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Surrogate-based modeling and dimension reduction techniques for multi-scale mechanics problems

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Abstract

Successful modeling and/or design of engineering systems often requires one to address the impact of multiple “design variables” on the prescribed outcome. There are often multiple, competing objectives based on which we assess the outcome of optimization. Since accurate, high fidelity models are typically time consuming and computationally expensive, comprehensive evaluations can be conducted only if an efficient framework is available. Furthermore, informed decisions of the model/hardware’s overall performance rely on an adequate understanding of the global, not local, sensitivity of the individual design variables on the objectives. The surrogate-based approach, which involves approximating the objectives as continuous functions of design variables from limited data, offers a rational framework to reduce the number of important input variables, i.e., the dimension of a design or modeling space. In this paper, we review the fundamental issues that arise in surrogate-based analysis and optimization, highlighting concepts, methods, techniques, as well as modeling implications for mechanics problems. To aid the discussions of the issues involved, we summarize recent efforts in investigating cryogenic cavitating flows, active flow control based on dielectric barrier discharge concepts, and lithium (Li)-ion batteries. It is also stressed that many multi-scale mechanics problems can naturally benefit from the surrogate approach for “scale bridging.”

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Abbreviations

b :

Vector of polynomial coefficients

C :

Cycling rate, where 1C is the rate required to charge/discharge the cell in one hour

D :

Characteristic length scale (m)

D s :

Solid diffusion coefficient (m2/s)

E :

Expectation value

f v :

Vapor mass fraction/Frequency of applied voltage (kHz)

F x :

x-directional Lorentzian force (mN/m)

F x,S :

Domain averaged x-directional Lorentzian force (mN/m)

F x,ST :

Time and domain averaged x-directional Lorentzian force (mN/m)

h :

Enthalpy (kJ/kg)

L :

Latent heat (kJ/kg)

m + :

Source term in cavitation model (1/s)

m − :

Sink term in cavitation model (1/s)

n p :

Particle number density of species p (1/m3)

N s :

Number of sampled design points

N RBF :

Number of neural basis functions

N v :

Number of design variables

P :

Pressure (N/cm2); power input due to the charge current through the upper electrode (W)

P v :

Saturation vapor pressure (N/cm2)

P diff :

L 2 norm between experiment and predicted pressure (N/cm2)

P T :

Time averaged power input due to the charge current through the upper electrode (W)

Pr :

Prandtl number

r f :

Positive-to-negative polarity time ratio of applied voltage waveform

R 2adj :

Adjusted coefficient of determination

R s,p :

Solid particle radius in positive electrode (μm)

s :

Vector of neuron position

S :

Area of computational domain for gas (m2)

S M :

Main sensitivity index

S T :

Total sensitivity index

t ∞ :

Reference time scale, t ∞ = D/U ∞ (s)

T :

Temperature (K) Period of applied voltage (s)

T diff :

L 2 norm between experiment and predicted temperature (K)

u :

Velocity (m/s)

u p :

Particle bulk velocity of species p, (u x,p , u y,p , u z,p ) (m/s)

U ∞ :

Reference velocity (m/s)

V :

Variance

V app :

Applied voltage to the upper electrode (kV)

x :

Vector of design variables; Space variable (m)

y :

Objective function

ŷ:

Surrogate approximation of objective function

ȳ:

Mean value of objective function

z :

Systematic departure

αl :

Liquid volume fraction

β:

Spread coefficient of radial basis neural network

ɛd :

Dielectric constant of insulator

μ:

Dynamic viscosity (kg/ms)

ρ:

Density (kg/m3)

σ:

Electronic conductivity (S/m)

σ∞ :

Cavitation number based on the free stream temperature

σa :

RMS error of polynomial response surface at sampled points

τ:

Dimensionless time

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Authors and Affiliations

  1. Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

    Wei Shyy

  2. University of Michigan, Ann Arbor, MI, USA

    Young-Chang Cho, Wenbo Du, Amit Gupta & Ann Marie Sastry

  3. National Taiwan Sun Yat-Sen University, Kaohsiung, Taiwan, China

    Chien-Chou Tseng

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  1. Wei Shyy
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  2. Young-Chang Cho
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  3. Wenbo Du
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  6. Ann Marie Sastry
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Correspondence to Wei Shyy.

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Shyy, W., Cho, YC., Du, W. et al. Surrogate-based modeling and dimension reduction techniques for multi-scale mechanics problems. Acta Mech Sin 27, 845–865 (2011). https://doi.org/10.1007/s10409-011-0522-0

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