Inverse Problems, Control and Shape Optimization PICOF ’12 Shape and Topology Optimization of Composite Materials with the level-set method Gabriel Delgado Gr´ egoire Allaire EADS-Innovation Works Center of Applied Mathematics - ´ Ecole Polytechnique Palaiseau, 3 April 2012 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 1 / 33
Outline Motivation 1 Problem description 2 Main methods in Topology Optimization 3 The algorithm and its implementation 4 Results 5 Conclusion 6 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 2 / 33
Outline Motivation 1 Problem description 2 Main methods in Topology Optimization 3 The algorithm and its implementation 4 Results 5 Conclusion 6 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 3 / 33
Motivation: Composite Materials in Aeronautics Figure: Composite structures of an A380 and evolution of the use of composites in Airbus. Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 4 / 33
Motivation: Laminated Composite Materials Characteristics Lamination of a sequence of unidirectionally reinforced plies. Each ply is typically a thin sheet of collimated fibers impregnated with a polymer matrix material. Strong resistance against severe environmental conditions. Less expensive and lighter than metallic alloys. Greatest benefit and drawback: Espace of design possibilities. Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 5 / 33
Motivation: Shape and Topology Optimization examples Figure: Topology optimized wing structure of the A380 and a shape optimized airfoil. Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 6 / 33
Outline Motivation 1 Problem description 2 Main methods in Topology Optimization 3 The algorithm and its implementation 4 Results 5 Conclusion 6 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 7 / 33
2D Multi-layered composite description Let be D ⊂ R 2 fixed and Ω = Ω i � � a laminated composite structure made from the superposition of N plies under plane stress state, of geometry Ω i ⊂ D , each one composed by two anisotropic elastic phases A i and A 0 � << 1 ). We consider the boundary of D made of two disjoint parts, � � A 0 � ( ∂ D = Γ D ∪ Γ N , | Γ D | � = 0 . Ω 1 A 0 Ω 2 A 90 Ω 3 A − 45 Ω 4 A 45 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 8 / 33
Physical model description On Ω i we denote by Σ i the interface between both phases. The characteristic function of Ω i is denoted by χ i ( x ) , so the composite elastic tensor of Ω at x is described by N N A i χ i + � � A 0 (1 − χ i ) . A χ = i =1 i =1 The strain and stress tensors are related to the displacement field u as 2 ( ∇ u + ∇ T u ) and σ ( u ) = A χ e ( u ) . The external charges are e ( u ) = 1 denoted by f (volume) and g (surface), and the displacement field u χ is the solution of the linearized elasticity system in D ⎧ − div ( A χ e ( u )) = f in D ⎨ u = 0 on Γ D Ae ( u ) n = g on Γ N ⎩ Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 9 / 33
Optimization problem Find the best composite structure Ω according to a criteria J (Ω , u χ ) , where u χ is the state variable (displacement) and Ω is the control variable, among a set of admissible shapes of layers U ad (manufacture and geometric constraints), under a given state of external charges ( f , g ) by changing the geometry of each biphasic layer Ω i { Ω i } i =1 ..N ∈ U ad , Ω i ⊂D J (Ω , u χ ) min Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 10 / 33
Outline Motivation 1 Problem description 2 Main methods in Topology Optimization 3 The algorithm and its implementation 4 Results 5 Conclusion 6 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 11 / 33
Main methods in Topology Optimization Homogenization The idea is to find the optimal distribution of material inside a structure, by authorizing intermediates densities ( 0 ≤ θ i ≤ 1 ) of each material. To do this, we define the set of homogenized tensors G θ , corresponding to the set of composite materials A ∗ made from the mixture of the phases A i in θ i proportion, where A ∗ is the law tensor representing the homogenized micro-structure. Unfortunately the main problem of this method is to find an explicit description of the set G θ . Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 12 / 33
Main methods in Topology Optimization Level set method Has as unique advantages to track topology changes and a clear and smooth boundary that can be easily managed. Problems that arise from a density approach like spurious eigenfrequencies and micro structure stress concentration are avoided. Coupled to the shape and topological derivative analysis, it makes the level set method a promising research direction in future applications on structures design. Let’s take a look of how it works. Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 13 / 33
General explanation: Level set method φ = 0 Ω 0 φ < 0 φ > 0 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 14 / 33
General explanation: Shape sensitivity J (Ω θ ) < J (Ω 0 ) Ω θ = ( Id + θ )(Ω 0 ) θ Boundary movement Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 15 / 33
General explanation: Topological sensitivity J (Ω ω θ ) < J (Ω θ ) Ω ω θ A Ω ω A ω Nucleation of an inhomogeneity Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 16 / 33
Outline Motivation 1 Problem description 2 Main methods in Topology Optimization 3 The algorithm and its implementation 4 Results 5 Conclusion 6 Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 17 / 33
Cost function and the adjoint problem Let’s take a general cost function J (Ω , u χ (Ω)) as � � J (Ω , u χ (Ω)) = j ( x , u χ (Ω)) dx + h ( x , u χ (Ω)) ds. D ∂ D This criteria can represent for example the compliance ( j ( x , u ) = f · u , h ( x , u ) = g · u ), the volume ( j = 1 ), a least square target displacement ( j ( x , u ) = | u − u 0 | 2 ), a stress dependent function ( j ( x , u ) = | σ ( u ) | 2 ), etc. Furthermore, in order to avoid the calculation of the explicit variation of u χ w.r.t. the domain, we introduce the adjoint state problem − div ( A χ e ( p χ )) = − j ′ ( x , u χ ) ⎧ in D ⎨ p χ = 0 on Γ D A χ e ( p χ ) n = − h ′ ( x , u χ ) on Γ N ⎩ Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 18 / 33
The level-set method We define the level set function φ i ( x ) on D ⊂ R 2 as x ∈ ∂ Ω i ∩ D , ⎧ φ i ( x ) = 0 ⇔ ⎨ φ i ( x ) < 0 ⇔ x ∈ Ω i , x ∈ ( D \ ¯ φ i ( x ) > 0 ⇔ Ω i ) . ⎩ From this definition we can easily deduce Outward normal vector n i = ∇ φ i / |∇ φ i | Curvature κ i = div n i This formulae have a meaning over all D and not only on Σ i . If the domain Ω i ( t ) evolves in a pseudo-time t ∈ R + according to a velocity field θ ( x , t ) , then the level set transport equation is ∂φ i ∂t + V|∇ φ i | = 0 . where V ( x , t ) = θ · n is the normal component of the advection field. Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 19 / 33
Level-set method: Reinitialization In order to regularize the level-set function (which may become too flat or too steep), we reinitialize it periodically by solving − ∂φ � in D × R + ∂t + sign ( φ 0 )(1 − |∇ φ | ) = 0 φ ( t = 0 , x ) = φ 0 ( x ) in D which admits as a stationary solution the signed distance to the initial interface { φ 0 = 0 } . Reinitialize the level function is really important to obtain a good approximation of the normal n i and the curvature κ i of Σ i . Question: How we find V that minimizes J (Ω , u χ ) ? Hint: Shape derivative. Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 20 / 33
Shape derivative: Variation of the cost function under boundary changes We are interested on the variations of the functional J (Ω , u χ (Ω)) when the position of the interface Σ i on each ply changes following a regular vector field θ . Let be ω 0 an open smooth set such that ω 0 ⊂ D . We denote by χ ω 0 the characteristic function of ω 0 , and we consider the variations on the form χ ω θ = χ ω 0 ◦ ( Id + θ ) , i.e. χ ω θ = χ ω 0 ◦ ( x + θ ( x )) . Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 21 / 33
Shape derivative Definition Let be the functional J ( ω ) : U ad → R . The shape derivative of J ( ω ) at echet derivative in W 1 , ∞ ( R 2 , R 2 ) at 0 of the ω 0 is defined as the Fr´ l’application θ → J ( ω 0 ◦ ( Id + θ )) , i.e. | o ( θ ) | J ( ω 0 ◦ ( Id + θ )) = J ( ω 0 ) + J ′ ( ω 0 )( θ ) + o ( θ ) with lim � θ � W 1 , ∞ = 0 θ → 0 where J ′ ( ω ) is a continuous linear form on W 1 , ∞ ( D ; R 2 ) . Theorem: Level-set adapted fixed mesh shape derivative � Ω i � Let be D h and Ω h = polygonal approximations of D and Ω . Given a h triangulation T Σ = { K l } l of D h , with K l ∩ Σ � = ∅ , then � � J ′ (Ω)( θ ) = A χ e ( u χ ) : e ( p χ )( ∇ · θ ) dx − A χ e ( u χ ) : e ( p χ ) θ · nds K l ∂K l � A χ {∇ ( e ( u χ )) · θ } : e ( p χ ) + A χ {∇ ( e ( p χ )) · θ } : e ( u χ ) dx + K l Gabriel Delgado (EADS-X) PICOF ’12 Palaiseau, 3 April 2012 22 / 33
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