DOCSLIB.ORG

Universid Ade De Sã O Pa

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Topological games and selection principles Matheus Duzi Ferreira Costa Dissertação de Mestrado do Programa de Pós-Graduação em Matemática (PPG-Mat) UNIVERSIDADE DE SÃO PAULO DE SÃO UNIVERSIDADE Instituto de Ciências Matemáticas e de Computação Instituto Matemáticas de Ciências SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Data de Depósito: Assinatura: ______________________ Matheus Duzi Ferreira Costa Topological games and selection principles Dissertation submitted to the Institute of Mathematics and Computer Sciences – ICMC-USP – in accordance with the requirements of the Mathematics Graduate Program, for the degree of Master in Science. EXAMINATION BOARD PRESENTATION COPY Concentration Area: Mathematics Advisor: Prof. Dr. Leandro Fiorini Aurichi USP – São Carlos September 2019 Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP, com os dados inseridos pelo(a) autor(a) Duzi, Matheus D812t Topological games and selection principles / Matheus Duzi; orientador Leandro Fiorini Aurichi. - - São Carlos, 2019. 166 p. Dissertação (Mestrado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2019. 1. Topological games. 2. Selection principles. 3. Covering properties. 4. Tightness properties. 5. Baire spaces. I. Fiorini Aurichi, Leandro , orient. II. Título. Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176 Matheus Duzi Ferreira Costa Jogos topológicos e princípios seletivos Dissertação apresentada ao Instituto de Ciências Matemáticas e de Computação – ICMC-USP, como parte dos requisitos para obtenção do título de Mestre em Ciências – Matemática. EXEMPLAR DE DEFESA Área de Concentração: Matemática Orientador: Prof. Dr. Leandro Fiorini Aurichi USP – São Carlos Setembro de 2019 This work is dedicated to all those who do not restrain themselves by the finitude of our earthly lives and, occasionally, decide to venture into the infinite abstract. ACKNOWLEDGEMENTS I express my gratitude to FAPESP for the financial support to this project (2017/09797-0). I thank my advisor Leandro F. Aurichi and everyone from the topology group of ICMC (officially awarded as "The best group of 2018") for all the help throughout these years with the topology and set theory seminars. Here, I particularly thank Henrique A. Lecco, who made the question that motivated the beginning of paper [Aurichi and Duzi 2019]. My thanks also goes to Piotr Szewczak and Boaz Tsaban for giving us access to the preliminary notes of [Szewczak and Tsaban 2019]. Finally, I am specially thankful for the closest friends and family who have stood beside me through the ups and downs of this incredible journey. I would not dare to name all of you, but make no mistake: your support will never be forgotten. ABSTRACT DUZI, M. Topological games and selection principles. 2019. 166 p. Dissertação (Mestrado em Ciências – Matemática) – Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos – SP, 2019. This paper is dedicated to the beginning of the development of a book introducing topological games and selection principles. Here, the classical games (such as the Banach-Mazur) and selection principles (such as the Rothberger or Menger properties) are presented. The most notable applications are also displayed – both the classical (such as the characterization of Baire spaces with the Banach-Mazur game) and the recent (such as the relation between the Menger property and D-spaces). In addition to the content for the book, a problem in finite combinatorics that was found in the study of positional strategies is presented (as well as a partial solution) together with some results regarding new variations of classical selection principles and games, which give rise to the characterization of some notable spaces. Keywords: topological games, selection principles, tightness properties, covering properties, Baire spaces. RESUMO DUZI, M. Jogos topológicos e princípios seletivos. 2019. 166 p. Dissertação (Mestrado em Ci- ências – Matemática) – Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos – SP, 2019. Este trabalho é dedicado ao início do desenvolvimento de um livro introdutório à jogos to- pológicos e princípios seletivos. Aqui, são apresentados os clássicos jogos (tais como o de Banach-Mazur) e princípios seletivos (tais como a propriedade de Rothberger ou de Menger). Também são exibidas as aplicações mais notáveis encontradas na literatura – tanto as mais tradicionais (tais como a caracterização de espaços de Baire com o jogo de Banach-Mazur), como as mais atuais (tais como a relação entre a propriedade de Menger e D-espaços). Além do conteúdo voltado para o livro, são apresentados um problema de combinatória finita (assim como uma solução parcial para tal) que foi encontrado com o estudo de estratégias posicionais e alguns resultados envolvendo novas variações de princípios de seleção e jogos clássicos, possibilitando a caracterização de alguns espaços notáveis. Palavras-chave: jogos topológicos, princípios seletivos, propriedades de tightness, propriedades de coberturas, espaços de Baire. CONTENTS Introduction...................................... 15 I BOOK 19 1 PRELIMINARIES ............................ 21 1.1 Set theory .................................. 21 1.2 Topology ................................... 22 2 INTRODUCING GAMES........................ 27 2.1 Finite games ................................. 28 2.2 Infinite games ................................ 29 2.3 Strategies ................................... 30 2.4 Positional strategies ............................. 37 2.5 The Banach-Mazur game ......................... 42 3 SELECTION PRINCIPLES....................... 49 3.1 Classes of selection principles ....................... 49 3.2 Closure properties .............................. 51 3.3 Covering properties ............................. 52 3.3.1 Menger spaces ................................ 52 3.3.2 Rothberger spaces .............................. 53 3.3.3 Hurewicz spaces ............................... 55 3.3.4 Alster spaces ................................. 58 3.3.5 The Scheepers Diagram .......................... 59 4 THE ASSOCIATED SELECTIVE GAMES .............. 67 4.1 Covering games ............................... 69 4.1.1 The Rothberger game ........................... 69 4.1.2 The Menger game .............................. 72 4.1.3 The Pawlikowski Theorem ......................... 81 4.1.4 The Hurewicz game ............................. 87 4.1.5 The Alster game ............................... 87 4.2 Closure games ................................ 90 5 SOME CONNECTIONS AND APPLICATIONS ........... 97 5.1 The Banach-Mazur game and Baire spaces ............... 97 5.2 The Rothberger game and Measure Theory .............. 100 5.2.1 Strong measure zero ............................ 100 5.2.2 Purely atomic measures .......................... 102 5.3 Productively Lindelöf spaces ........................ 105 5.4 Sieve completeness ............................. 107 5.5 D-spaces ................................... 109 5.6 Tightness games and countable tightness ................ 114 5.7 The space of real continuous functions ................. 117 6 DIAGRAMS ...............................123 II RESULTS 127 7 POSITIONAL STRATEGIES AND A PROBLEM IN FINITE COM- BINATORICS ..............................129 8 TOPOLOGICAL GAMES OF BOUNDED SELECTIONS . 133 8.1 Selection Principles ............................. 134 8.2 The associated games ........................... 135 8.3 The “modfin” and “mod1” variations ................... 139 8.4 The analogous to Pawlikowski’s and Hurewicz’s results ....... 147 8.5 The dual game ................................ 151 8.6 Conclusion .................................. 153 Bibliography .....................................159 15 INTRODUCTION Games have been an object of study for mathematicians since, arguably, the 17th century. It was only in the 20th century that, in the Scottish Book, the game henceforth known as the first infinite mathematical game was introduced: the Banach-Mazur game (for a brief history of games in mathematics we refer to [Telgársky 1987] or [Aurichi and Dias 2019]). Obviously, infinite games are not meant to be played. But what good is a game, if not for playing it? Infinite games are used to describe certain infinite combinatorial properties in an intuitive manner. This description usually happens in terms of winning strategies. The property of being a Baire space, for instance, is characterized by one of the players not having a winning strategy in the Banach-Mazur game, while if the other player has a winning strategy in this game we can go even further and conclude that the space is productively Baire. Another way to describe combinatorial properties is through selection principles. In essence, these are properties asserting that one can choose an element from each set of a sequence in order to obtain a significant object. Selection principles define classical properties, such as Rothberger or Menger spaces. Our initial goal was to start the development of a book introducing these infinite games and selection principles in the context of topology. Midway through this project, however, we have obtained some new results on the field, so we decided to divide this text into two parts, each with its respective chapters: Part I This
Recommended publications
  • Arxiv:Math/0412498V1

    Arxiv:Math/0412498V1

    A semifilter approach to selection principles Lubomyr Zdomsky February 8, 2020 Abstract In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal g is a lower bound of the additivity number of the σ-ideal generated by Menger subspaces of the Baire space, and under u < g every subset X of the real line with the property Split(Λ, Λ) is Hurewicz, and thus it is consistent with ZFC that the property Split(Λ, Λ) is preserved by unions of less than b subsets of the real line. Introduction In this paper we shall present two directions of applications of semifilters in selection principles on topological spaces. First, we shall consider preservation by unions of the Menger property. Trying to describe the σ-compactness in terms of open covers, K.Menger intro- duced in [Me] the following property, called the Menger property: a topological space X is said to have this property if for every sequence (un)n∈ω of open covers of X there exists a sequence (vn)n∈ω such that each vn is a finite subfamily of un and the collection {∪vn : n ∈ ω} is a cover of X. The class of Menger topological spaces, i.e. spaces having the Menger property appeared to be much wider than the class of σ-compact spaces (see [BT], [CP], [JMSS] and many others), but it has interesting properties itself and poses a number of open questions. One of them, namely the question about the value of additivity of corresponding σ-ideal, arXiv:math/0412498v1 [math.GN] 27 Dec 2004 will be discussed in this paper.
  • Arxiv:1603.03361V3 [Math.GN] 18 May 2016 Eeecs R O Eddfrtermidro Hspaper

    Arxiv:1603.03361V3 [Math.GN] 18 May 2016 Eeecs R O Eddfrtermidro Hspaper

    PRODUCTS OF MENGER SPACES: A COMBINATORIAL APPROACH PIOTR SZEWCZAK AND BOAZ TSABAN Abstract. We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in all but the most exotic models of real set theory. On the other hand, we establish pro- ductive properties for versions of Menger’s property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger’s property is strictly between Menger’s and Hurewicz’s classic properties. We include a number of open problems emerging from this study. 1. Introduction A topological space X is Menger if for each sequence U1, U2,... of open covers of the space X, there are finite subsets F1 ⊆ U1, F2 ⊆ U2, . whose union forms a cover of the space X. This property was introduced by Karl Menger [17], and reformulated as presented here by Witold Hurewicz [11]. Menger’s property is strictly between σ-compact and Lindelöf. Now a central notion in topology, it has applications in a number of branches of topology and set theory. The undefined notions in the following example, which are available in the indicated references, are not needed for the remainder of this paper. Example 1.1. Menger spaces form the most general class for which a positive solution of arXiv:1603.03361v3 [math.GN] 18 May 2016 the D-space problem is known [2, Corolarry 2.7], and the most general class for which a general form of Hindman’s Finite Sums Theorem holds [25].
  • Set Theory of Infinite Imperfect Information Games

    Set Theory of Infinite Imperfect Information Games

    SET THEORY OF INFINITE IMPERFECT INFORMATION GAMES BENEDIKT LOWEÄ Abstract. We survey the recent developments in the investigation of Blackwell determinacy axioms. It is generally believed that ordi- nary (Gale-Stewart) determinacy and Blackwell determinacy for in¯nite games are equivalent in a strong sense, but this conjecture (of Tony Mar- tin's) has not yet been proved. The paper is a snapshot of the current state-of-the-art knowledge in this area. 1. Introduction For most mathematicians, the term \game theory" evokes associations of applications in Economics and Computer Science: it is often associated with the Prisoner's Dilemma or other applications in the social sciences. Game theory is not perceived as an area of mathematics but rather as an area in which mathematics is applicable. In addition to this famous area of game theory that has been repeatedly honoured by the Nobel Prize for Economics, games also play an important r^ole in mathematics and computer science, often connected with logic. We use the name Logic & Games for the research ¯eld that connects techniques from logic in order to investigate games, or game theoretic methods in order to prove theorems of logic. This ¯eld itself is broad and has many sub¯elds that are as diverse as logic itself. One of its sub¯elds is the set theoretic study of in¯nite games. It was the Polish school of topologists and measure theorists that con- nected game theory to set theory. According to Steinhaus [St165, p. 464], Banach and Mazur knew in the 1930s that there is a non-determined in¯- nite game (constructed by a use of the axiom of choice) and that there is a connection between games and the Baire property.
  • In Presenting the Dissertation As a Partial Fulfillment of • The

    In Presenting the Dissertation As a Partial Fulfillment of • The

    thIne presentin requirementg sthe fo rdissertatio an advancen ads degrea partiae froml fulfillmen the Georgit oaf • Institute oshalf Technologyl make i,t availablI agreee thafort inspectiothe Librarn yan dof the materialcirculatiosn o fin thi accordancs type.e I witagrehe it stha regulationt permissios ngovernin to copgy bfromy th,e o rprofesso to publisr undeh fromr whos, ethis directio dissertation itn wa mas ywritten be grante, or,d sucin hhis copyin absenceg o,r bpublicatioy the Dean nis osolelf they foGraduatr scholarle Divisioy purposen whesn stooandd doe thast noanty involv copyineg potentia from, ol rfinancia publicatiol gainn of. ,I t thiiss under dis­­ bsertatioe allowen dwhic withouh involvet writtes npotentia permissionl financia. l gain will not 3/17/65 b I ! 'II L 'II LLill t 111 NETS WITH WELL-ORDERED DOMAINS A THESIS Presented to The Faculty of the Graduate Division by Gary Calvin Hamrick In Partial Fulfillment of the Requirements for the Degree Master of Science in Applied Mathematics Georgia Institute of Technology August 10, 1967 NETS WITH WELL-ORDERED DOMAINS Approved: j Date 'approved by Chairman: Z^tt/C7 11 ACKNOWLEDGMENTS I wish to thank ray thesis advisor, Dr. Roger D. Johnson, Jr., very much for his generous expenditure of time and effort in aiding me to pre­ pare this thesis. I further wish to thank Dr. William R. Smythe, Jr., and Dr. Peter B. Sherry for reading the manuscript. Also I am grateful to the National Science Foundation for supporting me financially with a Traineeship from January, 1966, until September, 1967. And for my wife, Jean, I hold much gratitude and affection for her forebearance of a thinly stocked cupboard while her husband was in graduate school.
  • HISTORICAL NOTES Chapter 1:. the Idea of Topologizing the Set of Continuous Functions from One Topological Space Into Another To

    HISTORICAL NOTES Chapter 1:. the Idea of Topologizing the Set of Continuous Functions from One Topological Space Into Another To

    HISTORICAL NOTES Chapter 1:. The idea of topologizing the set of continuous functions from one topological space into another topological space arose from the notions of pointwise and uniform convergence of sequences of functions. Apparently the work of Ascoli [1883], [1889] and Hadamard [1898] marked the beginning of function space theory. The topology of pointwise convergence and the topology of uniform convergence are among the first function space topologies considered in the early years of general topology. The I supremum metric topology was studied in Frechet [19061. The paper of Tychonoff [1935] showed that the (Tychonoff) product on the set RX is nothing but the topology of pointwise convergence. In 1945, Fox [19451 defined the compact-open topology. Shortly thereafter, Arens [1946] studied this topology, which he called k-topology. Among other things which Arens proved was the compact-open topology version of Theorem 1.2.3. Set-open topologies in a more general setting were studied by Arens and Dugundji [1951] in connection with the concepts of admissible and proper topologies. Theorem 1.2.5 is due to Jackson [1952], and Example 1.2.7 can be found in Dugundji [1968]. Chapter 1:. Admissible [i.e., conjoining) topologies were introduced by Arens [1946] and splitting (i.e., proper) topologies were studied by Arens and Dugundji [1951], where they proved Theorem 2.5.3. Proofs of Theorem 2.5.2 and Corollary 2.5.4.a can be found in Fox [1945]. Corollary 2.5.7 is apparently due to Jackson [1952]; and Morita [1956] proved Corollary 2.5.8.
  • Quasi-Arithmetic Filters for Topology Optimization

    Quasi-Arithmetic Filters for Topology Optimization

    Quasi-Arithmetic Filters for Topology Optimization Linus Hägg Licentiate thesis, 2016 Department of Computing Science This work is protected by the Swedish Copyright Legislation (Act 1960:729) ISBN: 978-91-7601-409-7 ISSN: 0348-0542 UMINF 16.04 Electronic version available at http://umu.diva-portal.org Printed by: Print & Media, Umeå University, 2016 Umeå, Sweden 2016 Acknowledgments I am grateful to my scientific advisors Martin Berggren and Eddie Wadbro for introducing me to the fascinating subject of topology optimization, for sharing their knowledge in countless discussions, and for helping me improve my scientific skills. Their patience in reading and commenting on the drafts of this thesis is deeply appreciated. I look forward to continue with our work. Without the loving support of my family this thesis would never have been finished. Especially, I am thankful to my wife Lovisa for constantly encouraging me, and covering for me at home when needed. I would also like to thank my colleges at the Department of Computing Science, UMIT research lab, and at SP Technical Research Institute of Sweden for providing a most pleasant working environment. Finally, I acknowledge financial support from the Swedish Research Council (grant number 621-3706), and the Swedish strategic research programme eSSENCE. Linus Hägg Skellefteå 2016 iii Abstract Topology optimization is a framework for finding the optimal layout of material within a given region of space. In material distribution topology optimization, a material indicator function determines the material state at each point within the design domain. It is well known that naive formulations of continuous material distribution topology optimization problems often lack solutions.
  • Vladimir V. Tkachuk Special Features of Function Spaces

    Vladimir V. Tkachuk Special Features of Function Spaces

    Problem Books in Mathematics Vladimir V. Tkachuk A Cp-Theory Problem Book Special Features of Function Spaces Problem Books in Mathematics Series Editors: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA For further volumes: http://www.springer.com/series/714 Vladimir V. Tkachuk ACp-Theory Problem Book Special Features of Function Spaces 123 Vladimir V. Tkachuk Departamento de Matematicas Universidad Autonoma Metropolitana-Iztapalapa San Rafael Atlixco, Mexico City, Mexico ISSN 0941-3502 ISBN 978-3-319-04746-1 ISBN 978-3-319-04747-8 (eBook) DOI 10.1007/978-3-319-04747-8 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014933677 Mathematics Subject Classification (2010): 54C35 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer.
  • Selective Covering Properties of Product Spaces, II: Gamma Spaces

    Selective Covering Properties of Product Spaces, II: Gamma Spaces

    SELECTIVE COVERING PROPERTIES OF PRODUCT SPACES, II: γ SPACES ARNOLD W. MILLER, BOAZ TSABAN, AND LYUBOMYR ZDOMSKYY Abstract. We study productive properties of γ spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: (1) Solving a problem of F. Jordan, we show that for every unbounded tower set X ⊆ R of cardinality ℵ1, the space Cp(X) is productively Fr´echet–Urysohn. In particular, the set X is productively γ. (2) Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova– Scheepers, we prove that, assuming the Continuum Hypothesis, there are γ spaces whose product is not even Menger. (3) Solving a problem of Scheepers–Tall, we show that the properties γ and Gerlits–Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel’ski˘ı duality to obtain results concerning local properties of function spaces and countable topo- logical groups. 1. Introduction For a Tychonoff space X, let Cp(X) be the space of continuous real-valued functions on X, endowed with the topology of pointwise convergence, that is, the topology inherited from the Tychonoff product RX . In their seminal paper [14], Gerlits and Nagy characterized the property that the space Cp(X) is Fr´echet–Urysohn—that every point in the closure of a set arXiv:1310.8622v4 [math.LO] 10 Oct 2018 is the limit of a sequence of elements from that set—in terms of a covering property of the domain space X.
  • Ultrafilters and Tychonoff's Theorem

    Ultrafilters and Tychonoff's Theorem

    Ultrafilters and Tychonoff's Theorem (October, 2015 version) G. Eric Moorhouse, University of Wyoming Tychonoff's Theorem, often cited as one of the cornerstone results in general topol- ogy, states that an arbitrary product of compact topological spaces is compact. In the 1930's, Tychonoff published a proof of this result in the special case of [0; 1]A, also stating that the general case could be proved by similar methods. The first proof in the gen- eral case was published by Cechˇ in 1937. All proofs rely on the Axiom of Choice in one of its equivalent forms (such as Zorn's Lemma or the Well-Ordering Principle). In fact there are several proofs available of the equiv- alence of Tychonoff's Theorem and the Axiom of Choice|from either one we can obtain the other. Here we provide one of the most popular proofs available for Ty- chonoff's Theorem, using ultrafilters. We see this approach as an opportunity to learn also about the larger role of ultra- filters in topology and in the foundations of mathematics, including non-standard analysis. As an appendix, we also include an outline of an alternative proof of Tychonoff's Theorem using a transfinite induction, using the two-factor case X × Y (previously done in class) for the inductive step. This approach is found in the exercises of the Munkres textbook. The proof of existence of nonprincipal ultrafilters was first published by Tarski in 1930; but the concept of ultrafilters is attributed to H. Cartan. 1. Compactness and the Finite Intersection Property Before proceeding, let us recall that a collection of sets S has the finite intersection property if S1 \···\ Sn 6= ? for all S1;:::;Sn 2 S, n > 0.
  • Topology Optimization for FDM Parts Considering the Hybrid Deposition Path Pattern

    Topology Optimization for FDM Parts Considering the Hybrid Deposition Path Pattern

    micromachines Article Topology Optimization for FDM Parts Considering the Hybrid Deposition Path Pattern Shuzhi Xu 1, Jiaqi Huang 2, Jikai Liu 2 and Yongsheng Ma 1,* 1 Department of Mechanical Engineering, University of Alberta, Edmonton, AB T2G 2G8, Canada; [email protected] 2 Center for Advanced Jet Engineering Technologies (CaJET), Key Laboratory of High Efficiency and Clean Mechanical Manufacture (Ministry of Education), School of Mechanical Engineering, Shandong University, Jinan 250100, China; [email protected] (J.H.); [email protected] (J.L.) * Correspondence: [email protected] Received: 11 June 2020; Accepted: 15 July 2020; Published: 22 July 2020 Abstract: Based on a solid orthotropic material with penalization (SOMP) and a double smoothing and projection (DSP) approach, this work proposes a methodology to find an optimal structure design which takes the hybrid deposition path (HDP) pattern and the anisotropic material properties into consideration. The optimized structure consists of a boundary layer and a substrate. The substrate domain is assumed to be filled with unidirectional zig-zag deposition paths and customized infill patterns, while the boundary is made by the contour offset deposition paths. This HDP is the most commonly employed path pattern for the fused deposition modeling (FDM) process. A critical derivative of the sensitivity analysis is presented in this paper, which ensures the optimality of the final design solutions. The effectiveness of the proposed method is validated through several 2D numerical examples. Keywords: solid orthotropic material with penalization; hybrid deposition paths; double smoothing and projection; fused deposition modeling 1. Introduction Additive manufacturing (AM) has gained fast development in research and industrial applications.
  • Compliance–Stress Constrained Mass Minimization for Topology Optimization on Anisotropic Meshes

    Compliance–Stress Constrained Mass Minimization for Topology Optimization on Anisotropic Meshes

    Research Article Compliance–stress constrained mass minimization for topology optimization on anisotropic meshes Nicola Ferro1 · Stefano Micheletti1 · Simona Perotto1 Received: 7 January 2020 / Accepted: 22 May 2020 / Published online: 11 June 2020 © Springer Nature Switzerland AG 2020 Abstract In this paper, we generalize the SIMPATY algorithm, which combines the SIMP method with anisotropic mesh adapta- tion to solve the minimum compliance problem with a mass constraint. In particular, the mass of the fnal layout is now minimized and both a maximum compliance and a maximum stress can be enforced as either mono- or multi-constraints. The new algorithm, named MSC-SIMPATY, is able to sharply detect the material–void interface, thanks to the anisotropic mesh adaptation. The presented test cases deal with three diferent scenarios, with a focus on the efect of the constraints on the fnal layouts and on the performance of the algorithm. Keywords Topology optimization · Stress constraint · Anisotropic mesh adaptation 1 Introduction in the literature driving topology optimization. Among these, we mention the density-based approaches [8, 10, Topology optimization is of utmost interest in diferent 46], the level-set methods [5, 53], topological derivative branches of industrial design, such as biomedical, space, procedures [49], phase feld techniques [12, 20], evolution- automotive, mechanical, architecture (see, e.g., [2, 13, ary approaches [54], homogenization [4, 9], performance- 19, 23, 57]). Similar formulations can also be adopted for based optimization [39]. We focus on the frst class and, the optimization of structures in diferent contexts, from in particular, on the SIMP (Solid Isotropic Material with fuid–structure interaction to the tailored design of mag- Penalization) method [9, 10, 46] where the material dis- netic or auxetic metamaterials [26, 36, 47, 56].
  • Morphology-Based Black and White Filters for Topology Optimization

    Morphology-Based Black and White Filters for Topology Optimization

    Downloaded from orbit.dtu.dk on: Sep 24, 2021 Morphology-based black and white filters for topology optimization Sigmund, Ole Published in: Structural and Multidisciplinary Optimization Link to article, DOI: 10.1007/s00158-006-0087-x Publication date: 2007 Document Version Early version, also known as pre-print Link back to DTU Orbit Citation (APA): Sigmund, O. (2007). Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 33(4-5), 401-424. https://doi.org/10.1007/s00158-006-0087-x General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Morphology-based black and white filters for topology optimization Ole Sigmund Department of Mechanical Engineering, Solid Mechanics Technical University of Denmark Nils Koppels Alle, B. 404, DK-2800 Lyngby, Denmark Tel.: (+45) 45254256, Fax.: (+45) 45931475 email: [email protected] Lyngby November 9, 2006 Abstract In order to ensure manufacturability and mesh-independence in density-based topology opti- mization schemes it is imperative to use restriction methods.