A non-probabilistic reliability-based topology optimization (NRBTO) method of continuum structures with convex uncertainties

Abstract

This paper develops a non-probabilistic reliability-based topology optimization (NRBTO) framework for continuum structures under multi-dimensional convex uncertainties. Combined with the solid isotropic material with penalization (SIMP) model and the set-theoretical convex method, the uncertainty quantification (UQ) analysis is firstly conducted to obtain mathematical approximations and boundary laws of considered displacement responses. By normalization treatment of the limit-state function, a new quantified measure of the non-probabilistic reliability is then defined and further deduced by the principle of the hyper-volume ratio. For circumventing optimization difficulties arising from large-scale design variables, the adjoint vector scheme for sensitivity analysis of the reliability index with respect to design variables are discussed as well. Numerical applications eventually illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

Appendices

Appendix 1

The mean value ${b}_j^c$ and variances $D\left({b}_j^I\right)$ of the convex model are defined as

$$ {b}_j^c=\left(\frac{b_j^U+{b}_j^L}{2}\right),\kern0.6em j=1,2,\dots m $$

(40)

$$ D\left({b}_j^I\right)={\left({b}_j^r\right)}^2={\left(\frac{b_j^U-{b}_j^L}{2}\right)}^2,\kern0.8000001em j=1,2,\dots m $$

(41)

In the above convex model, the mean value vector b^c is just the same as the central point of the ellipse as show in Fig. 3. The correlation coefficient is defined as the covariance for two uncertain parameters

$$ {\rho}_{b_i{b}_k}=\frac{\tan \left(\theta \right)}{1-{\tan}^2\left(\theta \right)}\left(D\left({b}_i^I\right)-D\left({b}_k^I\right)\right),\kern0.6em i=1,2,\dots m,k=1,2,\dots m $$

(42)

${\rho}_{b_i{b}_k}$ represents the degree of linear correlation of the uncertain parameters b_i and b_k. Based on the covariance and correlation coefficient, the correlation matrix is defined as

$$ {\boldsymbol{\uprho}}_{\mathbf{b}}=\left[\begin{array}{cccc}{\rho}_{b_1{b}_1}& {\rho}_{b_1b2}& \cdots & {\rho}_{b_1{b}_m}\\ {}{\rho}_{b_2{b}_1}& {\rho}_{b_2{b}_2}& \cdots & {\rho}_{b_2{b}_m}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}{\rho}_{b_m{b}_1}& {\rho}_{b_m{b}_2}& \dots & {\rho}_{b_m{b}_m}\end{array}\right] $$

(43)

The following expression

$$ {\displaystyle \begin{array}{l}{\left(\boldsymbol{b}-{\boldsymbol{b}}^{\mathbf{c}}\right)}^T{\left({\theta}^2{\boldsymbol{\rho}}_{\boldsymbol{b}}\right)}^{-1}\left(\boldsymbol{b}-{\boldsymbol{b}}^{\mathbf{c}}\right)\\ {}={\left(\boldsymbol{b}-{\boldsymbol{b}}^{\mathbf{c}}\right)}^T{\left({\theta}^2\left[\begin{array}{cccc}{\rho}_{b_1{b}_1}& {\rho}_{b_1b2}& \cdots & {\rho}_{b_1{b}_m}\\ {}{\rho}_{b_2{b}_1}& {\rho}_{b_2{b}_2}& \cdots & {\rho}_{b_2{b}_m}\\ {}\vdots & \vdots & \ddots & \vdots \\ {}{\rho}_{b_m{b}_1}& {\rho}_{b_m{b}_2}& \dots & {\rho}_{b_m{b}_m}\end{array}\right]\right)}^{-1}\left(\boldsymbol{b}-{\boldsymbol{b}}^{\mathbf{c}}\right)\le {\theta}^2\end{array}} $$

(44)

determines an ellipsoidal convex model. Then it yields

$$ {\left(\mathbf{b}-{\mathbf{b}}^{\mathbf{c}}\right)}^T\mathbf{W}\left({\mathbf{b}}^{\mathbf{r}},{\boldsymbol{\uprho}}_{\mathbf{b}}\right)\left(\mathbf{b}-{\mathbf{b}}^{\mathbf{c}}\right)\le {\theta}^2 $$

(45)

where $\mathbf{W}\left({\mathbf{b}}^{\mathbf{r}},{\boldsymbol{\uprho}}_{\mathbf{b}}\right)={\left({\theta}^2\left[\begin{array}{cccc}D\left({b}_1^I\right)& Cov\left({b}_1,{b}_2\right)& \cdots & Cov\left({b}_1,{b}_m\right)\\ {} Cov\left({b}_2,{b}_1\right)& D\left({b}_2^I\right)& \cdots & Cov\left({b}_2,{b}_m\right)\\ {}\vdots & \vdots & \ddots & \vdots \\ {} Cov\left({b}_m,{b}_1\right)& Cov\left({b}_m,{b}_2\right)& \dots & D\left({b}_m^I\right)\end{array}\right]\right)}^{-1}$.

Appendix 2

In order to find the reliability index R_i by (16), HV_safe and HV_total are need to solve firstly. For m = 2, as shows in Fig. 4a, it yields

$$ {R}_i={\left.\frac{S_{safe}}{S_{total}}\right|}_i $$

(46)

S_failure can be calculated by making use of the sector area formula and the triangle area formula, namely,

$$ {S}_{failure}={S}_{\sec tor}-{S}_{triangle}=\frac{2\operatorname{arccos}{d}_i}{2}-\frac{2\sqrt{1-{d}_i^2}\times {d}_i}{2}=\operatorname{arccos}{d}_i-{d}_i\sqrt{1-{d}_i^2} $$

(47)

then S_safe and the reliability index R_i can be obtained

$$ {S}_{safe}={S}_{total}-{S}_{failure}=\pi -\left(\operatorname{arccos}{d}_i-{d}_i\sqrt{1-{d}_i^2}\right) $$

(48)

and

$$ {R}_i={\left.\frac{S_{safe}}{S_{total}}\right|}_i=\frac{\pi -\left(\operatorname{arccos}{d}_i-{d}_i\sqrt{1-{d}_i^2}\right)}{\pi }=1-\frac{\operatorname{arccos}{d}_i-{d}_i\sqrt{1-{d}_i^2}}{\pi } $$

(49)

For m = 3, as shows in Fig. 4b, it yields

$$ {R}_i={\left.\frac{V_{safe}}{V_{total}}\right|}_i $$

(50)

similarly, V_failure can be calculated by making use of the spherical cap volume formula, namely,

$$ {V}_{failure}={V}_{cap}=\pi {\left(1-{d}_i\right)}^2\left(1-\frac{1-{d}_i}{3}\right) $$

(51)

then V_safe and the reliability index R_i can be obtained

$$ {V}_{safe}={V}_{total}-{V}_{failure}=\frac{4}{3}\pi \times {1}^3-\pi {\left(1-{d}_i\right)}^2\left(1-\frac{1-{d}_i}{3}\right)=\frac{4\pi -\pi {\left(1-{d}_i\right)}^2\left(2+{d}_i\right)}{3} $$

(52)

and

$$ {R}_i={\left.\frac{V_{safe}}{V_{total}}\right|}_i=\frac{\frac{1}{3}\left[4\pi -\pi {\left(1-{d}_i\right)}^2\left(2+{d}_i\right)\right]}{\frac{4}{3}\pi \times {1}^3}=1-\frac{2-3{d}_i+{d_i}^3}{4} $$

(53)

Combining (49) and (53), it yields

$$ {R}_i={\left.\frac{HV_{safe}}{HV_{total}}\right|}_i=\left\{\begin{array}{l}1-\frac{\operatorname{arccos}{d}_i-{d}_i\sqrt{1-{d}_i^2}}{\pi },m=2\\ {}1-\frac{2-3{d}_i+{d}_i^3}{4},m=3\end{array}\right. $$

(54)

this is exactly (17).

For multidimensional cases (m > 3), considering the complicated calculation process and numerous formulas, (18) is no longer derived in detail and the results in Ref (Chen 2008; Jiang et al. 2013) are directly listed.