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2022 | Buch

Tensorkalkül mit objektorientierten Matrizen für numerische Methoden in Mechanik und Ingenieurwissenschaften

Eine Grundlage für Tensor-/Matrix-Algorithmen der Finite-Elemente-Methode

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Die Intention des Buches ist es für die numerischen Algorithmen zur Berechnung von Tragstrukturen des Ingenieurwesens eine Synthese von klassischen Matrizen- und Tensormethoden einerseits und moderner Software-Technologie und objektorientierter Methoden anderseits vorzunehmen. Dabei ist es das Ziel, ein durchgängiges Methodenkonzept zu entwickeln, mit dem die theoretischen Modellierungsgrundlagen der Mechanik/Statik nahtlos in numerische Berechnungsprogramme umgesetzt werden können, ohne methodische Brüche in Teilbereichen überwinden zu müssen. Klassisches Beispiel dafür ist die symbolische Notation des Tensor- und Matrizenkalküls. Dieses ist traditionell stringent entwickelt, kompakt formuliert und auch anschaulich dargestellt. Jedoch der Anfänger erkennt i. a. nicht, wie diese Methodik auf digitalen Rechenanlagen in effizienten Programmcode umgesetzt werden kann. Der Schlüssel dazu ist bei der Modellierung der Tragstrukturen und beim Software-Design das objektorientierte Paradigma, das inzwischen im Ingenieurwesen hinlänglich untersucht, vgl. [Hartmann 2000] und weit verbreitet in der Software Entwicklung, wie z. B. mit den Programmiersprachen C++, Java u. a., angewendet wird.

Inhaltsverzeichnis

Frontmatter
Kapitel 1. Einleitung
Zusammenfassung
The main intent of the book is to provide a synthesis between classical matrix and tensor methods on the one hand and modern software technology on the other. For this purpose, the approach is used to develop a consistent method concept with the help of object-oriented methods, by which the theoretical modeling fundamentals of mechanics can be seamlessly converted into numerical computation programs for the solution of technical problems, without having to overcome methodological breaks in subareas. A classic example of this is the symbolic notation of the tensor and matrix calculus with its complex syntax and semantics, which appears to be less suitable for direct implementation in computer algorithms.
Udo F. Meißner
Kapitel 2. Grundlagen der Matrizenrechnung
Zusammenfassung
In this introductory chapter the most important basics of the classical matrix calculus are compiled, as found in [Zurmühl/Falk 1986], [Pestel/Leckie 1963]. The primary goal is to introduce the reader to the index notation for multi-dimensional matrices in order to base all arithmetic operations on it in the following context. The reader may become sufficiently familiar with this notation and gets used to the possibly strange notation, which differs considerably from the traditional symbolic notation of related publications.
Udo F. Meißner
Kapitel 3. Objektorientierte Matrizen
Zusammenfassung
In this chapter about object-oriented matrices, the focus is first placed on the structure and storage of multi-dimensional matrices, which are also mapped by applications on the digital von Neumann computers into the one-dimensional physical main memory. The section about hyper matrices explains this context and presents different representations for multi-dimensional matrices with the corresponding mathematical mapping rules.The representation of matrices is then followed by the exemplary implementation of the vectors and the multi-dimensional matrices in object-oriented matrix classes by use of the programming language C++.
Udo F. Meißner
Kapitel 4. Grundlagen der Tensorrechnung
Zusammenfassung
Tensor calculus, with its consistent formulation of invariants and transformations, has for years acquired a high significance in engineering. Thus, early works on the mechanics of load-bearing structures, such as [Green/Zerna 1954], which renewed the formulation of the theory, and compendia of mathematics on the tensor calculus in index notation, such as [Duschek/Hochrainer 1968], can serve as a basis at this point. In this chapter, the most important basics of tensor algebra are recapitulated in this context, especially to make the approach of the object-oriented matrix calculus comprehensible and to provide the ability to apply the presented methodology consistently.
Udo F. Meißner
Kapitel 5. Tensoranalysis für Finite Elemente
Zusammenfassung
In this chapter, the fundamentals of tensor analysis are compiled for use within the finite element method. As with the classical approaches, such as [Klingbeil 1966], the differential geometry of the spatial bodies is treated first, followed by the mechanical fundamentals for the deformation of elastic continua. The tensor representations are then specialized to the finite element approximations for the capture of the geometry and the description of the displacement fields.
Udo F. Meißner
Kapitel 6. Objektorientierte Numerik für Finite Elemente
Zusammenfassung
Finally, the application of the object-oriented matrix calculus is presented in two sections about a FEM-parallelogram and a FEM-triangle pannel element on the basis of typical C++ routines, which exemplify in the source code the calculations of the differential geometries, the matrices of the stiffness relationships and of the internal forces for both finite elements as well as the computation of the corresponding numerical results. These elements are a bilinear parallelogram element and a linear triangular element, each with abstracted elasticity parameters, so that the calculated results for deformations and stresses under tension and shear can be verified in the elementary way. The program code, which is kept simple and clear, is primarily intended to demonstrate here how the required matrix objects of the new matrix classes can be created and managed, and how the tensor/matrix arithmetic can be handled intuitively and clearly with the overloaded syntax for matrix operations.
Udo F. Meißner
Kapitel 7. Schlusswort
Zusammenfassung
With the present concept, a continuous synthesis from theoretical tensor notation and traditional matrix calculus to new classes of matrices and object-oriented algorithms for mechanics and engineering problems was accomplished. Conclusively, the advantage of this approach was illustrated by exemplary applications of the finite element method.
Udo F. Meißner
Backmatter
Metadaten
Titel
Tensorkalkül mit objektorientierten Matrizen für numerische Methoden in Mechanik und Ingenieurwissenschaften
verfasst von
Udo F. Meißner
Copyright-Jahr
2022
Electronic ISBN
978-3-658-39881-1
Print ISBN
978-3-658-39880-4
DOI
https://doi.org/10.1007/978-3-658-39881-1