Multiobjective Shape Optimization of Latticed Shells for Elastic Stiffness and Uniform Member Lengths

Makoto Ohsaki (Hiroshima University)

Shinnnosuke Fujita (Kanebako Struct. Eng.)

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

Tensor product Be’zier surface

Shape representation using Bezier surface

Tensor product Be’zier surface of order 3 i

ｊ

B : Bernstein basis function

Triangular patch Bezier surface

Shape representation using Bezier surface

ex). Triangular Bezier patch of order 4

5

Covariant component: subscript, underbarContravariant component: superscript, overbar

Definition of algebraic invariants of differential geometry

Covariant metric tensor

Covariant Hessian

Gradient

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

Two cases, and More convexity for larger absolute value of

Constraints to obtain a locally convex surface.

Locally convex surface

Constraint point

Locally convex surface with large stiffness

Optimum shape( )

Optimum shape( )

-14 -12 -10 -8 -6 -4 -2 0

-14

-12

-10

-8

-6

-4

-2

0 7 6.5 6

5.5 5

4.5 4

3.5 3

2.5 2

1.5 1

0.5 0

-0.5

-14 -12 -10 -8 -6 -4 -2 0

-14

-12

-10

-8

-6

-4

-2

0 6.5 6

5.5 5

4.5 4

3.5 3

2.5 2

1.5 1

0.5 0

-0.5

Initial shape

-14 -12 -10 -8 -6 -4 -2 0

-14

-12

-10

-8

-6

-4

-2

0 6

5.5 5

4.5 4

3.5 3

2.5 2

1.5 1

0.5

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

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

Locally cylindrical surface with large stiffness.

Optimum shape( )

Optimum shape( )

Initial shape

-14 -12 -10 -8 -6 -4 -2 0

-14

-12

-10

-8

-6

-4

-2

0 6

5.5 5

4.5 4

3.5 3

2.5 2

1.5 1

0.5

9 8.5 8

7.5 7

6.5 6

5.5 5

4.5 4

3.5 3

2.5 2

1.5 1

0.5 0

-0.5 -14 -12 -10 -8 -6 -4 -2 0

-14

-12

-10

-8

-6

-4

-2

0

8.5 8

7.5 7

6.5 6

5.5 5

4.5 4

3.5 3

2.5 2

1.5 1

0.5 -14 -12 -10 -8 -6 -4 -2 0

-14

-12

-10

-8

-6

-4

-2

0

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

11

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

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→

Developable surface

generating a developable surface

β2 vanishes at 25 points indicated by the dots in the figure.

-14 -12 -10 -8 -6 -4 -2 0

-14

-12

-10

-8

-6

-4

-2

0 8 7.5 7

6.5 6

5.5 5

4.5 4

3.5 3

2.5 2

1.5 1

0.5 0

-0.5 -1

Constraint points

14

Developable shape

15

16

17

18

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

19

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:

20

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

21

Triangular grids

Frame model Triangular Bezier patch

Design variables:Locations of control points

22

←strain energy

uniform member length

No feasible solution

←total length

←variance of member length

←total length

←strain energy

Fixed control points

Fixed supports

23

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

24

Fixed point

Move in horizontal dir.

Uniform member lengthCylindrical shapeSmall stiffness

25

Quadrilateral grid

Frame model Tensor product Bezier surface

26

Fixed point

Fixed support

27

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

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

29

Pareto optimal solutions from different initial solutions

30

Hexagonal Grid

Do not use parametric surfacesUse symmetry conditions

31

Fixed supports

32

Optimal shape: case 1

Initial

2.750f max min 536.2l l

0.08689f max min 0.002850l l

33

Optimal shape: case 3

Optimal shape: case 2

0.02636f max min 0.002579l l

0.05488f max min 0.005229l l

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. 

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.