OPTI 95 MIAMI OPTI 95 MIAMI OPTI 95 MIAMI 19- -21 September 1995 21 September 1995 19 19-21 September 1995 Second- -order Structural Optimization order Structural Optimization Second Second-order Structural Optimization Alvaro F. M. Azevedo Azevedo Alvaro F. M. Email: alvaro@fe.up.pt alvaro@fe.up.pt Email: Faculty of Engineering, University of Porto, PORTUGAL Faculty of Engineering, University of Porto, PORTUGAL
GENERAL PURPOSE GENERAL PURPOSE OPTIMIZATION METHOD OPTIMIZATION METHOD • Large scale optimization Large scale optimization • (> 1000 design variables) (> 1000 design variables) • Increased precision and reliability Increased precision and reliability • • Second Second- -order method order method •
NONLINEAR PROGRAMMING NONLINEAR PROGRAMMING ( ) Minimize f x ~ subject to ( ) ( ) 2 → = ≤ 0 g x + s 0 g x i i ~ ~ ~ ~ ( ) = 0 h x ~ ~ ~ • Variables / functions real and continuous • Symbolic manipulation of generalized polynomials ( ) − − Ex. 2 3 1 2 = − + − f x 59 . x x 31 . x 2 7 . x x x 18 . 1 4 2 1 3 5 ~ • Straightforward derivation and evaluation
p ( ) ( ) m ( ) ( ) ∑ ∑ • Lagrangian: g 2 h = + λ + + λ L X f x g x s h x k k k k k ~ ~ ~ ~ = = k 1 k 1 ( ) g h = λ λ • Variables: X s , , , x ~ ~ ~ ~ ~ • Stationary point of the Lagrangian: system of nonlinear equations ( ) g = 1,..., λ = i m 2 0 s i i ( ) 2 = 1,..., + = i m 0 g s i i ( ) ⇒ ∇ = L X 0 p ∂ m λ ∂ λ ∂ f g h ∑ ∑ ( ) ~ ~ g h = 1,..., + k + k = i n 0 k k ∂ ∂ ∂ x x x = = k 1 k 1 i i i ( ) h i = 0 = 1,..., i p
( ) ( ) − − q 1 q q 1 • Lagrange-Newton method: ∆ + ∇ = H X X L X 0 ~ ~ ~ ~ ~ (m) (m) (n) (p) ( ) ( ) 0 0 g 2 λ Diag Diag 2 s i (m) i ~ ~ ∂ g i 0 0 (m) ∂ x ~ ~ ~ = H j ∂ h j * (n) ∂ x i 0 (p) SYMMETRIC ~ 2 2 p 2 ∂ m ∂ ∂ f g h ∑ ∑ g h * + λ k + λ k k k ∂ ∂ ∂ ∂ ∂ ∂ x x x x x x = = k 1 k 1 i j i j i j
HESSIAN MATRIX SPARSITY PATTERN HESSIAN MATRIX SPARSITY PATTERN • Gaussian elimination • Conj. grad. method: . . . T T ∆ + ∇ = 0 H H X H L ~ ~ ~ ~ ~
Gaussian elimination Gaussian elimination • faster • more reliable • small pivots avoided • RAM requirements increase considerably with the number of variables Conjugate gradient method Conjugate gradient method • huge number of iterations • too slow in large problems • small RAM requirements
= ( x Z x ) scaling of all the variables scaling of all the variables i i i normalization of the constraints normalization of the constraints •Automatic Automatic • substitution of elementary eq. constraints substitution of elementary eq. constraints simplification of the nonlinear program simplification of the nonlinear program •Solution of the original NLP can be recovered Solution of the original NLP can be recovered • •Line search Line search •
NEWTOP COMPUTER PROGRAM (ANSI C) NEWTOP COMPUTER PROGRAM (ANSI C) - Input example: ### Main title of the nonlinear program Symmetric truss with two load cases (kN,cm) Min. +565.685*t5^2 + 100*t8^2; # truss volume (cm3) s.t.i.c. Min.area 4: -t4^2 + 0.15 < 0; s.t.e.c. Equil.16: +141.421*t5^2*disp16 - 100 = 0; END_OF_FILE
STRUCTURAL OPTIMIZATION STRUCTURAL OPTIMIZATION • Integrated formulation • In large scale problems the following transformation is advantageous: ≤ h 0 h 0 h = 0 → h = 0 ≤ h 0 h 0
•Truss sizing examples: Truss sizing examples: • – stress, displacement and side constraints – one load case •Desktop workstation: Desktop workstation: 256 MB RAM; 40 • 256 MB RAM; 40 MFlops MFlops •Computation time: Computation time: • → – small problems ( 100 bars) a few seconds → – medium problems (1000 bars) a few hours → – large problems (4000 bars) a few days
LARGE SCALE OPTIMIZATION EXAMPLE LARGE SCALE OPTIMIZATION EXAMPLE 3D truss sizing 3D truss sizing •Number of bars = 4 096 Number of bars = 4 096 • •Number of degrees of freedom = 3 135 Number of degrees of freedom = 3 135 • •Number of decision variables = 7 231 Number of decision variables = 7 231 • •Number of inequality constraints = 19 038 Number of inequality constraints = 19 038 • •No variable linking; no active set strategy No variable linking; no active set strategy •
BUILDING ROOF - - OPTIMAL SOLUTION OPTIMAL SOLUTION BUILDING ROOF Undeformed mesh mesh Undeformed
BUILDING ROOF - BUILDING ROOF - OPTIMAL SOLUTION OPTIMAL SOLUTION Deformed mesh Deformed mesh
NEWTOP ALGORITHM NEWTOP ALGORITHM ADVANTAGES ADVANTAGES • PRECISION PRECISION • • VERSATILITY VERSATILITY • • RELIABILITY RELIABILITY • • CAPACITY CAPACITY •
NEWTOP ALGORITHM NEWTOP ALGORITHM DRAWBACKS DRAWBACKS • EFFICIENCY ? EFFICIENCY ? • • INTEGRATED FORMULATION INTEGRATED FORMULATION • Too demanding when the n. design variables n. design variables is small is small Too demanding when the and the n. load cases n. load cases x x n. degrees of freedom n. degrees of freedom is high is high and the
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