Reliability Based Topology Optimization under Stochastic Excitation

Size: px

Start display at page:

Download "Reliability Based Topology Optimization under Stochastic Excitation"

Doreen Fletcher
5 years ago
Views:

1 The 11th US National Congress on Computational Mechanics Reliability Based Topology Optimization under Stochastic Excitation Junho Chun 07/26/2011 Advisors : Junho Song & Glaucio H. Paulino Department of Civil and Environmental Engineering University of Illinois, Urbana-Champaign

2 Contents Introduction Discrete Reliability based Summary Future research 2

3 Introduction Optimization in structural engineering Motivation Stochastic Process 3

0 Spectral Acceleration (cm/s 2 ) 60.0 50.0 40.0 30.0 20.

4 Optimization in structural engineering Introduction Structural elements Structure performance Spectral Acceleration (cm/s 2 ) Elevation (m) Period (s) Displacement [mm] 4

5 Introduction Motivation Application of Topology under stochastic process to building system design MEGA-BRACE DIAGONALS JOHN HANCOCK Courtesy of Skidmore, Owing and Merrill, LLP BBVA SSIGER 5

(gal) HYOGOKEN NANBU EQ - KOBE-JMA3.EW 1/17/1995 DT=0.02 Amax=617.

6 Introduction Stochastic process Random process Non-deterministic excitations Many possibilities the process might to go to - Earthquake excitations Acceleration (gal) HYOGOKEN NANBU EQ - KOBE-JMA3.EW 1/17/1995 DT=0.02 Amax=617.14gal x - time / DT = Wind loads 6

7 Discrete Representation of a Random Process Discrete representation Response Probability of failure 7

8 Discrete Representation of a Random Process Discrete representation Stochastic excitation can be discretized and represented in terms of a finite number of standard normal random variables Mean function, μ(t) Standard normal random variables, Deterministic basis functions, u = [ u 1 u n ] T s(t) = [ s 1 (t) s n (t) ] T - Dependent on the covariance structure of the process Karhunen Loeve expansion method etc. 8

9 Response of Linear System to Gaussian Excitation Linear system + Gaussian / Non-Gaussian process Duhamel s Integral - h(t) : the unit-impulse response function of the system Stochastic excitation f(t) 0 τ dτ t Response a(t) = [ a 1 (t) a n (t) ] T 9

10 Probability of Failure Limit state function Consider the set of realizations of f(t) that give rise to the event {x(t 0 ) x 0 } at time t = t 0, where x 0 is a selected threshold Realizations of u that satisfy the condition G : x 0 a(t 0 ) T u 0, failure event - In the space of u, these lie in a half space bounded by the hyper-plane, x 0 a(t 0 ) T u= 0 ά(t 0 ) u*(x x 0 x(t 0 ) 0 0, t 0 ) Failure domain β (x 0, t 0 ) Safe domain u 1 x 0 a(t 0 ) T = 0 Geometric response at time t 0v Reliability index - β(x 0,t 0 ) = x 0 / a(t 0 ) -β Failure probability - Pf [ x(t 0 ) > x 0 ] = Φ [ -β(x 0,t 0 )] 10

11 Probability of Failure FORM / SORM approximation To evaluate limit state surface of Non-Gaussian response or nonlinear system FORM (First Order Reliability Method) Linearizing limit state function in the standard normal space at an optimal point SORM (Second Order Reliability Method) Second order approximation of the limit state function u*(x 0, t 0 ) β (x 0, t 0 ) Safe domain SORM x 0 x(t 0 ) 0 Failure domain FORM u 1 x 0 a(t 0 ) T = 0 FORM and SORM approximations for non-gaussian response 11

12 Reliability Based Topology Optimization Concept CRBTO / SRBTO Formulation of the RBTO problem Formulation of the problem (on going research) 12

13 Reliability Based Topology Optimization Aim to obtain that maximizes the performance under the design constrains on the failure probabilities Deterministic Reliability based (RBTO) Nguyen, T.H., System reliability-based design and multi-resolution, PhD Thesis (2010) 13

14 Reliability Based Topology Optimization Component RBTO u 2 Failure domain Safe domain u 1 g 1 System RBTO u 2 Failure domain g n Safe domain u 1 u 2 Safe domain Failure domain u 1 14

15 Reliability Based Topology Optimization Formulation of the RBTO problem F 1 F 2 F 3 F 4 F 5 F 4 F 3 F 2 F 1 min : V(ρ) ρ g i (ρ,f) = d i 0 d i (ρ,f), i = 1,..,5 Solid domain Each of random variables : μ F =10 5 c.o.v ( σ F /μ F ) = 1/6 d i 0 = {1.25, 1.50, 1.75, 2.00, 2.20}, i=1,,5 Nguyen, T.H., J. Song, and G.H. Paulino (2010). Single-loop system reliability-based design using matrix-based system reliability method: theory and applications. J. of Mechanical Design, ASME, Vol. 132, ~11. 15

$Reliability Based Topology Optimization Results A B C Courtesy of Nguyen (2011) (A) (B) (C) DTO (μf =10 5 ) FORM-based CRBTO ( (μf =10 5 ) FORM-based SRBTO ( (μf =10 5 ) volfrac = 39.07% t P i = 0.$

16 Reliability Based Topology Optimization Results A B C Courtesy of Nguyen (2011) (A) (B) (C) DTO (μf =10 5 ) FORM-based CRBTO ( (μf =10 5 ) FORM-based SRBTO ( (μf =10 5 ) volfrac = 39.07% t P i = , P t sys = volfrac = 48.64% P t sys = , volfrac = 47.7% Nguyen, T.H., J. Song, and G.H. Paulino (2010). Single-loop system reliability-based design using matrix-based system reliability method: theory and applications. J. of Mechanical Design, ASME, Vol. 132, ~11. 16

Reliability Based Topology Optimization Formulation of the problem (on going) Design of a rectangular domain subjected to stochastic excitations L

17 Reliability Based Topology Optimization Formulation of the problem (on going) Design of a rectangular domain subjected to stochastic excitations L f(t) min : V(ρ) ρ s.t. : 1.5L Pf [ G: x 0 - a(t 0 ) T u 0 ] Pf target Φ [ -β(x 0,t 0 )] Φ [ -β target (x 0,t 0 )] x 0 : threshold value : time t 0 17

18 Reliability Based Topology Optimization Sensitivity Constraint function - The sensitivities with respect to the design variables, ρ obtained by applying the chain rule : Pf [ G: x 0 - a(t 0 ) T u 0 ] Pf target, where β(x 0,t 0 ) = x 0 / a(t 0 ) 18

19 Summary Discrete Representation of stochastic process, response was presented Probability of failure is obtained from discrete representation easily Reliability based for static loads problem was reviewed. Reliability based for stochastic loads case is on going research 19

20 Future Research Implementation with stochastic process Nonlinear system Earthquake ground motion modeling Sanaz Rezaeian, Armen Der Kiureghian (2010) Simulation of synthetic ground motions for specified earthquake and site characteristics The first passage probability Song, J., and A. Der Kiureghian (2006). Joint first-passage probability and reliability of systems under stochastic excitation. Journal of Engineering Mechanics. ASCE, 132(1), Application to multiple story building system 20

21 Thanks 21

System Reliability-Based Design Optimization of Structures Constrained by First Passage Probability

System Reliability-Based Design Optimization of Structures Constrained by First Passage Probability Junho Chun* University of Illinois at Urbana-Champaign, USA Junho Song Seoul National University, Korea