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Examples

Ready-to-follow tutorials demonstrating common Elementa workflows.

Example 1 — Parallel Plate Capacitor (2-D Electrostatics)

Goal: Compute the electric potential and field between two parallel conducting plates separated by vacuum.

Setup

Parameter Value
Plate width 0.1 m
Plate height 0.01 m
Gap 0.02 m
Voltage ±50 V

Steps

  1. New project → 2D, Electrostatics, Stationary.
  2. Parameters: W=0.1, H=0.01, gap=0.02.
  3. Geometry:
  4. Rectangle domain: dx=0.15, dy=0.1, cx=0, cy=0
  5. Rectangle plate_top: dx=W, dy=H, cx=0, cy=gap/2+H/2
  6. Rectangle plate_bot: dx=W, dy=H, cx=0, cy=-gap/2-H/2
  7. Mesh: element size 0.004.
  8. Boundary Conditions:
  9. plate_top boundaries → Electric Potential, value = 50
  10. plate_bot boundaries → Electric Potential, value = -50
  11. outer domain boundaries → Ground
  12. Solve.
  13. Results: surface plot of φ, arrow plot of E.

Expected Result

The potential varies linearly between the plates. The electric field magnitude in the gap is approximately:

\[|E| \approx \frac{\Delta V}{d} = \frac{100 \text{ V}}{0.02 \text{ m}} = 5000 \text{ V/m}\]

Example 2 — Heat Conduction Through a Wall (2-D Heat Transfer)

Goal: Compute the steady-state temperature distribution through a composite wall with fixed temperatures on each face.

Setup

Parameter Value
Wall thickness 0.3 m
Wall height 1.0 m
Inner temperature 293 K (20 °C)
Outer temperature 253 K (−20 °C)
Thermal conductivity 0.04 W/(m·K) (insulation)

Steps

  1. New project → 2D, Heat Transfer, Stationary.
  2. Parameters: L=0.3, H=1.0, k=0.04.
  3. Geometry: Rectangle wall: dx=L, dy=H, cx=0, cy=0.
  4. Physics config: set Thermal Conductivity to k.
  5. Mesh: element size 0.02.
  6. Boundary Conditions:
  7. Left edge (wall_edge_3) → Temperature, value = 293.15
  8. Right edge (wall_edge_1) → Temperature, value = 253.15
  9. Top/bottom edges → Thermal Insulation
  10. Solve.
  11. Results: surface plot of T, arrow plot of heat flux q.

Expected Result

A linear temperature gradient from 293 K to 253 K across the wall. Heat flux:

\[q = k \frac{\Delta T}{L} = 0.04 \times \frac{40}{0.3} \approx 5.3 \text{ W/m}^2\]

Example 3 — Transient Heating of a Block (2-D Heat Transfer, Time Dependent)

Goal: Simulate heating of a solid block from an initial temperature of 20 °C when one face is held at 100 °C.

Steps

  1. New project → 2D, Heat Transfer, Time Dependent.
  2. Parameters: L=0.1, rho=2700, cp=900, k=200 (aluminium).
  3. Geometry: Square block block: dx=L, dy=L.
  4. Physics config:
  5. Thermal Conductivity: k
  6. Density: rho
  7. Heat Capacity: cp
  8. Study → Time Dependent: t_start=0, t_end=100, dt=5.
  9. Boundary Conditions:
  10. Left face → Temperature, value = 373.15 (100 °C)
  11. Others → Thermal Insulation
  12. Solve.
  13. Results: Use the time slider in the plot panel to animate the temperature field evolution.

Example 4 — Sphere with Surface Charge (3-D Electrostatics)

Goal: Verify the analytical electric potential outside a uniformly charged sphere.

Steps

  1. New project → 3D, Electrostatics, Stationary.
  2. Parameters: R=0.05, R_ext=0.2.
  3. Geometry:
  4. Sphere inner: r=R
  5. Sphere outer: r=R_ext
  6. Boolean Difference domain = outer − inner
  7. Boundary Conditions:
  8. Inner sphere surface → Surface Charge Density, value = 1e-6
  9. Outer sphere surface → Ground
  10. Mesh: element size 0.02.
  11. Solve.
  12. Results: surface plot of φ, verify against Coulomb's law.