the materials computation center, university of illinois duane johnson and richard martin (pis), nsf...
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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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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.
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
Materials Computation Center, NSF DMR-03-25939Kraczek, Miller, Johnson, Haber, Nov. 2, 2006
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)
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
< Show movie, sample frame above >
Problem set-up:• IC: Heated by Gaussian pulse• Thermal BCs: insulated• Mechanical BCs: traction-free
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
t
x'
world lines of 2 displaced particles
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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 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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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 <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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
After1 pass
Materials Computation Center, NSF DMR-03-25939Kraczek, Miller, Johnson, Haber, Nov. 2, 2006
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
Materials Computation Center, NSF DMR-03-25939Kraczek, Miller, Johnson, Haber, Nov. 2, 2006
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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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
Materials Computation Center, NSF DMR-03-25939Kraczek, Miller, Johnson, Haber, Nov. 2, 2006
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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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)
vet
n0
n0
n0
Causal interface: Solution in Q depends on Q
Non-causal interface:Solution in Q and Q
interdependent
Materials Computation Center, NSF DMR-03-25939Kraczek, 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 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
Materials Computation Center, NSF DMR-03-25939Kraczek, Miller, Johnson, Haber, Nov. 2, 2006
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
Materials Computation Center, NSF DMR-03-25939Kraczek, 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