Chapter1 Calibration of System Parameters Under Model Uncertainty Ghina N. Absi and Sankaran Mahadevan Abstract This paper investigates the quantification of errors and uncertainty in Bayesian calibration of structural dynamics computational models, affected by choices in model fidelity. Since Bayesian calibration uses an MCMC approach for sampling the updated distributions, using a high-fidelity model for calibration can be prohibitively expensive. On the other hand, use of a low-fidelity model could lead to significant error in calibration and prediction. This paper investigates model parameter calibration with a low-fidelity model corrected using higher fidelity simulations, and the trade-off between accuracy and computational effort. Different fidelity models may have different mesh resolutions, physics assumptions, boundary conditions, etc. The application problem used is of a curved panel located in the vicinity of a hypersonic aircraft engine, subjected to acoustic, thermal and aerodynamic loads. Two models are used to calibrate the damping characteristics of the panel: frequency response analysis and full time history analysis, and the trade-off between accuracy and computational effort is examined. Keywords Multi-fidelity • Bayesian calibration • Hypersonic vehicle • Model uncertainty • Surrogate model 1.1 Introduction Finite element analysis (FEA) is commonly used in the dynamic simulation of engineering structures with complicated geometry and under complex loading conditions. However, construction of the FEA model is subjective, affected by the engineer’s assumptions. High-fidelity dynamic finite element analysis of complex systems is quite expensive, and considerable research has been done to construct cheaper and simpler surrogate models, equivalent static models, or reducedorder models. However, the errors and uncertainties increase with the reduction in model fidelity. Two principal qualities are desired in a functional finite element model [1]: (1) physical significance: the model should correctly represent how the mass, stiffness and damping are distributed, and (2) correctness, where the response from dynamics experiments is accurately predicted by the model. Mottershead and Friswell [2] group modeling errors into three types: (a) model form errors (due to assumptions regarding the underlying physics of the problem, especially with strongly nonlinear behavior), (b) model parameter uncertainty (due to assumptions regarding boundary conditions, parameters distributions and simplifying assumptions), and (c) model order errors (arising from the discretization of complex geometry and loading). Computationally efficient models have to be cheap enough to allow multiple repetitions of the simulations, but also retain precious information available from rigorous more expensive models. Many studies have concentrated on developing reduced-order models (ROM) to replace full fidelity dynamic analyses. McEwan [3, 4] proposed the Implicit Condensation (IC) method that included the non-linear terms of the equation of motion, but restricted the nonlinear function to cubic stiffness terms, and can only predict the displacements covered by the bending modes. Other methods explicitly include additional equations to calculate the membrane displacements in the ROM, such as those by Rizzi et al. [5, 6] and Mignolet et al. [7, 8]. Another direction of research to deal with inaccurate FEA is model updating. Direct updating methods have been proposed by computing closed-form solutions for the global stiffness and mass matrices using the structural equations of motion G.N. Absi • S. Mahadevan ( ) Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN, USA e-mail: sankaran.mahadevan@vanderbilt.edu H.S. Atamturktur et al. (eds.), Model Validation and Uncertainty Quantification, Volume 3: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04552-8__1, © The Society for Experimental Mechanics, Inc. 2014 1
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