Reliability-based design optimization under mixture of random, interval and convex uncertainties

Abstract

In view of the significant influences of multi-source uncertainties on structural safety, which generally exist in practical engineering (such as the dispersion of material, the uncertainty of external load and the error of processing technology), more academic research and engineering applications had paid attention to uncertainty in recent years. However, due to the complexity of the structural problems, there may be multiple uncertain parameters. Traditional methods of optimal design by single-source uncertainty model, particularly the one derived from probability theory, may no longer be feasible. This paper investigates a new formulation and numerical solution of reliability-based design optimization (RBDO) of structures exhibiting random and uncertain-but-bounded (interval and convex) mixed uncertainties. Combined with the non-probabilistic set-theory convex model and the classical probabilistic approach, the mathematical definition of hybrid reliability is firstly presented for a quantified measure of the safety margin. The reliability-based optimization incorporating such mixed reliability constraints is then formulated. The PSO algorithm is employed to improve the convergence and the stability in seeking the optimal global solution. Additionally, by introducing the general concept of the safety factor, the compatibility between the proposed hybrid RBDO technique and the safety factor-based model is further discussed. By virtue of the above two methods, two numerical examples of typical components (the cantilever structure and the truss structure) as well as one complex engineering example (the hypersonic wing structure) are performed, subjected to the strength or stiffness criteria. The accuracy and effectiveness of the present method are then demonstrated.

Appendix

Case (a): As is shown in Fig. 5a, when $Y_1$ is assigned to the maximum $\overline{Y_1 }$ or minimum $\underline{Y_1 }$, there is no intersection between the limit-state and the feasible domain, and the distance from the origin to the failure surface would be greater than unity. Hence, we have

$$\begin{aligned} \frac{M_Z +f({\varvec{x}})-b_1 \underline{Y_1 }}{\sqrt{(c_1 Z_1^r )^{2}+(c_2 Z_2^r )^{2}}}\ge \frac{M_Z +f({\varvec{x}})-b_1 \overline{Y_1 } }{\sqrt{(c_1 Z_1^r )^{2}+(c_2 Z_2^r )^{2}}}=\frac{M_Z +f({\varvec{x}})-b_1 \overline{Y_1 } }{N_Z }=D_1 ({\varvec{x}},\overline{Y_1 } )\ge 1 \end{aligned}$$

(37)

Then, the span of $f({\varvec{x}})$ is

$$\begin{aligned} f({\varvec{x}})\in \left[ {\left. {N_Z +b_1 \overline{Y_1 } -M_Z, +\infty } \right) } \right. \end{aligned}$$

(38)

Obviously, once the given ${\varvec{x}}$ satisfies (38), it is definitely safe with a reliability index $\eta \left( {{\varvec{d}},{\varvec{x}}} \right) =1$.

Case (b): As is shown in Fig. 5b, when $Y_1 =\underline{Y_1 }$, there is no intersection between the limit-state and the feasible domain of uncertain variables; when $Y_1 =\overline{Y_1 }$, the interference occurs, and the distance $D_1 ({\varvec{x}},\overline{Y_1 } )$ would be within an interval $[0,\;1]$, namely

$$\begin{aligned} D_1 ({\varvec{x}},\underline{Y_1 })\frac{M_Z +f({\varvec{x}})-b_1 \underline{Y_1 }}{N_Z }\ge 1, \quad \text {and }\;0\le D_1 ({\varvec{x}},\overline{Y_1 } )=\frac{M_Z +f({\varvec{x}})-b_1 \overline{Y_1 } }{N_Z }\le 1 \end{aligned}$$

(39)

It is apparent that

$$\begin{aligned} f({\varvec{x}})\in \left[ {\max (N_Z +b_1 \underline{Y_1 }-M_Z, b_1 \overline{Y_1 } -M_Z ), N_Z +b_1 \overline{Y_1 } -M_Z } \right] \end{aligned}$$

(40)

In this case, the reliability index $\eta \left( {{\varvec{d}},{\varvec{x}}} \right)$ is defined as the volume ratio between the transparent region and the total region. So, it reads

$$\begin{aligned} \eta \left( {{\varvec{d}},{\varvec{x}}} \right) =\frac{V_\mathrm{safe} }{V_\mathrm{total} }=1-\frac{V_\mathrm{failure} }{V_\mathrm{total} } \end{aligned}$$

(41)

where $V_\mathrm{total} =(\overline{Y_1 } -\underline{Y_1 })\pi$. By virtue of the integral operation, $V_\mathrm{failure}$ can be expressed as

$$\begin{aligned} V_\mathrm{failure} ={\int }_{Y_1^\alpha }^{\overline{Y_1 } } {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 \end{aligned}$$

(42)

where $S_1 ({\varvec{x}},Y_1 )$ stands for the area of the bow-shaped interference region which implies failure (its analytical expression has been deduced by our previous work in Ref. [48]); $Y_1^\alpha$ denotes the value of $Y_1$ when it satisfies $D_1 ({\varvec{x}},Y_1 )=1$, i.e.,

$$\begin{aligned} \frac{M_Z +f({\varvec{x}})-b_1 Y_1^\alpha }{N_Z}=1\Rightarrow Y_1^\alpha =\frac{f({\varvec{x}})-N_Z +M_Z }{b_1} \end{aligned}$$

(43)

Therefore, $\eta \left( {{\varvec{d}},{\varvec{x}}} \right)$ can be expanded as

$$\begin{aligned} \eta \left( {{\varvec{d}},{\varvec{x}}} \right) =1-\frac{V_\mathrm{failure} }{V_\mathrm{total} }=1-\frac{{\int }_{\frac{f({\varvec{x}})-N_Z +M_Z }{b_1 }}^{\overline{Y_1 } } {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi } \end{aligned}$$

(44)

Case (c): As is shown in Fig. 5c, differ from case (b), the distance from the origin to the failure surface is $D_2 ({\varvec{x}},\overline{Y_1 } )\in \left[ {0,\;1} \right]$ when $Y_1 =\overline{Y_1 }$. Hence, $V_\mathrm{failure}$ can be represented as

$$\begin{aligned} V_\mathrm{failure} ={\int }_{Y_1^\alpha }^{Y_1^\beta } {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 +{\int }_{Y_1^\beta }^{\overline{Y_1 } } {\left( {\pi -S_2 ({\varvec{x}},Y_1 )} \right) } \mathrm{d}Y_1 \end{aligned}$$

(45)

where $Y_1^\beta$ denotes the value of $Y_1$ when it satisfies $D_2 ({\varvec{x}},Y_1 )=0$, i.e.,

$$\begin{aligned} -\frac{M_Z +f({\varvec{x}})-b_1 Y_1^\beta }{N_Z }=0\Rightarrow Y_1^\beta =\frac{f({\varvec{x}})+M_Z }{b_1 } \end{aligned}$$

(46)

Substituting (45), (46) into (41), we obtain

$$\begin{aligned} \eta \left( {{\varvec{d}},{\varvec{x}}} \right)= & {} 1-\frac{V_\mathrm{failure} }{V_\mathrm{total} }=1-\frac{{\int }_{\frac{f({\varvec{x}})-N_Z +M_Z }{b_1 }}^{\frac{f({\varvec{x}})+M_Z }{b_1 }} {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 +{\int }_{\frac{f({\varvec{x}})+M_Z }{b_1 }}^{\overline{Y_1 } } {\left( {\pi -S_2 ({\varvec{x}},Y_1 )} \right) } \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi } \nonumber \\= & {} 1-\frac{\left( {\overline{Y_1 } -\frac{f({\varvec{x}})+M_Z }{b_1 }} \right) \pi }{(\overline{Y_1 } -\underline{Y_1 })\pi }-\frac{{\int }_{\frac{f({\varvec{x}})-N_Z +M_Z }{b_1 }}^{\frac{f({\varvec{x}})+M_Z }{b_1 }} {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi }+\frac{{\int }_{\frac{f({\varvec{x}})+M_Z }{b_1 }}^{\overline{Y_1 } } {S_2 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi } \nonumber \\= & {} \frac{f({\varvec{x}})-b_1 \overline{Y_1 } +M_Z }{(\overline{Y_1 } -\underline{Y_1 })b_1 }-\frac{{\int }_{\frac{f({\varvec{x}})-N_Z +M_Z }{b_1 }}^{\frac{f({\varvec{x}})+M_Z }{b_1 }} {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi }+\frac{{\int }_{\frac{f({\varvec{x}})+M_Z }{b_1 }}^{\overline{Y_1 } } {S_2 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi } \end{aligned}$$

(47)

Case (d): As is shown in Fig. 5d, similar to the cases discussed before, but the distance $D_2 ({\varvec{x}},\overline{Y_1})$ greater than unity, the expression of $V_\mathrm{failure}$ is changed as

$$\begin{aligned} V_\mathrm{failure} ={\int }_{Y_1^\alpha }^{Y_1^\beta } {S_1 ({\varvec{x}},Y_1 )} dY_1 +{\int }_{Y_1^\beta }^{Y_1^\gamma } {\left( {\pi -S_2 ({\varvec{x}},Y_1 )} \right) } dY_1 +\left( {\overline{Y_1 } -Y_1^\gamma } \right) \pi \end{aligned}$$

(48)

where $Y_1^\gamma$ denotes the value of $Y_1$ when it satisfies $D_2 ({\varvec{x}},Y_1 )=1$, i.e.,

$$\begin{aligned} -\frac{M_Z +f({\varvec{x}})-b_1 Y_1^\gamma }{N_Z }=1\Rightarrow Y_1^\gamma =\frac{f({\varvec{x}})+N_Z +M_Z }{b_1 } \end{aligned}$$

(49)

Substituting (48), (49) into (41), we obtain

$$\begin{aligned} \eta \left( {{\varvec{d}},{\varvec{x}}} \right)= & {} 1-\frac{V_\mathrm{failure} }{V_\mathrm{total} }\nonumber \\= & {} 1-\frac{{\int }_{\frac{f({\varvec{x}})-N_Z +M_Z }{b_1 }}^{\frac{f({\varvec{x}})+M_Z }{b_1 }} {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 +{\int }_{\frac{f({\varvec{x}})+M_Z }{b_1 }}^{\frac{f({\varvec{x}})+N_Z +M_Z }{b_1 }} {\left( {\pi -S_2 ({\varvec{x}},Y_1 )} \right) } \mathrm{d}Y_1 +\left( {\overline{Y_1 } -\frac{f({\varvec{x}})+N_Z +M_Z }{b_1 }} \right) \pi }{(\overline{Y_1 } -\underline{Y_1 })\pi } \nonumber \\= & {} 1-\frac{\left( {\frac{f({\varvec{x}})+N_Z +M_Z }{b_1 }-\frac{f({\varvec{x}})+M_Z }{b_1 }} \right) \pi }{(\overline{Y_1 } -\underline{Y_1 })\pi }-\frac{\left( {\overline{Y_1 } -\frac{f({\varvec{x}})+N_Z +M_Z }{b_1 }} \right) \pi }{(\overline{Y_1 } -\underline{Y_1 })\pi } \nonumber \\&\quad -\frac{{\int }_{\frac{f({\varvec{x}})-N_Z +M_Z }{b_1 }}^{\frac{f({\varvec{x}})+M_Z }{b_1 }} {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi }+\frac{{\int }_{\frac{f({\varvec{x}})+M_Z }{b_1 }}^{\frac{f({\varvec{x}})+N_Z +M_Z }{b_1 }} {S_2 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi } \nonumber \\= & {} \frac{f({\varvec{x}})-b_1 \underline{Y_1 }+M_Z }{(\overline{Y_1 } -\underline{Y_1 })b_1 }-\frac{{\int }_{\frac{f({\varvec{x}})-N_Z +M_Z }{b_1 }}^{\frac{f({\varvec{x}})+M_Z }{b_1 }} {S_1 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi }+\frac{{\int }_{\frac{f({\varvec{x}})+M_Z }{b_1 }}^{\frac{f({\varvec{x}})+N_Z +M_Z }{b_1 }} {S_2 ({\varvec{x}},Y_1 )} \mathrm{d}Y_1 }{(\overline{Y_1 } -\underline{Y_1 })\pi } \end{aligned}$$

(50)