Symmetry properties in structural optimization: some extensions

Abstract

In the present paper, some extensions of the previous theoretical results about the symmetry properties of structural optimization problems are reported. It is found that generally the condition of convexity can be relaxed to quasi-convexity in order to guarantee the existence of symmetry global optima. Furthermore, some new results about the symmetry properties of robust and discrete structural optimization problems are also presented. Numerous concrete examples illustrate the claims made in the present work explicitly.

Appendix A

In this appendix, we shall show that $\omega _1^2 ({\boldsymbol {A};\boldsymbol {k}})$ in (24) is a quasi-concave function of A.

Proof

It is sufficient to show that the superlevel set of $\omega _1^2 ({\boldsymbol {A};\boldsymbol {k}})$, that is $L_{\beta } =\left \{ {\left . \boldsymbol {A} \right |\omega _1^2 ({\boldsymbol {A};\boldsymbol {k}})\ge \beta } \right \}$ is a convex set for every $\beta \ge 0$. Due to (24), it yields that,

$$\begin{array}{l} \omega_1^2 ({\boldsymbol{A};\boldsymbol{k}}) \ge \beta \Leftrightarrow \dfrac{\boldsymbol{u}^{\top} \left({{\textbf{K}}(\boldsymbol{A})+ \sum\nolimits_{i=1}^{\textrm{nbs}} k_i {\textbf{K}}_i^{\textrm{bs}}} \right)\boldsymbol{u}}{\boldsymbol{u}^{\top} \left( {{\textbf{M}}(\boldsymbol{A})+{\textbf{M}}_0} \right)\boldsymbol{u}}\ge \beta ,\\ \forall \,\boldsymbol{u}\in \textrm{U}_{{\textrm ad}} ,\quad \boldsymbol{u}\ne {{\textbf 0}} \end{array}$$

(A.1)

and $L_{\beta }$ can be expressed as

$$ {\begin{array}{*{20}c} {L_{\beta} =\left\{ {\boldsymbol{A}\left| {\dfrac{\boldsymbol{u}^{\top} \left( {{\textbf{K}}\left( \boldsymbol{A} \right)+\sum\nolimits_{i=1}^{\textrm{nbs}} k_i {{\textbf K}}_i^{\textrm{bs}}} \right)\boldsymbol{u}}{\boldsymbol{u}^{\top} \left( {{\textbf{M}}\left( \boldsymbol{A} \right)+{\textbf{M}}_0} \right)\boldsymbol{u}}\ge \beta ,\forall \,\boldsymbol{u}\in \textrm{U}_{\textrm{ad}}} \right.} \right\}} \\ \\ = \displaystyle\bigcap\limits_{\boldsymbol{u} \in U_{\textrm ad}}\left\{\boldsymbol{A}\left|\dfrac{\boldsymbol{u}^{\top}({\textbf K}(\boldsymbol{A})+ \sum_{i=1}^{\textrm nbs}k_i {\textbf K}_i^{\textrm bs})\boldsymbol{u}}{\boldsymbol{u}^{\top}({\textbf M}(\boldsymbol{A})+ {\textbf M}_0)\boldsymbol{u}}\right.\geq \beta\right\} \end{array}} $$

(A.2)

For every fixed $\boldsymbol {u}^{\ast } \ne {{\textbf 0}}\in \textrm {U}_{\textrm {ad}}$,

$$ \frac{( {\boldsymbol{u}^{\ast}} )^{\top} ( {{\textbf{K}}(\boldsymbol{A})+ \sum\nolimits_{i=1}^{\textrm{nbs}} k_i {{\textbf K}}_i^{\textrm{bs}}})\boldsymbol{u}^{\ast}} {( {\boldsymbol{u}^{\ast}} )^{\top} ({{\textbf{M}}( \boldsymbol{A} )+{\textbf{M}}_0})\boldsymbol{u}^{\ast}} \ge \beta $$

(A.3)

implies (note that ${\textbf {M}}( \boldsymbol {A} )\succ 0$, ${\textbf {M}}_0 \succcurlyeq 0$, respectively)

$$ ( {\boldsymbol{u}^{\ast}} )^{\top} {\textbf{K}}( \boldsymbol{A} )\boldsymbol{u}^{\ast} \ge \beta ( {\boldsymbol{u}^{\ast}} )^{\top} {\textbf{M}}( \boldsymbol{A} )\boldsymbol{u}^{\ast} +t^{\ast} , $$

(A.4)

where

$$ t^{\ast} =\beta ( {\boldsymbol{u}^{\ast}} )^{\top} \left( {{\textbf{M}}_0 -\sum\nolimits_{i=1}^{\textrm{nbs}} {k_i {\textbf{K}}_i^{\textrm{bs}}} } \right)\boldsymbol{u}^{\ast} $$

(A.5)

is a constant scalar if $k_{i}$, $i = 1$, …, nbs are fixed. If K(A) and M(A) are all linear functions of A, (A.4) in fact represents a half space, which is obvious a convex set. Since the intersection of an infinite number of convex sets is still a convex set, we have $L_{\beta }$ is convex, which concludes the proof (Fig. 13). □

Fig. 13

$L_{\beta }$ is the intersection of an infinite number of half spaces