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

Makoto Ohsaki (Hiroshima University)

Shinnnosuke Fujita (Kanebako Struct. Eng.)

Background:Optimization of Shell Roofs

Structural performance Minimum strain energy＋

Cost performancePerformance measures：weight, volume, etc.

Constraints

Cost performanceMaterial cost, etc.

(Formulation is straightforward)

Aesthetic aspectRoundness, convexity,

Algebraic invariants of tensor algebra for

Constructability

Roundness, convexity, planeness, etc.

tensor algebra for differential geometry

yDevelopable surface → Reduce cost for scaffolding

Shape representation using Bezier surface

Tensor product Be’zier surface

B : Bernstein basis functionB : Bernstein basis function

ｊ

Tensor product Be’zier surface of order 3 ii

Shape representation using Bezier surface

Triangular patch Bezier surface

ex). Triangular Bezier patch of order 4

Definition of algebraic invariants of 

Covariant component: subscript, underbarC i i b

differential geometryContravariant component: superscript, overbar

Gradient

Covariant Hessian

Covariant metric tensor

55

Covariant metric tensor

55

Definitions of algebraic invariants of 

β1 : twice the mean curvature

differential geometryβ invariants

β1 : twice the mean curvature

β2 : the Gaussian curvature

γ1/ β0 : the curvature in the

steepest descent direction

γ3/ β0 : the curvature in the γ invariants

γ3/ β0 : t e cu vatu e t e

direction perpendicular to

the steepest descent the steepest descent

direction

: roundness measure

Relation between the invariants and surface shape

Local Local properties in the neighborhood of a point P on the surface are characterized by properties in the neighborhood of a point P on the surface are characterized by h i i f llh i i f ll

The contours in the neighbourhood of P are coaxial (part of) similar ellipses.

the invariants as follows:the invariants as follows:

The shape is locally concave (example : )The shape is locally convex. (example : )The shape is locally concave. (example : )

Th t i th i hb h d f P ( t f) i l h b l ( l )The contours in the neighbourhood of P are (part of) coaxial hyperbolas. (example : )

P is a critical point. (example : ) p ( p )

Relation between the invariants and surface shape

One of the principal curvatures is 0.The other principal curvature is positive. (example : )The other principal curvature is negative (example : )The other principal curvature is negative. (example : )

The shape is locally flat surface. (example : )

Relation between the invariants d f hand surface shape

Direction of gradient vector coincides with one of the principal direction,

and concave in one principal direction (example : )

Direction of gradient vector coincides with one of the principal direction, and the surface near P is locally cylindrical

>

and convex in one principal direction (example : ) <

Locally convex surfaceLocally convex surface

Constraint point

Constraints to obtain a locally convex surface.

Two cases, and More convexity for larger absolute value of

y

Initial shape

-6

-4

-2

0 6

5.5 5

4.5 4

3 5

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

-14

-12

-10

-8 3.5 3

2.5 2

1.5 1

0.5

St i 21 125 Max bending stress :7 9380

Max.compressive stress :7.1183Max.tensile stress :3.0838Max.vertical

disp :44 199Optimum shape( )

-4

-2

0 7 6.5 6

5.5 5

4 5

Strain energy:21.125 Max.bending stress :7.9380disp.:44.199

14 12 10 8 6 4 2 0

-14

-12

-10

-8

-6 4.5 4

3.5 3

2.5 2

1.5 1

0.5 0

-0.5Max.vertical

Max.compressive stress :3.0681Max.tensile stress :0.2700

Optimum shape( )

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

-2

0 6.5 6

5.5

Strain energy:1.8313 disp.:3.4742 Max.bending stress :0.5567

-12

-10

-8

-6

-4 5 4.5 4

3.5 3

2.5 2

1.5 1

0.5Max verticalMax.compressive stress :3.1871

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

-14 0.5 0

-0.5

Strain energy:2.9603

Max.verticaldisp.:5.4138 Max.bending stress :1.1442

Max.tensile stress :0.3651

Contour line

Locally convex surface with large stiffness

Locally cylindrical surfacey y

Constraint pointsConstraint points

Constraintsto obtain locally cylindrical and

fconvex surface.Two cases, and 0.025 More cylindricity and convexity for larger y y y gabsolute value

Initial shape

-6

-4

-2

0 6

5.5 5

4.5 4

3 5

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

-14

-12

-10

-8 3.5 3

2.5 2

1.5 1

0.5

St i 21 125 M b di t 7 9380

Max.compressive stress :7.1183Max.tensile stress :3.0838Max.vertical

disp :44 199Optimum shape( )

9 8.5 8

7.5 7

6.56

-4

-2

0

Strain energy:21.125 Max.bending stress :7.9380disp.:44.199

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

Max.compressive stress :3.2601Max tensile stress :0 3743Max.vertical

Optimum shape( )

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

8.5 8

7.57

-2

0

Strain energy:2.1157 Max.bending stress :1.3615Max.tensile stress :0.3743a .ve ca

disp.:3.3191

7 6.5 6

5.5 5

4.5 4

3.5 3

2.5 2

1 5

-12

-10

-8

-6

-4

Max.compressive stress :3.1842ili l 1.5

1 0.5 -14 -12 -10 -8 -6 -4 -2 0

-14

Strain energy:3.0851 Max.bending stress :1.0645Max.tensile stress :0.7586Max.vertical

disp.:4.7070Contour line

Locally cylindrical surface with large stiffness.

Multiobjective programming for d d tiff

Optimization problem

roundness and stiffness

Sum of αSum of αSum of αSum of α‐‐invariantsinvariants

■：point of measurement

Constraint approachConstraint approachMinimize strain energy Maximize roundness

1414

＋

1414

Find optimal solutions for different values of and

Multiobjective programming for d d tiff

s

Mroundness and stiffness

最大圧縮膜応力最大圧縮膜応力

最大引張膜応力最大引張膜応力

最大曲げ応力最大曲げ応力

最大鉛直変位最大鉛直変位dnes

s

ipal

stre

ss

Max. vertic

最大鉛直変位最大鉛直変位

Rou

nd

Max

. prin

ci

cal disp.

Strain energy

M

←stiffness roundness→

1515Deformed(×100)

Undeformed Deformed(×100)

Undeformed1515(×100)(×100)

Initial solutionPareto solutions

Developable surfaceDevelopable surface

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

generating a developable surface

the dots in the figure.Constraint points

developable surface

4

-2

0 8 7.5 7

6.5 6

-8

-6

-4 5.5 5

4.5 4

3.5 3

2.5

-14

-12

-10 2 1.5 1

0.5 0

-0.5-1

-14 -12 -10 -8 -6 -4 -2 0 1

Developable shapep p

17171717

18181818

19191919

20202020

Optimization of Latticed ShellOptimization of Latticed Shellpp

Structural Performanceminimizeminimize

Optimal shape with

large stiffness＋・Strain energy・Compliance

Non-structural Performance

Geomertical propertyUniform member length

Constructability・ Uniform member length・Minimum number of different joints

21212121

Performance measuresPerformance measures

Strain energy:Strain energy:

Variance of member length:Variance of member length:

Constraint on total member length:

：nodal displacement vector

2222

：stiffness matrix：number of members：length of kth member 2222：average ember length：total member length

Optimization ProblemOptimization Problempp

Multiobjective Optimization

( )f x ( )g xMinimize and

Multiobjective Optimization

0( )L Lxsubject tof(x): strain energy

Constraint Approachg(x): variance of member length

( )f x

( ) 0g xMinimize

subject to

( )g xMinimize

subject to ( )f fx

23230( )L Lx

( ) 0g xsubject to

0( )L Lx

subject to ( )f fx

2323

Triangular gridsTriangular gridsg gg g

Design variables:Locations of control points

24242424Frame model Triangular Bezier patch

Fixed control points

Fixed supports

←strain energy

←t t l l th

←variance of member length

←t t l l th

Fixed supports

2525uniform member length

←total length←total length

←strain energy

2525No feasible solutionNo feasible solution

Initial 1 3 2 0 0 5 0 05Initial          1.3 2.0                        0.5                   0.05

U if b l h S ll iUniform member length⇒ small stiffness

Small strain energy ⇒ unrealistic shape

ＶＶ ＶAllow small deviation of member length⇒ stiff and realistic shape

26261054mm 5.165mm 12.52mm 3045mm

262620.42mm 58.24mm 3.324mm 0.363mm

Fixed point

Move in horizontal dir.

2727

Uniform member lengthCylindrical shapeSmall stiffness 2727Small stiffness

Quadrilateral gridQuadrilateral gridgg

28282828Frame model Tensor product Bezier surface

Fixed pointFixed point

Fi d tFixed support

29292929

Initial8 226f

max min

8.226422.2

fl l

Large strain energySmall deviation of

member length

i

1.2700 07277

fl l

Small strain energy

max min 0.07277l l

3030

gyLarge deviation of

member length

0 4162f 3030

max min

0.41620.9248

fl l

Initial         0.032 0.08                     0.040                 0.005

Almost uniform member length

3131422.2mm 3.340mm 8.418mm 93.59mm

31310.840mm 3.792mm 1.665mm 0.144mm

Pareto optimal solutions from Pareto optimal solutions from ppdifferent initial solutionsdifferent initial solutions

32323232

Hexagonal GridHexagonal Gridgg

3333Do not use parametric surfaces3333Do not use parametric surfacesUse symmetry conditions

Fixed supports

34343434

Initial

2.750f max min 536.2l l

Optimal shape: case 1

353535350.08689f

max min 0.002850l l

Optimal shape: case 2

0.02636f max min 0.002579l l

Optimal shape: case 3

363636360.05488f max min 0.005229l l

ConclusionsConclusions• Multiobjective shape optimization of latticed shells.Multiobjective shape optimization of latticed shells.

– Objective functions: strain energy and variance of member lengths.

– Optimal shapes for triangular, quadrilateral, and hexagonal grids.C i h f i h l i bj i– Constraint approach for converting the multiobjective problem to a single objective problem.

• Feasible solution with uniform member lengths• Feasible solution with uniform member lengths⇒Minimize strain energy under uniform

member lengths.g• No feasible solution⇒Minimize variance of member length for g

specified strain energy. 

ConclusionsConclusions• Optimal shape of triangular grid with uniform• Optimal shape of triangular grid with uniform member lengths⇒ li d i l f i h il l i l⇒ cylindrical surface with equilateral triangles. 

• Optimal shapes of quadrilateral gridp p q g⇒ highly dependent on initial solution; 

bifurcation in objective function spacebifurcation in objective function space. • Various shapes with uniform member lengths for hexagonal grids.