Engineering a new architecture through barycentric element based ...
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Lauren L. Beghini¹, Alessandro Beghini², William F. Baker², Neil Katz², Glaucio H. Paulino¹ ¹Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, U.S.A. ²Skidmore, Owings & Merrill, LLP, Chicago, IL, U.S.A. Research Objecves Conclusions References To apply topology opmizaon to the field of structural engineering through high-rise building design Ulize manufacturing and layout constraints to make results more meaningful Address the importance of achieving a balance between engineering and architecture for efficient, sustainable design Historical examples of structures by architects with strong and innovave engineering concepts L.L . Beghini, A. Beghini, W. F. Baker, N. Katz, and G. H. Paulino. "Connecng architecture and engineering through structural topology opmizaon" Engineering Structures. Under revision. L.L . Stromberg, A. Beghini, W. F. Baker, and G. H. Paulino. "Topology opmizaon for braced frames: Combining connuum and beam/column elements." Engi- neering Structures. Vol 37, pp. 106-124, 2012. L.L . Stromberg, A. Beghini, W. F. Baker, and G. H. Paulino. "Applicaon of layout and topology opmizaon using paern gradaon for the conceptual design of buildings." Structural and Muldisciplinary Opmizaon. Vol 43, No. 2, pp. 165-180, 2011. Goal: overcome dichotomy between architectural aesthecs and engineering efficiency using topology opmizaon Introducon: Engineering and Architecture [1-3] Mulple websites Antonio Gaudi¹ Buckminster Fuller² Felix Candela³ Acknowledgements Naonal Science Foundaon Graduate Research Fellowship (GRFP) Skidmore, Owings, & Merrill, LLP Topology opmizaon using barycentric-elements can be a valuable tool to bridge the gap between engineering and architecture in the design industry. Moreover, re- sulng designs will be more efficient and sustainable, by opmizing the material consumpon. Opmal design problem: Example: Minimum Compliance Basic Topology Opmizaon Framework Other Criteria: Eigenmodes Deflecon (P-Δ) Buckling load Natural frequency min (, ) .. (, ) , , d du du f st g i k i = = … 0 1 for min (, ) (, ) () .. d du pu du Kdu p d d f g g V V st T = = ( ) - = ( ) - 1 2 Polygonal elements in nature: Bio-Inspired Design: Zendai Compeon (China) Picture of physical model using topology opmizaon results (courtesy of SOM) and resemblance to spider webs Clockwise: Giant’s Causeway (Ireland), honeycomb, giraffe’s skin, dragonfly wings [Mulple websites] Polygonal Finite Elements Applicaon of Barycentric Elements: Polygonal Meshes for Design 3D Voronoi mesh (CVT): Aſter Lloyd’s Algorithm Inial Voronoi Mesh Laplace shape funcons using natural neighbors: Polygonal shape funcon C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes. “Polygonal finite elements for topology opmiza- on: A unifying paradigm.” IJNME. Vol 82, No. 6, pp. 671-698, 2010. φ i i j j w w ( ( ) () ) x x x = ∑ ∈ J w s h i i i () () () x x x =
Transcript of Engineering a new architecture through barycentric element based ...
Lauren L. Beghini¹, Alessandro Beghini², William F. Baker², Neil Katz², Glaucio H. Paulino¹¹Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, U.S.A.
²Skidmore, Owings & Merrill, LLP, Chicago, IL, U.S.A.
Research Objectives
Conclusions
References
To apply topology optimization to the field of structural engineering through high-rise building designUtilize manufacturing and layout constraints to make results more meaningfulAddress the importance of achieving a balance between engineering and architecture for efficient, sustainable design
Historical examples of structures by architects with strong and innovative engineering concepts
L.L . Beghini, A. Beghini, W. F. Baker, N. Katz, and G. H. Paulino. "Connecting architecture and engineering through structural topology optimization" Engineering Structures. Under revision.L.L . Stromberg, A. Beghini, W. F. Baker, and G. H. Paulino. "Topology optimization for braced frames: Combining continuum and beam/column elements." Engi-neering Structures. Vol 37, pp. 106-124, 2012.L.L . Stromberg, A. Beghini, W. F. Baker, and G. H. Paulino. "Application of layout and topology optimization using pattern gradation for the conceptual design of buildings." Structural and Multidisciplinary Optimization. Vol 43, No. 2, pp. 165-180, 2011.
Goal: overcome dichotomy between architectural aesthetics and engineering efficiency using topology optimization
Introduction: Engineering and Architecture
[1-3] Multiple websitesAntonio Gaudi¹ Buckminster Fuller² Felix Candela³
AcknowledgementsNational Science Foundation Graduate Research Fellowship (GRFP)Skidmore, Owings, & Merrill, LLP
Topology optimization using barycentric-elements can be a valuable tool to bridge the gap between engineering and architecture in the design industry. Moreover, re-sulting designs will be more efficient and sustainable, by optimizing the material consumption.
Optimal design problem:
Example: Minimum Compliance
Basic Topology Optimization FrameworkOther Criteria:
EigenmodesDeflection (P-Δ)Buckling loadNatural frequency
min ( , )
. . ( , ) , ,d
d u
d u
f
s t g i ki = = …0 1for
min ( , )
( , )( )
. .d
d u p u
d u K d u pd d
f
gg V V
s t
T=
= ( ) −= ( ) −
1
2
Polygonal elements in nature:Bio-Inspired Design: Zendai Competition (China)
Picture of physical model using topology optimization results (courtesy of SOM) and resemblance to spider websClockwise: Giant’s Causeway (Ireland), honeycomb, giraffe’s skin, dragonfly wings [Multiple websites]
Polygonal Finite Elements
Application of Barycentric Elements: Polygonal Meshes for Design
3D Voronoi mesh (CVT):
After Lloyd’s AlgorithmInitial Voronoi Mesh
Laplace shape functions using natural neighbors:
Polygonal shape function
C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes. “Polygonal finite elements for topology optimiza-tion: A unifying paradigm.” IJNME. Vol 82, No. 6, pp. 671-698, 2010.
POLYGONAL FINITE ELEMENTS FOR TOPOLOGY OPTIMIZATION 681
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Figure 8. Sparsity pattern for the stiffness of polygonal mesh with 500 elements and 1002 nodes beforeRCM resequencing (left) and after resequencing (right). The bandwidth is reduced from 966 to 75.
Figure 9. For a convex polygon, every interior point is a natural neighbor to all the vertices. The geometricquantities si and hi used to define the Laplace shape functions are shown here.
Furthermore, it can be shown that these functions are linearly precise:
∑i∈I
xi�i (x)=x (8)
In this expression, xi represents the location of node qi . This property along with constant precision(partition of unity) ensures the convergence of the Galerkin method for second-order partial
Copyright � 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2010; 82:671–698DOI: 10.1002/nme
φii
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