FE design optimization under uncertainty as a Constrained - PowerPoint PPT Presentation

FE design optimization under uncertainty as a Constrained optimization problem under uncertainty Gert de Cooman, Etienne Kerre, Erik Quaeghebeur & Keivan Shariatmadar FUM & SYSTeMS Research Groups

Toy problem: two-component massless rod L Y 1 Y 2 a l 1 = ( 1 − x ) L l 2 = xL

Toy problem: two-component massless rod, tensile load L Y 1 Y 2 a l 1 = ( 1 − x ) L l 2 = xL d 1 d 2 Y 1 Y 2 F

Toy problem: two-component massless rod, tensile load L Y 1 Y 2 a l 1 = ( 1 − x ) L l 2 = xL d 1 d 2 Y 1 Y 2 F Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis FE analysis 3 nodes, boundary conditions [ c 1 + c 2 ][ d 1 ] [ 0 ] − c 2 c i = Y i a = , l i . − c 2 c 2 d 2 F d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis FE analysis 3 nodes, boundary conditions [ c 1 + c 2 ][ d 1 ] [ 0 ] − c 2 c i = Y i a = , l i . − c 2 c 2 d 2 F Solution solving the system (analytically) gives d 1 = FL 1 − x d 2 = d 1 + FL x Y 1 , Y 2 . a a d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis FE analysis 3 nodes, boundary conditions [ c 1 + c 2 ][ d 1 ] [ 0 ] − c 2 c i = Y i a = , l i . − c 2 c 2 d 2 F Solution solving the system (analytically) gives d 1 = FL 1 − x d 2 = d 1 + FL x Y 1 , Y 2 . a a Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL . d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component rod, tensile load: design optimization d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

Two-component rod, tensile load: design optimization Precisely known elastic moduli Y 1 and Y 2 This problem is ▶ a classical constrained optimization problem; ▶ considered ‘solved’. d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

Two-component rod, tensile load: design optimization Precisely known elastic moduli Y 1 and Y 2 This problem is ▶ a classical constrained optimization problem; ▶ considered ‘solved’. Uncertainty about elastic moduli Y 1 and Y 2 This problem is ▶ a constrained optimization problem under uncertainty; ▶ not well-posed as such. d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

Two-component rod, tensile load: design optimization Precisely known elastic moduli Y 1 and Y 2 This problem is ▶ a classical constrained optimization problem; ▶ considered ‘solved’. Uncertainty about elastic moduli Y 1 and Y 2 This problem is ▶ a constrained optimization problem under uncertainty; ▶ not well-posed as such. Approach: ▶ reformulate as a well-posed decision problem; ▶ solve the decision problem, i.e., derive a classical constrained optimization problem. d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

A constrained optimization problem under uncertainty Goal Maximize f ( x ) under the constraint that xRY . x optimization variable (values in X ) f objective function (from X to ℝ ) Y random variable (realizations y in Y ) R relation on X × Y .

A constrained optimization problem under uncertainty Goal Maximize f ( x ) under the constraint that xRY . x optimization variable (values in X ) f objective function (from X to ℝ ) Y random variable (realizations y in Y ) R relation on X × Y . Decision problem Find the optimal decisions x : ▶ associate a utility function with every decision z : { f ( z ) , zRy , G z ( y ) = with penalty value L < inf f ; L , z ∕ Ry , f ( x ) G z ( y ) f ( z ) f ( z ) L z x y zR z ∕ R

A constrained optimization problem under uncertainty Goal Maximize f ( x ) under the constraint that xRY . x optimization variable (values in X ) f objective function (from X to ℝ ) Y random variable (realizations y in Y ) R relation on X × Y . Decision problem Find the optimal decisions x : ▶ associate a utility function with every decision z : { f ( z ) , zRy , G z ( y ) = with penalty value L < inf f ; L , z ∕ Ry , f ( x ) G z ( y ) f ( z ) f ( z ) L z x y zR z ∕ R ▶ choose an optimality criterion, e.g., maximinity, maximality.

Uncertainty models & Optimality criteria Goal Find the optimal decisions x given an optimality criterion and the utility functions G z for all z in X .

Uncertainty models & Optimality criteria Lower and upper expectation With (almost) all uncertainty models correspond lower and upper expectation operators ( E and E ). Goal Find the optimal decisions x given an optimality criterion and the utility functions G z for all z in X .

Uncertainty models & Optimality criteria Lower and upper expectation With (almost) all uncertainty models correspond lower and upper expectation operators ( E and E ). Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility: E ( G x ) = sup E ( G z ) . z ∈ X Goal Find the optimal decisions x given an optimality criterion and the utility functions G z for all z in X .

Uncertainty models & Optimality criteria Lower and upper expectation With (almost) all uncertainty models correspond lower and upper expectation operators ( E and E ). Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility: E ( G x ) = sup E ( G z ) . z ∈ X Maximality Optimal x are undominated in pairwise comparisons with all other decisions: z ∈ X E ( G x − G z ) ≥ 0 . inf Goal Find the optimal decisions x given an optimality criterion and the utility functions G z for all z in X .

Uncertainty models & Optimality criteria Lower and upper expectation With (almost) all uncertainty models correspond lower and upper expectation operators ( E and E ). Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility: E ( G x ) = sup E ( G z ) . z ∈ X Maximality Optimal x are undominated in pairwise comparisons with all other decisions: z ∈ X E ( G x − G z ) ≥ 0 . inf Research results decision problem solutions for some combinations of uncertainty model and optimality criterion; more are on the way. Goal Find the optimal decisions x given an optimality criterion and the utility functions G z for all z in X .

Impact on FE design optimization under uncertainty ▶ An approach to dealing with uncertainty in FE design optimization in a uniform way. ▶ A toolbox of decision problem solutions.

Impact on FE design optimization under uncertainty ▶ An approach to dealing with uncertainty in FE design optimization in a uniform way. ▶ A toolbox of decision problem solutions. ▶ No reduction in the computational complexity; one faces ▶ an optimization problem to find the uncertainty-independent constraints (cf. doing an FE analysis under uncertainty), ▶ the resulting classical constrained optimization problem.