8 Three-dimensional Geometry Setup

8.1 Overview

This chapter extends the geometric foundation to three dimensions, demonstrating how to build and visualize 3D domains using RayTraceHeatTransfer.jl. The 3D approach uses surface elements (faces) to define enclosed volumes, with each face serving as a radiative boundary that can emit, absorb, and reflect thermal radiation.

The chapter introduces two fundamental 3D geometries:

Tetrahedron: The simplest 3D enclosed volume, formed by four triangular faces
Cube: A regular hexahedron with six quadrilateral faces, providing validation against known solutions

Each geometry demonstrates the construction of 3D domains from surface elements and their visualization using interactive 3D plotting.

8.2 What You’ll Learn

Building 3D enclosed volumes from surface faces
Understanding face orientation and the right-hand rule for surface normals
Visualizing 3D geometries with interactive plots
Setting up 3D domains for radiative transfer calculations

8.3 Concepts

3D Domain Construction

3D domains are built from surface faces that enclose volumes. Each face is defined by:

Vertices: Corner points defining the face geometry (triangles or quadrilaterals)
Orientation: Following the right-hand rule for outward-pointing surface normals
Properties: Temperature, heat flux, and surface emissivity specifications

Convexity Requirement

For accurate radiative heat transfer calculations, 3D enclosures must be convex. A convex enclosure means that any line segment connecting two points inside the volume remains entirely within the volume. This geometric constraint ensures:

Accurate view factor calculations between all surface elements
Proper radiative exchange without geometric obstruction
Convergent analytical solutions using the exchange factor methods

While non-convex geometries can be constructed and meshed using these tools, they require different approaches for radiative heat transfer analysis that are beyond the scope of this analytical method.

Face Ordering

The vertex ordering of faces follows the right-hand rule: when fingers curl from the first vertex toward subsequent vertices, the thumb points in the outward normal direction. This ensures consistent surface normal orientations for accurate view factor calculations.

Material Properties

Each face can be assigned:

Known temperature (T_in_w) or unknown temperature (-1.0 for known heat flux)
Known heat flux (q_in_w) when temperature is unknown
Surface emissivity (epsilon) for thermal radiation properties

8.4 Examples

8.4.1 Tetrahedron Domain

The tetrahedron represents the simplest 3D enclosed volume, formed by four triangular faces. We’ll construct a regular tetrahedron with vertices at (0,0,0), (1,0,0), (0,1,0), and (0,0,1).

First, define the geometry:

using RayTraceHeatTransfer
using GLMakie
GLMakie.activate!()
Makie.inline!(true)

# Define tetrahedron vertices
points = [
   0.0 0.0 0.0;  # Vertex 1
   1.0 0.0 0.0;  # Vertex 2
   0.0 1.0 0.0;  # Vertex 3
   0.0 0.0 1.0   # Vertex 4
]

# Define face connectivity (following right-hand rule)
faces = [
   1 2 3;  # Bottom face
   1 2 4;  # Front face
   1 3 4;  # Left face
   2 3 4   # Right face
]

Next, set up the domain properties and build and mesh the domain:

# Mesh fineness
Ndim = 5  # subdivisions per edge

# Surface properties
epsilon = ones(size(faces, 1))  # All surfaces are black bodies
q_in_w = zeros(size(faces, 1))  # No prescribed heat fluxes
T_in_w = [1000.0, 0.0, 0.0, -1.0]  # Mixed boundary conditions

# Create 3D domain
domain3D_tet = Domain3D_faces(points, faces, Ndim, q_in_w, T_in_w, epsilon)

Finally, visualize the meshed geometry:

# Create dual-view visualization
fig = Figure(backgroundcolor=RGBf(1.0, 1.0, 1.0), size=(1600, 800))

# Create two LScenes side by side
ax1 = LScene(fig[1,1], scenekw=(camera=cam3d!, show_axis=true))
ax2 = LScene(fig[1,2], scenekw=(camera=cam3d!, show_axis=true))

# Plot mesh in both views
plotMesh3D(ax1, domain3D_tet)
plotMesh3D(ax2, domain3D_tet)

# Set up opposing camera positions
cam1 = cameracontrols(ax1.scene)
cam2 = cameracontrols(ax2.scene)

# Set opposing eyeposition
update_cam!(ax2.scene, cam1.eyeposition[] .* -1, cam1.lookat[], cam1.upvector[])

# Adjust the zoom for the right-hand view
new_eye_pos = cam2.eyeposition[] * 0.6  # Multiply by < 1 to zoom in
update_cam!(ax2.scene, new_eye_pos, cam2.lookat[], cam2.upvector[])

fig

3D tetrahedron mesh: front and back

8.4.2 Cube Domain

The cube provides a more complex 3D geometry with six quadrilateral faces, offering validation against well-known analytical solutions for cubic enclosures.

Define the cube geometry:

# Define cube vertices
points = [
    0.0 0.0 0.0; # vertex 1
    0.0 0.0 1.0; # vertex 2
    0.0 1.0 0.0; # vertex 3
    0.0 1.0 1.0; # vertex 4
    1.0 0.0 0.0; # vertex 5
    1.0 0.0 1.0; # vertex 6
    1.0 1.0 0.0; # vertex 7
    1.0 1.0 1.0  # vertex 8
]

# Define face connectivity (6 quadrilateral faces)
faces = [
    1 2 4 3;  # Front face
    5 6 8 7;  # Back face
    1 5 7 3;  # Bottom face
    2 6 8 4;  # Top face
    3 4 8 7;  # Left face
    1 2 6 5   # Right face
]

Set the domain properties and build and mesh the domain:

# Mesh fineness
Ndim = 5  # subdivisions per edge

# Surface properties
epsilon = ones(size(faces, 1))  # All surfaces are black bodies
q_in_w = zeros(size(faces, 1))  # No prescribed heat fluxes
T_in_w = [1000.0, 0.0, 0.0, 0.0, 0.0, 0.0]  # One hot wall, others cold

# Create 3D domain
domain3D_cube = Domain3D_faces(points, faces, Ndim, q_in_w, T_in_w, epsilon)

Visualize the meshed cube:

# Create dual-view visualization
fig = Figure(backgroundcolor=RGBf(1.0, 1.0, 1.0), size=(1600, 800))

# Create two LScenes side by side
ax1 = LScene(fig[1,1], scenekw=(camera=cam3d!, show_axis=true))
ax2 = LScene(fig[1,2], scenekw=(camera=cam3d!, show_axis=true))

# Plot mesh in both views
plotMesh3D(ax1, domain3D_cube)
plotMesh3D(ax2, domain3D_cube)

# Set up opposing camera positions
cam1 = cameracontrols(ax1.scene)
cam2 = cameracontrols(ax2.scene)

# Set opposing eyeposition
update_cam!(ax2.scene, cam1.eyeposition[] .* -1, cam1.lookat[], cam1.upvector[])

# Adjust the zoom for the right-hand view
new_eye_pos = cam2.eyeposition[] * 0.6  # Multiply by < 1 to zoom in
update_cam!(ax2.scene, new_eye_pos, cam2.lookat[], cam2.upvector[])

fig

3D cube mesh: front and back

8.5 Key Considerations

Face Orientation: Proper vertex ordering ensures outward-pointing surface normals, which are essential for accurate radiative calculations.

Mesh Resolution: The Ndim parameter controls surface subdivision density, affecting both accuracy and computational cost.

Boundary Conditions: Mixed temperature and flux boundary conditions enable modeling of realistic thermal systems with both prescribed temperatures and radiative equilibrium surfaces.

The 3D geometric framework established here provides the foundation for exchange factor calculations and analytical radiative transfer solutions in subsequent chapters.

# Three-dimensional Geometry Setup ```{julia} #| label: env #| include: false import Pkg # Activate the book project Pkg.activate(abspath(joinpath(@__DIR__, ".."))) # adjust if your cell lives deeper if get(ENV, "CI", "false") == "true" # --- CI on GitHub Pages: use newest *registered* version within [compat] --- # Keep RayTraceHeatTransfer in [deps] with a [compat] like: RayTraceHeatTransfer = "0.3" Pkg.instantiate() Pkg.update("RayTraceHeatTransfer") # float to latest allowed version else # --- Local dev: always track the latest main branch --- url = "https://github.com/NikoBiele/RayTraceHeatTransfer.jl" try Pkg.rm("RayTraceHeatTransfer"; force=true) catch end Pkg.add(Pkg.PackageSpec(name="RayTraceHeatTransfer", url=url, rev="main")) Pkg.instantiate() end using GLMakie GLMakie.activate!() Makie.inline!(true) using RayTraceHeatTransfer ``` ## Overview This chapter extends the geometric foundation to three dimensions, demonstrating how to build and visualize 3D domains using RayTraceHeatTransfer.jl. The 3D approach uses surface elements (faces) to define enclosed volumes, with each face serving as a radiative boundary that can emit, absorb, and reflect thermal radiation. The chapter introduces two fundamental 3D geometries: - **Tetrahedron**: The simplest 3D enclosed volume, formed by four triangular faces - **Cube**: A regular hexahedron with six quadrilateral faces, providing validation against known solutions Each geometry demonstrates the construction of 3D domains from surface elements and their visualization using interactive 3D plotting. ## What You'll Learn - Building 3D enclosed volumes from surface faces - Understanding face orientation and the right-hand rule for surface normals - Visualizing 3D geometries with interactive plots - Setting up 3D domains for radiative transfer calculations ## Concepts **3D Domain Construction** 3D domains are built from surface faces that enclose volumes. Each face is defined by: - **Vertices**: Corner points defining the face geometry (triangles or quadrilaterals) - **Orientation**: Following the right-hand rule for outward-pointing surface normals - **Properties**: Temperature, heat flux, and surface emissivity specifications **Convexity Requirement** For accurate radiative heat transfer calculations, 3D enclosures must be **convex**. A convex enclosure means that any line segment connecting two points inside the volume remains entirely within the volume. This geometric constraint ensures: - Accurate view factor calculations between all surface elements - Proper radiative exchange without geometric obstruction - Convergent analytical solutions using the exchange factor methods While non-convex geometries can be constructed and meshed using these tools, they require different approaches for radiative heat transfer analysis that are beyond the scope of this analytical method. **Face Ordering** The vertex ordering of faces follows the right-hand rule: when fingers curl from the first vertex toward subsequent vertices, the thumb points in the outward normal direction. This ensures consistent surface normal orientations for accurate view factor calculations. **Material Properties** Each face can be assigned: - Known temperature (T_in_w) or unknown temperature (-1.0 for known heat flux) - Known heat flux (q_in_w) when temperature is unknown - Surface emissivity (epsilon) for thermal radiation properties ## Examples ### Tetrahedron Domain The tetrahedron represents the simplest 3D enclosed volume, formed by four triangular faces. We'll construct a regular tetrahedron with vertices at (0,0,0), (1,0,0), (0,1,0), and (0,0,1). First, define the geometry: ```{julia} #| output: false using RayTraceHeatTransfer using GLMakie GLMakie.activate!() Makie.inline!(true) # Define tetrahedron vertices points = [ 0.0 0.0 0.0; # Vertex 1 1.0 0.0 0.0; # Vertex 2 0.0 1.0 0.0; # Vertex 3 0.0 0.0 1.0 # Vertex 4 ] # Define face connectivity (following right-hand rule) faces = [ 1 2 3; # Bottom face 1 2 4; # Front face 1 3 4; # Left face 2 3 4 # Right face ] ``` Next, set up the domain properties and build and mesh the domain: ```{julia} #| output: false # Mesh fineness Ndim = 5 # subdivisions per edge # Surface properties epsilon = ones(size(faces, 1)) # All surfaces are black bodies q_in_w = zeros(size(faces, 1)) # No prescribed heat fluxes T_in_w = [1000.0, 0.0, 0.0, -1.0] # Mixed boundary conditions # Create 3D domain domain3D_tet = Domain3D_faces(points, faces, Ndim, q_in_w, T_in_w, epsilon) ``` Finally, visualize the meshed geometry: ```{julia} #| warning: false #| fig-cap: "3D tetrahedron mesh: front and back" # Create dual-view visualization fig = Figure(backgroundcolor=RGBf(1.0, 1.0, 1.0), size=(1600, 800)) # Create two LScenes side by side ax1 = LScene(fig[1,1], scenekw=(camera=cam3d!, show_axis=true)) ax2 = LScene(fig[1,2], scenekw=(camera=cam3d!, show_axis=true)) # Plot mesh in both views plotMesh3D(ax1, domain3D_tet) plotMesh3D(ax2, domain3D_tet) # Set up opposing camera positions cam1 = cameracontrols(ax1.scene) cam2 = cameracontrols(ax2.scene) # Set opposing eyeposition update_cam!(ax2.scene, cam1.eyeposition[] .* -1, cam1.lookat[], cam1.upvector[]) # Adjust the zoom for the right-hand view new_eye_pos = cam2.eyeposition[] * 0.6 # Multiply by < 1 to zoom in update_cam!(ax2.scene, new_eye_pos, cam2.lookat[], cam2.upvector[]) fig ``` ### Cube Domain The cube provides a more complex 3D geometry with six quadrilateral faces, offering validation against well-known analytical solutions for cubic enclosures. Define the cube geometry: ```{julia} #| output: false # Define cube vertices points = [ 0.0 0.0 0.0; # vertex 1 0.0 0.0 1.0; # vertex 2 0.0 1.0 0.0; # vertex 3 0.0 1.0 1.0; # vertex 4 1.0 0.0 0.0; # vertex 5 1.0 0.0 1.0; # vertex 6 1.0 1.0 0.0; # vertex 7 1.0 1.0 1.0 # vertex 8 ] # Define face connectivity (6 quadrilateral faces) faces = [ 1 2 4 3; # Front face 5 6 8 7; # Back face 1 5 7 3; # Bottom face 2 6 8 4; # Top face 3 4 8 7; # Left face 1 2 6 5 # Right face ] ``` Set the domain properties and build and mesh the domain: ```{julia} #| output: false # Mesh fineness Ndim = 5 # subdivisions per edge # Surface properties epsilon = ones(size(faces, 1)) # All surfaces are black bodies q_in_w = zeros(size(faces, 1)) # No prescribed heat fluxes T_in_w = [1000.0, 0.0, 0.0, 0.0, 0.0, 0.0] # One hot wall, others cold # Create 3D domain domain3D_cube = Domain3D_faces(points, faces, Ndim, q_in_w, T_in_w, epsilon) ``` Visualize the meshed cube: ```{julia} #| warning: false #| fig-cap: "3D cube mesh: front and back" # Create dual-view visualization fig = Figure(backgroundcolor=RGBf(1.0, 1.0, 1.0), size=(1600, 800)) # Create two LScenes side by side ax1 = LScene(fig[1,1], scenekw=(camera=cam3d!, show_axis=true)) ax2 = LScene(fig[1,2], scenekw=(camera=cam3d!, show_axis=true)) # Plot mesh in both views plotMesh3D(ax1, domain3D_cube) plotMesh3D(ax2, domain3D_cube) # Set up opposing camera positions cam1 = cameracontrols(ax1.scene) cam2 = cameracontrols(ax2.scene) # Set opposing eyeposition update_cam!(ax2.scene, cam1.eyeposition[] .* -1, cam1.lookat[], cam1.upvector[]) # Adjust the zoom for the right-hand view new_eye_pos = cam2.eyeposition[] * 0.6 # Multiply by < 1 to zoom in update_cam!(ax2.scene, new_eye_pos, cam2.lookat[], cam2.upvector[]) fig ``` ## Key Considerations **Face Orientation**: Proper vertex ordering ensures outward-pointing surface normals, which are essential for accurate radiative calculations. **Mesh Resolution**: The Ndim parameter controls surface subdivision density, affecting both accuracy and computational cost. **Boundary Conditions**: Mixed temperature and flux boundary conditions enable modeling of realistic thermal systems with both prescribed temperatures and radiative equilibrium surfaces. The 3D geometric framework established here provides the foundation for exchange factor calculations and analytical radiative transfer solutions in subsequent chapters.