PDF Multi-fidelity Information Fusion for Structural Dynamics Model Calibration

Mark McDonald, thank you for all your comments and feedback that allowed this study (and the researcher!) to reach its potential. I cannot thank my father and mother enough, who gave me all the tools to break all barriers.

Introduction

Overview

First, it takes advantage of the information available in the high-fidelity simulation to improve the low-fidelity surrogate model, and then uses the improved low-fidelity surrogate model for calibration. This extension also considers the effect of the data on the uncertainty in the experimental data.

Research Objectives

It takes advantage of the information available in the high-fidelity simulation to first improve the low-fidelity surrogate model and then uses the improved low-fidelity surrogate model for calibration. A subset of high-fidelity simulation data is first used to correct the low-fidelity surrogate model.

Dissertation Organization

Background

Calibration of Damping
Bayesian Calibration
Surrogate Model – Polynomial Chaos Expansion
Error Inclusion in Calibration
Kullback–Leibler divergence
Methodology Evaluation
Summary

2), and the damping coefficient is directly defined as the ratio of the maximum damping force (fD,max) to the maximum velocity (vmax. The likelihood function is based on the joint PDF of observations yDi, conditional on , θ σobs and εd .

Multi-Fidelity Approach to Dynamics Model Calibration

Introduction

4 shows a notional diagram of how the surrogate model error and the model misfit term can vary with model fidelity. The proposed approach uses a low-fidelity surrogate model corrected with high-fidelity simulations for system parameter calibration.

Multi-Fidelity Calibration Method

The Bias Correction Method
The Pre-Calibration Method

Update the parameters of low-fidelity model as well as the discrepancy term with the high-fidelity simulation results, that is, use the relationship. The low-fidelity surrogate model was pre-calibrated with high-fidelity simulations and then recalibrated with the experimental data (LFcorr).

Numerical Example

Problem Description
Results
Discussion

7 shows the reverse side of one of the discrepancy expressions (at strain gage location SG4) for illustration purposes. This implies that the experimental output at SG1 for high temperatures did not significantly affect the calibration result due to the high uncertainty in the strain gauge reading. Calibration of both model parameters and discrepancy between the low-fidelity model with high-fidelity simulations allowed information from both models to be preserved with respect to the parameters.

Most importantly, the results demonstrate the fusion of information from low- and high-fidelity models in the proposed methodology, which combines different high- and low-fidelity sensitivities. Using the proposed calibration method, the final calibration results take advantage of the sensitivity of the low- and high-resolution models to different parameters. Not only were the results in favor of using the improved low-fidelity model in the calibration process, but its convergence was faster than the low- and high-fidelity calibrations with experimental data.

Fig. 5: Curved panel dimensions and strain gage locations (units: cm)

Conclusion

However, more elaborate representations of model differences, such as input dependent random field (following the Kennedy and O'Hagan approach) could also be included in the proposed formulation; such implementations are limited by the amount of data available. In the presence of limited data, the choice of priors also affects the calibration result. The calibration becomes more complicated when the model parameters are input-dependent (eg, load-dependent damping), and Chapter 4 extends the proposed multi-fidelity approach to account for this dependence.

Input-Dependence Effects in Dynamics Model Calibration

Introduction

In experiments, it is difficult to replicate the natural phenomena in laboratory settings while maintaining the quality of the recorded data, especially in the presence of high temperature, which can negatively affect the recording devices such as strain gauges. Complex interactions can result in non-linearity and input dependence being active at the same time, so there is a need to distinguish between these effects in the calibration process. The purpose of this chapter is to extend the approach of fusion of information from models of different levels of fidelity to Bayesian calibration of input-dependent model parameters.

We consider geometric nonlinearity (effect of large deformations) and material nonlinearity (non-linear stress-strain relationships that can also be temperature dependent) in the modeling phase, which is separated from the input dependence of the parameters. We also include the effect of different types of inputs on the sensors (i.e. strain gauges) and use realistic experimental data to illustrate the advantages of this approach. The remainder of this chapter is organized as follows: Section 4.2 proposes the extension of the multi-fidelity Bayesian calibration approach to input-dependent parameter calibration in the presence of .

Multi-Fidelity Calibration Method for Input-dependent System Parameters

Model Calibration with Input-Dependent Parameters
Dynamics Model Calibration
Multi-Fidelity Calibration Method

The dependence of the input variables on the parameters θ as well as the measurement standard deviation σobs and the discrepancy εd are represented by their hyperparameters (respectively). In this thesis, slice sampling (described in Section 2.2) is used to calculate the posteriors of the hyperparameters. The input dependence of the damping parameters must be considered, as well as the input effects on the sensor performance.

Manufacturers typically use a fourth order correction of the observed error for the effect of temperature on the voltage [78]. The dependence of the observed error standard deviation on the temperature can be modeled as exponential, and written as Define the priors of the hyperparameters of i(X) and the difference between the LF and HF modelsD2,1(X), denoted λθi and.

Fig. 18: Bayesian Network (a) without input-dependence, and (b) with input-dependence

Numerical Example

Results
Discussion

An important concern in the calibration exercise is to account for the non-linear behavior of the panel. A sensitivity analysis of the low-fidelity and high-fidelity surrogate models using the first-order Sobol index [79] showed that the strain output is more sensitive to the friction damping parameters. The posterior using the corrected low-fidelity model also converges close to the posterior MAP estimate that comes from the calibration using the low-fidelity model.

If the input settings in Section 4.3.2 are set to T = 70F (ambient temperature) and P = 140dB (lowest acoustic load), the posteriors of the calibration quantities reflect the results in Section 3.3.2. 23 and 24 show that after accounting for geometric and material non-linearity in the corrected low-fidelity model, the effect of the acoustic loading input on the calibration parameters is negligible. The inclusion of the temperature effect on the strain gauges in the calibration of damping provides an implicit.

Fig. 21: Temperature distribution in experimental setup (left) and ANSYS model (right)

Conclusion

Most importantly, this numerical example illustrates a systematic approach to Bayesian model calibration in the presence of geometric and material nonlinearity, which allows investigation of temperature and acoustic loading effects on frictional damping, material damping, misfit between models with reliability of low and high, model shape error between models and experiments, and observed error. In the experimental direction, since strain gauges are affected differently in the presence of non-uniform temperature loads, and measurement errors in turn affect the posterior distributions, it is important to optimize sensor locations in order to maximize information gain in calibration process.

Simulation Resource Optimization for Multi-Fidelity Model Calibration

Introduction
Optimization formulation
Numerical Example

Results
Discussion

Conclusion

It is also possible to start with the high-fidelity training points and select the low-fidelity superset using some optimization criteria. We choose high-fidelity posterior runs that maximize the KL divergence between the forward and backward coupling until convergence. The objective of the optimization is to select high-confidence runs on the input values that maximize the KL divergence between the joint prior and joint posterior distributions of the quantities involved in the correction using the low-confidence surrogate.

The output of the selected run is then added to the initial high-fidelity data. This synthetic high-fidelity model is used in the optimization framework to select the new high-fidelity point that maximizes the KL divergence. We can see that Post.11 converges to the posterior calculated using all available high-fidelity simulations (46).

Fig. 29: KL divergence values between subsequent optimization iterations for the joint posteriors of the calibration parameters

Sensor Configuration Optimization

Introduction
Multi-Fidelity Sensor Location Optimization
Numerical Example

Results

Discussion
Conclusion

In the presence of large data sets, the Fisher information determinant affects the information entropy [101]. We consider the effect of inputs on both the model parameters and the sensor behavior. The optimization objective is to select the sensor configuration that maximizes the KL divergence between the joint prior distribution of the calibration quantities and the model errors and their joint posterior distribution.

We also assume that the standard deviation of the observed error is the same at all sensor locations due to the large number of sensors. However, this methodology can be applied to the case of a continuous candidate set by finding the optimal location coordinates. This chapter developed a sensor configuration optimization method that selects the number and location of sensors to maximize information gain in system parameter calibration.

Table 7: Sensor optimization results over multiple temperature setting Number

Conclusion

Summary of Accomplishments

The proposed approach was extended to the calibration of input-dependent system parameters, model parameters, and measurement error uncertainty by considering the functional relationships between input variables and calibration unknowns. The results showed a clear influence of temperature on parameters related to damping, as well as measurement errors of strain measurement. The multi-fidelity information fusion approach was then optimized to maximize information retention in the calibration exercise.

Model calibration is further improved by selecting sensor number and sensor locations to maximize information gain during calibration. The results of applying the proposed method to the curved panel problem showed promising results, although additional computational resources would be required to achieve robust convergence. Although illustrated for a structural dynamic problem, the proposed approach is applicable to problems in any discipline where multiple reliability models are available.

Future Work

O'Hagan, Bayesian Calibration of Computer Models, Journal of the Royal Statistical Society: Series B (Statistical Methodology. Pericchi, The Intrinsic Bayes Factor for Model Selection and Prediction, Journal of the American Statistical Association. Katafygiotis, Bayesian Modal Updating with Complete Input and Noisy Measurements with Incomplete Responses, Journal of Engineering Mechanics.

Au, Fast Bayesian FFT Method for Ambient Modal Identification with Separated Modes, Journal of Engineering Mechanics. Wasserman, The Selection of Prior Distributions by Formal Rules, Journal of the American Statistical Association. Yang, Kalibrering af funktionelle parametre i ionkanalmodeller af hjerteceller, Journal of the American Statistical Association.