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Numerical Homogenization for Linear Elasticity in Translation Invariant Spaces

  • Composite materials are used in many modern tools and engineering applications and consist of two or more materials that are intermixed. Features like inclusions in a matrix material are often very small compared to the overall structure. Volume elements that are characteristic for the microstructure can be simulated and their elastic properties are then used as a homogeneous material on the macroscopic scale. Moulinec and Suquet [2] solve the so-called Lippmann-Schwinger equation, a reformulation of the equations of elasticity in periodic homogenization, using truncated trigonometric polynomials on a tensor product grid as ansatz functions. In this thesis, we generalize their approach to anisotropic lattices and extend it to anisotropic translation invariant spaces. We discretize the partial differential equation on these spaces and prove the convergence rate. The speed of convergence depends on the smoothness of the coefficients and the regularity of the ansatz space. The spaces of translates unify the ansatz of Moulinec and Suquet with de la Vallée Poussin means and periodic Box splines, including the constant finite element discretization of Brisard and Dormieux [1]. For finely resolved images, sampling on a coarser lattice reduces the computational effort. We introduce mixing rules as the means to transfer fine-grid information to the smaller lattice. Finally, we show the effect of the anisotropic pattern, the space of translates, and the convergence of the method, and mixing rules on two- and three-dimensional examples. References [1] S. Brisard and L. Dormieux. “FFT-based methods for the mechanics of composites: A general variational framework”. In: Computational Materials Science 49.3 (2010), pp. 663–671. doi: 10.1016/j.commatsci.2010.06.009. [2] H. Moulinec and P. Suquet. “A numerical method for computing the overall response of nonlinear composites with complex microstructure”. In: Computer Methods in Applied Mechanics and Engineering 157.1-2 (1998), pp. 69–94. doi: 10.1016/s00457825(97)00218-1.
  • Kompositmaterialien werden häufig für moderne Werkstoffe und Anwendungen im Ingenieurwesen verwendet. Einschlüsse oder Fasern sind dabei häufig viel kleiner als das Werkstück selbst. Für Referenzelemente, die charakteristisch für das Komposit sind, können die makroskopischen elastischen Eigenschaften berechnet und als homogenes Material auf der Makroebene eingesetzt werden. Basierend auf den Elastizitätsgleichungen der periodischen Homogenisierung lösen Moulinec und Suquet [2] die sogenannte Lippmann-Schwinger-Gleichung. Sie diskretisieren die Gleichung dabei auf einem Tensorproduktgitter mittels trigonometrischer Polynome als Ansatzfunktionen. In dieser Dissertation verallgemeinern wir ihr Gitter auf anisotrope Muster und erweitern ihren Ansatzraum auf anisotrope translationsinvariante Räume, für die wir einen Konvergenzbeweis führen. Diese Funktionenräume beinhalten zudem de la Vallée Poussin-Mittel und periodische Box-Splines. Letztere sind eine Generalisierung der konstanten Finiten Elemente von Brisard und Dormieux [1]. Für hochaufgelöste Daten führen wir Mischregeln ein, die Rechungen auf einem gröberen Gitter ermöglichen und dazu Informationen des feinen Gitters verwenden. Wir demonstrieren die Möglichkeiten von anisotropen Gittern, Translateräumen und Mischregeln an zwei- und dreidimensionalen Beispielen. Literatur [1] S. Brisard und L. Dormieux. “FFT-based methods for the mechanics of composites: A general variational framework”. In: Computational Materials Science 49.3 (2010), S. 663–671. doi: 10.1016/j.commatsci.2010.06.009. [2] H. Moulinec und P. Suquet. “A numerical method for computing the overall response of nonlinear composites with complex microstructure”. In: Computer Methods in Applied Mechanics and Engineering 157.1-2 (1998), S. 69–94. doi: 10.1016/s00457825(97)00218-1.

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Metadaten
Author:Dennis Merkert
URN:urn:nbn:de:hbz:386-kluedo-53355
Advisor:Bernd Simeon
Document Type:Doctoral Thesis
Language of publication:English
Date of Publication (online):2018/07/17
Year of first Publication:2018
Publishing Institution:Technische Universität Kaiserslautern
Granting Institution:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2018/06/29
Date of the Publication (Server):2018/07/18
Tag:FFT; Lippmann-Schwinger equation; composites; homogenization; micromechanics; pattern; sampling; translation invariant spaces
GND Keyword:Homogenisierung <Mathematik>; Diskrete Fourier-Transformation; Elastizität; Lineare partielle Differentialgleichung; Harmonische Analyse; Mikrostruktur
Page Number:III, 102
Faculties / Organisational entities:Kaiserslautern - Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):42-XX FOURIER ANALYSIS / 42Bxx Harmonic analysis in several variables (For automorphic theory, see mainly 11F30) / 42B05 Fourier series and coefficients
42-XX FOURIER ANALYSIS / 42Bxx Harmonic analysis in several variables (For automorphic theory, see mainly 11F30) / 42B35 Function spaces arising in harmonic analysis
42-XX FOURIER ANALYSIS / 42Bxx Harmonic analysis in several variables (For automorphic theory, see mainly 11F30) / 42B37 Harmonic analysis and PDE [See also 35-XX]
45-XX INTEGRAL EQUATIONS / 45Axx Linear integral equations / 45A05 Linear integral equations
65-XX NUMERICAL ANALYSIS / 65Txx Numerical methods in Fourier analysis / 65T40 Trigonometric approximation and interpolation
65-XX NUMERICAL ANALYSIS / 65Txx Numerical methods in Fourier analysis / 65T50 Discrete and fast Fourier transforms
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Bxx Elastic materials / 74B05 Classical linear elasticity
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Exx Material properties given special treatment / 74E30 Composite and mixture properties
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Mxx Special kinds of problems / 74M25 Micromechanics
74-XX MECHANICS OF DEFORMABLE SOLIDS / 74Sxx Numerical methods [See also 65-XX, 74G15, 74H15] / 74S25 Spectral and related methods
Licence (German):Creative Commons 4.0 - Namensnennung, nicht kommerziell, keine Bearbeitung (CC BY-NC-ND 4.0)