Time and spacetime finite element methods

1
Time and spacetime finite element methods for
atomistic, continuum and coupled simulations of
solids Students aBrent Kraczek,
bScott T. Miller, PIs
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
_at_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
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
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
Atomistic-continuum coupling

• Objective Develop coupling formalism for solid
  mechanics that
• Treats different scales
• with appropriate methods
• Allows refinement/coarsening of scales
• in both space and time
• Maintains compatibility and
• balance of momentum and energy
• Consistently handles thermal fields and/or
  changes in d.o.f.
• Is O(N) and parallelizable for dim1
• 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

5
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, 1953247-3273 (2006)

Problem Shock-loading of plate with crack at
middle (symmetry reduced to ¼ plate)
Figure shows mesh onlyphysical results reflected
in mesh refinement
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Spacetime FE (SDG) Flux balance laws

• Information passed between elements via flux
  conditions on M and e
• Flux-balance on M enforces linear momentum
  balance
• Flux-balance on e 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

Q
?Q
7
MCV Non-Fourier thermal model

• Thermal transfer at atomistic scale is through
  vibrationshyperbolic
• Standard heat equation based on Fouriers law is
  parabolic
• 1. ( Maxwell (1867), Cattaneo (1948), Vernotte
  (1958) )

• MCV1 modification to Fouriers law
• Yields hyperbolic heat equation
• Parameter t is relaxation time
• Appropriate for short time and/or length scales

• Fouriers law
• Yields parabolic heat equation
• Infinite propagation speed
• Appropriate in most cases

8
Spacetime FE for generalized thermoelasiticy

• 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

9
Thermoelastic problem Laser pulse heating
Laser pulse modeled as a Gaussian-type heat source

• Animation
• Color field shows temperature
• Height field shows velocity magnitude
• lt Show movie, sample frame above gt

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

10
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
Atomistic TFE Energy error
Problem Single particle in non-linear potential
well (Lennard-Jones oscillator) representative of
future MD use

• 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

12
Atomistic TFE Trajectory error
Problem Traveling pulse in 100 atom chain, w/
N-nn linear spring interaction

• Linear springs allow direct comparison with
  analytic solution
• Convergence rate for trajectory error in 100 atom
  chain is 2p
• (p polynomial order)

13
Coupled atomistic-continuum system

• Underlying mathematical model is time/spacetime
  FE
• Coupling time/spacetime methods through flux
  compatibility at GAC
• 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)

14
Coupled atomistic-continuum system

• Continuum compatibility relations (kinematic and
  momentum)

• v and s determined
• implicitly from values on
• both sides of interface.
• To supply flux conditions from atomistics,
• homogenize atomic velocities at boundary ltvgtA
• solve for forces on atoms as initially
  undetermined forces
• Momentum balanced explicitly Energy balance will
  depend on ltvgtA

15
Coupled system Results in 1d
Atomistic 200 atoms 54 dof
Continuum 40 elements 5x5 dof
Coupled 20 elements, 5x5 dof 100 atoms,
54 dof
Initial
After 1 pass
16
Coupled system Total energy error
A
B
C
Consider this configuration
A. Pulse begins in continuum region B. Pulse
fully in atomistic region C. Pulse fully in
continuum region

• Energy error reflects position of pulse in region

17
Coupled system Momentum balance
A. Pulse begins in continuum region B. Pulse
fully in atomistic region C. Pulse fully in
continuum region
C
B
A

• Total momentum 10-10
• Component momentum reflects pulse passing through
  coupling boundaries

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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
Blank

• Extra slides for explanation follow

20
Atomistic vs. continuum models of solids 1d

• Analogous continuum system
• r mass density,
• C elastic modulus
• lt no minimum scale gt
• L length
• Stress
• Continuum equation of motion
• Wave speed

• Atomistic mass-spring system
• m atomic mass,
• K atomic interaction (spring constant)
• a lattice spacing (interatomic distance)
• N masses -gt length LNa
• Force
• Atomistic equation of motion
• Wave speed (phase velocity)

r, C
ui1
ui-1
ui
21
Spacetime FE (SDG) Continuum fields

• Define
• Strain-velocity
• Stress-momentum
• M and e follow characteristics of wave equation
  allows causal meshing

Non-causal interface Solution in Qb and
Qa interdependent
Causal interface Solution in Qb depends on Qa
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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 implicitlycausal 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.

Non-causal interface Solution in Qb and
Qa interdependent
Causal interface Solution in Qb depends on Qa
23
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
24
Atomistic TFE Energy error
Linear spring interaction allows exact
integration of force ? energy error for
iterated solution is machine-precision noise