Calculations

Calculations

• Spalding's law of the wall

  y⁺ as a function of u⁺.

  y⁺ = u⁺ + e^(−κB) (e^(κu⁺) − 1 − κu⁺ − (κu⁺)² ÷ 2 − (κu⁺)³ ÷ 6)

  Flow study: turbulence model and wall functions flow-turb.js

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• Friction velocity

  Solves the wall law for the friction velocity from the speed at a distance from the wall.

  find u_τ with spalding(U ÷ u_τ) = y u_τ ÷ ν

  Flow study: turbulence model and wall functions flow-turb.js

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• Inverse wall law

  u⁺ as a function of y⁺.

  Flow study: turbulence model and wall functions flow-turb.js

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• Wall cells: normal and wall distance

  Each wall cell's wall normal and its true distance to the wall, and the model's starting fields.

  n = Σ wᵢ nᵢ ÷ |Σ wᵢ nᵢ|; y = Σ wᵢ c_n (q c_n) ÷ Σ wᵢ c_n over links with c_n > 0.3, y ≥ 0.05

  Flow study: turbulence model and wall functions flow-turb.js

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• Wall distance of every cell

  The distance from each fluid cell to the nearest wall point.

  d(m) = |v(n) − c| carried from a settled neighbor n along the link c

  Flow study: turbulence model and wall functions flow-turb.js

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• Model update in a cell

  The wall function in wall cells and one pseudo-time step of the model equation elsewhere.

  ν̃_new = (ν̃ + Δt × rhs) ÷ (1 + Δt c_w1 f_w ν̃ ÷ d²); wall cell: ν̃ = κ u_τ y

  Flow study: turbulence model and wall functions flow-turb.js

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• Pseudo-time step

  The model's time step from the advection and diffusion limits.

  Δt = max(0.05, min(every, 0.5 ÷ u_max, 0.8 σ ÷ (6 (ν + ν̃_max))))

  Flow study: turbulence model and wall functions flow-turb.js

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• First-cell y⁺ statistics

  The minimum, maximum, mean and percentiles of y⁺ over the wall cells.

  percentile p = sorted[⌊p × n⌋]

  Flow study: turbulence model and wall functions flow-turb.js

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• Wall force correction for slip

  The term added to the momentum-exchange force on walls that carry a slip velocity.

  F += Σ u_w (f_in − f_out)

  Flow study: turbulence model and wall functions flow-turb.js

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• Streamlines

  Traces lines along the flow from seed points for drawing.

  x ← x + ds × v(x + ½ ds v̂) ÷ |v|

  Simulation and flow result plots flow.js

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• Slice of a field

  Copies one layer of cells of a field for the cut-plane picture.

  Simulation and flow result plots flow.js

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• Field for coloring

  Picks the values and the color range for the cut plane.

  Simulation and flow result plots flow-ui.js

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• Plot mesh

  Splits the part's triangles until no edge is longer than the cell size, for drawing a field.

  midpoint = (a + b) ÷ 2

  Simulation and flow result plots fea.js

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• One subdivision step

  New face, edge and vertex points of the cage refined once.

  smooth vertex = Q ÷ n + 2R ÷ n + (n − 3) v ÷ n; crease vertex = (a + 6v + b) ÷ 8; edge = mean of its ends and the two face points

  Freeform bodies: subdivision surface and cage edits form.js

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• Limit position of a vertex

  Where a vertex ends up on the smooth surface.

  smooth: (n² v + 4 Σ e + Σ f) ÷ (n (n + 5)); crease: (a + 4v + b) ÷ 6; corner: v

  Freeform bodies: subdivision surface and cage edits form.js

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• Limit surface normal at a vertex

  The unit normal of the smooth surface at an interior vertex.

  A = 1 + cos(2π ÷ n) + cos(π ÷ n) √(2 (9 + cos(2π ÷ n))); normal = t₁ × t₂ ÷ |t₁ × t₂|

  Freeform bodies: subdivision surface and cage edits form.js

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• Bézier patch of one quad

  The 16 control points of the bicubic patch for one face of the refined cage.

  b = M G Mᵀ with rows of M: (1, 4, 1, 0) ÷ 6, (0, 4, 2, 0) ÷ 6, (0, 2, 4, 0) ÷ 6, (0, 1, 4, 1) ÷ 6; across a sharp edge: 2 × edge − inner

  Freeform bodies: subdivision surface and cage edits form.js

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• Point on a patch

  Evaluates a bicubic Bézier patch at (u, v).

  S(u, v) = Σᵢ Σⱼ Bᵢ(u) Bⱼ(v) b_ij, B = ((1 − u)³, 3u(1 − u)², 3u²(1 − u), u³)

  Freeform bodies: subdivision surface and cage edits form.js

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• Normal of a patch

  The unit normal of a patch at (u, v).

  n = ∂S/∂u × ∂S/∂v, normalized

  Freeform bodies: subdivision surface and cage edits form.js

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• Refinement level for conversion

  How many times the cage is refined before it is turned into patches.

  level = max(1 if any face is not a quad, ⌈largest semi-sharp weight − 1e-12⌉), at most 4

  Freeform bodies: subdivision surface and cage edits form.js

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• How well the patches meet

  The largest gap and the largest angle between normals across shared patch edges.

  gap = max |p₀ − p₁|; angle = acos(clamp(n₀ · n₁, −1, 1)) × 180 ÷ π

  Freeform bodies: subdivision surface and cage edits form.js

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• Fitting a cage to a target surface

  Moves cage vertices until their limit points lie on a sphere or torus.

  p ← p + (proj(L) − L), L the vertex's limit point

  Freeform bodies: subdivision surface and cage edits form.js

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• Torus cage

  The vertices of a torus-shaped starting cage.

  ((R + r cos b) cos a, r sin b, (R + r cos b) sin a), a = 2π i ÷ n, b = 2π j ÷ m

  Freeform bodies: subdivision surface and cage edits form.js

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• Mirror partners

  Finds each vertex's partner across the symmetry plane.

  q = p mirrored; partner when |v − q| ≤ 1e-6 × size

  Freeform bodies: subdivision surface and cage edits form.js

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• Rebuilding the mirrored half

  Keeps one side of the symmetry plane and mirrors it.

  mirror: x' = 2 × at − x

  Freeform bodies: subdivision surface and cage edits form.js

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• Moving, rotating and scaling vertices

  Applies a 4 × 4 transform to the selected vertices, with an optional soft falloff and mirrored motion.

  p' = p + w (M p − p); soft weight w = (1 − (d ÷ R)²)²

  Freeform bodies: subdivision surface and cage edits form.js

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• Extruding faces

  Moves a region of faces along its averaged normals and adds side walls.

  new vertex = v + dist × (Σ face normals at v) ÷ |Σ face normals at v|

  Freeform bodies: subdivision surface and cage edits form.js

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• Face normal

  The area-weighted normal of a cage face with any number of corners.

  n_x = Σ (p_y − q_y)(p_z + q_z), and cyclic

  Freeform bodies: subdivision surface and cage edits form.js

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• Flattening vertices onto a plane

  Projects the selected vertices onto a given plane or their best-fit plane.

  p' = p − n ((p − o) · n)

  Freeform bodies: subdivision surface and cage edits form.js

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• Best-fit plane normal

  The direction in which a set of centered points varies least.

  A = Σ p pᵀ; normal = eigenvector of the smallest eigenvalue

  Freeform bodies: subdivision surface and cage edits form.js

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• Adding a face through points

  Adds a face, reusing existing vertices that are close to the given points.

  reuse a vertex when |q − p| ≤ 1e-6

  Freeform bodies: subdivision surface and cage edits form.js

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• Bridging two loops

  Joins two loops of equal size with rows of quads.

  offset = argmin Σ |aᵢ − b₍offset − i₎|²; row vertex = a + (b − a) × r ÷ segments

  Freeform bodies: subdivision surface and cage edits form.js

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• Smooth surface for display

  The limit positions of the refined cage as a triangle mesh.

  Freeform bodies: display mesh form.js

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• Tolerance and roughness text

  The text of a tolerance value or an Ra value on the drawing.

  text = value rounded to 4 decimals

  Drawing annotations: tolerance frames, datums and surface finish gdt.js

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• Circle and arc points

  The points of a circle or arc in a symbol.

  (c_x + r cos t, c_y + r sin t), t = a₀ + sweep × i ÷ n

  Drawing annotations: tolerance frames, datums and surface finish gdt.js

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• Annotation layout

  The frame, compartments, leader, datum triangle and surface texture legs of one annotation.

  frame height = 2h; datum triangle height = side × √3 ÷ 2; texture legs at 60°: offset = leg ÷ tan(π ÷ 3)

  Drawing annotations: tolerance frames, datums and surface finish gdt.js

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• Point on a B-spline curve

  The point of a curve at a parameter, from its poles and knots.

  C(t) = Σ Nⱼ(t) × Pⱼ over the p + 1 poles of the span

  Surface through a network of curves gordon.js

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• Point on a B-spline surface

  The point of the surface at (u, v), from its pole grid and two knot vectors.

  S(u, v) = Σᵢ Σⱼ Nᵢ(u) × Nⱼ(v) × Pᵢⱼ

  Surface through a network of curves gordon.js

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• Interpolation knots by averaging

  The knot vector used to interpolate points at given parameters.

  inner knot j = (t_j + … + t_(j+p−1)) ÷ p

  Surface through a network of curves gordon.js

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• Band solve for interpolation

  Factors the collocation matrix once and solves for the poles of every curve that shares its parameters.

  N × P = Q, with N banded; LU without pivoting, then forward and back substitution

  Surface through a network of curves gordon.js

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• Degree raising and knot refinement

  Re-expresses a curve with a higher degree or more knots without changing its shape.

  new poles = solve(N_fine(g), C(g)), g = Greville points of the finer knot vector

  Surface through a network of curves gordon.js

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• Merged knot vector

  The smallest knot vector that contains the knot vectors of several curves.

  Surface through a network of curves gordon.js

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• Joining exact curves end to end

  Makes one B-spline from a chain of lines and splines, with a corner knot at each joint.

  knot = a + (b − a) × u for each piece's inner knots; joint knots repeated d times

  Surface through a network of curves gordon.js

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• Sketch chains as world curves

  Turns the chosen lines, arcs, circles and splines of a sketch into curves over 0 to 1, one per chain.

  share of a piece = its length ÷ total length; arc length = r × sweep; circle = 2π × r

  Surface through a network of curves gordon.js

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• Closest points of two segments

  The nearest pair of points on two straight segments, used to find where two curves may cross.

  s = (b·f − c·e) ÷ (a·e − b²), t = (b·s + f) ÷ e, both clamped to 0…1

  Surface through a network of curves gordon.js

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• Crossing of two curves

  The parameters on a profile and a guide where they meet, and the gap between them there.

  minimize |A(s) − B(t)|² by Gauss–Newton; point = (A(s) + B(t)) ÷ 2; gap = |A(s) − B(t)|

  Surface through a network of curves gordon.js

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• All crossings of a profile and a guide

  Finds every place two curves come within the gap tolerance, so the network can be checked to be a grid.

  Surface through a network of curves gordon.js

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• Common parameters of a curve family

  The shared parameter at which every profile (or guide) meets each curve of the other family.

  t_q = mean over the curves of (x_q − x_first) ÷ (x_last − x_first)

  Surface through a network of curves gordon.js

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• Monotone reparametrization

  A smooth increasing map from the common parameters to one curve's own crossing parameters.

  m_k = (w₁ + w₂) ÷ (w₁ ÷ d_(k−1) + w₂ ÷ d_k); cubic Hermite on each interval

  Surface through a network of curves gordon.js

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• Distance from a point to a curve

  How far a point of the fitted surface or curve is from the true curve near a given parameter range.

  golden-section search for the minimum of |g(t) − q|

  Surface through a network of curves gordon.js

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