Symmetry analysis for structural optimization problems involving reliability measure and bi-modulus materials

Abstract

In the present work, some symmetry theorems for structural optimization problems involving reliability measure and bi-modulus materials are established. In the first part of this paper, two types of symmetric reliability-based structural optimization problems are discussed. It is proved that, for the first type symmetric RBDO problems, the existence of symmetric global optimal solutions can be guaranteed even through the probability density function (PDF) is non-symmetric. While for the second type symmetric RBDO problems, more severe conditions on PDF are required to ensure the existence of symmetric global optimal solutions. In the second part of the paper, the convexity of the compliance of truss structures involving bi-modulus materials is confirmed firstly. Based on this result, it is found that for both deterministic and robust symmetric compliance minimization of truss structures involving bi-modulus materials, symmetric global optima always exist. The theoretical results are also demonstrated through some academic examples.

Appendix A

In this appendix, we shall prove the fact that

$$ \left({\mathbf{\mathcal{R}}}^i{\boldsymbol{u}}^{*},{\mathbf{\mathcal{R}}}^i{\boldsymbol{z}}^{*}\right)\in \mathrm{A}\mathrm{r}\mathrm{g}\kern0.5em { \max}_{{\mathcal{U}}^{ad}\times {\mathcal{Z}}^{ad}}\left(-2\Pi \left(\boldsymbol{u},\boldsymbol{z};{\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right)\right). $$

The augmented Lagrangian function of the optimization in (38) is

$$ L\left(\boldsymbol{u},\boldsymbol{z},\boldsymbol{\lambda}; \boldsymbol{A}\right)=-{\boldsymbol{u}}^{\top }{\mathbf{K}}_{\mathbf{uu}}^{+}\left(\boldsymbol{A}\right)\boldsymbol{u}+2{\boldsymbol{u}}^{\top }{\mathbf{K}}_{\mathbf{uz}}\left(\boldsymbol{A}\right)\boldsymbol{z}-{\boldsymbol{z}}^{\top }{\mathbf{K}}_{\mathbf{zz}}\left(\boldsymbol{A}\right)\boldsymbol{z}+2{\boldsymbol{f}}^{\top}\boldsymbol{u}+{\boldsymbol{\lambda}}^{\top}\boldsymbol{z}, $$

Where λ ∈ Λ and Λ = {(λ ₁, …, λ _n )^⊤|λ _i  ≥ 0, i = 1, …, n}.

Accordingly, the Karush-Kuhn-Tucker (K-K-T) conditions of $\underset{{\mathcal{U}}^{ad}\times {\mathcal{Z}}^{ad}}{ \max}\left(-2\Pi \left(\boldsymbol{u},\boldsymbol{z};\boldsymbol{A}\right)\right)$ can be written as:

$$ \left\{\begin{array}{c}\hfill {\mathbf{K}}_{\mathbf{uu}}^{+}\left(\boldsymbol{A}\right){\boldsymbol{u}}^{*}-{\mathbf{K}}_{\mathbf{uz}}\left(\boldsymbol{A}\right){\boldsymbol{z}}^{*}-\boldsymbol{f}=\mathbf{0},\hfill \\ {}\hfill {\mathbf{K}}_{\mathbf{zz}}\left(\boldsymbol{A}\right){\boldsymbol{z}}^{*}-{\mathbf{K}}_{\mathbf{uz}}^{\top}\left(\boldsymbol{A}\right){\boldsymbol{u}}^{*}+{\boldsymbol{\lambda}}^{*}=\mathbf{0},\hfill \\ {}\hfill \kern0.75em {\lambda}_i^{*}{z}_i^{*}=0,{\lambda}_i^{*}\ge 0,{z}_i^{*}\le 0,i=1,..,n.\hfill \end{array}\right. $$

(51)

On the other hand, the K-K-T conditions of the optimization problem $\underset{{\mathcal{U}}^{ad}\times {\mathcal{Z}}^{ad}}{ \max}\Pi \left(\boldsymbol{u},\boldsymbol{z};{\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right)$ are:

$$ \left\{\begin{array}{c}\hfill {\mathbf{K}}_{\mathbf{uu}}^{+}\left({\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right){\boldsymbol{u}}_R^{*}-{\mathbf{K}}_{\mathbf{uz}}\left({\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right){\boldsymbol{z}}_R^{*}-\boldsymbol{f}=\mathbf{0},\hfill \\ {}\hfill {\mathbf{K}}_{\mathbf{zz}}\left({\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right){\boldsymbol{z}}_R^{*}-{\mathbf{K}}_{\mathbf{uz}}^{\top}\left({\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right){\boldsymbol{u}}_R^{*}+{\boldsymbol{\lambda}}_R^{*}=\mathbf{0},\hfill \\ {}\hfill {\left({\lambda}_R^{*}\right)}_i{\left({z}_R^{*}\right)}_i=0,{\left({\lambda}_R^{*}\right)}_i\ge 0,{\left({z}_R^{*}\right)}_i\le 0,i=1,..,n.\hfill \end{array}\right. $$

(52)

Then with use of the following relations

$$ \left\{\kern0.75em \begin{array}{c}\hfill {\mathbf{K}}_{\mathbf{u}\mathbf{u}}^{+}\left({\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right)={\mathbf{R}}_{\mathbf{u}}^{\top }{\mathbf{K}}_{\mathbf{u}\mathbf{u}}^{+}\left(\boldsymbol{A}\right){\mathbf{R}}_{\mathbf{u}},\hfill \\ {}\hfill {\mathbf{K}}_{\mathbf{u}\mathbf{z}}\left({\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right)={\mathbf{R}}_{\mathbf{u}}^{\top }{\mathbf{K}}_{\mathbf{u}\mathbf{z}}\left(\boldsymbol{A}\right){\mathbf{R}}_{\mathbf{z}},\hfill \\ {}\hfill {\mathbf{K}}_{\mathbf{z}\mathbf{z}}\left({\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right)={\mathbf{R}}_{\mathbf{z}}^{\mathbf{\top}}{\mathbf{K}}_{\mathbf{z}\mathbf{z}}\left(\boldsymbol{A}\right){\mathbf{R}}_{\mathbf{z}},\hfill \end{array}\right. $$

(53)

(52) can be simplified as:

$$ \left\{\begin{array}{c}\hfill {\mathbf{R}}_{\mathbf{u}}^{\top }{\mathbf{K}}_{\mathbf{u}\mathbf{u}}^{+}\left(\boldsymbol{A}\right){\mathbf{R}}_{\mathbf{u}}{\boldsymbol{u}}_R^{*}-{\mathbf{R}}_{\mathbf{u}}^{\top }{\mathbf{K}}_{\mathbf{u}\mathbf{z}}\left(\boldsymbol{A}\right){\mathbf{R}}_{\mathbf{z}}{\boldsymbol{z}}_R^{*}-\boldsymbol{f}=\mathbf{0},\hfill \\ {}\hfill {\mathbf{R}}_{\mathbf{z}}^{\mathbf{\top}}{\mathbf{K}}_{\mathbf{z}\mathbf{z}}\left(\boldsymbol{A}\right){\mathbf{R}}_{\mathbf{z}}{\boldsymbol{z}}_R^{*}-{\mathbf{R}}_{\mathbf{z}}^{\top }{\mathbf{K}}_{\mathbf{u}\mathbf{z}}\left(\boldsymbol{A}\right){\mathbf{R}}_{\mathbf{u}}{\boldsymbol{u}}_R^{*}+{\boldsymbol{\lambda}}_R^{*}=\mathbf{0},\hfill \\ {}\hfill {\left({\lambda}_R^{*}\right)}_i{\left({z}_R^{*}\right)}_i=0,{\left({\lambda}_R^{*}\right)}_i\ge 0,{\left({z}_R^{*}\right)}_i\le 0,i=1,..,n.\hfill \end{array}\right. $$

(54)

Since the external force is symmetric, i.e., f = R ^⊤ _u f, it can be verified immediately that the ${\boldsymbol{u}}_R^{*}={\mathbf{R}}_{\mathbf{u}}^{\top }{\boldsymbol{u}}^{*}\in {\mathcal{U}}^{ad},\ {\boldsymbol{z}}_R^{*}={\mathbf{R}}_{\mathbf{z}}^{\top }{\boldsymbol{z}}^{*}\in {\mathcal{Z}}^{ad}$ and λ ^* _R  = R ^⊤ _z λ* ∈ Λ do satisfy (54) by taking (52) into consideration. Finally, since $\Pi \left(\boldsymbol{u},\boldsymbol{z};{\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right)$ is strictly convex in u and z jointly, then it can be concluded that $\left({\mathbf{\mathcal{R}}}^i{\boldsymbol{u}}^{*},{\mathbf{\mathcal{R}}}^i{\boldsymbol{z}}^{*}\right)\in \mathrm{A}\mathrm{r}\mathrm{g}{ \max}_{{\mathcal{U}}^{ad}\times {\mathcal{Z}}^{ad}}\left(-2\Pi \left(\boldsymbol{u},\boldsymbol{z};{\mathbf{\mathcal{R}}}^i\boldsymbol{A}\right)\right)$