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Error-lumped inverse uncertainty quantification of automotive heat exchangers (HEXs) using large-scale database from system level tests

  • Industrial Application Paper
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Abstract

Reliability-based design optimization (RBDO) utilizing computer simulations can lead to a highly reliable optimum design. However, conventional RBDO methods require full statistical information of input variables to estimate reliabilities of engineering systems or components, which is not easy to obtain in most engineering applications. In this paper, an uncertainty quantification method with mean-correlated simulations is proposed to estimate the statistical information of input variables from corresponding system response distributions obtained using large-scale test database. The proposed approach employs an error-lumped inverse method and kernel density estimation (KDE) for the uncertainty quantification. All possible errors such as measurement error, simulation error, and error by input variable difference are lumped into one to minimize residual errors of responses. Because quantified uncertainties using the error-lumped inversed method could be too scattered, distribution correction is proposed to reduce effective range of input variables while maintaining response distributions. Numerical and engineering examples show that the proposed approach can well estimate uncertainties of input variables using mean-correlated simulation and system response distributions obtained from test results.

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Abbreviations

A f :

Overlapped area of two probability density functions

b :

Bandwidth for kernel density estimation

E :

Elasticity of cantilever beam

e :

Lumped error of system response for single-input and single-response physics

e min :

Minimum value of e.

e i :

Lumped error of ith system response

e m :

Measuring error of system response for single-input and single-response physics

\( {e}_i^m \) :

Measuring error of ith system response

e s :

Simulation error of system response for single-input and single-response physics

\( {e}_i^s \) :

Simulation error of ith system response

e UF :

Geometrically averaged error of system responses

\( {e}_{UF}^{\mathrm{min}} \) :

Minimum value of eUF

F b :

Fin buckling force of heat exchangers

\( {F}_b^r \) :

Required specification of fin buckling force for product design

F p :

Fin density of heat exchangers

F x :

Axial force of cantilever beam

F y :

Vertical force of cantilever beam

f(x):

Probability density function of variable x

\( \hat{f}(x) \) :

Probability density function of variable x calculated from kernel density estimation

h(x):

Response simulator of system for single-input and single-response physics

h i(x):

Response simulator of ith system

K(•):

Kernel function for kernel density estimation

L :

Length of cantilever beam

M :

Number of simulators

N :

Number of samples

n :

Exponent of Af which amplifies validation metric for more detailed comparison

P f :

Failure probability

\( {P}_f^{\ast } \) :

Failure probability using the assumed uncertainty as Gaussian distribution

\( \dot{Q} \) :

Heat dissipation of heat exchangers

\( {\dot{Q}}^r \) :

Required specification of heat dissipation for product design

S j, 0 :

Initial effective range of jth variable

S j, α :

Effective range of jth variable with significant level α

T :

Thickness of cantilever beam

V :

Validation metric

V α :

Validation metric with significant level α

V α(δ):

Validation metric of deflection with significant level α

V α(τ):

Validation metric of stress with significant level α

V α(ω):

Validation metric of first natural frequency with significant level α

\( {\overline{V}}_{\alpha } \) :

Averaged validation metric of Vα(δ),Vα(τ), and Vα(ω)

W :

Width of cantilever beam

w j(•):

Weight function for probability distribution function of jth input variable

x :

Exact value of input variable for single-input and single-response physics

x j :

Exact value of jth input variable

\( {x}_j^{LB} \) :

Lower bound of jth input variable

\( {x}_j^{UB} \) :

Upper bound of jth input variable

\( \overline{x} \) :

Nominal value of input variable for single-input and single-response physics

\( {\overline{x}}_j \) :

Nominal value of jth input variable

Δx j :

Difference between exact value and nominal value of jth input variable

\( \hat{x} \) :

Factorized value of input variable for single-input and single-response physics

\( {\hat{x}}_j \) :

Factorized value of jth input variable

\( {\hat{x}}_{j,1-\alpha } \) :

Factorized value of jth input variable at significant level 1-α

\( {\hat{x}}_{j,\alpha } \) :

Factorized value of jth input variable at significant level α

y :

Exact value of system response for single-input and single-response physics

y i :

Exact value of ith system response corresponding exact input values

y m :

Measured value of system response for single-input and single-response physics

\( {y}_i^m \) :

Measured value of ith system response

α :

Significant level

β :

Required validation metric level of calculated distribution

ΔP air :

Air-side pressure drop of heat exchangers

\( \Delta {P}_{air}^r \) :

Required specification of air-side pressure drop for product design

δ :

Deflection of cantilever beam

ε :

Difference between measured and simulated value of system response for single-input and single-response physics

ε i :

Difference between measured value and simulated value of ith system response

ε m :

Difference by measuring error of system response for single-input and single-response physics

\( {\varepsilon}_i^m \) :

Difference by measuring error of ith system response

ε s :

Difference by simulation error of system response for single-input and single-response physics

\( {\varepsilon}_i^s \) :

Difference by simulation error of ith system response

ε x :

Difference of system response by input variables for single-input and single-response physics

\( {\varepsilon}_i^x \) :

Difference of ith system response by input variables

ε V :

Difference between the exact distribution and the distribution predicted from sampling

\( {\varepsilon}_V^{\mathrm{average}} \) :

Average of differences from kernel density estimation

\( {\varepsilon}_V^{DKW} \) :

Difference calculated from Dvoretzky-Kiefer-Wolfowitz inequality

\( {\varepsilon}_V^{\mathrm{min}} \) :

Minimum of differences from kernel density estimation

\( {\varepsilon}_V^{\mathrm{max}} \) :

Maximum of differences from kernel density estimation

\( {\varepsilon}_V^{SE} \) :

Difference calculated from standard error of mean prediction

μ :

Mean value of distribution

μ j :

Mean value of weight function wj(•) forjth input variable

ρ :

Density of cantilever beam

σ :

Standard deviation of distribution

σ j :

Standard deviation of weight function wj(•) forjth input variable

θ L :

Louver angle of heat exchangers

τ :

Stress of cantilever beam

ω :

First natural frequency of cantilever beam

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Authors and Affiliations

  1. Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 34141, South Korea

    Seongmin Pae, Hwisang Jo & Ikjin Lee

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  1. Seongmin Pae
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  2. Hwisang Jo
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  3. Ikjin Lee
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Correspondence to Ikjin Lee.

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The authors declare that they have no conflict of interest.

Replication of results

Matlab codes for the lumped-error inverse uncertainty quantification and distribution correction in Section 3 are uploaded on https://github.com/idolab/HEX_IUQ. Unfortunately, the engineering application is related to the simulation model and system level test data are restricted so that it cannot be shared. Overall concepts and algorithms can be validated through the numerical example.

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Responsible Editor: Xiaoping Du

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Pae, S., Jo, H. & Lee, I. Error-lumped inverse uncertainty quantification of automotive heat exchangers (HEXs) using large-scale database from system level tests. Struct Multidisc Optim 64, 2709–2724 (2021). https://doi.org/10.1007/s00158-021-02946-8

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  • DOI: https://doi.org/10.1007/s00158-021-02946-8

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