@article{Legatiuk,
  author    = {Legatiuk, Dmitrii},
  title     = {Mathematical Modelling by Help of Category Theory: Models and Relations between Them},
  series = {mathematics},
  volume    = {2021},
  journal   = {mathematics},
  number    = {volume 9, issue 16, article 1946},
  publisher = {MDPI},
  address   = {Basel},
  doi       = {10.3390/math9161946},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20210817-44844},
  pages     = {17},
  abstract  = {The growing complexity of modern practical problems puts high demand on mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice is becoming particularly important. Methods for model comparison and model choice typically used in practical applications nowadays are computationbased, and thus time consuming and computationally costly. Therefore, it is necessary to develop other approaches to working abstractly, i.e., without computations, with mathematical models. An abstract description of mathematical models can be achieved by the help of abstract mathematics, implying formalisation of models and relations between them. In this paper, a category theory-based approach to mathematical modelling is proposed. In this way, mathematical models are formalised in the language of categories, relations between the models are formally defined and several practically relevant properties are introduced on the level of categories. Finally, an illustrative example is presented, underlying how the category-theory based approach can be used in practice. Further, all constructions presented in this paper are also discussed from a modelling point of view by making explicit the link to concrete modelling scenarios.},
  subject      = {Kategorientheorie},
  language  = {en}
}
@article{LegatiukWeiszPatrault,
  author    = {Legatiuk, Dmitrii and Weisz-Patrault, Daniel},
  title     = {Coupling of Complex Function Theory and Finite Element Method for Crack Propagation Through Energetic Formulation: Conformal Mapping Approach and Reduction to a Riemann-Hilbert Problem},
  series = {Computational Methods and Function Theory},
  volume    = {2021},
  journal   = {Computational Methods and Function Theory},
  publisher = {Springer},
  address   = {Heidelberg},
  doi       = {10.1007/s40315-021-00403-7},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20210805-44763},
  pages     = {1 -- 23},
  abstract  = {In this paper we present a theoretical background for a coupled analytical-numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical-numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann-Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.},
  subject      = {Angewandte Mathematik},
  language  = {en}
}