Reliability-based topology optimization using a new method for sensitivity approximation - application to ground structures

Abstract

This paper proposes an efficient gradient-based optimization approach for reliability-based topology optimization of structures under uncertainties. Our objective is to find the optimized topology of structures with minimum weight which also satisfy certain reliability requirements. In the literature, those problems are primarily performed with approaches that use a first-order reliability method (FORM) to estimate the gradient of the probability of failure. However, these approaches may lead to deficient or even invalid results because the gradient of probabilistic constraints, calculated by first order approximation, might not be sufficiently accurate. To overcome this issue, a newly developed segmental multi-point linearization (SML) method is employed in the optimization approach for a more accurate estimation of the gradient of failure probability. Meanwhile, this implementation also improves the approximation of the probability evaluation at no extra cost. In general, adoption of the SML method leads to a more accurate and robust approach. Numerical examples show that the new approach, based on the SML method, is numerically stable and usually provides optimized structures that have more of the desired features than conventional FORM-based approaches. The present approach typically does not lead to a fully stressed design, and thus this feature will be verified by numerical examples.

Appendix: Nomenclature

Abbreviations

c.o.v.:

Coefficient of variation

CDF:

Cumulative Distribution Function

CONLIN:

Convex Linearization Method

FORM:

First Order Reliability Method

HLRF:

Hassofer-Lind-Rackwitz-Fiessler (Algorithm)

KKT:

Karush-Kuhn-Tucker (Optimality Conditions)

MCS:

Monte Carlo Simulation

MMA:

Method of Moving Asymptotes

MPP:

Most Probable Point

PDF:

Probability Density Function

PMA:

Performance Measure Approach

RBDO:

Reliability-Based Design Optimization

RBTO:

Reliability-Based Topology Optimization

RIA:

Reliability Index Approach

SML:

Segmental Multi-point Linearization

SORM:

Second Order Reliability Method

Symbols

$\overline {G}^{j}$ :

The affine function describing hyperplane segment i

$\overline {S}_{j}$ :

Hyperplane segments of a piecewise linearized limit state surface

β :

Reliability index

β ^t :

Target reliability index

δΩ:

Change of failure domain

e _i :

Orthonormal basis of space

R :

Rotational matrix

u :

Transformed random variables

u ^∗ :

Most likely failure point (design point)

x :

Design variables

Ω,Ω^′ :

Failure domain and failure domain after design update

Φ,φ :

CDF and PDF of standard normal distribution

G :

Limit state function in transformed random space

G ^e :

Equivalent limit state function

h _k :

Deterministic constraints

k :

User defined parameters for the fitting scheme

n :

Number of random variables

p :

Number of fitting points

P _f :

Failure probability

$P^{t}_{f}$ :

Target failure probability

R :

Reliability measured in probability

R ^t :

Target reliability

S :

Limit state surface

W _j :

Weight for contribution of segment j