Construction and sharp consistency estimates for atomistic/continuum coupling methods: a 2D model problem

Transcription

1 Construction and sharp consistency estimates for atomistic/continuum coupling methods: a 2D model problem Lei Zhang with: Christoph Ortner Mathematical Institute University of Oxford

2 Point Defects in Crystal [Wikipedia]

3 determines their strength. You CT simulations using the fixed-parameter SW potential, will discuss the crystallographic two different TB schemes [22,23], and our hybrid aspectsscheme. of plastic deformation in Atomistic Com direct diagonalization would be approximately an or more detail in your A4 Material To be able to perform several fully quantum-mechanical Failure course. of magnitude slower. Improving on the accuracy of simulations we used a 64 Si atom cubic cell. As the cell is classical potentials is crucial to simulate this exten Figure 2. the exact TB forces to be used so small, here we compute!"#a $b a) Time: 2.98 fitpsby direct diagonalization b) Time: ps defect. LOTF simulations on this system reveal in the LOTF of the 3.02 periodic topology for the free energy minimum at 900 K wh system Hamiltonian. We note that the large difference!, differs from that obtained in the same conditions (and between the results obtained by fitting the scheme on the 0 Kunder [26]) from the SW model, with a square of ato Single crystals usually deform very easily %30 4 two different quantum models islowdue to the different values of stress. Since they have a nearly adjacent to an antiphase defect (red atom formed predictions of these models in accurate quantitative ideal structure, once thecomdeformation begins at Fig. 4), and a different defect migration pathway. Wh one point, it can continue the crystal. (111) putations, reproduced by the present method. This throughout emthe TB stress levels for plastic deformation are model used [23] may still not capture ev phasizes scheme can at best be (112)the fact that the present Typical in the range between 1 and 10 MPa,relevant much too feature of the phenomenon under study, th expected to reproduce the results of the QM model that it low for most engineering purposes. results indicate the need to enforce electronic-struct is given [24], but can in no way improve its accuracy. Most engineering alloys are precision. used in Indeed, in dislocation dynamics the inaccur To further elucidate how the scheme adaptsform to the local polycrystalline (Figure 3). This means that canonical phase space sampling provided by class 930#0 environment, we report in Fig. 3 the the up of a great eachtime pieceevolution of metal isofmade potentials is regarded as the main source of discrepa 205 Korsunsky; lecture notes Figure 3. number of single crystals, or grains, each having Csanyi etal;jphys.:condens.matter(2005) ARTICLEparameters IN PRESS Cijk at a dimerized Si[100] surface at room between theory and experiment [26]. al. / J. Mech. Phys. Solids 55 (2007) Dobson & Luskin (grain boundary) c) Time: 3.15Inps Time: 3.47 (crackremain tip inps temperature. the bulk these angled) parameters atsilicon) 4 M=Numbers of atom (edge dislocation) all times close to their!1=3 equilibrium value (left). However, on the reconstructed surface, the equilibrium Current computationa angle on the lower side of the buckled dimers (inset) is Empirical potent lowered to almost 90". The corresponding parameters learn this by switching to a value close to zero, and Semi-empirical p flip back and forth between zero and!1=3 as the buckling direction varies with time (right). Quantum mecha Moving to a problem where a fully quantum approach Nonconvex potential would be practically unfeasible, we simulate the gliding motion at 900 K of an opposing pair of 30" partial Can be modeled by c dislocations in Si, using a 4536 atom unit cell. We flag Csanyi et al; PRL 93; 2004 for QM Gavini; treatment all atoms within 7.0 A of Miller,Tadmor the dangling(2003) PhD Thesis ctron-density around the vacancy on (1 0 0) plane; (b) contours of groundfig. 4 (color). Equilibrium structure of the Use Si 30"ato par AtC method: (dislocation in silicon) Figurebonds 6. Four snapshots of theatoms) opening (111)[1 1 0] crack systembdry) simulated using LOTF. Atoms in (undercoordinated formed during dislocancy on (1 1 1) plane. (crack & grain (vacancy in aluminium) dislocation kink from a 900 K hybrid simulation (only tion motion,quantum corresponding to two disjoint QM of red were treated mechanically using the abregions initio SIESTA package.left The quantum region glide plane atoms are shown). One undercoordinated atomby ( freedom elsewhere ariation of the vacancy formation with sample size.the diffusing kink of each atoms,energy which follow Defects in Crystals

4 Scales in Materials Modelling Current computational limits: Empirical potentials (EAM): atoms Semi-empirical potentials (TB): 10 6 atoms Almost ab initio (KS-DFT): 10 4 atoms Schrödinger Equation: a few electrons Density of atoms in typical materials: atoms/mm 3 Goal: QM MM FEM Rountree et al, Annu.Rev.Mater.Res.2002

5 u Application: Crack at Grain Boundary Applications of Coarse-Graining [Iglesias/Leiva, Acta Mater. 06] complicated defects, defect interaction, crack growth, growth,... 8m atoms, void15000 repatoms Application: Shear in Twinned Aluminium B Langwallner (Oxford University) [Iglesia & Leiva, 2006], [Miller et al, 1998], The Quasicontinuum Method [QC software tutorial, Miller & Tadmor]

6 Atomistic Mechanics (0T statics) Atomistic body: N atoms at positions y = (y n ) N n=1 Rd N Total energy of configuration y: min Ea tot (y) := E a (y) + P a (y) E a = interaction potential, P a = potential of external frcs

7 Atomistic Mechanics (0T statics) Atomistic body: N atoms at positions y = (y n ) N n=1 Rd N Total energy of configuration y: min Ea tot (y) := E a (y) + P a (y) E a = interaction potential, P a = potential of external frcs Crystallisation:[Theil, Theil/Harris,... ]

8 The Quasicontinuum Idea (a) (b) Molecular statics problem: y a argmin E tot a (Y ) Coarse grained problem: y h argmin E tot a (Y h ) where mesh T h resolves the defect, and Y h = Y P 1 (T h ).

9 Atomistic Stored Energy: E a (y) = Cauchy Born Approximation x L Cauchy Born Stored Energy: E c (y) = W ( y) dv, Ω V ( y(x + r) y(x); r R ) where W (F) = V ({Fr; r R}).

10 Atomistic Stored Energy: E a (y) = Cauchy Born Approximation x L Cauchy Born Stored Energy: E c (y) = W ( y) dv, Ω V ( y(x + r) y(x); r R ) where W (F) = V ({Fr; r R}). Theorem: [Similar to result by E/Ming; 2007] Let y a argmin Ea tot be sufficiently smooth globally, then there exists y c argmin Ec tot such that y a y c L 2 C ( 3 y a L y a 2 ) L 4 If there are no defects, then the Cauchy Born model is a highly accurate continuum approximation.

11 Atomistic/Continuum Coupling: First Attempt (a) (b) (c) E a (y h ) E ac (y h ) := ω x V x + W ( y h ) dx x L Ω c a

12 Atomistic/Continuum Coupling: First Attempt Fails the patch test: δe a (y F ) = 0 and δe c (y F ) = 0, but δe qce (y F ) 0! (a) (b) (c) E a (y h ) E ac (y h ) := ω x V x + W ( y h ) dx x L Ω c a Error: x Component x 10 3 Error: y Component

13 Alternative Approaches 1 Energy-based coupling: interface correction 2 Force-based coupling: FeAt: Kohlhoff, Schmauder, Gumbsch (1989, 1991) Dead-load GF removal: Shenoy, Miller, Rodney, Tadmor, Phillips, Ortiz (1999) AtC: Parks, Gunzburger, Fish, Badia, Bochev, Lehoucq, et al. (2007,... ) CADD: Shilkrot & Curtin & Miller (2002,... ) Stress-based coupling: Makridakis/Ortner/Süli (2011)... 3 Blending methods: Belytschko & Xiao (2004) Klein & Zimmerman (2006) Parks, Gunzburger, Fish, Badia, Bochev, Lehoucq, et al. (2008)...

14 A Priori Error Analysis Framework: Let y a argmin Ea tot, y ac argmin Eac tot, then... (standard ideas)... [y a y ac ] L 2 CONSISTENCY STABILITY = δe a(y a ) δe ac (y a ) H 1 inf u L 2 =1 δ 2 E ac (y a )u, u 3 Steps: 1 CONSISTENCY: δe ac (y) δe a (y), u 2 y L a 2 (Ω c) 2 STABILITY: δ 2 E ac (y)u, u C stab u 2 L 2 3 REGULARITY: bounds on 2 y a, note that r a for defects

15 Consistency Numerical Analysis Literature on Energy-Based Coupling: 1D, NN, variational analysis: [Blanc/LeBris/Legoll, 2005] 1D, Error estimates for QCE and QNL: [Dobson/Luskin, 2009], [Ming/Yang, 2010], [Ortner, 2010], [Ortner/Wang, 2011] 1D, Sharp Stability Analysis, Linear Regime: [Dobson/Luskin/Ortner, 2010] 1D, finite range: [Li/Luskin, preprint] 1D, EAM potentials: [Li/Luskin, 2011] 1D, blending methods: [Luskin/van Koten, preprint] 2D, pair interactions, defects: [Ortner/Shapeev, preprint] 2D, first order consistency for general finite-range interactions: [Ortner, preprint] 2D, multi-body interactions: [Ortner/Zhang, preprint] Analysis for Force-based Coupling: Dobson & Luskin (2008); Ming (2009); Dobson & Luskin & Ortner (2009, 2010, 2010); Makridakis & Ortner & Süli (2010, preprint); Dobson & Ortner & Shapeev (preprint), Lu & Ming (manuscript) Analysis for Multi-Lattices: Dobson &Elliott & Luskin & Tadmor (2007); Abdulle & Lin & Shapeev (preprint)

16 Patch Test Consistency implies First-order Consistency? Suppose the interface potentials Ṽx are fitted numerically, s.t. E ac becomes patch test consistent: δe ac (y F ) = 0 F R d d Does this automatically imply that E ac is first-order consistent? Theorem: [First-order Consistency] Suppose δe ac passes the patch test, V finite range multi-body potential + technical conditions + d = 1; or d = 2, Ω a connected; or d = 3, flat interface, homogeneous Ṽx then δeac (y) δe a (y), u h h 2 y L 2 (Ω c) u h L 2 If E ac is also stable then this would imply y a y ac L 2 h 2 y a L 2 (Ω c)

17 Consistency in 1d no ghost forces = consistent in a negative Sobolev space. A/C method E ac has no ghost forces if F ac (y F ; x) = E ac(y) y(x) = 0 F > 0 x L. Theorem: [sharp estimate in 1d] Assume that the A/C energy E ac has no ghost forces, then there exists a constant C depending only on M 2 (µ) and M 3 (µ), such that sup δe ac (y) δe a (y), u C ( y l p (I) + y l p (C) + y 2 u R L l (C)). 2p u l p =1

18 Consistency in 1d no ghost forces = consistent in a negative Sobolev space. 1 A/C stress function δe ac(y), u = x L Σac(y; x)u (x), where F ac(y; x) = Σ ac(y; x + 1) Σ ac(y; x). Similarly Σ a(y; x), Σ c(y; x). 2 2nd order error for Σ c(y; x) (by symmetry, V(b)=V(-b)) Σa(y; x) Σ c(y; x) C ( y l (x+r) + y 2 l (x R)). 3 Σ ac(y; x) = Σ a(y; x) for x A; Σ ac(y; x) = Σ c(y; x) for x C, 4 F ac(y F ; x) = 0 Σ ac(y F ; x) = Σ a(y F ; x) = Σ c(y F ; x), 5 Σ ac and Σ a are Lipschitz continous when V is smooth Σac(y; x) Σ a(y; x) Σac(y; x) Σ ac(y F ; x) + Σa(y F ; x) Σ a(y; x) C F y l (I ± ) Σac(y; x) Σ a(y; x) C y l ((x R) I) 6 use δe ac(y) δe a(y), u = x L (Σac(y; x) Σa(y; x))u (x) to complete the proof.

19 Construction of General A/C Schemes E ac (y h ) = V x + Ṽ x + Vx c x L a x L i x L c Construct Ṽ s.t. δe ac(y F ) = 0 for all F R d d. General Construction: [Shimokawa et al, 2004; E/Lu/Yang, 2006] Ṽ x = V ( g x,r ; r R ) g x,r = s R x C x,r,s g s Find C x,r,s s.t. δe ac (y F ) = 0 geometric conditions only! F Explicit constructions for flat interfaces: [E/Lu/Yang, 2006] Explicit constructions for 2D general interface: [Ortner/Zhang; preprint] In general: compute C x,r,s numerically in preprocessing 2d, NN, multibody potential, triagular lattice

20 Construction of General A/C Schemes General Construction: E ac (y h ) = V x + Ṽ x + Vx c x L a x L i x L c Construct Ṽ s.t. δe ac(y F ) = 0 for all F R d d. Ṽ x = V ( g x,r ; r R ) g x,r = s R x C x,r,s g s Find C x,r,s s.t. δe ac (y F ) = 0 geometric conditions only! F 2. Patch Test Consistency 0 = δe ac(y F ), u = V F,r C x,r,sd su x L r R s R = (C x as,r,sv F,r C x,r,sd r V F,r )u(x) x L r R s R 1. Local Energy Consistency Ṽ (y F ) = V (y F ) r = s R x C x,r,ss. (a) (C x s,r,sv F,r C x,r,sv F,r ) = 0. (b) r R s R Solve (a) + (b) + B.C. in L a and L c to obtain C x,r,s for x L i. unknowns: L I R 2, eqns: 2 L I R.

21 Construction of General A/C Schemes E ac (y h ) = V x + Ṽ x + Vx c x L a x L i x L c Construct Ṽ s.t. δe ac(y F ) = 0 for all F R d d. General Construction: Flat interface Interface with corner C x,r,r for NN interaction, multibody potential, one-sided construction. 1. works for general interface in 2d 2. preprocessing for longer interaction range

22 Consisency of The Schemes E ac (y h ) = V x + Ṽ x + Vx c x L a x L i x L c Construct Ṽ s.t. δe ac (y F ) = 0 for all F R d d. Theorem: [Ortner/Zhang] There exists a constant C that depends only on M 2 (V ) and M 3 (V ) such that δe ac (y) δe a (y) U 1,p C ( 3 y l p (Ω c) + 2 y 2 l 2p (Ω c) + 2 y l p (Ω I )).

23 Stress Function in 2D Define atomistic and continuum stress, δe a (y), u = T T T Σ a (y; T ) : T u δe c (y), u = T Σ c (y; T ) : T u T T index conventions: δe a(y), u 6 = V x,j D j u(x) x L j=1 ( 1 = T 2 T T 6 ) V xt,j,j a j : T u j=1 Since D j u(x) = 1 2 ( T x,j u a j + Tx,j 1 u a j ).

24 Stress Function in 2D Define atomistic and continuum stress, δe a (y), u = T T δe c (y), u = T T T Σ a (y; T ) : T u T Σ c (y; T ) : T u δe a(y), u 6 = V x,j D ju(x) (D ju(x) = D j+3u(x + a j)) x L j=1 1 6 = T V xt 2 T,j,j a j : T u T T j=1 }{{} =Σ 1 a (y;t ) 1 6 1( = T VxT,j,j V xt 2 T 2,j+3,j+3) a j : T u T T j=1 }{{} =Σ 2 a (y;t ) index conventions:

25 Stress Function in 2D Define atomistic and continuum stress, Similarly, δec(y), u δe a (y), u = T T δe c (y), u = T T = T W ( T y) : T u(t ) T T 1 6 = T j V ( T ya) a j 2 T T T j=1 }{{} =Σ 1 c (y;t )= W ( T y) : T u 1 6 1( ) = T VT,j + V TT,j,j aj : T u 2 T 2 T T j=1 }{{} =Σ 2 c (y;t ) T Σ a (y; T ) : T u T Σ c (y; T ) : T u index conventions:

26 Stress Function in 2D Define atomistic and continuum stress, δe a (y), u = T T T Σ a (y; T ) : T u δe c (y), u = T Σ c (y; T ) : T u T T index conventions: stress functions are not unique Σ 1 a(y; T ) W ( T y) = O( D 2 y ) Σ 2 a(y; T ) Σ 2 c(y; T ) = O( D 2 y 2 + D 3 y ) Theorem: [Consistency for CB energy] δea(y) δe c(y) U 1,p C( 3 y l p + 2 y 2 l 2p ).

27 Discrete Divergence Free Tensor Field The stress function is not unque, up to a divergence free tensor field, namely, σ such that σ : T u = 0 T T Characterization of div-free field, [Arnold/Falk, Polthier/Preuß] Lemma: v piecewise constant vector on T, v divergence free, i.e., v T u = 0 T T iff a function w N 1 (T ), such that v = J w, where J is the counter-clockwise rotation by π/2. N 1 (T ) is Crouzeix Raviart finite element space. For basis ζ f N 1(T ) associated with f = T 1 T 2, J ζ f T u = Jn 1 T1 u + Jn 2 T2 u = u 1 u 2 + u 2 u 1 = 0 T T

28 Characterization of stress function If an A/C energy E ac satisfies patch test consistency, 0 = δe ac (y F ), u = T T T Σ ac (y F ; T ) : T u Lemma: a function ψ(f, T ) N 1 (T ) 2, such that Σ ac (y F ; T ) = W (F) + J ψ(f; T ) For general deformation y, deformation gradient average for patch ω f = (T 1 T 2 ), F f (y) = y dx, Corrector function: ˆψ(y; ) = f F ψ( F f (y); m f ) ζf Define the modified stress function, Σ ac (y; T ) := Σ ac (y; T ) J ˆψ(y; T ), for T T. Σ ac(y F ; T ) = W (F) = Σ a(y F ; T ) Σ ac(y; T ) = Σ a(y; T ), T Ω A and Σac(y; T ) = Σ c(y; T ), T Ω C ω f

29 Proof of Consistency δe ac (y) δe a (y), u = T T (Σ ac (y; T ) Σ a (y; T )) : u = T T ( Σ ac (y; T ) Σ a (y; T )) : u 1 T Ω A, Σac(y; T ) = Σ a(y; T ), 2 T Ω C, Σac(y; T ) Σ a(y; T ) = Σ c(y; T ) Σ a(y; T ), 2nd order consistency 3 T Ω I, Let y T = y y(t ), we have, Σac(y; T ) Σ a(y; T ) Σac(y; T ) W ( y(t )) + W ( y(t )) Σ a(y; T ) = Σac(y; T ) Σac(y T ; T ) + Σ a(y T ; T ) Σ a(y; T ) C y(t ) y l C D 2 y δe ac(y) δe a (y) U 1,p C ( 3 y lp (Ω c) + 2 y 2 l 2p (Ω c) + 2 y lp (Ω I )).

30 Numerical Experiment Test Problem: single vacancy in the triangular lattice, Vacancy defect in 269,100 atom cell atomic spacings Eac Ea / Ea E0 QCE B-QCE (DoF) 1 (DoF) 2 GR-AC yac ya 1,2/ ya id 1,2 QCE B-QCE (DoF) 1/2 GR-AC (DoF) yac ya 1, / ya x 1, QCE B-QCE (DoF) 1 GR-AC # DoFs # DoFs 10 3 # DoFs Error in Energy H 1 Error W 1, Error

31 Outlook on A/C Methods Summary Ghost force removal Patch test consistency consistency Construction of practical energy-based a/c methods Sharp consistency error estimates through stress based formulation

32 Outlook on A/C Methods Summary Ghost force removal Patch test consistency consistency Construction of practical energy-based a/c methods Sharp consistency error estimates through stress based formulation Open Problems Consistency in 3D: proof or counterexample Stability A/C coupling at surfaces Implementation, benchmarks, applications

33 Outlook on A/C Methods Summary Ghost force removal Patch test consistency consistency Construction of practical energy-based a/c methods Sharp consistency error estimates through stress based formulation Open Problems Consistency in 3D: proof or counterexample Stability A/C coupling at surfaces Implementation, benchmarks, applications Major Open Problems A/C methods for multi-lattices A/C methods for Coulomb interaction A/C methods for electronic structure models (done only for insulators)