Structural Topology Optimization Method for Morphogenesis of Dendriforms

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1 Open Journal of Cvl Engneerng, 216, 6, ISSN Onlne: ISSN Prnt: Structural Topology Optmzaton Method for Morphogeness of Dendrforms Xrong Peng Cvl Engneerng College, Hunan Cty Unversty, Yyang, Chna How to cte ths paper: Peng, X.R. (216) Structural Topology Optmzaton Method for Morphogeness of Dendrforms. Open Journal of Cvl Engneerng, 6, Receved: July 25, 216 Accepted: September 2, 216 Publshed: September 5, 216 Copyrght 216 by author and Scentfc Research Publshng Inc. Ths wor s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.). Open Access Abstract The topology optmzaton method of contnuum structures s adopted for the morphogeness of dendrforms durng the conceptual desgn phase. The topology optmzaton model wth mnmzng structural stran energy as objectve and subject to structural weght constrant s establshed by the ndependent contnuous mappng method (ICM) whch s a popular and effcent method for the topology optmzaton of contnuum structures. Ths optmzaton model s an optmzaton problem wth a sngle constrant and can be solved by the teraton formula establshed based on the saddle condton. Tang the morphogeness of a plane dendrform as an example, the nfluences on topologes of the dendrform are dscussed for several factors such as the rato of the reserved weght to the total weght, the stffness and the geometry shape of the roof structure, the heght of the desgn area, and so on. And several examples of applcaton scenaros are presented, too. Numercal examples show that the proposed structural topology optmzaton method for the morphogeness of dendrforms s feasble. It can provde dversform topologes for the conceptual desgn of dendrforms. Keywords Dendrforms, Topology Optmzaton, Morphogeness 1. Introducton Dendrforms, whch was put forward frst by a German Fre Otto n the 196s, s a nd of bonc structure desgned based on the shapes and mechancal characterstcs of natural trees. The Stuttgart arport termnal (Fgure 1(a)) n Germany bult n 1991 s a typcal engneerng desgned by Fre Otto. The roof supported by dendrforms wth three layers of branches provdes a broad space for the arport termnal [1]. Dendrforms have advantages such as the reasonable paths transmttng loads, the broad space DOI: /ojce September 5, 216

2 (a) (b) (c) (d) Fgure 1. Engneerng buldngs adopted dendrforms. (a) Stuttgart Arport Termnal, Stuttgart, Germany [1]; (b) Changsha ralway staton n Chna [3]; (c) Qatar natonal conventon center [9]; (d)tote restaurant, Bombay [2]. acheved by a few bars. Many large-span space structures, such as arports, ralway statons, publc centers, and so on, adopted dendrforms as supported structures. For example, the Changsha ralway staton for hgh-speed trans n Chna (Fgure 1(b)), the Qatar natonal conventon center (Fgure 1(c)) and the Tote restaurant n Bombay (Fgure 1(d)), etc. The morphogeness of the dendrforms s the most mportant problem durng ts structural desgn for ts many branches and complcated form. The heght, the layer, the number and the locaton of branches need to be desgned. The sutable supported locaton of the roof structure needs to be determned. Thus, every component of the dendrform conforms to the optmal paths of transmttng loads; and the functonal requrements of the buldng are also met. There are three nds of methods for the morphogeness of dendrforms: the expermental methods, the geometrc methods and the numercal methods. Expermental methods have the wet thread method, dry thread method, the beaded thread method and so on. The applcaton of the expermental methods s restrcted for ther results nfluenced by model scales [1]. Geometrc methods adopt the fractal theory to generate the geometrc shapes of dendrforms. Gawell generated geometrc shapes of dendrforms based on the L-system fractal theory and ntroduced an engneerng applcaton on the Tote restaurant n Bombay [2]. The dendrforms generated by the fractal method are focused on only the shape characterstcs of the trees. Ther geometrc confguratons can be mproved by the structure optmzaton technology to consder the mechancs characterstcs of the trees [3]. Wth the development of the numercal analyss and structural optmzaton technology, numercal methods are appled to generate 527

3 the forms of dendrforms, and researches and applcatons on ths aspect become the hotspots. Von Buelow put forward a method of generatng the dendrforms by usng a genetc algorthm to fnd the shortest paths [4]. Yue Wu put forward a reverse hangng method for form fndng of dendrforms [5]. Qan Zhang studed the form fndng of dendrforms based on the sldng cable element [6]. Usng the optmzaton method of seleton structures, Changyu Cu proposed the form fndng method of dendrforms based senstvty [7]. Meanwhle, he mproved the evolutonary structural optmzaton method (ESO) method, whch was a topology optmzaton method of contnuum structures, to generate the optmal topology of dendrforms [8]. Sasa appled the mproved ESO method to desgn the Qatar natonal conventon center (Fgure 1(c)) [9]. The more ratonal desgn can be acheved by the form fndng method based on the topology optmzaton of contnuum structures because the optmal topologes wth seleton forms can be obtaned, and t s unnecessary to specfy some pror data such as the heght, the layers and the numbers of branches. But the ESO method [1] adopted at past has some shortcomngs such as the low effcency for ts too many optmzaton teratons, dfferent optmal topologes obtaned by tang dfferent deletng rate, and unstable algorthm [11]. The ndependent contnuous mappng (ICM) method, one of the topology optmzaton methods of contnuum structures, has a hgh solvng effcency because t establshes an optmzaton model and solves the model by the dual sequental quadratc programmng method [12]. In ths paper, a morphogeness method of dendrforms s presented based on the ICM method whch has hgh effcency. Some factors affected the forms of dendrforms are dscussed and some conclusons are followed, whch are useful for the topology desgn of dendrforms. Fnally, several applcaton examples are presented to llustrate the feasblty of the proposed method. 2. Topology Optmzaton Method for Morphogeness of Dendrforms 2.1. Topology Optmzaton Model for Morphogeness of Dendrforms The morphogeness of dendrforms s used usually durng the conceptual desgn phase. The usual way s to desgn a structural topology wth maxmum stffness under the vertcal loads actng on the roof structures. Thus, t can be formulated as a topology optmzaton problem of the contnuum structure, namely: under the specfed consumpton of materal, wthn the specfed desgn area, optmzng the topology to maxmze the structural stffness under the specfed roof loads. Because the maxmum structural stffness s equvalent to the mnmum structural stran energy, the topology optmzaton model for generatng a dendrform bols down to the topology optmzaton problem wth mnmzng structural stran energy objectve subject to structural weght (or volume) constrant, as shown n Equaton (1): N Fnd t E Mae e( t ) mn (1) s.t. W ( t ) W t 1 528

4 where, t s the topology desgn varable. e s structural stran energy, W s structural weght Modelng and Solvng of Topology Optmzaton Problem wth Mnmzng Stran Energy Objectve Subject to Weght Constrant For the topology optmzaton problem wth mnmzng structural stran energy as an objectve subject to a specfed weght constrant, the optmzaton model can be establshed by the ICM method as the followng process. The dscrete topology varables wth values or 1 are extended to the contnuous topology varables wth values n the nterval [, 1] by the approxmaton of the step functon. The element weght and stffness matrx are dentfed by the flter functons of weght and stffness respectvely [12]: = ( ), = f ( t ) w f t w w (2) where w and are the weght and stffness matrx of the -th element. w and are the nherent weght and nherent stffness matrx of the -th element. The flter functons of weght and stffness can be taen as power functons: w w ( ) =, f ( t ) t α f t t α where the power α w and α can tae 1 and 3, respectvely. Thus, the elemental stran energy can be expressed as: where ( ) ( ) t α α ( ) ( ) ( ) ( ) α t t T ( ) T α α 2 2t t ( ) = (3) 1 e = u u = u u = e (4) and e are the topology varable and the stran energy of the -th element at the -th teraton respectvely. The elemental weght can be expressed as: w = t w (5) αw Therefore, the topology optmzaton model establshed by the ICM method s wrtten as: N Fnd t E N ( ) ( α ) ( ) α Mae t e t mn =1 (6) N αw s.t. t w W =1 t 1 To prevent the stffness matrx to appear sngular whle the topology varable taes value, a small value t s adopted to replace wth, and t can be taen as t =.1. Because of the Equaton (4) s an optmzaton problem wth a sngle constrant, the constrant must to be taen as the equalty constrant. Otherwse the problem wll be an unconstraned problem and become a meanngless problem. Note ( ) ( α ) ( ) A = t e, I = t < t < 1 = 1,, N, then, Equaton (6) s wrtten { } and defne the actve set ( ) a 529

5 as: N Fnd t E α α Mae A t + A t mn Ia Ia αw αw (7) s.t. t w + t w = W Ia Ia t t 1 The augmented Lagrangan functon of the problem s: w w L( t, λ) = A t α + A t α + λ t α w + t α w W (8) Ia Ia Ia Ia The saddle pont for the above functon tang the extremum condton s: L t = α A t + α λwt = (9) α+ 1 αw 1 w From t, we obtan: ( ) ( α+ αw) ( ) ( α + α ) 1 1 w t = α A α w 1 λ (1) w Substtute Equaton (1) nto the equalty constrant condton of Equaton (7) we obtan ( ) 1 ( ) w 1 ( ) ( w) w w A ww α + α 1 α + α α λ α = W t α w (11) Ia Ia 1 ( ) ( α+ αw ) αw 1 λ = W t w w αa ( αww ) Substtute Equaton (12) nto Equaton (1), we have 1 ( α + α ) w (12) Ia Ia t W t w A ( w ) w A ( w ) a a 1 ( α ) 1 ( ) + αw α + α α w w = α αw α αw I I (13) Consderng the nterval constrants of topology desgn varables, namely ( ) ( 1) t t < t ( + 1) t = t t < t < 1 ( t > 1) (14) Update the actve set, and return to Equaton (13) to calculate t. Termnate the teraton loop whle the actve set s unchanged. The optmal result s the soluton of Equaton (6). Modfy the structure accordng to the Equaton (2), and enter nto the next teraton. Iterate untl the followng convergence crtera s met: ( ) ( ) ( + 1 ( 1) e e e ) e + ε are structural stran energy of the prevous and current tera- where ton. ε ( ) e and ( 1) e + = (15) s the convergence accuracy, and ε =.1 s adopted here. * t 53

6 3. Morphogeness of Dendrforms 3.1. Example 1: The Morphogeness of Plane Dendrforms As showed n Fgure 2, the desgn area s a rectangular wth szes of 1 m 7.5 m. A unform vertcal dstrbuton load q = 1 N/m s appled on the upper border. An edge wth thcness of.3 m s used as the roof structure and s the non-desgn area. A fxed regon wth wdth.4 m located at the center of the bottom border s regarded as the root of the dendrform. The structural materal s steel wth the elastc modulus E = MPa and the Posson s rato.3. Under a specfed constrant of the weght rato, the optmal topology of the dendrform s obtaned by mnmzng the structural stran energy, namely maxmzng the structural stffness. The weght rato s defned as the rato of reserved weght to the ntal total weght. Fgure 3 shows the optmal topologes of dendrforms whle the weght rato s changed and the stffness of the roof structure s unchanged. Optmal topologes of dendrforms are dfferent whle the consumpton of materal s changed. Wth the ncrease of the materal, the topology of dendrforms s more complcated. Fgure 4 shows the optmal topologes whle the stffness of the roof structures s changed and the weght rato (1%) s unchanged. For the convenence of dealng wth model, the dfferent stffness of roof structures s smulated by settng dfferent elastc modulus of the materal of the roof structure, rather than changng ts geometrc szes. Data lsted n Fgure 4 are the elastc modulus of materals. It can be seen from Fgure 4 that the branches of optmal topology of dendrforms are decreased wth the ncrease of the stffness of roof structures, and are reduced to a pllar n the extreme case. Fgure 2. Degn condtons of a plane dendrform. (a) (b) (c) Fgure 3. Optmal topologes of dendrforms under dfferent weght rato (wth the ncrease of the materal, the topology s more complcated). (a) weght rato 5%; (b) weght rato 1%; (c) weght rato 2%. 531

7 (a) (b) (c) (d) Fgure 4. Optmal topologes of dendrforms under dfferent stffness of the roof structure (branches of optmal topology are decreased wth the ncrease of the stffness of roof structures, and s reduced to a pllar n the extreme case). (a) E = MPa; (b) E = MPa; (c) E = MPa; (d) E = MPa. Fgure 5 shows the optmal topologes whle the heght of the desgn area s changed and the stffness of the roof structures (E = MPa) and the weght rato (1%) are unchanged. In the cases that the heght s small (Fgures 5(a)-(c)), the man trun of the dendrform wll not appear. In the cases that the heght s large enough (Fgures 5(c)-(f)), the man trun appears; and wth the ncrease of the heght of the desgn area, the optmal topologes of branches of the dendrforms are unchanged, only the heght of the man trun s ncreased. Fgure 6 shows the nfluence of the geometrc shapes of roof structures on the optmal topology. An example of the roof structure wth a slope shape s showed n Fgure 6(a). The optmal topology s dfferent wth that of the horzontal roof structure. The bg branches are leaned to the hgh sde of the roof. An example of the roof structure wth an arc shape s showed n Fgure 6(b). A for shape branch appears, and t s not the case wth all bnary branch form Example 2: Morphogeness of Dendrforms for the Bearng Seleton of Walls Adoptng dendrforms as bearng seleton structures of walls, not only the loads actng on the walls can be transmtted effectvely along the branches of dendrforms, but also a beautful vsual can be acheved. The morphogeness of plane dendrforms can present dverse optons. Fgure 7 s an engneerng adoptng plane dendrforms as the bearng seleton of walls, Jangwan Cheng, n Chongqng, Chna [13]. As showed n the left fgure of Fgure 8, the desgn area s four walls along a square wth szes of 1 m 1 m and the heghts of 8 m. A unform vertcal dstrbuton load q = 1 N/m s appled on the upper borders of the walls. Along the loadng edges, the structures wth heghts of.2 m are used as the beams of the walls to carry the loads, and are non-desgn area. Fxed regons wth the wdth of.2 m at the four corners of the bottom of the walls are taen as the roots of dendrforms. Structural materal s steel, and the materal propertes are same wth those n Example 1. The weght rato of 1% s specfed as a constrant, and the mnmzng structural stran energy s taen as the objectve. The optmal topology s shown n the rght fgure of Fgure 8. Branches of dendrforms are stretched n the two vertcal planes of each corner. In each sde of the walls, branches of dendrforms are stretched and ntersected at the mddle part of the wall, 532

8 (a) (b) (c) (d) (e) (f) Fgure 5. Optmal topologes of dendrforms under dfferent heghts of desgn area (n the cases that the heght s small (Fgures 5(a)-(c)), the man trun wll not appear; n the cases that the heght s large enough (Fgures 5(c)-(f)), the man trun appears; and wth the ncrease of the heght of the desgn area, the optmal topologes of branches of the dendrforms are unchanged, only the heght of the man trun s ncreased). (a) 5 m; (b) 7.5 m; (c) 1 m; (d) 12.5 m; (e) 15 m; (f) 2 m. (a) (b) Fgure 6. Optmal topologes of dendrforms under dfferent shapes of roof structures (left fgure: meshes of fnte elements; rght fgure: optmal topology). Dfferent shapes of roof structures lead to dfferent topologes. (a) Roof structure wth slope shape; (b) Roof structure wth arc shape. Fgure 7. Bearng seleton of walls of Jangwan Cheng, Chongqng, Chna [13]. Fgure 8. Morphogeness of dendrforms for the bearng seleton of walls (left fgure: meshes of fnte elements; rght fgure: optmal topology). An arch structure s formed whch are wdely used n engneerng for ts wonderful mechancal performance. 533

9 and an arch structure s formed whch are wdely used n engneerng for ts wonderful mechancal performance Example 3: Morphogeness of 3-D Dendrforms As showed n the left fgure of Fgure 9, the desgn regon s a cube wth szes of 2 m 2 m 2 m. A unform vertcal dstrbuton load q = 1 N/m 2 s appled on the upper sde of the cube. Along the upper sde, a layer wth the thcness of.5 m s taen as the roof structure, and s specfed as non-desgn regon. An area wth szes of 1m 1m n the mddle part of the bottom of the cube s fxed and taen as the root of the dendrform. Structural materal s steel, and the materal propertes are same wth those n Example 1. The constrant and objectve of the model are same wth those n Example 2. The optmal topology s shown n the rght fgure of Fgure 9. It s a form wth four branches at each node, and the branch heght of the lower layer s greater than that of the upper layer Example 4: Morphogeness of Multple Plane Dendrforms As showed n Fgure 1(a), the desgn area s a rectangular wth szes of 5 m 7.5 m. A unform vertcal dstrbuton load q = 1 N/m s appled on the upper border. An Fgure 9. Morphogeness of 3d dendrforms (left fgure: meshes of fnte elements; rght fgure: optmal topology). (a) (b) Fgure 1. Morphogeness of multple plane dendrforms. (a) Desgn condtons of multple plane dendrforms; (b) Optmal topology of multple plane dendrforms. 534

10 edge wth thcness of.3 m s used as the roof structure and s the non-desgn area. The structure s dvded nto four spans and 5 fxed ponts are set. Each fxed regon has the wdth of.3m and s regarded as the roots of the dendrforms. Structural materal s steel, and the materal proper-tes are same wth those n Example 1. The constrant and objectve of the model are same wth those n Example 2. The optmal topology s shown n Fgure 1(b). Because the spans are not equal, forms of dendrforms are asymmetrc. The adjacent dendrforms also stretch toward the mddle part of the spans, and arch structures are formed. 4. Conclusons 1) It s showed from the numercal examples that t s feasble to generate dendrforms by the topology optmzaton method of contnuum structures. Dverse optons can be provded by the morphogeness of dendrforms based on the topology optmzaton method durng the conceptual desgn phase. Comparng wth the dendrforms generatng by the fractal methods whch don t consder the mechancal performance of dendrforms, the forms generated by the presented method acheve the maxmum structural stffness. Comparng wth those methods based on mechanc models whch need to specfy desgn parameters such as the heghts and the number of layers, branches at each node, and so on, the presented method can provde larger desgn space and see more optmum topology of dendrforms because t s not necessary to specfy desgn parameters. 2) The stffness of the roof structures has sgnfcant effects on the optmal topology of dendrforms. Therefore, the roof structure should be analyzed together wth the desgn area and be nvolved n the optmzaton process, and ts stffness should be smulated accurately. 3) The rato of the structural weght has sgnfcant effects on the optmal topology of dendrforms. By settng a proper weght rato to mae the stffness of the dendrform be smlar to that of the roof structure, an deal topology can be acheved. 4) The desgn regon and the geometrc shape of the roof structure have sgnfcant effects on the optmal topology of dendrforms. The parameters should be specfed and smulated accurately durng the conceptual desgn phase. Acnowledgements Ths wor was supported by Natural Scence Foundaton n Hunan provnce of Chna (216JJ616) and by Department of Educaton n Hunan Provnce of Chna (15C247). References [1] Rann, I.M. and Sassone, M. (214) Tree-Inspred Dendrforms and Fractal-Le Branchng Structures n Archtecture: A Bref Hstorcal Overvew. Fronters of Archtectural Research, 3, [2] Gawell, E. (213) Non-Eucldean Geometry n the Modelng of Contemporary Archtectural Forms. The Journal of Polsh Socety for Geometry and Engneerng Graphcs, 24,

11 [3] L, R. (214) Research on Branchng Structures Based on Fractal Theory. Barbn Insttute of Technology, Harbn. (In Chnese) [4] von Buelow, P. (27) A Geometrc Comparson of Branchng Structures n Tenson and Compresson versus Mnmal Paths. Proceedng of IASS 27, Unversty IUAV of Vence, Vence, 252. [5] Wu, Y., Zhang, J.L. and Cao, Z.G. (211) Form-Fndng Analyss and Engneerng Applcaton of Branchng Structures. Journal of Buldng Structures, 32, (In Chnese) [6] Zhang, Q., Chen, Z.H., Wang, X.D. and Lu, H.B. (215) Form-Fndng of Tree Structures Based on Sldng Cable Element. Journal of Tanjn Unversty (Scence and Technology), 48, (In Chnese) [7] Cu, C.Y., Jang, B.S. and Cu, G.Y. (213) The Senstvty-Based Morphogeness Method for Framed Structures. Chna Cvl Engneerng Journal, 46, 1-8. [8] Cu, C.Y. and Yan, H. (26) An Advanced Structural Morphogeness Technque Extended Evolutonary Structural Optmzaton Method and Its Engneerng Applcatons. Chna Cvl Engneerng Journal, 39, [9] Sasa, M. (27) Morphogeness of Flux Structure. AA Publcatons, London. [1] Huang, X.D. and Xe, Y.-M. (21) A Further Revew of ESO Type Methods for Topology Optmzaton. Structural and Multdscplnary Optmzaton, 41, [11] Sgmund, O. and Maute, K. (213) Topology Optmzaton Approaches A Comparatve Revew. Structural Multdscplnary Optmzaton, 48, [12] Su, Y.K. and Ye, H.L. (213) Contnuum Topology Optmzaton Methods ICM. Scence Press, Bejng, [13] Lu, X. (214) Morphogeness Research for Branchng Seleton Structures of Wall. Barbn Insttute of Technology, Harbn, 6-7. Submt or recommend next manuscrpt to SCIRP and we wll provde best servce for you: Acceptng pre-submsson nqures through Emal, Faceboo, LnedIn, Twtter, etc. A wde selecton of journals (nclusve of 9 subjects, more than 2 journals) Provdng 24-hour hgh-qualty servce User-frendly onlne submsson system Far and swft peer-revew system Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the number of cted artcles Maxmum dssemnaton of your research wor Submt your manuscrpt at: 536