Design of homogenized microstructure using stress-based topology optimization

D

ESIGN OF MICROSTRUCTURES USING STRESS

–

BASED

TOPOLOGY OPTIMIZATION

Maxime Collet

, Matteo Bruggi

, Lise Noël

, Simon Bauduin

, Pierre Duysinx

1

_{University of Liège, Aerospace and Mechanical Engineering Department, Liège, Belgium}

Politecnico di Milano, Civil and Environmental Engineering Department,Milan, Italy

M

OTIVATION

Microstructural design (MD) using topology optimization along with ho-mogenization to compute equivalent elastic properties is used to conceive optimal microstructures addressing specific needs:

max K s.t. V ≤ 0, 5: K_optimized = 0, 6734

Accounting for the stress level within the optimized layouts ensures the

structural integrity of the microstructure.

S

ENSITIVITY ANALYSIS OF THE STRESSES

Sensitivity along with the qp-relaxation (Bruggi,2008)a: ∂σ_{V M,e} ∂x_k = δek(p − q)x p−q−1 e σV M,e + ∂σ_{V M,e} ∂x_k x p−q e

Adjoint approach (active set : Na selected constraints < N total constraints)

∂σ_{V M,e} ∂x_k = λ T ∂K ∂x_k u − λ T  pxp−1_k f_e,0i 

with the adjoint problem and force vector:

Kλ = − h uT_e M0_eu_e
−1/2 M0_eu_e iT f_e,0i = n1 X i=1 n2 X j=1 w_iw_jB_eT H0_eεi₀detJ

a_{Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology}

optimization.Struct Multidisc Optim 36:125-141

E

XAMPLE

1: C

LASSICAL

B

ULK MAXIMIZATION

The maximization of the bulk modulus under stress constraints with σ_y0 = 1, 4N/m2

max K s.t. V ≤ 0.5 AND σ ≤ σ_y0: K_optimized = 0, 6719

• _{Similar topology compared to the classical solution but with a}

controlled stress level ( _σσ0

y ≤ 1)

• _{More square shaped compared to the classical solution} • _{Lower optimized bulk modulus}

E

XAMPLE

2: Bronzi di Riace

SEISMIC INSULATOR

Bronzi di Riace statue with its marble seismic insulator:

Currently: Four marble balls used to isolate the statues in case of earth-quakes. Design requirements are high bulk and low shear (under pre-scribed value). A topology optimization can be formulated:

max

0<x_min≤x_e≤1 K

s.t. K(x)ui = f (x)i; i = 1, 2

G ≤ G∗

hσ_{V M,e}(x)i ≤ σ_y0

• _{Two load cases !}

• _{Starting point: Hole with}

void at the center of the RVE

• σ_y0 = 1.8N/m2 _{and G ≤} 10−3N/m2

• No stress constraints : Koptimized = 0, 3665

• With stress constraints : Koptimized = 0, 4376

• _{Control of the local stress constraints ( σ}σ0

y ≤ 1)

• _{Change in the topology of the optimized layouts} • _{Existence of local optima !}

C

ONCLUSION AND

P

ERSPECTIVES

• _{Control of the stress level introduced in a topology optimization}

framework for microstructural design (sensitivity analysis)

• _{The optimization might be steered toward better optimum =⇒}

Presence of local minima

• _{Speed up the computation ! CPU requirement is still to high !} • _{Extension to 3D manufacturable microstructures}

A

CKNOWLEDGEMENTS

The first author, Maxime Collet, would like to acknowledge the Belgian National Fund for Scientific research (FRIA) for its financial support. Part of this work was done when the first author was spending a doctoral stay at Politecnico di Milano