Modeling Crack Patterns by Modified STIT Tessellations

Random planar tessellations are presented which are generated by subsequent division of their polygonal cells. The purpose is to develop parametric models for crack patterns appearing at length scales which can change by orders of magnitude in areas such as nanotechnology, materials science, soft matter, and geology. Using the STIT tessellation as a reference model and comparing with phenomena in real crack patterns, three modifications of STIT are suggested. For all these models a simulation tool, which also yields several statistics for the tessellation cells, is provided on the web. The software is freely available via a link given in the bibliography of this article. The present paper contains results of a simulation study indicating some essential features of the models. Finally, an example of a real fracture pattern is considered which is obtained using the deposition of a thin metallic film onto an elastomer material – the results of this are compared to the predictions of the model.

Regenerative processes for Poisson zero polytopes

Let (Mt:t>0) be a Markov process of tessellations of ℝℓ, and let (𝒞t:t>0) be the process of their zero cells (zero polytopes), which has the same distribution as the corresponding process for Poisson hyperplane tessellations. In the present paper we describe the stationary zero cell process (at𝒞at:t∈ℝ),a>1, in terms of some regenerative structure and we show that it is a Bernoulli flow. An important application is to STIT tessellation processes.

Line Segments in the Isotropic Planar Stit Tessellation

This paper presents a powerful characterisation for the structure of internal vertices of the STIT'sI-segments. The characterisation allows certain mathematical analyses to be performed easily. We demonstrate this by deriving new results for various topological properties of the tessellation: for example, the numbers of various types of edge and cell side within the typicalI-segment. The characterisation also provides a tool for the calculations of metric properties of the tessellation; many new length distributions and frame-coverage results are given.

Line Segments in the Isotropic Planar Stit Tessellation

This paper presents a powerful characterisation for the structure of internal vertices of the STIT's I-segments. The characterisation allows certain mathematical analyses to be performed easily. We demonstrate this by deriving new results for various topological properties of the tessellation: for example, the numbers of various types of edge and cell side within the typical I-segment. The characterisation also provides a tool for the calculations of metric properties of the tessellation; many new length distributions and frame-coverage results are given.

STIT tessellations are Bernoulli and standard

Let (Yt:t>0) be a STIT tessellation process and a>1. We prove that the random sequence (anYan:n∈ℤ) is a non-anticipating factor of a Bernoulli shift. We deduce that the continuous time process (atYat:t∈ℝ) is a Bernoulli flow. We use the techniques and results in Martínez and Nagel [Ergodic description of STIT tessellations. Stochastics 84(1) (2012), 113–134]. We also show that the filtration associated to the non-anticipating factor is standard in Vershik’s sense.

Mixing properties for STIT tessellations

The so-called STIT tessellations form a class of homogeneous (spatially stationary) tessellations inRdwhich are stable under the nesting/iteration operation. In this paper we establish the mixing property for these tessellations and give the decay rate of P(A∩M= ∅,ThB∩M= ∅) / P(A∩Y= ∅)P(B∩Y= ∅) − 1, whereAandBare both compact connected sets,his a vector ofRd,This the corresponding translation operator, andMis a STIT tessellation.

Mixing properties for STIT tessellations

The so-called STIT tessellations form a class of homogeneous (spatially stationary) tessellations in Rd which are stable under the nesting/iteration operation. In this paper we establish the mixing property for these tessellations and give the decay rate of P(A ∩ M = ∅, ThB ∩ M = ∅) / P(A ∩ Y = ∅)P(B ∩ Y = ∅) − 1, where A and B are both compact connected sets, h is a vector of Rd, Th is the corresponding translation operator, and M is a STIT tessellation.