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Probabilistic measures for assessing appropriateness of robust design optimization solutions

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Abstract

Robust design optimization (RDO) is a popular framework for addressing uncertainties in the design of engineering systems by considering different statistical measures, typically the mean and standard deviation of the system response. RDO can lead to a wide range of different candidate designs, establishing a different compromise between these competing objectives. This work introduces a new robustness measure, termed probability of dominance, for assessing the appropriateness of each candidate design. This measure is defined as the likelihood that a particular design will outperform the rival designs within a candidate set. Furthermore, a multi-stage implementation is introduced to facilitate increased versatility/confidence in the decision-making process by considering the comparison among smaller subsets within the initial larger set of candidate designs. For enhancing the robustness in these comparisons the impact of prediction errors, introduced to address potential differences between the real (i.e. as built) system and the numerical model adopted for it, is also addressed. This extends to proper modeling of the influence of the prediction error, including selection of its probability model, as well as evaluation of its impact on the probability of dominance and on the RDO formulation itself. Two illustrative examples are presented, the first considering the design of a tuned mass damper (TMD) for vibration mitigation of harmonic excitations and the second a topology optimization problem for minimum compliance. Extensive comparisons are presented in these two examples and the discussions demonstrate the power of the proposed approach for assessing the designs’ robustness.

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Abbreviations

γ :

Parameter associated with definition of prediction error variance

Δ:

Set of candidate designs

ε :

Log of error when it is incorporated through a multiplicative formulation

Θ:

Uncertain Space

θ :

uncertain model parameters

{θ j}:

Set of samples from p(θ)

λ :

Decay rate for exponential correlation function

μ(x):

Mean for system model

μ s (x):

Mean for real system

[μ o σ ο ]:

Utopia point (minimum mean and standard deviation) for system model

[μ s  σ s ]:

Utopia point (minimum mean and standard deviation) for real system

v l :

Weight for defining norm for exponential correlation

ρ ik :

Correlation function for the prediction errors associated with designs x i and x k

Σ d il :

Covariance matrix for the vector composed of the error for i th design subtracting the errors from each of the designs in S d il

σ(x):

Standard deviation for system model

σ e :

Standard deviation for prediction error

σ s (x):

Standard deviation for real system

d :

Dimension of subsets of Δ under consideration

D d (x i ):

Degree of dominance of design x i against the d-dimensional subsets

D t :

Threshold for acceptable dominance level

Dh d il (θ):

Vector of the different between performance for i th design and the performance of the designs in S d il

Dl d il (θ):

Version of Dh d il (θ) utilizing log of performance

Ε[.]:

Expectation for random variable

e :

Prediction error

F G [.]:

Gaussian cumulative distribution function

f(x):

Objective function

h(x, θ):

Performance function for system model

h s (x):

Performance function for real model

I(A i , θ):

Indicator function for dominance of x i against set A for given θ

. i :

Subscript i. Characteristic associated with the i th design

M d il :

Margin of dominance for design x i within set S d il

m :

Number of designs in Δ

N :

Number of samples for stochastic simulation

n θ :

Dimension of θ

n x :

Dimension of x

P D (x i |A):

Probability of dominance of design x i over set A

P D (x i |A,θ):

Probability of dominance of design x i over set A given the system configuration θ

\( {\tilde{P}}_D\left({\mathbf{x}}_i\Big|A\right) \) :

Probability of dominance of design x i over set A calculated through Monte Carlo simulation

p(θ):

Probability density function for θ

p(e):

Probability model for prediction error

S d :

Set of all possible d-dimensional subsets of Δ

S d i :

Set of d-dimensional subsets of Δ including design x i

S d il :

l th subset of d-dimensional subsets of Δ including design x i

w :

Weight used in the RDO formulation

X :

Admissible design space

\( \underset{\bar{\mkern6mu}}{\mathbf{x}} \) :

Ordered of candidate designs based on probability of dominance P D (x i |Δ)

x :

Design variable vector

x dli k :

k th design within the S d il set

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Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, University of Notre Dame, 365 Fitzpatrick Hall, Notre Dame, IN, 46556, USA

    Juan Camilo Medina & Alexandros Taflanidis

  2. Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, IN, 46556, USA

    Alexandros Taflanidis

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  1. Juan Camilo Medina
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  2. Alexandros Taflanidis
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Correspondence to Alexandros Taflanidis.

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Medina, J.C., Taflanidis, A. Probabilistic measures for assessing appropriateness of robust design optimization solutions. Struct Multidisc Optim 51, 813–834 (2015). https://doi.org/10.1007/s00158-014-1160-5

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  • DOI: https://doi.org/10.1007/s00158-014-1160-5

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