Cmd_puc2018.pdf

COMPUTATIONAL METAMATERIAL DESIGN Víctor D. Fachinotti

Pontificia Universidad Católica de Chile, April-May 2018

OUTLINE • • • • • •

Computational metamaterial design Microscale analysis Multiscale problem as a macroscopic one with inhomogeneous material Macroscopic thermo-mechanical response as a function of microstructure Material design as an optimization problem Applications: • Optimization of the mechanical response under thermal loads • Optimization of the thermal response using free material optimization (FMO) • Heat flux manipulation • Design of easy-to-make devices using discrete material optimization (DMO) • Design of easiest-to-make devices using topology optimization • Mechanical cloaking • Advantages of computational metamaterial design • Perspectives 2

METAMATERIAL DESIGN • MATERIAL DESIGN: to modify the microstructure of the material in a macroscopic piece in order to obtain an optimal response of the piece • METAMATERIAL: the so-designed material, usually having extraordinary effective properties: • optical or acoustical camouflage /invisibility • negative Poisson ratio • negative thermal conductivity, thermal camouflage, etc.

3

Metamaterial with negative Poisson ratio made by dip-in direct-laser-writing optical lithography (Bückmann et al., Advanced Materials, 2012)

Metamaterial that twitsts under compression (Frenzel et al., Science, 2017) 4

Metamaterial made of PVC and PDMS for cloaking elastic waves (Stenger et al., PRL 2012)

Metamaterial as a laminate of copper (A) and polyurethane (B) for heat flux inversion (Narayana & Sato, PRL 2013)

5

MACROSCOPIC BODY WITH VARIABLE MICROSTRUCTURE • Let the microstructure vary throughout the macroscopic domain, being sampled at a series of points 𝑿𝛼 • Each 𝑿𝛼 has its own Representative Volume Element (RVE)

6

COMPUTATIONAL METAMATERIAL DESIGN • Computational Metamaterial Design involves the computational solution of a series of multiscale problems for changing microstructure

micro-scale analysis (at each RVE)

Effective properties

MACROSCALE ANALYSIS (AT THE BODY W)

Macroscopic response

until finding the optimal macroscopic response

7

QUANTITATIVELY CHARACTERIZED MICROSTRUCTURE • Let the RVE at any sampling point 𝑿𝛼 ∈ Ω be characterized by a finite (𝛼) (𝛼) number of scalar (micro)parameters 𝑝1 , 𝑝2 , …

Ex.: Narayana & Sato’s heat flux inverter (PRL 2012)

Ω 𝛼

𝛼

 Effective properties at 𝑿𝛼 ∈ Ω = 𝑓(𝑝1 , 𝑝2 , … ) 8

MACROSCOPIC BODY WITH VARIABLE QUANTITATIVELY CHARACTERIZED MICROSTRUCTURE RVE caracterized by 𝒑 1 RVE caracterized by 𝒑 2 9

MICROSCALE ANALYSIS

10

MICROSCALE ANALYSIS • Goal: determination of the effective properties as analytical functions of the microparameters Microscale analysis Analytical 𝜶 𝒑𝟏

𝜶 , 𝒑𝟐

,…

Experimental +RSM

𝐞𝐟𝐟 𝐩𝐫𝐨𝐩 𝐚𝐭 𝐗 𝜶 𝜶 𝜶 = 𝒇(𝒑𝟏 , 𝒑𝟐 , …)

Numerical +RSM 11

ANALYTICAL MICROSCALE ANALYSIS: LAMINATE Effective anisotropic conductivity

𝑑𝐴 𝑘𝐴 + 𝑑𝐵 𝑘𝐵 + 𝑑𝐶 𝑘𝐶 𝑘𝜆𝜆 = 𝑑𝐴 + 𝑑𝐵 + 𝑑𝐶 𝑑𝐴 + 𝑑𝐵 + 𝑑𝐶 𝑘𝜏𝜏 = 𝑑𝐴 𝑑𝐵 𝑑𝐶 + + 𝑘𝐴 𝑘𝐵 𝑘𝐶 𝑘𝑥𝑥 = 𝑘𝜆𝜆 cos 2 𝜃 + 𝑘𝜏𝜏 sin2 𝜃 𝑘𝑦𝑦 = 𝑘𝜆𝜆 sin2 𝜃 + 𝑘𝜏𝜏 cos 2 𝜃 𝑘𝑥𝑦 = 𝑘𝑦𝑥 = (𝑘𝜆𝜆 −𝑘𝜏𝜏 )cos 𝜃 sin 𝜃

12

EXPERIMENTAL+NUMERICAL MICROSCALE ANALYSIS: PAPER • Using upscaling techniques, discrete element simulations and X-ray microtomography of the geometry of wood fibers and their bonds and the architecture of the fibrous network, Marulier (PhD thesis, 2013) determined the homogenized elastic moduli: 𝑪orth = 1.14 × 109 𝜙 − 0,02 2 𝑨 𝑎 ⟹ 𝑪𝑥𝑦 = 𝚯 𝜃 𝑪orth 𝜙, 𝑎 𝚯 𝜃 𝑇 – 𝜙: fiber content – 𝑨(𝑎): fiber orientation tensor (response surface from experiments), 𝑎: orientation intensity – 𝚯 𝜃 : serves to rotates from 𝜆𝜏 to 𝑥𝑦, 𝜃: angle between the 𝑥 and 𝜆 * Collaboration with S. Le Corre (LTN Nantes) and L. Orgéas (LCNRS Grenoble) 13

NUMERICAL MICROSCALE ANALYSIS: CANCELLOUS BONE • Using FEM for a geometrically parameterized cell, Kowalczyk (2006) determined the homogenized elastic moduli: ′ 𝐶𝑖𝑗𝑘𝑙 = 𝑓 𝑡𝑐 , 𝑡𝑣 , 𝑡ℎ ′ ⟹ 𝐶𝑖𝑗𝑘𝑙 = 𝑅𝑚𝑖 𝑅𝑛𝑗 𝑅𝑝𝑘 𝑅𝑞𝑙 𝐶𝑚𝑛𝑝𝑞

– 𝑡𝑐 , 𝑡𝑣 , 𝑡ℎ : geometric parameters – 𝑹(𝜓1 , 𝜓2 , 𝜓3 ): 3D rotation tensor, 𝝍: rotation vector * Collaboration with A. Cisilino & L. Colabella (INTEMA, Argentina) 14

NUMERICAL MICROSCALE ANALYSIS: SOLID WITH INCLUSIONS • Using FEM on RVEs with variable 𝑏 and ℎ, we determined the effective thermomechanical properties 𝑘𝑖𝑗 = 𝑘𝑖𝑗 𝑏, ℎ

𝐶𝑖𝑗𝑘𝑙 = 𝐶𝑖𝑗𝑘𝑙 𝑏, ℎ 𝜕𝜎𝑖𝑗 𝜕𝑇

= 𝑑𝑖𝑗 (𝑏, ℎ)

GRIDS FROM FEM PARAMETRIC ANALYSIS

POLYNOMIAL RESPONSE SURFACES

* Fachinotti, Toro, Sánchez & Huespe, Int. J. Solids & Struct. 2015 15

REDUCTION OF THE MULTISCALE PROBLEM • Once you know the effective material properties as functions of the microparameters 𝒑 from the microscale analysis, the multiscale problem becomes a classic macroscopic problem with inhomogeneous material properties

16

MACROSCOPIC THERMO-MECHANICAL RESPONSE AS A FUNCTION OF MICROSTRUCTURE

17

THERMOMECHANICAL RESPONSE IN A BODY WITH VARIABLE MICROSTRUCTURE

18

THERMOMECHANICAL RESPONSE AS A FUNCTION OF MICROSTRUCTURE • Given the microstructure 𝑷 = 𝒑 1 , 𝒑(2) , … througout Ω: 1) solve the steady state FEM heat equation: 𝛻𝝓

𝑇

𝒌 𝒑 𝛻𝝓𝑑𝑣

𝝓𝑇 𝑞 wall 𝑑𝑠

𝑻=

Ω

𝜕Ωq 𝑲ther 𝑷

𝑸

⟹ 𝑻 = 𝑲ther 𝑷 −𝟏 𝑸 = 𝑻 𝑷 2) solve the FEM equilibrium equation: 𝑩𝑇 𝑪 𝒑 𝑩 𝑑𝑣 𝑼 = Ω

𝚽 𝑇 𝒕wall 𝑑𝑠 − 𝜕Ω𝜎

𝑲mech 𝑷

𝑩𝑇 𝝈,𝑇 𝒑 Δ𝑇 𝑷 𝑑𝑣 = 𝟎 Ω

𝑭mech −𝟏 [𝑭

𝑭ther 𝑷

⟹ 𝑼 = 𝑲mech 𝑷 mech − 𝑭ther 𝑷 ] = 𝑼 𝑷 • The macroscopic thermo-mechanical response is the nonlinear function ℛ = 𝑓 𝑼 𝑷 , 𝑻 𝑷 , 𝑷 = ℛ(𝑷)

19

MATERIAL DESIGN AS AN OPTIMIZATION PROBLEM • To design a material consists of finding the optimal set

𝑷opt

=

opt (1) (1) (2) (2) 𝑝1 , 𝑝2 , … , 𝑝1 , 𝑝2 , …

such that ℛ 𝑷opt = min ℛ 𝑷 𝑷

subject to 𝑎𝑖 ≤ 𝑃𝑖 ≤ 𝑏𝑖 Bound constraints 𝑐𝑖 𝑷 ≥ 0 Inequality constraints 𝑑𝑖 𝑷 = 0 Equality constraints • This is a nonlinear constrained optimization problem with a usually large number of design variables 20

MATERIAL DESIGN FOR OPTIMAL MACROSCOPIC MECHANICAL RESPONSE UNDER THERMAL LOADS with S. Toro, P. Sánchez & A. Huespe (CIMEC)

21

THERMAL DEFLECTION OF A CANTILEVER PLATE T=50°C 0.3 m T=0°C 3m

STEEL E=2e11Pa n=0.3 a=1e-5/°C k=36.5W/(m°C)

COPPER E=1.2e11Pa n=0.34 a=1.7e-5/°C k=384W/(m°C) 22

MAXIMAL DEFLECTION To maximize the deflection of the tip of the beam consists of finding 𝑷opt

= 𝑏

1

,ℎ

1

,𝑏

2

,ℎ

2

,…

opt

such that

−𝑢𝑦(𝑷opt) = min(−𝑢𝑦 (𝑷)) 𝑷

𝑿

𝑏

𝑛

1

𝑛

Periodic RVE

ℎ

𝑛

uy Copper Steel 23

MAXIMAL DEFLECTION: OPTIMIZATION PROBLEM To maximize the tip deflection, we solve the nonlinear constrained optimization problem min (−𝑢𝑦 (𝑷)) 2𝑁

𝑷∈ℝ

subject to 0≤ℎ 0≤𝑏

𝑛 𝑛

≤1 ≤1

𝑛 = 1,2, … , 𝑁 = 287 = #𝑛𝑜𝑑𝑒𝑠

EFFECTIVE PROPERTIES AS FUNCTIONS OF MICROSTRUCTURE Grids from FEM microscale homogenization analysis Polynomial response surfaces

25

MAXIMAL DEFLECTION: OPTIMAL SOLUTION Thickness of the vertical layers

Thickness of the horizontal layers

(100% copper)

(100% steel)

26

MAXIMAL DEFLECTION: VERTICAL DISPLACEMENTS Copper beam

Optimal beam

uy = 1.361 uy,copper

27

MINIMAL DEFLECTION: OPTIMIZATION PROBLEM To minimize the tip deflection, we solve the nonlinear constrained optimization problem min (𝑢𝑦 (𝑷)) 2𝑁

𝑷∈ℝ

subject to 0≤ℎ 0≤𝑏

𝑛 𝑛

≤1 ≤1

𝑛 = 1,2, … , 𝑁 = 287 = #𝑛𝑜𝑑𝑒𝑠

MINIMAL DEFLECTION: OPTIMAL SOLUTION Thickness of the vertical layers

Thickness of the horizontal layers

(100% copper)

(100% steel)

29

MINIMAL DEFLECTION: VERTICAL DISPLACEMENTS Steel beam

Optimal beam

uy = 0.527 uy,steel 30

MATERIAL DESIGN FOR OPTIMAL MACROSCOPIC THERMAL RESPONSE USING FREE MATERIAL OPTIMIZATION (FMO)

with S. Giusti (GIDMA, UTN Córdoba, Argentina)

31

FREE MATERIAL OPTIMIZATION OF THE THERMAL RESPONSE • FREE MATERIAL OPTIMIZATION (FMO): the design variables are the effective properties themselves 1

1

2

2

𝑛

𝑛

𝑛

• For 𝑷 = 𝑘𝑥𝑥 , 𝑘𝑦𝑦 , 𝑘𝑥𝑥 , 𝑘𝑦𝑦 , … (with 𝑘𝑥𝑥 , 𝑘𝑦𝑦 , and 𝑘𝑥𝑦 = 0 being the effective conductivities at node 𝑛), let us find 𝑷opt = arg min 𝑷

2 (𝑇 (𝑷) − 200°C) 𝑖 𝑖∈𝐴𝐵

subject to 𝑛 , 𝑛 ≤1 0.001 ≤ 𝑘𝑥𝑥 𝑘𝑦𝑦

32

INITIAL TEMPERATURE DISTRIBUTION Initial guess: 𝑘𝑥𝑥 = 𝑘𝑦𝑦 = 0.5

33

OPTIMAL DISTRIBUTIONS OF CONDUCTIVITIES kxx

kyy 34

TEMPERATURE FOR THE OPTIMAL SOLUTION

35

DETERMINATION OF THE MICROSTRUCTURE • Knowing the optimal macroscopic 𝑘𝑥𝑥 and 𝑘𝑦𝑦 at a point of the mesh, a topology optimization problem is solved to determine the microstructure we need to achieve such 𝑘𝑥𝑥 and 𝑘𝑦𝑦 • Topology optimization using the topology derivative approach

36

TOPOLOGY OPTIMIZATION AT THE MICROSCALE 3

6

9

2

5

8

1

4

3

6

2

5

1

4

9

8

7

7

37

COMPUTATIONAL METAMATERIAL DESIGN FOR HEAT FLUX MANIPULATION with I. Peralta, A. Ciarbonetti (CIMEC)

38

MANIPULATING THE HEAT FLUX

Prescribed boundary temperature

𝒒

3

𝑿

3

𝒒

2

𝑿

𝒒

1

𝑿

1

2

𝛀

• Given 𝒒 𝑞 as the desired heat flux at 𝑿 you have to find 𝑷 such that −𝒌 𝒑 grad 𝑇 𝑷

Prescribed boundary flux

𝑿𝑞

=𝒒

𝑞

𝑞

, 𝑞 = 1,2, … , 𝑁𝑞 ,

for 𝑖 = 1,2, … , 𝑁𝑞 39

HEAT FLUX MANIPULATION AS AN OPTIMIZATION PROBLEM • In order to perform the given task as well as possible, let us solve the nonlinear optimization problem 1 min feasible 𝑷 𝑁𝑞

𝑁𝑞 𝑞=1

−𝒌 𝒑 grad 𝑇 𝑷

𝑿𝑞

−𝒒

𝑞

2

MSE 𝑷

subject to constraints accounting for, at least, the feasibility of the microstructure.  Maybe, MSE 𝑷 ≠ 0 for feasible 𝑷  We’ll find the “optimal” feasible 𝑷

40

DESIGN OF A HEAT FLUX CONCENTRATION AND CLOAKING DEVICE • To find 𝑷opt = [𝑑 1 , 𝜃 in Ωdevice ) such that

𝑷opt

= arg

1

1 min 𝑷 𝑁𝑞

,…, 𝑑

𝑞

𝑁

,𝜃

𝑁 ]opt

(𝑁 = 1896 is the # elems

−𝒌 𝒑 grad 𝑇 𝑷

𝑿𝑞

−𝒒

𝑞

2

with 𝒒

𝑞

= 5𝒒0 in Ωconc 1

2

𝒒 𝑞 = 𝒒0 in Ωcloak and Ωcloak subject to the box constraints 0 ≤ 𝑃2𝑒−1 ≡ 𝑑

0 ≤ 𝑃2𝑒 ≡ 𝜃

𝑒

𝑒

≤1

≤𝜋 41

HEAT FLUX CONCENTRATION AND CLOAKING: OPTIMAL METAMATERIAL DISTRIBUTION Fraction of copper

Fraction of PDMS

Orientation

42

HEAT FLUX CONCENTRATION AND CLOAKING: OPTIMAL CONDUCTIVITY DISTRIBUTION

43

HEAT FLUX CONCENTRATION AND CLOAKING: OPTIMAL TEMPERATURE DISTRIBUTION

* Peralta, Fachinotti & Ciarbonetti, Scientific Reports, Jan. 2017 (http://www.nature.com/articles/srep40591)

44

EASY-TO-MAKE HEAT FLUX MANIPULATING DEVICES USING DISCRETE MATERIAL OPTIMIZATION (DMO) with I. Peralta (CIMEC)

45

MULTIPHASE TOPOLOGY OPTIMIZATION • The material at the element Ω 𝑒 is either one of 𝑀 predefined, candidate materials with conductivities 𝒌1 , 𝒌2 , … , 𝒌𝑀 • Each material maybe a metamaterial itself 𝑒

• The design variables for Ω 𝑒 are the fractions 𝑓𝑚 of each material 𝑚 = 1,2, … , 𝑀 • The conductivity at Ω 𝑒 is defined by the mixture law 𝑒 𝑒 𝑒 𝒌 𝑒 = 𝑓1 𝒌1 + 𝑓2 𝒌2 + ⋯ + 𝑓𝑀 𝒌𝑀 • We must use an optimization algorithm driving to optimal solutions 𝑒 𝑒 with 𝑓𝑚 = 1 or 𝑓𝑚 = 0 • An integer optimization algorithm (e.g. GA) is unaffordable given the large number of design variables 46

DISCRETE MATERIAL OPTIMIZATION • Using the Discrete Material Optimization (DMO) approach proposed by Stegmann & Lund (IJNME, 2005), we define: 𝑒 𝑓𝑚

=

∗ (𝒑 𝑒 ) 𝑓𝑚 ∗ 𝑒 𝑖 𝑓𝑖 𝒑

with 𝑓𝑚∗ 𝒑

𝑒

=

= 𝑓𝑚 (𝒑 𝑒 𝜌𝑚

𝑝

𝑒

) 𝑀 𝑗=1,𝑗≠𝑚

• The design variables for the finite element Ω

𝑒

𝑒

𝜌𝑖 : artificial density of material 𝑚 at Ω • 𝑝 ≥ 3, like in SIMP for Topology Optimization 𝑒

• This definition forces 𝜌𝑗≠𝑖 → 0 when 𝜌𝑖

𝑒

1−

are 𝑝 𝑒

𝑒

𝑝

𝑒 𝜌𝑚 𝑒

𝑒

𝑒

= [𝜌1 , 𝜌2 , … , 𝜌𝑀 ]

, like in Topology Optimization

→1

• It doesn’t need a constraint (one per finite element!) to make

𝑖 𝑓𝑖

𝑒

=1 47

DESIGN OF A HEAT FLUX CONCENTRATION AND CLOAKING DEVICE USING DMO 1

1

1

1896

• To find 𝑷opt = [𝜌1 , 𝜌2 , 𝜌3 ,…, 𝜌1 𝑷opt

= arg

1 min 𝑷 𝑁𝑞

𝑞

1896

, 𝜌2

−𝒌 𝒑 grad 𝑇 𝑷

1896 opt ]

, 𝜌3

𝑿𝑞

−𝒒

𝑞

such that

2

subject to the box constraints 0 ≤ 𝑃𝑖 ≤ 1 with 𝒒

𝑞

= 5𝒒0 in Ωconc

𝒒

𝑞

= 𝒒0 in Ωcloak and Ωcloak

1

2

48

HEAT FLUX CONCENTRATION AND CLOAKING USING DMO: OPTIMAL METAMATERIAL DISTRIBUTION

49

HEAT FLUX CONCENTRATION AND CLOAKING USING DMO: FULLY DISCRETE METAMATERIAL DISTRIBUTION

50

HEAT FLUX CONCENTRATION AND CLOAKING USING DMO: OPTIMAL TEMPERATURE DISTRIBUTION

* Peralta & Fachinotti, Scientific Reports, July 2017 (http://www.nature.com/articles/s41598-017-06565-6)

51

EVEN EASIER-TO-MAKE HEAT FLUX MANIPULATING DEVICES USING TOPOLOGY OPTIMIZATION with A. Ciarbonetti, I. Peralta (CIMEC), I. Rintoul (INTEC) 52

TOPOLOGY OPTIMIZATION • The material at the element Ω 𝑒 is either one of two predefined candidate materials with conductivities 𝒌1 , 𝒌𝟐 • Each material is isotropic • There is only one design variable for Ω of material 1

𝑒

: the artificial density 𝜌

𝑒

• The conductivity at Ω 𝑒 is defined using SIMP (Solid Isotropic Material with Penalization) 𝒌

𝑒

= 𝜌

• A priori, using 𝑝 ≥ 3, 𝜌

𝑒

𝑒

𝑝

𝒌1 + 1 − 𝜌

→ 0 or 𝜌

𝑒

𝑒

𝑝

𝒌2

→1 for the optimal solution

53

DESIGN OF HEAT FLUX INVERTER USING TOPOLOGICAL OPTIMIZATION • To find 𝑷opt = [𝜌 𝑷𝑜𝑝𝑡

= arg

1

,…, 𝜌

1 min 𝑷 𝑁𝑞

𝑞

4000 opt ]

such that

−𝒌 𝒑 grad 𝑇 𝑷

𝑿𝑞

−𝒒

𝑞

2

subject to the box constraints 0 ≤ 𝜌(𝑒) ≤ 1 with

𝒒

𝑞

= −𝒒0 in Ωinvert

54

HEAT FLUX INVERTER: TOPOLOGY OPTIMIZATION SOLUTION

55

HEAT FLUX INVERTER: BLACK AND WHITE FILTERING • For manufacturability, regions with intermediate material fractions (“grey zones”) must be avoided • Black and white filters (Sigmund 2007) serve to this end • Here, a simple a posteriori b&w filter is preferred: material fraction greater than 𝑤 ∗ is taken to 1; otherwise, it is taken to 0

56

HEAT FLUX INVERTER: TOPOLOGY OPTIMIZATION SOLUTION + BLACK AND WHITE FILTERING

57

HEAT FLUX INVERTER: TOPOLOGY OPTIMIZATION WITH AND WITHOUT BLACK AND WHITE FILTERING WITHOUT B&W FILTER

WITH B&W FILTER

58

HEAT FLUX INVERTER: TOPOLOGY OPTIMIZATION WITH AND WITHOUT BLACK AND WHITE FILTERING WITHOUT B&W FILTER

WITH B&W FILTER

59

HEAT FLUX INVERTER: EXPERIMENTAL VALIDATION Computationally designed device

60

HEAT FLUX INVERTER: EXPERIMENTAL VALIDATION

61

HEAT FLUX INVERTER: EXPERIMENTAL VALIDATION

V. Fachinotti et al., “Optimization-based design of easy-to-make devices for heat flux manipulation”, Int. J. Ther. Sci., 2018

62

HEAT FLUX INVERTER: COMPARISON WITH NARAYANA AND SATO´S INVERTER • Accomplishment of the inversion task 𝑞invert = −𝑘agar

𝑇𝐷 −𝑇𝐶 𝐶𝐷

= −𝛼𝑞0

𝐶

𝛼 = 0.997

𝛼 = 0.778

𝛼 = 0.774

S. Narayana & Y. Sato, “Heat Flux Manipulation with Engineered Thermal Materials”, Physical Review Letters 2012

𝐷

𝛼 = 0.395 63

MECHANICAL CLOAKING with I. Peralta (CIMEC)

64

MECHANICAL CLOAKING Displacement 𝒖

Displacement 𝒖0

Ωdev Ωincl

Ωcloak

• To cloak the inclusion Ωincl , let us design the material in Ωdev such that the displacement in Ωcloak ressembles 𝒖0 • The cloaking task consists of finding the material distribution 𝑷 in Ωdev such that 𝒖 𝒙𝑖 , 𝑷 = 𝒖0 𝒙𝑖 ∀𝒙𝑖 ∈ Ωcloak

65

MECHANICAL CLOAKING AS AN OPTIMIZATION PROBLEM Let us accomplish the cloaking task as well as possible by solving the nonlinear constrained optimization problem 1 min 𝒖 𝒙𝑖 , 𝑷 − 𝒖0 𝒙𝑖 2 𝑷 𝑁cloak 𝒙𝑖 ∈Ωcloak

subject to bound, equality and/or inequality constraints

66

CLOAKING OF A HOLE IN A PLATE • The displacement in a nylon plate is originally 𝒖0 • Once the plate is holed, let us design a device Ωdev to cloak the hole Ωincl • Using DMO, 1

1

15084

𝑷 = 𝜌Al 𝜌PTFE … 𝜌Al

15084

𝜌PTFE

• The optimal 𝑷 is the solution of 1 min 𝑷 19972

𝒖 𝒙𝑖 , 𝑷 − 𝒖0 𝒙𝑖 𝒙𝑖 ∈Ωcloak

subject to 0 ≤ 𝑃𝑖 ≤ 1

2

PTFE or aluminum

Nylon

𝑖 = 1,2, … , 30168 67

CLOAKING OF A HOLE IN A PLATE

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PERSPECTIVES • Robustness • instabilities (checkerboard) • grey zones • convergence • 3D • Multiobjective optimization • Applications • Isolation: to deviate the heat flux from the zones where it is undesired, to drive it to somewhere where it maybe useful • Optimization of Austempered Ductile Iron (with B. Tourn) • Mechanical properties depend on the thermal history • Topology and heat treatment optimization to make a macroscopic piece have a given mechanical response • Metamaterials for wind turbine blades (with A. Albanesi) • Fabrication, patents 69

ACKNOWLEDGEMENTS • European Research Council, Grant Agreement n. 320815 “COMPDES-MAT: Advanced Tools for Computational Design of Engineering Materials”, leaded by X. Oliver (CIMNE-UPC), 2013-2017. • National Littoral University (UNL), Santa Fe, Argentina, research project CAI+D 2016 50420150100087LI, “Metamaterials: Computational Design, Thermal, Mechanical and Acoustic Applications, and Prototyping”, 2017-2019. • National Agency for the Promotion of Science and Technology of Argentina (ANPCyT), research project PICT 2016-2673 “Computational Design of Metamaterials”, 2017-2020.

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