1. The Materials Computation Center, University of Illinois Duane Johnson and Richard Martin (PIs), NSF DMR-03-25939 www.mcc.uiuc.edu Time and spacetime finite element methods for atomistic, continuum and coupled simulations of solids Students: a Brent Kraczek, b Scott T. Miller, PI’s : a,c Duane D. Johnson, b Robert B. Haber, University of Illinois at Urbana-Champaign, Departments of a Physics, b Mechanical Science and Engineering, and c Materials Science and Engineering { kraczek, smiller5, duanej, r-haber }@uiuc.edu Support: Materials Computation Center, UIUC, NSF ITR grant DMR-0325939 and Center for Process Simulation and Design, NSF ITR grant DMR-0121695 The Materials Computation Center is supported by the National Science Foundation.

2. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Meeting MCC objectives This project achieves objects of MCC mission through Collaborative work involving calculations in atomistic, continuum and coupled systems Involves two students with different backgrounds Development new algorithms and codes in each problem type Collaboration between 2 NSF centers, MCC and CPSD (Center for Process Simulation and Design) Codes to be made available through software archive

3. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Atomistic and continuum methods Atomistic Coupled ODEs with discrete –mass, momentum –position, velocity Fixed number of d.o.f., treated individually This severely limits size and/or duration of simulation May be refined in time Non-local interactions “Correct” description of defects Continuum PDE with continuous fields –mass, momentum density –displacement, velocity, thermal Representative subset of d.o.f. optimized for problem size and accuracy May be refined in space, time Local stress/strain Need to address explicitly cohesion, plasticity, etc.

4. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Atomistic-continuum coupling Objective: Develop coupling formalism for solid mechanics that 1.Treats different scales with appropriate methods 2.Allows refinement/coarsening of scales in both space and time 3.Maintains compatibility and balance of momentum and energy 4.Consistently handles thermal fields and/or changes in # d.o.f. 5.Is O(N) and parallelizable for dim≥1 6.Accomplishes all this within a consistent mathematical framework These objectives partially fulfilled by focusing on time integration using Time/spacetime finite element methods in atomistic/continuum Coupling via fluxes defined within these finite element models

5. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Continuum formulation: Spacetime finite elements Spacetime discontinuous Galerkin (SDG) finite element (FE) method 1 Solves wave equation in solids in n-spatial-dim x t O(N) solution via causal meshing Captures complex behavior of wave propagation, including shock loading Enables different temporal scales for different spatial portions of problem 1. R. Abedi, et al., CMAME, 195:3247-3273 (2006) x y y x t Figure shows mesh only—physical results reflected in mesh refinement Problem: Shock-loading of plate with crack at middle (symmetry reduced to ¼ plate)

6. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Information passed between elements via flux conditions on M and  –Flux-balance on M enforces linear momentum balance –Flux-balance on  enforces compatibility –Energy flux on element boundary may be written as ⇒ compatibility and momentum balance imply energy balance. Fluxes will also be used in atomistic-continuum coupling Spacetime FE (SDG): Flux balance laws Q ∂Q

7. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Thermal transfer at atomistic scale is through vibrations—hyperbolic Standard heat equation based on Fourier’s law is parabolic 1. ( Maxwell (1867), Cattaneo (1948), Vernotte (1958) ) MCV: Non-Fourier thermal model MCV 1 modification to Fourier’s law –Yields hyperbolic heat equation –Parameter  is relaxation time –Appropriate for short time and/or length scales Fourier’s law –Yields parabolic heat equation –Infinite propagation speed –Appropriate in most cases

8. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Use SDG for coupled wave equation and MCV heat equation Constitutive equations include –MCV equation for heat flux evolution –Stress tensor with additional term linear in temperature Enforce balance of energy through new boundary fluxes: –Total energy flux –MCV heat flux Spacetime FE for generalized thermoelasiticy

9. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Thermoelastic problem: Laser pulse heating Laser pulse modeled as a Gaussian-type heat source Animation: Color field shows temperature Height field shows velocity magnitude Problem set-up: IC: Heated by Gaussian pulse Thermal BCs: insulated Mechanical BCs: traction-free

10. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Atomistic time FE for molecular dynamics Time finite element (TFE) method for atomistic system compatible with continuum spacetime finite element Divide problem into simultaneous solution on successive time intervals: Discretize trajectories in position, velocity in suitable basis (eg. Lagrange interpolation functions) High order convergence for trajectory and energy error world lines of 2 displaced particles

11. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Atomistic TFE: Energy error Machine precision noise for sufficient refinement Number of force evaluations per time step depends on –Number of Gauss points used (Ng) –Number of iterations required Problem: Single particle in non-linear potential well (Lennard-Jones oscillator) representative of future MD use

12. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Atomistic TFE: Trajectory error Linear springs allow direct comparison with analytic solution Convergence rate for trajectory error in 100 atom chain is 2p (p = polynomial order) Problem: Traveling pulse in 100 atom chain, w/ N-nn linear spring interaction

13. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Coupled atomistic-continuum system Underlying mathematical model is time/spacetime FE Coupling time/spacetime methods through flux compatibility at  AC Currently implemented for 1d with 1 st NN atom at boundary Division of solution space into continuum and atomistic regions remains constant ⇒ Implemented for atomistic TFE with linear springs and VVerlet for linear springs and non-linear Morse potential (all 1NN) t2t2 t1t1  AC Model system Continuum region Atomistic region v*  v C  C A F A

14. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Coupled atomistic-continuum system Continuum compatibility relations (kinematic and momentum) v* and  * determined implicitly from values on both sides of interface. To supply flux conditions from atomistics, –homogenize atomic velocities at boundary A –solve for forces on atoms as initially undetermined forces Momentum balanced explicitly; Energy balance will depend on A t2t2 t1t1  AC Model system Continuum region Atomistic region v*  v C  C A F A

15. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Coupled system: Results in 1d Atomistic 200 atoms 5+4 dof Continuum 40 elements 5x5 dof Coupled 20 elements, 5x5 dof 100 atoms, 5+4 dof Initial After 1 pass

16. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Coupled system: Total energy error Energy error reflects position of pulse in region Consider this configuration A B C A. Pulse begins in continuum region B. Pulse fully in atomistic region C. Pulse fully in continuum region

17. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Coupled system: Momentum balance Total momentum ~10 -10 Component momentum reflects pulse passing through coupling boundaries A B C A. Pulse begins in continuum region B. Pulse fully in atomistic region C. Pulse fully in continuum region

18. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Conclusions We have developed a set of mathematically consistent FE element tools for atomistic, continuum and coupled atomistic-continuum simulations Spacetime finite element (Spacetime Discontinuous Galerkin) developed for continuum wave equation –O(N) with causal meshing and excellent shock capturing ability –Thermoelasticity handled through non-Fourier heat model Time finite element developed for highly accurate molecular dynamics Coupled atomistic-continuum simulations achieved through flux conditions at At-C interface. Model/testing codes to be posted on software archive

19. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Analogous continuum system  = mass density, C = elastic modulus L = length Stress Continuum equation of motion Wave speed Atomistic vs. continuum models of solids: 1d Atomistic mass-spring system m = atomic mass, K = atomic interaction (spring constant) a = lattice spacing (interatomic distance) N masses -> length L=Na Force Atomistic equation of motion Wave speed (phase velocity) uiui u i+1 u i-1  C

20. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Define –Strain-velocity –Stress-momentum M and  follow characteristics of wave equation— allows causal meshing Spacetime FE (SDG): Continuum fields M= -p  = E M ( , p)  (v, E)   v etet n0n0 n0n0 n0n0 Causal interface: Solution in Q  depends on Q  Non-causal interface: Solution in Q  and Q  interdependent

21. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Spacetime FE: Causal meshing Goal: Mesh space to obtain O(N) solution by taking advantage of wave characteristics Algorithm Pitch “tents” —patches of tetrahedra in 2d x t —causally advancing solution Solve a patch implicitly—causal separation is between patches Refine or coarsen as necessary, taking special care to ensure progress R. Abedi, et al., Proc. 20th Ann. ACM Symp. on Comp. Geometry, 300-309, 2004. Causal interface: Solution in Q  depends on Q  Non-causal interface: Solution in Q  and Q  interdependent

22. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Atomistic TFE: force evaluations v. time step Fix number of atoms, initial condition and total run duration 100 atom chain in 1d with pulse IC of width ~7 atoms Total time = 200 a/c 1nn linear spring interaction

23. Materials Computation Center, NSF DMR-03-25939 Kraczek, Miller, Johnson, Haber, Nov. 2, 2006 Atomistic TFE: Energy error Linear spring interaction allows exact integration of force ⇒ energy error for iterated solution is machine-precision noise