Contributed Talk

Best practice in metamodeling for data derived from civil engineering application

Maria Steiner, Tom Lahmer
Bauhaus-Universitaet Weimar, Germany


Frequent civil engineering problems exist that lead to a high computation time for determining the optimal deployment of construction materials. Application of mathematical models is useful in reducing the costs and thus in keeping the computation time low without incurring a loss in model quality. With response surface methods, also called metamodels, calculations and forecasts can be completed faster because of a reduced amount of numerical or experimental data required. The aim of this contribution is at first to analyze the applicability of metamodels to civil engineering problems where the data come either from mathematical models or experimental tests, in particular in the field of structural health monitoring. Furthermore, it seeks to illustrate which metamodel is the best for various specific application fields. In the literature two different types of metamodels can be found, which differ significantly in their properties and thus in their applicability. At first, deterministic methods are mentioned. In this area Polynomial Regression [1, 2], Radial Basis Function Interpolation [3], and Moving Least Squares [4] are specific examples. On the other hand, the Stochastic Process Model (Kriging) and its variations [5, 6] are part of the second group of models, the stochastic methods. Furthermore, it should be noted that there is a distinction between interpolation and regression possible. For noisy measurement data regressive methods can be more advantageous because of a possible smoothing. With respect to the decision which metamodel is to be used, the different computational effort, flexibility, predictive ability, and robustness of all available models should be observed. To infer to a general applicability for response surface methods used in civil engineering, the models are analyzed with several examples. As one of them a variance based sensitivity analysis [7] of a model representing a triaxial soil test is considered. In order to receive the data, various finite element methods are used. A large amount of data is required to determine sufficiently accurate and robust values; therefore, the computation time magnifies with increasing number of input parameters. Even for a comparatively simple material model, Mohr Coulomb, with five parameters a computation time of often more then 20 hours is necessary for a normal computation power. With application of metamodels, for example linear regression, the cost can be extremely reduced and it leads to a similar solution. However, there are some open points that need to be discussed: The necessary amount of the data to produce the various metamodels, the trade-off between model complexity and accuracy, and the criteria for optimal model selection. The decision of which method is used is coupled to these criteria and may vary depending on the situation. In order to provide a general overview, it is necessary to classify the engineering models based on their underlying partial differential equations. There is a separation in elliptic, parabolic and hyperbolic equations possible, which is related to the different engineering fields of the problems. It is obvious that for example static systems and problems related to the heat equation differ visibly. Thus it is probable that also different metamodels are optimal for each application field. For testing purposes, fundamental solutions, which exist in each equation field, can be used. Particularly in the simplified one dimensional case they provide explicit and simple to evaluate functions. Furthermore, it may happen that no single model emerges as the optimal method for some engineering tasks. In this case it is necessary to discuss a combination of several response surface methods. In conclusion, it is possible to increase the use of metamodels by providing an overview of the relation between these models and the corresponding classes of civil engineering problems and thereby improve the construction of engineering models.

[1] R. Myers: Response surface methodology. Allyn and Bacon, 1971.
[2] G.E.P. Box and N.R. Draper: Empirical model-building and response surfaces. JohnWiley and Sons, 1987.
[3] M.D. Buhmann: Radial basis functions: theory and implementation. Cambridge University Press, 2003.
[4] P. Lancaster and K. Salkauskas: Surface generated by moving least squares methods. Mathematics of Computation, 37, 141–158, 1981.
[5] A. Forrester, A. Sobester, A. Keane: Engineering design via surrogate modelling: A practical guide. Progress in Aerospace Sciences, Wiley, 40–50, 2008.
[6] D.G. Krige: A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa, 52, 119-139, 1951.
[7] A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, Mi. Saisana, S. Tarantola: Global Sensitivity Analysis. The Primer. John Wiley & Sons, 155–167, 2008.

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