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Time-dependent reliability-based design optimization considering aleatory and epistemic uncertainties

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Abstract

Directly solving time-dependent reliability-based design optimization (TRBDO) with aleatory and epistemic uncertainties is time-demanding, which limits its engineering application. By treating aleatory and epistemic uncertainties with probability and evidence variables respectively, an advanced decoupling method named sequential optimization and unified time-dependent reliability analysis (SOUTRA) is proposed in this work. By the SOUTRA, the original nested optimization process is solved by a sequence of unified time-dependent reliability analysis, updated reliability index target estimation and deterministic optimization. Only few numbers of the unified time-dependent reliability analysis are required to derive the optimum by the SOUTRA; thus, it is highly efficient. Furthermore, in order to construct the deterministic optimization, a new probability transformation method named focal element midpoint (FEM) is established to convert the evidence variable into a random one. FEM can avoid the issues of uniformity approach and equal areas method, and both are used in the existing probability transformation. Several numerical and engineering applications are introduced to illustrate the effectiveness of the proposed SOUTRA.

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Abbreviations

RBDO:

Reliability-based design optimization

TRBDO:

Time-dependent reliability-based design optimization

SOUTRA:

Sequential optimization and unified time-dependent reliability analysis

FEM:

Focal element midpoint

SORA:

Sequential optimization and reliability assessment

SLA:

Single-loop approach

KKT:

Karush-Kuhn-Tucker

MPTP:

Minimum performance target point

AMV:

Advanced mean value

AAMV:

Adjusted advanced mean value

MMV:

Modified mean value

HDMV:

Hybrid descent mean value

TIEM:

Time-invariant equivalent method

T-SORA:

Time-dependent sequential optimization and reliability assessment

T-SLA:

Time-dependent single-loop approach

FARM:

First-order approximate reliability analysis method

DNOM:

Double nested optimization method

FD:

Frame of discernment

BPA:

Basic probability assignment

PDF:

Probability density function

Bel:

Belief measure

Pl:

Plausibility measure

iTRPD:

Improved time-variant reliability analysis–based stochastic process discretization

SOUTRA_UA:

SOUTRA combined with uniformity approach

SOUTRA_EAM:

SOUTRA combined with equal areas method

α = {α1, α2}:

FD with two mutually exclusive basic elements

Ω(α):

Power set of FD

∅:

Empty set

m :

BPA

A:

Focal element

m(A):

BPA of focal element A

B:

Proposition

S:

Safety region

W :

Evidence input vector

\( \tilde{\mathbf{W}} \) :

Transformed random variable vector of W

W j :

jth evidence variable

\( {\tilde{W}}_j \) :

Transformed random variable of Wj

X :

Random design vector

P :

Random parameter vector

Z = [X, P]:

Random input variable vector

Y(t):

Stochastic process vector

d :

Deterministic design parameter vector

n W :

Size of the evidence input variables

\( {n}_{W_j} \) :

Number of focal element of Wj

n X :

Size of the random design vector

n P :

Size of the random parameter vector

n z :

Size of the random input variable vector

n Y :

Size of the stochastic process vector

n d :

Size of the deterministic design parameter vector

Ω(Wj):

Power set of FD of Wj

Ω(W):

Joint possible set

\( {\mathrm{A}}_j^{i_j} \) :

ijth focal element in Ω(Wj)

Al :

lth focal elements of Ω(W)

\( {\mathrm{A}}_j^{i_jL},{\mathrm{A}}_j^{i_jU} \) :

Lower bound and upper bound of \( {\mathrm{A}}_j^{i_j} \)

\( {\mathrm{A}}_j^{i_jS} \) :

Dispersion of the focal element \( {\mathrm{A}}_j^{i_j} \)

\( {\mathrm{A}}_j^{i_jM} \) :

Midpoint of the focal element \( {\mathrm{A}}_j^{i_j} \)

\( {m}_j\left({\mathrm{A}}_j^{i_j}\right) \) :

BPA of \( {\mathrm{A}}_j^{i_j} \)

m(Al):

BPA of Al

n JFM :

Number of joint focal element in Ω(W)

t ∈ [t0, te]:

Time parameter

g(Z, Y(t), W, t):

Performance function

gi(d, Z, Y(t), W, t):

ith probabilistic constraint

Gi(·):

ith constraint in standard normal space

R:

Time-dependent reliability

Bel(B):

Belief measure of B

Bel(S):

Unified time-dependent Bel

\( {\mathrm{Bel}}^{g_i}\left(\mathrm{S}\right) \) :

Unified time-dependent Bel of ith constraint

Bell(S):

Sub-belief corresponds to focal element Al

Belil(S):

Sub-belief corresponds to focal element Al and ith constraint

Pl(B):

Plausibility measure of B

Pl(S):

Unified time-dependent Pl

Pll(S):

Sub-plausibility corresponds to focal element Al

f(d, μX):

Objective function

\( {f}_{\mathrm{min}}^{(0)}\left(\mathbf{d},{\boldsymbol{\upmu}}_{\mathbf{X}}\right) \) :

Initial value of objective function

\( {f}_{\mathrm{min}}^{(k)}\left(\mathbf{d},{\boldsymbol{\upmu}}_{\mathbf{X}}\right) \) :

Objective value in kth iteration

\( {f}_{\mathrm{min}}^{final}\left(\mathbf{d},{\boldsymbol{\upmu}}_{\mathbf{X}}\right) \) :

Finial optimum of objective function

\( {f}_{{\tilde{W}}_j}^{UA}\left({\tilde{w}}_j\right) \) :

PDF based on uniformity approach

\( {f}_{{\tilde{W}}_j}^{EAM}\left({\tilde{w}}_j\right) \) :

PDF based on equal areas method

\( {f}_{{\tilde{W}}_j}^{FEM}\left({\tilde{w}}_j\right) \) :

PDF based on focal element midpoint

\( {\delta}_j\left({\tilde{w}}_j\right) \) :

Indicator function

Φ(·):

Standard normal cumulative distribution function

Φ−1(·):

Inverse function of Φ(·)

\( {\mathrm{R}}_i^{tar} \) :

Reliability target of ith probabilistic constraint

\( {\beta}_i^{tar} \) :

Equivalent reliability index target

\( {\beta}_i^{tar(k)} \) :

Updated reliability index target in kth iteration

βi(t0):

Reliability index of ith constraint function at t0

\( {\beta}_i^{(k)}\left({t}_0\right) \) :

Reliability index of ith constraint function at t0 in kth iteration

β iUT :

Equivalent unified time-dependent reliability index

βilmin(t∗):

Minimum instantaneous reliability index at t∗

dL, dU :

Lower bound and upper bound of d

μ X :

Mean vector of X

\( {\boldsymbol{\upmu}}_X^L,{\boldsymbol{\upmu}}_X^U \) :

Lower bound and upper bound of μX

μ P :

Mean vector of P

\( {\boldsymbol{\upmu}}_{\mathbf{Y}\left({t}_0\right)} \) :

Mean vector of Y(t0)

\( {\boldsymbol{\upmu}}_{\tilde{\mathbf{W}}} \) :

Mean vector of \( \tilde{\mathbf{W}} \)

d (0) :

Initial value of d

d (k) :

Estimated d in kth iteration

d final :

Finial optimum of d

\( {\boldsymbol{\upmu}}_{\mathbf{S}}^{(0)} \) :

Initial value of μS

\( {\boldsymbol{\upmu}}_{\mathbf{S}}^{\left(k-1\right)} \) :

Mean vector of inputs in (k − 1)th iteration

\( {\boldsymbol{\upmu}}_{\mathbf{X}}^{(k)} \) :

Estimated μX in kth iteration

\( {\boldsymbol{\upmu}}_{\mathbf{X}}^{final} \) :

Finial optimum of μX

\( {\mathbf{u}}_{i\mathrm{MPTP}}^{(0)} \) :

Initial value of MPTP

\( {\mathbf{u}}_{i\mathrm{MPTP}}^{\left(k-1\right)} \) :

MPTP in standard normal space

\( {\mathbf{s}}_{i\mathrm{MPTP}}^{\left(k-1\right)} \) :

MPTP in original probability space

\( \mathbf{U}={\left\{{\mathbf{U}}_{\mathbf{X}},{\mathbf{U}}_{\mathbf{P}},{\mathbf{U}}_{\mathbf{Y}\left({t}_0\right)},{\mathbf{U}}_{\tilde{\mathbf{W}}}\right\}}^{\mathrm{T}} \) :

Standard normal random variable vector

\( \mathbf{u}={\left\{{\mathbf{u}}_{\mathbf{X}},{\mathbf{u}}_{\mathbf{P}},{\mathbf{u}}_{\mathbf{Y}\left({t}_0\right)},{\mathbf{u}}_{\tilde{\mathbf{W}}}\right\}}^{\mathrm{T}} \) :

Realization of standard normal random vector

\( {\boldsymbol{\upmu}}_{\mathbf{S}}={\left\{{\boldsymbol{\upmu}}_{\mathbf{X}},{\boldsymbol{\upmu}}_{\mathbf{P}},{\boldsymbol{\upmu}}_{\mathbf{Y}\left({t}_0\right)},{\boldsymbol{\upmu}}_{\tilde{\mathbf{W}}}\right\}}^{\mathrm{T}} \) :

Mean vector of inputs

\( {\mathbf{M}}_i^{(k)} \) :

Shifting vector of kth iteration

\( {\varepsilon}_0^f \) :

Relative error threshold value

\( {\varepsilon}_f^{(k)} \) :

Relative error in kth iteration

Δ:

Ordinate of first support point

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Funding

This work was supported by the National Major Science and Technology Projects of China (Grant 2017-IV-0009-0046), the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant CX201931), and the China Scholarship Council (No. 201906290125).

Author information

Authors and Affiliations

  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, China

    Yan Shi & Zhenzhou Lu

  2. Energy Department, Politecnico di Milano, Milan, Italy

    Yan Shi, Jiayan Zhou & Enrico Zio

  3. College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing, China

    Jiayan Zhou

  4. MINES Pairs Tech, PSL Research University, CRC, Sophia Antipolis, France

    Enrico Zio

  5. Department of Nuclear Engineering, College of Engineering, Kyung Hee University, Seoul, South Korea

    Enrico Zio

Authors
  1. Yan Shi
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  2. Zhenzhou Lu
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  3. Jiayan Zhou
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  4. Enrico Zio
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Corresponding author

Correspondence to Zhenzhou Lu.

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The authors declare that they have no conflict of interest.

Replication of results

To further understand the proposed method for TRBDO with probability and evidence variables and replicate the solutions presented in this paper, the MATLAB codes of the proposed SOUTRA for the numerical example are provided as the supplementary material. Overall concepts and algorithms can be validated and extended through the numerical example.

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Responsible Editor: Yoojeong Noh

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Highlights

• A new method named SOUTRA is proposed to solve TRBDO with mixed uncertainties.

• A new probability transformation method called FEM is established.

• Original nested optimization is solved by a sequence of optimization process.

• Only few numbers of unified time-dependent reliability analysis are required.

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Shi, Y., Lu, Z., Zhou, J. et al. Time-dependent reliability-based design optimization considering aleatory and epistemic uncertainties. Struct Multidisc Optim 62, 2297–2321 (2020). https://doi.org/10.1007/s00158-020-02691-4

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  • DOI: https://doi.org/10.1007/s00158-020-02691-4

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