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MathidiaMathematics Academy
Advanced mathematical modelling
Lesson 61 of 75
Introduction

Multiscale modelling

Objective. Connect behaviour across temporal and spatial scales.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Multiscale modelling connects behaviour across different temporal or spatial scales, such as linking fast microscopic dynamics to slow macroscopic behaviour, often via homogenisation that replaces rapidly varying microstructure with an effective averaged parameter. For example, a material with periodic microstructure of conductivity alternating between k₁ and k₂ can be replaced macroscopically by an effective conductivity, such as the harmonic mean for layers in series: k_eff = 2k₁k₂/(k₁+k₂). This effective parameter lets a coarse simulation reproduce the material's bulk response without resolving every microscopic layer, provided the scales are well separated.

Worked example

In a multiscale model of a layered material, the microscale conductivities are k₁ = 4 and k₂ = 12 arranged in series. Using the effective conductivity formula k_eff = 2k₁k₂/(k₁+k₂), find the homogenised macroscopic conductivity.

4122
Bar chart of the values 4, 12, 2
  1. 1Compute the numerator 2k₁k₂ = 2 × 4 × 12 = 96.
  2. 2Compute the denominator k₁+k₂ = 4 + 12 = 16.
  3. 3Divide: k_eff = 96/16.
  4. 4Simplify 96/16 = 6, the effective macroscopic conductivity.

Answer: 6

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Common mistake. A common mistake is applying a homogenised effective parameter when the microscale and macroscale are not well separated, which gives inaccurate predictions because fine-scale fluctuations still significantly affect the coarse-scale behaviour.

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