The Materials Computation Center, University of Illinois Duane Johnson and Richard Martin (PIs), NSF DMR-03-25939 Time and spacetime finite.

The Materials Computation Center, University of IllinoisDuane 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: aBrent Kraczek, bScott T. Miller, PI’s : a,cDuane D. Johnson, bRobert B. Haber,

University of Illinois at Urbana-Champaign,Departments of aPhysics, bMechanical Science and Engineering, and

cMaterials 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.

Materials Computation Center, NSF DMR-03-25939Kraczek, 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


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.


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


Continuum formulation: Spacetime finite elements

• Spacetime discontinuous Galerkin (SDG) finite element (FE) method1

• 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)


• 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


• 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

MCV1 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


• 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


Thermoelastic problem: Laser pulse heating

Laser pulse modeled as a Gaussian-type heat source

Animation: • Color field shows temperature• Height field shows velocity magnitude

< Show movie, sample frame above >

Problem set-up:• IC: Heated by Gaussian pulse• Thermal BCs: insulated• Mechanical BCs: traction-free


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

t

x'

world lines of 2 displaced particles


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


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


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 1st 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)

)1( td ⊗

t2

t1

AC

Model system

Continuumregion

Atomisticregion

v*

v C

C<v>A

FA


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 <v>A – solve for forces on atoms as initially undetermined forces

• Momentum balanced explicitly; Energy balance will depend on <v>A

)1( td ⊗

t2

t1

AC

Model system

Continuumregion

Atomisticregion

v*

v C

C<v>A

FA


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

After1 pass


Coupled system: Total energy error

• Energy error reflects position of pulse in region

Consider this configuration

A BC

A. Pulse begins in continuum regionB. Pulse fully in atomistic regionC. Pulse fully in continuum region


Coupled system: Momentum balance

• Total momentum ~10-10

• Component momentum reflects pulse passing through coupling boundaries

A B CA. Pulse begins in continuum regionB. Pulse fully in atomistic regionC. Pulse fully in continuum region


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


Analogous continuum system

= mass density, C = elastic modulus< no minimum scale >L = length

Stress

Continuum equation of motion

Wave speed

xxx uC

u ,,

1

==&&

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)

ui ui+1ui-1

C

ii uKF Δ−= xCu

x

uC ,=

∂∂

=

( ) ( )[ ]

⎟⎠

⎞⎜⎝

⎛ +−⋅=

−+−=

−+

−+

211

11

2

/

a

uuu

am

Ka

uuuum

Ku

iii

iiii&&

N

n

ak

ka

km

Kc

π2,

2sin

2=⋅=

m

Ka

Cc ==

am K


• 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)

vet

n0

n0

n0

Causal interface: Solution in Q depends on Q

Non-causal interface:Solution in Q and Q

interdependent


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 progressR. 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


Atomistic TFE: force evaluations v. time step

Fix number of atoms, initial condition and total run duration100 atom chain in 1d with pulse IC of width ~7 atomsTotal time = 200 a/c1nn linear spring interaction


Atomistic TFE: Energy error

Linear spring interaction allows exact integration of force ⇒ energy error for iterated solution is machine-precision noise

The Materials Computation Center, University of Illinois Duane Johnson and Richard Martin (PIs), NSF DMR-03-25939 Time and spacetime finite.

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