad \forall (\xi^1, \xi^2) \in \bar{\Omega},
$$

where $\vec{r}_1(\xi^1)$ defines the directrix of the cylindrical surface, the directrix being a plane curve lying on the plane spanned by the orthonormal fixed vectors $\vec{e}_1$ and $\vec{e}_2$, and symmetric with respect to the generator $\xi^1 = 0$ of the surface (Figs. 1.1 and 1.2) ;

$\xi^1$ is the arc length measured along the plane curve from the axis of symmetry ($\xi^1 = 0$) on the surface, the range of $\xi^1$ being $[-s_o, s_o]$, $s_o > 0$;

$\vec{e}_3$ is the vector orthonormal to $\vec{e}_1$ and $\vec{e}_2$ such that $\{\vec{e}_1, \vec{e}_2, \vec{e}_3\}$ form a fixed orthonormal basis in $\mathcal{E}^3$, and consequently, $\vec{e}_3$ is also orthogonal to $\vec{r}_1(\xi^1)$ $\forall \xi^1 \in [-s_o, s_o]$, i.e.

$$
(1.1.5) \quad \vec{r}_1 \cdot \vec{e}_3 = 0 \quad \forall \xi^1 \in [-s_o, s_o].
$$

The range of Latin indices i,j,k,... is 1,2,3, whereas that of Greek indices
α,β,γ,... is 1,2.
---

The Einstein summation convention with respect to any index occuring twice
both as a superscript and a subscript will always be used.

1.2. Local Base Vectors and Their Geometric Properties

Let $\vec{a}_1, \vec{a}_2, \vec{a}_3$ be the **covariant unit base vectors** at any point of the surface $S \cup \partial S$, and $\vec{a}, \vec{a}_1, \vec{a}_2, \vec{a}_3$ the corresponding contravariant unit base vectors at the same point of $S \cup \partial S$. Then, the base vectors $\vec{a}_1$ and $\vec{a}_2$ are defined as follows :

$$
(1.2.1) \quad \left\{
\begin{array}{l}
\vec{a}_1 = \vec{\phi}, \quad \vec{e}_1 = \displaystyle\frac{\partial \vec{\phi}}{\partial \xi^1} = \displaystyle\frac{\partial \vec{r}_1(\xi^1)}{\partial \xi^1} = \vec{\tau}, \\
\\
\vec{a}_2 = \vec{\phi}, \quad \vec{e}_2 = \displaystyle\frac{\partial \vec{\phi}}{\partial \xi^2} = \vec{e}_3,
\end{array}
\right.
$$

where $\vec{\tau}$ is the unit vector in the direction of the tangent to the directrix
defined by $\vec{r}_1(\xi^1)$ and perpendicular to $\vec{e}_3$, i.e., $\vec{e}_3 \cdot \vec{\tau} = 0$. Since $\vec{r}_1(\xi^1)$ defines a plane curve,

$$ (1.2.2) \qquad \frac{\partial \vec{\tau}}{\partial \xi^1} = -K \vec{n}, \quad (\text{Fig. 1.2}) $$

where $\vec{n}$ is the unit vector normal to $\vec{\tau}$, lying on the plane of $\vec{r}_1(\xi^1)$, i.e.,

$$ \vec{n} \cdot \vec{\tau} = \vec{n} \cdot \vec{e}_3 = 0 \Rightarrow \vec{n} = \vec{a}_3, $$

$K$ is the curvature of the directrix $\vec{r}_1(\xi^1)$ and is a function of $\xi^1$ only, i.e.,

$$ (1.2.3) \qquad \left\{ \begin{array}{l} K = K(\xi^1) = \frac{1}{R}, \\ R = R(\xi^1) \text{ being the radius of curvature of the directrix.} \end{array} \right. $$

Finally, one gets from (1.2.1) and (1.2.2) :

$$ (1.2.4) \qquad \left\{ \begin{array}{l} \vec{a}_3 = -R \vec{a}_{1,1} \quad \text{; and} \\[1em] \begin{aligned} \vec{a}_{1,1} &= -\frac{1}{R} \vec{a}_3, \quad \vec{a}_{2,1} = \frac{\partial \vec{e}_3}{\partial \xi^1} = \vec{0}, \\ \vec{a}_{1,2} &= \vec{0}, \quad \vec{a}_{2,2} = \frac{\partial \vec{e}_3}{\partial \xi^2} = \vec{0}. \end{aligned} \end{array} \right. $$

The covariant coefficients $a_{\alpha\beta}$ and the corresponding contravariant coefficients $a^{\alpha\beta}$ of the first fundamental form for the surface S are defined from (1.2.1) as follows :

$$ (1.2.5) \qquad a_{\alpha\beta} = \vec{a}_{\alpha} \cdot \vec{a}_{\beta} \Rightarrow a_{11} = a_{22} = 1, a_{12} = a_{21} = 0 $$

$$ (1.2.6) \qquad \Rightarrow (a_{\alpha\beta}) = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = (\delta_{\alpha\beta}) \Rightarrow a = |(a_{\alpha\beta})| = 1 $$

$$ (1.2.7) \qquad \Rightarrow a^{\alpha\beta} = a_{\alpha\beta}. $$

Then, it follows obviously that $\vec{a}^i = \vec{a}_i$.
The coefficients $b_{\alpha\beta}$ of the second fundamental form for the cylindrical surface S are defined by: $b_{\alpha\beta} = \vec{a}_{\alpha} \cdot \vec{a}_{\beta}$. Then, from (1.2.4), (1.2.5) and (1.2.7), one gets:

$$ (1.2.8) \quad \left\{ \begin{array}{l} b_{11} = -\frac{1}{R}, \quad b_{12} = b_{21} = b_{22} = 0, \\ b_1^1 = -\frac{1}{R}, \quad b_2^1 = b_1^2 = b_2^2 = 0. \end{array} \right. $$

From (1.2.1) and (1.2.4), it is obvious that all Christoffel symbols $\Gamma^\gamma_{\alpha\beta} = \vec{a}_{\alpha,\beta} \cdot \vec{a}^\gamma = \vec{a}_{\alpha,\beta} \cdot \vec{a}_\gamma$ vanish and, consequently, covariant derivatives over the surface S are equal to ordinary partial derivatives, i.e.

$$ (1.2.9) \quad u_{\alpha|\beta} = u_{\alpha,\beta} - \Gamma^\gamma_{\alpha\beta} \quad u_\gamma = u_{\alpha,\beta}. $$

## 2. - STRAIN TENSOR AND TENSOR OF CHANGE OF CURVATURE

For shallow cylindrical shells in the sense of KOITER [13] - [15], the strain tensor $\gamma_{\alpha\beta}$ and the tensor of change of curvature $\rho_{\alpha\beta}$ are given by:

$$ (2.1.1) \quad \left\{ \begin{array}{l} \gamma_{\alpha\beta} = \frac{1}{2} (u_{\alpha|\beta} + u_{\beta|\alpha}) - b_{\alpha\beta} u_3 = \frac{1}{2} (u_{\alpha,\beta} + u_{\beta,\alpha}) - b_{\alpha\beta} u_3 ; \\ \rho_{\alpha\beta} = u_3|_{\alpha\beta} = u_{3,\alpha\beta}, \end{array} \right. $$

where

$$ (2.1.2) \quad \vec{u} = (u_1, u_2, u_3) = u_1 \vec{a}_1 + u_2 \vec{a}_2 + u_3 \vec{a}_3 $$

is the displacement vector field over the middle surface S of the shallow cylindrical shell, since $u_i = u^i$ for the cylindrical surface S.

Then, the covariant components of the tensors $\gamma_{\alpha\beta}$ and $\rho_{\alpha\beta}$ can be explicitly written as follows:

$$ (2.1.3) \quad \left\{ \begin{array}{l} \gamma_{11} = u_{1,1} + \frac{u_3}{R} ; \quad \gamma_{12} = \gamma_{21} = \frac{1}{2} (u_{1,2} + u_{2,1}) ; \quad \gamma_{22} = u_{2,2} ; \\ \rho_{11} = u_{3,11} ; \quad \rho_{12} = \rho_{21} = u_{3,12} ; \quad \rho_{22} = u_{3,22} . \end{array} \right. $$
Using (1.2.6) and (1.2.7), one can deduce the following relations :

$$
(2.1.4) \quad \left\{
\begin{aligned}
\gamma_1^1 &= \gamma^{11} = \gamma_{11} \ , \quad \gamma_2^1 = \gamma_1^2 = \gamma^{12} = \gamma^{21} = \gamma_{12} \ , \quad \gamma_2^2 = \gamma^{22} = \gamma_{22} ; \\
\rho_1^1 &= \rho^{11} = \rho_{11} \ , \quad \rho_2^1 = \rho_1^2 = \rho^{12} = \rho^{21} = \rho_{12} \ , \quad \rho_2^2 = \rho^{22} = \rho_{22}.
\end{aligned}
\right.
$$

Thus,

$$
(2.1.5) \quad \left\{
\begin{aligned}
\gamma_{\beta}^{\alpha} \gamma_{\alpha}^{\beta} &= (\gamma_{11})^2 + 2(\gamma_{12})^2 + (\gamma_{22})^2 = \\
&= \left(u_{1,1} + \frac{u_3}{R}\right)^2 + \frac{1}{2} \left(u_{1,2} + u_{2,1}\right)^2 + \left(u_{2,2}\right)^2 ;
\end{aligned}
\right.
$$

$$
(2.1.6) \quad \gamma_{\alpha}^{\alpha} \gamma_{\beta}^{\beta} = (\gamma_{11} + \gamma_{22})^2 = \left(u_{1,1} + u_{2,2} + \frac{u_3}{R}\right)^2 ;
$$

$$
(2.1.7) \quad \rho_{\beta}^{\alpha} \rho_{\alpha}^{\beta} = (\rho_{11})^2 + 2(\rho_{12})^2 + (\rho_{22})^2 = (u_{3,11})^2 + 2(u_{3,12})^2 + (u_{3,22})^2 ;
$$

$$
(2.1.8) \quad \rho_{\alpha}^{\alpha} \rho_{\beta}^{\beta} = (\rho_{11} + \rho_{22})^2 = (u_{3,11} + u_{3,22})^2.
$$

Consequently,

$$
(2.1.9) \quad \left\{
\begin{aligned}
& (1-\nu) \gamma_{\beta}^{\alpha} \gamma_{\alpha}^{\beta} + \nu \gamma_{\alpha}^{\alpha} \gamma_{\beta}^{\beta} = (1-\nu) \left[ \left(u_{1,1} + \frac{u_3}{R}\right)^2 + \frac{1}{2}\left(u_{1,2} + u_{2,1}\right)^2 + \left(u_{2,2}\right)^2 \right] \\
& \qquad + \nu \left[ \left(u_{1,1} + u_{2,2} + \frac{u_3}{R}\right)^2 \right];
\end{aligned}
\right.
$$

$$
(2.1.10) \quad \left\{
\begin{aligned}
& (1-\nu) \rho_{\beta}^{\alpha} \rho_{\alpha}^{\beta} + \nu \rho_{\alpha}^{\alpha} \rho_{\beta}^{\beta} = (1-\nu) \left[ (u_{3,11})^2 + 2(u_{3,12})^2 + (u_{3,22})^2 \right] + \\
& \qquad + \nu \left[ (u_{3,11} + u_{3,22})^2 \right].
\end{aligned}
\right.
$$

3. - VARIATIONAL PROBLEM OF SHALLOW NONCIRCULAR CYLINDRICAL SHELL

3.1. The potential energy of deformation of shallow noncircular cylindrical
shells is given by [2], [15] :
$$
(3.1.1) \quad \left\{
\begin{aligned}
a(\vec{v}, \vec{v}) &= \int_S \frac{Et}{1-\nu^2} [\{(1-\nu) \gamma_\beta^\alpha(\vec{v}) \gamma_\alpha^\beta(\vec{v}) + \nu \gamma_\alpha^\alpha(\vec{v}) \gamma_\beta^\beta(\vec{v})\} + \\
&\quad \frac{t^2}{12} \{(1-\nu) \rho_\beta^\alpha(\vec{v}) \rho_\alpha^\beta(\vec{v}) + \nu \rho_\alpha^\alpha(\vec{v}) \rho_\beta^\beta(\vec{v})\}] \, dS \\
&= \int_\Omega \frac{Et}{1-\nu^2} [\{(1-\nu) \gamma_\beta^\alpha(\vec{v}) \gamma_\alpha^\beta(\vec{v}) + \nu \gamma_\alpha^\alpha(\vec{v}) \gamma_\beta^\beta(\vec{v})\} + \\
&\quad \frac{t^2}{12} \{(1-\nu) \rho_\beta^\alpha(\vec{v}) \rho_\alpha^\beta(\vec{v}) + \nu \rho_\alpha^\alpha(\vec{v}) \rho_\beta^\beta(\vec{v})\}] \, d\xi
\end{aligned}
\right.
$$

where

$$
(3.1.2) \quad \left\{
\begin{aligned}
dS &= \sqrt{a} \, d\xi^1 \, d\xi^2 = d\xi^1 \, d\xi^2, && \text{since } a=1 \ (1.2.6), \\
d\xi &= d\xi^1 \, d\xi^2,
\end{aligned}
\right.
$$

E is the Young's modulus,
t is the thickness of the shell,

$$
(3.1.3) \quad \vec{v} = (v_1, v_2, v_3) = v_1 \vec{a}_1 + v_2 \vec{a}_2 + v_3 \vec{a}_3
$$

is the admissible displacement vector field over S, belonging to an admissible space $\vec{V}$ to be made precise later.

The potential energy of the external force $\vec{f}$ acting on the shallow cylindrical shell is given by [7], [15] :

$$
(3.1.4) \quad \left\{
\begin{aligned}
f(\vec{v}) &= \int_S \vec{f} \cdot \vec{v} \, dS \\
&= \int_\Omega (f_1 v_1 + f_2 v_2 + f_3 v_3) \, d\xi, && \text{(see (3.1.2))}
\end{aligned}
\right.
$$

where

$$
(3.1.5) \quad \vec{f} = (f_1, f_2, f_3) = f_1 \vec{a}_1 + f_2 \vec{a}_2 + f_3 \vec{a}_3 \quad \text{such that } f_i \in L^2(\Omega) \, ;
$$

$\vec{v}$ defined by (3.1.3) belongs to an admissible space $\vec{V}$ to be defined now.
### 3.2. Admissible Space $\vec{v}$ of Displacement Vector Fields $\vec{v}$

Here, the case of shallow noncircular cylindrical shell clamped along its straight and curvilinear edges defined by $\xi^1 = \pm s_o$, $\xi^2 = \pm l$ (Figs. 1.1 and 1.2) respectively, is considered.

Then, the boundary conditions, which the components $v_\alpha$, $v_3$ of the displacement vector field $\vec{v}$ considered over $\bar{\Omega}$ must satisfy, are given by:

$$ (3.2.1) \qquad \left\{ \begin{array}{l} v_\alpha = 0 \text{ on } \Gamma, \\ v_3 = \partial_n v_3 = 0 \text{ on } \Gamma, \end{array} \right. $$

where $\partial_n v_3$ is the derivative of $v_3$ in the direction of the unit outer normal vector $\vec{n}$ at the boundary $\Gamma$.

But

$$ (3.2.2) \qquad \partial_n v_3 = 0 \text{ on } \Gamma \Rightarrow v_{3,1} = v_{3,2} = 0 \text{ on } \Gamma. $$

Hence, the components $v_\alpha, v_3$ must satisfy the boundary conditions (3.2.1) in order that displacement fields $\vec{v}$ are kinematically admissible ones.

Let $\vec{w}$ be a product Sobolev space defined by:

$$ (3.2.3) \qquad \vec{w} = H^1(\Omega) \times H^1(\Omega) \times H^2(\Omega), $$

where

$H^m(\Omega)$ is the usual Sobolev space defined by [1], [16], [21]:

$$ (3.2.4) \qquad H^m(\Omega) = \{v : \partial^\alpha v \in L^2(\Omega), |\alpha| \le m\}, \quad (m=1,2), $$

such that the inner product $\langle \cdot, \cdot \rangle_{m,\Omega}$, the norm $\| \cdot \|_{m,\Omega}$ and the semi-norm $|\cdot|_{m,\Omega}$ in $H^m(\Omega)$ are given by: $\forall (u,v) \in H^m(\Omega) \times H^m(\Omega)$,

$$ (3.2.5) \qquad \langle u, v \rangle_{m, \Omega} = \sum_{|\alpha| \le m} \int_{\Omega} \partial^\alpha u \partial^\alpha v \, d\xi, $$
$$
(3.2.6) \quad \forall v \in H^m(\Omega), \|v\|_{m,\Omega} = \left( \sum_{|\alpha| \le m} \int_{\Omega} (\partial^\alpha v)^2 d\xi \right)^{1/2},
$$

$$
(3.2.7) \quad \forall v \in H^m(\Omega), |v|_{m,\Omega} = \left( \sum_{|\alpha|=m} \int_{\Omega} (\partial^\alpha v)^2 d\xi \right)^{1/2}.
$$

In particular, the following notation will be used :

$$
(3.2.8) \quad \|v\|_{L^2(\Omega)}^2 = \|v\|_{L^2(\Omega_o, \Omega)}
$$

Then, the inner product $((\cdot, \cdot))_{\tilde{W}}$ and the norm $\|\cdot\|_{\tilde{W}}$ in the space $\tilde{W}$ are
defined by : $\forall (u, v) \in \tilde{W} \times \tilde{W}$,

$$
(3.2.9) \quad ((u, \tilde{v}))_{\tilde{W}} = \langle u_1, v_1 \rangle_{1, \Omega} + \langle u_2, v_2 \rangle_{1, \Omega} + \langle u_3, v_3 \rangle_{2, \Omega},
$$

$$
(3.2.10) \quad \forall \tilde{v} \in \tilde{W}, \quad \|\tilde{v}\|_{\tilde{W}}^2 = \|v_1\|_{1, \Omega}^2 + \|v_2\|_{1, \Omega}^2 + \|v_3\|_{2, \Omega}^2.
$$

Now, the admissible space $\tilde{V}$ of displacements $\tilde{v}$ can be defined as follows :

$$
(3.2.11) \quad \left\{
\begin{array}{@{}l@{}}
\tilde{V} = \{ \tilde{v} : \tilde{v} \in \tilde{W}, v_\alpha = 0 \text{ on } \Gamma, v_3 = \partial_n v_3 = 0 \text{ on } \Gamma \} \\
\qquad = H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega),
\end{array}
\right.
$$

which follows from (3.2.1)-(3.2.3).

3.3. *Variational Problem* (P)

The *bilinear form* corresponding to the potential energy of deformation
$a(\vec{u}, \vec{u})$ in (3.1.1) is given by :

$$
(3.3.1) \quad \left\{
\begin{array}{@{}l@{}}
\displaystyle a(\vec{u}, \vec{v}) = \int_{\Omega} \frac{\mathrm{E}\tau}{1-\nu^2} \left[ \{(1-\nu)\gamma_{\beta}^{\alpha}(\vec{u})\gamma_{\alpha}^{\beta}(\vec{v}) + \nu\gamma_{\alpha}^{\alpha}(\vec{u})\gamma_{\beta}^{\beta}(\vec{v})\} \right. \\
\displaystyle \qquad \left. + \frac{t^2}{12} \{(1-\nu)\rho_{\beta}^{\alpha}(\vec{u})\rho_{\alpha}^{\beta}(\vec{v}) + \nu\rho_{\alpha}^{\alpha}(\vec{u})\rho_{\beta}^{\beta}(\vec{v})\} \right] \, \mathrm{d}\xi \\[2ex]
\forall \vec{u} = (u_1, u_2, u_3) \in \vec{V} = H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega), \\[2ex]
\forall \vec{v} = (v_1, v_2, v_3) \in \vec{V} = H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega).
\end{array}
\right.
$$
Then, the variational problem of clamped shallow noncircular shell
is given by :

$$
(3.3.2) \quad (\mathbf{P}) \quad \left\{
\begin{array}{l}
\text{Find } \vec{u} = (u_1, u_2, u_3) \in \vec{V} = H_0^1(\Omega) \times H_0^1(\Omega) \times H_0^2(\Omega) \\
\text{such that} \\
\quad a(\vec{u}, \vec{v}) = f(\vec{v}) \quad \forall \vec{v} \in \vec{V}.
\end{array}
\right.
$$

4. - NORM EQUIVALENCE

The strain tensor $\gamma_{\alpha\beta}$ (2.1.1) can be written in the following form :

$$
(4.1.1) \qquad \gamma_{\alpha\beta} = \gamma_{\beta\alpha} - b_{\alpha\beta} u_3, \qquad (o)
$$

where

$$
(4.1.2) \qquad \gamma_{\alpha\beta} = \frac{1}{2} (u_{\alpha}|_{\beta} + u_{\beta}|_{\alpha}) = \frac{1}{2} (u_{\alpha,\beta} + u_{\beta,\alpha}) \text{ by virtue of (1.2.9).}
$$

The displacement vector $\vec{u}$ is decomposed as follows :

$$
(4.1.3) \qquad \vec{u} = \vec{u}_o + u_3 \vec{a}_3 ,
$$

where

$$
\vec{u}_o = u_1 \vec{a}_1 + u_2 \vec{a}_2 , \quad u_\alpha \in H_o^1(\Omega) , \quad u_3 \in H_o^2(\Omega),
$$

such that

$$
(4.1.4) \qquad \gamma_{\alpha\beta}(\vec{u}_o) = \frac{1}{2} (u_{\alpha,\beta} + u_{\beta,\alpha}).
$$

THEOREM 4.1.1 : Let $\vec{u}_o \in H_o^1(\Omega) \times H_o^1(\Omega)$ and its norm be defined by :

$$
(4.1.5) \qquad \|\vec{u}_o\|^2 = \|u_1\|^2_{H_o^1(\Omega)} + \|u_2\|^2_{H_o^1(\Omega)}.
$$

Then, the mapping
$$
(4.1.6) \quad \vec{u}_o \rightarrow \left( \int_{\Omega} \left[ (\gamma_{11}(\vec{u}_o))^{(o)} + (\gamma_{12}(\vec{u}_o))^{(o)} + (\gamma_{22}(\vec{u}_o))^{(o)} \right]^2 d\xi \right)^{1/2}
$$

is a norm equivalent to the original norm (4.1.5) over the space
$H_o^1(\Omega) \times H_o^1(\Omega)$, i.e., $\exists$ positive constants $C_\alpha > 0$ independent of $\vec{u}_o$ such that

$$
(4.1.7) \quad \left\{
\begin{array}{l}
C_1 \|\vec{u}_o\|_{H_o^1(\Omega) \times H_o^1(\Omega)}^2 \le \int_{\Omega} [(\gamma_{11}(\vec{u}_o))^{(o)} + (\gamma_{12}(\vec{u}_o))^{(o)} + (\gamma_{22}(\vec{u}_o))^{(o)}]^2 d\xi \\
\le C_2 \|\vec{u}_o\|_{H_o^1(\Omega) \times H_o^1(\Omega)}^2 .
\end{array}
\right.
$$

Proof : (i) The second inequality in (4.1.7) will be proved first.

From (4.1.4), one has $\forall \vec{u}_o \in H_o^1(\Omega) \times H_o^1(\Omega)$,

$$
\gamma_{11}(\vec{u}_o) = u_{1,1}, \quad \gamma_{12}(\vec{u}_o) = \frac{1}{2}(u_{1,2} + u_{2,1}), \quad \gamma_{22}(\vec{u}_o) = u_{2,2}.
$$

Then,

$$
\left\{
\begin{array}{l}
\displaystyle \left(\gamma_{11}(\vec{u}_o)\right)^2 + \left(\gamma_{12}(\vec{u}_o)\right)^2 + \left(\gamma_{22}(\vec{u}_o)\right)^2 = \left(u_{1,1}\right)^2 + \frac{1}{4}\left(u_{1,2}+u_{2,1}\right)^2 + \left(u_{2,2}\right)^2 \\
\le \left[\left(u_{1,1}\right)^2 + \left(u_{1,2}\right)^2 + \left(u_{2,1}\right)^2 + \left(u_{2,2}\right)^2\right]
\end{array}
\right.
$$

Thus, the second inequality is proved with $C_2=1$.

(ii) Now, the first inequality in (4.1.7) will be proved. Put

$$
(4.1.8) \quad \left\{
\begin{array}{l}
b(\vec{u}_o, \vec{v}_o) = \int_{\Omega} [(\gamma_{11}(\vec{u}_o))^{(o)} (\gamma_{11}(\vec{v}_o))^{(o)} + (\gamma_{12}(\vec{u}_o))^{(o)} (\gamma_{12}(\vec{v}_o))^{(o)} + (\gamma_{22}(\vec{u}_o))^{(o)} (\gamma_{22}(\vec{v}_o))^{(o)}] \, d\xi \\
\forall \vec{u}_o \in H_o^1(\Omega) \times H_o^1(\Omega), \quad \forall \vec{v}_o \in H_o^1(\Omega) \times H_o^1(\Omega) \quad \forall \Omega \subset \Sigma^2.
\end{array}
\right.
$$

Then, $b(\vec{u}_o, \vec{v}_o) = b(\vec{v}_o, \vec{u}_o)$, and $\forall \vec{u}_o \in H_o^1(\Omega) \times H_o^1(\Omega)$,

$$
(4.1.9) \quad b(\vec{u}_o, \vec{u}_o) = \int_{\Omega} \left[ (\gamma_{11}(\vec{u}_o))^{(o)} + (\gamma_{12}(\vec{u}_o))^{(o)} + (\gamma_{22}(\vec{u}_o))^{(o)} \right]^2 d\xi .
$$

Since $\Omega \subset \Sigma^2$, the Korn's inequality can be applied to this functional
$b(\vec{u}_o, \vec{u}_o)$ [6], [10], [18], [21] i.e. $\forall \vec{u}_o \in H_o^1(\Omega) \times H_o^1(\Omega)$, $\exists$ a positive
constant $C_o > 0$ such that
$$
(4.1.10) \quad \left\{
\begin{aligned}
b(\vec{u}_o, \vec{u}_o) &\ge C_o \int_{\Omega} \left[ \left(\frac{\partial u_1}{\partial \xi}\right)^2 + \left(\frac{\partial u_2}{\partial \xi}\right)^2 + \left(\frac{\partial u_1}{\partial \xi^2}\right)^2 + \left(\frac{\partial u_2}{\partial \xi^2}\right)^2 \right] d\xi \\
&= C_o \left( \|\partial_1 u_1\|_{0,\Omega}^2 + \|\partial_2 u_2\|_{0,\Omega}^2 + \|\partial_2 u_1\|_{0,\Omega}^2 + \|\partial_1 u_2\|_{0,\Omega}^2 \right),
\end{aligned}
\right.
$$

where

$$
\partial_{\alpha} u_{\beta} = \frac{\partial u_{\beta}}{\partial \xi^{\alpha}} = u_{\beta, \alpha}.
$$

Since $u_\alpha \in H_o^1(\Omega)$, the *Poincaré-Friedrichs' inequality* can be applied, i.e.,
$\forall u_\alpha \in H_o^1(\Omega)$, $\exists$ a positive constant $C_3 > 0$ independent of $u_\alpha$ such that

$$
(4.1.11) \quad \| u_{\alpha} \|_{0,\Omega}^2 \le C_3 ( \| \partial_1 u_{\alpha} \|_{0,\Omega}^2 + \| \partial_2 u_{\alpha} \|_{0,\Omega}^2 ).
$$

Combining the inequalities (4.1.10) and (4.1.11), the following inequality
is obtained :

$$
(4.1.12) \quad \left\{
\begin{aligned}
\| \vec{u}_o \|_{H_o^1(\Omega)}^2 \quad &\le C_4 [ \| \partial_1 u_1 \|_{0,\Omega}^2 + \| \partial_2 u_1 \|_{0,\Omega}^2 + \| \partial_1 u_2 \|_{0,\Omega}^2 + \\
&\qquad \qquad \qquad \qquad \qquad \qquad + \| \partial_2 u_2 \|_{0,\Omega}^2 ] \le C_5 b(\vec{u}_o, \vec{u}_o),
\end{aligned}
\right.
$$

where $C_4, C_5 > 0$ are positive constants independent of $u_\alpha$.

$\Rightarrow$ ∃ a positive constant $C_1 > 0$ independent of $\vec{u}_o$ such that

$$
C_1 \| \vec{u}_o \|_{H_o^1(\Omega) \times H_o^1(\Omega)}^2 \le b(\vec{u}_o, \vec{u}_o) = \int_{\Omega} \left[ \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^{2} \binom{o}{\gamma_{\alpha\beta}} (\vec{u}_o)^2 \right] d\xi.
$$

Thus, the theorem is proved. ■

**THEOREM 4.1.2** : Let $u_3 \in H_o^2(\Omega)$. Then, the mapping

$$
(4.1.13) \quad u_3 \to \left( \int_{\Omega} \left[ (\rho_{11}(u_3))^2 + (\rho_{12}(u_3))^2 + (\rho_{22}(u_3))^2 \right] d\xi \right)^{1/2},
$$

where $\rho_{\alpha\beta}(u_3) = u_3, \alpha\beta$ (2.1.1), is a norm equivalent to the original norm
over space $H_o^2(\Omega)$, i.e. $\exists$ positive constants $\alpha_1, \alpha_2 > 0$ independent of $u_3$
such that $\forall u_3 \in H_o^2(\Omega)$

$$
(4.1.14) \quad \alpha_1 \|u_3\|_{2,\Omega}^2 \le \int_{\Omega} \left[ (\rho_{11}(u_3))^2 + (\rho_{12}(u_3))^2 + (\rho_{22}(u_3))^2 \right] d\xi \le \alpha_2 \|u_3\|_{2,\Omega}^2 .
$$
**Proof**: (i) the second inequality in (4.1.14) is directly obtained with $\alpha_2=1$.

(ii) For the proof of the first inequality in (4.1.14), one utilizes
the fact that

$$
(4.1.15) \quad u_3 \in H_o^2(\Omega) \Rightarrow u_3 \in H_o^1(\Omega), \partial_\alpha u_3 \in H_o^1(\Omega)
$$

in order to invoke repeatedly the Poincaré-Friedrichs' inequality, i.e.
∃ positive constants α₃, α₄, α₅ > 0 independent of u₃ such that

$$
(4.1.16) \quad \| \partial_{\alpha} u_3 \|_{0,\Omega}^2 \le \alpha_3 [ \| \partial_1 (\partial_{\alpha} u_3) \|_{0,\Omega}^2 + \| \partial_2 (\partial_{\alpha} u_3) \|_{0,\Omega}^2 ] ,
$$

$$
(4.1.17) \quad \left\{
\begin{array}{l}
\|u_3\|_{0,\Omega}^2 \le \alpha_4 ( \|\partial_1 u_3\|_{0,\Omega}^2 + \|\partial_2 u_3\|_{0,\Omega}^2 ) \\
\le \alpha_5 [ \|u_{3,11}\|_{0,\Omega}^2 + \|u_{3,12}\|_{0,\Omega}^2 + \|u_{3,22}\|_{0,\Omega}^2 ] ,
\end{array}
\right.
$$

(by virtue of (4.1.16)).

Combining (4.1.16) and (4.1.17), one gets

$$
\|u_3\|_{2,\Omega}^2 \le \alpha_6 \int_{\Omega} [(\rho_{11}(u_3))^2 + (\rho_{12}(u_3))^2 + (\rho_{22}(u_3))^2] d\xi,
$$

$\alpha_6 > 0$ being independent of $u_3$, from which the result follows with
$0 < \alpha_1 < \frac{1}{\alpha_6}$, $\alpha_1$ being independent of $u_3$. $\blacksquare$

5.- EXISTENCE AND UNIQUENESS OF SOLUTION OF THE VARIATIONAL PROBLEM (P)

5.1 $\vec{v}$-Ellipticity of the Bilinear Form $a(\vec{u}, \vec{v})$

**THEOREM 5.1.1 :** Let

$$
(5.1.1) \quad \left\{
\begin{array}{l}
\gamma_{\alpha\beta} = \frac{1}{2} (u_{\alpha,\beta} u_{\beta,\alpha} + u_{\alpha,\beta} u_{\beta,\alpha}) - b_{\alpha\beta} u_3, \\
\rho_{\alpha\beta} = u_{3,\alpha\beta},
\end{array}
\right.
$$

$$
(5.1.2) \quad \text{where } b_{11} \neq 0, b_{12} = b_{21} = b_{22} = 0 \quad (1.2.8).
$$
Then, $\forall \vec{u} = (u_1, u_2, u_3) \in \vec{V} = H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega)$, $\exists$ a positive constant $\alpha_o$ independent of $u$ such that the following inequality holds :

$$ \int_{\Omega} \left[ \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^{2} (\gamma_{\alpha\beta}^2 + \rho_{\alpha\beta}^2) d\xi \ge \alpha_o ||u||_{H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega)}^2 \right] $$

**Proof :** The following well-known inequality will be utilized :

$\forall a \in \mathbb{R}, \forall b \in \mathbb{R},$

$$ (5.1.3) \qquad (a+b)^2 \ge (1-\epsilon_1)a^2 - \left(\frac{1}{\epsilon_1} - 1\right) b^2 \quad \text{for } \epsilon_1 \in ]0,1[. $$

(Here the value of $\epsilon_1$ has been restricted within $]0,1[$ in order to keep the coefficient of $a^2$ strictly positive, although the inequality is valid for all $\epsilon_1 > 0$.)

From (4.1.2), (5.1.1) and (5.1.3), the following inequality is obtained :

$$ \left\{ \begin{array}{l} \gamma_{\alpha\beta}^2 = \left[ \frac{1}{2} (u_{\alpha,\beta} + u_{\beta,\alpha}) - b_{\alpha\beta} u_3 \right]^2 = (\gamma_{\alpha\beta} - b_{\alpha\beta} u_3)^2 \\ \ge (1-\epsilon_1)(\gamma_{\alpha\beta})^2 - \left(\frac{1}{\epsilon_1} - 1\right) (b_{\alpha\beta} u_3)^2 \quad \forall \epsilon_1 \in ]0,1[ \\ \Rightarrow (\gamma_{11}^2 + \gamma_{12}^2 + \gamma_{22}^2) \ge \left[ (1-\epsilon_1)(\gamma_{11})^2 - \left(\frac{1}{\epsilon_1} - 1\right) (b_{11}u_3)^2 + (\gamma_{12})^2 + (\gamma_{22})^2 \right] \\ \Rightarrow (\gamma_{11}^2 + \gamma_{12}^2 + \gamma_{22}^2) \ge \left[ (1-\epsilon_1) \sum_{\substack{\alpha,\beta=1 \\ \alpha \le \beta}}^2 (\gamma_{\alpha\beta})^2 - \left(\frac{1}{\epsilon_1} - 1\right) b_{11}^2 u_3^2 \right] . \end{array} \right. $$

Put

$$ \begin{gathered} \sup_{(\xi^1, \xi^2) \in \Omega} \quad b_{11}^2 (\xi^1, \xi^2) = m > 0 \\ (5.1.5) \qquad \Rightarrow \quad -b_{11}^2 \ge -m. \end{gathered} $$

Then, one gets from Theorem 4.1.1, (5.1.4) and (5.1.5) :

$$ (5.1.6) \qquad \int_{\Omega} (\gamma_{11}^2 + \gamma_{12}^2 + \gamma_{22}^2) d\xi \ge (1-\epsilon_1) C_1 ||\vec{u}_o||_{H_o^1(\Omega) \times H_o^1(\Omega)}^2 - (\frac{1}{\epsilon_1} - 1)m ||u_3||_{2,\Omega}^2, $$
$$ C_1 > 0, \quad \epsilon_1 \in ]0,1[. $$
Again, from the Theorem 4.1.2, one gets :

$$
(5.1.7) \quad \int_{\Omega} (\rho_{11}^2 + \rho_{12}^2 + \rho_{22}^2) d\xi \ge \alpha_1 \|u_3\|_{2,\Omega}^2 , \quad \alpha_1 > 0.
$$

Combining (5.1.6) and (5.1.7), the following inequality is obtained :

$$
(5.1.8) \quad \int_{\Omega} \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^{2} (\gamma_{\beta}^2 + \rho_{\alpha\beta}^2) d\xi \ge (1-\epsilon_1) C_1 \|\vec{u}_o\|_{H_o^1(\Omega) \times H_o^1(\Omega)}^2 + [\alpha_1 - (\frac{1}{\epsilon_1} - 1)m] \|u_3\|_{2,\Omega}^2
$$

Now, $\epsilon_1 \in ]0,1[$ will be chosen in such a way that the expression in the square bracket on the right-hand side is strictly positive, i.e.

$$
\begin{align*}
(5.1.9) \quad & [\alpha_1 - (\frac{1}{\epsilon_1} - 1)m] > 0 \\
\Rightarrow & \epsilon_1 > \frac{m}{m+\alpha_1}, \quad (m > 0, \alpha_1 > 0)
\end{align*}
$$

$$
\text{where } 0 < \frac{m}{m+\alpha_1} < 1.
$$

Hence, for any choice of $\epsilon_1$ satisfying the inequality :

$$
(5.1.10) \quad 0 < \frac{m}{m+\alpha_1} < \epsilon_1 < 1,
$$

the inequality (5.1.9) will be satisfied.

Let $K_1 > 0$ be the value of the expression in the square bracket for such a choice of $\epsilon_1$, i.e.,

$$
(5.1.11) \quad K_1 = [\alpha_1 - (\frac{1}{\epsilon_1} - 1)m] \quad \text{for } 0 < \frac{m}{m+\alpha_1} < \epsilon_1 < 1.
$$

Then, from (5.1.8) and (5.1.11), the following inequality is obtained :

$$
(5.1.12) \quad \left\{
\begin{array}{ll}
\displaystyle \int_{\Omega} \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^{2} (\gamma_{\alpha\beta}^2 + \rho_{\alpha\beta}^2) d\xi \ge (1-\epsilon_1) C_1 \|\vec{u}_o\|_{H_o^1(\Omega) \times H_o^1(\Omega)}^2 & + K_1 \|u_3\|_{2,\Omega}^2 \\
\\
\phantom{\displaystyle \int_{\Omega} \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^{2} (\gamma_{\alpha\beta}^2 + \rho_{\alpha\beta}^2) d\xi \ge} \ge \alpha_o \|\vec{u}_o\|_{H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega)}^2 & ,
\end{array}
\right.
$$

where $\alpha_o$ is a positive constant independent of $\vec{u}$, and may be chosen, for example, $\min\{K_1, (1-\epsilon_1)C_1\}$. Thus, the theorem is proved. $\blacksquare$
**THEOREM 5.1.2** : The bilinear form $a(\cdot, \cdot)$ of the variational problem $(\mathcal{P})$ (3.3.2) of shallow noncircular cylindrical shells defined by (3.1.1) is $\vec{v}$-elliptic, i.e., ∃ a positive constant α > 0 independent of $\vec{u}$ such that
$$
\forall \vec{u} \in \vec{V} = H_0^1(\Omega) \times H_0^1(\Omega) \times H_0^2(\Omega),
$$

$$
(5.1.13) \quad a(\vec{u}, \vec{u}) \geq \alpha \|\vec{u}\|^2_{\vec{V}}.
$$

**Proof** : For the sake of simplicity, but not without loss of generality,
E,t,v are assumed to be constant .
Put

$$
(5.1.14) \quad a(\vec{u}, \vec{u}) = \phi_1(\vec{u}) + \phi_2(\vec{u}),
$$

where

$$
(5.1.15) \quad \phi_1(\vec{u}) = \frac{\mathrm{E}\mathrm{t}}{1-\nu^2} \int_{\Omega} [(1-\nu)\gamma_{\beta}^{\alpha} \gamma_{\alpha}^{\beta} + \nu \gamma_{\alpha}^{\alpha} \gamma_{\beta}^{\beta}] d\xi,
$$

$$
(5.1.16) \quad \phi_2(\vec{u}) = \frac{\mathrm{E}\mathrm{t}}{12(1-\nu^2)} \int_{\Omega} [(1-\nu)\rho_{\beta}^{\alpha} \rho_{\alpha}^{\beta} + \nu \rho_{\alpha}^{\alpha} \rho_{\beta}^{\beta}] d\xi.
$$

But from (2.1.6) and (2.1.8), one has : $\forall \vec{u} \in \vec{V}$,

$$
(5.1.17) \quad \left\{
\begin{array}{l}
\nu \gamma_{\alpha}^{\alpha} \gamma_{\beta}^{\beta} \ge 0, \\
\nu \rho_{\alpha}^{\alpha} \rho_{\beta}^{\beta} \ge 0.
\end{array}
\quad (0 \le \nu < 1/2)
\right.
$$

Thus, from (2.1.5), (2.1.7) and (5.1.17), it follows :

$$
(5.1.18) \quad \left\{
\begin{array}{l}
\phi_1(\vec{u}) \ge \alpha_1 \int_{\Omega} \left( \sum_{\substack{\alpha,\beta=1 \\ \alpha \le \beta}}^{2} \gamma_{\alpha\beta}^2 \right) d\xi, \\
\\
\phi_2(\vec{u}) \ge \alpha_2 \int_{\Omega} \left( \sum_{\substack{\alpha,\beta=1 \\ \alpha \le \beta}}^{2} \rho_{\alpha\beta}^2 \right) d\xi,
\end{array}
\right.
$$

where $\alpha_1, \alpha_2$ are positive constants independent of $\vec{u}$.
Then, utilizing the Theorem (5.1.2) and the inequalities (5.1.18), one gets the result :

$$
\begin{align*}
a(\vec{u}, \vec{u}) &\ge \alpha_3 \int_{\Omega} \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^{2} (\gamma_{\alpha\beta}^2 + \rho_{\alpha\beta}^2) d\xi \quad (\alpha_3 = \min\{\alpha_1, \alpha_2\} > 0) \\
&\ge \alpha_3 \alpha_o \|\vec{u}\|_{H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega)}^2 \\
&\ge \alpha \|\vec{u}\|_{\vec{V}}^2,
\end{align*}
$$

where $\alpha$ is a positive constant independent of $\vec{u}$. ■

5.2. Continuity of Bilinear Form $a(\cdot, \cdot)$ on $\vec{V} \times \vec{V}$

**THEOREM 5.2.1** : The bilinear form $a(\cdot, \cdot)$ of the variational problem $(\rho)$
(3.3.2) defined by (3.1.1) is continuous on $\vec{V} \times \vec{V}$, i.e. $\exists$ a positive
constant $M > 0$ such that $\forall (\vec{u}, \vec{v}) \in \vec{V} \times \vec{V}$,

$$
(5.2.1) \quad |a(\vec{u}, \vec{v})| \le M \|\vec{u}\|_{\vec{V}} \|\vec{v}\|_{\vec{V}}.
$$

*Proof* : From (2.1.4), one has

$$
(5.2.2) \quad \left\{
\begin{aligned}
\gamma_{\beta}^{\alpha}(\vec{u}) \gamma_{\alpha}^{\beta}(\vec{v}) &= \sum_{\alpha, \beta=1}^{2} \gamma_{\alpha\beta}(\vec{u}) \gamma_{\alpha\beta}(\vec{v}) \\
\rho_{\beta}^{\alpha}(\vec{u}) \rho_{\alpha}^{\beta}(\vec{v}) &= \sum_{\alpha, \beta=1}^{2} \rho_{\alpha\beta}(\vec{u}) \rho_{\alpha\beta}(\vec{v}),
\end{aligned}
\right.
$$

Then, from (3.1.1) and (5.2.2), one can deduce : $\exists K_2 > 0$ such that

$$
(5.2.3) \quad \left\{
\begin{aligned}
|a(\vec{u}, \vec{v})| &\le K_2 \left( \sum_{\alpha, \beta=1}^{2} \left[ \left| \int_{\Omega} \gamma_{\alpha\beta}(\vec{u}) \gamma_{\alpha\beta}(\vec{v}) d\xi \right| + \left| \int_{\Omega} \rho_{\alpha\beta}(\vec{u}) \rho_{\alpha\beta}(\vec{v}) d\xi \right| \right] \right. \\
&\qquad + \left| \int_{\Omega} (\gamma_{11}(\vec{u}) + \gamma_{22}(\vec{u})) (\gamma_{11}(\vec{v}) + \gamma_{22}(\vec{v})) d\xi \right| \\
&\qquad \left. + \left| \int_{\Omega} (\rho_{11}(\vec{u}) + \rho_{22}(\vec{u})) (\rho_{11}(\vec{v}) + \rho_{22}(\vec{v})) d\xi \right| \right\} .
\end{aligned}
\right.
$$
But $\vec{v} \in H_0^1(\Omega) \times H_0^1(\Omega) \times H_0^2(\Omega) \Rightarrow \gamma_{\alpha\beta}(\vec{v}) \in L^2(\Omega), \rho_{\alpha\beta}(\vec{v}) \in L^2(\Omega)$.

Hence, introducing the following notations :

$$|\gamma_{\alpha\beta}(\cdot)| = \left( \int_{\Omega} (\gamma_{\alpha\beta}(\cdot))^2 d\xi \right)^{1/2}, \quad |\rho_{\alpha\beta}(\cdot)| = \left( \int_{\Omega} (\rho_{\alpha\beta}(\cdot))^2 d\xi \right)^{1/2},$$

$$|\gamma_{11}(\cdot)+\gamma_{22}(\cdot)| = \left( \int_{\Omega} (\gamma_{11}(\cdot)+\gamma_{22}(\cdot))^2 d\xi \right)^{1/2},$$

$$|\rho_{11}(\cdot)+\rho_{22}(\cdot)| = \left( \int_{\Omega} (\rho_{11}(\cdot)+\rho_{22}(\cdot))^2 d\xi \right)^{1/2},$$

and applying the Cauchy-Schwarz inequality in $L^2(\Omega)$, one can deduce from
(5.2.3) : $\forall (\vec{u}, \vec{v}) \in \vec{V} \times \vec{V}$,

$$\begin{aligned} |a(\vec{u}, \vec{v})| &\le K_3 [ \langle |\gamma_{11}(\vec{u})|, |\gamma_{12}(\vec{u})|, |\gamma_{22}(\vec{u})|, |\rho_{11}(\vec{u})|, |\rho_{12}(\vec{u})|, |\rho_{22}(\vec{u})| \rangle , \\ & \quad (|\gamma_{11}(\vec{v})|, |\gamma_{12}(\vec{v})|, |\gamma_{22}(\vec{v})|, |\rho_{11}(\vec{v})|, |\rho_{12}(\vec{v})|, |\rho_{22}(\vec{v})|) \gg_{\mathbb{R}^6} 6 \\ & \quad + \langle |\gamma_{11}(\vec{u})+\gamma_{22}(\vec{u})|, |\rho_{11}(\vec{u})+\rho_{22}(\vec{u})| \rangle , \\ & \quad (|\gamma_{11}(\vec{v}), \gamma_{22}(\vec{v})|, |\rho_{11}(\vec{v})+\rho_{22}(\vec{v})|) \gg_{\mathbb{R}^2} 2 \]$$

where $\langle \cdot, \cdot \rangle$ denotes inner product in $\mathbb{R}^n$, $n \ge 2$,

$K_3$ is a positive constant independent of $\vec{u}, \vec{v}$.

$$\begin{aligned} |a(\vec{u}, \vec{v})| &\le K_3 \left[ \left\{ \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^2 (|\gamma_{\alpha\beta}(\vec{u})|^2 + |\rho_{\alpha\beta}(\vec{u})|^2) \right\}^{1/2} \right. \\ & \qquad \left. \cdot \left\{ \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^2 (|\gamma_{\alpha\beta}(\vec{v})|^2 + \right. \right. \\ & \qquad \left. \left. + |\rho_{\alpha\beta}(\vec{v})|^2) \right\}^{1/2} + \left( \sum_{\alpha=1}^2 \gamma_{\alpha\alpha}(\vec{u})^2 + \sum_{\alpha=1}^2 \rho_{\alpha\alpha}(\vec{u})^2 \right)^{1/2} \left( \sum_{\alpha=1}^2 \gamma_{\alpha\alpha}(\vec{v})^2 + \right. \right. \\ & \qquad \left. \left. \left( \sum_{\alpha=1}^2 \rho_{\alpha\alpha}(\vec{v})^2 \right)^{1/2} \right) \right] \end{aligned}$$
$$
(5.2.4) \quad \left\{
\begin{aligned}
& |a(\vec{u}, \vec{v})| \le K_4 \left[ \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^2 (|\gamma_{\alpha\beta}(\vec{u})|^2 + |\rho_{\alpha\beta}(\vec{u})|^2) \right]^{1/2} \\
& \quad \{ \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^2 (|\gamma_{\alpha\beta}(\vec{v})|^2 + |\rho_{\alpha\beta}(\vec{v})|^2) \}^{1/2} \}, \\
& \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^2 |\rho_{\alpha\beta}(\vec{v})|^2
\end{aligned}
\right.
$$

since

$$
\begin{align*}
|\gamma_{11}(\cdot) + \gamma_{22}(\cdot)|^2 &\le 2 \left( \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^2 |\gamma_{\alpha\beta}(\cdot)|^2 \right), \\
|\rho_{11}(\cdot) + \rho_{22}(\cdot)|^2 &\le 2 \left( \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^2 |\rho_{\alpha\beta}(\cdot)|^2 \right).
\end{align*}
$$

Then, utilizing the inequality : $\forall \vec{v} \in \vec{V}$,

$$
(\gamma_{\alpha\beta}(\vec{v}))^2 = (\gamma_{\alpha\beta} - b_{\alpha\beta}v_3)^2 \le 2[(\gamma_{\alpha\beta})^2 + b_{\alpha\beta}^2 v_3^2] ; \rho_{\alpha\beta}(\vec{v}) = v_3, \alpha\beta
$$

and the second inequalities in (4.1.7) and (4.1.14), one can easily deduce :

∃ a positive constant K₅ > 0 such that ∀$\vec{v} \in \vec{V}$,

$$
(5.2.5) \quad \left[ \sum_{\substack{\alpha, \beta=1 \\ \alpha \le \beta}}^{2} \left( |\gamma_{\alpha\beta}(\vec{v})|^2 + |\rho_{\alpha\beta}(\vec{v})|^2 \right) \right]^{1/2} \le K_5 ||\vec{v}||_{\vec{V}}^2.
$$

Then, from (5.2.4) and (5.2.5), the result follows :

$$
|a(\vec{u}, \vec{v})| \le K_4 [K_5 \|\vec{u}\|_{\vec{V}} \|K_5 \|\vec{v}\|_{\vec{V}}] \le M \|\vec{u}\|_{\vec{V}} \|\vec{v}\|_{\vec{V}},
$$

M being a positive constant. ■

5.3. Continuity of the Linear Form $f(\cdot)$ on $\vec{V}$

**THEOREM 5.3.1** : The linear form $f(\cdot)$ of the variational problem $(\rho)$
(3.3.2) defined by (3.1.4) is continuous on $\vec{V}$, i.e., ∃ a constant
$C_0 \ge 0$ independent of $\vec{v}$ such that $\forall \vec{v} \in \vec{V}$,

$$
(5.3.1) \quad |f(\vec{v})| \le C_0 ||\vec{v}||_{\vec{V}}.
$$

**Proof** : Applying Cauchy-Schwarz inequality, one gets from (3.1.4) and
(3.1.5) : $\forall \vec{v} \in \vec{V}$
$$
(5.3.2) \quad \begin{aligned}
|f(\vec{v})| & \le \sum_{i=1}^{3} ( \|f_i\|_{0,\Omega} \|v_i\|_{0,\Omega} ) \\
& \le ( \sum_{i=1}^{3} \|f_i\|_{0,\Omega}^2 )^{1/2} ( \sum_{i=1}^{3} \|v_i\|_{0,\Omega}^2 )^{1/2} \\
& \le C_o \|\vec{v}\|_{L^2(\Omega) \times L^2(\Omega) \times L^2(\Omega)} \le C_o \|\vec{v}\|_{H_o^1(\Omega) \times H_o^1(\Omega) \times H_o^2(\Omega)} ,
\end{aligned}
$$

where

$$
C_o \geq ( \sum_{i=1}^{3} \|f_i\|_{0,\Omega}^2 )^{1/2} \geq 0, \text{ since } f_i \in L^2(\Omega) \quad (3.1.5);
$$

$$
\|\vec{v}\|_{L^2(\Omega) \times L^2(\Omega) \times L^2(\Omega)}^2 = \sum_{i=1}^{3} \|v_i\|_{0,\Omega}^2 .
$$

5.4. Existence and Uniqueness of Solution of the Variational Problem ($\mathcal{P}$).

**THEOREM 5.4.1** : The variational problem ($\mathcal{P}$) defined in (3.3.2) has a unique solution.

**Proof** : Since the bilinear form $a(\cdot, \cdot)$ in (3.3.2) is $\vec{V}$-elliptic (Theorem 5.1.2.) and continuous on $\vec{V} \times \vec{V}$ (Theorem 5.2.1), and the linear form $f(\cdot)$ in (3.3.2) is continuous on $\vec{V}$ (Theorem 5.3.1), according to Lax-Milgram lemma [21], there is one and only one solution of the variational problem ($\mathcal{P}$) (3.3.2).
**Remark 5.4.1** : All the theorems proved in Sections 4 and 5 for shallow *noncircular* cylindrical shells are available in the case of shallow *circular* cylindrical shells without any change in the proofs, since for the circular case, all $b_{\alpha\beta}$ in (1.2.8) become constant and for "m" in (5.1.5), one can choose $1/R^2$, $R$ being the (constant) radius of curvature of circular cylindrical surface. ■

**Remark 5.4.2** : All the results given in Sections 2-5 for cylindrical shells, shallow in the sense of W.T. Koiter, can be extended directly to the cases of *open* cylindrical shells [4] for which *Donnell's approximations* are valid. It is wellknown that Donnell's approximations are valid, even when cylindrical shells are *not* shallow in the *literal sense*. ■
## 6. - CONFORMING FINITE ELEMENT ANALYSIS

### 6.1. Finite Element Spaces

For the conforming finite element approximation of the continuous problem $(\mathcal{P})$ (3.3.2), the rectangular domain $\bar{\Omega}$ is subdivided into triangles $K$ by means of a **triangulation** $\mathcal{T}_h$, and the corresponding **finite elements** are denoted by $(K, P_K, \Sigma_K)$, where $P_K$ is a **finite dimensional space of functions** over $K$ and $\Sigma_K$ is the **set of degrees of freedom** of the triangle $K$ [6].

**Remark 6.1.1** : Subdivision of $\bar{\Omega}$ into rectangles to construct corresponding finite elements may as well be done. ■

Let $h_K$ denote the **diameter** of $K$ and
$\rho_K$ be the **diameter** of the circle inscribed on $K$.

A family of triangulations $\mathcal{T}_h$ will be called **regular**, if the following two conditions are satisfied :

(i) There is a positive constant $\sigma > 0$ such that

$$6.1.1) \quad \forall h > 0, \forall K \in \mathcal{T}_h, \quad \frac{h_K}{\rho_K} \le \sigma;$$

(ii)

(6.1.2) The quantity $h = \max_{K \in \mathcal{T}_h} h_K$ tends to zero.

Then, the corresponding family of finite elements $(K, P_K, \Sigma_K)$ is called **regular**. Corresponding to each triangulation $\mathcal{T}_h$, a product finite element space $\tilde{X}_h = X_{h_1} \times X_{h_2} \times \dots \times X_{h_K}$ is introduced in the following way [7] :

Space $X_{h_1}$ is the set of functions $v_h$ satisfying the following properties :

(i) On every $K \in \mathcal{T}_h$, the functions $v_h$ belong to space $P_K \subset H^1(K)$ ;

(ii) On every $K \in \mathcal{T}_h$, the functions $v_h$ are determined by their values and possibly also by the values of their derivatives occurring in the set of degrees of freedom $\Sigma_K$ ;

(iii) The functions $v_h$ are such that $X_{h_1} \subset \mathcal{C}^0(\bar{\Omega})$.

Space $X_{h_2}$ is the set of functions $w_h$ such that :

a) On each $K \in \mathcal{T}_h$, the functions $w_h \in P_K \subset H^2(K)$ ;

b) On each $K \in \mathcal{T}_h$, the functions $w_h$ are determined by their values and the values of the derivatives occurring in the set of degrees of freedom $\Sigma_K$ ;
c) The functions $w_h$ are such that the inclusion $X_{h2} \subset C^{(1)}(\bar{\Omega})$ holds.

Then, a suitable subspace $\tilde{V}_h = V_{h1} \times V_{h1} \times V_{h2}$ of the space $\tilde{X}_h$ is constructed
by taking into account the boundary conditions involved in the definition
of the space $\tilde{V}$ such that

$$
(6.1.3) \qquad \tilde{V}_h \subset \tilde{V},
$$

from which the nomenclature, Conforming Finite Element method follows,
and the corresponding approximation is called Internal Approximation.
Thus, $V_{h1}$ and $V_{h2}$ can be defined as follows :

$$
(6.1.4) \quad V_{h1} = \{ v_h \in X_{h1} ; v_h |_{\Gamma} = 0 \} ,
$$

$$
(6.1.5) \quad V_{h2} = \{ w_h \in X_{h2} ; w_h |_{\Gamma} = \partial_n w_h |_{\Gamma} = 0 \}.
$$

The condition (6.1.3) follows from the inclusions $V_{h1} \subset H_o^1(\Omega)$ and $V_{h2} \subset H_o^2(\Omega)$,
which are proved in the Theorems 2.1.1 and 2.1.2 in [6].

6.2. Discrete Problem

The discrete problem ($\mathcal{P}_h$) corresponding to the continuous problem ($\mathcal{P}$)
(3.3.2) is defined as follows :

Find a function $\tilde{u}_h \in \tilde{V}_h$ such that

$$
(6.2.1) \quad (\mathcal{P}_h) \quad a(\tilde{u}_h, \tilde{v}_h) = f(\tilde{v}_h) \quad \forall \tilde{v}_h \in \tilde{V}_h,
$$

where $a(\cdot, \cdot)$ and $f(\cdot)$ are defined by (3.1.1) and (3.1.4) respectively.

By virtue of the inclusion $\tilde{V}_h \subset \tilde{V}$, the bilinear form $a(\cdot, \cdot)$ is $\tilde{V}_h$-elliptic
and continuous over $\tilde{V}_h \times \tilde{V}_h$, and the linear form $f(\cdot)$ is also continuous
over $\tilde{V}_h$. Therefore, according to Lax-Milgram lemma, the discrete problem
$(\mathcal{P}_h)$ (6.2.1) has a unique solution.
Estimation of the Error $||\vec{u}-\vec{u}_h||_{\vec{V}}$

Thus, the problem $(\rho_h)$ reduces to the estimation of the error $||\vec{u}-\vec{u}_h||_{\vec{V}}$,
where $\vec{u} \in \vec{V}$ is the solution of the continuous problem $(\rho)$ (3.3.2),
$\vec{u}_h \in \vec{V}_h$ is the solution of the discrete problem $(\rho_h)$ (6.2.1),
$||\cdot||_{\vec{V}}$ is the norm in the space $\vec{V}$.

If the integrals involved in the definition of $a(\cdot, \cdot)$ and $f(\cdot)$ can be
calculated exactly, then an error estimate can be obtained with the help
of the following theorem [7], [3].

**THEOREM 6.2.1** : Let $(K, P'_K, \Sigma'_K)$ and $(K, P''_K, \Sigma''_K)$ be two families of "affine" or "almost affine" [6] regular finite elements.

Let $V_{h1}$ and $V_{h2}$ be two finite element spaces defined by (6.1.4) and (6.1.5)
respectively.

Furthermore, suppose that the following inclusions are satisfied :

$$
P_{m_1} (K) \subset P'_K , \text{ where } m_1 \in \mathbb{N} , m_1 \ge 1,
$$

$$
P_{m_2} (K) \subset P''_K, \text{ where } m_2 \in \mathbb{N}, m_2 \ge 2,
$$

$$
H^{m+1}_{\alpha} (K) \subset C^{s\alpha} (K) , \alpha = 1,2,
$$

where $s_1$ (resp. $s_2$) is the highest order of the partial derivatives occurring
in the set of degrees of freedom $\Sigma'_K$ (resp. $\Sigma''_K$) of the triangle $K$.

Then, if the solution $\vec{u} = (u_1, u_2, u_3)$ of the continuous problem $(\rho)$ (3.3.2)
belongs to the space $H^{m_1+1}(\Omega) \times H^{m_1+1}(\Omega) \times H^{m_2+1}(\Omega)$, there exists a constant
$C$ independent of $h$ such that

$$
\| \vec{u} - \vec{u}_h \|_{\vec{V}} \le C h^{\min(m_1, m_2-1)} \| \vec{u} \|_{H^{m_1+1}(\Omega) \times H^{m_1+1}(\Omega) \times H^{m_2+1}(\Omega)},
$$
where $\vec{u}_h$ is the solution of the discrete problem $(\rho_h)$ (6.2.1).

**Proof** : Ciarlet's Theorem 3.1.6 in [6] valid for affine regular families of finite elements can be extended to the case of "almost-affine" families, utilizing his definition of "almost-affine" family of finite elements in [6], i.e.

∃ constants $C_1, C_2$ independent of $h_K$
such that for all finite elements K belonging to each of the two families
and for all functions

$$
v_{\alpha} \in H^{\frac{m_1+1}{2}}(\Omega) \text{ and}
$$

$$
v_3 \in H^{\frac{m_2+1}{2}} (\Omega),
$$

the following estimates hold :

$$
\| v_{\alpha} - \pi' v_{\alpha} \|_{1, K} \le C_1 h_K^{\frac{m_1}{2}} |v_{\alpha}|_{m_1+1, K},
$$

$$
\| v_3 - \pi'' v_3 \|_{2, K} \le C_2 h_K^{\frac{m_2-1}{2}} |v_3|_{m_2+1, K},
$$

$$
\text{where } \pi' v_{\alpha} = \pi_{h_1} v_{\alpha} |_{K},
$$

$$
\pi'' v_3 = \pi h_2 v_3 |_{K},
$$

$\pi h_1$ is the $V_{h_1}$ -<u>interpolation operator</u>,

$\pi h_2$ is the $V_{h_2}$ - <u>interpolation operator</u>.

By virtue of the regularity of the family of finite elements, the quantity
$h = \max_{K \in \mathcal{C}_h} h_K$ tends to zero (6.1.2).
Then, the following inequality is obtained :

$$
\begin{align*}
\inf_{\substack{\vec{v} \\ \vec{v}_h \in \mathcal{V}_h}} \| \vec{u} - \vec{v}_h \|_{\vec{V}} &\le \| \vec{u} - \vec{\pi}_h \vec{u} \|_{\vec{V}} \\
&= \left( \sum_{K \in \mathcal{C}_h} \left[ \| u_1 - \vec{\pi}_h u_1 \|_{1, K}^2 + \| u_2 - \vec{\pi}_h u_2 \|_{1, K}^2 + \| u_3 - \vec{\pi}_h u_3 \|_{2, K}^2 \right] \right)^{1/2} \\
&\le C_3 h^{\min\{m_1, m_2-1\}} (|u_1|_{m_1+1, \Omega+}^2 |u_2|_{m_1+1, \Omega+}^2 |u_3|_{m_2+1, \Omega+}^2)^{1/2},
\end{align*}
$$

where $\vec{\pi}_h \vec{u} = (\vec{\pi}_h u_1, \vec{\pi}_h u_2, \vec{\pi}_h u_3)$ denotes the $\vec{v}_h$-interpolate of the function $\vec{u} \in \vec{V}$.
Now, applying Cea's lemma [6], the result is obtained :

$$
\begin{equation}
\begin{aligned}
\| \vec{u} - \vec{u}_h \|_{\vec{V}} &\le C_4 \inf_{\vec{v}_h \in \vec{V}_h} \| \vec{u} - \vec{v}_h \|_{\vec{V}} \\
&\le C_h \min\{m_1, m_2 - 1\} (|u_1|_{m_1+1, \Omega}^2 + |u_2|_{m_1+1, \Omega}^2 + |u_3|_{m_2+1, \Omega}^2)^{1/2}
\end{aligned}
\tag{6.2.1}
\end{equation}
$$

**Remark 6.2.1** : In the error estimate $||\vec{u}-\vec{u}_h||_{\vec{V}}$ (6.2.2) of the conforming finite element approximation of the discrete problem $(\rho_h)$ (3.3.2), it has been assumed that the integrals involved in the definitions of $a(\cdot, \cdot)$ and $f(\cdot)$ have been calculated exactly. But rarely this assumption can be realized in practice. Hence, one must take recourse to numerical integration for the approximate evaluation of these integrals in $a(\cdot, \cdot)$ and $f(\cdot)$. Then, the following new discrete problem $(\rho_h^*)$ is to be solved by the conforming finite element method :

Find $\vec{u}_h^* \in \vec{V}_h$ such that

$$
(6.2.3) \quad (\rho_h^*) \quad a_h(\vec{u}^*, \vec{v}_h) = f_h(\vec{v}_h) \quad \forall \vec{v}_h \in \vec{V}_h,
$$

where $a_h(\cdot, \cdot)$ and $f_h(\cdot)$ are new bilinear and linear forms obtained by evaluating approximately $a(\cdot, \cdot)$ and $f(\cdot)$ respectively by means of a suitable quadrature formula [6], [3];

$\vec{v}_h$ is the admissible finite element space defined earlier.

For the existence and uniqueness of conforming finite element solution of this new discrete problem $(\rho_h^*)$, one must prove that the bilinear form $a_h(\cdot, \cdot)$ is **uniformly** $\vec{v}_h$-elliptic with respect to $h$, i.e.

$$
(6.2.4) \quad \exists \tilde{\alpha} > 0, \forall h > 0, \forall \vec{v}_h \in \vec{V}_h, |a_h(\vec{v}_h^*, \vec{v}_h^*)| \ge \tilde{\alpha} \|v_h\|_{\vec{V}}^2,
$$

$h$ being sufficiently small.

Now, if $a_h(\cdot, \cdot)$ is uniformly $\vec{v}_h$-elliptic and there exists a unique solution $\vec{u}_h^* \in \vec{V}_h$ of the discrete problem $(\rho_h^*)$ (6.2.3), then the error estimate $||\vec{u}-\vec{u}_h^*||_{\vec{V}}$ is given by the following theorem [6], [7], [3] :
**THEOREM 6.2.2** : Let the bilinear forms $a_h(\cdot, \cdot)$ of a family of discrete problems $(\rho_h^*)$ (6.2.3) be uniformly $\vec{v}_h$-elliptic in the sense of (6.2.4). Then, $\exists$ a positive constant $C>0$ independent of $h$ such that

$$ \|\vec{u}-\vec{u}_h^*\|_{\vec{v}} \le C \left\{ \begin{aligned} & \inf_{\vec{v}_h \in \vec{v}_h^*} \left[ \|\vec{u}-\vec{v}_h\|_{\vec{v}} + \sup_{\vec{w}_h \in \vec{w}_h^*} \frac{|a(\vec{v}_h, \vec{w}_h)-a_h(\vec{v}_h, \vec{w}_h)|}{\|\vec{w}_h\|_{\vec{v}}} \right] \\ & + \sup_{\vec{w}_h \in \vec{w}_h^*} \frac{|f(\vec{w}_h)-f_h(\vec{w}_h)|}{\|\vec{w}_h\|_{\vec{v}}} \end{aligned} \right\}, $$

where $\vec{u} \in \vec{v}$, $\vec{u}_h^* \in \vec{v}_h^*$ are the solutions of the continuous problem $(\rho)$ (3.3.2) and the discrete problem $(\rho_h^*)$ (6.2.3) respectively.

**Proof** : The triangular inequality gives

$$ \|\vec{u}-\vec{u}_h^*\|_{\vec{v}} \le \|\vec{u}-\vec{v}_h\| + \|\vec{v}_h-\vec{u}_h^*\| \quad \forall \vec{v}_h \in \vec{v}_h^*. $$

But by virtue of the uniform $\vec{v}_h$-ellipticity of $a_h(\cdot, \cdot)$, $\exists \tilde{\alpha} > 0$ independent of $h$ such that $\forall \vec{v}_h \in \vec{v}_h^*$,

$$ \begin{aligned} \tilde{\alpha} \|\vec{u}_h^* - \vec{v}_h\|_{\vec{v}}^2 &\le a_h(\vec{u}_h^* - \vec{v}_h, \vec{u}_h^* - \vec{v}_h) \\ &= a(\vec{u} - \vec{v}_h, \vec{u}_h^* - \vec{v}_h) + [a(\vec{v}_h, \vec{u}_h^* - \vec{v}_h) \\ &\quad - a_h(\vec{v}_h, \vec{u}_h^* - \vec{v}_h)] + [f_h(\vec{u}_h^* - \vec{v}_h) - f(\vec{u}_h^* - \vec{v}_h)]. \end{aligned} $$

Again, the continuity of $a(\cdot, \cdot)$ (5.2.1) implies $\forall \vec{v}_h \in \vec{v}_h^*$,

$$ \begin{aligned} \tilde{\alpha} \|\vec{u}_h^* - \vec{v}_h\|_{\vec{v}} \le M \|\vec{u} - \vec{v}_h\|_{\vec{v}} &+ \frac{|a(\vec{v}_h, \vec{u}_h^* - \vec{v}_h) - a_h(\vec{v}_h, \vec{u}_h^* - \vec{v}_h)|}{\|\vec{u}_h^* - \vec{v}_h\|_{\vec{v}}} + \\ &+ \frac{|f_h(\vec{u}_h^* - \vec{v}_h) - f(\vec{u}_h^* - \vec{v}_h)|}{\|\vec{u}_h^* - \vec{v}_h\|_{\vec{v}}} \\ \le M &\|\vec{u} - \vec{v}_h\|_{\vec{v}} + \sup_{\vec{w}_h \in \vec{w}_h^*} \frac{|a(\vec{v}_h, \vec{w}_h) - a_h(\vec{v}_h, \vec{w}_h)|}{\|\vec{w}_h\|_{\vec{v}}} + \\ &+ \sup_{\vec{w}_h \in \vec{w}_h^*} \frac{|f_h(\vec{w}_h) - f(\vec{w}_h)|}{\|\vec{w}_h\|_{\vec{v}}}, \end{aligned} $$

where $M>0$ and is independent of $h$.
$$
\begin{align*}
\Rightarrow \|\vec{u}-\vec{u}_h\|_{\vec{V}} &\le (1+\frac{M}{\tilde{\alpha}}) \|\vec{u}-\vec{v}_h\|_{\vec{V}} + \frac{1}{\tilde{\alpha}} \sup_{\vec{w}_h \in \vec{V}_h} \frac{|a(\vec{v}_h, \vec{w}_h)-a_h(\vec{v}_h, \vec{w}_h)|}{\|\vec{w}_h\|_{\vec{V}}} \\
&\quad + \frac{1}{\tilde{\alpha}} \sup_{\vec{w}_h \in \vec{V}_h} \frac{|f_h(\vec{w}_h)-f(\vec{w}_h)|}{\|\vec{w}_h\|_{\vec{V}}}
\end{align*}
$$

Now, taking the infimum with respect to $\vec{v}_h \in \vec{V}_h$, the result is obtained. ■

**Remark 6.2.2** : In the proof of the Theorem 6.2.2, the assumption of the uniform $\vec{v}_h$-ellipticity of $a_h(\cdot, \cdot)$ has been made. For the proof of this, one can use the general result in [3], which is also applicable to the case of shallow noncircular cylindrical shells. But a direct proof of the uniform $\vec{v}_h$-ellipticity for the present problem is envisaged in a future publication. ■
ACKNOWLEDGEMENT

My heartiest thanks are due to a colleague and friend of mine at the
Laboria, Michel BERNADOU who took all pains to read the first version
of the manuscript of this report and gave valuable suggestions for its
improvement. I express my gratitude to Professor P.G. CIARLET who
made accessible to me all his results which I have liberally used in
the present report. Then, I must thank Professor Paul ROUGEE, University
of Paris North, for giving me the opportunity to have interesting dis-
cussions with him on some of the results of mine. Finally, I express
my gratitude to Mrs. F. Weber for her sincere cooperation and careful
typing of the manuscript.
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