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Structural optimization oriented time-dependent reliability methodology under static and dynamic uncertainties

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Abstract

Uncertainty with characteristics of time-dependency, multi-sources and small-samples extensively exists in the whole process of structural design. Associated with frequent occurrences of material aging, load varying, damage accumulating, traditional reliability-based design optimization (RBDO) approaches by combination of the static assumption and the probability theory will be no longer applicable when dealing with the design problems for lifecycle structural models. In view of this, a new non-probabilistic time-dependent RBDO method under the mixture of time-invariant and time-variant uncertainties is investigated in this paper. Enlightened by the first-passage concept, the hybrid reliability index is firstly defined, and its solution implementation relies on the technologies of regulation and the interval mathematics. In order to guarantee the stability and efficiency of the optimization procedure, the improved ant colony algorithm (ACA) is then introduced. Moreover, by comparisons of the models of the safety factor-based design as well as the instantaneous RBDO design, the physical means of the proposed optimization policy are further discussed. Two numerical examples are eventually presented to demonstrate the validity and reasonability of the developed methodology.

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Acknowledgements

The authors would like to thank the National Nature Science Foundation of China (No. 11372025, 11432002, 11602012), the 111 Project (No. B07009), the Defense Industrial Technology Development Program (No. JCKY2016601B001, JCKY2016205C001), and the China Postdoctoral Science Foundation (No. 2016 M591038) for the financial supports. Besides, the authors wish to express their many thanks to the reviewers for their useful and constructive comments.

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Authors and Affiliations

  1. Institute of Solid Mechanics, Beihang University, Beijing, 100083, China

    Lei Wang, Xiaojun Wang & Zhiping Qiu

  2. China Academy of Launch Vehicle Technology R&D Center, Beijing, 100076, China

    Di Wu

  3. Faculty of Mechanical Engineering & Mechanics, Ningbo University, Ningbo, Zhejiang, 315211, China

    Menghui Xu

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  1. Lei Wang
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  2. Xiaojun Wang
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Correspondence to Lei Wang.

Appendix A

Appendix A

  1. (a)

    Firstly, the common approach for determination of the smallest parametric set (i.e., one hyper-rectangular ‘box’ in essence) containing multi-dimensional sample data is discussed. In consideration of the case that the uncertain parameters α l (l = 1, 2, …, s) constitute an s-dimensional parametric space, and we have limited information corresponding to these parameters, manifested by a set containing S sample points, namely, \( {\alpha}_l^{(S)}=\left\{{a}_l(1),{\alpha}_l(2),\dots, {\alpha}_l(S)\right\} \). Thus, the expression of the transformation matrix reads

$$ \mathbf{T}\left(\boldsymbol{\uptheta} \right)=\left({\boldsymbol{\updelta}}_1,{\boldsymbol{\updelta}}_2,\cdots, {\boldsymbol{\updelta}}_{\mathbf{s}}\right) $$
(30)

where θ = (θ l ) (l = 1, 2, …, s − 1) denotes the vector of rotation angles related to the original coordinate space, and

$$ {\boldsymbol{\updelta}}_{\mathbf{l}}=\left\{\begin{array}{c}\hfill {\mathbf{0}}_{\mathbf{l}-2}\hfill \\ {}\hfill {\tilde{\boldsymbol{\updelta}}}_{\mathbf{l}}\hfill \end{array}\right\} $$
(31)

where 0 l − 2 means a column vector with l − 2 zero items, \( {\tilde{\boldsymbol{\updelta}}}_{\mathbf{l}} \) is deduced by

$$ {\tilde{\boldsymbol{\updelta}}}_{\mathbf{l}}=\left(\begin{array}{c}\hfill -\sin {\theta}_{l-1}\hfill \\ {}\hfill \cos {\theta}_{l-1}\cos {\theta}_l\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill \cos {\theta}_{l-1}\sin {\theta}_l\cdots \sin {\theta}_{s-2}\cos {\theta}_{s-1}\hfill \\ {}\hfill \cos {\theta}_{l-1}\sin {\theta}_l\cdots \sin {\theta}_{s-2}\sin {\theta}_{s-1}\hfill \end{array}\right) $$
(32)

By utilizing of the transformation matrix T(θ), each point of the original sample set \( {\alpha}_l^{(S)} \) will be modified and be further rewritten as \( {\beta}_l^{(S)} \) (from α-space to β-space). In order to determine the smallest interval set to envelope full samples, the boundary laws of the s-dimensional ‘box’ should be examined by

$$ \left|\boldsymbol{\upbeta} -{\boldsymbol{\upbeta}}^{\mathbf{c}}\right|\le {\boldsymbol{\upbeta}}^{\mathbf{r}} $$
(33)

where the center vector \( {\boldsymbol{\upbeta}}^{\mathbf{c}}={\left({\beta}_1^c,{\beta}_2^c,\cdots, {\beta}_s^c\right)}^T \) and the semi-axis vector \( {\boldsymbol{\upbeta}}^{\mathbf{r}}={\left({\beta}_1^r,{\beta}_2^r,\cdots, {\beta}_s^r\right)}^T \). The components \( {\beta}_l^c \) and \( {\beta}_l^r \) are given by

$$ {\beta}_l^c=\frac{1}{2}\left(\underset{S}{\max}\left({\beta}_l^{(S)}\right)+\underset{S}{\min}\left({\beta}_l^{(S)}\right)\right)\kern0.5em \mathrm{and}\kern0.5em {\beta}_l^r=\frac{1}{2}\left(\underset{S}{\max}\left({\beta}_l^{(S)}\right)-\underset{S}{\min}\left({\beta}_l^{(S)}\right)\right) $$
(34)

Accordingly, the hyper-volume of the ‘box’ as enclosed in (33) is obtained by

$$ {V}_{hyper}=\prod_{l=1}^s{\left(2{\beta}_l^r\right)}^s $$
(35)

which is a function of the rotation angles θ l . Consequently, it can be defined that the best interval set or the hyper-rectangle among all possible boxes is the one which contains all given sample points and processes the minimum volume, i.e.,

$$ {V}_{hyper}^{\ast }=\underset{{\boldsymbol{\uptheta}}^{\ast}}{\min }{V}_{hyper}\left({\theta}_1,{\theta}_2,\dots, {\theta}_{s-1}\right) $$
(36)

Certainly, once the minimum \( {V}_{hyper}^{\ast } \) is gained, the optimal design parameters \( {\theta}_l^{\ast } \) may actually reflect the correlationship between the uncertain variables α l and α l + 1.

  1. (b)

    In terms of a specific interval process Y j (t), the correlation properties between Y j (t 1) and Y j (t 2) with any instant times t 1 and t 2 are what we really concerned. Hence, the above optimization procedure can be simplified and be executed by following steps:

  1. i)

    Collect all the sample curves \( {y}_j^{(S)}(t)=\left\{{y}_j\left(t,1\right),{y}_j\Big(t,2\Big),\dots, {y}_j\Big(t,S\Big)\right\} \) and conduct the time-discretization operation with small increment Δt as

$$ {y}_j^{(S)}\left({t}_1\right)=\left\{{y}_j\left({t}_1,1\right),{y}_j\Big({t}_1,2\Big),\dots, {y}_j\Big({t}_1,S\Big)\right\} $$
(37)

and

$$ {y}_j^{(S)}\left({t}_2\right)=\left\{{y}_j\left({t}_2,1\right),{y}_j\Big({t}_2,2\Big),\dots, {y}_j\Big({t}_2,S\Big)\right\} $$
(38)

where t 1 = j 1 Δt and t 2 = j 2 Δt (j 1, j 2 = 1, 2, …).

  1. ii)

    Construct a two-dimensional sample space (Y j (t 1)-Y j (t 2)) originating from (37) and (38), and accomplish the aforementioned optimization solution to get the smallest rotary rectangular domain (as illustrated in Fig. 1), with rotation angel \( {\theta}_j^{\ast}\left({t}_1,{t}_2\right) \) and the two semi-axes \( {\beta}_j^r\left({t}_1\right) \) and \( {\beta}_j^r\left({t}_2\right) \).

  2. iii)

    By virtue of the characteristic parameters of the smallest interval set, the covariance function \( {Cov}_{Y_j}\left({t}_1,{t}_2\right) \) can be given by

$$ {Cov}_{Y_j}\left({t}_1,{t}_2\right)=\frac{{\left({\beta}_j^r\left({t}_1\right)\right)}^2-{\left({\beta}_j^r\left({t}_2\right)\right)}^2}{3}\cdot \sin {\theta}_j^{\ast}\left({t}_1,{t}_2\right)\cdot \cos {\theta}_j^{\ast}\left({t}_1,{t}_2\right) $$
(39)

Then, the auto-correlation coefficient function ρ j (t 1, t 2) arrives at

$$ {\rho}_j\left({t}_1,{t}_2\right)=\frac{Cov_{Y_j}\left({t}_1,{t}_2\right)}{Y_j^r\left({t}_1\right)\cdot {Y}_j^r\left({t}_2\right)} $$
(40)
  1. iv)

    By traversing all values of counting indexes j 1 and j 2, properties of autocorrelation of the interval process vector Y j (t) can be eventually confirmed.

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Wang, L., Wang, X., Wu, D. et al. Structural optimization oriented time-dependent reliability methodology under static and dynamic uncertainties. Struct Multidisc Optim 57, 1533–1551 (2018). https://doi.org/10.1007/s00158-017-1824-z

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