Abstract This paper introduces a precise and easy to use method for defining the microstructure of heterogeneous solids. This method is based on the concept of Heterogeneous Composite Bézier Hyperpatch, and allows to accurately represent the primary material proportions, as well as the size and shape of the material phases. The solid microstructure is modelled using two functions: a material distribution function (to compute the portion of the solid volume occupied by each primary material), and a modified Perlin noise function that determines the shape and size of each primary material phase. With this method, the position and orientation of the solid in the modeling space does not affect the portion of its volume that is occupied by each primary material, nor the shape and size of the phases. However, the solid microstructure is coherently and automatically modified when the shape of the solid is edited. Regarding continuity, this method allows to define to which extent continuity (both in shape and material distribution) has to be preserved at the junction of the cells that compose the solid. This makes modeling geometrically complex figures very easy.

The paper deals with the state estimation of nonlinear stochastic dynamic systems. The stress is laid on the assessment of the estimate error, which is caused by the violation of the estimator design assumptions. The assessment is based on measures comparing estimators actual working conditions and the assumptions under which the estimators have been proposed. In particular, the measures of nonlinearity and non-Gaussianity are discussed. The measures are briefly introduced and selected typical representatives are detailed with respect to their implementation. Performance of the measures is evaluated in the framework of the Gaussian filters in a numerical study.