c 2014 Xiaojia(Shelly) Zhang

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1 c 2014 Xaoja(Shelly) Zhang

2 MACRO-ELEMENT APPROACH FOR TOPOLOGY OPTIMIZATION OF TRUSSES USING A GROUND STRUCTURE METHOD BY XIAOJIA(SHELLY) ZHANG THESIS Submtted n partal fulfllment of the requrements for the degree of Master of Scence n Cvl Engneerng n the Graduate College of the Unversty of Illnos at Urbana-Champagn, 2014 Urbana, Illnos Advser: Professor Glauco H. Paulno

3 Abstract In ths thess, the generaton of ntal ground structures for generc domans n two and three dmensons are dscussed. Two methods of dscretzaton are compared: Vorono-based dscretzatons and structured quadrlateral dscretzatons. In addton, a smple and effectve member generaton approach s proposed: the Macro-element approach; whch can be mplemented wth both types of dscretzaton. The features of the approach are dscussed: effcent generaton of ntal ground structures; reducton n matrx bandwdth for global stffness matrx; fner control of bar connectvty; and reducton of overlapped bars. Numercal examples are presented whch dsplay the features of the proposed approach, and hghlght the comparson wth lterature results and tradtonal ground structure generaton methods.

4 To my mother, Yanhua and my father, Jguo.

5 Acknowledgements Frst and foremost, I would lke to express my deepest grattude to my advsor, Professor Glauco H. Paulno, wthout whom ths work would not have been possble. Hs tremendous energy about research, constant support and encouragement have had a profound mpact on my professonal lfe. Sncere thanks are due to Professor Gholamreza Mesr, who helped me tremendously durng my undergraduate and graduate years, every dscusson wth hm was nsprng. Ths thess would not have been possble wthout the contrbuton of Professor Adeldo Soares Ramos Jr. whose gudance and suggestons were nstrumental n ths work. I would also lke to thank all of my colleagues, Cameron Talsch, Lauren Beghn, Arun Gan, Tomas Zegard, Sofa Leon, Danel Sprng, Junho Chun, Evguen Flpov, Heng Ch, Maryam Edn, Tuo Zhao, Ke Lu, Ludmar Lma de Aguar, Adeldo Soares Ramos Jr., Lus Arnaldo, Peng We for ther help and support. I own my specal grattude to Sofa Leon and Tomas Zegard, for donatng ther tme gvng me nvaluable helps durng my graduate lfe; Heng Ch, for many frutful dscussons and suggestons durng my graduate study; Cameron Talsch, for all the nspraton durng each dscusson; and Danel Sprng, for helpng me polsh and proof read ths thess. Addtonally, I also thank my frends: Aoba Wang, Chen Ma, Xao Ma for ther contnued support and encouragement. I am extremely thankful for the camaradere and kndness they have provded throughout ths tme. Fnally, I wsh to thank my parents, for ther uncondtonal love and support. No words can descrbe my apprecaton for everythng they have done for me. It s wth great pleasure that I dedcate ths thess to them. v

6 Table of Contents LIST OF FIGURES v LIST OF TABLES x NOMENCLATURE x LIST OF ABBREVIATIONS x Chapter 1 INTRODUCTION Motvaton Background on Ground Structure Method Thess Organzaton Chapter 2 PROBLEM FORMULATION Optmzaton Formulaton Implementaton Aspects Chapter 3 GROUND STRUCTURE GENERATION Base Mesh Member Generaton Approach Chapter 4 EXAMPLES AND VERIFICATIONS Macro-Element Approach: Comparson wth Analytcal Soluton Macro-Element Approach: Applcatons to Complex Domans Macro-Element Approach: Comparson wth Contnuum Structural Optmzaton Soluton Chapter 5 CONCLUSIONS Appendx A SENSITIVITY ANALYSIS Appendx B KARUSH-KUHN-TUCKER (KKT) CONDITIONS Appendx C OPTIMALITY CRITERIA (OC) METHOD REFERENCES v

7 LIST OF FIGURES 1.1 Optmzaton problems usng tradtonal GSM wth sold lnes for allowed connectons and dashed lnes for volated connectons: (a) concave doman; (b) convex doman wth separated desgn regons Generaton of ground structure for level 1 connectvty: (a) problem doman and boundary condtons; (b) ntal ground structure; (c) optmal topology Generaton of ground structure for level 2 connectvty: (a) ntal ground structure; (b) optmal topology Generaton of ground structure for full level connectvty: (a) ntal ground structure; (b) optmal topology Topology optmzaton wth and wthout overlappng bars n a box doman: (a) problem doman and boundary condtons; (b) optmal topology wth overlappng connectons; (c) optmal topology wth non-overlappng connectons; (d) bar areas dstrbuton wth overlappng bars; (e) bar areas dstrbuton wth non-overlappng bars Mchell s soluton n a box doman: (a) doman wth boundary condtons; (b) known analytcal soluton [1] Dscretzaton technques for formng the base mesh: (a) structured quadrlateral mesh; (b) Vorono-based mesh Macro-element approach usng a structured quadrlateral element wth 4 nodes nserted per edge: (a) sngle structured quadrlateral element; (b) sngle element wth nserted addtonal nodes on each edge; (c) all possble connectons wthn the element Macro-element approach usng a structured quadrlateral mesh wth 4 nodes nserted per edge: (a) box doman dscretzed nto 8 elements and boundary condtons; (b) box doman wth 4 nodes nserted on each edge; (c) all possble connectons wthn each element Macro-element approach usng Vorono-based element wth 4 nodes nserted per edge: (a) sngle Vorono-based element; (b) sngle element wth nserted addtonal nodes on each edge; (c) all possble connectons wthn the element v

8 3.5 Macro-element approach usng Vorono-based mesh wth 4 nodes nserted per edge: (a) half rng doman and boundary condtons; (b) half rng doman wth 1 node nserted on each edge; (c) all possble connectons wthn each element Macro-element approach usng structured quadrlateral mesh n 3D tower doman: (a) tower doman and boundary condtons; (b) 2 nodes nserted on front surface edges and 1 node nserted on sde surface edges; (c) all possble connectons wthn each element surface Optmzaton problems usng Macro-element approach to correctly generate ntal ground structures wth vald connectons: (a) concave doman; (b) convex doman wth separated desgn domans (a) Wrench doman wth a boxed zoom-n regon; (b) possble connectons usng the classc GSM; (c) possble connectons usng the Macro-element approach Approxmaton of Mchell s soluton n a box doman usng both structured quadrlateral meshes and Vorono-based meshes: (a) topology obtaned from the tradtonal GSM wth 2000 structured quadrlateral elements; (b) topology obtaned from the tradtonal GSM wth 800 Vorono-based elements; (c) topology obtaned from the Macro-element approach usng 120 structured quadrlateral elements wth 7 nodes nserted along each edge; (d) topology obtaned from the Macro-element approach usng 240 Vorono-based elements wth 7 nodes nserted along each edge Wrench doman example wth Vorono-based mesh usng classc GSM and Macro-element approach: (a) wrench doman and boundary condtons; (b) dscretzaton usng 1000 Vorono-based elements; (c) dscretzaton usng 260 Vorono-based elements wth 7 addtonal nodes nserted along each edge; (d) fnal topology usng the full level classc GSM; (e) fnal topology usng the Macro-element approach Global stffness matrx after RCM algorthm from Wrench example usng: (a) tradtonal full level GSM; (b) Macroelement approach Serpentne doman example wth Quadrlateral mesh and Vorono-based mesh: (a) Serpentne doman and boundary condtons; (b) quadrlateral dscretzaton; (c) Voronobased dscretzaton v

9 4.5 Fnal topologes for Serpentne doman example usng classc GSM and Macro-element approach: (a) full level ground structure method usng 2781 structured quadrlateral elements; (b) full level ground structure method usng 1250 Vorono-based elements; (c) the Macro-element approach usng 380 structured quadrlateral elements wth 5 addtonal nodes nserted along each edge; (d) the Macro-element approach usng 250 Vorono-based elements wth 4 addtonal nodes nserted along each edge Hook doman example for contnuum structural optmzaton and for truss optmzaton usng GSM wth the Macroelement approach: (a) doman and boundary condtons; (b) fnal topology from contnuum optmzaton usng Poly- Top (R =2.0, v = 0.3); (c) fnal topology from truss optmzaton usng the GSM wth the Macro-element approach (400 Vorono-based elements wth 5 addtonal nodes nserted per edge) v

10 LIST OF TABLES 4.1 Numercal nformaton for Example Numercal nformaton for wrench doman Bandwdth and profle comparson of the stffness matrx for Wrench doman wth dfferent bar generaton method Numercal nformaton for the serpentne problem x

11 Nomenclature a a max a mn a o b C(a) E f g(a) K K 0 L l M move N n tol u V max y q α j Cross-sectonal areas vector Maxmum area for member Mnmum area for member Average area Geometry vector of member Objectve functon Young s Modulus for member External force vector Constrant functon Global stffness matrx Constant element stffness matrx Member s length vector Length for member Number of bars Move lmt Number of nodes Unt vector n axal drecton of the member Tolerance Nodal dsplacement vector Maxmum materal volume Intermedate varable vector Number of load cases Weghtng factor of load case j x

12 λ Adjont varable φ, β Lagrange multplers Ψ η Potental energy of the member Numercal dampng factor n OC x

13 Lst of Abbrevatons 2D 3D FEM GSM KKT OC Two-Dmensonal Three-Dmensonal Fnte Element Method Ground Structure Method Karush-Kuhn-Tucker Optmalty Crtera x

14 Chapter 1 Introducton 1.1 Motvaton The ground structure method (GSM) [2, 3, 4], s a technque for truss layout optmzaton. The generaton of the ntal ground structure, as a crucal component of the ground structure method, has so far receved lttle attenton n lterature. Smth [5] proposed an approach for the generaton of ground structures, usng unstructured meshes to represent the desgn doman. However, the approach he proposes requres addtonal preprocessng steps; ncludng the decomposton of desgn elements and the generaton of boundary faces. In ths thess, a smple and effectve approach s proposed for generaton of ground structures, namely the Macro-element approach. Ths approach s capable of generatng ground structures for desgn domans of non-trval geometres wth ease, and does not requre any addtonal nformaton on the outer and nner boundares of the doman. Ths approach s proposed n a general settng t may be combned wth any type of dscretzaton, ncludng quadrlateral and Vorono-based elements n two or three dmensonal domans. The elastc formulaton s adopted n ths thess [3]. One lmtaton assocated wth the tradtonal GSM s that, for concave domans Ω, as llustrated n Fgure 1.1(a), or for separated desgn domans, Ω 1 and Ω 2, as llustrated n Fgure 1.1(b); one needs to verfy that connectons don t fall outsde the boundary (for concave domans), or cross the border of separated desgn domans. Wth the approach presented n ths thess, these problems are naturally avoded. 1.2 Background on Ground Structure Method In GSM, the structural doman s frst dscretzed by nodes, and then the nodes are connected by the truss members. For a full-level ground structure, 1

15 (a) (b) Fgure 1.1: Optmzaton problems usng tradtonal GSM wth sold lnes for allowed connectons and dashed lnes for volated connectons: (a) concave doman; (b) convex doman wth separated desgn regons. all nodes n the doman are connected to one another. Ths leads to a fully populated global stffness matrx, whch adds to the computatonal cost [6]. Alternatvely, defntons of dfferent connectvty levels have been descrbed n the lterature [7]. The basc dea behnds the dfferent levels s that, n general, long bars are not needed n the ground structure. Thus, many bars n full-leveled ground structures are unused n the optmzaton process [8]. In addton, lower levels of connectvty may reduce the computatonal cost assocated wth these unused bars n the optmzaton. However, one problem wth the levelng method s that the assgnment of a suffcent connectvty level s problem dependent, makng t mpractcal to defne a general ground structure level for all problems. Dfferent connectvty levels can result n dfferent topologes for the same problem, as llustrated n Fgure 1.2, Fgure 1.3 and Fgure 1.4: fnal topology from level 2 ground structure n Fgure 1.3(b) has the fewest number of bars, whle topology from full level n Fgure 1.4(b) provdes the best soluton n terms of complance. In the current work, only full level ground structures are consdered n classc GSM. Overlappng bars are undesrable n the GSM and can be addressed durng the generaton process. Alternatvely, the overlappng members can be removed after the generaton process s complete. Fgure 1.5 shows a smple example whch hghlghts the mportance of removng overlappng bars. The soluton of ths problem s trval: one straght horzontal member carres all the load to the supports. When overlappng bars are not removed, the member szes are not equal, as can be seen n Fgure 1.5(d) (soluton s not 2

16 (a) (b) (c) Fgure 1.2: Generaton of ground structure for level 1 connectvty: (a) problem doman and boundary condtons; (b) ntal ground structure; (c) optmal topology (a) (b) Fgure 1.3: Generaton of ground structure for level 2 connectvty: (a) ntal ground structure; (b) optmal topology. 3

17 (a) (b) Fgure 1.4: Generaton of ground structure for full level connectvty: (a) ntal ground structure; (b) optmal topology. unque). However, when overlappng bars are removed, all member szes are equal, as shown n Fgure 1.5(e). Thus, t s mportant to remove overlappng members n order to obtan meanngful results n even the smplest examples. In addton, ths lowers the total number of bars n the model and decreases the computatonal cost of the method. In the present work, overlappng bars are removed durng the bar generaton process. Mchell [1] derved analytcal solutons for many nterestng optmzaton problems. He showed that, for a structure to be optmal, all members should be fully stressed. Ths leads to the requrement that all tensle and compressve bar pars should ntersect each other orthogonally (for problems wthout materal or geometrc non-lneartes). Therefore, f we assume all bars to have the same stress lmts, the orthogonalty n pars of bars should appear n the optmal solutons[9]. Fgure 1.6(a) shows an optmzaton problem for a smply supported beam wth ts dscrete Mchell s soluton shown n Fgure 1.6(b). Mchell s soluton s actually contnuous soluton (nfntely dense members), when usng the tradtonal GSM, the Mchell s soluton can be approxmated by a very fne ntal ground structure. However, ths comes wth a hgh computatonal cost. In Chapter 3, the optmzaton problem n Fgure 1.6(a) s nvestgated wth hghly refned ground structures usng both the tradtonal GSM and the proposed approach. 1.3 Thess Organzaton The thess s organzed as follows: n Chapter 2, the ground structure optmzaton formulaton s revewed for complance mnmzaton usng only the 4

18 (a) Normalzed bar area (b) Bar number (d) Normalzed bar area (c) Bar number (e) Fgure 1.5: Topology optmzaton wth and wthout overlappng bars n a box doman: (a) problem doman and boundary condtons; (b) optmal topology wth overlappng connectons; (c) optmal topology wth nonoverlappng connectons; (d) bar areas dstrbuton wth overlappng bars; (e) bar areas dstrbuton wth non-overlappng bars. 5

19 (a) (b) Fgure 1.6: Mchell s soluton n a box doman: (a) doman wth boundary condtons; (b) known analytcal soluton [1]. 6

20 cross-sectonal areas of the bars as the desgn varables. Then, dscretzaton methods and the proposed approach are ntroduced n Chapter 3, namely the Macro-element Approach, whch are used n the generaton of ntal ground structures, and followed by a dscusson of ts attrbutes for optmzaton of trusses usng the Ground Structure Method. Several numercal examples are presented n Chapter 4 to hghlght the propertes of the new approach. Fnally, summary and concluson are gven wth suggestons of the extenson of ths work n the future. 7

21 Chapter 2 Problem Formulaton 2.1 Optmzaton Formulaton The formulaton used n the present work consders equlbrum and compatblty condtons, whch explctly s known as elastc analyss [10]. The ntal ground structure s gven wth N nodes and M members. The equlbrum state of the system can be descrbed by: Ku = f (2.1) where, for the case of a two-dmensonal problem, f R 2N s the external force vector, u R 2N s the dsplacement vector and K R 2N 2N s the stffness matrx. For a full level ground structure n a convex doman, before removng overlappng bars, the relaton between M and N s M = N(N 1) 2. The stffness matrx K may be expressed as [3]: M K(a) = a K 0, =1 K 0 = E l b b T (2.2) where K 0 s a constant matrx assocated to each member n global coordnates, E and l are Young s Modulus and length of member, respectvely. Moreover, a R M s a vector of desgn varables (areas of bars) for the optmzaton problem, and b s a vector descrbng the drecton n the orentaton of member wth the form: b =. n (). n (). (2.3) 8

22 where n () s a unt vector n the axal drecton of the member. Here, the optmzaton problem s defned as one whch fnds the set of desgn varables that mnmzes complance of the structure, subject to equlbrum and volume constrants. Because the man purpose of ths thess s to explore the connectvty generaton n the ground structure, the smplest dsplacement based formulaton s adopted, wth a small postve lower bound mposed on the desgn varables a. In that case, the stffness K wll be postve defnte for the whole feasble set of desgn varables. The convexty of the problem can also be shown and the exstence of soluton s guaranteed [11]. The problem statement wth multple load cases can be formulated as [4]: s.t. C(a) = mn a q j=1 α j f T j u j (a) g(a) = a T L V max 0 a mn a a max = 1 : M (2.4) where u j s the soluton of Equaton (2.1), C(a) s the objectve functon, q s the number of load cases, α j s the weghtng factor of load case j, g(a) s the constrant functon, a and L are the vectors of area and length, respectvely, V max s the maxmum materal volume, and a max lower bounds, respectvely. and a mn are the upper and Ths formulaton allows computng the element stffness matrces only once. The global stffness matrx can then be assembled from the element stffness matrces and current desgn varables (member cross-sectonal areas). Ths facltates a more effcent mplementaton. To avod a sngular tangent stffness matrx n the soluton of the structural lnear equaton (2.1), we prevent zero element areas by usng a small lower bound, a mn. The upper and lower bounds are defned by a max = 10 3 a 0 and a mn = 10 2 a 0, respectvely; where a 0 s the average area. a 0 = V max L (2.5) 9

23 2.2 Implementaton Aspects The concepts n ths thess have been mplemented n a complete truss layout optmzaton solver n MATLAB R. The mplementaton conssts of two components: ground structure generaton and optmzaton. The ground structure generaton process ncludes base mesh generaton and connectvty generaton. Three alternatves for generatng the base mesh for the doman geometry are employed: generatng a Vorono-based mesh usng the mesh generator for polygonal elements named PolyMesher [12], generatng a structured quadrlateral mesh usng an ntrnsc subroutne, or mportng an unstructured mesh from elsewhere. Here, the man dea of the ground structure generaton s to produce non-overlappng connectvty usng the base mesh. The testng for mutual overlapped connectons s an addtonal procedure for the connectvty generaton for the tradtonal GSM. The optmzaton process contans three components: solvng the structural problem for a set of gven desgn varables, computng the senstvtes of the desgn varables, and updatng the desgn varables based on the Optmalty Crtera (OC). The detals of the senstvty calculatons, Karush-Kuhn-Tucker (KKT) condtons and OC are provded n the appendces. 10

24 Chapter 3 Ground Structure Generaton 3.1 Base Mesh In ths work, two types of mesh are used to dscretze the doman: structured quadrlateral mesh and Vorono-based mesh, as shown n Fgure 3.1. The need for dfferent types of mesh s problem dependent. A full level ground structure, usng a structured quadrlateral mesh, provdes pars of orthogonal bars by constructon, but they tend to be orented n a lmted number of drectons. In addton, when the doman s complex, ths meshng procedure s generally tedous and complcated. Because the stffness matrx s densely populated n the case of the full level GSM, the assocated computatonal cost s qute hgh [6]. Based on these ssues, the use of the Vorono-based mesh s proposed. Vorono-based meshes easly dscretze non-convex domans and have been shown to be advantageous n contnuum topology optmzaton [13, 12]. The seeds of Vorono-based meshes are ntally generated randomly, then terated to algn unformly usng the Centrod Vorono Tessellaton (CVT) method. After extractng node and element nformaton, the optmzaton procedure s the same as wth structured quadrlateral meshes. Vorono-based meshes, as opposed to structured quadrlateral meshes, can generate bars wth a greater number of drectons, because of ts random node dstrbuton, but provdes fewer orthogonal pars of bars; whch s not desrable accordng to the Mchell s soluton [1]. When the doman s concave or contans holes, the Vorono-based grd s the preferred dscretzaton n ths work. However, for these domans, generatng a full level, non-overlappng ntal ground structure, s complcated. As the generated bars have to le entrely wthn the doman, not all connectons are feasble. Therefore, addtonal nformaton on the outer and nner boundares of the doman s needed wth respect to the tradtonal GSM. 11

25 (a) (b) Fgure 3.1: Dscretzaton technques for formng the base mesh: (a) structured quadrlateral mesh; (b) Vorono-based mesh. Ths ssue s naturally solved by the member generaton approach that s proposed, whch s further dscussed n Secton Member Generaton Approach In ths secton, a new approach for generatng ntal ground structures usng structured quadrlateral and Vorono-based meshes n two or threedmensonal domans s presented. A Macro-element approach s proposed to overcome some of the dffcultes n ground structure generaton, as dscussed above. The resultng generaton of the ground structure, usng ths technque, s almost fully automated and effcent Macro-element Approach The basc dea behnd ths method s to nsert equally spaced nodes on each edge of each element, then connectons are only generated wthn each element. An llustraton of the Macro-element approach usng a structured quadrlateral mesh and a Vorono-based mesh s shown n Fgure 3.2, Fgure 3.3, Fgure 3.4 and Fgure 3.5. Dfferent scenaros were consdered, to show the flexblty of ths method: four equally spaced nodes were nserted on each edge for the structured quadrlateral mesh n Fgure 3.2 and Fgure 3.3; three addtonal nodes were nserted per edge for the sngle Vorono-based mesh n Fgure 3.4; and one node was nserted per edge for the quarter rng doman example. Ths method can also be used n three dmensonal (3D) 12

26 (a) (b) (c) Fgure 3.2: Macro-element approach usng a structured quadrlateral element wth 4 nodes nserted per edge: (a) sngle structured quadrlateral element; (b) sngle element wth nserted addtonal nodes on each edge; (c) all possble connectons wthn the element. domans, as shown n Fgure Attrbutes and Propertes The propertes of the Macro-element approach relate to the ground structure generaton process, optmzaton process and fnal topology aspect. In the ground structure generaton process, the proposed approach avod nvald connectons outsde the boundary (for concave domans) by constructon, as llustrated n Fgure 3.7(a). Ths can also be used to prevent connectons across separated desgn domans, Ω 1 and Ω 2 as n Fgure 3.7(b). Therefore, the bars can be generated wthout the addtonal step of detectng boundares or checkng for feasble connectons. For the case of concave domans, the ntal ground structure can be generated effcently as long as the doman s dscretzed and the element connectvty matrx s known. Ths approach s effcent n terms of the ntal ground structure generaton. Another aspect of the approach s n the optmzaton process. The global stffness matrx usng the two new approach wll have a reduced matrx bandwdth. Ths advantage becomes mportant when the problem sze s large. After nodes are nserted n the edge of each element, the Reverse Cuthll McKee (RCM) algorthm [14] s used to renumber the nodes. Snce bars are only generated n each element, the bandwdth for the global stffness matrx of the problem wll be reduced accordngly. In terms of fnal topology, the Macro-element approach offers more control n the bar dstrbuton by provdng a fner control of the connectvty. 13

27 (a) (b) (c) Fgure 3.3: Macro-element approach usng a structured quadrlateral mesh wth 4 nodes nserted per edge: (a) box doman dscretzed nto 8 elements and boundary condtons; (b) box doman wth 4 nodes nserted on each edge; (c) all possble connectons wthn each element. 14

28 (a) (b) (c) Fgure 3.4: Macro-element approach usng Vorono-based element wth 4 nodes nserted per edge: (a) sngle Vorono-based element; (b) sngle element wth nserted addtonal nodes on each edge; (c) all possble connectons wthn the element. Whether a long straght bar s desred or a curved bar s needed between two nodes, the proposed approach can be used to obtan the desred connectvty, as llustrated n Fgure 3.8(b). The standard GSM, on the other hand, ether has lmted connectons or very dense ground structures. Furthermore, the proposed method does not generate overlappng bars n the doman. For the Macro-element approach, overlappng bars appear only n the (refned) element boundary, and can be effcently and systematcally removed. Therefore, the problem of fndng and removng these bars becomes a reduced local problem wth an assocated lower computatonal cost. The proposed approach s general enough to be used wth any type of elements, ncludng quadrlateral elements and Vorono-based elements. Ths technque can handle dfferent types of domans, concave or convex and n two or three-dmensons. Also, snce the Macro-element approach s ndependent from the optmzaton formulatons, t s flexble and can be extended to other applcatons wth ease. 15

29 (a) (b) (c) Fgure 3.5: Macro-element approach usng Vorono-based mesh wth 4 nodes nserted per edge: (a) half rng doman and boundary condtons; (b) half rng doman wth 1 node nserted on each edge; (c) all possble connectons wthn each element. 16

30 (a) (b) (c) Fgure 3.6: Macro-element approach usng structured quadrlateral mesh n 3D tower doman: (a) tower doman and boundary condtons; (b) 2 nodes nserted on front surface edges and 1 node nserted on sde surface edges; (c) all possble connectons wthn each element surface. 17

31 (a) (b) Fgure 3.7: Optmzaton problems usng Macro-element approach to correctly generate ntal ground structures wth vald connectons: (a) concave doman; (b) convex doman wth separated desgn domans. (a) (b) (c) Fgure 3.8: (a) Wrench doman wth a boxed zoom-n regon; (b) possble connectons usng the classc GSM; (c) possble connectons usng the Macroelement approach. 18

32 Chapter 4 Examples and Verfcatons In ths secton, three examples wll be presented to demonstrate the varous features of the proposed approach. All examples are nvestgated usng the same volume constrant: V max = A Ω t, where A Ω s the area of the doman; stoppng crtera: tol = 10 8 ; move value: move = (a max a mn ) 100 and dampng factor for the OC update scheme: η = 0.7. The Young s modulus for all the bars s taken to be E 0 = and the ntal guess of bar areas s chosen as: a ntal = 0.7 a 0. The three examples use the Macroelement approach wth Vorono-based elements and quadrlateral elements, and nclude a comparson wth the tradtonal GSM. 4.1 Macro-Element Approach: Comparson wth Analytcal Soluton In ths example, the man dea s to are both the tradtonal GSM and the Macro-element approach to approxmate the Mchell s soluton shown n Fgure 1.6. Both structured quadrlateral and Vorono-based dscretzatons are used n solvng the box doman wth the tradtonal GSM and the Macroelement approach. Key features of the results are presented n Table 4.1. The fnal topologes, usng the full level tradtonal GSM wth a dense structured quadrlateral mesh and a Vorono-based mesh are llustrated n Fgure 4.1(a) and Fgure 4.1(b), respectvely. The fnal topologes from these dense meshes are smlar to the analytcal soluton shown n Fgure 1.6(b), but contan multple layers along the boundary lnes. By usng the proposed Macro-element approach, the topologes are very close to the analytcal soluton, as shown n Fgure 4.1(c) and Fgure 4.1(d), for the structured quadrlateral and Vorono-based meshes, respectvely. Hence, usng the proposed method, we can get a good approxmaton to the analytcal soluton. 19

33 Quad Vorono (a) (b) Quad Vorono (c) (d) Fgure 4.1: Approxmaton of Mchell s soluton n a box doman usng both structured quadrlateral meshes and Vorono-based meshes: (a) topology obtaned from the tradtonal GSM wth 2000 structured quadrlateral elements; (b) topology obtaned from the tradtonal GSM wth 800 Vorono-based elements; (c) topology obtaned from the Macro-element approach usng 120 structured quadrlateral elements wth 7 nodes nserted along each edge; (d) topology obtaned from the Macro-element approach usng 240 Vorono-based elements wth 7 nodes nserted along each edge. Mesh GSM Number of Bars Complance Quadrlateral Mesh Tradtonal (Fgure 4.1(a), (c)) Macro-element Vorono-based Mesh Tradtonal (Fgure 4.1(b), (d)) Macro-element Table 4.1: Numercal nformaton for Example

34 GSM Number of Bars Complance Value Tradtonal GSM Macro-Element Table 4.2: Numercal nformaton for wrench doman. 4.2 Macro-Element Approach: Applcatons to Complex Domans In the second and thrd examples, we apply the Macro-element approach to both the Wrench and Serpentne domans ntroduced n Talsch et al. [12]. These are non-trval domans and showcase the capabltes of ths approach for arbtrary domans. Agan, the fnal topologes are compared wth those obtaned usng the tradtonal GSM Wrench Doman wth Vorono-based Mesh The frst problem consders the Wrench doman, wth a Vorono-based dscretzaton, as shown n Fgure 4.2. The dea s to compare the fnal topology and the number of bars that both methods produce under smlar computatonal cost. In the tradtonal GSM, 1000 Vorono-based elements were used to dscretze the doman, as llustrated n Fgure 4.2(b). A full level connectvty was generated wthn the doman. Overlappng bars were removed durng the bar generatng process. The fnal topology s shown n Fgure 4.2(d). The boundary lnes around the rght hole n the fnal topology are not smooth, and the bars n the mddle of the doman are not detaled. The base mesh used n the Macro-element approach s shown n Fgure 4.2(c); dscretzed by 260 Vorono-based elements wth 7 addtonal nodes nserted along each edge. The bars were only connected wthn each element. The tme spent for mesh generaton s smlar to that for the full-level GSM, and the fnal topology usng the Macro-element approach s shown n Fgure 4.2(e). Qualtatvely, f we compare the results from the two methods, the Macro-element approach results n a clear and crsp soluton around the rght hole, and the bars nsde the doman are smoother than those obtaned usng tradtonal GSM. In addton, the soluton obtaned usng the Macro-element approach exhbts the close-to-orthogonal pars of bars. 21

35 (a) (b) (c) (d) (e) Fgure 4.2: Wrench doman example wth Vorono-based mesh usng classc GSM and Macro-element approach: (a) wrench doman and boundary condtons; (b) dscretzaton usng 1000 Vorono-based elements; (c) dscretzaton usng 260 Vorono-based elements wth 7 addtonal nodes nserted along each edge; (d) fnal topology usng the full level classc GSM; (e) fnal topology usng the Macro-element approach. 22

36 GSM Normalzed max. sem-bandwdth Normalzed profle Full Level Macro-element Table 4.3: Bandwdth and profle comparson of the stffness matrx for Wrench doman wth dfferent bar generaton method. A comparson of the number of bars n the ntal ground structures and the fnal complance values between the tradtonal GSM and the Macroelement approach s presented n Table 4.2. Ths showcases the ablty of the Macro-element approach to handle large problems wth an assocated low computatonal cost n ground structure generaton. For the case of concave domans, the tme spent n the bar generaton process s greatly reduced n the Macro-element approach, because there s no need to detect the boundary or search for a large number of overlappng bars. The source of the computatonal effcency of the Macro-element approach s apparent when we compare the bandwdth and profle of the stffness matrx for both methods. The normalzed maxmum sem-bandwdth and normalzed profle are computed as: normalzed maxmum sembandwdth = max.sembandwdth 2N (4.1) where N s number of nodes. normalzed profle = profle N (2N + 2 ) + 1 (4.2) Both the bandwdth and the profle of the global stffness matrces are sgnfcantly reduced by the usng Macro-element approach, as shown n Table 4.3. A vsual comparson of global stffness matrx s shown n Fgure 4.3. The reducton n the bandwdth and profle of the global stffness matrx becomes ncreasngly sgnfcant, as sze of the stffness matrx ncreases Serpentne Doman wth Structured Quadrlateral Mesh and Vorono-based Mesh In ths problem, the Serpentne doman s consdered wth dfferent dscretzatons for the tradtonal GSM and the Macro-element approach. Ths s to obtan a smlar number of bars so that we may compare the fnal topolo- 23

37 (a) (b) Fgure 4.3: Global stffness matrx after RCM algorthm from Wrench example usng: (a) tradtonal full level GSM; (b) Macro-element approach. ges from each method. To show the compatblty and flexblty of the Macro-element approach wth dfferent types of dscretzaton, two dfferent dscretzatons are used,.e. structured quadrlateral meshes and Vonorobased meshes. The geometry, load and support condtons of the serpentne doman are llustrated n Fgure 4.4(a). The structured quadrlateral mesh was generated by ABAQUS, whle the Vorono-based mesh was generated by PolyMesher. In order to mantan a smlar number of bars, meshes wth dfferent levels of refnement were used for the tradtonal GSM and the Macro-element approach. For a structured quadrlateral dscretzaton, the tradtonal GSM used 2781 elements whle the Macro-element approach used 380 elements wth 5 addtonal nodes nserted per edge. For a Voronobased dscretzaton, the tradtonal GSM used 550 Vorono elements and the Macro-element approach used 250 elements wth 4 nodes nserted per edge. The fnal topologes from the two approach are shown n Fgure 4.5. The number of bars, computatonal cost and complance values are summarzed n Table 4.4. Both the tradtonal GSM and the Macro-element approach result n smlar complance values and optmzed topologes. However, the tradtonal 24

38 (a) (b) (c) Fgure 4.4: Serpentne doman example wth Quadrlateral mesh and Vorono-based mesh: (a) Serpentne doman and boundary condtons; (b) quadrlateral dscretzaton; (c) Vorono-based dscretzaton. Number of Bars Complance Quadrlateral Mesh Tradtonal (Fgure 4.5(a), (c)) Macro-element Vorono-based Mesh Tradtonal (Fgure 4.5(b), (d)) Macro-element Table 4.4: Numercal nformaton for the serpentne problem. 25

39 Quad Vorono (a) (b) Quad Vorono (c) (d) Fgure 4.5: Fnal topologes for Serpentne doman example usng classc GSM and Macro-element approach: (a) full level ground structure method usng 2781 structured quadrlateral elements; (b) full level ground structure method usng 1250 Vorono-based elements; (c) the Macro-element approach usng 380 structured quadrlateral elements wth 5 addtonal nodes nserted along each edge; (d) the Macro-element approach usng 250 Vorono-based elements wth 4 addtonal nodes nserted along each edge. 26

40 GSM requres addtonal boundary nformaton to generate a vald ntal ground structure and overlappng bars were nvolved. In the Macro-element approach, snce we only connect bars wthn each element, there s no need to detect the boundary or search for a large number of overlappng bars because the overlappng bars were removed effcently and systematcally. Ths showcases the capablty of the proposed method to obtan smlar topologes as the tradtonal GSM, whle offerng smple and effectve ground structure generaton. Further, f we compare the results from a structured quadrlateral mesh and a Vorono-based mesh, they yeld almost the same truss layouts wth smlar topology and bar dstrbutons. However, for complex domans such as the Serpentne doman, Vorono-based meshes can dscretze the doman wth ease. Furthermore, for a coarse mesh dscretzaton on a non-trval doman, a Vorono-based mesh can provde more drectons for the bars to span and hence can result n a better soluton when compared to results obtaned usng a structured quadrlateral mesh. 4.3 Macro-Element Approach: Comparson wth Contnuum Structural Optmzaton Soluton In ths example, the fnal topology of the Hook problem usng the GSM wth the Macro-element approach s compared wth a densty based optmzaton soluton usng PolyTop [15], as shown n Fgure 4.6. Both solutons are obtaned usng Vorono-based dscretzatons. Qualtatvely, both topologes seem to converge to the same soluton and have smlar trats, such as a fan feature n the center. Ths suggests that the GSM usng the Macro-element approach can offer accurate and promsng desgns. 27

41 (a) (b) (c) Fgure 4.6: Hook doman example for contnuum structural optmzaton and for truss optmzaton usng GSM wth the Macro-element approach: (a) doman and boundary condtons; (b) fnal topology from contnuum optmzaton usng PolyTop (R =2.0, v = 0.3); (c) fnal topology from truss optmzaton usng the GSM wth the Macro-element approach (400 Voronobased elements wth 5 addtonal nodes nserted per edge). 28

42 Chapter 5 Conclusons In ths thess, the generaton of ntal ground structures for generc 2D and 3D domans has been dscussed and explored. Two types of dscretzaton are used, structured quadrlateral dscretzatons and Vorono-based dscretzatons. They offer alternatve approaches for ground structure generaton. Further, a bar generaton approach has been presented: the Macro-element approach. Ths approach can effcently generate an ntal ground structure. It avods nvald connectons outsde the boundary of concave domans, reduces the bandwdth of the global stffness matrx, provdes more control over the bar connectvty, and reduces the generaton of overlappng bars. From Example 4.1, the proposed Macro-element approach s shown to yeld smlar numercal (dscretzed) results to the exact soluton. In addton, the Macro-element approach converges to a smlar soluton to that n the tradtonal full-level GSM, at a lower computatonal cost, and offers a smple alternatve way of generatng ntal ground structures. Example 4.2 hghlghts the features of the Macro-element approach for domans wth complex geometres. Furthermore, the topology from the Macro-element approach n Example 4.3 shows excellent agreement wth the topology obtaned from contnuum structural optmzaton. Ths work offers room for future extensons. They nclude explorng the Macro-element for 3D Vorono tessellatons and nvestgatng the plastc formulaton approach. 29

43 Appendx A Senstvty Analyss Usng the Nested formulaton [3], the senstvty of the objectve functon can be calculated by the adjont method: C = f T u(a)+λ T [K(a)u(a) f] (A.1) Takng the dervatve of the objectve functon wth respect to the desgn varables yeld the relaton: C = f T u(a) [ K(a) + λ T u(a) K(a) u(a) ] a a a a (A.2) If we assume that the equlbrum condton s satsfed mplctly, the vector λ can have any value. By choosng λ = K(a) 1 f = u, we can elmnate the terms n Equaton (A.2) contanng u(a) ( a. Fnally, the senstvty of the ) objectve functon has the form: C = u T K(a) u (A.3) a a The senstvty of the volume constrant can be calculated as: g a = L (A.4) 30

44 Appendx B Karush-Kuhn-Tucker (KKT) Condtons Because the optmzaton problem n Equatons (2.4) s convex, the KKT condtons are both necessary and suffcent optmalty condtons. To defne them, we start wth the Lagrangan: Therefore n L(a, φ) = C(a)+φ( a L V max ) =1 (B.1) L a (a, φ ) 0, f a = a max (B.2) L a (a, φ ) = 0, f a mn < a < a max (B.3) L a (a, φ ) 0, The dervatve the Lagrangan s gven as: f a = a mn (B.4) L (a, φ) = C(a) + φl (B.5) a a After substtuton of Equaton (A.3) nto Equaton (B.5) and, subsequently, Equatons (B.2) through (B.4), we obtan the followng resultng KKT condtons at the pont (a, φ ): u T K(a) u φ, f a = a max a (B.6) u T K(a) u = φ, f a mn < a < a max a (B.7) 31

45 u T K(a) u φ, f a = a mn a (B.8) Based on Equaton (B.7), we obtan constant specfc stran energy at the optmum n the lnear case, whch corresponds to a fully stressed desgn wth Ψ = (σ 2 ) /E o, where Ψ s the stran energy of member. 32

46 Appendx C Optmalty Crtera (OC) Method The optmzaton n ths work s solved by the OC algorthm.thus the outlne of OC algorthm s presented n ths Appendx. Ths algorthm can be derved by replacng the objectve and constrants functons wth the approxmatons on the current desgn pont usng an ntermedate varable. In such way, a sequence of separable and explct sub-problems s generated to approxmate of the orgnal problem. In ths context, we lnearzed the objectve functon usng exponental ntermedate varables as [16]: y = ( a a mn a max a mn ) p (C.1) C(a) = Ĉ(a) = C ( y(a k ) ) + ( ) T C ( y(a) y(a k ) ) (C.2) y a=a k Then, after substtuton of ( ) C y = ( ) C a j a=a k a j y and substtuton of the a=a k Equaton (C.1) n the Equaton (C.2), the followng equatons are obtaned: Ĉ(a) = C ( y(a k ) ) + n =1 C a a=a k 1 ( a k p a mn ) [( a a mn ) p 1] a k a mn (C.3) 33

47 s.t. mn a Ĉ(a) g(a) = a T L V max = 0 a mn a a max = 1 : M (C.4) By means of the Lagrangan dualty, ths problem can be solved wth: L(a, γ) = Ĉ(a) + β g(a) (C.5) where β s a Lagrangan multpler and the optmalty condtons are gven as: L (a, β) = Ĉ(a) +β g(a) a a a = C a a=a k ( a a mn ) p 1 +β L a k a mn = 0 (C.6) L β = at L V max = 0 Solvng the Equaton (C.6) for a (β) to obtan: a (β) = a = a mn + (B (β)) 1 ( ) 1 p a k a mn (C.7) (C.8) and substtutng n Equaton (C.7), the Lagrange multpler β s obtaned, for example, va b-secton method where B s defned as: B = C a a=a k β L (C.9) To calculate β and a, the box constrants need to be satsfed, and thus the next desgn pont a new s defned as: a new = a +, a, a, 34 a a + a a otherwse (C.10)

48 where the a + and a are the bounds for the search regon defned by: a = max(a mn, a k move) (C.11) a + = mn(a max, a k + move) (C.12) n whch the varable move s the move lmt usually specfed as a fracton of a max a mn. In the convex examples, presented n ths work, a fast convergence s obtaned usng values for move larger than a max a mn. The quantty η = 1 1 p s usually called a numercal dampng factor and for p = 1, a recprocal approxmaton s obtaned. The p values can be estmated usng dfferent approaches. In ths work a two pont approxmaton approach s used based on the work of Fadel et al. [17] and presented by Groenwold and Etman [16]. In ths approach, the estmaton of p (k) s: p (k) = 1 + ln ( ) C a a=a k 1/ C a a=a k ln ( ) (C.13) a k 1 /a k where ln(.) s the natural logarthm. At the frst step we use p = 1 and we restrct 15 p 0.1 for the subsequent teratons. The convergence crtera used s: where tol s the tolerance. ( a k max a k 1 ) 1 + a k 1 tol (C.14) 35

49 References [1] A. G. M. Mchell, The Lmts of Economy of Materal n Frame- Structures, Phlosophcal Magazne, vol. 8, pp , [2] W. S. Dorn, R. E. Gomory, and H. J. Greenberg, Automatc desgn of optmal structures, Journal de Mecanque, vol. 3, pp , [3] P. W. Chrstensen and A. Klarbrng, An Introducton to Structural Optmzaton. Sprnger, [4] M. P. Bendsø e and O. Sgmund, Topology optmzaton: Theory, Methods, and Applcatons. Berln, Germany: Sprnger, [5] O. Smth, Generaton of ground structures for 2D and 3D desgn domans, Engneerng Computatons, vol. 15, no. 4, pp , [6] M. T. Heath, Scentfc Ccomputng: An Introductory Survey, 2nd ed. McGraw-Hll, 1997, vol [7] T. Sokół, A 99 lne code for dscretzed Mchell truss optmzaton wrtten n Mathematca, Structural and Multdscplnary Optmzaton, vol. 43, no. 2, pp , [8] M. Glbert and A. Tyas, Layout optmzaton of large-scale pn-jonted frames, Engneerng Computatons, vol. 20, no. 8, pp , [9] M. Ohsak, Optmzaton of Fnte Dmensonal Structures. CRC Press, [10] U. Krsch, Optmal Topologes of Structures, Appled Mechancs Revews, vol. 42, pp , [11] K. Svanberg, On local and global mnma n structural optmzaton, n New Drectons n Optmum Structural Desgn, E. Gallhager, R. H. Ragsdell, and O. C. Zenkewcz, Eds. Chchester: John Wley and Sons, 1984, pp [12] C. Talsch, G. H. Paulno, A. Perera, and I. F. M. Menezes, Poly- Mesher: a general-purpose mesh generator for polygonal elements wrtten n Matlab, Structural and Multdscplnary Optmzaton, vol. 45, no. 3, pp , Jan [13] C. Talsch, G. H. Paulno, A. Perera, and I. F. M. Menezes, Polygonal fnte elements for topology optmzaton : A unfyng paradgm, no. December 2009, pp ,

50 [14] E. Cuthll and J. McKee, Reducng the Bandwdth of Sparse Symmetrc Matrces, Proceedngs of the th natonal conference of the ACM, pp , [15] C. Talsch, G. H. Paulno, A. Perera, and I. F. M. Menezes, PolyTop: a Matlab mplementaton of a general topology optmzaton framework usng unstructured polygonal fnte element meshes, Structural and Multdscplnary Optmzaton, vol. 45, no. 3, pp , Jan [16] A. A. Groenwold and L. F. P. Etman, On the equvalence of optmalty crteron and sequental approxmate optmzaton methods n the classcal topology layout problem, Internatonal Journal for Numercal Methods n Engneerng, no. May 2007, pp , [17] G. M. Fadel, M. F. Rley, and B. J. M, Two pont exponental approxmaton method for structural optmzaton, Structural Optmzaton, vol. 124, no. 2, pp , [18] W. Achtzger, On smultaneous optmzaton of truss geometry and topology, Structural and Multdscplnary Optmzaton, vol. 33, no. 4-5, pp , Jan [19] W. Achtzger and M. Stolpe, Global optmzaton of truss topology wth dscrete bar areas Part II: Implementaton and numercal results, Computatonal Optmzaton and Applcatons, vol. 44, no. 2, pp , Dec [20] W. Achtzger and M. Stolpe, Truss topology optmzaton wth dscrete desgn varables Guaranteed global optmalty and benchmark examples, Structural and Multdscplnary Optmzaton, vol. 34, pp. 1 20, [21] A. Klarbrng and N. Strömberg, A note on the mn-max formulaton of stffness optmzaton ncludng non-zero prescrbed dsplacements, Structural and Multdscplnary Optmzaton, vol. 45, no. 1, pp , June [22] T. Hagshta and M. Ohsak, Topology optmzaton of trusses by growng ground structure method, Structural and Multdscplnary Optmzaton, vol. 37, no. 4, pp , Apr [23] M. Ohsak and N. Katoh, Topology optmzaton of trusses wth stress and local constrants on nodal stablty and member ntersecton, Structural and Multdscplnary Optmzaton, vol. 29, no. 3, pp , Oct

51 [24] M. Hagshta, T. Ohsak, Topology optmzaton of trusses by growng ground structure method, Structural and Multdscplnary Optmzaton, pp , [25] G. I. N. Rozvany, Structural Desgn va Optmalty Crterahe: Prager Approach to Structural Optmzaton. Sprnger, [26] G. I. N. Rozvany, Exact analytcal solutons for some popular benchmark problems n topology optmzaton, Structural Optmzaton, vol. 15, pp , [27] G. I. N. Rozvany and T. Sokół, Exact truss topology optmzaton: allowance for support costs and dfferent permssble stresses n tenson and compresson-extensons of a classcal soluton by Mchell, Structural and Multdscplnary Optmzaton, vol. 45, no. 3, pp , Nov [28] O. Smth, Topology optmzaton of trusses wth local stablty constrants and multple loadng condtons a heurstc approach, Structural optmzaton, vol. 13, pp , [29] T. Sokół and G. I. N. Rozvany, New analytcal benchmarks for topology optmzaton and ther mplcatons. Part I: b-symmetrc trusses wth two pont loads between supports, Structural and Multdscplnary Optmzaton, vol. 46, no. 4, pp , Mar [30] T. Sokół and G. I. N. Rozvany, Exact least-volume trusses for two symmetrc pont loads and unequal permssble stresses n tenson and compresson, Structural and Multdscplnary Optmzaton, vol. 47, no. 1, pp , Dec [31] K. Svanberg, On the convexty and concavty of complances, Structural and Multdscplnary Optmzaton, vol. 7, pp ,

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