acase@tulane.edu / 1 / Xiao Guan

The object of this paper is to present the properties and some of the applications of the Elliptic Integrals.

<p dir="ltr">This thesis investigates the large deflection behavior of cantilever beams under various configurations and loading conditions. The primary objective is to uset an analytical model using elliptic integrals to solve the second-order non-linear differential equations that govern the deflection of these beams. The analytical model is implemented in Python and compared against Finite Element Analysis (FEA) results obtained from ANSYS, ensuring the accuracy and reliability of the model. The study examines multiple beam configurations, including straight and inclined beams, with both free and fixed tip slopes. Sensitivity analysis is conducted to assess the impact of key parameters, such as Young’s modulus, beam height, width, and length, on the deflection behavior. This analysis reveals critical insights into how variations in material properties and geometric dimensions affect beam performance. A detailed error analysis using Root Mean Square Error (RMSE) is performed to compare the analytical model's predictions with the FEA results. The error analysis highlights any discrepancies, demonstrating the robustness of the analytical approach. The results show that the analytical model, based on elliptic integrals, closely matches the FEA results across a range of configurations and loading scenarios. The insights gained from this study can be applied to optimize the design of cantilever beams in various engineering applications, including prosthetics, robotics, and structural components. Overall, this research provides a comprehensive understanding of the large deflection behavior of cantilever beams and offers a reliable analytical tool for engineers to predict beam performance under different conditions. The integration of Python-based numerical methods with classical elliptic integral solutions presents a useful approach that enhances the precision and applicability of beam deflection analysis.</p>

Submitted by JOSE VINICIUS ZAPTE BERGAMO (vinni.zapte@gmail.com) on 2018-05-21T01:27:15Z No. of bitstreams: 1 Versão Final.pdf: 1512028 bytes, checksum: 03a7fa4505560dd5c8c218ebc20d5c7a (MD5) / Rejected by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br), reason: A ficha catalográfica deve ser solicitada à biblioteca, pelo site: http://ib.rc.unesp.br/#!/biblioteca/biblioteca/ , clicar em Serviços, Ficha Catalográfica. A ficha catalográfica só pode ser elaborada por um bibliotecário. De acordo com a Resolução CFB nº 184/2017 de 29/09/2017 – na Ficha catalográfica deve constar o nome do Bibliotecário/CRB, e ser elaborada de acordo com as normas vigentes segundo à AACR2. É proibido perante a lei (Art. 297 – Código Penal) qualquer alteração documental, sem autorização do Bibliotecário responsável. DA FALSIDADE DOCUMENTAL: (I) FALSIDADE DE DOCUMENTO PUBLICO ART. 297: Falsificar, no todo ou em parte, documento publico, ou alterar documento publico verdadeiro: Pena – reclusão, de dois a seis anos, e multa. DOCUMENTO PUBLICO: é aquele elaborado por funcionário publico, de acordo com as formalidades, e desempenho de suas funções. Art. 232, CPP - Consideram-se documentos quaisquer escritos, instrumentos ou papéis, públicos ou particulares. Obs: O arquivo da dissertação também está com várias páginas em branco. Favor removê-las. on 2018-05-21T16:44:27Z (GMT) / Submitted by JOSE VINICIUS ZAPTE BERGAMO (vinni.zapte@gmail.com) on 2018-05-22T21:00:36Z No. of bitstreams: 1 Versão final.pdf: 2094478 bytes, checksum: d2ae82de50952a7c6fd4a2c3bcfafa7a (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-05-23T11:37:05Z (GMT) No. of bitstreams: 1 bergamo_jvz_me_rcla.pdf: 2053307 bytes, checksum: 136b1ac8c78bcede6e781522c69ee3c6 (MD5) / Made available in DSpace on 2018-05-23T11:37:05Z (GMT). No. of bitstreams: 1 bergamo_jvz_me_rcla.pdf: 2053307 bytes, checksum: 136b1ac8c78bcede6e781522c69ee3c6 (MD5) Previous issue date: 2018-04-24 / É bem conhecido que em Mecânica Analítica muitos problemas integráveis não tem primitivas escritas em forma de funções elementares, tais como: corpo rígido assimétrico em rotação livre; pêndulo esférico, entre outros. O uso de funções elípticas faz-se necessário para se buscar soluções analíticas desses problemas. Neste trabalho, faremos primeiramente uma revisão da teoria dessas funções adotando como referência alguns textos clássicos. Feito isso, estudaremos a formulação de problemas de dinâmica, a saber o pêndulo simples e o pião simétrico. Por fim, com as integrais desses problemas em mãos, iremos determinar suas soluções com o uso das funções elípticas de Jacobi e Weierstrass. / It is well known that in Analytical Mechanics many simple integrable problems cannot be written in terms of elementary functions, such as: rigid asymmetrical body in free rotation, spherical pendulum, among others. The use of elliptic functions becomes necessary in order to obtain analytical solutions of these problems. In this work, we present a review of the theory of these functions accordingly to some classical texts. In the sequence, we study two problems of mechanics: the simple pendulum and the symmetrical top. Finally, we will determine the solutions to these problems using of the Jacobi and Weierstrass elliptic functions.

This thesis addresses issues related to the simulation and optimization of the injection molding of short fiber-reinforced plastics (SFRPs). The injection molding process is modeled by a two phase flow problem. The simulation of the two phase flow is accompanied by the solution of the Folgar-Tucker equation (FTE) for the simulation of the moments of fiber orientation densities. The FTE requires the solution of the so called 'closure problem'', i.e. the representation of the 4th order moments in terms of the 2nd order moments. In the absence of fiber-fiber interactions and isotropic initial fiber density, the FTE admits an analytical solution in terms of elliptic integrals. From these elliptic integrals, the closure problem can be solved by a simple numerical inversion. Part of this work derives approximate inverses and analytical inverses for special cases of fiber orientation densities. Furthermore a method is presented to generate rational functions for the computation of arbitrary moments in terms of the 2nd order closure parameters. Another part of this work treats the determination of effective material properties for SFRPs by the use of FFT-based homogenization methods. For these methods a novel discretization scheme, the 'staggered grid'' method, was developed and successfully tested. Furthermore the so called 'composite voxel'' approach was extended to nonlinear elasticity, which improves the approximation of material properties at the interfaces and allows the reduction of the model order by several magnitudes compared to classical approaches. Related the homogenization we investigate optimal experimental designs to robustly determine effective elastic properties of SFRPs with the least number of computer simulations. Finally we deal with the topology optimization of injection molded parts, by extending classical SIMP-based topology optimization with an approximate model for the fiber orientations. Along with the compliance minimization by topology optimization we also present a simple shape optimization method for compensation of part warpage for an black-box production process.:Acknowledgments v Abstract vii Chapter 1. Introduction 1 1.1 Motivation 1 1.2 Nomenclature 3 Chapter 2. Numerical simulation of SFRP injection molding 5 2.1 Introduction 5 2.2 Injection molding technology 5 2.3 Process simulation 6 2.4 Governing equations 8 2.5 Numerical implementation 18 2.6 Numerical examples 25 2.7 Conclusions and outlook 27 Chapter 3. Numerical and analytical methods for the exact closure of the Folgar-Tucker equation 35 3.1 Introduction 35 3.2 The ACG as solution of Jeffery's equation 35 3.3 The exact closure 36 3.4 Carlson-type elliptic integrals 37 3.5 Inversion of R_D-system 40 3.6 Moment tensors of the angular central Gaussian distribution on the n-sphere 49 3.7 Experimental evidence for ACG distribution hypothesis 54 3.8 Conclusions and outlook 60 Chapter 4. Homogenization of SFRP materials 63 4.1 Introduction 63 4.2 Microscopic and macroscopic model of SFRP materials 63 4.3 Effective linear elastic properties 65 4.4 The staggered grid method 68 4.5 Model order reduction by composite voxels 80 4.6 Optimal experimental design for parameter identification 93 Chapter 5. Optimization of parts produced by SFRP injection molding 103 5.1 Topology optimization 103 5.2 Warpage compensation 110 Chapter 6. Conclusions and perspectives 115 Appendix A. Appendix 117 A.1 Evaluation of R_D in Python 117 A.2 Approximate inverse for R_D in Python 117 A.3 Inversion of R_D using Newton's/Halley's method in Python 117 A.4 Inversion of R_D using fixed point method in Python 119 A.5 Moment computation using SymPy 120 A.6 Fiber collision test 122 A.7 OED calculation of the weighting matrix 123 A.8 OED Jacobian of objective and constraints 123 Appendix B. Theses 125 Bibliography 127 / Diese Arbeit befasst sich mit Fragen der Simulation und Optimierung des Spritzgießens von kurzfaserverstärkten Kunststoffen (SFRPs). Der Spritzgussprozess wird durch ein Zweiphasen-Fließproblem modelliert. Die Simulation des Zweiphasenflusses wird von der Lösung der Folgar-Tucker-Gleichung (FTE) zur Simulation der Momente der Faserorientierungsdichten begleitet. Die FTE erfordert die Lösung des sogenannten 'Abschlussproblems'', d. h. die Darstellung der Momente 4. Ordnung in Form der Momente 2. Ordnung. In Abwesenheit von Faser-Faser-Wechselwirkungen und anfänglich isotroper Faserdichte lässt die FTE eine analytische Lösung durch elliptische Integrale zu. Aus diesen elliptischen Integralen kann das Abschlussproblem durch eine einfache numerische Inversion gelöst werden. Ein Teil dieser Arbeit leitet approximative Inverse und analytische Inverse für spezielle Fälle von Faserorientierungsdichten her. Weiterhin wird eine Methode vorgestellt, um rationale Funktionen für die Berechnung beliebiger Momente in Bezug auf die Abschlussparameter 2. Ordnung zu generieren. Ein weiterer Teil dieser Arbeit befasst sich mit der Bestimmung effektiver Materialeigenschaften für SFRPs durch FFT-basierte Homogenisierungsmethoden. Für diese Methoden wurde ein neuartiges Diskretisierungsschema 'staggerd grid'' entwickelt und erfolgreich getestet. Darüber hinaus wurde der sogenannte 'composite voxel''-Ansatz auf die nichtlineare Elastizität ausgedehnt, was die Approximation der Materialeigenschaften an den Grenzflächen verbessert und die Reduzierung der Modellordnung um mehrere Größenordnungen im Vergleich zu klassischen Ansätzen ermöglicht. Im Zusammenhang mit der Homogenisierung untersuchen wir optimale experimentelle Designs, um die effektiven elastischen Eigenschaften von SFRPs mit der geringsten Anzahl von Computersimulationen zuverlässig zu bestimmen. Schließlich beschäftigen wir uns mit der Topologieoptimierung von Spritzgussteilen, indem wir die klassische SIMP-basierte Topologieoptimierung um ein Näherungsmodell für die Faserorientierungen erweitern. Neben der Compliance-Minimierung durch Topologieoptimierung stellen wir eine einfache Formoptimierungsmethode zur Kompensation von Teileverzug für einen Black-Box-Produktionsprozess vor.:Acknowledgments v Abstract vii Chapter 1. Introduction 1 1.1 Motivation 1 1.2 Nomenclature 3 Chapter 2. Numerical simulation of SFRP injection molding 5 2.1 Introduction 5 2.2 Injection molding technology 5 2.3 Process simulation 6 2.4 Governing equations 8 2.5 Numerical implementation 18 2.6 Numerical examples 25 2.7 Conclusions and outlook 27 Chapter 3. Numerical and analytical methods for the exact closure of the Folgar-Tucker equation 35 3.1 Introduction 35 3.2 The ACG as solution of Jeffery's equation 35 3.3 The exact closure 36 3.4 Carlson-type elliptic integrals 37 3.5 Inversion of R_D-system 40 3.6 Moment tensors of the angular central Gaussian distribution on the n-sphere 49 3.7 Experimental evidence for ACG distribution hypothesis 54 3.8 Conclusions and outlook 60 Chapter 4. Homogenization of SFRP materials 63 4.1 Introduction 63 4.2 Microscopic and macroscopic model of SFRP materials 63 4.3 Effective linear elastic properties 65 4.4 The staggered grid method 68 4.5 Model order reduction by composite voxels 80 4.6 Optimal experimental design for parameter identification 93 Chapter 5. Optimization of parts produced by SFRP injection molding 103 5.1 Topology optimization 103 5.2 Warpage compensation 110 Chapter 6. Conclusions and perspectives 115 Appendix A. Appendix 117 A.1 Evaluation of R_D in Python 117 A.2 Approximate inverse for R_D in Python 117 A.3 Inversion of R_D using Newton's/Halley's method in Python 117 A.4 Inversion of R_D using fixed point method in Python 119 A.5 Moment computation using SymPy 120 A.6 Fiber collision test 122 A.7 OED calculation of the weighting matrix 123 A.8 OED Jacobian of objective and constraints 123 Appendix B. Theses 125 Bibliography 127