Structural optimization under dynamic reliability constraints utilizing probability density evolution method and metamodels in augmented input space

Abstract

An effective method for solving a class of dynamic-reliability-based design optimization (DRBDO) problems is proposed in the present paper. Failure probability functions and their sensitivities with respect to the design variables are estimated in the framework of the probability density evolution method (PDEM). In particular, a PDEM-based metamodel-refined approach is defined in an augmented input space to improve the efficiency of failure probability estimations and sensitivity analyses. Moving trust regions are imposed on the augmented input space to ensure the accuracy of the metamodel. To solve the optimization problems, the PDEM-based metamodel-refined approach is embedded into a feasible direction interior point scheme. In this scheme, a feasible search direction is first obtained by solving the perturbed Karush–Kuhn–Tucker (KKT) conditions. Then, a line search technique, which is consistent with the PDEM-based metamodel-refined approach, is employed to speed up the convergence of the optimization process. The results of the numerical examples indicate that the proposed method is a competitive choice for solving a class of DRBDO problems with a small number of reliability and structural analyses.

Appendices

Appendix A

The numerical solution of the GDEE includes the following four steps.

Step A.1: Discretize the probability-assigned space $\Omega _{{\mathbf{\Theta}}}$ with a representative point set selected by an optimal point selection strategy, e.g., the GF-discrepancy minimization-based method (Chen and Zhang 2013; Chen et al. 2016; Chen and Chan 2019). Denote the representative point set by ${\mathcal{P}}_{{{\text{sel}}}} = \left\{ {\left. {\left( {{\boldsymbol{\theta}}_{s} ,P_{s} } \right)} \right|s = 1,2, \cdots ,n_{{{\text{sel}}}} } \right\}$, where $n_{{{\text{sel}}}}$ is the total number of representative points, ${\boldsymbol{\theta}}_{s} = \left( {\theta_{s1} ,\theta_{s2} , \cdots ,\theta_{{sn_{{\text{r}}} }} } \right)^{{\mathsf{T}}}$ is the $s$-th representative point, and $P_{s}$ is the assigned probability associated with the $s$-th representative point.

Step A.2: Evaluate velocity responses, namely, $\dot{Y}\left( {{\boldsymbol{\theta}}_{s} {,}t,{\mathbf{x}}} \right)$ in Eq.(9) or $\dot{W}\left( {{\boldsymbol{\theta}}_{s} ,t_{{\text{v}}} ,{\mathbf{x}}} \right)$ in Eq.(13) at each representative point ${\boldsymbol{\theta}}_{s} ,s = 1,2, \cdots ,n_{{{\text{sel}}}}$. When the PDEM is used directly, the velocity responses are obtained by exact response analyses. On the other hand, in the framework of the PDEM-based metamodel-refined algorithm, the velocity responses are obtained by the proposed metamodel in terms of the extreme value of structural responses (Zhou et al. 2019).

Step A.3: For each representative point ${\boldsymbol{\theta}}_{s}$, substitute the corresponding velocity in the GDEE, i.e., Eq.(9) or Eq.(13). The equation can be solved by the FDM with the total variation diminishing (TVD) scheme (Li and Chen 2009). Note that, for a representative point ${\boldsymbol{\theta}}_{s}$, the initial condition of the GDEE, i.e., Eq. (10) or Eq. (14), is $\left. {p_{{Y{\mathbf{\Theta}}}} \left( {y,{\boldsymbol{\theta}}_{s} {,}t;{\mathbf{x}}} \right)} \right|_{t = 0} = \delta \left( {y - y_{0} } \right)P_{s}$ or $\left. {p_{{W{\mathbf{\Theta}}}} \left( {w,{\boldsymbol{\theta}}{,}t_{{\text{v}}} ;{\mathbf{x}}} \right)} \right|_{{t_{{\text{v}}} = 0}} = \delta \left( w \right)P_{s}$, respectively.

Step A.4: By synthesizing the results of the GDEE at all representative points, the instantaneous PDF of the quantity of interest can be expressed as follows:

$$p_{Y} \left( {y{,}\,t;\,{\mathbf{x}}} \right)\, = \,\int\limits_{{\Omega _{{\mathbf{\Theta}}} }}^{{}} {p_{{Y{\mathbf{\Theta}}}} \left( {y,\,{\boldsymbol{\theta}}{,}\,t;\,{\mathbf{x}}} \right){\text{d}}} {\boldsymbol{\theta}}\, \approx \,\sum\limits_{s = 1}^{{n_{{{\text{sel}}}} }} {p_{{Y{\mathbf{\Theta}}}} \left( {y,\,{\boldsymbol{\theta}}_{{\text{s}}} {,}\,t;\,{\mathbf{x}}} \right)}$$

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or

$$p_{{\tilde{Y}_{{{\text{ext}}}} }} \left( {\tilde{y};{\mathbf{x}}} \right)\, = \,p_{W} \left( {w,\,t_{{\text{v}}} \, = \,1;\,{\mathbf{x}}} \right)\, = \,\int\limits_{{\Omega _{{\mathbf{\Theta}}} }}^{{}} {p_{{W{\mathbf{\Theta}}}} \left( {w,\,{\boldsymbol{\theta}}{,}\,t_{{\text{v}}} \, = \,1;\,{\mathbf{x}}} \right){\text{d}}} {\boldsymbol{\theta}}\, \approx \,\sum\limits_{s = 1}^{{n_{{{\text{sel}}}} }} {p_{{W{\mathbf{\Theta}}}} \left( {w,\,{\boldsymbol{\theta}}_{s} {,}\,t_{{\text{v}}} \, = \,1;\,{\mathbf{x}}} \right)}.$$

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Appendix B

A damped BFGS updating scheme is employed in the present formulation to update the approximate Hessian matrix of the Lagrangian during the optimization process (Herskovits and Santos 1997; Nocedal and Wright 2006; Jensen et al. 2013).

The matrix ${\mathbf{B}}^{\left( k \right)}$ is initialized as the identity matrix, that is,

$${\mathbf{B}}^{\left( 0 \right)} = {\mathbf{I}}.$$

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The update of the matrix ${\mathbf{B}}^{\left( k \right)}$ relies on two vectors ${\mathbf{s}}^{\left( k \right)}$ and ${\boldsymbol{\gamma}}_{0}^{\left( k \right)}$ defined as follows:

$$\begin{gathered} {\mathbf{s}}^{\left( k \right)} \, = \,{\mathbf{x}}^{{\left( {k + 1} \right)}} \, - \,{\mathbf{x}}^{\left( k \right)} \hfill \\ {\boldsymbol{\gamma}}_{0}^{\left( k \right)} \, = \,\nabla {\kern 1pt} f\left( {{\mathbf{x}}^{{\left( {k + 1} \right)}} } \right)\, + \,\sum\limits_{{j = 1}}^{{n_{{\text{p}}} }} {\lambda_{{{\text{p}}j}}^{{\left( {k + 1} \right)}} \nabla h_{j} \left( {{\mathbf{x}}^{{\left( {k + 1} \right)}} } \right)} \, + \,\sum\limits_{l = 1}^{{n_{{\text{s}}} }} {\lambda_{{{\text{s}}l}}^{{\left( {k + 1} \right)}} \nabla g_{l} \left( {{\mathbf{x}}^{{\left( {k + 1} \right)}} } \right)} \hfill \\ \end{gathered}$$

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In addition, a scalar $\phi^{\left( k \right)}$ is defined as

$${\phi}^{\left( k \right)} \, = \,\left\{ \begin{array}{*{20}c} 1, &{\text{if}} \, \left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} {\boldsymbol{\gamma}}^{\left( k \right)} \, \ge \,{0}{\text{.2}}\left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} {\mathbf{B}}^{\left( k \right)} {\mathbf{s}}^{\left( k \right)} \hfill \\ \frac{{0.8\left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} \mathbf{B}^{\left( k \right)} {\mathbf{s}}^{\left( k \right)} }}{{\left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} \mathbf{B}^{\left( k \right)} {\mathbf{s}}^{\left( k \right)} \, - \,\left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} {\boldsymbol{\gamma}}^{\left( k \right)} }},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} &{\text{otherwise}} \hfill \\ \end{array} \right.$$

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Then, the matrix ${\mathbf{B}}^{\left( k \right)}$ is updated as

$${\mathbf{B}}^{{\left( {k + 1} \right)}} \, = \,{\mathbf{B}}^{\left( k \right)} \, + \,\frac{{{\boldsymbol{\gamma}}^{\left( k \right)} \left[ {{\boldsymbol{\gamma}}^{\left( k \right)} } \right]^{{\mathsf{T}}} }}{{\left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} {\boldsymbol{\gamma}}^{\left( k \right)} }}\, - \,\frac{{{\mathbf{B}}^{\left( k \right)} {\mathbf{s}}^{\left( k \right)} \left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} {\mathbf{B}}^{\left( k \right)} }}{{\left[ {{\mathbf{s}}^{\left( k \right)} } \right]^{{\mathsf{T}}} {\mathbf{B}}^{\left( k \right)} {\mathbf{s}}^{\left( k \right)} }},$$

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where

$${\boldsymbol{\gamma}}^{\left( k \right)} \, = \,\phi^{\left( k \right)} {\boldsymbol{\gamma}}_{0}^{\left( k \right)} \, + \,\left( {1\, - \,\phi^{\left( k \right)} } \right){\mathbf{B}}^{\left( k \right)} {\mathbf{s}}^{\left( k \right)}.$$

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It is noted that the updated matrix ${\mathbf{B}}^{{\left( {k + 1} \right)}}$ is a symmetric positive definite matrix. Validation calculations have shown that, with this choice of approximate Hessian matrix, the proposed optimization scheme exhibits good convergence properties.