multiobjective shape optimization of shells for … • multiobjective shape optimization of...
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Multiobjective Shape Optimization of Latticed Shells for Elastic Stiffness and Uniform Member Lengths
Makoto Ohsaki (Hiroshima University)
Shinnnosuke Fujita (Kanebako Struct. Eng.)
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Constructability
Cost performance
Aesthetic aspect
Material cost, etc.
Developable surface → Reduce cost for scaffolding
Roundness, convexity, planeness, etc.
Performance measures:weight, volume, etc.
(Formulation is straightforward)
Algebraic invariants of tensor algebra for differential geometry
Structural performance Minimum strain energy+
Constraints
Background:Optimization of Shell Roofs
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Tensor product Be’zier surface
Shape representation using Bezier surface
Tensor product Be’zier surface of order 3 i
j
B : Bernstein basis function
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Triangular patch Bezier surface
Shape representation using Bezier surface
ex). Triangular Bezier patch of order 4
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Covariant component: subscript, underbarContravariant component: superscript, overbar
Definition of algebraic invariants of differential geometry
Covariant metric tensor
Covariant Hessian
Gradient
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β1 : twice the mean curvature
β2 : the Gaussian curvature
γ1/ β0 : the curvature in the
steepest descent direction
γ3/ β0 : the curvature in the
direction perpendicular to
the steepest descent
direction
: roundness measure
Definitions of algebraic invariants of differential geometry
γ invariants
β invariants
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Two cases, and More convexity for larger absolute value of
Constraints to obtain a locally convex surface.
Locally convex surface
Constraint point
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Locally convex surface with large stiffness
Optimum shape( )
Optimum shape( )
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0 7 6.5 6
5.5 5
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0.5 0
-0.5
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0 6.5 6
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Initial shape
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Strain energy:21.125 Max.bending stress :7.9380
Max.compressive
stress :7.1183Max.tensile stress :3.0838
Strain energy:1.8313
Strain energy:2.9603
Max.verticaldisp.:3.4742
Max.verticaldisp.:5.4138
Max.compressive stress :3.0681
Max.bending stress :0.5567Max.tensile stress :0.2700
Max.compressive stress :3.1871
Max.bending stress :1.1442Max.tensile stress :0.3651
Max.verticaldisp.:44.199
Contour line
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Constraintsto obtain locally cylindrical and convex surface.
Locally cylindrical surface
Constraint points
Two cases, and 0.025 More cylindricity and convexity for larger absolute value
Constraint points
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Locally cylindrical surface with large stiffness.
Optimum shape( )
Optimum shape( )
Initial shape
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9 8.5 8
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3.5 3
2.5 2
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0.5 0
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Strain energy:21.125 Max.bending stress :7.9380
Max.compressive
stress :7.1183Max.tensile stress :3.0838Max.verticaldisp.:44.199
Strain energy:2.1157 Max.bending stress :1.3615
Max.compressive stress :3.2601Max.tensile stress :0.3743Max.vertical
disp.:3.3191
Strain energy:3.0851 Max.bending stress :1.0645
Max.compressive stress :3.1842Max.tensile stress :0.7586Max.vertical
disp.:4.7070Contour line
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Optimization problem
■:point of measurement
+
Constraint approachMinimize strain energy Maximize roundness
Find optimal solutions for different values of and
Sum of α‐invariants
Multiobjective programming for roundness and stiffness
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12Deformed(×100)
最大圧縮膜応力
最大引張膜応力
最大曲げ応力
最大鉛直変位
Undeformed Deformed(×100)
Undeformed
Initial solutionPareto solutions
Strain energy
Rou
ndne
ss
Max
. prin
cipa
l stre
ss
Max. vertical disp.
Multiobjective programming for roundness and stiffness
←stiffness roundness→
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Developable surface
generating a developable surface
β2 vanishes at 25 points indicated by the dots in the figure.
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-0.5 -1
Constraint points
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Developable shape
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Optimization of Latticed Shell
Structural Performanceminimize
Optimal shape with
large stiffness+Non-structural Performance
Geomertical property
Constructability・ Uniform member length・Minimum number of different joints
・Strain energy・Compliance
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Performance measures
Strain energy:
Variance of member length:
:nodal displacement vector:stiffness matrix:number of members:length of kth member:average ember length:total member length
Constraint on total member length:
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Optimization Problem
0( )L Lx
( )f x ( )g xMinimize and
subject to
0( )L Lx
( )f x
( ) 0g xMinimize
subject to
0( )L Lx
( )g xMinimize
subject to ( )f fx
Multiobjective Optimization
Constraint Approach
f(x): strain energyg(x): variance of member length
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Triangular grids
Frame model Triangular Bezier patch
Design variables:Locations of control points
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←strain energy
uniform member length
No feasible solution
←total length
←variance of member length
←total length
←strain energy
Fixed control points
Fixed supports
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Initial 1.3 2.0 0.5 0.05
1054mm 5.165mm 12.52mm 3045mm
20.42mm 58.24mm 3.324mm 0.363mm
V
Uniform member length⇒ small stiffness
V VAllow small deviation of member length⇒ stiff and realistic shape
Small strain energy ⇒ unrealistic shape
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Fixed point
Move in horizontal dir.
Uniform member lengthCylindrical shapeSmall stiffness
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Quadrilateral grid
Frame model Tensor product Bezier surface
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Fixed point
Fixed support
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Initial
Large strain energySmall deviation of
member length
Small strain energyLarge deviation of
member length
max min
8.226422.2
fl l
max min
1.2700.07277
fl l
max min
0.41620.9248
fl l
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28422.2mm 3.340mm 8.418mm 93.59mm
0.840mm 3.792mm 1.665mm 0.144mm
Initial 0.032 0.08 0.040 0.005
Almost uniform member length
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Pareto optimal solutions from different initial solutions
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Hexagonal Grid
Do not use parametric surfacesUse symmetry conditions
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Fixed supports
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Optimal shape: case 1
Initial
2.750f max min 536.2l l
0.08689f max min 0.002850l l
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Optimal shape: case 3
Optimal shape: case 2
0.02636f max min 0.002579l l
0.05488f max min 0.005229l l
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Conclusions• Multiobjective shape optimization of latticed shells.
– Objective functions: strain energy and variance of member lengths.
– Optimal shapes for triangular, quadrilateral, and hexagonal grids.
– Constraint approach for converting the multiobjective problem to a single objective problem.
• Feasible solution with uniform member lengths⇒Minimize strain energy under uniform
member lengths.• No feasible solution
⇒Minimize variance of member length for specified strain energy.
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Conclusions• Optimal shape of triangular grid with uniform member lengths⇒ cylindrical surface with equilateral triangles.
• Optimal shapes of quadrilateral grid⇒ highly dependent on initial solution;
bifurcation in objective function space. • Various shapes with uniform member lengths for hexagonal grids.