The Yee Lattice Foundations

Computational Wave Architects: Lesson 1

Mastering the spatial and temporal discretization of Maxwell's Equations for FDTD simulation.

Maxwell's Foundation

To simulate waves, we must discretize the time-dependent Maxwell's curl equations in isotropic, non-dispersive media:

\[ \frac{\partial \mathbf{H}}{\partial t} = -\frac{1}{\mu} \nabla \times \mathbf{E} \] \[ \frac{\partial \mathbf{E}}{\partial t} = \frac{1}{\epsilon} \nabla \times \mathbf{H} \]

The Problem:

How do we calculate derivatives on a computer?
How can we maintain accuracy while minimizing memory?
Solution: Finite-Difference Time-Domain (FDTD)

The Yee Cell: Spatial Staggering

Kane Yee's 1966 breakthrough: E and H components are staggered in space by half-grid cells.

E

Located on the edges of the cell

H

Located in the center of the faces

Why? Central-difference approximations become 2nd order accurate!

Spatial Dual-Grid Architecture

Leapfrog Integration: Temporal Staggering

E

t = n

H

t = n + 1/2

E

t = n + 1

// Update H field using previous E

H[n+1/2] = H[n-1/2] + (dt/μ) * (∇ × E[n])

// Update E field using new H

E[n+1] = E[n] + (dt/ε) * (∇ × H[n+1/2])

1D TE Formulation

Consider a wave traveling in Z, with \( E_x \) and \( H_y \):

E-Field Update

\[ E_x|_{k}^{n+1} = E_x|_{k}^{n} - \frac{\Delta t}{\epsilon \Delta z} \left( H_y|_{k+1/2}^{n+1/2} - H_y|_{k-1/2}^{n+1/2} \right) \]

"The future E depends on its current state and the spatial difference of the neighboring magnetic fields."

H-Field Update

\[ H_y|_{k+1/2}^{n+1/2} = H_y|_{k+1/2}^{n-1/2} - \frac{\Delta t}{\mu \Delta z} \left( E_x|_{k+1}^{n} - E_x|_{k}^{n} \right) \]

"The magnetic field updates half-way through the step, driven by the electric field gradient."

The Path Forward

Implement

Translate these equations into your preferred matrix-based language.

Discrete Grid

Define dx, dt, and the array sizes carefully. Remember the indexing shift.

Execute

Inject a Gaussian pulse and watch the birth of your first numerical wave.

"How do we chop space-time without breaking physics?"

Wave Engine Architects

The Yee Lattice Foundations

The Speed of Information

Infinite Space Simulation

Complex Material Modeling

Device Design Mastery

Yee Lattice Foundations Slides

Maxwell Discretization Worksheet

Yee Lattice Teacher Guide

Stability and Dispersion Slides

Stability and Dispersion Worksheet

Stability and Dispersion Teacher Guide

Infinite Space Slides

PML Implementation Worksheet

Infinite Space Teacher Guide

Complex Material Slides

ADE Material Worksheet

Complex Material Teacher Guide

Device Design Slides

Final Project Guide

Final Project Rubric

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