Keywords: random fields, Monte Carlo simulation, finite element analysis, uncertainty quantification, pattern recognition, scale-space theory, parameterisation.

System behaviour is influenced by a high number of uncertainties. A key source of uncertainty is the scatter of physical boundary conditions and system characteristics. To predict more reliably the real system's behaviour by means of simulation, probabilistic models can be created that account for aleatory uncertainties. Local effects of uncertainty can be modelled by random fields in the n-dimensional feature space [1]. Random fields describing geometry, loads or material properties may possess a highly complex structure. Homogeneous Gaussian random fields are not well suited for the representation of structural features such as shape or connectivity [2].

We have identified four general constraints for the parameterisation of highly structured morphologies: similarity, structure preservation, locality, and reversibility. Similarity means that two similar random fields ought to have similar parameter sets. Structure preservation of features is essential for a reproduction of features in different samples. Locality means that a sample's deviation in one specific region leads to a locally limited deviation of the parameters in the parameter space. To synthesise samples for the simulation an inverse transformation is required.

In this paper we develop a new multi-scale method for a pattern recognition based parameterisation of random fields: The approximated differential scale-space (ADSS) that is based on the so called scale-space theory [3]. The ADSS is based upon a convolution with a Gaussian filter on multiple scales. The multi-scale approach enables the detection of features of different orders of magnitude. For each scale a blob detection finds the relevant features yielding a set of parameters. The ADSS leads to a set of parameters that describes the features of the random field separated by location and frequency.

In one- and two-dimensional application examples it is shown that the ADSS fulfils the four requirements mentioned above. As a result of the independence of the parameters it is possible to synthesise samples on any scale thus accounting for different levels of detail being taken into account. The proposed ADSS can be used for applications of any dimensionality as there is no methodical restriction to the number of dimensions.

1

E. Vanmarcke, "Random Fields: Analysis and Synthesis", MIT Press, Cambridge, USA, 1983. doi:10.1115/1.3269255

2

L. Graham-Brady, X.F. Xu, "Simulation and Classification of Random Multiphase Materials through Short-Range Correlation", 7th AIAA Non-Deterministic Approaches Conference, Newport, USA, May 1-5, 2006.

3

T. Lindeberg, "Scale-Space Theory in Computer Vision", Kluwer Academic Publishers, Boston, USA, 1994.

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