multiscale-modeling

name	multiscale-modeling
description	Multiscale modeling — length scales (nano to macro), molecular dynamics (LAMMPS), DFT (VASP/Quantum ESPRESSO), representative volume element (RVE), homogenization, concurrent vs. hierarchical coupling, FEA-MD coupling (Quasicontinuum), crystal plasticity (CPFEM), micromechanics (Mori-Tanaka, Eshelby), CALPHAD thermodynamics.
metadata	{"priority":7,"promptSignals":{"phrases":["multiscale modeling","molecular dynamics simulation","crystal plasticity","homogenization","RVE representative volume element","DFT materials"],"minScore":3}}

Multiscale Modeling in Mechanics — Complete Skill

Length Scale Hierarchy

Scale levels in materials science:

Scale	Size	Methods	Information
Quantum/Electronic	0.1–10 nm	DFT, tight-binding	Electronic structure, bonding
Atomistic	1–100 nm	MD, Monte Carlo	Defects, interfaces, diffusion
Dislocation/Crystal	100 nm–10 μm	DD, CPFEM	Plastic slip, grain effects
Micro/Meso	10–1,000 μm	RVE-homogenization	Phase distribution, composite
Macro/Continuum	> 1 mm	FEA, continuum	Structural response

Coupling strategies: Sequential (hierarchical): scale N informs scale N+1 via parameters (most common; no bidirectional) Concurrent: regions solved simultaneously at different scales; expensive; used at crack tips (QC method) Information passing: MD → Potential parameters → Mesoscale → Constitutive laws → FEA

Density Functional Theory (DFT)

Physics

Kohn-Sham equations: [-ℏ²/2m ∇² + V_ext(r) + V_H(r) + V_xc(r)] φ_i(r) = ε_i φ_i(r) [single-electron Schrödinger analog] V_H = Hartree potential (electron-electron); V_xc = exchange-correlation (many-body approx)

Exchange-correlation functional: GGA (Generalized Gradient Approximation): most common; PBE functional (Perdew-Burke-Ernzerhof) LDA (Local Density Approximation): simpler; less accurate for surfaces and molecules Hybrid (HSE06): includes exact exchange → better bandgaps; more expensive

Output: total energy E, electron density ρ(r), forces on atoms, elastic constants, band structure

Computational cost: DFT scales as O(N³) with number of electrons N; N > 1,000 atoms → prohibitive without special methods Linear-scaling DFT: O(N) methods exist; less accurate

DFT Software

VASP (Vienna Ab initio Simulation Package): most widely used; plane-wave basis; licensed; excellent documentation Quantum ESPRESSO: open source; plane-wave DFT; community support ABINIT: open source; all-electron options; response functions SIESTA: localized basis; linear scaling; good for large biomolecules

Key DFT inputs: KPOINTS: k-point mesh (Monkhorst-Pack); denser mesh → more accurate; converge energies vs. mesh Plane-wave cutoff: ENCUT ≥ 1.3 × ENMAX (VASP); higher → more accurate; converge total energy Pseudopotentials/PAW potentials: from VASP database; POTCAR file

DFT deliverables for multiscale: Elastic constants (C_ij) → macroscale material properties Stacking fault energies → dislocation mobility models Grain boundary energies → microstructure evolution Point defect energies → diffusion coefficients

Molecular Dynamics (MD)

Theory

Classical equations of motion: m_i × r̈_i = -∂U/∂r_i [Newton; U = potential energy = Σ pair/many-body interactions]

Interatomic potentials: EAM (Embedded Atom Method): for metals; F(ρ_e) + 0.5 × Σ φ(r_ij) [density + pair; works well for FCC/BCC metals] Tersoff/REBO: covalent materials (C, Si, Ge); environment-dependent bond order ReaxFF: reactive; bonds can form/break; for reactive simulations (oxidation, crack, combustion) LAMMPS AIREBO: hydrocarbons; more accurate than Tersoff for C-H bonds

Integration: Velocity-Verlet: r_{n+1} = r_n + v_n Δt + F_n/(2m) Δt²; v_{n+1} = v_n + (F_n + F_{n+1})/(2m) Δt Time step Δt: 1–5 fs for metal; 0.1–0.5 fs for reactive/ReaxFF; must be < 1/10 of fastest vibration period

Ensembles: NVE (microcanonical): constant N, V, E; pure dynamics; no thermostat NVT (canonical): constant N, V, T; Nosé-Hoover thermostat; damps to target T NPT (isothermal-isobaric): constant N, P, T; Parrinello-Rahman barostat; for materials at P and T

MD Applications for Multiscale

Elastic constants from MD: Deformation-based: apply strain → compute stress → fit C_ij = σ_ij / ε_kl Fluctuation method: C_ij from strain fluctuations in NPT ensemble Target: agreement within 10–20% of DFT → validates potential quality

Dislocation mobility: Apply stress → track dislocation velocity v_d vs. applied stress σ → fit v_d = M × (σ - σ_Peierls)^m Pass M (mobility) and m (exponent) to crystal plasticity model

Interface properties: Work of separation W_sep = 2 × γ_surface - γ_interface [for crack on interface] W_sep input to cohesive zone model in macro FEA for delamination

Crack tip atomistics (concurrent): MD core region around crack tip + continuum elasticity far field CADD (Coupled Atomistic and Discrete Dislocation) or Quasicontinuum method

LAMMPS Setup Example

# LAMMPS input for NiAl EAM tension test
units metal
boundary p p p  # periodic
atom_style atomic

# Read geometry
read_data nickel_Al.dat

# Pair style
pair_style eam/alloy
pair_coeff * * NiAlH_jea.eam.alloy Ni Al

# Equilibrate at 300 K
velocity all create 300 87287 dist gaussian
fix 1 all npt temp 300 300 0.1 iso 0 0 1.0
run 10000

# Deform to compute elastic modulus
fix 2 all deform 1 x erate 1e-9  # strain rate 1e-9 /fs
compute stress all stress/atom NULL
variable press equal -(c_stress[1]+c_stress[2]+c_stress[3])/(3*vol)
run 100000

Crystal Plasticity Finite Element Method (CPFEM)

Kinematics

Multiplicative decomposition: F = F_e × F_p [deformation gradient F = elastic × plastic; multiplicative, not additive] Velocity gradient: L = Ḟ × F⁻¹ = L_e + F_e × L_p × F_e⁻¹

Plastic velocity gradient: L_p = Σ_α γ̇_α × (m_α ⊗ n_α) [α = slip system; γ̇_α = slip rate; m_α = slip direction; n_α = slip plane normal]

Slip rate (power-law flow rule): γ̇_α = γ̇₀ × |τ_α / τ_c_α|^(1/m) × sign(τ_α) [τ_α = resolved shear stress; τ_c_α = critical resolved shear stress; m = rate sensitivity; γ̇₀ = reference rate]

Hardening: τ̇_c_α = Σ_β h_αβ × |γ̇_β| [h_αβ = hardening matrix; h_αα = self-hardening; h_αβ (α≠β) = latent hardening] h_αα = h₀ × sech²(h₀ × γ / (τ_sat - τ₀)) [Voce hardening; h₀, τ_sat, τ₀ = fit to experiment]

Slip systems: FCC (Cu, Al, Ni): {111}<110> slip; 12 slip systems BCC (Fe, W): {110}<111>; 12 slip systems + {112}<111>; 12 = 24 total HCP (Ti, Mg): {0001}<11-20> basal; {10-11}<1-10> prismatic; pyramidal; complex

CPFEM Software

DAMASK (Düsseldorf Advanced Material Simulation Kit): open source; spectral solver (FFT-based); very efficient for periodic microstructures Abaqus UMAT: user subroutine for crystal plasticity; flexible; integrated with Abaqus FEA DREAM.3D: microstructure generation → inputs to CPFEM

Representative Volume Element (RVE) and Homogenization

RVE Concept

RVE (Representative Volume Element): Smallest volume element that statistically represents microstructure Contains: sufficient grains, phases, defects to capture mean and variance RVE size rule: L_RVE ≥ 10 × d_grain (diameter) for reliable averages (statistical RVE)

Homogenization: compute effective properties from microstructure C_eff = ⟨C(x)⟩ + correction [effective stiffness from micro field average]

Voigt and Reuss Bounds

Voigt (upper bound — equal strain): C_Voigt = Σ f_i × C_i [f_i = volume fraction; linear average of stiffness] Assumes equal strain in all phases → stiffer estimate

Reuss (lower bound — equal stress): S_Reuss = Σ f_i × S_i; C_Reuss = S_Reuss⁻¹ [average compliance; stiffer phases carry less load] True effective stiffness: C_Reuss ≤ C_eff ≤ C_Voigt

Hashin-Shtrikman bounds: tighter bounds using variational principles; require only volume fractions + phase properties

Mori-Tanaka (MT) Method

For ellipsoidal inclusions in matrix: C_eff = C_m + f_i × (C_i - C_m) × (I + S × C_m⁻¹ × (C_i - C_m))⁻¹ × A_MT S = Eshelby tensor (depends on inclusion shape); A_MT = concentration tensor

Eshelby tensor for sphere: S_1111 = (7 - 5ν_m) / (15(1 - ν_m)) [for isotropic matrix] Sphere: S isotropic; fiber: S anisotropic

MT accuracy: Good for: f_i < 0.4 (inclusion volume fraction); random orientation Poor for: high f_i; touching inclusions; platelet geometries

Computational Homogenization (FEA on RVE)

Approach: Apply macroscopic strain ε_macro as boundary conditions on RVE Solve RVE with periodic boundary conditions (PBC) → get micro-fields Average stress: σ_macro = ⟨σ(x)⟩ = (1/V) ∫ σ(x) dV Extract C_eff from σ_macro = C_eff × ε_macro for 6 load cases

Periodic boundary conditions: u(x + L) - u(x) = ε_macro × L [periodicity; L = RVE edge vector] Ensures macroscopic strain applied without artificial boundary effects

CALPHAD (Thermodynamics)

CALPHAD (CALculation of PHAse Diagrams): Compute thermodynamic equilibrium phases from first principles + experimental databases THERMO-CALC, Pandat (commercial software)

Gibbs energy: G = G° + ΔH_mix + T × ΔS_mix [mixing terms from interaction parameters] Phase equilibrium: minimize total G of multi-component system → stable phases and compositions

Application to multiscale: determine microstructure composition and phases → input to RVE construction → input to CPFEM ICME (Integrated Computational Materials Engineering): CALPHAD → microstructure → CPFEM → continuum FEA → product design

Standards and References

Source	Scope
Feyel & Chaboche (2000) Comp. Meth. Appl. Mech. Eng.	FE² concurrent multiscale
Asaro & Lubarda "Mechanics of Solids and Materials"	Crystal plasticity reference
DAMASK documentation	damask.mpie.de
LAMMPS documentation	lammps.sandia.gov
Kohn & Sham (1965) Phys. Rev.	DFT formulation
Eshelby (1957) Proc. Roy. Soc.	Inclusion problem

Output

Provide: scale of interest (electronic/atomistic/mesoscale/continuum), coupling strategy (sequential/concurrent/information passing), DFT setup (if applicable: code VASP/QE, functional PBE, k-point mesh, cutoff eV, output elastic C_ij), MD setup (if applicable: code LAMMPS, potential EAM/ReaxFF, ensemble NVT/NPT, timestep [fs], T [K], strain rate [/s]), CPFEM setup (if applicable: slip system count, hardening law, rate sensitivity m, τ₀ and τ_sat [MPa]), RVE parameters (size [μm], grain count, f_i [volume fractions], PBC vs. homogeneous BC), homogenization method (Voigt/Reuss bounds [GPa], Mori-Tanaka C_eff [GPa], computational C_eff [GPa from FEA on RVE]), effective properties passed to next scale, and applicable reference (Feyel 2000, Asaro & Lubarda, LAMMPS/VASP docs).