Refine
Has Fulltext
- yes (13) (remove)
Document Type
- Conference Proceeding (8)
- Article (5)
Institute
- Professur Angewandte Mathematik (3)
- In Zusammenarbeit mit der Bauhaus-Universität Weimar (2)
- Institut für Bauinformatik, Mathematik und Bauphysik (IBMB) (2)
- Professur Informatik im Bauwesen (2)
- Graduiertenkolleg 1462 (1)
- Institut für Strukturmechanik (ISM) (1)
- Junior-Professur Komplexe Tragwerke (1)
- Professur Modellierung und Simulation - Konstruktion (1)
Keywords
- Computerunterstütztes Verfahren (6)
- Angewandte Informatik (5)
- Angewandte Mathematik (5)
- Architektur <Informatik> (2)
- Building Information Modeling (2)
- CAD (2)
- Data, information and knowledge modeling in civil engineering; Function theoretic methods and PDE in engineering sciences; Mathematical methods for (robotics and) computer vision; Numerical modeling in engineering; Optimization in engineering applications (2)
- OA-Publikationsfonds2022 (2)
- Aerodynamik (1)
- Aeroelastizität (1)
Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures
(2022)
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.
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.
It is well-known that the solution of the fundamental equations of linear elasticity for a homogeneous isotropic material in plane stress and strain state cases can be equivalently reduced to the solution of a biharmonic equation. The discrete version of the Theorem of Goursat is used to describe the solution of the discrete biharmonic equation by the help of two discrete holomorphic functions. In order to obtain a Taylor expansion of discrete holomorphic functions we introduce a basis of discrete polynomials which fulfill the so-called Appell property with respect to the discrete adjoint Cauchy-Riemann operator. All these steps are very important in the field of fracture mechanics, where stress and displacement fields in the neighborhood of singularities caused by cracks and notches have to be calculated with high accuracy. Using the sum representation of holomorphic functions it seems possible to reproduce the order of singularity and to determine important mechanical characteristics.