System reliability-based design optimization with interval parameters by sequential moving asymptote method

Abstract

Reliability-based design optimization (RBDO) offers a powerful tool to deal with the structural design with heterogeneous interval parameters concurrently. However, it is time-consuming in the practical engineering design. Therefore, a novel sequential moving asymptote method (SMAM) is proposed to improve the computational efficiency for convex model in this study, in which the nested double-loop optimization problem is decoupled to a sequence of deterministic suboptimization problems based on the method of moving asymptotes. In addition, the sensitivity of reliability index is derived, so the finite difference for the nested optimization loop can be avoided to tremendously improve the computational efficiency. Then, the accuracy of the SMAM is proved based on the error analysis. Furthermore, the Kreisselmeier-Steinhauser (KS) function is used to assemble the multiple constraints to deal with the parallel and series RBDO problems. One benchmark mathematical example, three numerical examples, and one complex civil engineering example, i.e., tower crane, are tested to demonstrate the efficiency of the proposed method by comparison with other existing methods, and the results indicate that SMAM offers a general and effective tool for non-probabilistic reliability analysis and optimization.

Appendix

In the proposed SMAM, we develop the approximate sensitivity analysis method to substitute the FDM in order to promote the efficiency, and the corresponding computational error should be performed. By using the Taylor expansion, we can expand the performance function at the MCP, which is formulated as follows:

$$ g\left({\mathbf{d}}^m,{\mathbf{q}}^m\right)=g\left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)+{\nabla}_{\mathbf{q}}g{\left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)}^{\mathrm{T}}\left({\mathbf{q}}^m-{\mathbf{q}}^{\ast}\right)+O\left({\left\Vert \left({\mathbf{q}}^m-{\mathbf{q}}^{\ast}\right)\right\Vert}^2\right) $$

(39)

$$ {\nabla}_{\mathbf{d}}g\left({\mathbf{d}}^m,{\mathbf{q}}^m\right)={\nabla}_{\mathbf{d}}g\left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)+{\nabla}_{\mathbf{d}\mathbf{q}}^2g{\left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)}^{\mathrm{T}}\left({\mathbf{q}}^m-{\mathbf{q}}^{\ast}\right)+O\left({\left\Vert \left({\mathbf{q}}^m-{\mathbf{q}}^{\ast}\right)\right\Vert}^2\right) $$

(40)

where ∇²g denotes the second-order derivative of function g. Then, the sensitivity of approximate non-probabilistic reliability index with respect to design variables in SMAM can be written as follows:

$$ {\displaystyle \begin{array}{c}{\nabla}_{d^m}\hat{\eta}=\frac{\nabla_{d^m}g\left({d}^m,{q}^m\right)}{{\left\Vert {\nabla}_{q^m}g\left({d}^m,{q}^m\right)\right\Vert}_{\frac{p}{p-1}}}\\ {}=\left(\frac{\nabla_dg\left({d}^m,{q}^{\ast}\right)+{\nabla}_{dq}^2g{\left({d}^m,{q}^{\ast}\right)}^{\mathrm{T}}\left({q}^m-{q}^{\ast}\right)+O\left({\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert}^2\right)}{{\left\Vert {\nabla}_qg\left({d}^m,{q}^{\ast}\right)\right\Vert}_{\frac{p}{p-1}}}\right)\times \frac{{\left\Vert {\nabla}_qg\left({d}^m,{q}^{\ast}\right)\right\Vert}_{\frac{p}{p-1}}}{{\left\Vert {\nabla}_qg\left({d}^k,{q}^m\right)\right\Vert}_{\frac{p}{p-1}}}\end{array}} $$

(41)

In (41), the term $\frac{{\left\Vert {\nabla}_{\mathbf{q}}g\left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)\right\Vert}_{\frac{p}{p-1}}}{{\left\Vert {\nabla}_{\mathbf{q}}g\left({\mathbf{d}}^k,{\mathbf{q}}^m\right)\right\Vert}_{\frac{p}{p-1}}}$ can be expanded as follows:

$$ {\displaystyle \begin{array}{c}\frac{{\left\Vert {\nabla}_qg\left({d}^m,{q}^{\ast}\right)\right\Vert}_{\frac{p}{p-1}}}{{\left\Vert {\nabla}_qg\left({d}^m,{q}^m\right)\right\Vert}_{\frac{p}{p-1}}}={\left(\frac{\sum \limits_{i=1}^n{\left|{\nabla}_qg\left({q}_i^{\ast}\right)+{\nabla}_{qq}^2g{\left({q}_i\right)}^{\mathrm{T}}\left({q}_i^m-{q}_i^{\ast}\right)+O\left({\left\Vert \left({q}_i^m-{q}_i^{\ast}\right)\right\Vert}^2\right)\right|}^{\frac{p}{p-1}}}{\sum \limits_{i=1}^n{\left|{\nabla}_qg\left({q}_i^{\ast}\right)\right|}^{\frac{p}{p-1}}}\right)}^{\frac{p-1}{p}}\\ {}\le {\left(1+\frac{\sum \limits_{i=1}^{nq}{\left|{\nabla}_{qq}^2g{\left({q}_i\right)}^{\mathrm{T}}\left({q}_i^m-{q}_i^{\ast}\right)+O\left({\left\Vert \left({q}_i^m-{q}_i^{\ast}\right)\right\Vert}^2\right)\right|}^{\frac{p}{p-1}}}{\sum \limits_{i=1}^{nq}{\left|{\nabla}_qg\left({q}_i^{\ast}\right)\right|}^{\frac{p}{p-1}}}\right)}^{\frac{p-1}{p}}\\ {}=1+O\left(\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert \right)\end{array}} $$

(42)

Hence, ${\nabla}_{{\mathbf{d}}^m}\hat{\eta}$ satisfies:

$$ {\displaystyle \begin{array}{c}{\nabla}_d\eta =\frac{\nabla_dg\left({d}^m,{q}^{\ast}\right)}{{\left\Vert {\nabla}_qg\left({d}^m,{q}^{\ast}\right)\right\Vert}_{\frac{p}{p-1}}}\\ {}=\left(\frac{\nabla_dg\left({d}^m,{q}^{\ast}\right)+{\nabla}_{dq}^2g{\left({d}^m,{q}^{\ast}\right)}^{\mathrm{T}}\left({q}^m-{q}^{\ast}\right)+O\left({\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert}^2\right)}{{\left\Vert {\nabla}_qg\left({d}^m,{q}^{\ast}\right)\right\Vert}_{\frac{p}{p-1}}}\right)\times \left(1+O\left(\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert \right)\right)\\ {}={\nabla}_d\hat{\eta}+\frac{\nabla_{dq}^2g{\left({d}^m,{q}^{\ast}\right)}^{\mathrm{T}}\left({q}^m-{q}^{\ast}\right)}{{\left\Vert {\nabla}_qg\left({d}^m,{q}^{\ast}\right)\right\Vert}_{\frac{p}{p-1}}}+O\left({\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert}^2\right)\\ {}={\nabla}_d\hat{\eta}+O\left({q}^m-{q}^{\ast}\right)\end{array}} $$

(43)

For reliability index, it can be expanded by Taylor expansion.

$$ \eta \left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)=\hat{\eta}\left({\mathbf{d}}^m,{\mathbf{q}}^m\right)+{\left({\nabla}_{\mathbf{d}}\hat{\eta}\left({\mathbf{d}}^m,{\mathbf{q}}^m\right)\right)}^T\left(\mathbf{d}-{\mathbf{d}}^m\right)+O\left(\left\Vert {\left(\mathbf{d}-{\mathbf{d}}^m\right)}^2\right\Vert \right) $$

(44)

$$ \eta \left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)=\eta \left({\mathbf{d}}^m,{\mathbf{q}}^m\right)+{\left({\nabla}_{\mathbf{d}}\eta \left({\mathbf{d}}^m,{\mathbf{q}}^m\right)\right)}^T\left({\mathbf{d}}^m-{\mathbf{d}}^{\ast}\right)+O\left(\left\Vert {\left({\mathbf{d}}^m-{\mathbf{d}}^{\ast}\right)}^2\right\Vert \right) $$

(45)

$$ {\nabla}_{\mathbf{d}}\eta \left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)={\nabla}_{\mathbf{d}}\hat{\eta}\left({\mathbf{d}}^m,{\mathbf{q}}^m\right)+O\left(\left\Vert \left({\mathbf{q}}^m-{\mathbf{q}}^{\ast}\right)\right\Vert \right) $$

(46)

$$ {\nabla}_{\mathbf{d}}\hat{\eta}\left({\mathbf{d}}^m,{\mathbf{q}}^m\right)={\nabla}_{\mathbf{d}}\eta \left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)-O\left(\left\Vert \left({\mathbf{q}}^m-{\mathbf{q}}^{\ast}\right)\right\Vert \right) $$

(47)

$$ {\nabla}_{\mathbf{d}}\hat{\eta}\left({\mathbf{d}}^m,{\mathbf{q}}^m\right)={\nabla}_{\mathbf{d}}\eta \left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)+{\left({\nabla}_{\mathbf{d}\mathbf{q}}^2\hat{\eta}\left({\mathbf{d}}^m,{\mathbf{q}}^{\ast}\right)\right)}^T\left({\mathbf{q}}^{\ast }-{\mathbf{q}}^m\right)+O\left(\left\Vert {\left({\mathbf{q}}^m-{\mathbf{q}}^{\ast}\right)}^2\right\Vert \right) $$

(48)

Considering q^∗ is the optimal solution, which satisfies ∇_qη(d^m, q^∗) = 0, so the error of reliability constraint can be computed as follows:

$$ {\displaystyle \begin{array}{c}{\eta}_j-\left({\hat{\eta}}_j^m+{\left({\nabla}_{d^m}{\hat{\eta}}_j\right)}^T\left(d-{d}^m\right)\right)\\ {}={\eta}_j-\left({\hat{\eta}}_j^m+{\left({\nabla}_d\eta \left({d}^m,{q}^{\ast}\right)-O\left({q}^m-{q}^{\ast}\right)\right)}^T\left(d-{d}^m\right)\right)\\ {}={\eta}_j-{\hat{\eta}}_j^m-\left({\nabla}_d\eta \left({d}^m,{q}^{\ast}\right)\left(d-{d}^m\right)-O\left(\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert \left\Vert d-{d}^m\right\Vert \right)\right)\\ {}={\left({\nabla}_d\hat{\eta}\left({d}^m,{q}^m\right)-{\nabla}_d\eta \left({d}^m,{q}^{\ast}\right)\right)}^T\left(d-{d}^m\right)+O\left(\left\Vert {\left(d-{d}^m\right)}^2\right\Vert \right)+O\left(\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert \left\Vert d-{d}^m\right\Vert \right)\\ {}=O\left(\left\Vert {\left(d-{d}^m\right)}^2\right\Vert \right)+O\left(\left\Vert \left({q}^m-{q}^{\ast}\right)\right\Vert \left\Vert d-{d}^m\right\Vert \right)\end{array}} $$

(49)