Calculations

Calculations

• Clean-up after an event

  Merges corners that met, removes edges folded onto each other as ridges, splits a loop a corner has cut, and closes loops with no area left.

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-skeleton.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Shared part of two folded edges

  The stretch two opposite edges on one line have in common, which becomes a ridge.

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-skeleton.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Face boundaries chained into polygons

  Joins the boundary pieces of a face into closed loops.

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-skeleton.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Collinear points removed from a face

  Drops points of a face polygon that lie on the line between their neighbors.

  removed when |u × v| ÷ max(|u|, |v|) ≤ tolerance

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-skeleton.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Roof lines joined end to end

  Joins the pieces of one ridge, hip or valley into a single line.

  joined when ends meet within tolerance and cos θ ≥ 1 − 1e-9

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-skeleton.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Faces kept

  Builds each face's plan polygon and drops slivers.

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-skeleton.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Polygon area

  The signed plan area of a polygon.

  A = ½ Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-skeleton.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Distance from a point to a segment

  The shortest distance from a plan point to a segment.

  t = clamp((p − a)·d ÷ |d|², 0, 1); distance = |p − a − d t|

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Eave line and edge speeds

  Moves each wall line out to the eave and sets how fast and from what height each roof edge rises.

  out = overhang − fascia thickness; speed = 1 ÷ tan(pitch); start = plate − out × tan(pitch)

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Slope and pitch of each face

  The slope and the pitch in degrees stored on each roof face.

  slope = 1 ÷ speed; pitch = atan(1 ÷ speed) in degrees

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Ridge, hip or valley

  Decides what each roof line is and measures its length.

  len = √(Δx² + Δy² + Δz²)

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Eave and rake edges

  Finds the face edges that lie on the eave line, tells level eaves from sloped rakes, and measures them.

  top height = face height + thickness ÷ cos(pitch)

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Soffit height

  The level of the soffit under each pitched eave.

  z_eave = plate − out × tan(pitch); z = min(z_eave, z_eave + thickness ÷ cos(pitch) − fascia)

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Roof area and line totals

  The roof's sloped area, its plan area, and the total length of ridges, hips, valleys, eaves and rakes.

  area = Σ |plan area of face| ÷ cos(pitch)

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Outline moved out by the overhangs

  The polygon with each edge moved out by its own distance.

  corner = p₀ + u_prev × ((p₁ − p₀) × u) ÷ (u_prev × u)

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Eave line crosses itself

  Checks that the eave polygon does not cross itself and runs anticlockwise.

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Roof height over a point

  The height of the roof's underside above a plan point.

  Roof faces, ridges and areas from an outline addons/trusscraft/lib/bim/res-roofplan.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Edges split where they cross or overlap

  Finds where two wall edges cross, touch or run along each other.

  t = (c − a) × s ÷ (r × s); u = (c − a) × r ÷ (r × s)

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Points matched on a grid

  Decides when two split points are the same point.

  key = (round(x ÷ 0.001), round(y ÷ 0.001))

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Boundary pieces

  Keeps the pieces that have a wall on one side and open space on the other.

  test points = middle ± n × min(0.5, length ÷ 4)

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Next piece at a joint

  Chooses which boundary piece a loop follows where several meet.

  a = (angle back − angle of piece), brought into (0, 2π]

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Room area and perimeter

  Tells rooms from islands and gives each room its net area and full perimeter.

  area = loop area − Σ hole areas; perimeter = loop + Σ hole perimeters

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Polygon area

  The signed area of a loop.

  A = ½ Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Polygon perimeter

  The length round a loop.

  P = Σ √(Δx² + Δy²)

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Polygon centroid

  The center of area of a loop.

  C = Σ (a + b)(a × b) ÷ (3 Σ a × b)

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Label point inside a room

  A point well inside a room for its label.

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Point inside a polygon

  Tests whether a plan point lies inside a polygon.

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Repeated and collinear points removed

  Tidies a loop before its area is taken.

  collinear when |ab × bc| ≤ 1e-7 × (|ab| |bc| + 1e-12)

  Rooms and building outline from walls addons/trusscraft/lib/bim/model-rooms.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • PartCraft
  64-bit

• Velocity beyond a wall

  The value used just outside the region so that the velocity at a wall equals the wall's own.

  u_ghost = 2 × u_wall − u_inside

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Value at a cell face

  The value carried across a cell face by the flow.

  central ½(a + b) when |u| h ÷ ν ≤ 2, else the upwind value

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Time step

  The largest stable time step for the current velocities.

  dt = min(0.4 h ÷ u_max, 0.2 h² ÷ ν, 0.8 ν ÷ u_max²)

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Momentum step for u

  The predicted x velocity at each face after one time step.

  u* = u + dt (−convection + ν ∇²u + source)

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Pressure equation

  Sets up the pressure equation that makes the velocity field divergence-free.

  ∇²p = (∇·u*) ÷ dt

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Pressure solve

  Solves the pressure equation by conjugate gradient.

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Velocity correction

  Removes the pressure gradient from the predicted velocities.

  u = u* − dt (p_right − p_left) ÷ h

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Tracer transport

  Moves a tracer with the flow and lets it diffuse, one time step.

  limiter φ(r) = (r + |r|) ÷ (1 + |r|); C′ = C − dt (∇·flux) + dt × source

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Run to a steady state

  Steps the flow until it stops changing or a limit is reached.

  change = max |u′ − u| ÷ dt

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Velocity at cell centers

  The velocity at the center of each cell.

  u_c = ½(u_left + u_right); v_c = ½(v_bottom + v_top)

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Velocity at a point

  The velocity at any point of the region.

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Stream function

  The stream function at the cell corners.

  ψ(i, j) = ψ(i, j − 1) + U(i, j − 1) × h_y

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• VTK export

  Writes velocity, pressure and tracer fields as a legacy ASCII VTK file.

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Mixing time

  The time at which a tracer's spread falls below a target.

  CV = √(Σ (c − mean)² ÷ n) ÷ mean

  2D flow solver (finite volume) cfd/fv.js

  • CadCraft
  64-bit

• Lattice directions and weights

  The nine directions of the lattice and the weight of each.

  w = 4/9, 1/9 (×4), 1/36 (×4)

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Mixer set-up

  Creates the grid, the solid mask, the force field and the starting distributions.

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Equilibrium distributions

  The equilibrium the flow and the tracer relax toward.

  f_eq = w ρ (1 + 3 c·u + 4.5 (c·u)² − 1.5 u²); g_eq = w C (1 + 3 c·u)

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Collision step

  Density, velocity and tracer value of each cell, and the relaxation of its distributions.

  ρ = Σ f; u = (Σ f c + ½ F) ÷ ρ; f ← f − (f − f_eq) ÷ τ + source

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Outlet condition

  Sets the last column of a flow-through box from its neighbor.

  f_out = f_eq(1, u_nb) + (f_nb − f_eq(ρ_nb, u_nb))

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Velocity at a point

  The velocity reported at a position in the grid.

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Result fields

  A copy of the velocity, speed, density or tracer field for reading or plotting.

  speed = √(u_x² + u_y²)

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Mixing time

  The number of steps until the tracer is evenly spread.

  deviation = max |C − mean|; reached when deviation ≤ level × starting deviation

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit