Shape Op(miza(on for Interface Design: Using the level set method for the design of mul(- phase elas(c and thermoelas(c materials and interfaces

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1 Shape Op(miza(on for Interface Design: Using the level set method for the design of mul(- phase elas(c and thermoelas(c materials and interfaces N. Vermaak 1, G. Michailidis 2, G. Parry 3, R. Estevez 3, F. Jouve 4, G. Allaire 2, Y. Bréchet 3 1 Lehigh University USA; 2 Ecole Polytechnique CMAP; 3 Univ. Grenoble SIMAP; 4 Univ. Paris Diderot (Paris 7) LJLL; Colloque Filling Gaps in Materials Space : Methods and Applica(ons

2 Outline Background and mo(va(on for level set based op(miza(on and interface design Use of the level set based op(miza(on for benchmark problems of material design Beyond benchmark problems: use of the level set based op(miza(on for design of materials including interface effects Natasha Vermaak & Georgios Michailidis 2/26

3 Filling Gaps in Materials Space: Methods and Applica(ons Mike Ashby, Designing architectured materials Scripta Materialia 68 (2013) 4 7 Combina(ons of two or more materials or of materials and space, configured in such a way as to have acributes not offered by any one material alone Natasha Vermaak & Georgios Michailidis 3/26

4 Filling Gaps in Materials Space: Methods and Applica(ons Mike Ashby, Designing architectured materials Scripta Materialia 68 (2013) 4 7 Combina(ons of two or more materials or of materials and space, configured in such a way as to have acributes not offered by any one material alone hcp://en.wikipedia.org/wiki/topology Natasha Vermaak & Georgios Michailidis 4/26

Filling Gaps in Materials Space: Methods and Applica(ons Mike Ashby, Designing architectured materials Scripta Materialia 68 (2013) 4 7 Combina(ons of two or more materials or of

5 Filling Gaps in Materials Space: Methods and Applica(ons Mike Ashby, Designing architectured materials Scripta Materialia 68 (2013) 4 7 Combina(ons of two or more materials or of materials and space, configured in such a way as to have acributes not offered by any one material alone hcp://en.wikipedia.org/wiki/topology Natasha Vermaak & Georgios Michailidis 5/26

6 Interface Modeling Example for Metals Gibbs Ideal Interface A B Guggenheim Interface Zone A B atom species 1 species 2 Sharp interface boundary on atomic scale (semiconductors by MBE) Smooth or graded broad transi(ons (or thin layers of new compounds) created by interdiffusion or surface reac(ons Physics and Chemistry of Interfaces, Hans- Jürgen Buc, Karlheinz Graf, Michael Kappl, Wiley, 2003 Understanding Solids: The Science of Materials, R. J. D. Tilley, Wiley, 2004 Natasha Vermaak & Georgios Michailidis 6/26

7 Interface Modeling!"#$ d % Young s &'()*)+,"-$!"x$,.,0!"x$,.,/,0 Material A Material B x Interpolation 12(34,"50$ #,.,/,0 #,.,0 N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Brechet, Material Interface Effects on the Topology Op(miza(on of Mul(- Phase Structures Using A Level Set Method, Structural and Mul(disciplinary Op(miza(on, (In review: Struct. Mul(dis. Opt.). G. Allaire, C. Dapogny, G. Delgado, G. Michailidis, Mul(- phase structural op(miza(on via a level- set method, (In review: ESAIM: Cont., Opt. Calc. Var.) Natasha Vermaak & Georgios Michailidis 7/26

8 Op(mizing structures: standard interface modeling h MINIMIZE COMPLIANCE g w 2 1 Property value, P 1.0 Monotonic!" 0 " Interface Zone, d # (a) Young s Modulus. P A = 1 P B = COMPLIANCE 3.25 (c) Iterations (b) Final design on the full-domain. 1, 10, 25, 40. N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Brechet, Material Interface Effects on the Topology Op(miza(on of Mul(- Phase Structures Using A Level Set Method, Structural and Mul(disciplinary Op(miza(on, (In review: Struct. Mul(dis. Opt.). G. Allaire, C. Dapogny, G. Delgado, G. Michailidis, Mul(- phase structural op(miza(on via a level- set method, (In review: ESAIM: Cont., Opt. Calc. Var.) Natasha Vermaak & Georgios Michailidis 8/26

9 Op(mizing structures: including interface eﬀects 20 terface Effects Non monotonic PA= 1 PB= 0.1 mval = 2PA mval = 5PA 5.0 PA= 1 PB= 0.1 Property value, P Property value, P !" 0 Interface Zone, d# 1 B= 0.1 mval = 5PA PA= 1 PB= Interface Zone, d# " 0.1 N. Ve (b) Final design the full-domain. (a)onyoung s Modulus. (b) COMPLIANCE: 1.0 Property value, P 5.0!" 19 " (a) Young s Modulus. Non monotonic COMPLIANCE: 2.1 A= s. Non monotonic Young s Modulus "!" 0 Interface Zone, d# (a) Young s (b) Final design on the full-domain. (c)modulus. Iterations " 0.1 1, 50, 75, 90. (b) Final design on the full-domain. (c) Iterations 1, 5 N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire, Y. Brechet, Material Interface Eﬀects on the Topology Op(miza(on of Mul(- Phase Structures Using A Level Set Method, Structural and Mul(disciplinary Op(miza(on, (In review: Struct. Mul(dis. Opt.). G. Allaire, C. Dapogny, G. Delgado, G. Michailidis, Mul(- phase structural op(miza(on via a level- set method, (In review: ESAIM: Cont., Opt. Calc. Var.) October 15, 2013 Natasha Vermaak & Georgios Michailidis Fig. 8:applied Case for another non-monotonic interface sc ase for non-monotonic interface interpolation in the built-in-beam problem (Figureinterpolation 4). Results we 9/26

10 Outline Background and mo(va(on for level set based op(miza(on and interface design Use of the level set based op(miza(on for benchmark problems of material design Beyond benchmark problems: use of the level set based op(miza(on for design of materials including interface effects Natasha Vermaak & Georgios Michailidis 10/26

11 Benchmark Materials Design Problems * Ceramics not included C A Steeves, S L dos Santos e Lucato, M Y He, E An(nucci, J W Hutchinson and A G Evans. Concepts for structurally robust materials that combine low thermal expansion with high s(ffness. Journal of the Mechanics and Physics of Solids, 55(9), pp , Natasha Vermaak & Georgios Michailidis 11/26

Some benchmark problems Designing with 3 phases for target CTE using a unit cell approach Lakes 1996 high CTE low CTE Ex 1: Target CTE proper(es ( α 11 * = - α 22 * ) and shear proper(es, with given

constraints on the cons(tuent materials. AFRL 2006 UCSB 2007 Smooth interpolaeon of material properees across material interfaces Lakes, R.S., 1996.

12 Some benchmark problems Designing with 3 phases for target CTE using a unit cell approach Lakes 1996 high CTE low CTE Ex 1: Target CTE proper(es ( α 11 * = - α 22 * ) and shear proper(es, with given volume inequality constraints on the cons(tuent materials Sigmund & Torquato 1996 Ex 2: Target elas(c and CTE proper(es, while constraining for minimum shear s(ffness and given volume inequality constraints on the cons(tuent materials. AFRL 2006 UCSB 2007 Smooth interpolaeon of material properees across material interfaces Lakes, R.S., Cellular solid structures with unbounded thermal expansion. J. Mater. Sci. Lec. 15 (6), Sigmund, O., Torquato, S., Composites with extremal thermal expansion coefficients. Appl. Phys. Lec. 69 (21), Sigmund, O., Torquato, S., Design of materials with extreme thermal expansion using a three- phase topology op(miza(on method. J. Mech. Phys. Sol. 45 (6), M.P. Bendsoe, O. Sigmund, (2003). Topology Op(miza(on: Theory, Methods and Applica(ons, Springer, Berlin. Fujii, D., Chen, B.C. Kicuchi, N. (2001). Composite material design of two- dimensional structures using the homogeniza(on design method, Int J. Num. Meth. Eng 50, C A Steeves, S L dos Santos e Lucato, M Y He, E An(nucci, J W Hutchinson and A G Evans. Concepts for structurally robust materials that combine low thermal expansion with high s(ffness. Journal of the Mechanics and Physics of Solids, 55(9), pp , X. Wang, Y. Mei, M. Wang, Level- set method for design of mul(- phase elas(c and thermoelas(c materials, Int J of Mechanics and Materials in Design (2004) 1: G. Allaire, C. Dapogny, G. Delgado, G. Michailidis, Mul(- phase structural op(miza(on via a level- set method, (In review: ESAIM: Cont., Opt. Calc. Var.) low CTE high CTE Natasha Vermaak & Georgios Michailidis 12/26

13 Outline Background and mo(va(on for level set based op(miza(on and interface design Use of the level set based op(miza(on for benchmark problems of material design Beyond benchmark problems: use of the level set based op(miza(on for design of materials including interface effects Natasha Vermaak & Georgios Michailidis 13/26

14 Shape/topology op(miza(on The art of structure is where to put the holes. ~Robert Le Ricolais ( ) Natasha Vermaak & Georgios Michailidis 14/26

15 Shape/topology op(miza(on Numerical Algorithm 1. Ini(alize the shape, Ω 0 2. Iterate un(l convergence for Evaluate the advec(on velocity (via shape gradient) Transport the shape by to obtain a new shape Ω k+1 Gregoire Alliaire, Shape and Topology Op(miza(on, Ecole Polytechnique, hcp:// Natasha Vermaak & Georgios Michailidis 15/26

16 The level set method Method for tracking evolving interfaces S. Osher, UCLA, hcp:// J.A. Sethian, Berkeley,hcp://math.berkeley.edu/~sethian/level_set.html Natasha Vermaak & Georgios Michailidis 16/26

Mech. Engrg. 193 (2004). G. Allaire, C. Dapogny, G. Delgado, G.

17 The level set method Mul(- phase descrip(on Using m level set func(ons, up to n = 2 m different phases can be described. M. Wang and X. Wang, Color level sets: a mul(- phase method for structural topology op(miza(on with mul(ple materials, Comput. Methods Appl. Mech. Engrg. 193 (2004). G. Allaire, C. Dapogny, G. Delgado, G. Michailidis, Mul(- phase structural op(miza(on via a level- set method, (In review: ESAIM: Control, Op(misa(on and Calculus of Varia(ons) Natasha Vermaak & Georgios Michailidis 17/26

Unit Cell & Periodic Homogeniza(on Periodic domain Periodic unit cell The unit cell problem Homogenized Elas(city tensor hcp://www.cmap.polytechnique.fr/~allaire/homog/lect1.pdf ALLAIRE G.

18 Unit Cell & Periodic Homogeniza(on Periodic domain Periodic unit cell The unit cell problem Homogenized Elas(city tensor hcp:// ALLAIRE G., Shape op(miza(on by the homogeniza(on method, Springer Verlag, New York (2002). TARTAR, L., The general theory of homogeniza(on. A personalized introduc(on. Lecture Notes of the Unione Matema(ca Italiana, 7. Springer- Verlag, Bologna (2009). Natasha Vermaak & Georgios Michailidis 18/26

19 1) Target CTE and shear s(ffness proper(es With volume inequality constraints Isotropic Cons(tuents: E 1 = E 0 /2 = v = 0.3 E 0 = 0.91 Gpa v = 0.3 α 1 th11 = α1 th22 = 10.0 µm/ C α 0 th11 = α0 th22 = 1.0 µm/ C Objec(ve homogenized proper(es: A T 1212 = 0.04 GPa A tht 11 = 0.4 KPa/ C A tht 22 = KPa/ C Constraints: phase 0 volume ra(o 0.25 phase 1 volume ra(o 0.34 Addi(onal Parameters: Mesh of 100 x 100 Q1 elements. Total iters 200. Natasha Vermaak & Georgios Michailidis 19/26

1) Results E 1 = E 0 /2 = 0.455 v = 0.3 E 0 = 0.91 Gpa v = 0.3 α 1 th11 = α1 th22 = 10.

0 µm/ C Op(mized unit- cell has homogenized proper(es that converge near targets Ini(aliza(on

20 1) Results E 1 = E 0 /2 = v = 0.3 E 0 = 0.91 Gpa v = 0.3 α 1 th11 = α1 th22 = 10.0 µm/ C α 0 th11 = α0 th22 = 1.0 µm/ C Op(mized unit- cell has homogenized proper(es that converge near targets Ini(aliza(on Unit cell a{er convergence 16 unit cells Thermal Expansion: Homogenized(*) and Target(T) Elas(c tensor Homogenized(*) and Target(T) Natasha Vermaak & Georgios Michailidis 20/26

21 2) Target elas(c, CTE proper(es With minimum shear s(ffness and volume inequality constraints Isotropic Cons(tuents: E 1 = E 0 /2 = v = 0.3 E 0 = 0.91 Gpa v = 0.3 α 1 th11 = α1 th22 = 5.0 µm/ C α 0 th11 = α0 th22 = 1.0 µm/ C Objec(ve homogenized proper(es: A T 1111 = AT 2222 = 0.05 GPa A tht 11 = AthT 22 = 0.1 KPa/ C Constraints: phase 0 volume ra(o 0.2 phase 1 volume ra(o 0.24 minimum homogenized shear s(ffness A* Gpa Addi(onal Parameters: Mesh of 100 x 100 Q1 elements. Total iters 200. Natasha Vermaak & Georgios Michailidis 21/26

2) Results E 1 = E 0 /2 = 0.455 v = 0.3 E 0 = 0.91 Gpa v = 0.3 α 1 th11 = α1 th22 = 5.

22 2) Results E 1 = E 0 /2 = v = 0.3 E 0 = 0.91 Gpa v = 0.3 α 1 th11 = α1 th22 = 5.0 µm/ C α 0 th11 = α0 th22 = 1.0 µm/ C Op(mized unit- cell has homogenized proper(es that converge near targets Ini(aliza(on Unit cell a{er convergence 16 unit cells Thermal Expansion: Homogenized(*) and Target(T) Elas(c tensor Homogenized(*) and Target(T) Natasha Vermaak & Georgios Michailidis 22/26

23 Beyond benchmark problems Designing with 3 phases for target CTE using a unit cell approach Including material interface effects Ex 1:!"#$ Target elas(c and CTE proper(es, while constraining for minimum shear s(ffness and given volume inequality constraints on the cons(tuent materials. d %!"x$,.,0 Young s &'()*)+,"-$ Isotropic Cons(tuents:!"x$,.,/,0 Material A Material B x E 1 = E 0 /2 = v = 0.3 α 1 th11 = α1 th22 = 5.0 µm/ C Interpolation 12(34,"50$ E 0 = 0.91 GPa v = 0.3 α 0 th11 = α0 th22 = 1.0 µm/ C #,.,/,0 #,.,0 Natasha Vermaak & Georgios Michailidis 23/26

24 1) Target elas(c, CTE proper(es With minimum shear s(ffness and volume inequality constraints Interface property transi(on: E 0 = 0.91 E 1 = α int = 10 α 0 = 1 α 1 = 5 Objec(ve homogenized proper(es: A T 1111 = AT 2222 = 0.05 GPa A tht 11 = AthT 22 = 0.1 KPa/ C Constraints: phase 0 volume ra(o 0.2 phase 1 volume ra(o 0.24 minimum homogenized shear s(ffness A* Gpa Addi(onal Parameters: Mesh of 100 x 100 Q1 elements. Total iters 200. Natasha Vermaak & Georgios Michailidis 24/26

455 16 unit cells α int = 10 α 0 = 1 α 1 = 5 Thermal Expansion:

25 1.1) Results Op(mized proper(es converge near targets Ini(aliza(on E 0 = 0.91 Unit cell a{er convergence E 1 = unit cells α int = 10 α 0 = 1 α 1 = 5 Thermal Expansion: Homogenized(*) and Target(T) Elas(c tensor Homogenized(*) and Target(T) Natasha Vermaak & Georgios Michailidis 25/26

1.2) Results Opt. prop. converge near targets E 0 = 0.91 E int = 0.

26 1.2) Results Opt. prop. converge near targets E 0 = 0.91 E int = E 1 = α 0 = 1 α 1 = 5 α int = 0.5 Ini(aliza(on Unit cell a{er convergence 16 unit cells Thermal Expansion: Homogenized(*) and Target(T) Elas(c tensor Homogenized(*) and Target(T) Natasha Vermaak & Georgios Michailidis 26/26

27 Shape Op(miza(on for Interface Design: Using the level set method for the design of mul(- phase elas(c and thermoelas(c materials and interfaces N. Vermaak 1, G. Michailidis 2, G. Parry 3, R. Estevez 3, F. Jouve 4, G. Allaire 2, Y. Bréchet 3 1 Lehigh University USA; 2 Ecole Polytechnique CMAP; 3 Univ. Grenoble SIMAP; 4 Univ. Paris Diderot (Paris 7) LJLL; Colloque Filling Gaps in Materials Space : Methods and Applica(ons

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29 Mul(func(onal Structural Design Natasha Vermaak & Georgios Michailidis 29/30

30 Mul(func(onal Structural Design Natasha Vermaak & Georgios Michailidis 30/30

CSMA ème Colloque National en Calcul des Structures mai 2017, Presqu île de Giens (Var)

CSMA ème Colloque National en Calcul des Structures mai 2017, Presqu île de Giens (Var) CSMA 207 3ème Colloque National en Calcul des Structures 5-9 mai 207, Presqu île de Giens (Var) TOPOLEV: Topological optimization using level-set method D. Lachouette, Ph. Conraux 2, G. Allaire 3, F. Jouve