Topology optimization of non-linear, dissipative structures and materials Ivarsson, Niklas 2020 Document Version: Publisher's PDF, also known as Version of record Link to publication Citation for published version (APA): Ivarsson, N. (2020). Topology optimization of non-linear, dissipative structures and materials. Division of solid mechanics, Lund University. 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LUND UNIVERSITY PO Box 117 221 00 Lund +46 46-222 00 00 TOPOLOGY OPTIMIZATION OF NON-LINEAR, DISSIPATIVE STRUCTURES AND MATERIALS niklas ivarsson Solid Doctoral Thesis Mechanics Department of Construction Sciences Solid Mechanics ISRN LUTFD2/TFHF-20/1063-SE(1-249) ISBN: 978-91-7895-484-1 (print) ISBN: 978-91-7895-485-8 (pdf) Topology optimization of non-linear, dissipative structures and materials Doctoral Thesis by Niklas Ivarsson Copyright �c 2020 by Niklas Ivarsson Printed by Media-Tryck AB, Lund, Sweden For information, address: Division of Solid Mechanics, Lund University, Box 118, SE-221 00 Lund, Sweden Homepage: http://www.solid.lth.se Till Lars& Cecilia Preface This thesis is the result of my doctoral studies carried out at the Division of Solid Mechanics at Lund University. I would like to take this opportunity to express my gratitude to the people who have helped me during this time and in my work. First of all, I would like to thank my supervisor, Prof. Mathias Wallin, whose guidance and support made this thesis possible. I am very grateful to Mathias for sharing his knowledge with me and for his persistent help during this work. I would also like to thank Prof. Matti Ristinmaa, Prof. Daniel Tortorelli and Ass. Prof. Oded Amir for all their help in this PhD project. My current and former colleagues at the Division of Solid Mechanics all deserve thanks for making my time at the division a very pleasant experience. Special thanks to Anna Dalklint for all the valuable discussions on topology optimization, and to Dr. Stefanos Athanasopoulos for all the advice and help, and also for the fun times out of the office. I also thank the Royal Physiographic Society of Lund and Sk˚anska Ingenj¨orsklubben for granting me scholarships that made attendance at conferences possible. The support from my friends and loved ones outside of work must not be forgotten. Thanks to the guys of “onsdags¨ol”for all the fun times that helped me take my mind off work when I needed to. Many thanks to my good friends, Dr. Jakob Helander and Dr. Filip Jans˚aker, for all their support over the past years. I also thank Josephine for her support and for making me very happy. Finally, I thank my brothers and my loving parents for literally everything they have done for me. Malm¨o, June 2020 Niklas Ivarsson i Abstract Topology optimization is a mathematical tool forfinding optimal distributions of material phases within a design domain. It is commonly used in early design stages to generate conceptual structural layouts. Ultimately, designs generated by topology optimization consist of distinct material phases with enhanced performance. Most research on topology optimization methods focuses on linear elastic mechanical problems with the objective to maximize stiffness. Whereas linear problems are well understood, methods for solving geometrically and/or material non-linear, transient, path-dependent problems are less so. This thesis consists of a general introductory part and six appended papers in which topology optimization frameworks that account for non-linear response are presented. The introductory part gives an overview of topology optimization methods, motivates optimiz- ation for non-linear problems and details the specific solution methods for the non-linear, transient and path-dependent frameworks on which the included papers are based on. All papers but Paper E present non-linear optimization frameworks that employ hyper- elastic material models and are based onfinite strain theory. In Papers A and B, topology optimization methods for multiple material phases and for tangent stiffness maximization, respectively, are established. It is shown that an arbitrary number of phases can be in- cluded in the optimization and that the optimized distribution of the phases depends on the load magnitude. It is also shown that the definition of the stiffness measure used in topology optimization of non-linear elastic structures is of great importance. In Papers C, D and F, inelastic constitutive models are included in the optimization frameworks. Rate-dependent, i.e. viscous, inertial andfinite strain effects are combined with topology optimization to generate maximum energy absorbing structures. Both macroscopic struc- tures and microstructural designs are presented, wherein plastic work is optimized. It is shown that micro and macro response can be tailored for a specific load magnitude, loading rate and load path. The developed scheme in Paper F enables optimization of structures for maximum stiffness while constraining the maximum specific plastic work. Since signi- ficant plasticity occurs at design domain boundaries, a regularization technique for explicit control of boundary effects in topology optimization problems, presented in Paper E, is utilized. This optimization approach shows that by constraining the specific plastic work, stiff designs with smooth stress distributions are generated. iii Sammanfattning Matematiska optimeringsmetoder ¨ar kraftfulla verktyg f¨or effektiv design av lastb¨arande strukturer. Genom optimering kan en given m¨angd material omf¨ordelas f¨oratt f¨orb¨attra strukturers h˚allfasthet, exempelvis f¨or att uppn˚amaximal styvhet. Konventionella met- oder, d¨ar designf¨orslag iterativt utv¨arderas och f¨orb¨attras baserat p˚am¨ansklig intuition, ¨ar ofta ineffektiva och tidskr¨avande. F¨oratt f¨orkorta denna process och m¨ojligg¨ora optimal design kan topologioptimering anv¨andas. Topologioptimering l¨oser ett optimeringsprob- lem baserat p˚amatematiska beskrivningar av strukturens geometri. Beskrivningen till˚ater ¨andringar g¨allande strukturens form, tjocklek och sammankoppling av dess komponenter. L¨osningen till optimeringsproblemet ger konceptuella designer med maximerad ¨onskv¨ard egenskap, samtidigt som specifika villkor p˚adesignen, s˚asom begr¨ansad materialanv¨and- ning, uppfylls. Deflesta etablerade topologioptimeringsmetoder ¨arbaserade p˚ateori som endast g¨aller under begr¨ansade f¨orh˚allanden; antaganden om materialegenskaper och yttre lasters stor- lek och hastighet g¨ors f¨or att kunna etablera en f¨orenklad fysikalisk modell vilket medf¨or en f¨orenkling av det matematiska optimeringsproblemet. Dessa antaganden begr¨ansar dock anv¨andningsomr˚adena f¨or topologioptimering. Vid optimering av exempelvis krocks¨akra fordon, skyddsutrustning och st¨otd¨ampare kan lokalt stor och plastisk irreversibel deform- ation av materialet f¨orekomma. Dessutom sker deformationen ofta under ett v¨aldigt kort tidsspann. S˚aledesfinns det behov f¨or topologioptimeringsmetoder baserade p˚afysikaliska modeller som innefattar stora plastiska t¨ojningar samt irreversibla och transienta processer. S˚adana metoder har etablerats i denna avhandling. I avhandlingens f¨orsta del presenteras metoder som har utvecklats f¨or att maximera strukturers styvhet under olinj¨ar elastisk materialrespons och stora t¨ojningar. Modellerna m¨ojligg¨or design av strukturer uppbyggda avflera material och visar att den optimerade strukturen ¨ar starkt beroende av lastniv˚an. Definitionen av sj¨alva styvhetsm˚attet visas ¨aven vara avg¨orande f¨or den optimerade designen d˚aresponsen ¨ar olinj¨ar. I den senare delen av avhandlingen presenteras ett verktyg f¨or att optimera energi- absorption av designer b˚ade p˚amakroskopisk och mikroskopisk niv˚a, d¨ar irreversibla och transienta effekter inkluderas. Detta optimeringsverktyg genererar optimerade designer med markant h¨ogre energiabsorption i j¨amf¨orelse med metoder som antar sm˚at¨ojningar och l˚angsamma lasthastigheter. Slutligen presenteras en optimeringsmetod som m¨ojligg¨or maximering av strukturell styvhet samtidigt som generering av plastisk energi begr¨ansas. Optimeringsmetoden kan anv¨andas f¨or att undvika utmattning och skador p˚astrukturen. I anknytning till denna metod har ¨aven ett koncept utvecklats f¨or att kontrollera randeffekter i optimeringspro- cessen; detta f¨or att eliminera artifakter som annars kan uppkomma vid designdom¨anens yttre r¨ander och d¨armed p˚averka strukturens prestanda. v List of appended papers This doctoral thesis is based on the following manuscripts: Paper A Mathias Wallin, Niklas Ivarsson and Matti Ristinmaa Large strain phase-field based multi-material