A primal dual variational formulation and a multi-duality principle for a non-linear model of plates

TL;DR

This paper introduces a new primal dual variational formulation and a multi-duality principle for a non-linear plate model, providing conditions for global optimality and analyzing duality gaps.

Contribution

It develops a novel primal dual formulation and multi-duality principle for the Kirchhoff-Love non-linear plate model, including global optimality conditions.

Findings

Established a duality principle suitable for negative definite membrane stress tensors.

Developed a primal dual variational formulation with sufficient conditions for global optimality.

Proved there is no duality gap between primal and dual formulations in a local extremal context.

Abstract

This article develops a new primal dual formulation for the Kirchhoff-Love non-linear plate model. At first we establish a duality principle which includes sufficient conditions of global optimality through the dual formulation. At this point we highlight this first duality principle is specially suitable for the case in which the membrane stress tensor is negative definite. In a second step, from such a general principle, we develop a primal dual variational formulation which also includes the corresponding sufficient conditions for global optimality. The results are based on standard tools of convex analysis and on a well known Toland result for D.C. optimization. Finally, in the last section, we present a multi-duality principle and qualitative relations between the critical points of the primal and dual formulations. We formally prove there is no duality gap between such primal and dual formulations in a local extremal context.

Equations

$$\hat{\text{u}}=\{{\hat{u}}_{\alpha },{\hat{u}}_3\}={\hat{u}}_{\alpha }{\text{a}}_{\alpha }+{\hat{u}}_3{\text{a}}_3.$$

$${\hat{u}}_{\alpha }(x_1,x_2,x_3)=u_{\alpha }(x_1,x_2)-x_{3}w(x_1,x_2{)}_{,\alpha }$$

$$\text{ and }{\hat{u}}_3(x_1,x_2,x_3)=w(x_1,x_2).$$

$$U$$

$$\Lambda (u)=\{\gamma (u),\kappa (u)\},$$

$${\gamma }_{\alpha \beta }(u)=\frac{u_{\alpha ,\beta }+u_{\beta ,\alpha }}{2}+\frac{w_{,\alpha }{w}_{,\beta }}{2},$$

$${\kappa }_{\alpha \beta }(u)=-w_{,\alpha \beta }.$$

$$N_{\alpha \beta }(u)=H_{\alpha \beta \lambda \mu }{\gamma }_{\lambda \mu }(u),$$

$$M_{\alpha \beta }(u)=h_{\alpha \beta \lambda \mu }{\kappa }_{\lambda \mu }(u),$$

$$(G\circ \Lambda )(u)=\frac{1}{2}{\int }_{\Omega }{N}_{\alpha \beta }(u){\gamma }_{\alpha \beta }(u)dx+\frac{1}{2}{\int }_{\Omega }{M}_{\alpha \beta }(u){\kappa }_{\alpha \beta }(u)dx$$

$$F_1(u)=⟨w,P{⟩}_{L^2(\Omega )}+⟨u_{\alpha },P_{\alpha }{⟩}_{L^2(\Omega )},$$

$$J(u)=(G\circ \Lambda )(u)-F_1(u)$$

$$w_{,\alpha \beta }=\frac{{\partial }^{2}w}{\partial x_{\alpha }\partial x_{\beta }},$$

$$u_{\alpha ,\beta }=\frac{\partial u_{\alpha }}{\partial x_{\beta }},$$

$$N_{\alpha \beta ,1}=\frac{\partial N_{\alpha \beta }}{\partial x_1},$$

$$N_{\alpha \beta ,2}=\frac{\partial N_{\alpha \beta }}{\partial x_2}.$$

$${\gamma }_{\alpha \beta }(u)=\frac{1}{2}(u_{\alpha ,\beta }+u_{\beta ,\alpha })+\frac{1}{2}{w}_{,\alpha }{w}_{,\beta },$$

$${\kappa }_{\alpha \beta }(u)=-w_{,\alpha \beta },\forall u\in U,\alpha ,\beta \in \{1,2\}$$

$$J(u)=G_1(\gamma (u))+G_2(\kappa (u))-⟨u,f{⟩}_{L^2},$$

$$G_1(\gamma (u))=\frac{1}{2}{\int }_{\Omega }{H}_{\alpha \beta \lambda \mu }{\gamma }_{\alpha \beta }(u){\gamma }_{\lambda \mu }(u)dx,$$

$$G_2(\kappa (u))=\frac{1}{2}{\int }_{\Omega }{h}_{\alpha \beta \lambda \mu }{\kappa }_{\alpha \beta }(u){\kappa }_{\lambda \mu }(u)dx,$$

$$⟨u,f{⟩}_{L^2}=⟨w,P{⟩}_{L^2}+⟨u_{\alpha },P_{\alpha }{⟩}_{L^2},$$

$$f=(P_1,P_2,P)\in L^2(\Omega ;{\mathbb{R}}^3).$$

$$w=-(h_{\alpha \beta \lambda \mu }{D}_{\lambda \mu }^{\ast }{D}_{\alpha \beta }{)}^{-1}(\text{ div }Q+\text{ div }{z}^{\ast }-P),$$

$$\text{ div }Q+\text{ div }{z}^{\ast }-P=-h_{\alpha \beta \lambda \mu }{w}_{,\alpha \beta \lambda \mu }$$

$$w\in U_2.$$

$$G_2^{\ast }(z^{\ast },Q)$$

$${\tilde{G}}_1^{\ast }(-Q,N)$$

$$\{N_{\alpha \beta }+K_{\alpha \beta }\}$$

$$\{\overline{N_{\alpha \beta }+K_{\alpha \beta }}\}=\{N_{\alpha \beta }+K_{\alpha \beta }{\}}^{-1}.$$

$$F^{\ast }(z^{\ast })$$

Full text

At first we establish a duality principle which includes sufficient conditions of global optimality through the dual formulation. At this point we highlight this first duality principle is specially suitable for the case in which the membrane stress tensor is negative definite. In a second step, from such a general principle, we develop a primal dual variational formulation which also includes the corresponding sufficient conditions for global optimality. The results are based on standard tools of convex analysis and on a well known Toland result for D.C. optimization.

Finally, in the last section, we present a multi-duality principle and qualitative relations between the critical points of the primal and dual formulations. We formally prove there is no duality gap between such primal and dual formulations in a local extremal context.

1 Introduction

In this work we develop a new primal dual variational formulation for the Kirchhoff-Love non-linear plate model. We emphasize the results here presented may be applied to a large class of non-convex variational problems.

At this point we start to describe the primal formulation.

Let $\Omega\subset\mathbb{R}^{2}$ be an open, bounded, connected set which represents the middle surface of a plate of thickness $h$. The boundary of $\Omega$, which is assumed to be regular (Lipschitzian), is denoted by $\partial\Omega$. The vectorial basis related to the cartesian system $\{x_{1},x_{2},x_{3}\}$ is denoted by $(\textbf{a}_{\alpha},\textbf{a}_{3})$, where $\alpha=1,2$ (in general Greek indices stand for 1 or 2), and where $\textbf{a}_{3}$ is the vector normal to $\Omega$, whereas $\textbf{a}_{1}$ and $\textbf{a}_{2}$ are orthogonal vectors parallel to $\Omega.$ Also, n is the outward normal to the plate surface.

The displacements will be denoted by

[TABLE]

The Kirchhoff-Love relations are

[TABLE]

Here $-h/2\leq x_{3}\leq h/2$ so that we have $u=(u_{\alpha},w)\in U$ where

[TABLE]

It is worth emphasizing that the boundary conditions here specified refer to a clamped plate.

We define the operator $\Lambda:U\rightarrow Y_{1}\times Y_{1}$, where $Y_{1}=Y^{*}_{1}=L^{2}(\Omega;\mathbb{R}^{2\times 2})$, by

[TABLE]

[TABLE]

[TABLE]

The constitutive relations are given by

[TABLE]

[TABLE]

where: $\{H_{\alpha\beta\lambda\mu}\}$ and $\{h_{\alpha\beta\lambda\mu}=\frac{h^{2}}{12}H_{\alpha\beta\lambda\mu}\}$, are symmetric positive definite fourth order tensors. From now on, we denote $\{\overline{H}_{\alpha\beta\lambda\mu}\}=\{H_{\alpha\beta\lambda\mu}\}^{-1}$ and $\{\overline{h}_{\alpha\beta\lambda\mu}\}=\{h_{\alpha\beta\lambda\mu}\}^{-1}$.

Furthermore $\{N_{\alpha\beta}\}$ denote the membrane stress tensor and $\{M_{\alpha\beta}\}$ the moment one. The plate stored energy, represented by $(G\circ\Lambda):U\rightarrow\mathbb{R}$ is expressed by

[TABLE]

and the external work, represented by $F_{1}:U\rightarrow\mathbb{R}$, is given by

[TABLE]

where $P,P_{1},P_{2}\in L^{2}(\Omega)$ are external loads in the directions $\textbf{a}_{3}$, $\textbf{a}_{1}$ and $\textbf{a}_{2}$ respectively. The potential energy, denoted by $J:U\rightarrow\mathbb{R}$ is expressed by:

[TABLE]

Finally, we also emphasize from now on, as their meaning are clear, we may denote $L^{2}(\Omega)$ and $L^{2}(\Omega;\mathbb{R}^{2\times 2})$ simply by $L^{2}$, and the respective norms by $\|\cdot\|_{2}.$ Moreover, unless otherwise indicated, derivatives are always understood in the distributional sense, $\mathbf{0}$ may denote the zero vector in appropriate Banach spaces and, the following and relating notations are used:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Here we emphasize the general Einstein convention of sum of repeated indices holds throughout the text, unless otherwise indicated.

Remark 1.1**.**

About the references, details on the Sobolev involved may be found in [1]. Mandatory references are the original results of Telega and his co-workers in [2, 3, 13, 9]. About convex analysis, the results here developed follow in some extent [4], for which the main references are [8, 14].

We emphasize, our results complement, in some sense, the original ones presented in [2, 3, 13, 9].

Finally, existence results for models in elasticity including the plate model here addressed are developed in [5, 6, 7]. Similar problems are addressed in [10, 11].

2 The first duality principle

Theorem 2.1**.**

Let $\Omega\subset\mathbb{R}^{2}$ be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by $\partial\Omega$. Let $U=U_{1}\times U_{2}$ where $U_{1}=W_{0}^{1,2}(\Omega;\mathbb{R}^{2})$ and $U_{2}=W_{0}^{2,2}(\Omega)$. We recall that $Y_{1}=Y_{1}^{*}=L^{2}(\Omega;\mathbb{R}^{2\times 2})$ and define $Y_{2}=Y_{2}^{*}=L^{2}(\Omega;\mathbb{R}^{2})$,

[TABLE]

[TABLE]

and $J:U\rightarrow\mathbb{R}$ by,

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

where

[TABLE]

In the next lines we shall denote

[TABLE]

if

[TABLE]

and

[TABLE]

Define also $G_{2}^{*}:Y_{2}^{*}\times Y_{2}^{*}\rightarrow\mathbb{R},$ by

[TABLE]

and $\tilde{G}_{1}^{*}:Y_{2}^{*}\times Y_{1}^{*}\rightarrow\mathbb{R}\cup\{+\infty\}$ where

[TABLE]

if

[TABLE]

is positive definite.

Here we have denoted,

[TABLE]

We denote also,

[TABLE]

if $\{K_{\alpha\beta}\}$ is positive definite, where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

for some

[TABLE]

At this point we also define,

[TABLE]

where,

[TABLE]

Moreover, we denote,

[TABLE]

and define

[TABLE]

Assume $u_{0}\in U$ is such that $\delta J(u_{0})=\mathbf{0}$ and $N_{0}\in B^{*}$, where

[TABLE]

Under such hypotheses,

[TABLE]

where

[TABLE]

[TABLE]

and where

[TABLE]

and

[TABLE]

Proof.

From the general result in Toland [14], we have

[TABLE]

$\forall u\in U,\;Q\in Y_{2}^{*},\;N\in A^{*}.$

Therefore,

[TABLE]

$\forall u\in U,\;Q\in Y_{2}^{*},\;N\in A^{*}.$

Summarizing,

[TABLE]

$\forall u\in U,\;Q\in Y_{2}^{*},\;N\in A^{*},$ so that

[TABLE]

Suppose now $u_{0}\in U$ is such that

[TABLE]

and $N_{0}\in B^{*}.$

Observe that, from (14),

[TABLE]

so that

[TABLE]

Hence,

[TABLE]

Moreover, from $\delta J(u_{0})=\mathbf{0}$, we have

[TABLE]

where, as above indicated

[TABLE]

and

[TABLE]

From (15),

[TABLE]

so that from this and the inversion of (16), we have

[TABLE]

so that

[TABLE]

that is,

[TABLE]

where

[TABLE]

Also, from (15) and (16) we obtain

[TABLE]

Furthermore,

[TABLE]

This last equation corresponds to

[TABLE]

Moreover

[TABLE]

so that

[TABLE]

which means

[TABLE]

Finally, from

[TABLE]

we get

[TABLE]

Summarizing, we have obtained

[TABLE]

At this point we shall obtain a standard correspondence between the primal and dual formulations.

First, we recall that from

[TABLE]

we have

[TABLE]

From

[TABLE]

we obtain

[TABLE]

Finally, from

[TABLE]

and

[TABLE]

we get

[TABLE]

Joining the pieces, we obtain

[TABLE]

Moreover, since $N_{0}\in A^{*}$, we have

[TABLE]

From this, (13) and (22), we obtain

[TABLE]

The proof is complete. ∎

3 The primal dual formulation and related duality principle

At this point we present the main result of this article, which is summarized by the next theorem.

Theorem 3.1**.**

Consider the notation and context of the last theorem. Assume those hypotheses, more specifically suppose $\delta J(u_{0})=\mathbf{0}$ and $N_{0}\in B^{*}$, where

[TABLE]

[TABLE]

and

[TABLE]

Recall also that

[TABLE]

where,

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

Under such assumptions and notation, denoting also

[TABLE]

we have

[TABLE]

where,

[TABLE]

where generically we have denoted

[TABLE]

if $\eta={\nabla}^{2}\hat{w}$ and $\hat{w}\in W^{2,2}_{0}(\Omega).$

Proof.

Observe that

[TABLE]

$\forall Q\in\hat{Y}_{2}^{*},\;N\in A^{*}.$

Also, such an infimum is attained through the equation

[TABLE]

that is

[TABLE]

that is

[TABLE]

where

[TABLE]

Hence,

[TABLE]

From

[TABLE]

and

[TABLE]

we obtain

[TABLE]

Let $v\in W^{2,2}_{0}(\Omega)$ be such that $Q=\nabla v$.

From the last equation

[TABLE]

so that

[TABLE]

Thus

[TABLE]

so that

[TABLE]

Thus

[TABLE]

Moreover, from $\delta J(u_{0})=\mathbf{0}$ we have

[TABLE]

so that

[TABLE]

and therefore

[TABLE]

Also,

[TABLE]

so that

[TABLE]

From these last results and from the last theorem, we may obtain

[TABLE]

From this, also from the last theorem and from (33), we finally get

[TABLE]

The proof is complete.

∎

4 A multi-duality principle for non-convex optimization

Our final result is a multi-duality principle, which is summarized by the following theorem.

Theorem 4.1**.**

Considering the notation and statements of the plate model addressed in the last sections, assuming a not relabeled finite dimensional approximate model, in a finite elements or finite differences context, let $J_{1}:U\times Y_{1}^{*}\times Y_{2}^{*}\rightarrow\mathbb{R}$ be a functional where

[TABLE]

and where

[TABLE]

Define also,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

and where

[TABLE]

Moreover

[TABLE]

where

[TABLE]

We also define,

[TABLE]

[TABLE]

and

[TABLE]

Let $u_{0}\in U$ be such that $\delta J(u_{0})=\mathbf{0}$ and define

[TABLE]

and

[TABLE]

Under such hypotheses,

if $\delta^{2}J(u_{0})>\mathbf{0}$ and $N_{0}\in E^{*}$, defining

[TABLE]

and

[TABLE]

where $r_{1}>0$ is such that

[TABLE]

in $B_{r_{1}}(u_{0})$, we have

[TABLE]

[TABLE]

and if $K>0$ is sufficiently big,

[TABLE]

and there exist $r,\;r_{2}>0$ such that

[TABLE] 2. 2.

If $N_{0}\in A^{*}_{+}$, defining

[TABLE]

and

[TABLE]

then

[TABLE]

[TABLE]

and

[TABLE] 3. 3.

If $\delta^{2}J(u_{0})<\mathbf{0}$ so that $N_{0}\in A^{*}_{-}$, defining

[TABLE]

where

[TABLE]

we have that

[TABLE]

[TABLE]

and there exist $r,\;r_{1},\;r_{2}>0$ such that

[TABLE]

Proof.

From the assumption $N_{0}\in E^{*}$ we have that

[TABLE]

where such a supremum is attained through the equation

[TABLE]

Moreover, there exists $r>0,\;r_{2}>0$ such that

[TABLE]

and (we justify that the first infimum $Q$ in this equation (41) is well defined in the next lines)

[TABLE]

Observe the concerning extremal in $Q$ is attained through the equation,

[TABLE]

Hence, from

[TABLE]

from the implicit function theorem and chain rule, we get

[TABLE]

Therefore,

[TABLE]

From this we shall denote

[TABLE]

Let us now show that the first infimum in $Q$ in (41) is well defined.

Recall again that,

[TABLE]

where such a supremum is attained through the equation

[TABLE]

that is,

[TABLE]

Taking the variation of this last equation in $Q_{\rho}$ we have

[TABLE]

that is

[TABLE]

so that

[TABLE]

Hence, if $K>0$ is sufficiently big, we obtain

[TABLE]

Therefore the first infimum in $Q$ in (41) is well defined.

Also, from (41) and the second order necessary condition for a local minimum, we obtain

[TABLE]

Assume now again $\delta J(u_{0})=\mathbf{0}$ and $N_{0}\in A^{*}_{+}.$

Recall that

[TABLE]

Observe that if $N\in A^{*}_{+}$, by direct computation we may obtain

[TABLE]

Therefore $J_{3}(u,Q)$ is convex since is the supremum of a family of convex functionals.

Similarly as above we may obtain

[TABLE]

and

[TABLE]

so that

[TABLE]

Moreover,

[TABLE]

Hence,

[TABLE]

From these last results we may write,

[TABLE]

From this, similarly as above, we may obtain

[TABLE]

Finally, suppose now $\delta^{2}J(u_{0})<\mathbf{0},$ so that $N_{0}\in A^{*}_{-}.$

From this we obtain

[TABLE]

where, as previously indicated,

[TABLE]

Here,

[TABLE]

Denoting,

[TABLE]

also from $N_{0}\in A^{*}_{-}$ and from

[TABLE]

[TABLE]

(the proofs of such results are very similar to those of the corresponding cases developed above), there exist $r,r_{1},r_{2}>0$ such that for $N\in A^{*}_{-}\cap B_{r_{2}}(N_{0})$, we have

[TABLE]

and

[TABLE]

The proof is complete. ∎

5 Conclusion

In this article we have developed a new primal dual variational formulation and a multi-duality principle applied to a non-linear model of plates.

About the primal dual formulation, we emphasize such a formulation is concave so that it is very interesting from a numerical analysis point of view.

Finally, the results here presented may be also developed in a similar fashion for a large class of problems, including non-linear models in elasticity and other non-linear models of plates and shells.