Calculations

Calculations

• Lattice units from physical ones

  Chooses the lattice speed, viscosity, relaxation time and time step for a cell size, speed and fluid.

  Re_cell = U dx ÷ ν; u = min(0.05, √(0.03 ÷ k), 0.5 Re_cell); τ = 3 u ÷ Re_cell + ½; dt = u dx ÷ U

  Flow study: lattice Boltzmann solver flow.js

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• Face normal and area

  The area-weighted outward normal of a face, its area and how flat it is.

  area = Σ |nₜ| ÷ 2; flatness = |Σ nₜ| ÷ (2 × area)

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Area of an opening's holes

  The area of the holes in a flat face, which is the opening's true area.

  loop area = ½ Σ (pₐ × p_b) · n; holes = Σ −area over clockwise loops

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Lid cells over an opening

  Finds the cells just behind a flat face that cover its holes.

  raster area = hole pixels × (h ÷ 2)²

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Wall distance along a link

  Where the link from a fluid cell to a solid cell crosses the part's surface.

  t = (e₂ · (tv × e₁)) ÷ (e₁ · (d × e₂))

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Frontal area

  The area of the part seen along the flow, used as the reference area for drag and lift coefficients.

  area = covered pixels × pixel size², pixel size = larger extent ÷ 1024

  Flow study: fluid cells, openings and wall distances flow-domain.js

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• Automatic cell size

  Picks the cell size that gives about a target number of cells in the grid box.

  h = ∛(volume ÷ target), rounded up to 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 or 10 times a power of ten

  Flow study: fluid cells, openings and wall distances flow-study.js

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• Cell count estimate

  The number of grid cells a study will use.

  n = ⌈extent ÷ h − 1e-6⌉ + margin cells, per axis

  Flow study: fluid cells, openings and wall distances flow-study.js

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• Lift direction

  The direction lift is measured along, made square to the flow.

  l = l₀ − (l₀ · d) d, then divided by its length

  Flow study: fluid cells, openings and wall distances flow-study.js

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• Inlet flow and speed

  Each inlet's flow rate, mean speed and the speed set on its lattice links.

  Q = v × A (or the given Q); U = Q ÷ A; link speed = Q × (A_face ÷ A) ÷ (lattice link area × dx²); D = √(4A ÷ π)

  Flow study: results in physical units flow-worker.js

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• Speed and pressure scales

  The factors between lattice and physical speed and pressure, and the density set at each outlet.

  c_u = dx ÷ dt; c_p = ρ c_u²; ρ_outlet = 1 + 3 (p − p_ref) ÷ c_p

  Flow study: results in physical units flow-worker.js

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• Second steady-state test

  Watches the pressure drop or the drag force between checks before the run is accepted as steady.

  |q − q_prev| ÷ max(|q|, 1e-30) < 1e-4 on two checks in a row

  Flow study: results in physical units flow-worker.js

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• Mass flow, mean pressure and pressure drop

  The mass and volume flow through each opening, its mean pressure, and the pressure drop between inlets and outlets.

  ṁ = lattice mass × ρ dx³ ÷ dt; p = p_ref + (mean ρ − 1) ÷ 3 × c_p; Δp = flow-weighted mean inlet p − outlet p

  Flow study: results in physical units flow-worker.js

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• Force, drag and lift coefficients

  The force on the part in newtons and, for external flow, drag, lift and their coefficients.

  F = lattice force × ρ dx⁴ ÷ dt²; C_d = F · d ÷ (½ ρ U² A)

  Flow study: results in physical units flow-worker.js

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• Reynolds number

  The Reynolds number of the study.

  Re = U √(4A ÷ π) ÷ ν (internal); Re = U L ÷ ν (external)

  Flow study: results in physical units flow-worker.js

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• Inflow eddy viscosity

  The turbulence level given to inflows in a turbulent run.

  ν_t = max(3ν, √1.5 × 0.09^0.25 × I × u × ℓ), I = 5 % (internal) or 1 % (external)

  Flow study: results in physical units flow-worker.js

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• Mass balance

  The share of the inflow that does not leave through the outlets.

  balance = (ṁ_in − ṁ_out) ÷ ṁ_in

  Flow study: results in physical units flow-ui.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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• Making a curve family compatible

  Gives every profile (or every guide) the same degree and knots, exactly when possible and by fitting otherwise.

  exact: raise degree, merge knots; otherwise: cubic interpolation of C(r(s)), points doubled until error ≤ fitTol

  Surface through a network of curves gordon.js

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• Boolean sum of the three surfaces

  Adds the surface through the profiles and the surface through the guides and subtracts the surface through the grid points.

  P = Lu + Lv − T, pole by pole

  Surface through a network of curves gordon.js

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• Tangency to a plane along a boundary

  Makes the surface leave a boundary curve tangent to a plane and reports the angle it made with the plane before.

  second pole row: q ← a + (d − n × (d·n)), d = q − a; angle = asin(|d·n| ÷ |d|)

  Surface through a network of curves gordon.js

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• Deviation from the input curves

  The largest distance between the finished surface and any profile or guide, shown in the panel.

  dev = max over 401 samples per curve of |S(s) − C(r(s))|

  Surface through a network of curves gordon.js

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• Fold check

  Refuses a surface whose normal flips between neighboring sample points.

  n = (S(u + h, v) − S(u − h, v)) × (S(u, v + h) − S(u, v − h)), h = 1e-7; fold if n₁·n₂ < 0

  Surface through a network of curves gordon.js

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• Extrude height and start

  The height of the extruded tool and where it starts along the sketch normal, for blind, two-sided, mid-plane and through-all ends.

  two-sided: h = depth + depth₂; mid-plane: z₀ = −h ÷ 2; through all: h = extent + 1 + 0.01 × (max − min)

  Feature tool dimensions on the mesh kernel kernel.js

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• Revolve profile in axis coordinates

  Turns each profile point into a radius from the axis and a height along it before the profile is revolved.

  r = max(0, d·n); z = d·u; d = p − a, u = unit axis, n = unit normal to the axis on the profile's side

  Feature tool dimensions on the mesh kernel kernel.js

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• Plane, cylinder or cone swept by a profile line

  Classifies the surface a straight profile line sweeps and, for a cone, finds its apex and slope.

  slope = (r₂ − r₁) ÷ (z₂ − z₁); apex height = z₁ − r₁ ÷ slope

  Feature tool dimensions on the mesh kernel kernel.js

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• Hole depth

  The depth of a drilled hole, including a through hole's length past the body.

  through: depth = extent on the drilling side + 1 + 0.01 × (max − min)

  Feature tool dimensions on the mesh kernel kernel.js

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• Countersink cone height

  The height of a countersink from its diameter, the hole diameter and the included angle.

  h = (R − r) ÷ tan(angle × π ÷ 360)

  Feature tool dimensions on the mesh kernel kernel.js

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• Thread zone of a tapped hole

  The inner and outer radius of a tapped hole's thread, used when parts are checked against each other.

  r₀ = min(drill, d − 1.082532 × p) ÷ 2; r₁ = d ÷ 2

  Feature tool dimensions on the mesh kernel kernel.js

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• Circular pattern step

  The angle between copies of a circular pattern and the rotation applied to each copy.

  full turn: step = 360 ÷ count; otherwise step = angle ÷ (count − 1); copy i turned by step × i

  Feature tool dimensions on the mesh kernel kernel.js

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• Extent of the body along a normal

  How far the body reaches in front of and behind a sketch plane, for through-all cuts and through holes.

  range = min and max of (corner − o)·n over the 8 corners of the bounding box

  Feature tool dimensions on the mesh kernel kernel.js

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• Reference stored for a picked flat face

  Chooses the stable face reference for a picked flat triangle, or stores a fixed plane when none matches.

  match: c.n·n > 1 − 1e-6 and |(p − c.o)·c.n| < 1e-4 × max(1, |p|); else o rounded to 9 decimals, n to 12

  Feature tool dimensions on the mesh kernel kernel.js

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• Arc through three points

  Center, radius, start angle and signed sweep of an outline arc given by its start, middle and end.

  c_x = (|a|²(m_y − b_y) + |m|²(b_y − a_y) + |b|²(a_y − m_y)) ÷ d, d = 2 (a_x(m_y − b_y) + m_x(b_y − a_y) + b_x(a_y − m_y))

  Circuit board outline, holes and part boxes kicad.js

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• Arc as straight segments

  The points that replace an arc or circle of the board outline.

  steps = max(2, ⌈|sweep| ÷ 7.5°⌉); pₖ = c + r (cos aₖ, sin aₖ), aₖ = a₀ + sweep × k ÷ steps

  Circuit board outline, holes and part boxes kicad.js

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• Joining outline pieces into loops

  Chains lines and arcs of the edge layer into closed outlines.

  Circuit board outline, holes and part boxes kicad.js

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• Footprint point on the board

  Moves a point of a footprint (a pad, a corner of its box) to board coordinates.

  x = at_x + l_x cos a − l_y sin a; y = −at_y + l_x sin a + l_y cos a; l_y = −p_y, a = rotation × π ÷ 180

  Circuit board outline, holes and part boxes kicad.js

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• Board thickness parameter

  The part parameter that drives the board's extrusion depth.

  board_t = thickness rounded to 4 decimals

  Circuit board outline, holes and part boxes kicad.js

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