Optimization of bodies with locally periodic microstructure - PowerPoint PPT Presentation

IRF’2009 3 rd International Conference on Integrity, Reliability & Failure Porto, 20-24 July 2009 Optimization of bodies with locally periodic microstructure Cristian Barbarosie CMAF, Universidade de Lisboa joint work with Anca-Maria Toader CMAF, Universidade de Lisboa

Goal : Optimize the macroscopic properties (e.g. compliance) of a two-dimensional body made of a linearly elastic material and presenting locally periodic (quasi-periodic) microscopic perforations. IRF’2009 Cristian Barbarosie 1

Main ingredient : a code for optimizing the properties of a periodic microstruc- ture, by varying the shape and topology of the hole in the periodicity cell C. Barbarosie, Structural Optimization 1997, Computational Mechanics 2003 – FORTRAN code, triangular finite element mesh on the two-dimensional torus, shape optimization through mesh variation C. Barbarosie, A.-M. Toader, Struct Multidisc Optim 2009, Shape and topology optimization for periodic problems, Part I and Part II – topology optimization added (hole nucleation), improved remeshing Finite element code for the macroscopic analysis, collaboration of Paulo Vieira (C++, libMesh) and of S´ ergio Lopes (FreeFem++) Interface (written in Python) between macroscopic and microscopic codes A similar approach, with a different treatment of the cellular problem : P. Coelho, P. Fernandes, J. Guedes, H. Rodrigues, Struct Multidisc Optim 2008 IRF’2009 Cristian Barbarosie 2

Linear elasticity   − div ( C ∇ u ) = f in Ω − ( C ijkl u k,l ) ,j = f i in Ω   u = 0 on Γ D u i = 0 on Γ D ( C ∇ u ) n = g on Γ N ( C ijkl u k,l ) n j = g i on Γ N   Non-homogeneous material: C depends on x ∈ Ω. Small scale ε . Sequence C ε of tensor fields.  − div ( C ε ∇ u ε ) = f in Ω  u ε = 0 on Γ D ( C ε ∇ u ε ) n = g on Γ N  C ε ijkl ∈ L ∞ (Ω) Which is the behaviour of the body (macroscopic behaviour) ? IRF’2009 Cristian Barbarosie 3

Definition We say that a sequence C ε of elastic tensors H -converges to C H if, for any f ∈ L 2 (Ω) and for any g ∈ H 1 / 2 (Γ N ), the solution u ε ∈ H 1 (Ω) of problem  − div ( C ε ∇ u ε ) = f in Ω  u ε = 0 on Γ D ( C ε ∇ u ε ) n = g on Γ N  converges, weakly in H 1 (Ω), to the solution u of problem  − div ( C H ∇ u ) = f in Ω  u = 0 on Γ D ( C H ∇ u ) n = g on Γ N  IRF’2009 Cristian Barbarosie 4

For given α and β , 0 < α < β , we define T α,β (Ω) as the space of all tensor functions C whose components belong to L ∞ (Ω) and such that � Cξ � ≤ β � ξ � , ∀ ξ matrix � Cξ, ξ � ≥ α � ξ � 2 , ∀ ξ symmetric matrix Theorem The H -convergence defines a metrizable topology on T α,β (Ω). The H -convergence has local character. Moreover, T α,β (Ω) is a compact space when endowed with the H -topology. IRF’2009 Cristian Barbarosie 5

Periodic microstructure: C ε ( x ) = C ∗ ( x ε ), x ∈ Ω. C ∗ : R n → R n × n × n × n periodic tensor field G periodicity group: C ∗ ( x + � v ) = C ∗ ( x ) , ∀ x ∈ Ω , ∀ � v ∈ G IRF’2009 Cristian Barbarosie 6

For a periodic microstructure, the homogenized tensor C H is constant in Ω and can be defined in terms of cellular problems. For an arbitrary matrix A ∈ R n × n � in R n − div ( C ∗ ∇ w ) = 0 w ( x ) = Ax + ϕ ( x ) , ϕ G− periodic v , ∀ y ∈ R n , ∀ � w ( y + � v ) = w ( y ) + A� v ∈ G 1 1 � � C H A = A = ∇ w A C ∗ ∇ w A | Y | | Y | Y Y 1 � � C H A, B � = � C ∗ ∇ w A , ∇ w B � | Y | Y IRF’2009 Cristian Barbarosie 7

in R n � − div ( C ∗ ( A + ∇ ϕ A )) = 0 ϕ A G− periodic in R n � − div ( C ∗ ∇ ϕ A ) = div ( C ∗ A ) ϕ A G− periodic 1 � C H A = C ∗ ( A + ∇ ϕ A ) | Y | Y 1 � � C H A, B � = � C ∗ ( A + ∇ ϕ A ) , ( B + ∇ ϕ B ) � | Y | Y � − div ( C ∗ ∇ ϕ A ) = div ( C ∗ A ) in Y ϕ A G− periodic IRF’2009 Cristian Barbarosie 8

Quasi-periodic microstructure: C ε ( x ) = C ∗ ( x, x ε ), x ∈ Ω. C ∗ : R n × R n → R n × n × n × n pattern tensor field, periodic in the second argument v ) = C ∗ ( x, y ) , ∀ x ∈ Ω , ∀ y ∈ R n , ∀ � G periodicity group: C ∗ ( x, y + � v ∈ G IRF’2009 Cristian Barbarosie 9

For a quasi-periodic microstructure, the homogenized tensor C H is no longer constant. It each point x ∈ Ω, the tensor C H ( x ) is defined in terms of a different set of cellular problems, given by the pattern tensor field C ∗ ( x, · ) : in R n � − div y ( C ∗ ( x, y ) ∇ y w ) = 0 w ( y ) = Ay + ϕ ( y ) , ϕ G− periodic 1 1 � � C H ( x ) A = A = ∇ w A C ∗ ( x, · ) ∇ w A | Y | | Y | Y Y 1 � � C H ( x ) A, B � = � C ∗ ( x, · ) ∇ w A , ∇ w B � | Y | Y IRF’2009 Cristian Barbarosie 10

Macroscopic problem  − div ( C H ∇ u ) = f in Ω  u = 0 on Γ D ( C H ∇ u ) n = g on Γ N  Objective functional (e.g.) � � � � C H ∇ u, ∇ u � Φ = gu = 2 gu − Γ N Γ N Ω Constraint on the volume of material : � V = θ Ω IRF’2009 Cristian Barbarosie 11

Chain of dependencies : C H ( x ) C ∗ ( x, y ) → → Φ δC H δC ∗ → → δ Φ Macroscopic level : � � � C H ∇ u, ∇ u � Φ = 2 gu − Γ N Ω � � � � δC H ∇ u, ∇ u � − 2 � C H ∇ u, ∇ δu � δ Φ = 2 gδu − Γ N Ω Ω � � δC H ∇ u, ∇ u � δ Φ = − Ω � δV = δθ Ω IRF’2009 Cristian Barbarosie 12

Microscopic level – fixed x ∈ Ω : Isotropic elastic tensor C ∗ ( x, y ), of Lam´ e coefficients λ and µ . Shape variations : Y T 1 � � D S C H A, B � = � � 2 µ � e ( w A ) , e ( w B ) � + λ tr( e ( w a ))tr( e ( w B )) � � τ,� n � | Y | ∂T � D S C H = S � � n � τ,� ∂T IRF’2009 Cristian Barbarosie 13

Topology variations : λ + 2 µ � D T C H A, B � ( y ) = − π � 4 µ � e ( w A ) , e ( w B ) � + | Y | λ + µ λ 2 + 2 λµ − µ 2 � tr( e ( w A )) tr( e ( w B )) ( y ) µ D T C H ( y ) = T ( y ) IRF’2009 Cristian Barbarosie 14

The macroscopic domain Ω is divided in finite elements K 1 , K 2 , . . . , K n . In each K e , a periodic pattern tensor field C ∗ e ( y ) is considered, which gives rise to the homogenized elastic tensor C H e , with local material density θ e . Each periodic microstructure C ∗ e is optimized, by using the chain of derivatives presented above. Shape optimization and topology optimization steps (at the cellular level) are alternated. IRF’2009 Cristian Barbarosie 15

For the case of (microscopic) shape variations, in each macroscopic finite element K e consider a deformation field � τ e . The corresponding changes in the homogenized tensor C H e and in the local density θ e are: 1 1 � � δC H e = S e � � n � δθ e = � � n � τ e ,� τ e ,� | Y | | Y | ∂T e ∂T e This gives, for the macroscopic functional : � � � δC H ∇ u, ∇ u � + Λ δ (Φ + Λ V ) = − δθ Ω Ω � � � � � = − � S e , ∇ u ⊗ ∇ u �� � n � + Λ | K e | � � n � τ e ,� τ e ,� ∂T e K e ∂T e e e � � = | K e | γ e � � τ e ,� n � ∂T e e γ e are computable functions, depending on the macroscopic strain ∇ u and on the microscopic solutions w A in the periodicity cell in K e . For a steepest descent method, simply choose deformation fields � τ e = − γ e � n . IRF’2009 Cristian Barbarosie 16

For (microscopic) topology variations, pick a macroscopic finite element K e ; consider a virtual nucleation of a small hole at location y in the periodicity cell Y . Then the corresponding variation in the lagrangean is � � � δC H ∇ u, ∇ u � + Λ δ (Φ + Λ V ) = − δθ Ω Ω � ∇ u ⊗ ∇ u � ρ 2 + Λ | K e | ρ 2 = | K e | η e ( y ) ρ 2 = −� T e ( y ) , K e Find the minimum point of the scalar function η e defined in the periodicity cell Y . If the minimum value is negative, nucleate a small hole at the minimum point. IRF’2009 Cristian Barbarosie 17

Control flow diagram Initiate process with (globally) periodic microstructure compute homogenized perform a shape optimization elastic coefficients or topology optimization step and material density in each periodicity cell no solve macroscopic elastic problems(s) yes stop compute sensitivities convergence ? � ∇ u ⊗ ∇ u K e IRF’2009 Cristian Barbarosie 18

Data flow diagram CONTROL SCRIPT Python homogenized elastic sensitivities coefficients homogenized elastic sensitivities optmization parameters coefficients � ∇ u ⊗ ∇ u densities K e MACROSCOPIC ANALYSIS MICROSCOPIC OPTIMIZATION (shape, topology) C++ libmesh FreeFem++ FORTRAN IRF’2009 Cristian Barbarosie 19

Lagrangean 200 ��� ��� ��� ��� ��� ��� ��� ��� 180 ��� ��� ��� ��� ��� ��� ��� ��� 160 ��� ��� ��� ��� ��� ��� ��� ��� 140 ��� ��� ��� ��� ��� ��� ��� ��� 0 50 100 150 200 ��� ��� ��� ��� ��� ��� 1.4 compliance volume 1.2 40 1 30 0.8 0.6 0 50 100 150 200 0 50 100 150 200 IRF’2009 Cristian Barbarosie 20