Calculations

Calculations

• Solver tolerances of a buckling study

  How tightly the pre-stress solve and each inner solve are converged.

  pre-stress: 1e-10; each K⁻¹ solve: 1e-9

  Buckling load factors and thermal stress setup buckling.js

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• Temperature rise at each node

  The temperature change that drives thermal stress: a uniform rise, or the temperature field minus the stress-free temperature.

  ΔT = T − T_ref, T_ref = 20 when not given

  Buckling load factors and thermal stress setup buckling.js

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• Time step for a transient thermal load

  Which saved time step of a transient run is nearest the requested time.

  step = round(time ÷ (t ÷ steps)), held in [0, steps]; time = step × dt

  Buckling load factors and thermal stress setup buckling.js

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• Voxel model volume and mass

  The volume and mass of the analysis model listed with the study.

  volume = elements × h³; mass = volume × density (tonne/mm³) × 1000

  Buckling load factors and thermal stress setup buckling.js

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• Numbers written to the deck

  Formats every coordinate, material value, load and penalty for the input deck.

  text = v.toExponential(10)

  CalculiX deck export and result import calculix.js

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  64-bit

• Material cards

  Writes elastic constants, density and the two-point plastic curve of each part's material.

  density (tonne/mm³) = density (kg/m³) × 1e-12; plastic: (σ_y, 0) and (σ_y + H, 1)

  CalculiX deck export and result import calculix.js

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  64-bit

• Support directions

  Writes each held direction, and a local axis system for supports on slanted faces.

  axis when ||v| − 1| < 1e-9; slanted: local x = n, local y = unit(t − (t · n) n)

  CalculiX deck export and result import calculix.js

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• Nodal loads

  Writes the nodal forces of the study as concentrated loads.

  one line per non-zero force component

  CalculiX deck export and result import calculix.js

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• Load increments

  The first and smallest load increment given to the other program.

  first = 1 ÷ steps; smallest = 1 ÷ (64 × steps); total and largest = 1

  CalculiX deck export and result import calculix.js

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• Reading the result file

  Reads node positions and result values from the text result file.

  value = parseFloat(12-character field, D exponent read as E)

  CalculiX deck export and result import calculix.js

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  64-bit

• Default penalty stiffness

  The contact stiffness used when none is given.

  kn = 20 × E ÷ h

  Contact between parts in stress analysis contact.js

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• Closest point on a triangle

  The point of a flat triangle nearest a given point, as weights of the three corners.

  region tests on d₁ … d₆ = dot products of edges with corner-to-point vectors; inside: v = v_b ÷ (v_a + v_b + v_c)

  Contact between parts in stress analysis contact.js

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• Closest point on a curved face

  The nearest point of a six-node master face, in the face's own coordinates.

  best of 4 flat sub-triangles (corners and midsides); (L₁, L₂) = weights × reference coordinates

  Contact between parts in stress analysis contact.js

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• Point and normal on a face

  The position and outward unit normal of a six-node face at given face coordinates.

  x = Σ N_k x_k; n = unit(Σ ∂N_k/∂L₁ x_k × Σ ∂N_k/∂L₂ x_k)

  Contact between parts in stress analysis contact.js

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• Integration weights on the slave face

  The area each contact point on a slave face stands for.

  w = w_q × |∂x/∂L₁ × ∂x/∂L₂| ÷ 2

  Contact between parts in stress analysis contact.js

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• Pairing slave points with the master

  Finds the master point each slave point faces, and reports whether the pairing changed.

  paired when d² ≤ cutoff²; moved when the face changes or |ΔL| > 2e-3

  Contact between parts in stress analysis contact.js

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• Gap, contact force and stiffness

  The normal gap at a contact point and the force and stiffness it adds.

  g = n · (x_s − x_m); force = kn × w × (−g) when g < 0; stiffness = kn × w × c cᵀ ⊗ n nᵀ; pressure = force ÷ w

  Contact between parts in stress analysis contact.js

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• Bonded tie

  Holds a slave point to the master point it started nearest to.

  rel = (x_s − x_m) − (x_s − x_m) at the first call; force = −kt × w × rel

  Contact between parts in stress analysis contact.js

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• Machined part: stock, cutting, holes, handling

  The cost rows of a part cut from a block of stock.

  stock = (size + 2 × allowance) per axis; stock cost = stock volume × density × 1e-9 × price per kg; cutting minutes = (stock volume − volume) ÷ removal rate × finishing; cost = minutes ÷ 60 × rate per hour

  Manufacturing cost estimate cost.js

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• Printed part: filament and machine time

  The cost rows of a 3D-printed part.

  printed volume = volume × fill; filament = printed volume × density × 1e-9 × price per kg; hours = printed volume ÷ volume per hour

  Manufacturing cost estimate cost.js

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• Cost per part and rounding

  The cost of each part at the given quantity, the share of each row, and the rounded figures returned.

  total = unit + setup ÷ quantity; share = row ÷ total; returned amounts = round to 0.01

  Manufacturing cost estimate cost.js

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• Rounding to two decimals

  Rounds an amount or a listed quantity to hundredths.

  r2(v) = round(v × 100) ÷ 100

  Manufacturing cost estimate cost.js

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• Number of holes

  Counts the holes the part has, for the time per hole.

  hole features + circles in cut sketches, each × (pattern count − 1) more for patterns

  Manufacturing cost estimate cost.js

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  Exact

• Stock block

  The size and position of the stock around the part.

  offset mode: min = box min − (side, side, bottom), max = box max + (side, side, top); cylinder: r = ½ × box diagonal in XY + side

  Part and stock in machine coordinates cam.js

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• Work origin and frame change

  Moves points from model coordinates to the work frame (G54) and back.

  work = (p · x, p · y, p · z) − origin

  Part and stock in machine coordinates cam.js

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• Mesh in the work frame and snap tolerance

  Copies the part's mesh into work coordinates and sets how close a mesh point must be to an exact edge to be taken as that edge.

  tol = 1e-4 + 2e-6 × largest model dimension

  Part and stock in machine coordinates cam.js

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  32-bit in part

• Where a vertical line meets the part

  The heights at which a vertical line through (x, y) crosses the part's surface.

  weights l₁, l₂, l₃ from the triangle's XY corners; z = l₁ z₁ + l₂ z₂ + l₃ z₃

  Part and stock in machine coordinates cam.js

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  32-bit in part

• Top facet and its slope

  The height and slope of the highest mesh triangle under a point, used by 3D finishing.

  g_x = ((z₂ − z₁)(y₃ − y₁) − (z₃ − z₁)(y₂ − y₁)) ÷ d₂; g_y likewise

  Part and stock in machine coordinates cam.js

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  32-bit in part

• Material below a point

  Whether there is part material just under a height at (x, y), used to tell a through hole from a blind one.

  first surface below z faces up

  Part and stock in machine coordinates cam.js

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  32-bit in part

• Circles of round edges

  The center and radius of each circular edge lying in a plane square to the tool.

  full circle: c = (a + mid) ÷ 2, r = |a − mid| ÷ 2; arc: circle through a, mid, b

  Part and stock in machine coordinates cam.js

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• Circle through three points

  The center and radius of the circle through three 2D points.

  d = 2 (a_x (b_y − c_y) + b_x (c_y − a_y) + c_x (a_y − b_y)); u_x = (|a|² (b_y − c_y) + |b|² (c_y − a_y) + |c|² (a_y − b_y)) ÷ d

  Part and stock in machine coordinates cam.js

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• Arc from start, middle and end

  The signed sweep of an arc about a known center that passes from a start through a middle point to an end.

  counterclockwise when mod(a_m − a₀, 2π) < mod(a₁ − a₀, 2π); sweep = ± mod(a₁ − a₀, 2π)

  Part and stock in machine coordinates cam.js

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• Exact outline of a flat face

  Turns a face's mesh boundary into lines and arcs whose ends are the exact edge ends.

  mesh corner within tol of an edge end → that end; mesh run within tol of an edge circle → arc on that circle

  Part and stock in machine coordinates cam.js

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• Hole position, size and depth

  The center, radius, top and bottom of a round face along the tool axis, and whether it is a hole and goes through.

  c, r from the exact cylinder; top, bottom = mesh z range snapped to an edge end within tol

  Part and stock in machine coordinates cam.js

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• Chaining picked edges into a loop

  Orders picked edges into one closed outline and makes each start exactly on the previous end.

  ends join when closer than tol; then next.start = previous.end

  Part and stock in machine coordinates cam.js

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• Length of a line or arc

  The length of one piece of an outline or toolpath.

  line: |e − s|; arc: |sweep| × r

  2D curve offsets for toolpaths cam-offset.js

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• Point along a piece

  The point a given fraction of the way along a line or arc.

  line: s + (e − s) t; arc: c + r (cos, sin)(a₀ + sweep × t)

  2D curve offsets for toolpaths cam-offset.js

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• Is an angle on an arc

  Whether a direction from the center falls within an arc, and how far along.

  u = mod(±(a − a₀), 2π); on the arc when u ≤ |sweep| + eps; t = u ÷ |sweep|

  2D curve offsets for toolpaths cam-offset.js

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• Distance from a point to a line or arc

  The shortest distance from a point to one piece of an outline.

  line: |q − (s + d × clamp(((q − s) · d) ÷ |d|², 0, 1))|; arc: ||q − c| − r| when the direction is on the arc, else the nearer end

  2D curve offsets for toolpaths cam-offset.js

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• Signed area of a loop

  The area enclosed by a loop of lines and arcs, positive when counterclockwise.

  A = ½ Σ [line: s × e; arc: r c_x (sin a₁ − sin a₀) − r c_y (cos a₁ − cos a₀) + r² × sweep]

  2D curve offsets for toolpaths cam-offset.js

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• Winding number

  How many times the loops wind round a point, which says whether the point is inside.

  count signed crossings of a ray to +x; arc crossing at x = c_x ± √(r² − (q_y − c_y)²)

  2D curve offsets for toolpaths cam-offset.js

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• Line and line

  The point where two line pieces cross, or the ends of their overlap when they lie on one line.

  t_a = ((B.s − A.s) × s) ÷ (r × s); parallel when |r × s| ≤ 1e-12 |r| |s|

  2D curve offsets for toolpaths cam-offset.js

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• Line and arc

  The points where a line piece meets an arc.

  a t² + b t + c = 0; t = (−b ± √(b² − 4ac)) ÷ 2a

  2D curve offsets for toolpaths cam-offset.js

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• Arc and arc

  The points where two arcs meet.

  a = (r_A² − r_B² + d²) ÷ 2d; h = √(r_A² − a²); p = c_A + a u ± h n

  2D curve offsets for toolpaths cam-offset.js

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• Raw offset of a loop

  Moves every piece of a loop sideways by the offset and joins the corners with arcs.

  line: s, e + d × left normal; arc: r ∓ d; corner arc radius d, sweep = atan2(t_a × t_b, t_a · t_b)

  2D curve offsets for toolpaths cam-offset.js

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• Splitting pieces at intersections

  Cuts every offset piece where it meets another.

  cut at parameter t unless within 1e-5 mm of an end or of the previous cut

  2D curve offsets for toolpaths cam-offset.js

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  64-bit

• Offset of a region

  The outline at a fixed distance inside or outside a region, which is the path of the tool center.

  keep a piece when dist(midpoint, original outline) ≥ d − 1e-6 and the midpoint is on the offset side

  2D curve offsets for toolpaths cam-offset.js

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• Chaining pieces into loops

  Joins the kept pieces end to start into closed loops.

  next piece starts within 4e-5 mm of the current end; sharpest left turn wins; then next.start = current.end

  2D curve offsets for toolpaths cam-offset.js

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• Merging collinear lines and co-circular arcs

  Joins neighboring pieces that lie on the same line or circle into one.

  lines: |u × v| < 1e-10 and end within 1e-9 mm of the line; arcs: centers and radii within 1e-9 mm, same direction

  2D curve offsets for toolpaths cam-offset.js

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• Splitting arcs to half turns

  Cuts arcs so none sweeps more than 180°, as G-code controls expect.

  n = ceil(|sweep| ÷ π); piece i from a₀ + sweep × i ÷ n

  2D curve offsets for toolpaths cam-offset.js

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