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

Y1 Y2 a L l1 = (1−x)L l2 = xL

Toy problem: two-component massless rod, tensile load

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Toy problem: two-component massless rod, tensile load

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

FE analysis 3 nodes, boundary conditions

[c1 +c2 −c2 −c2 c2 ][d1 d2 ] = [0 F ] , ci = Yia

li .

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

FE analysis 3 nodes, boundary conditions

[c1 +c2 −c2 −c2 c2 ][d1 d2 ] = [0 F ] , ci = Yia

li .

Solution solving the system (analytically) gives

d1 = FL

a 1−x Y1 ,

d2 = d1 + FL

a x Y2 .

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component massless rod, tensile load: FE analysis

FE analysis 3 nodes, boundary conditions

[c1 +c2 −c2 −c2 c2 ][d1 d2 ] = [0 F ] , ci = Yia

li .

Solution solving the system (analytically) gives

d1 = FL

a 1−x Y1 ,

d2 = d1 + FL

a x Y2 .

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that d2 < D.

Two-component rod, tensile load: design optimization

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Two-component rod, tensile load: design optimization

Precisely known elastic moduli Y1 and Y2 This problem is

▶ a classical constrained optimization problem; ▶ considered ‘solved’.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Two-component rod, tensile load: design optimization

Precisely known elastic moduli Y1 and Y2 This problem is

▶ a classical constrained optimization problem; ▶ considered ‘solved’.

Uncertainty about elastic moduli Y1 and Y2 This problem is

▶ a constrained optimization problem under uncertainty; ▶ not well-posed as such.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < Da FL.

Two-component rod, tensile load: design optimization

Precisely known elastic moduli Y1 and Y2 This problem is

▶ a classical constrained optimization problem; ▶ considered ‘solved’.

Uncertainty about elastic moduli Y1 and Y2 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.

Y1 Y2 a L l1 = (1−x)L l2 = xL Y1 Y2 F d1 d2

Goal Maximize x under the constraint that 1−x

Y1 + x Y2 < 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:

Gz(y) = { f(z), zRy, L, z∕Ry,

with penalty value L < inff;

f(x) x z f(z) Gz(y) y zR z∕R L f(z)

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:

Gz(y) = { f(z), zRy, L, z∕Ry,

with penalty value L < inff;

f(x) x z f(z) Gz(y) y zR z∕R L f(z)

▶ 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 Gz 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 Gz 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(Gx) = sup

z∈X

E(Gz).

Goal Find the optimal decisions x given an optimality criterion and the utility functions Gz 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(Gx) = sup

z∈X

E(Gz).

Maximality Optimal x are undominated in pairwise comparisons with all

• ther decisions:

inf

z∈X E(Gx −Gz) ≥ 0.

Goal Find the optimal decisions x given an optimality criterion and the utility functions Gz 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(Gx) = sup

z∈X

E(Gz).

Maximality Optimal x are undominated in pairwise comparisons with all

• ther decisions:

inf

z∈X E(Gx −Gz) ≥ 0.

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 Gz 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;

• ne faces

▶ an optimization problem to find the uncertainty-independent constraints

(cf. doing an FE analysis under uncertainty),

▶ the resulting classical constrained optimization problem.