In structural analysis with multivariate random fields, the underlying distribution functions, the autocorrelations, and the crosscorrelations require an extensive quantification. While those parameters are difficult to measure in experiments, a lack of knowledge is included. Therefore, polymorphic uncertainty models are attained by involving uncertainty models with epistemic characteristic for the quantification of the stochastic models in this contribution. Three extensions for random fields with polymorphic uncertainty modeling are introduced. Interval probability based random fields, fuzzy probability based random fields, and structural dependent autocorrelations for random fields are shown. Applications for engineering problems are shown for each extension, where uncertainty analysis of structures with different materials is performed. In this contribution, a damage simulation of a concrete beam with interval valued parametrization of stochastic models, an application for porous media in a multiphysical structural analysis with fuzzy valued parametrization and an uncertainty analysis with structural dependent autocorrelations for timber structures are presented.

Deterministic design and a priori parameters are used in traditional optimization approaches. The material characteristics of solid wood are not deterministic in reality. Hence, realistic optimization and simulation methods need to take the uncertainties of parameters into account. The uncertainty characteristics of wood are mainly originated in natural variation. In addition to this, incertitudes from lack of knowledge are inherent. Accordingly, the aleatoric approach of randomness can be expanded to a polymorphic uncertainty model. Fuzzy probability based randomness is used in this work. Therefore, the epistemic approach of fuzziness is taken into account. The distribution functions of random variables are parametrized by fuzzy variables. So coupling of both, aleatoric and epistemic uncertainties, is involved.Interactions of fuzzy variables and crosscorrelations of random variables are considered among and within the parameters. Crosscorrelated random fields are used to represent spatial variation of material parameters. The autocovariance structures are modeled structurally dependent on the tree trunk axes. FEM results are applied as basic solutions of a loaded timber structure. A local orthotropic material formulation with respect to specifically located tree trunk axes is used. The optimal positions of the tree trunk axes for each wooden log are examined as design parameters. Polymorphic uncertainty is used to describe a priori parameters. The developed methods for uncertainty analysis are embedded in an automated and parallelized optimization processing. An analysis of a two-tier glulam beam, according to a purlin of a timber roof construction, is shown as numerical example for the optimization framework.

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The uncertainty characteristics of wood are mainly affected by natural variation. Out of this, the traditional approach of stochastic variables can be expanded to a polymorphic uncertainty model. Therefore, e.g. fuzzy probability based randomness is used by extending stochastic variables with fuzzy variables in parameterization concerning the distribution functions, see [1] and [2]. The coupling of both aleatoric and epistemic uncertainty is involved in the uncertainty analysis.The FEM is applied as basic solution of particular load situations on the focused timber structure. A local orthotropic formulation is used and the properties are evaluated on each integration point with respect to the tree trunk axis.In this contribution, an approach to polymorphic uncertainty modeling for timber structures is introduced. According to [3], models representing the spatial variation and interdependencies of material parameters are necessary for a realistic representation in numerical simulation. For this purpose, on the one hand interactions between fuzzy variables, on the other hand correlations among random variables are considered. Random fields are utilized to capture spatially varying material properties in context with the discretization of FE. Approaches to both spatially and structurally depending autocorrelations along with crosscorrelations based on [4] are presented.The preliminary steps aim at an optimization in design of timber structures, provided that polymorphic uncertain design as well as a priori parameters are considered. The developed tools for uncertainty analysis and the basic FEM solution are prepared as a basis for an automated optimization processing, whereas they are preferably parallelized, incorporating methods for reducing the numerical effort. Results of the uncertainty analysis of a timber structure are shown exemplary.

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