Optimization of trusses under uncertain loads. Felipe Alvarez , - - PowerPoint PPT Presentation

Optimization of trusses under uncertain loads.

Felipe Alvarez∗, Miguel Carrasco†, Benjamin Ivorra †† *Universidad de Chile, Departamento de Ingenier´ ıa Matem´ atica

†Universidad de los Andes, Departamento de Ingenier´

ıa

††Universidad de Complutense de Madrid, Departamento de Matem´

aticas Aplicadas.

Optimal design of trusses. – p. 1/4

• Optimization of trusses
• Standard minimum compliance truss design.
• Instabilities and the standard multiload model.
• Random Perturbations: Minimizing the expected

compliance.

• Numerical examples.
• Random Perturbations: Minimizing the variance.
• Numerical examples.

Optimal design of trusses. – p. 2/4

Optimization of trusses

Optimal design of trusses. – p. 3/4

Introduction

• Definition.
• Problem: find the best structure able to carry a

external nodal force.

• Constraints: mechanical equilibrium , total volume

and others . . .

• Minimize the compliance.

Optimal design of trusses. – p. 4/4

Minimum compliance truss design

min

λ,u

1 2fTu

(D)

s.t. K(λ)u = f λ ∈ ∆m

1 2fT u is called Compliance.

Optimal design of trusses. – p. 5/4

Mathematical formulation

• λ ∈ Rm; m number of bars.
• n grades of freedom.
• f ∈ Rn nodal load vector.
• u ∈ Rn nodal displacements.

min

λ,u

1 2f Tu s.t. K(λ)u = f λ ∈ ∆m

• ∆m = {λ ∈ Rm | λ ≥ 0,

m

• i=1

λi = 1}.

• K(λ) ∈ Rn×n, stiffness matrix.

Optimal design of trusses. – p. 6/4

Stiffness Matrix

K(λ) =

m

• i=1

λiKi

where

Ki = bibT

i ∈ Rn×n

Ki is dyadic.

Optimal design of trusses. – p. 7/4

Stiffness Matrix

K(λ) =

m

• i=1

λiKi

where

Ki = bibT

i ∈ Rn×n

Ki is dyadic. bi = √Ei li γi ∈ Rn

• Ei: Young modulus.
• li: length of bar i.
• γi: cosines/sines vector.

Optimal design of trusses. – p. 7/4

Ground structure approach

1 2 3 4 5 6 −1 −0.5 0.5 1 1.5

f Optimal design of trusses. – p. 8/4

Ground structure approach

1 2 3 4 5 6 −1 −0.5 0.5 1 1.5

f

1 2 3 4 5 6 −1 −0.5 0.5 1 1.5

f Optimal design of trusses. – p. 8/4

Ground structure approach

f

Optimal design of trusses. – p. 9/4

Ground structure approach

f f

• Nodal positions are fixed in the reference configuration.
• We use a mesh full of nodes and bars.

Optimal design of trusses. – p. 9/4

Ground structure approach

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 −1 −0.5 0.5 1 1.5 2 2.5 3 X Y

f

Typically a large number of bar vanish at the optimum.

Optimal design of trusses. – p. 10/4

min

λ∈Rm,u∈Rn

1 2fT u

(D)

s.t. K(λ)u = f λ ∈ ∆m

1 2fT u is independent of u satisfying K(λ)u = f.

Optimal design of trusses. – p. 11/4

Minimax Formulation

(D) min

λ∈∆m{1

2fTu | K(λ)u = f} (D) min

λ∈∆m max x∈Rn{fT x − 1

2xTK(λ)x}

• (Minimax Theorem)

(P) min

x∈Rn max 1≤i≤m{1

2xT Kix − fTx}

Optimal design of trusses. – p. 12/4

Instabilities and the standard multiload model

Optimal design of trusses. – p. 13/4

Example

The design problem (D) may produce unsatisfactory results in respect to mechanical stability (see Ben-Tal and Nemirovsky 1997)

Optimal design of trusses. – p. 14/4

Example

The design problem (D) may produce unsatisfactory results in respect to mechanical stability (see Ben-Tal and Nemirovsky 1997)

1 2 3 4 5 6

Optimal design of trusses. – p. 14/4

Example

The design problem (D) may produce unsatisfactory results in respect to mechanical stability (see Ben-Tal and Nemirovsky 1997)

1 2 3 4 5 6

Optimal design of trusses. – p. 14/4

Multiload model

Standard multiload model

min

λ∈Rm

1 2

k

• j=1

αjfjTuj s.t. K(λ)uj = fj, j = 1, . . . , k λ ∈ ∆m

We minimize a weighted average of the compliances.

Optimal design of trusses. – p. 15/4

Multiload model

Defining

ˆ f = (α1f1T, . . . , αkfkT)T ∈ Rn(k+1), ˆ Ki = diag(α1Ki, . . . , αkKi) ∈ Rn(k+1)×n(k+1), ˆ K(λ) =

m

• i=1

λi ˆ Ki,

Then multiload model can be written as (D). remark: ˆ

K = bibT

i .

Optimal design of trusses. – p. 16/4

Random Perturbations: Minimizing the expected compliance.

Optimal design of trusses. – p. 17/4

Random load model

Let ξ ∈ Rn be a perturbation on the load vector f.

Ψ(ξ, λ) =     

1 2(f + ξ)Tx

if λ ∈ ∆m and u ∈ Rn such that

K(λ)u = f + ξ, +∞

• therwise

Ψ: (Rn, B(Rn)) → (R ∪ {+∞}, ¯ B(R))

Results to be proper, lower semicontinuous and convex.

Optimal design of trusses. – p. 18/4

Random load model

For each λ ∈ ∆m

Ψ(·, λ): (Rn, B(Rn)) → (R ∪ {+∞}, ¯ B(R)) ξ → Ψ(ξ, λ)

is measurable.

Optimal design of trusses. – p. 19/4

Random load model

Minimum expected compliance design problem

(D;

min

λ∈∆m

We assume that ξ is a random variable corresponding to an uncertain nodal load perturbation

ξ : (Ω, A) → (Rn, B(Rn)) ω → ξ(ω)

Optimal design of trusses. – p. 20/4

Example: discrete perturbation

1.

(D;

min

λ∈Rm

1 2

k

• j=1

αjfjT uj s.t. K(λ)uj = fj, i = 1, . . . , k λ ∈ ∆m (D;

αj =

Optimal design of trusses. – p. 21/4

Continuous case

Optimal design of trusses. – p. 22/4

Continuous case

Theorem 0.1. Let ξ : Ω → Rn be a continuous random variable with mean vector

Var(ξ) = PP T , with P ∈ Rn×k.

Then the corresponding minimum expected compliance design problem (D;

(D;

min

λ∈Rm

1 2fTu

mean load

+ 1 2 Trace(P T U)

• variance

s.t. K(λ)u = f K(λ)U = P λ ∈ ∆m

Optimal design of trusses. – p. 23/4

Continuous case

Remark pj, j = 1, . . . , k, columns of P Defining

ˆ f = (fT, p1T, . . . , pkT)T ∈ Rn(k+1), ˆ Ki = diag(Ki, Ki, . . . , Ki) ∈ Rn(k+1)×n(k+1), ˆ K(λ) =

m

• i=1

λi ˆ Ki,

Then (D;

Optimal design of trusses. – p. 24/4

example Random multidimensional independent perturbation

ξ = k

j=1 εjdj, where

dj ∈ Rn,

j.

(D;

min

λ∈Rm

1 2fTu + 1 2

r

• j=1

σjdjT uj s.t. K(λ)u = f K(λ)uj = σjdj j = 1, . . . , k λ ∈ ∆m.

Optimal design of trusses. – p. 25/4

Numerical results

Optimal design of trusses. – p. 26/4

Toy example

Toy example, Ben-Tal and Nemirowski 1997

0.5 1 1.5 2 2.5 −0.5 0.5 1 1.5

1 2 3 4 5 6

0.5 1 1.5 2 2.5 −0.5 0.5 1 1.5 0.5 1 1.5 2 2.5 −0.5 0.5 1 1.5

(a)

0.5 1 1.5 2 2.5 −0.5 0.5 1 1.5

(b)

Toy example, Ben-Tal and Nemirowski 1997 – p. 27/4

Problem compl. mean max compl. min compl. Standard single load

∄ +∞ +∞

(a) 0.025 0.053 0.013 (b) 0.023 0.051 0.012

Toy example, Ben-Tal and Nemirowski 1997 – p. 28/4

Electricity mast

Electricity mast, Achtziger et al. 1992

Electricity mast, Achtziger et al. 1992 – p. 29/4

Electricity mast

−6 −4 −2 2 4 6 8 10 12 5 10 15 −6 −4 −2 2 4 6 8 10 12 5 10 15 −6 −4 −2 2 4 6 8 10 12 5 10 15

Electricity mast, Achtziger et al. 1992 – p. 30/4

Dome

Dome 3D – p. 31/4

Dome

Dome 3D – p. 32/4

Dome

Dome 3D – p. 33/4

Random Perturbations: Minimizing the variance.

Dome 3D – p. 34/4

Stochastic model including variance

We consider

min

λ∈Rm α

Dome 3D – p. 35/4

Stochastic model including variance

We consider

min

λ∈Rm α

Theorem 0.2. ξ ∼ N(0, PP T ). Taking for simplicity α = 0 and

β = 1. (D;

λ∈Rm

1 2

• Trace((P TU)2) + 2fTUUT f
• s.t. K(λ)u = f

K(λ)U = P λ ∈ ∆m

Apparent harder to solve.

Dome 3D – p. 36/4

Michel 3D

Michel 3D – p. 37/4

Michel 3D

Michel 3D – p. 38/4

Michel 3D – p. 39/4

Michel 3D – p. 40/4

Michel 3D – p. 41/4

Michel 3D – p. 42/4

Michel 3D – p. 43/4

Boxplot

1 −2 −1.5 −1 −0.5 0.5 1 1.5 2 Values Column Number

Figure 1: Boxplot representation

– p. 44/4

Boxplot

1−0 1−0.01 1−0.1 1−0.25 1−0.5 1−0.75 1−1 0.75−1 0.5−1 0.25−1 0.1−1 20 30 40 50 60 70 80 90 100 110 120 Compliance Coefficient values α−β

– p. 45/4

Conclusions

• Classical model (single load) don’t get good

results .

• Random model helps to avoid this problem.
• It’s equivalent to multiload one, but another

interpretations in weight factors holds.

• Continuous Model.
• We have to know the influence of each

perturbation.

• Increase the dimension of the problem.
• Highly non linear.

– p. 46/4

Optimization of trusses under uncertain loads.

Felipe Alvarez∗, Miguel Carrasco†, Benjamin Ivorra †† *Universidad de Chile, Departamento de Ingenier´ ıa Matem´ atica

†Universidad de los Andes, Departamento de Ingenier´

ıa

††Universidad de Complutense de Madrid, Departamento de Matem´

aticas Aplicadas.

– p. 47/4

Optimization of trusses under uncertain loads.

Felipe Alvarez∗, Miguel Carrasco†, Benjamin Ivorra †† *Universidad de Chile, Departamento de Ingenier´ ıa Matem´ atica

†Universidad de los Andes, Departamento de Ingenier´

ıa

††Universidad de Complutense de Madrid, Departamento de Matem´

aticas Aplicadas.

– p. 48/4

Boxplot

– p. 49/4