TY - JOUR A1 - Legatiuk, Dmitrii A1 - Weisz-Patrault, Daniel T1 - 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 JF - Computational Methods and Function Theory N2 - 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. KW - Angewandte Mathematik KW - Finite-Elemente-Methode KW - Rissausbreitung KW - Modellierung KW - Bruchmechanik KW - fracture mechanics KW - crack propagation KW - coupling KW - energetic approach Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20210805-44763 UR - https://link.springer.com/article/10.1007/s40315-021-00403-7 VL - 2021 SP - 1 EP - 23 PB - Springer CY - Heidelberg ER - TY - CHAP A1 - Legatiuk, Dmitrii A1 - Bock, Sebastian A1 - Gürlebeck, Klaus ED - Gürlebeck, Klaus ED - Lahmer, Tom ED - Werner, Frank T1 - THE PROBLEM OF COUPLING BETWEEN ANALYTICAL SOLUTION AND FINITE ELEMENT METHOD T2 - Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 04 - 06 2012, Bauhaus-University Weimar N2 - This paper is focused on the first numerical tests for coupling between analytical solution and finite element method on the example of one problem of fracture mechanics. The calculations were done according to ideas proposed in [1]. The analytical solutions are constructed by using an orthogonal basis of holomorphic and anti-holomorphic functions. For coupling with finite element method the special elements are constructed by using the trigonometric interpolation theorem. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Computerunterstütztes Verfahren Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-27730 UR - http://euklid.bauing.uni-weimar.de/ikm2012 SN - 1611-4086 ER - TY - JOUR A1 - Legatiuk, Dmitrii T1 - Mathematical Modelling by Help of Category Theory: Models and Relations between Them JF - mathematics N2 - 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. KW - Kategorientheorie KW - Modellierung KW - Modellierungsmethode KW - OA-Publikationsfonds2021 Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20210817-44844 UR - https://www.mdpi.com/2227-7390/9/16/1946?type=check_update&version=1 VL - 2021 IS - volume 9, issue 16, article 1946 PB - MDPI CY - Basel ER - TY - JOUR A1 - Kavrakov, Igor A1 - Legatiuk, Dmitrii A1 - Gürlebeck, Klaus A1 - Morgenthal, Guido T1 - A categorical perspective towards aerodynamic models for aeroelastic analyses of bridge decks JF - Royal Society Open Science N2 - Reliable modelling in structural engineering is crucial for the serviceability and safety of structures. A huge variety of aerodynamic models for aeroelastic analyses of bridges poses natural questions on their complexity and thus, quality. Moreover, a direct comparison of aerodynamic models is typically either not possible or senseless, as the models can be based on very different physical assumptions. Therefore, to address the question of principal comparability and complexity of models, a more abstract approach, accounting for the effect of basic physical assumptions, is necessary. This paper presents an application of a recently introduced category theory-based modelling approach to a diverse set of models from bridge aerodynamics. Initially, the categorical approach is extended to allow an adequate description of aerodynamic models. Complexity of the selected aerodynamic models is evaluated, based on which model comparability is established. Finally, the utility of the approach for model comparison and characterisation is demonstrated on an illustrative example from bridge aeroelasticity. The outcome of this study is intended to serve as an alternative framework for model comparison and impact future model assessment studies of mathematical models for engineering applications. KW - Brücke KW - Aerodynamik KW - Aeroelastizität KW - bridge KW - abstract modelling KW - category theory KW - bridge aerodynamics KW - bridge aeroelasticity KW - aerodynamic models KW - model complexity KW - OA-Publikationsfonds2019 Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20190314-38656 UR - https://royalsocietypublishing.org/doi/10.1098/rsos.181848 IS - Volume 6, Issue 3 ER - TY - JOUR A1 - Gürlebeck, Klaus A1 - Legatiuk, Dmitrii A1 - Webber, Kemmar T1 - Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures JF - Mathematics N2 - The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions. KW - Mauerwerk KW - Elastizitätstheorie KW - Mathematische Modellierung KW - quaternionic analysis KW - mathematical modelling KW - operator calculus KW - model comparison KW - micropolar elasticity KW - OA-Publikationsfonds2022 Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20220721-46726 UR - https://www.mdpi.com/2227-7390/10/10/1670 VL - 2022 IS - Volume 10, issue 10, article 1670 SP - 1 EP - 22 PB - MDPI CY - Basel ER - TY - JOUR A1 - Gürlebeck, Klaus A1 - Legatiuk, Dmitrii A1 - Nilsson, Henrik A1 - Smarsly, Kay T1 - Conceptual modelling: Towards detecting modelling errors in engineering applications JF - Mathematical Methods in Applied Sciences N2 - Rapid advancements of modern technologies put high demands on mathematical modelling of engineering systems. Typically, systems are no longer “simple” objects, but rather coupled systems involving multiphysics phenomena, the modelling of which involves coupling of models that describe different phenomena. After constructing a mathematical model, it is essential to analyse the correctness of the coupled models and to detect modelling errors compromising the final modelling result. Broadly, there are two classes of modelling errors: (a) errors related to abstract modelling, eg, conceptual errors concerning the coherence of a model as a whole and (b) errors related to concrete modelling or instance modelling, eg, questions of approximation quality and implementation. Instance modelling errors, on the one hand, are relatively well understood. Abstract modelling errors, on the other, are not appropriately addressed by modern modelling methodologies. The aim of this paper is to initiate a discussion on abstract approaches and their usability for mathematical modelling of engineering systems with the goal of making it possible to catch conceptual modelling errors early and automatically by computer assistant tools. To that end, we argue that it is necessary to identify and employ suitable mathematical abstractions to capture an accurate conceptual description of the process of modelling engineering systems. KW - Angewandte Mathematik KW - Angewandte Informatik KW - Ingenieurwissenschaften KW - Modellierung KW - engineering KW - abstraction KW - modelling KW - formal approaches KW - type theory Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20200110-40614 UR - https://onlinelibrary.wiley.com/doi/abs/10.1002/mma.5934 SP - 1 EP - 10 ER - TY - JOUR A1 - Cerejeiras, Paula A1 - Kähler, Uwe A1 - Legatiuk, Anastasiia A1 - Legatiuk, Dmitrii T1 - Discrete Hardy Spaces for Bounded Domains in Rn JF - Complex Analysis and Operator Theory N2 - Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in Rn. On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered. KW - Dirac-Operator KW - Randwertproblem KW - Funktionentheorie KW - discrete Dirac operator KW - discrete monogenic functions KW - discrete boundary value problems Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20210804-44746 UR - https://link.springer.com/article/10.1007/s11785-020-01047-6 VL - 2021 IS - Volume 15, article 4 SP - 1 EP - 32 PB - Springer CY - Heidelberg ER -