TY - CHAP A1 - Al-Yasiri, Zainab A1 - Gürlebeck, Klaus ED - Gürlebeck, Klaus ED - Lahmer, Tom T1 - ON BOUNDARY VALUE PROBLEMS FOR P-LAPLACE AND P-DIRAC EQUATIONS T2 - Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar N2 - The p-Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p-Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p-Laplace equation for 2 < p < 3 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p-Laplace equation into the p-Dirac equation. This equation will be solved iteratively by using a fixed point theorem. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Building Information Modeling KW - Computerunterstütztes Verfahren KW - 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 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-27928 SN - 1611-4086 ER - TY - CHAP A1 - Pastohr, Henry A1 - Kornadt, Oliver A1 - Gürlebeck, Klaus T1 - Numerische Untersuchungen zum Thermischen Strömungsverhalten im Aufwindkraftwerk T1 - Numerische und analytische Untersuchungen zum Strömungsverhalten im Aufwindkraftwerk N2 - Das Aufwindkraftwerk ist eine thermo- hydrodynamische Maschine zur Elektroenergiegewinnung, bestehend aus einem Treibhaus, einem Kamin und einer oder mehreren Turbinen. In dieser Studie wurden numerische Ergebnisse zum thermischen Strömungsverhalten in einem Aufwindkraftwerk unter der Berücksichtigung der Teilmodelle Erdboden, Kollektor, Atmosphäre, Umlenkung, Kamin und Turbine erhaltenden. Hierzu wurden die stationären Grundgleichungen der Thermofluiddynamik auf strukturierten, körperangepassten und rotationssymmetrischen Gittern unter Beachtung aller Rand- und Kopplungsbedingungen numerisch mit dem finite Volumenverfahren gelöst. Besonderes Augenmerk wurde dabei auf die Kalibrierung des Modells im Ruhezustand, auf die numerische Simulation, auf den Einfluss der Strahlung, auf die Betrachtung der Turbine, auf das Dichtemodell sowie auf den turbulenten Strömungszustand gelegt. Die erhaltenen Ergebnisse werden durch Approximationen 2. Ordnung, Gitterunabhängigkeit und durch einen sehr geringen Abbruchfehler charakterisiert. Für 4 verschiedene Einstrahlungen wurden die Verläufe von Temperatur und Geschwindigkeit im Aufwindkraftwerk erhalten. Zusätzlich sind für Vergleichszwecke der Massenstrom, der Temperaturhub, die Leistung an der Turbine und der Wirkungsgrad der Anlage bestimmt wurden. Aufbauend auf den Berechnungen in dieser Arbeit und den numerischen und analytischen Berechnungen in [1] können nun erweiterte Parameterstudien und instationäre Simulationen zum Aufwindkraftwerk durchgeführt werden. KW - Aufwindkraftwerk KW - Strömungsmechanik Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20111215-3436 ER - TY - CHAP A1 - Gürlebeck, Klaus A1 - Bock, Sebastian A1 - Falcao, M. Irene T1 - Applications of Bergman kernel functions N2 - In this paper we revisit the so-called Bergman kernel method (BKM) for solving conformal mapping problems. This method is based on the reproducing property of the Bergman kernel function. The main drawback of this well known technique is that it involves an orthonormalization process and thus is numerically unstable. This difficulty can be, in some cases, overcome by using the Maple system, which makes no use of numeric quadrature. We illustrate this implementation by presenting a numerical example. The construction of reproducing kernel functions is not restricted to real dimension 2. Results concerning the construction of Bergman kernel functions in closed form for special domains in the framework of hypercomplex function theory suggest that BKM can also be extended to mapping problems in higher dimensions, particularly 3-dimensional cases. We describe such a generalized BKM-approach and present numerical examples obtained by the use of specially developed software packages for quaternions. KW - Konforme Abbildung KW - Kernel Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20111215-2994 ER - TY - JOUR A1 - Lahmer, Tom A1 - Bock, Sebastian A1 - Hildebrand, Jörg A1 - Gürlebeck, Klaus T1 - Non-destructive identification of residual stresses in steel under thermal loadings JF - Inverse Problems in Science and Engineering N2 - Non-destructive identification of residual stresses in steel under thermal loadings KW - Angewandte Mathematik KW - Stochastik KW - Strukturmechanik Y1 - 2016 SP - 1 EP - 17 ER - TY - CHAP A1 - Bock, Sebastian A1 - Gürlebeck, Klaus ED - Gürlebeck, Klaus ED - Könke, Carsten T1 - A Coupled Ritz-Galerkin Approach Using Holomorphic and Anti-holomorphic Functions N2 - The contribution focuses on the development of a basic computational scheme that provides a suitable calculation environment for the coupling of analytical near-field solutions with numerical standard procedures in the far-field of the singularity. The proposed calculation scheme uses classical methods of complex function theory, which can be generalized to 3-dimensional problems by using the framework of hypercomplex analysis. The adapted approach is mainly based on the factorization of the Laplace operator EMBED Equation.3 by the Cauchy-Riemann operator EMBED Equation.3 , where exact solutions of the respective differential equation are constructed by using an orthonormal basis of holomorphic and anti-holomorphic functions. KW - Architektur KW - CAD KW - Computerunterstütztes Verfahren Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170327-29281 UR - http://euklid.bauing.uni-weimar.de/ikm2006/index.php_lang=de&what=papers.html ER - TY - CHAP A1 - Nguyen, Manh Hung A1 - Gürlebeck, Klaus ED - Gürlebeck, Klaus ED - Lahmer, Tom ED - Werner, Frank T1 - ON M-CONFORMAL MAPPINGS AND GEOMETRIC PROPERTIES 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 - Monogenic functions play a role in quaternion analysis similarly to that of holomorphic functions in complex analysis. A holomorphic function with nonvanishing complex derivative is a conformal mapping. It is well-known that in Rn+1, n ≥ 2 the set of conformal mappings is restricted to the set of Möbius transformations only and that the Möbius transformations are not monogenic. The paper deals with a locally geometric mapping property of a subset of monogenic functions with nonvanishing hypercomplex derivatives (named M-conformal mappings). It is proved that M-conformal mappings orthogonal to all monogenic constants admit a certain change of solid angles and vice versa, that change can characterize such mappings. In addition, we determine planes in which those mappings behave like conformal mappings in the complex plane. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Computerunterstütztes Verfahren Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-27833 UR - http://euklid.bauing.uni-weimar.de/ikm2012 SN - 1611-4086 ER - TY - CHAP A1 - Hommel, Angela A1 - Gürlebeck, Klaus ED - Gürlebeck, Klaus ED - Lahmer, Tom T1 - THE RELATIONSHIP BETWEEN LINEAR ELASTICITY THEORY AND COMPLEX FUNCTION THEORY STUDIED ON THE BASIS OF FINITE DIFFERENCES T2 - Digital Proceedings, International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering : July 20 - 22 2015, Bauhaus-University Weimar N2 - 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. KW - Angewandte Informatik KW - Angewandte Mathematik KW - Building Information Modeling KW - Computerunterstütztes Verfahren KW - 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 Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20170314-28010 SN - 1611-4086 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 - Al-Yasiri, Zainab Riyadh Shaker A1 - Mutashar, Hayder Majid A1 - Gürlebeck, Klaus A1 - Lahmer, Tom ED - Shafiullah, GM T1 - Damage Sensitive Signals for the Assessment of the Conditions of Wind Turbine Rotor Blades Using Electromagnetic Waves JF - Infrastructures N2 - One of the most important renewable energy technologies used nowadays are wind power turbines. In this paper, we are interested in identifying the operating status of wind turbines, especially rotor blades, by means of multiphysical models. It is a state-of-the-art technology to test mechanical structures with ultrasonic-based methods. However, due to the density and the required high resolution, the testing is performed with high-frequency waves, which cannot penetrate the structure in depth. Therefore, there is a need to adopt techniques in the fields of multiphysical model-based inversion schemes or data-driven structural health monitoring. Before investing effort in the development of such approaches, further insights and approaches are necessary to make the techniques applicable to structures such as wind power plants (blades). Among the expected developments, further accelerations of the so-called “forward codes” for a more efficient implementation of the wave equation could be envisaged. Here, we employ electromagnetic waves for the early detection of cracks. Because in many practical situations, it is not possible to apply techniques from tomography (characterized by multiple sources and sensor pairs), we focus here on the question of whether the existence of cracks can be determined by using only one source for the sent waves. KW - Windkraftwerk KW - Rotorblatt KW - Elektrostatische Welle KW - MATLAB KW - wind turbine rotor blades KW - electromagnetic waves KW - crack detection KW - Empire XPU 8.01 KW - Matlab KW - OA-Publikationsfonds2022 Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:gbv:wim2-20220831-47093 UR - https://www.mdpi.com/2412-3811/7/8/104 VL - 2022 IS - Volume 7, Issue 8 (August 2022), article 104 PB - MDPI CY - Basel 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 -