Calculations

Calculations

• Sketch curves as kernel edges

  Turns the lines, arcs, circles and splines of a sketch into the kernel's edges for a loft.

  arc: start, middle and end points c + r × (cos, sin) at a₀, a₀ + sweep ÷ 2, a₀ + sweep, then an arc through the three points

  Surface features built with the kernel surface.js

  • PartCraft
  64-bit

• Network surface as a kernel face

  Hands the surface fitted through a network of curves to the kernel and sets the tolerance its seams are sewn with.

  face tolerance 1e-9; sewing tolerance = max(1e-7, 1e-7 × size)

  Surface features built with the kernel surface.js

  • PartCraft
  64-bit

• Area of a surface body

  The area of each surface body left after a rebuild.

  area = Σ over the faces of OpenCascade's surface area

  Surface features built with the kernel surface.js

  • PartCraft
  64-bit

• Display mesh of a surface body

  The triangles used to draw each surface body.

  linear deflection = 2e-4 × diagonal; angular deflection = 2π ÷ 64

  Surface display meshes, curvature map and zebra stripes surface.js

  • PartCraft
  32-bit, display only

• Curvature per vertex

  The discrete Gaussian and mean curvature at each vertex of a triangle mesh, for the curvature map.

  K = (2π − Σ angles) ÷ A; H = |Σ (cot α + cot β)(pᵢ − pⱼ)| ÷ 4A, A = one third of the area of the triangles at the vertex

  Surface display meshes, curvature map and zebra stripes surface.js

  • PartCraft
  32-bit, display only

• Zebra stripe value

  Whether a surface normal reflects a light or a dark stripe.

  r = d − 2(d · n)·n; stripe = ⌊(atan2(r_x, r_z) ÷ π + 1) × stripes ÷ 2⌋ mod 2

  Surface display meshes, curvature map and zebra stripes surface.js

  • PartCraft
  32-bit, display only

• Basis functions and derivatives

  The B-spline basis functions and their derivatives at a parameter.

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Point and derivatives of a NURBS curve

  A rational curve's point and first two derivatives at a parameter.

  C = A ÷ w; C′ = (A′ − w′·C) ÷ w; C″ = (A″ − 2w′·C′ − w″·C) ÷ w

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Point and derivatives of a line, circle or ellipse

  A profile curve's point and first two derivatives at a parameter.

  circle and ellipse: C = c + a·cos t·x + b·sin t·y

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Point and derivatives of a B-spline surface

  A B-spline surface's point and all derivatives up to second order at (u, v).

  homogeneous sums A[k][l] = Σ Nᵤ⁽ᵏ⁾·Nᵥ⁽ˡ⁾·w·P; S[k][l] by the quotient rule with binomial factors 1, 2, 1

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Closest point on a sphere, torus or cone

  The nearest point, the normal there and the distance, in closed form.

  sphere: c + r·n̂, d = | |p − c| − r |; torus: tube center m = c + R·ê, then m + r·n̂; cone: τ = ((ρ − r₀)·k + t) ÷ (1 + k²)

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• One-dimensional Newton minimizer

  Refines the parameter of the nearest point on a profile curve.

  Δt = −f′ ÷ f″ (or one sample spacing downhill when f″ ≤ 0); stop when |Δt| ≤ 1e-15 × (|b − a| + |t|)

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Start points for the minimizer

  Samples the distance along a curve and runs Newton from the two best samples.

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Closest point on a surface of revolution

  The nearest point on a profile turned about an axis.

  minimize ½((ρ(v) − ρ_p)² + (t(v) − t_p)²) over the profile parameter v

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Closest point on an extruded surface

  The nearest point on a curve swept along a direction.

  minimize ½ |across(p − C(u))|²; P = C(u) + ((p − C(u)) · d)·d

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Closest point on a B-spline surface

  The nearest point on a B-spline face, kept inside the face's parameter box.

  Newton on ∇½|S(u, v) − p|² = 0 with the Hessian [Sᵤ·Sᵤ + Sᵤᵤ·r, Sᵤ·Sᵥ + Sᵤᵥ·r; …]

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Smallest radius of curvature near a point

  The radius of curvature the mesher uses to size elements on a curved face.

  sphere, torus: r; cone: ρ × √(1 + k²); else 1 ÷ max |Δn| ÷ moved over three directions 60° apart

  Exact surfaces for the tetrahedral mesher surfaces.js

  • PartCraft
  64-bit

• Sketch plane from the file

  Reads a sketch plane's origin, normal and x direction and checks that x lies in the plane.

  |x̂ · n̂| ≤ 1e-6

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Sketch points into the part's plane

  Builds the file's sketch frame and maps each sketch point into the frame of the plane the sketch is put on.

  x = unit(x − n·(x · n)); y = n × x; (u, v) → world = o + u·x + v·y → plane = ((w − o′) · x′, (w − o′) · y′)

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Distance from a point to a segment

  The distance from a position to a straight piece of an edge.

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

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Nearest edge to a stored position

  Finds which edge of the part built so far a fillet or chamfer in the file refers to, and how far the stored position is from it.

  minimum over the edge polyline segments of the point-to-segment distance

  Import of a parametric export file swimport.js

  • PartCraft
  32-bit in part

• Distance from a point to a triangle

  The distance from a position to one display triangle of a face.

  inside the triangle: |(p − a) · n̂|; else the least distance to its three sides

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Nearest face to a stored position

  Finds which face of the part built so far a feature in the file refers to.

  minimum over the display triangles of the point-to-triangle distance

  Import of a parametric export file swimport.js

  • PartCraft
  32-bit in part

• Flat face lying in a sketch's plane

  Looks for a named flat face of the part in the plane a sketch was drawn on, so the sketch can follow that face.

  | |n_face · n| − 1 | < 1e-9 and |(o − o_face) · n_face| < 1e-7 × max(1, |o|)

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Base plane and world axis tests

  Decides whether a plane is one of the three base planes.

  | |n̂ · n_base| − 1 | < 1e-9 and |o · n_base| < 1e-9 × max(1, |o|)

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Dimension turned into a parameter

  Gives a dimension that other equations use a parameter of its own, holding its value.

  expression text = formatNumber(value) (12 significant digits, expr.js)

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Equation result against the stored value

  Warns when an imported equation would give a different number from the value the file holds.

  |v − held| > 1e-6 × max(1, |held|)

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Volume check

  Compares the volume of the rebuilt part with the volume recorded in the file.

  same when |V − V_file| ≤ 1e-4 × max(1, V_file)

  Import of a parametric export file swimport.js

  • PartCraft
  64-bit

• Number formatter

  Writes one number as macro text.

  text = toPrecision(v, 15); |v| < 1e-14 → 0

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Millimeters to meters

  Converts a length or a coordinate from millimeters to the meters the macro uses.

  m = round(mm × 10⁶) ÷ 10⁹

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Degrees to radians

  Converts an angle in degrees to the radians the macro uses.

  rad = deg × π ÷ 180

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Unit-vector component

  Writes a component of a sketch plane's normal or x direction.

  u = round(v × 10¹²) ÷ 10¹²

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Sketch dimension value and text position

  Works out the value of each sketch dimension in meters or radians and where its text sits.

  length: value ÷ 1000; angle: value × π ÷ 180; arc radius = |start − center|

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Pick point on a face

  Finds a point inside a face, used by the macro to select that face.

  centroid of the largest triangle = (a + b + c) ÷ 3, moved onto the plane: p − ((p − o)·n) n

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Countersink depth and taper

  The depth and half angle of the tapered cut that stands in for a countersink.

  depth = (D − d) ÷ 2 ÷ tan(angle × π ÷ 360); taper = angle ÷ 2 × π ÷ 180

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Expected volume and box after each step

  Writes the part's volume and bounding box after a feature, for the macro to compare with what it built.

  volume in m³ = volume in mm³ × 10⁻⁹; box corners through the millimeter-to-meter conversion

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Body check tolerances in the macro

  The comparison the macro makes between the body it built and the expected volume and box.

  |v − vol| ≤ 10⁻⁴ × |vol| + 10⁻¹⁵; |box − expected| ≤ 10⁻⁵ m + 0.02 × diagonal

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Final volume comparison

  Writes the finished part's volume into the macro and the test that reports whether the rebuilt part matches.

  |v − vol| ≤ 10⁻⁴ × vol, v = reported volume × 10⁹ (mm³)

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• Numbers in equations

  Writes a parameter expression as equation text for the macro, including its numeric constants.

  constant → formatNumber(value); pow(a, b) → (a) ^ (b); log(x) → log(x) ÷ log(10)

  Numbers written into the VBA macro export swmacro.js

  • PartCraft
  64-bit

• 64-point rule on the tetrahedron

  Builds the integration points and weights used for mass and body loads.

  x = u, y = v(1 − u), z = w(1 − u)(1 − v); weight = wₐ w_b w_c ÷ 8 × (1 − u)² (1 − v)

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Shape functions of the 10-node tetrahedron

  The ten shape functions and their derivatives at a point of the element.

  corner: N = L(2L − 1); midside: N = 4 LᵢLⱼ

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Jacobian and physical derivatives

  The element's Jacobian determinant at a point and the shape-function derivatives in x, y and z.

  J = Σ xₐ ∂Nₐ/∂ξ; det J by cofactors; ∂N/∂x = J⁻ᵀ ∂N/∂ξ

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Element stiffness

  The 30 × 30 stiffness matrix of one element and its volume.

  Kₑ = Σₚ wₚ det Jₚ E · Bᵀ D B; volume = Σₚ wₚ det Jₚ

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Element mass and shape-function integrals

  The consistent mass matrix of one element and ∫N dV for each node, used for gravity.

  Mₑ = Σₚ ρ wₚ det Jₚ NₐN_b; ∫Nₐ dV = Σₚ wₚ det Jₚ Nₐ

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Stress and strain at a Gauss point

  The six strains and six stresses at one Gauss point from the element's displacements.

  ε = B u; σ = E · D (ε − α ΔT [1 1 1 0 0 0])

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• 6-point rule on a triangle face

  The points and weights used to integrate loads, areas and heat exchange over boundary faces.

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Shape functions of the 6-node face

  The six shape functions of a boundary face and their derivatives.

  corner: N = L(2L − 1); midside: N = 4 LᵢLⱼ

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Integration over a boundary face

  The area-weighted normal and position at each rule point of a curved boundary face.

  n dA = (∂x/∂L₁ × ∂x/∂L₂) × w ÷ 2

  Quadratic tetrahedron element matrices tetfea.js

  • PartCraft
  64-bit

• Area element on a face for heat terms

  The area weight at each rule point of a 6-node face, for heat exchange and heat loads.

  dA = |∂x/∂L₁ × ∂x/∂L₂| × w ÷ 2

  Quadratic tetrahedron element matrices tet-elements.js

  • PartCraft
  64-bit

• Element conductivity matrix

  The 10 × 10 heat conduction matrix of one element.

  Kₑ = Σₚ wₚ det Jₚ k ∇Nₐ·∇N_b

  Quadratic tetrahedron element matrices tet-elements.js

  • PartCraft
  64-bit