Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics Fr´ ed´ eric Legoll IMA, U of Minnesota (Minneapolis) and ENPC Paris joint work with Xavier Blanc (Universit´ e Paris 6) and Claude Le Bris (CERMICS, ENPC). http://www.ima.umn.edu/ ∼ legoll F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.1/24
Outline of the talk Some motivations for multiscale methods A prototypical 1D multiscale method Analysis of the method: case of a convex interatomic potential Lennard-Jones case F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.2/24
Nanoindentation simulation Tadmor, Miller, Phillips, Ortiz, J. of Material Research, 1999 ( www.qcmethod.com ) F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.3/24
Paradigm: study nanoscale localized phenomena Nanoindenter (25 ˚ A) Non smooth deformation Smooth deformation 1000 ˚ A 2000 ˚ A Large computational domain; Expected deformation: non-smooth in some small region of the solid. Coupling an (accurate) atomistic model with a (cheap) continuum mechanics model. F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.4/24
The atomistic model Reference configuration (1D): Ω = (0 , L ) ⊂ R u i Current position of atom i : Atomic lattice parameter: h, with Nh = L 1 u j − u i � � E µ ( u 0 , . . . , u N ) = � Energy per particle: W h 2 N i � = j Wh ( ξ ) � u j − u i � W h ( u j − u i ) = W ξ h h Atomistic model (assuming Nearest Neighbour interactions): N − 1 � u i +1 − u i N E µ ( u 0 , . . . , u N ) = h � − h u i f ( i h ) � � W L h L i =0 i =0 E µ ( u 0 , . . . , u N ) , u 0 = 0 , u N = a, u i +1 > u i � � inf → Intractable! F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.5/24
Continuum mechanics model X. Blanc, C. Le Bris, P .-L. Lions (ARMA 2002): if u is smooth enough, � � W ( u ′ ( x )) dx − h → 0 E µ ( u (0) , u ( h ) , . . . , u ( Nh )) = E M ( u ) = lim f ( x ) u ( x ) dx Ω Ω Continuum model (elastic energy density derived from atomistic model). � More generally, W CM ( F ) = 1 / 2 W ( F · k ) . k ∈ Z 3 ,k � =0 E M ( u ) , u ∈ H 1 (Ω) , u (0) = 0 , u ( L ) = a, u ′ > 0 a.e. on Ω � � inf What if deformation is not smooth in the whole domain? Use different models in the different do- mains. F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.6/24
Coupled model: a first attempt � W ( u ′ ( x )) − f ( x ) u ( x ) dx E c ( u ) := Ω M ( u ) � u i +1 − u i � � − u i f ( ih ) + h W h i ∈ Ω µ ( u ) Ω M ( u ) = subdomain where u is smooth, � where Ω µ ( u ) = subdomain where u is non-smooth. Highly nonlinear problem → remove the link between u and the partition of Ω F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.7/24
The natural coupled model For any partition Ω = Ω M ∪ Ω µ with Ω M = ∪ j ( a j h, b j h ) : Continuum mech. a j +1 h Atomistic model a j h b j h � W ( u ′ ( x )) − f ( x ) u ( x ) dx E c ( u ) := Ω M � u i +1 − u i � � � u i f ( ih ) + h W − h h i,ih ∈ Ω µ i, [ ih,ih + h ] ⊂ Ω µ Balance between numerical efficiency / precision u | Ω M ∈ H 1 (Ω M ) , u | Ω µ ≡ ( u i ) ih ∈ Ω µ , E c ( u ) , inf u a j = u (( a j h ) + ) , u b j = u (( b j h ) − ) , u (0) = 0 , u ( L ) = a, u ↑ F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.8/24
The coupled problem after discretization Discretization of the continuum mechanics term on a mesh of size H ≫ h : �� � � � � U k N ′ E H � � U, u | Ω µ := W k ( x ) dx − U k f ( x ) N k ( x ) dx c Ω M Ω M k k � u i +1 − u i � � � u i f ( ih ) + h W − h h i,ih ∈ Ω µ i, [ ih,ih + h ] ⊂ Ω µ Questions: How to choose the partition? Idea: the set Ω M should consist of all the zones of regularity of u µ Is E c a good definition for the coupled energy? We will show that inf E H c is not always the discretized version of inf E c ! F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.9/24
Two different cases for the potential Convex interatomic potential W ; The Lennard-Jones case. 3 W LJ 2 1 0 -1 0 1 2 3 F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.10/24
Convex case: definition of the partition f ∈ C 0 (Ω); W ∈ C 2 ( R ) with 0 < α ≤ W ′′ ( z ) and | W ′ ( z ) | ≤ β | z − 1 | The atomistic, macroscopic and coupled problems are well-posed. W convex = ⇒ elliptic regularity: { singularities of u } = { singularities of f } Assume f ∈ C 0 (Ω) . The interval ( ih, ih + h ) is said to be regular if f ′ ∈ L 1 ( ih, ih + h ) , � f ′ � L 1 ( ih,ih + h ) ≤ hκ f and � f � L ∞ ( ih,ih + h ) ≤ κ f L � � Set Ω M := ∪ ( ih, ih + h ) which are regular = ∪ j ( a j h, b j h ) Partition just depends on f ! F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.11/24
Estimates between u c and u µ (convex case) With previous definition of partition, ∃ h 0 such that, for all h ≤ h 0 , � � − u i +1 − u i u i +1 − u i � � µ µ c c sup � ≤ C 1 hκ f , � � h h � � i ∈ Ω µ � � u ′ c − (Π c u µ ) ′ � L ∞ (Ω M ) ≤ C 1 hκ f , � ≤ C 2 hκ f , � � u i c − u i � sup � u c − Π c u µ � L ∞ (Ω M ) ≤ C 2 hκ f , µ i ∈ Ω µ | I c − I µ | ≤ C 3 hκ f . Π c : affine interpolation operator F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.12/24
The Lennard-Jones case W LJ 2 ∂ x W LJ z 12 − 2 1 W ∗∗ W LJ ( z ) := z 6 LJ 1 W ′ LJ (1) = 0 0 W ′′ LJ ( r c ) = 0 -1 0 1 2 3 F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.13/24
Macroscopic problem ( f ≡ 0 ) Natural variational space: � � u ∈ W 1 , 1 (Ω) , 1 u ′ ∈ L 12 (Ω) , u ′ > 0 a.e. , u (0) = 0 , u ( L ) = a X M ( a ) = � L � a � W LJ ( u ′ ( x )) dx : inf E M = LW ∗∗ E M ( u ) = LJ L 0 3 If a > L : inf { E M ( u ) , u ∈ X M ( a ) } = LW LJ (1) 2 Problem has no minimizers in X M ( a ) . 1 u ′ “Minimizers” u M are s.t. M has u M ( x ) Dirac masses (“crack” nucleation). 0 0 1 2 x a 0 0 L F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.14/24
Macroscopic problem � � u ∈ D ′ (Ω) , u ′ = Du + � v i δ x i , Du ∈ L 1 (Ω) , x i ∈ Ω SBV (Ω) = . i ∈ N � � E M ( u ) , u ∈ SBV (Ω) , 1 Du ∈ L 12 (Ω) , u ′ > 0 a.e. , u (0) = 0 , u ( L ) = a inf When f ≡ 0 : If a ≤ L : u M ( x ) = ax/L . If a > L : infinity of solutions, u M = x + � i v i H ( x − x i ) . Crack location is not determined (because NN interaction and f ≡ 0 ). Results can be generalized to the case f � = 0 , f ∈ L 1 (Ω) : ∃ θ M s.t. – if a ≤ θ M , ∃ ! solution, which is smooth; – if a > θ M , “crack”. F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.15/24
The atomistic problem ( f ∈ C 0 (Ω) ) � u i +1 − u i � N − 1 N � � u i f ( i h ) , u 0 = 0 , u N = a, u ↑ � � inf E µ ( u ) = h W LJ − h h i =0 i =0 There exists a threshold θ µ such that: – if a ≤ θ µ , unique minimizer; – if a > θ µ and h small enough: one or many minimizers, smooth everywhere except on a single bond ( i µ , i µ + 1) : u ( x ) u i µ +1 − u i µ C µ µ ∼ h → 0 h (“crack”) h i µ + 1 i µ u i +1 − u i � � ∀ i � = i µ , → h → 0 0 µ µ See L. Truskinovsky, 1996. x F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.16/24
Natural micro-macro approach Suppose f ≡ 0 , a > L (crack case): For any partition Ω = Ω M ∪ Ω µ , � u i +1 − u i � � � W LJ ( u ′ ( x )) dx + h E c ( u ) = W LJ h Ω M i, [ ih,ih + h ] ⊂ Ω µ E c ( u ) , u | Ω M ∈ SBV (Ω M ) , u | Ω µ = ( u i ) ih ∈ Ω µ , inf u a j = u (( a j h ) + ) , u b j = u (( b j h ) − ) , u (0) = 0 , u ( L ) = a, u ↑ There exist minimizers u c . u ′ c has Dirac masses in Ω M ! F . Legoll, Multimat workshop, Paris, March 14-16th: Coupling atomistic with continuum mechanics – p.17/24
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