Derived Quantities Guide

Derived Quantities Guide

Overview

Derived quantities are computed from raw simulation output to extract physical meaning. This guide explains each quantity and its physical interpretation.

Volume Fraction

Definition: Fraction of domain with field value above/below threshold.

Formula: V_f = N_above / N_total

Physical Meaning:

• For order parameter φ ∈ [0,1]: fraction of phase with φ > 0.5
• For concentration: fraction above solubility limit
• For temperature: fraction above/below critical temperature

Usage:

python derived_quantities.py --input field.json --quantity volume_fraction \
    --field phi --threshold 0.5 --json

Interpretation:

• V_f = 0.5: Equal phase fractions (equilibrium for symmetric systems)
• V_f → 0 or 1: Predominantly single phase
• Time evolution of V_f: Phase transformation kinetics

Interface Area/Length

Definition: Surface area (3D) or perimeter (2D) of iso-surface at threshold.

Method: Marching squares/cubes approximation

• Count cells where threshold is crossed
• Multiply by average cell size

Physical Meaning:

• Interfacial area controls reaction/diffusion rates
• Decreases during coarsening (curvature-driven)
• Increases during nucleation/growth

Usage:

python derived_quantities.py --input field.json --quantity interface_area \
    --field phi --threshold 0.5 --json

Interpretation:

• A(t) ∝ t^(-1/3): Normal coarsening (2D)
• A(t) ∝ t^(-1/2): Interface-controlled growth
• Sudden increase: New phase nucleation

Gradient Magnitude

Definition: |∇φ| = √[(∂φ/∂x)² + (∂φ/∂y)² + ...]

Method: Central finite differences

• Interior: (φᵢ₊₁ - φᵢ₋₁) / (2Δx)
• Boundaries: Forward/backward differences

Physical Meaning:

• High gradients: Sharp interfaces, steep concentration profiles
• Low gradients: Smooth fields, equilibrium regions
• Maximum gradient: Interface width indicator

Usage:

python derived_quantities.py --input field.json --quantity gradient_magnitude \
    --field phi --json

Interpretation:

• Max|∇φ| ≈ 1/η for phase-field (η = interface width)
• Mean|∇φ| decreases during coarsening
• High gradients in wrong regions: Numerical instability

Integral / Total Mass

Definition: ∫φ dV ≈ Σφᵢ × ΔV

Physical Meaning:

• Total amount of conserved quantity
• Should be constant for Cahn-Hilliard dynamics
• May change for Allen-Cahn (non-conserved)

Usage:

python derived_quantities.py --input field.json --quantity integral \
    --field concentration --json

Conservation Check:

• |M(t) - M(0)| / M(0) < 10⁻¹⁰: Well-conserved
• Drift > 10⁻⁶: Possible discretization issue

Total Variation

Definition: TV(φ) = ∫|∇φ| dV ≈ Σ|φᵢ₊₁ - φᵢ|

Physical Meaning:

• Measures total "roughness" of field
• Related to interface area for binary fields
• Regularization term in image processing

Usage:

python derived_quantities.py --input field.json --quantity total_variation \
    --field phi --json

Interpretation:

• TV decreases: Smoothing/coarsening
• TV increases: Sharpening/nucleation
• Constant TV: Steady state

Centroid

Definition: Center of mass weighted by field values

Formula:

• x_c = ∫x φ dV / ∫φ dV
• y_c = ∫y φ dV / ∫φ dV

Physical Meaning:

• Center of precipitate or inclusion
• Drift indicates asymmetric growth
• Multiple centroids for multi-particle systems

Usage:

python derived_quantities.py --input field.json --quantity centroid \
    --field phi --json

Quantity Selection Guide

Physical Units

All quantities assume consistent units based on grid spacing: