Boundary conditions for molecular dynamics

Transcription

1 Xiantao Li Department of Mathematics Penn State University Collaborators Weinan E and Jerry Z. Yang Princeton University AtC Workshop: Sandia National Lab, Mar 21, 26

2 Motivations Boundary Reflection Fracture simulation in triangular lattice (Holian et. al. 1996)

3 Motivations Boundary Reflection Fracture simulation in BCC iron (Machova and Ackland 1998)

4 Motivations Boundary condition The role of boundary conditions: 1. Prevent phonon reflection 2. External loading 3. Maintain the system temperature 4. Couple with continuum models MD MD Boundary Conditions

5 Motivations Boundary condition The role of boundary conditions: 1. Prevent phonon reflection 2. External loading 3. Maintain the system temperature 4. Couple with continuum models MD PDE Boundary Conditions

6 Motivations Outline General formulation s From zero temperature to finite temperature Summary

7 General formulation Mori - Zwanzig s view Generalized Langevin equation (GLE), da t dt = iω(t)a Θ(t s)a(s)ds + R(t). A = (q, p): smaller number of variables Θ(t): correlation matrix R(t): from unresolved degrees of freedom R(t)A() =

8 General formulation Random forcing For system embedded in a linear heat bath, R(t) is a stationary Gaussian process. The second fluctuation dissipation theorem: R(t)R() T = p() 2 Θ(t) = k B T Θ(t). the correlation function determines the random force for T =, R = ; Θ(t) provides zero temperature BC.

9 General formulation Existing work GLE formulations One dimensional case: Adelman and Doll 1974, 1975 Multi dimensional case: Tully 198 Zero temperature BC Exact BC: Cai, de Koning, Bulatov and Yip 2 Approx. BC: E and Huang 21, 22 General Exact BC: Wagner, Karpov, Liu, Park et al 24, 25 Approx. BC for general system: E and Li 26

10 Exact boundary conditions: u i (t) = j t θ j (s)u i+j (t s)ds. n Boundary condition

11 Exact boundary condition: a 1D example u (t) = t β = J 2(2t) t β(t s)u 1 (s)ds, j u u 1 u 2

12 Exact boundary condition: a 1D example u (t) = t β = J 2(2t) t β(t s)u 1 (s)ds, j u u 1 u 2

13 Exact boundary condition: a 1D example.8 u (t) = t β = J 2(2t) t β(t s)u 1 (s)ds, j β(t) t

14 Boundary conditions: exact vs approximate Exact boundary conditions: nonlocal in both space and time numerical implementation is expensive premature truncation leads to large reflection Local boundary condition: Variational BC (E and Huang 21 and 22, E and Li 26) u i (t) = j J t A j (s)u i+j (t s)ds, J finite. Local in both space and time Minimize phonon reflection Optimal on a given stencil (J and (, t ))

15 Variational boundary condition Main components: 1. Reflection coefficients 2. Energy flux 3. Variational formulation u j = e i(r j k I ω st) ε s (k I ) + c R ss ei(r j k R ω s (kr )t) ε s(k R )

16 Variational boundary condition Main components: 1. Reflection coefficients 2. Energy flux 3. Variational formulation

17 Variational boundary condition Main components: 1. Reflection coefficients 2. Energy flux 3. Variational formulation

18 Variational boundary condition Main components: 1. Reflection coefficients 2. Energy flux 3. Variational formulation

19 Variational boundary condition Main components: 1. Reflection coefficients 2. Energy flux 3. Variational formulation J R = s BZ l css R l 2 ωs 2 ( ω s n)dk.

20 Variational boundary condition Main components: 1. Reflection coefficients 2. Energy flux 3. Variational formulation min I [{α j}; n] {α j } css R l 2 ( ω s n)dk s BZ l

21 Choosing the stencil J The space stencil J, only involves a small number of atoms translational invariant along the boundary respect some crystal symmetry

22 Choosing the stencil J The space stencil J, only involves a small number of atoms translational invariant along the boundary respect some crystal symmetry

23 Choosing the stencil J The space stencil J, only involves a small number of atoms translational invariant along the boundary respect some crystal symmetry

24 Choosing the stencil J The space stencil J, only involves a small number of atoms translational invariant along the boundary respect some crystal symmetry

25 Choosing the stencil J The space stencil J, only involves a small number of atoms translational invariant along the boundary respect some crystal symmetry

26 Example I: 1D chain R(k) = 1 j 1 j t e ijk a j (τ)e iωτ dτ e ijk t. a j (τ)e iωτ dτ Exact kernel M=2 M=1 M=5.2 The approximate kernels can be quite different from the exact ones VBC leads to much less reflection than the exact BC truncated to the same interval time history kernels

27 Example I: 1D chain R(k) = 1 j 1 j t e ijk a j (τ)e iωτ dτ e ijk t. a j (τ)e iωτ dτ VBC: M=2 Exact BC: M=2 Exact BC: M=2 R.5.4 The approximate kernels can be quite different from the exact ones VBC leads to much less reflection than the exact BC truncated to the same interval k phonon reflection

28 Example II: comparing different stencils Selection of the stencil

29 Example II: comparing different stencils J = 4

30 Example II: comparing different stencils J = 8

31 Example II: comparing different stencils J = 17

32 Example II: comparing different stencils J =4 J =8 J = J R M Total reflection on various stencil J = 17

33 Application: fracture simulation in BCC Iron fixed boundary condition variational boundary condition

34 Finite temperature boundary conditions From zero to finite temperature: GLE Boundary condition: U(r, t) = j t A j (t s)u(r + r j, s)ds. r J the boundary atoms those atoms at the boundary that have been removed from the system, r + r j J 1 : the interior atoms next to the boundary that are directly influenced by the boundary atoms. For r J 1, f(r, t) = V r + fex, f ex : the force due to the boundary atoms, f ex = r k J D k U(r k, t).

35 Finite temperature boundary conditions Zero temperature GLE Adding the external force: ü(r, t) = m t β m (s)u(r + r m, t s)ds V r. GLE in the standard form, ü(r, t) = m Θ m ()u(r + r m, t) t Θ m (s) u(r + r m, t s)ds V r + R(t) Θ m (t) = t β m (s)ds.

36 Finite temperature boundary conditions Solving GLE Represent the noise by a Fourier integral: R(t) = ˆR(ω)e iωt dω. The Fourier coefficients are Gaussian random variables and, R(t)R() T = Θ(t) = ˆR(ω)ˆR(η) = ˆΘ(ω)δ(ω + η). ˆΘ(ω): power spectrum the random noise is determined by the power spectrum

37 Finite temperature boundary conditions Approximation of the power spectrum ˆΘ ˆΘ C ( E + CẼ ) 1. Θ: computed from approximate boundary condition Θ : computed from exact boundary condition C: reflection coefficients. The error is controlled by the reflection coefficients!

38 T Finite temperature boundary conditions Example I: one dimensional chain 1.4 x x v(t)v() t t System temperature Velocity correlation

39 T Finite temperature boundary conditions Example II: 2D triangular lattice v(t) v() time t System temperature Velocity correlation

40 Summary Summary 1. General formulation from Mori - Zwanzig s viewpoint 2. Determine the random forcing from the correlation function 3. Variational formulation to obtain the correlation function 4. Reflection coefficients as a measure of the accuracy 5. Applications to various systems Reference: X. Li and W. E, Variational boundary conditions for molecular dynamics simulations of solids at low temperature, comm. comp. phys. 1 (26),