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Structural topology optimization under constraints on instantaneous failure probability

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Abstract

Accurate prediction of stochastic responses of a structure caused by natural hazards or operations of non-structural components is crucial to achieve an effective design. In this regard, it is of great significance to incorporate the impact of uncertainty into topology optimization of structures under constraints on their stochastic responses. Despite recent technological advances, the theoretical framework remains inadequate to overcome computational challenges of incorporating stochastic responses to topology optimization. Thus, this paper presents a theoretical framework that integrates random vibration theories with topology optimization using a discrete representation of stochastic excitations. This paper also discusses the development of parameter sensitivity of dynamic responses in order to enable the use of efficient gradient-based optimization algorithms. The proposed topology optimization framework and sensitivity method enable efficient topology optimization of structures under stochastic excitations, which is successfully demonstrated by numerical examples of structures under stochastic ground motion excitations.

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Abbreviations

f(t):

Stationary Gaussian input process

v :

Vector of n uncorrelated standard normal random variables

s(t):

Vector of deterministic basis functions

h f (·):

Impulse-response function of a filter

h s (·):

Impulse-response function of a system

u(t):

Displacement time history

a(t):

Vector of deterministic basis functions

u 0 :

Threshold value

E f :

Failure event

P(E f ):

Failure probability

P target f :

Target failure probability

Φ[·]:

Cumulative distribution function of the standard normal distribution

β:

Reliability index

βtarget :

Target reliability index

d :

Vector of deterministic design variables

\( {\tilde{\rho}}_e\left(\mathbf{d}\right) \) :

Element density

p :

Stiffness penalization parameter

q :

Mass penalization parameter

D(·):

Elastic tensor determined by material density function

D 0 :

Elasticity tensor of the solid material

E 0 :

Young’s modulus of the solid phase

K e :

Element stiffness matrix

M e :

Element mass matrix

K :

Global stiffness matrix

M :

Global mass matrix

C :

Global damping matrix

f :

Global load vector

ü :

Global acceleration vector

\( \dot{\mathbf{u}} \) :

Global velocity vector

u :

Global displacement vector

λ :

Adjoint variable vector

Φ0 :

Power spectral density of the white noise process

ω f :

Predominant frequency of a random process (a filter)

ζ f :

Bandwidth of a random process (a filter)

Δ i /L i :

Inter-story drift rations of the i-th floor evaluated at specified points

L i :

i-th floor height

Δ i :

i-th floor drift evaluated at a specified point

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Acknowledgements

The authors gratefully acknowledge funding provided by the National Science Foundation (NSF) through project CMMI 1234243. We also acknowledge support from the Raymond Allen Jones Chair at the Georgia Institute of Technology. The second author also acknowledges the support from the Integrated Research Institute of Construction and Environmental Engineering at Seoul National University, and the National Research Foundation of Korea (NRF) Grant (No. 2015R1A5A7037372), funded by the Korean Government (MSIP). Any opinion, finding, conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors.

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Correspondence to Glaucio H. Paulino.

Appendix: Derivation of (11)

Appendix: Derivation of (11)

To derive (11) with a uniform time step, the convolution integral in (9) is carried out for discrete time intervals. An entry of the vector a(t j ) can be written as follows:

$$ {a}_i\left({t}_j\right)={\displaystyle {\int}_0^{t_j}{s}_i\left(\uptau \right){h}_s\left({t}_j-\uptau \right)d\uptau} $$
(A.1)

where

$$ {t}_i=i\varDelta t,\ {t}_j=j\varDelta t $$
(A.2)

Then, one can consider the following three cases:

  1. Case 1:

    i = j

$$ \begin{array}{c}{a}_i\left({t}_i\right)={\displaystyle {\int}_0^{t_i}\sqrt{2\uppi {\Phi}_0\varDelta t}\cdot {h}_f\left(\uptau -{t}_i\right){h}_s\left({t}_i-\uptau \right)d\uptau}\\ {}=\sqrt{2\uppi {\Phi}_0\varDelta t}\cdot {\left.\left({\displaystyle \int {h}_f\left(\uptau -{t}_i\right){h}_s\left({t}_i-\uptau \right)d\uptau}\right)\right|}_{\uptau ={t}_i}\left( \because {h}_f\left(\uptau -{t}_i\right)=0\ \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0\le \uptau <{t}_i\right)\end{array} $$
(A.3)
  1. Case 2:

    i > j

$$ \begin{array}{c}{a}_i\left({t}_j\right)={\displaystyle {\int}_0^{t_j}\sqrt{2\uppi {\Phi}_0\varDelta t}\cdot {h}_f\left(\uptau -{t}_i\right){h}_s\left({t}_j-\uptau \right)d\uptau}\\ {}=0\ \left( \because {h}_f\left(\uptau -{t}_i\right)=0\ \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em 0\le \uptau <{t}_i\right)\end{array} $$
(A.4)
  1. Case 3:

    i < j

$$ \begin{array}{c}{a}_i\left({t}_j\right)={\displaystyle {\int}_0^{t_j}\sqrt{2\uppi {\Phi}_0\varDelta t}\cdot {h}_f\left(\uptau -{t}_i\right){h}_s\left({t}_j-\uptau \right)d\uptau}\\ {}={\displaystyle {\int}_{t_i}^{t_j}\sqrt{2\uppi {\Phi}_0\varDelta t}\cdot {h}_f\left(\uptau -{t}_i\right){h}_s\left({t}_j-\uptau \right)d\uptau}\ \left(\because {h}_f\left(\uptau -{t}_i\right)=0\ \mathrm{f}\mathrm{o}\mathrm{r}\ 0\le \uptau <{t}_i\right)\\ {}={\displaystyle {\int}_0^{t_j-{t}_i}\sqrt{2\uppi {\Phi}_0\varDelta t}\cdot {h}_f\left(\tilde{\uptau}\right){h}_s\left({t}_j-{t}_i-\tilde{\uptau}\right)d\tilde{\uptau}}\\ {}={\displaystyle {\int}_0^{c\varDelta t}\sqrt{2\uppi {\Phi}_0\varDelta t}\cdot {h}_f\left(\tilde{\uptau}\right){h}_s\left(c\varDelta t-\tilde{\uptau}\right)d\tilde{\uptau}},\ c=j-i\end{array} $$
(A.5)

Thus,

$$ {a}_i\left({t}_j\right)={a}_i\left(j\varDelta t\right)=\left\{\begin{array}{c}\hfill {a}_n\left({t}_n\right)\kern3em i=j\kern5.75em \hfill \\ {}\hfill \begin{array}{l}0\kern4.75em i>j\\ {}{a}_{n-c}\left({t}_n\right)\kern2.25em i<j,\ c=j-i\ \end{array}\hfill \end{array}\right. $$
(A.6)

Therefore, the uniform step size (t n -t n-1 = ∆t, t n  = t 0) in the convolution integral leads to the following expression:

$$ {a}_i\left({t}_j\right)={a}_i\left(j\varDelta t\right)={a}_{n+i-j}\left({t}_0\right),\kern0.5em i=1,2,\dots, n,\kern1em j=i,\dots, n $$
(A.7)

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Chun, J., Song, J. & Paulino, G.H. Structural topology optimization under constraints on instantaneous failure probability. Struct Multidisc Optim 53, 773–799 (2016). https://doi.org/10.1007/s00158-015-1296-y

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