Multiscale Materials Modeling Ladder

Multiscale Ladder

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System size —

Simulated time —

Compute budget —

Estimated cost

Particles / cells

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Scaling

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Wall-clock on cluster

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Feasibility

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How it works

Computational materials science is a ladder of methods, each valid over a different span of length and time, because cost grows brutally with system size:

DFT:        cost ∝ N³   (electrons, N ≲ few 100 atoms)
MD:         cost ∝ N·T  (atoms + LJ pair forces, N ≲ 10⁶–10⁹)
Mesoscale:  cost ∝ M·T  (phase-field grid cells, coarse)
Continuum:  cost ∝ E·T  (finite-element mesh, coarsest)

Move the system size slider and the atom/cell count in each method scales with the box volume you chose; move simulated time and the number of steps scales with it. The wall-clock estimate multiplies those together with each method's per-step cost, using rough real numbers (DFT: ~10⁻³ s per SCF step per atom³ term at small N, MD: ~10⁻⁷ s per atom per step, mesoscale: ~10⁻⁸ s per cell per step). Raise the compute budget slider and the feasibility bar shows which method finishes inside it — this is exactly the tradeoff that forces multiscale workflows to hand off: run DFT to parameterize an interatomic potential, run MD on that potential to get diffusion or elastic constants, then feed those numbers into a phase-field or continuum model to reach engineering length and time scales no atomistic method could ever reach directly.

DFT — electron density isosurface around a handful of atoms; exact quantum mechanics, but cost explodes as N³ so it tops out around a few hundred atoms and picoseconds.
Molecular dynamics — Lennard-Jones particles integrated with Newton's laws; scales roughly linearly in atoms and steps, reaching micron boxes and nanoseconds.
Mesoscale / phase-field — a coarse grid of grains/domains evolving by curvature-driven or Cahn-Hilliard-like dynamics; each "cell" already represents thousands of atoms, reaching microns to millimeters and seconds.
Continuum / FEM — a deforming mesh with bulk elastic/thermal properties; cheapest of all, used from millimeters up to full parts.

DFT — electron density nucleus / atom electron density

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