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Research Papers

Model Selection Among Physics-Based Models

[+] Author and Article Information
S. Mahadevan

e-mail: sankaran.mahadevan@vanderbilt.edu
Department of Civil and
Environmental Engineering,
Vanderbilt University
, Nashville, TN 37235

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received January 15, 2012; final manuscript received August 1, 2012; published online January 7, 2013. Assoc. Editor: Michael Kokkolaras.

J. Mech. Des 135(2), 021003 (Jan 07, 2013) (15 pages) Paper No: MD-12-1046; doi: 10.1115/1.4023155 History: Received January 15, 2012; Revised August 01, 2012

The optimal solution of a design optimization problem is dependent on the predictive models used to evaluate the objective and constraints. Since different models give different predictions and can yield different design decisions, when more than one model is available, the choice of model used to represent the objectives/constraints of the design becomes important. This paper addresses the problem of model selection among physics-based models during the prediction stage, which is in contrast to model selection during the calibration and validation stages, and therefore affects design under uncertainty. Model selection during calibration addresses the problem of selecting the model that is likely to provide the best generalization of the calibration data over the entire domain. Model selection during the validation stage examines the validity of a calibrated model by comparing its predictions against the validation data. This paper presents an approach that allows for model selection during the prediction stage, which selects the “best” model for each prediction point. The proposed approach is based on the estimation of the model prediction error under stationary/nonstationary uncertainty. By selecting the best model at each prediction point, the proposed approach partitions the input domain of the models into nonoverlapping regions. The effects of measurement noise, sparseness of validation data, and model prediction uncertainty are included in deriving a probabilistic selection criterion for model selection. The effects of these uncertainties on the classification errors are analyzed. The proposed approach is demonstrated for the problem of selecting between two parametric models for energy dissipation in a mechanical lap joint under dynamic loading, and for the problem of selecting fatigue crack growth models.

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Figures

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Fig. 1

Classification-based model selection. Assigning each point in the domain to the best performing model results in a partition of the domain. The contour represents the classification boundary that partitions the domain. The region in the domain where M1 yields better predictions is marked by circles. The region corresponding to M2 is marked by +.

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Fig. 2

Partition of the domain using Pr[Δe∧12>0] for the 2D problem. Model 1 is selected in regions where Pr[Δe∧12>0]=1. Regions where Pr[Δe∧12>0]=0, model 2 is selected.

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Fig. 3

Effect of model error estimation uncertainties on the selection criterion

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Fig. 4

Model error estimates for 1D example - Stationary Uncertainty

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Fig. 5

Model prediction and prediction uncertainties for 1D example

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Fig. 6

Model error estimates for 1D example - Non-Stationary Uncertainty

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Fig. 7

Classification boundaries and probabilities for 1D example

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Fig. 8

Blurring of the classification boundary due to sparsity of validation points (2D example function)

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Fig. 9

Misclassification rate as a function of number of samples. Two different probability levels α (Eq. (7)) are considered.

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Fig. 10

Misclassification rate as a function of measurement noise. Two different α levels are considered. x axis shows the log(σn2).

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Fig. 11

Iwan model. The Iwan model is a parallel system of spring-slider elements, each with a spring constant k and slider strength φi. The displacement of each element is xi and the displacement of the entire system is u.

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Fig. 12

Smallwood model. Force versus displacement curve, when the applied force is a harmonic function.

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Fig. 13

Iwan and Smallwood models. Data and model prediction errors (Units: F0: lb, Da, Error: in. lb).

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Fig. 14

Iwan and Smallwood models with nonstationary measurement errors (Units: F0: lb, e∧: in. lb)

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Fig. 15

Iwan and Smallwood models: Effect of sparseness of data (Units: F0: lb, e∧: in. lb)

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Fig. 16

Crack growth behavior in metals

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Fig. 17

Model prediction errors (Units: ΔK—Ksi in0.5, errors—in.)

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Fig. 18

Crack growth models. In panel A, bands around each error estimate correspond to ±5σ (Units: ΔK—Ksi in0.5, errors—in.).

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Fig. 19

Finite element meshes of the full model, submodel, and initial crack front

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