A Duality Principle and a Concerning Convex Dual Formulation Suitable for Non-convex Variational Optimization

Preprint

Article

A Duality Principle and a Concerning Convex Dual Formulation Suitable for Non-convex Variational Optimization

Altmetrics

Downloads

411

Views

792

Comments

Fabio Botelho^*

This version is not peer-reviewed

Submitted:

12 January 2023

Posted:

12 January 2023

You are already at the latest version

Alerts

Abstract

This article develops a duality principle and a related convex dual formulation suitable for a large class of models in physics and engineering. The results are based on standard tools of functional analysis, calculus of variations and duality theory. In particular, we develop applications to a model in non-linear elasticity.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

MSC: 49N15

1. Introduction

In this article we establish a duality principle and a related convex dual formulation for a large class of models in non-convex optimization.

More specifically, the main duality principle is applied to a model in non-linear elasticity.

Such results are based on the works of J.J. Telega and W.R. Bielski [2,3,10,11] and on a D.C. optimization approach developed in Toland [12].

About the other references, details on the Sobolev spaces involved are found in [1]. Related results on convex analysis and duality theory are addressed in [4,5,6,7,9]. Finally, the model in non-linear elasticity here presented may be found in [8].

Remark.

In this text we adopt the standard Einstein convention of summing up repeated indices unless otherwise indicated.

At this point we start to describe the primal and dual variational formulations.

Let

Ω \subset R^{3}

be an open, bounded, connected set with a regular (Lipschitzian) boundary denoted by

\partial Ω .

For the primal formulation, consider a functional

J : V \to R

where

\begin{matrix} J (u) & = & \frac{1}{2} \int_{Ω} H_{i j k l} e_{i j} (u) e_{k l} (u) d x - {〈 u_{i}, f_{i} 〉}_{L^{2}} . \end{matrix}

(1)

Here

{H_{i j k l}}

is a fourth order symmetric positive definite tensor and

{e_{i j} (u)} = \{\frac{1}{2} (u_{i, j} + u_{j, i}) + \frac{1}{2} (u_{m, i} u_{m, j})\},

where

u = (u_{1}, u_{2}, u_{3}) \in V = W_{0}^{1, 4} (Ω; R^{3})

denotes the field of displacements resulting from the action of the external forces

f = (f_{1}, f_{2}, f_{2}) \in L^{2} (Ω; R^{3})

on the elastic solid comprised by

Ω \subset R^{3} .

Moreover, denoting

Y = Y^{*} = L^{2} (Ω; R^{3 \times 3}),

the stress tensor

σ \in Y^{*}

is defined by

{σ_{i j} (u)} = {H_{i j k l} e_{k l} (u)} .

At this point we define the functionals

F_{1} : V \times Y \to R,

F_{2} : V \to R

and

G : V \to R

\begin{matrix} F_{1} (u, σ) & = & - \sum_{i = 1}^{3} \frac{K}{2} \int_{Ω} {(u_{i, i})}^{2} d x \\ + \sum_{i = 1}^{3} \frac{K_{1}}{2} \int_{Ω} {(σ_{i j, j} + {(σ_{i m} u_{m, j})}_{, j} + f_{i})}^{2} d x \\ + \frac{K_{2}}{2} \int_{Ω} u_{i, j} u_{i, j} d x - {〈 u_{i}, f_{i} 〉}_{L^{2}}, \end{matrix}

(2)

for appropriate positive real constants,

K, K_{1}, K_{2}

to be specified,

F_{2} (u) = \frac{K_{2}}{2} \int_{Ω} u_{i, j} u_{i, j} d x,

and

G (u) = \frac{1}{2} \int_{Ω} H_{i j k l} e_{i j} (u) e_{k l} (u) d x + \sum_{i = 1}^{3} \frac{K}{2} \int_{Ω} {(u_{i, i})}^{2} d x .

Here, it is worth highlighting that

F_{1} (u, σ) - F_{2} (u) + G (u) = J (u) + \sum_{i = 1}^{3} \frac{K_{1}}{2} \int_{Ω} {(σ_{i j, j} + {(σ_{i m} u_{m, j})}_{, j} + f_{i})}^{2} d x, \forall u \in V, σ \in Y^{*} .

Furthermore, we define the functionals

F_{1}^{*} : {[Y^{*}]}^{3} \to R

F_{2}^{*} : Y^{*} \to R

and

G^{*} : {[Y^{*}]}^{2} \to R

F_{1}^{*} (σ, Q, \tilde{Q}) = sup_{u \in V} {- {〈 u_{i, j}, σ_{i j} 〉}_{L^{2}} - {〈 u_{i, j}, Q_{i j} 〉}_{L^{2}} + {〈 u_{i, j}, {\tilde{Q}}_{i j} 〉}_{L^{2}} - F_{1} (u, σ)},

\begin{matrix} F_{2}^{*} (\tilde{Q}) & = & sup_{v_{2} \in Y} \{{〈 {(v_{2})}_{i j}, {\tilde{Q}}_{i j} 〉}_{L^{2}} - \frac{K_{2}}{2} \int_{Ω} {(v_{2})}_{i j} {(v_{2})}_{i j} d x\} \\ = & \frac{1}{2 K_{2}} \int_{Ω} {\tilde{Q}}_{i j} {\tilde{Q}}_{i j} d x, \end{matrix}

(3)

and

G^{*} (σ, Q) = sup_{(v_{1}, v_{2}) \in Y \times Y} {{〈 {(v_{1})}_{i j}, σ_{i j} 〉}_{L^{2}} + {〈 {(v_{2})}_{i j}, Q_{i j} 〉}_{L^{2}} - \hat{G} (v_{1}, v_{2})},

where

\begin{matrix} \hat{G} (v_{1}, v_{2}) & = & \frac{1}{2} \int_{Ω} H_{i j k l} ({(v_{1})}_{i j} + \frac{1}{2} {(v_{2})}_{m i} {(v_{2})}_{m j}) ({(v_{1})}_{k l} + \frac{1}{2} {(v_{2})}_{m k} {(v_{2})}_{m l}) d x \\ + \sum_{i = 1}^{3} \frac{K}{2} \int_{Ω} {({(v_{2})}_{i i})}^{2} d x, \end{matrix}

(4)

so that

G^{*} (σ, Q) = \frac{1}{2} \int_{Ω} (\bar{σ_{i j}^{K}}) Q_{m i} Q_{m j} d x + \frac{1}{2} \int_{Ω} {\bar{H}}_{i j k l} σ_{i j} σ_{k l} d x,

σ \in B^{*}

where

B^{*} = {σ \in Y^{*} : ∥ σ_{i j} ∥_{\infty} \leq K / 8, \forall i, j \in {1, 2, 3} and {σ_{i j}} < - ε I_{d}},

for some small parameter

ε > 0

and where

I_{d}

denotes the

3 \times 3

identity matrix. Observe that such a definition for

B^{*}

corresponds to the case of negative definite stress tensors, which refers to compression in a solid mechanics context.

Here

{σ_{i j}^{K}} = [\begin{matrix} σ_{11} + K & σ_{12} & σ_{13} \\ σ_{21} & σ_{22} + K & σ_{23} \\ σ_{31} & σ_{32} & σ_{33} + K \end{matrix}]

(5)

\bar{σ_{i j}^{K}} = {σ_{i j}^{K}}^{- 1},

and

{{\bar{H}}_{i j k l}} = {H_{i j k l}}^{- 1}

in an appropriate tensor sense.

At this point we define

J^{*} (σ, Q, \tilde{Q}) = - F_{1}^{*} (σ, Q, \tilde{Q}) + F_{2}^{*} (\tilde{Q}) - G^{*} (σ, Q) .

Specifically for

K_{2} ≫ K_{1} ≫ K ≫ max {1 / ε^{2}, 1, K_{3}, ∥ H_{i j k l} ∥},

we define

D^{*} = {Q \in Y^{*} : ∥ Q_{i j} ∥_{\infty} \leq K_{3}, \forall i, j \in {1, 2, 3}} .

By direct computation, we may obtain

\{\frac{\partial^{2} J^{*} (σ, Q, \tilde{Q})}{\partial σ \partial Q}\} < 0,

and

\{\frac{\partial^{2} J^{*} (σ, Q, \tilde{Q})}{\partial {\tilde{Q}}^{2}}\} > 0,

B^{*} \times D^{*} \times Y^{*},

so that

J^{*}

is concave in

(σ, Q)

and convex in

\tilde{Q}

B^{*} \times D^{*} \times Y^{*} .

2. The main duality principle and a related convex dual variational formulation

Our main duality principle is summarized by the following theorem.

Theorem 1.

Considering the statements and definitions of the previous section, suppose

(\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) \in B^{*} \times D^{*} \times Y^{*}

is such that

δ J^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) = 0 .

Let

u_{0} \in V

be such that

{(u_{0})}_{i, j} = \frac{\partial F_{2}^{*} (\tilde{Q})}{\partial {\tilde{Q}}_{i j}} .

Under such hypotheses, we have

\begin{matrix} J (u_{0}) & = & min_{u \in V} \{J (u) + \sum_{i = 1}^{3} \frac{K_{1}}{2} \int_{Ω} {({\hat{σ}}_{i j, j} + {({\hat{σ}}_{i m} u_{m, j})}_{, j} + f_{i})}^{2} d x\} \\ = & inf_{\tilde{Q} \in Y^{*}} \{sup_{(σ, Q) \in B^{*} \times D^{*}} J^{*} (σ, Q, \tilde{Q})\} \\ = & J^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) . \end{matrix}

(6)

Proof.

Observe that there exists

\hat{u} \in V

such that, defining

H (u, σ, Q, \tilde{Q}) = - {〈 u_{i, j}, σ_{i j} 〉}_{L^{2}} - {〈 u_{i, j}, Q_{i j} 〉}_{L^{2}} + {〈 u_{i, j}, {\tilde{Q}}_{i j} 〉}_{L^{2}} - F_{1} (u, σ),

we have

\frac{\partial H (\hat{u}, \hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial u} = 0

and

F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) = H (\hat{u}, \hat{σ}, \hat{Q}, \hat{\tilde{Q}}) .

Moreover, from the variation in of

J^{*}

\tilde{Q}

, we obtain

- \frac{\partial F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial {\tilde{Q}}_{i j}} + \frac{\partial F_{2}^{*} (\hat{\tilde{Q}})}{\partial {\tilde{Q}}_{i j}} = 0,

where

\begin{matrix} \frac{\partial F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial {\tilde{Q}}_{i j}} \\ = & \frac{\partial H (\hat{u}, \hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial {\tilde{Q}}_{i j}} \\ + \frac{\partial H (\hat{u}, \hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial u} \frac{\partial \hat{u}}{\partial {\tilde{Q}}_{i j}} \\ = & {\hat{u}}_{i, j} . \end{matrix}

(7)

From such last two equations we get

{(u_{0})}_{i, j} = \frac{\partial F_{2}^{*} (\hat{\tilde{Q}})}{\partial {\tilde{Q}}_{i j}} = \frac{\partial F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial {\tilde{Q}}_{i j}} = {\hat{u}}_{i, j},

so that from the concerning boundary conditions,

u_{0} = \hat{u} .

On the other hand, from the variation of

J^{*}

in Q we have

- \frac{\partial F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial Q_{i j}} - \frac{\partial G^{*} (\hat{σ}, \hat{Q})}{\partial Q_{i j}} = 0,

so that

{(u_{0})}_{i, j} - \bar{{\hat{σ}}_{i m}^{K}} {\hat{Q}}_{m j} = 0,

and therefore

{\hat{Q}}_{i j} = {\hat{σ}}_{i m} {(u_{0})}_{m, j} + K δ_{i j} {(u_{0})}_{i, j} .

Finally, from the variation of

J^{*}

σ

we obtain

- \frac{\partial F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial σ_{i j}} - \frac{\partial G^{*} (\hat{σ}, \hat{Q})}{\partial σ_{i j}} = 0,

so that

{(u_{0})}_{i, j} + \frac{1}{2} {(u_{0})}_{m, i} {(u_{0})}_{m, j} - {\bar{H}}_{i j k l} {\hat{σ}}_{k l} = 0 .

Thus, since

{H_{i j k l}}

is symmetric, we get

{\hat{σ}}_{i j} = H_{i j k l} e_{k l} (u_{0}) .

From these last results and from

\frac{\partial H (\hat{u}, \hat{σ}, \hat{Q}, \hat{\tilde{Q}})}{\partial u} = 0

we obtain

{\hat{σ}}_{i j, j} + {({\hat{σ}}_{i m} {(u_{0})}_{m, j})}_{, j} + f_{i} = 0, \forall i \in {1, 2, 3},

so that

δ J (u_{0}) = 0 .

Finally, from such last results and the Legendre transform properties, we have

\begin{matrix} F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) \\ = & - {〈 {(u_{0})}_{i, j}, {\hat{σ}}_{i j} 〉}_{L^{2}} - {〈 {(u_{0})}_{i, j}, {\hat{Q}}_{i j} 〉}_{L^{2}} + {〈 {(u_{0})}_{i, j}, {\hat{\tilde{Q}}}_{i j} 〉}_{L^{2}} - F_{1} (u_{0}, \hat{σ}), \end{matrix}

(8)

F_{2}^{*} (\hat{\tilde{Q}}) = {〈 {(u_{0})}_{i, j}, {\hat{\tilde{Q}}}_{i j} 〉}_{L^{2}} - F_{2} (u_{0}),

and

G^{*} (\hat{σ}, \hat{Q}) = {〈 {(u_{0})}_{i, j}, {\hat{σ}}_{i j} 〉}_{L^{2}} + {〈 {(u_{0})}_{i, j}, {\hat{Q}}_{i j} 〉}_{L^{2}} - G (u_{0}) .

From these results, we obtain

\begin{matrix} J^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) & = & - F_{1}^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) + F_{2}^{*} (\hat{\tilde{Q}}) - G^{*} (\hat{σ}, \hat{Q}) \\ = & F_{1} (u_{0}, \hat{σ}) - F_{2} (u_{0}) + G (u_{0}) \\ = & J (u_{0}) . \end{matrix}

(9)

Joining the pieces, we have got

\begin{matrix} J (u_{0}) & = & min_{u \in V} \{J (u) + \sum_{i = 1}^{3} \frac{K_{1}}{2} \int_{Ω} {({\hat{σ}}_{i j, j} + {({\hat{σ}}_{i m} u_{m, j})}_{, j} + f_{i})}^{2} d x\} \\ = & inf_{\tilde{Q} \in Y^{*}} \{sup_{(σ, Q) \in B^{*} \times D^{*}} J^{*} (σ, Q, \tilde{Q})\} \\ = & J^{*} (\hat{σ}, \hat{Q}, \hat{\tilde{Q}}) . \end{matrix}

(10)

The proof is complete.

□

Remark.

A similar result is valid if we would define

B^{*} = {σ \in Y^{*} : ∥ σ_{i j} ∥_{\infty} \leq K / 8, \forall i, j \in {1, 2, 3} and {σ_{i j}} > ε I_{d}} .

This case refers to a positive definite tensor

{σ_{i j}}

and the previous case to a negative definite one.

3. A closely related primal-dual variational formulation for a similar model

In this section we present a new primal-dual variational formulation for a closely related model of plates.

At this point we start to describe the primal formulation.

Let

Ω \subset R^{2}

be an open, bounded, connected set which represents the middle surface of a plate of thickness h. The boundary of

Ω

, which is assumed to be regular (Lipschitzian), is denoted by

\partial Ω

. The vectorial basis related to the cartesian system

{x_{1}, x_{2}, x_{3}}

is denoted by

(a_{α}, a_{3})

, where

α = 1, 2

(in general Greek indices stand for 1 or 2), and where

a_{3}

is the vector normal to

Ω

, whereas

a_{1}

and

a_{2}

are orthogonal vectors parallel to

Ω .

Also,

n

is the outward normal to the plate surface.

The displacements will be denoted by

\hat{u} = {{\hat{u}}_{α}, {\hat{u}}_{3}} = {\hat{u}}_{α} a_{α} + {\hat{u}}_{3} a_{3} .

The Kirchhoff-Love relations are

\begin{matrix} {\hat{u}}_{α} (x_{1}, x_{2}, x_{3}) = u_{α} (x_{1}, x_{2}) - x_{3} w {(x_{1}, x_{2})}_{, α} \\ and {\hat{u}}_{3} (x_{1}, x_{2}, x_{3}) = w (x_{1}, x_{2}) . \end{matrix}

(11)

Here

- h / 2 \leq x_{3} \leq h / 2

so that we have

u = (u_{α}, w) \in U

where

\begin{matrix} U & = & \{(u_{α}, w) \in W^{1, 2} (Ω; R^{2}) \times W^{2, 2} (Ω), \\ u_{α} = w = \frac{\partial w}{\partial n} = 0 on \partial Ω\} \\ = & W_{0}^{1, 2} (Ω; R^{2}) \times W_{0}^{2, 2} (Ω) . \end{matrix}

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

We define the operator

Λ : U \to Y \times Y

, where

Y = Y^{*} = L^{2} (Ω; R^{2 \times 2})

, by

Λ (u) = {γ (u), κ (u)},

γ_{α β} (u) = \frac{u_{α, β} + u_{β, α}}{2} + \frac{w_{, α} w_{, β}}{2},

κ_{α β} (u) = - w_{, α β} .

The constitutive relations are given by

N_{α β} (u) = H_{α β λ μ} γ_{λ μ} (u),

(12)

M_{α β} (u) = h_{α β λ μ} κ_{λ μ} (u),

(13)

where:

{H_{α β λ μ}}

and

\{h_{α β λ μ} = \frac{h^{2}}{12} H_{α β λ μ}\}

, are symmetric positive definite fourth order tensors. From now on, we denote

{{\bar{H}}_{α β λ μ}} = {H_{α β λ μ}}^{- 1}

and

{{\bar{h}}_{α β λ μ}} = {h_{α β λ μ}}^{- 1}

Furthermore

{N_{α β}}

denote the membrane force tensor and

{M_{α β}}

the moment one. The plate stored energy, represented by

(G \circ Λ) : U \to R

is expressed by

(G \circ Λ) (u) = \frac{1}{2} \int_{Ω} N_{α β} (u) γ_{α β} (u) d x + \frac{1}{2} \int_{Ω} M_{α β} (u) κ_{α β} (u) d x

(14)

and the external work, represented by

F : U \to R

, is given by

F (u) = {〈 w, P 〉}_{L^{2}} + {〈 u_{α}, P_{α} 〉}_{L^{2}},

(15)

where

P, P_{1}, P_{2} \in L^{2} (Ω)

are external loads in the directions

a_{3}

a_{1}

and

a_{2}

respectively. The potential energy, denoted by

J : U \to R

is expressed by:

J (u) = (G \circ Λ) (u) - F (u)

More explicitly, recalling that

γ_{α β} (u) = \frac{u_{α, β} + u_{β, α}}{2} + \frac{1}{2} w_{, α} w_{, β},

we have

\begin{matrix} J (u) \\ = & \frac{1}{2} \int_{Ω} H_{α β λ μ} (\frac{u_{α, β} + u_{β, α}}{2} + \frac{1}{2} w_{, α} w_{, β}) (\frac{u_{λ, μ} + u_{μ, λ}}{2} + \frac{1}{2} w_{, λ} w_{, μ}) d x \\ + \frac{1}{2} \int_{Ω} h_{α β λ μ} w_{, α β} w_{, λ μ} d x - {〈 u_{α}, P_{α} 〉}_{L^{2}} - {〈 w, P 〉}_{L^{2}} \\ = & - {〈 γ_{α β} (u), N_{α β} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} H_{α β λ μ} γ_{α β} (u) γ_{λ μ} (u) d x \\ + \frac{1}{2} \int_{Ω} h_{α, β, λ, μ} w_{, α β} w_{, λ μ} d x + {〈\frac{u_{α, β} + u_{β, α}}{2} + \frac{1}{2} w_{, α} w_{, β}, N_{α β}〉}_{L^{2}} \\ - {〈 u_{α}, P_{α} 〉}_{L^{2}} - {〈 w, P 〉}_{L^{2}} \\ \geq & inf_{v \in Y} \{{〈 v_{α β}, N_{α β} 〉}_{L^{2}} + \frac{1}{2} \int_{Ω} H_{α β λ μ} v_{α β} v_{λ μ} d x\} \\ + inf_{{u_{α}} \in {[W_{0}^{1, 2} (Ω)]}^{2}} \{\frac{1}{2} \int_{Ω} h_{α β λ μ} w_{, α β} w_{, λ μ} d x \\ + {〈\frac{u_{α, β} + u_{β, α}}{2} + \frac{1}{2} w_{, α} w_{, β}, N_{α β}〉}_{L^{2}} - {〈 u_{α} P_{α} 〉}_{L^{2}} - {〈 w, P 〉}_{L^{2}}\} \\ \geq & - \frac{1}{2} \int_{Ω} {\bar{H}}_{α β λ μ} N_{α β} N_{λ μ} d x \\ + \frac{1}{2} \int_{Ω} h_{α, β, λ, μ} w_{, α β} w_{, λ μ} d x + {〈\frac{1}{2} w_{, α} w_{, β}, N_{α β}〉}_{L^{2}} - {〈 w, P 〉}_{L^{2}} \\ = & J_{1}^{*} (u, N), \end{matrix}

(16)

\forall u \in U, N \in B^{*},

where

B^{*} = B_{1}^{*} \cap B_{2}^{*}

B_{1}^{*} = {N \in Y^{*} : N_{α β, β} + P_{α} = 0, in Ω},

B_{2}^{*} = {N \in Y^{*} {: ∥ N ∥}_{\infty} \leq K},

for an appropriate constant

K > 0

At this point, we also define

V_{1} = \{u \in U : ∥ w_{, α β} ∥_{\infty} \leq K_{3}, \forall α, β \in {1, 2}\},

for an appropriate constant

K_{3} > 0

We highlight the constants

K_{3} > 0

and

K > 0

must be such that the restrictions which define

B_{2}^{*}

and

V_{1}

are not active at a concerning critical point.

Here we present the following primal-dual formulation suitable for an optimization of the original primal variational formulation

J_{2}^{*} (u, N) = J_{1}^{*} (u, N) + \frac{K_{1}}{2} \int_{Ω} {({(h_{α β λ μ} w_{, λ μ})}_{, α β} - {(N_{α β} w_{, β})}_{, α} - P)}^{2} d x .

More specifically,

\begin{matrix} J_{2}^{*} (u, N) & = & - \frac{1}{2} \int_{Ω} {\bar{H}}_{α β λ μ} N_{α β} N_{λ μ} d x \\ + \frac{1}{2} \int_{Ω} h_{α, β, λ, μ} w_{, α β} w_{, λ μ} d x + {〈\frac{1}{2} w_{, α} w_{, β}, N_{α β}〉}_{L^{2}} \\ - {〈 w, P 〉}_{L^{2}} \\ + \frac{K_{1}}{2} \int_{Ω} {({(h_{α β λ μ} w_{, λ μ})}_{, α β} - {(N_{α β} w_{, β})}_{, α} - P)}^{2} d x . \end{matrix}

(17)

We may observe that for

K_{1} \approx O (\frac{1}{4 ∥ H ∥ K_{3}^{2}})

and

K_{3} > 0

sufficiently small,

J_{2}^{*}

is convex in u and concave in N and on

V_{1} \times B^{*} .

Finally, we may also define

J_{3}

J_{3} (u) = sup_{N \in B^{*}} J_{2}^{*} (u, N) .

We observe that

J_{3}

has a large region of convexity around any critical point.

4. A duality principle for a related model in phase transitions

In this section we present a duality principle for a related model in phase transition.

Let

Ω = [0, 1] \subset R

and consider a functional

J : V \to R

where

J (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{1}{2} \int_{Ω} u^{2} d x,

and where

V = {u \in W^{1, 4} (Ω) : u (0) = 0 and u (1) = 1 / 2} .

A global optimum point is not attained for J so that the problem of finding a global minimum for J has no solution.

Anyway, one question remains, how the minimizing sequences behave close the infimum of J.

From the Ekeland variational principle the equation

δ J (u_{0}) \approx 0,

may be approximately satisfied by points for which J is arbitrarily close to its infimum.

We intend to use duality theory to approximately solve such a global optimization problem.

At this point we define,

F : V \to R

and

G : V \to R

F (u) = \frac{1}{2} \int_{Ω} {({(u^{'})}^{2} - 1)}^{2} d x + \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x,

and

G (u) = - \frac{1}{2} \int_{Ω} u^{2} d x + + \frac{K}{2} \int_{Ω} {(u^{'})}^{2} d x,

so that

J (u) = F (u) - G (u), \forall u \in V .

Observe that if

K > 0

is large enough, both F and G are convex.

Denoting

Y = Y^{*} = L^{2} (Ω)

we also define the polar functionals

F^{*} : Y^{*} \to R

and

G^{*} : Y^{*} \to R

F^{*} (v^{*}) = sup_{u \in V} {{〈 u, v^{*} 〉}_{L^{2}} - F (u)},

and

G^{*} (v^{*}) = sup_{u \in V} {{〈 u, v^{*} 〉}_{L^{2}} - G (u)} .

From the standard Toland result in [12] for D.C. optimization, we may obtain

inf_{u \in U} J (u) = inf_{u \in U} {F (u) - G (u)} = inf_{v^{*} \in Y^{*}} {G^{*} (v^{*}) - F^{*} (v^{*})} .

In fact, we may also obtain

inf_{u \in U} J (u) = inf_{(u, v^{*}) \in U \times Y^{*}} {G^{*} (v^{*}) - {〈 u, v^{*} 〉}_{L^{2}} + F (u)} .

With such results in mind, we define a primal dual variational formulation for the primal problem, represented by

J_{1}^{*} : V \times Y^{*} \to R

, where

J_{1}^{*} (u, v^{*}) = G^{*} (v^{*}) - {〈 u, v^{*} 〉}_{L^{2}} + F (u) .

Having defined such a functional, we may obtain numerical results by solving a sequence of convex auxiliary sub-problems, through the following algorithm.

Set $K ≫ 1$ . and $0 < ε ≪ 1 .$
Choose $u_{1} \in V,$ such that $∥ u_{1} ∥_{1, \infty} ≪ K / 4 .$
Set $n = 1 .$
Calculate $v_{n}^{*}$ solution of equation:

$\frac{\partial J_{1}^{*} (u_{n}, v_{n}^{*})}{\partial v^{*}} = 0,$

that is

$\frac{\partial G^{*} (v_{n}^{*})}{\partial v^{*}} - u_{n} = 0,$

so that

$v_{n}^{*} = \frac{\partial G (u_{n})}{\partial u} .$
Calculate $u_{n + 1}$ by solving the equation:

$\frac{\partial J_{1}^{*} (u_{n + 1}, v_{n}^{*})}{\partial u} = 0,$

that is

$- v_{n}^{*} + \frac{\partial F (u_{n + 1})}{\partial u} = 0 .$
If $∥ u_{n} - u_{n + 1} ∥ \leq ε$ , then stop, else set $n : = n + 1$ and go to item 4.

We have obtained numerical results for

K = 10000000

. For the solution

u_{0}

obtained please see Figure 1.

5. Conclusion

In this article we have developed a convex dual variational formulation suitable for non-convex variational primal formulations.

It is worth highlighting, the results may be applied to a large class of models in physics and engineering.

We also emphasize the duality principle here presented is applied to a model in non-linear elasticity. In a future research, we intend to extend such results for other related models of plates and shells.

References

R.A. Adams and J.F. Fournier, Sobolev Spaces, 2nd edn. (Elsevier, New York, 2003).
W.R. Bielski, A. Galka, J.J. Telega, The Complementary Energy Principle and Duality for Geometrically Nonlinear Elastic Shells. I. Simple case of moderate rotations around a tangent to the middle surface. Bulletin of the Polish Academy of Sciences, Technical Sciences, Vol. 38, No. 7-9, 1988.
W.R. Bielski and J.J. Telega, A Contribution to Contact Problems for a Class of Solids and Structures, Arch. Mech., 37, 4-5, pp. 303-320, Warszawa 1985.
F.S. Botelho, Functional Analysis, Calculus of Variations and Numerical Methods in Physics and Engineering, CRC Taylor and Francis, Florida, 2020.
F.S. Botelho, Variational Convex Analysis, Ph.D. thesis, Virginia Tech, Blacksburg, VA -USA, (2009).
F. Botelho, Topics on Functional Analysis, Calculus of Variations and Duality, Academic Publications, Sofia, (2011).
F. Botelho, Functional Analysis and Applied Optimization in Banach Spaces, Springer Switzerland, 2014.
P.Ciarlet, Mathematical Elasticity, Vol. I – Three Dimensional Elasticity, North Holland Elsivier (1988).
R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, (1970).
J.J. Telega, On the complementary energy principle in non-linear elasticity. Part I: Von Karman plates and three dimensional solids, C.R. Acad. Sci. Paris, Serie II, 308, 1193-1198; Part II: Linear elastic solid and non-convex boundary condition. Minimax approach, ibid, pp. 1313-1317 (1989).
A.Galka and J.J.Telega Duality and the complementary energy principle for a class of geometrically non-linear structures. Part I. Five parameter shell model; Part II. Anomalous dual variational priciples for compressed elastic beams, Arch. Mech. 47 (1995) 677-698, 699-724.
J.F. Toland, A duality principle for non-convex optimisation and the calculus of variations, Arch. Rat. Mech. Anal., 71, No. 1 (1979), 41-61.

Figure 1. solution

u_{0} (x)

through the primal dual formulation for a large

K > 0

Figure 1. solution

u_{0} (x)

through the primal dual formulation for a large

K > 0

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

A Duality Principle and a Concerning Convex Dual Formulation Suitable for Non-convex Variational Optimization

Abstract

1. Introduction

2. The main duality principle and a related convex dual variational formulation

3. A closely related primal-dual variational formulation for a similar model

4. A duality principle for a related model in phase transitions

5. Conclusion

References

MDPI Initiatives

Important Links

Subscribe