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

Optimization of trusses under uncertain loads. Felipe Alvarez ∗ , Miguel Carrasco † , Benjamin Ivorra †† ıa Matem´ *Universidad de Chile, Departamento de Ingenier´ 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 1 2 f T u ( D ) min λ,u s.t. K ( λ ) u = f λ ∈ ∆ m 2 f T u is called Compliance . 1 Optimal design of trusses. – p. 5/4

Mathematical formulation 1 2 f T u min • λ ∈ R m ; m number of bars. λ,u • n grades of freedom. s.t. K ( λ ) u = f • f ∈ R n nodal load vector. λ ∈ ∆ m • u ∈ R n nodal displacements. m • ∆ m = { λ ∈ R m | λ ≥ 0 , � λ i = 1 } . i =1 • K ( λ ) ∈ R n × n , stiffness matrix . Optimal design of trusses. – p. 6/4

Stiffness Matrix m � K ( λ ) = λ i K i i =1 where K i = b i b T i ∈ R n × n K i is dyadic. Optimal design of trusses. – p. 7/4

Stiffness Matrix m � K ( λ ) = λ i K i i =1 where K i = b i b T i ∈ R n × n K i is dyadic. √ E i γ i ∈ R n b i = l i • E i : Young modulus. • l i : length of bar i. • γ i : cosines/sines vector. Optimal design of trusses. – p. 7/4

Ground structure approach 1.5 1 0.5 0 f −0.5 −1 0 1 2 3 4 5 6 Optimal design of trusses. – p. 8/4

Ground structure approach 1.5 1 0.5 0 f −0.5 −1 0 1 2 3 4 5 6 1.5 1 0.5 0 f −0.5 −1 0 1 2 3 4 5 6 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 3 2.5 2 1.5 Y 1 f 0.5 0 −0.5 −1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 X Typically a large number of bar vanish at the optimum. Optimal design of trusses. – p. 10/4

1 2 f T u ( D ) min λ ∈ R m ,u ∈ R n s.t. K ( λ ) u = f λ ∈ ∆ m 2 f T u is independent of u satisfying K ( λ ) u = f. 1 Optimal design of trusses. – p. 11/4

Minimax Formulation λ ∈ ∆ m { 1 2 f T u | K ( λ ) u = f } ( D ) min x ∈ R n { f T x − 1 2 x T K ( λ ) x } ( D ) λ ∈ ∆ m max min � (Minimax Theorem) 1 ≤ i ≤ m { 1 2 x T K i x − f T x } ( P ) x ∈ R n max min 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) 4 6 5 1 2 3 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) 4 6 5 1 2 3 Optimal design of trusses. – p. 14/4

Multiload model Standard multiload model k 1 � α j f jT u j min 2 λ ∈ R m j =1 s.t. K ( λ ) u j = f j , j = 1 , . . . , k λ ∈ ∆ m We minimize a weighted average of the compliances. Optimal design of trusses. – p. 15/4

Multiload model Defining f = ( α 1 f 1 T , . . . , α k f kT ) T ∈ R n ( k +1) , ˆ K i = diag( α 1 K i , . . . , α k K i ) ∈ R n ( k +1) × n ( k +1) , ˆ m � ˆ λ i ˆ K ( λ ) = K i , i =1 Then multiload model can be written as ( D ). remark: ˆ K � = b i b T 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 ∈ R n be a perturbation on the load vector f.  if λ ∈ ∆ m and u ∈ R n such that 1 2 ( f + ξ ) T x   Ψ( ξ, λ ) = K ( λ ) u = f + ξ,   + ∞ otherwise Ψ: ( R n , B ( R n )) → ( 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 ( R ∪ { + ∞} , ¯ ( R n , B ( R n )) B ( R )) Ψ( · , λ ): → �→ Ψ( ξ, λ ) ξ is measurable. Optimal design of trusses. – p. 19/4

Random load model P ) E ξ [Ψ( ξ, λ )] Minimum expected compliance design problem ( D ; min λ ∈ ∆ m We assume that ξ is a random variable corresponding to an uncertain nodal load perturbation ( R n , B ( R n )) (Ω , A ) → ξ : �→ ξ ( ω ) ω Optimal design of trusses. – p. 20/4

Example: discrete perturbation P with finite support Sop( P ) = { ξ 1 , . . . , ξ k } . P ) 1. k 1 � α j f jT u j ( D ; min 2 λ ∈ R m j =1 s.t. K ( λ ) u j = f j , P ) is the standard multiload model where i = 1 , . . . , k P ( ξ = ξ j ) and f j = f + ξ j , j = 1 , . . . , k. λ ∈ ∆ m ( D ; α j = Optimal design of trusses. – p. 21/4

Continuous case Optimal design of trusses. – p. 22/4

Continuous case E ( ξ ) = 0 and Variance–covariance matrix Theorem 0.1. Let ξ : Ω → R n be a continuous random variable P ) is given by with mean vector Var( ξ ) = PP T , with P ∈ R n × k . P ) Then the corresponding minimum expected compliance design problem ( D ; 1 + 1 2 Trace( P T U ) 2 f T u ( D ; min λ ∈ R m � �� � � �� � mean load variance s.t. K ( λ ) u = f K ( λ ) U = P λ ∈ ∆ m Optimal design of trusses. – p. 23/4

Continuous case Remark p j , j = 1 , . . . , k, columns of P Defining f = ( f T , p 1 T , . . . , p kT ) T ∈ R n ( k +1) , ˆ K i = diag( K i , K i , . . . , K i ) ∈ R n ( k +1) × n ( k +1) , ˆ m P ) may be written as a multiload model. � ˆ λ i ˆ K ( λ ) = K i , i =1 Then ( D ; Optimal design of trusses. – p. 24/4

example Random multidimensional independent perturbation E ( ε j ) = 0 and Var( ε j ) = σ 2 ξ = � k j =1 ε j d j , where P ) d j ∈ R n , j . r 2 f T u + 1 1 � σ j d jT u j ( D ; min 2 λ ∈ R m j =1 s.t. K ( λ ) u = f K ( λ ) u j = σ j d j 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 1.5 1.5 4 5 6 1 1 0.5 0.5 0 0 1 2 3 −0.5 −0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 1.5 1.5 (a) (b) 1 1 0.5 0.5 0 0 −0.5 −0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Toy example, Ben-Tal and Nemirowski 1997 – p. 27/4

Problem compl. max min mean compl. 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 15 15 15 10 10 10 5 5 5 0 0 0 −6 −4 −2 0 2 4 6 8 10 12 −6 −4 −2 0 2 4 6 8 10 12 −6 −4 −2 0 2 4 6 8 10 12 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 E ξ [Ψ( ξ, λ )] + β Var[Ψ( ξ, λ )] . We consider min λ ∈ R m α Dome 3D – p. 35/4

Stochastic model including variance E ξ [Ψ( ξ, λ )] + β Var[Ψ( ξ, λ )] We consider min λ ∈ R m α Theorem 0.2. ξ ∼ N (0 , PP T ) . Taking for simplicity α = 0 and P var ) min β = 1 . 1 � � Trace(( P T U ) 2 ) + 2 f T UU T f ( D ; 2 λ ∈ R m 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 2 1.5 1 0.5 Values 0 −0.5 −1 −1.5 −2 1 Column Number – p. 44/4 Figure 1: Boxplot representation

Boxplot 120 110 100 90 80 Compliance 70 60 50 40 30 20 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 Coefficient values α − β – p. 45/4