Topics in Molecular Modeling: II I . Absorbing boundary conditions

1. Topics in Molecular Modeling: III. Absorbing boundary conditions Xiantao Li (xiantao.li@gmail.com) Department of Mathematics, Pennsylvania State University

2. Outline 1D example Reflection coefficients Local approximations Multi-dimensional problems Dirichletto Neumann map

3. A 1D example The equations of motion Linearization on the left Assume that the left half is at mechanical equilibrium initially: Solve these equation using as boundary condition. The displacement of the atom at the boundary (Halpern 1982)

4. A 1D example The effective model This provides the effective boundary condition. The same procedure can be applied to the right boundary. The approach can be extended to problems with multiple neighbor interactions. The boundary condition is known as the absorbing BC.

5. Reflection Analysis Taking the Fourier transform in space, The linear model becomes: is the dispersion relation. Assume For an approximate BC, measures the quality of the ABC.

6. Examples of approximate ABCs Fixed BC: Then Complete reflection. Neumann BC: . Then Again complete reflection. First order BC: Then

7. Further challenges • There are a lot of further challenges • In 2D or 3D, the domain geometry is more complicated. • How do we design approximate ABCs that are automatically stable? • What if the initial displacement and velocity in the bath are not zero?

8. Domain partition & Partial harmonic approximation Molecular statics [Wu & Li, MSMSEE 2017] Molecular dynamics [Wu & Li, J. Comp. Mech. 2017]

9. Formulation using Dirichlet to Neumann (DtN) Map. Static Exterior Problem Continuum Discrete Lattice model (Free space) Lattice Green’s function Boundary atom equation Evaluation (coupling with nonlinear atomistic models for the microstructures) • Continuum elastostatics • The Green’s function in • Integral equation (Cruse, 1974) • Evaluation (if needed) • Very successful in fracture mechanics

10. Atomistic/based boundary element method: The discrete DtN map B:the exterior domain; : atoms at the inner boundary; outer boundary. Partition of the variables: ; the variables in B. Further partition: (Due to the short range interactions) is the solution at the inner boundary (filled circles) DtN map: The traction at the boundary: (involves open circles).

11. Domain Decomposition Due to the presence of defects, follows a nonlinear model serves as the boundary condition for the reduced problem in The DtN map: In practice, we alternative between the two equations until convergence (Dirichlet-Neumann coupling). Atomistic-based boundary element method (ABEM) (Li 2012).

12. Computing the matrices in the DtN map • K is defined based on the lattice Green’s function • The Green’s function can be expressed as a Fourier integral • Integration can be done by k-point sampling. The results can be tabulated. • For large distance, the integral is difficult to compute. • But it can be approximated by continuum Green’s functions Duffin & Shelly (1958) Martinsson & Rodin (2002)

13. Discrete DtN map More efficient than flexible BC (Sinclair 1973, Trinkle 2009, etc) Extension to quantum mechanics (Li, Lin and Lu 2016). Compared to boundary element methods, no need to discretize the boundary, no quadrature and no singular integrals. Nonlinear problems can be handled, iteratively.

14. Example: Dislocation Dipole • Aluminum, EAM • Two dislocations with opposite Burger’s vectors • Uniform Shear • Full Model: 0.9M atoms • ABEM: 1380 nodes

15. Crack propagation (Wu and Li, MSMSE 2017) • Further crack extension • 10 microns by 10 microns • Billions of atoms • Further extension (microns) • Only the atoms at the tips are included. • The crack faces are treated as remote boundary with quadrature

16. Crack propagation (snapshots)

17. Dynamics problems: Absorbing Boundary Conditions (ABCs) • Model in the bath • The domain can be multi-connected or have corners. • Goal of the reduction: • Eliminate the surrounding atoms • Avoid wave reflections around the defects • Obtain results as if the simulation is done over the entire domain • DtN Map: An impulse/response representation of the ABC • Formulation (Laplace transform)

18. Dynamic DtN map ABEM -> In the time-domain Explicit formulas available for flat boundaries (half-space). The combined model with is an integro-differential equation. Direct implementation can still be expensive due to the time integration.

19. Partially harmonic approximation The presence of defects does not allow the assumption of zero initial displacement For multi-body interactions, the linearization is more difficult. We assume that the bath is initially at mechanical equilibrium relative to the interior: This defines the new reference state. The operator can be computed from the ABEM. Approximation of the total energy:

20. The computation of the DtN map • For each is computable from • The matrices are determined by • Fourier integral in the near field • Continuum limit in the far-field: Exponential integrals. • corresponds to the static (time-independent) problem. • But in the time domain is not easy to compute.

21. Local boundary conditions(Wu and Li, Computational Mechanics 2018) • Rational approximation (Padé is possible, but complicated) • First-order approximation: • Interpolation: • In the time-domain: . • Theorem (Stability, Wu and Li 2018) The dynamics of is energy stable.

22. Second order boundary conditions Rational approximation In time-domain: . Interpolation: Theorem (Stability) The dynamics of is energy stable if .

23. Example: Dynamics of dislocation dipole Zeroth order First order Second order

24. Finite temperature case • If , then the DtN becomes, • is a stationary Gaussian process. • Approximate BC: • is a Gaussian white noise.