Topology design of trusses using a voronoi- based ground ...
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Topology design of trusses using a voronoi- based ground structure method 10th World Congress on Structural and Multidisciplinary Optimization Xiaojia (Shelly) Zhang, Sushant Maheshwari, Adeildo Ramos Jr., Glaucio H. Paulino May 21 st , 2013
Transcript of Topology design of trusses using a voronoi- based ground ...
Topology design of trusses using a voronoi-based ground structure method
10th World Congress on Structural and Multidisciplinary Optimization
Xiaojia (Shelly) Zhang, Sushant Maheshwari,Adeildo Ramos Jr., Glaucio H. Paulino
May 21st, 2013
Introduction & Motivation
Topology Optimization of Trusses□ Limited Resources□ Extreme Structures□ Functionality□ Architectural and Structural Value
Michell Solutions□ Equal stresses in all members□ Orthogonality between two bars
having opposite stresses
Michell cantilever solution
Michell, 1904 2
Introduction & Motivation
Full Ground Structure(GS) Method (Structured Mesh)
Solution obtained from a structured mesh of 2040 elements
□ Constrained geometry of full structured mesh can bias the orientation of members, leading to sub-optimal designs
□ Difficult to generate for non-convex domain□ Computational cost is quite high
Need a new approach!
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Introduction & Motivation
Practical applications of GS Method
□ Single load vs. Multiple load cases
Beghini, 2013 (Doctoral dissertation-UIUC)
Collaboration work of Prof. Paulino’s research group and SOM
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Outline
Introduction and Motivation
Formulation
Voronoi-based GS method
Macro Element based GS method
Use of GS in design of Lotte tower
Concluding Remarks
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Nested Formulation
Single load case
minaFTu(a)
s. t. k(a)u = f
a jl j £Vmax
j=1
n
å
a j ³ 0
minxFTu(x)
s. t. ljxj-V
max
j=1
n
å £ 0
xj ,min
£ xj£ x
j ,max
u(x) = k(x)-1 f
M.P.Bendsoe O.SigmundTopology optimization Theory, Methods and Applications, Springer, 2003, New York
minx
Fi
Tui(x)
i=1
m
å
s. t. ljxj-V
max
j=1
n
å £ 0
xj ,min
£ xj£ x
j ,max
ui(x) = k(x)-1 fi
Multiple load case
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Different Types of Meshes
Structured Mesh Voronoi-based Mesh
7C. Talischi, G. H. Paulino, A. Pereira , Ivan F. M. MenezesPolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struct Multidisc Optim (2012) 45:309–328
Algorithm to Eliminate Overlapping Bars
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q
Case 1
Case 2
cos(q ) =1±0.01 Do not generate connectivity
Concave Domain using Voronoi-based Mesh
Full level# Bars: 9,123Time: 1,291 sec
9C. Talischi, G. H. Paulino, A. Pereira , Ivan F. M. MenezesPolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struct Multidisc Optim (2012) 45:309–328
Concave Domain using Voronoi-based Mesh
Full level# Bars: 9,348Time: 932 sec
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Motivation for new approach!
C. Talischi, G. H. Paulino, A. Pereira , Ivan F. M. MenezesPolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struct Multidisc Optim (2012) 45:309–328
Macro-element Approach: Structured Mesh
Insert additional nodes Connect all nodes within each macro-element
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Initial Mesh
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Initial Mesh
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Macro-element Approach: Voronoi-based Mesh
Insert additional nodes Connect all nodes within each macro-element
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Macro-element Approach: Better Connectivity
Standard Macro-element
13C. Talischi, G. H. Paulino, A. Pereira , Ivan F. M. MenezesPolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab, Struct Multidisc Optim (2012) 45:309–328
Macro-element Approach: Easier Bar Generation
Fewer overlapping bars
Avoid connectivity outside boundary for concave domains
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Macro-element Approach: Lower Computational Cost
Global stiffness matrix
0
2.5
Standard Macro-element
Scal
e u
pComputation cost ~10,000 bars
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Macro-element approach: Rectangular Domain
Smoother boundary lines
E = 2*108 n = 0.3
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10
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
10
Full level
0 2 4 6 8 10 12
1
2
3
4
5
6
7
8
9
4 Nodes/Edge
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Standard Macro-element
Objective 1.718e-02 1.686e-02
# of Bars 6,146 22,800
Time 1,019 sec 1,126 sec
Macro-element approach: Wrench Domain
Orthogonal pairs of bars Smooth contour lines
E = 2*108
n = 0.3
Full level 8 Nodes/Edge
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Standard Macro-element
# of Bars 32,324 317,637
Time 9,932 sec 9,864 sec
Macro-element approach: Hook Domain
Clear interior details and better shape
Macro-element5 Nodes/Edge
StandardFull level
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Standard Macro-element
# of Bars 26,753 206,424
Time 2,344 sec 2,371 sec
Macro-element: Single Load vs. Multiple Load
E = 2*108 n = 0.3
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Multiple load case
Structured Voronoi-based
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Single load case
Multiple load case
Single load case
Use of GS in design of Lotte tower
Lotte Tower(Courtesy SOM)
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Use of GS in design of Lotte tower
Case A: Wind loading about one axis and symmetry (Single load case)
Front Stromberg et. al, 2010
Side
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Use of GS in design of Lotte tower
Case A: Wind loading about one axis and symmetry (Single load case)
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Use of GS in design of Lotte tower
Case B: Wind loading about two axis and symmetry (Multiple load case)
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(Courtesy SOM)
Future Applications of GS
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30 St. Mary Axe, Londonhttp://en.wikipedia.org/wiki/30_St_Mary_Axe
Aldar Headquarter, AbuDhabihttp://en.wikipedia.org/wiki/Aldar_headquarters_building
Conclusions
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Structured & Voronoi tessellations offer alterative approaches for ground structure generation.
Voronoi-based meshes are ideal to generate ground structures in non-convex domains.
Macro-element approach□ Efficient and accurate w.r.t. the full ground structure method
(either Voronoi-based or Cartesian based)□ Seams to satisfy isoperimetric property□ Promising for actual engineering design
Macro-patch approach for Lotte Tower leads to a diagrid-like structure, which is a typical design tool.
Both Macro-element and macro-patch approach lead to viable multiple load case solutions.
