Calculations

Calculations

• Backslash for dense matrices

  Solves A x = b, or the least-squares problem when A is not square.

  CraftScript array arithmetic cs-tensor.js

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• Number of elements in a range

  How many values a:s:b has.

  n = ⌊(b − a) ÷ s + max(3ε × max(|a|, |b|) ÷ |s|, 1e-10)⌋ + 1

  CraftScript array arithmetic cs-tensor.js

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• Values of a range

  The values a, a + s, a + 2s … up to b.

  x_k = a + k × s

  CraftScript array arithmetic cs-tensor.js

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• Sum, product, mean, maximum and minimum

  Reduces an array along one dimension.

  mean = (Σ xₖ) ÷ n

  CraftScript array arithmetic cs-tensor.js

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• Unit factors

  The factors that turn the given force, length and torque units into newtons, millimeters and newton millimeters.

  Design-it sizing and hand checks design-it.js

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• Arguments to working units

  Converts the given force, lengths and torque to N, mm and N·mm.

  value × unit factor

  Design-it sizing and hand checks design-it.js

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• Rounding to stock sizes and to reported decimals

  Rounds a thickness up to a stock plate or a 0.5 mm step, and rounds reported values.

  r2(v) = round(100 v) ÷ 100; r3(v) = round(1000 v) ÷ 1000

  Design-it sizing and hand checks design-it.js

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• Bolt positions in the flange

  The heights of the two bolts and the flange height.

  y_b = t + 1.5 × hole; spacing = clamp(round(a ÷ 10) × 5, 30, 100); H = y_t + 1.5 × hole

  Design-it sizing and hand checks design-it.js

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• Bracket checks

  Bending and shear stress, tip deflection and bolt loads of the bracket, each with its limit and safety factor.

  σ = 6 F a ÷ (b t²); τ = 1.5 F ÷ (b t); δ = F a³ ÷ (3 E I), I = b t³ ÷ 12; T = M y₂ ÷ (y₁² + y₂²); (T ÷ T_all)² + (V ÷ V_all)² ≤ 1

  Design-it sizing and hand checks design-it.js

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• Bracket sizing, volume and mass

  The thickness for strength and for the deflection limit, the bolt size, and the bracket's volume and mass.

  t = √(6 M ÷ (b σ_allow)); t = ∛(4 F a³ ÷ (E b δ)); V = b (L t + (H − t) t) − 2 (π ÷ 4) hole² t; mass = V × density × 1e-6

  Design-it sizing and hand checks design-it.js

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• Shaft checks

  Von Mises stress, torsional shear and twist of a round shaft.

  σ' = 32 ÷ (π d³) × √(M² + ¾T²); τ = 16 T ÷ (π d³); θ = T L ÷ (G J), G = E ÷ (2(1 + ν)), J = π d⁴ ÷ 32

  Design-it sizing and hand checks design-it.js

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• Shaft sizing, volume and mass

  The required diameter, the standard size above it, and the shaft's volume and mass.

  d = ∛(32 SF √(M² + ¾T²) ÷ (π S_y)); V = (π ÷ 4) d² L

  Design-it sizing and hand checks design-it.js

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• The built part

  Sketches and extrudes the sized bracket or shaft as part features.

  density (g/cm³) = density (kg/m³) ÷ 1000

  Design-it sizing and hand checks design-it.js

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• Edges and facet normals

  Finds the mesh's edges, each facet's unit normal, and which edges are sharp.

  sharp when n₁ · n₂ < cos 20°

  Drawing sheet views, dimensions and PDF drawing.js

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• One projected view with hidden lines

  Projects the mesh onto a view plane, finds the view's extents, and splits each drawn edge into visible and hidden parts.

  (u, v, w) = (p · right, p · up, p · dir)

  Drawing sheet views, dimensions and PDF drawing.js

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• Merging collinear segments

  Joins touching or overlapping segments on the same line and removes hidden lines under visible ones.

  Drawing sheet views, dimensions and PDF drawing.js

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• Section hatching

  Hatch lines at an angle inside the cut outline.

  Drawing sheet views, dimensions and PDF drawing.js

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• Clip to the detail circle

  Keeps the parts of segments inside the detail view's circle.

  A t² + B t + C = 0; t = (−B ± √(B² − 4AC)) ÷ 2A

  Drawing sheet views, dimensions and PDF drawing.js

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• Holes for the hole table

  Collects each hole's axis, diameter and callout from the kernel's cylindrical faces.

  diameter = 2 × smallest radius on the axis

  Drawing sheet views, dimensions and PDF drawing.js

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• Hole table rows

  The X and Y of each hole measured from the view's lower-left corner, and its tag.

  x = o · right − u₀; y = o · up − v₀

  Drawing sheet views, dimensions and PDF drawing.js

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• Sheet scale

  Picks the drawing scale for a paper size.

  first s in the ISO 5455 series with width × s ≤ 0.82 × area width and height × s ≤ 0.82 × area height

  Drawing sheet views, dimensions and PDF drawing.js

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• Dimension text

  The text of an overall dimension or a hole table position.

  text = value rounded to 2 decimals

  Drawing sheet views, dimensions and PDF drawing.js

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• Overall dimensions

  The width and height on the front view and the depth on the top view.

  width = u₁ − u₀; height = v₁ − v₀

  Drawing sheet views, dimensions and PDF drawing.js

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• Numbers in the PDF

  Writes one coordinate or size into the PDF content.

  Drawing sheet views, dimensions and PDF drawing.js

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• PDF page content

  Writes the sheet's lines, fills and texts in millimeters under a millimeter to point transform.

  1 mm = 72 ÷ 25.4 pt

  Drawing sheet views, dimensions and PDF drawing.js

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• Section of a faceted mesh

  The section view's visible lines, cut outline, extents and the cutting plane's trace.

  Drawing section view of a faceted part drawing.js

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• Binary STL

  Writes each triangle's unit normal and three corners.

  n = (v₁ − v₀) × (v₂ − v₀) ÷ |(v₁ − v₀) × (v₂ − v₀)|

  Part export to STL, 3MF, DXF and STEP export.js

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• Numbers in a 3MF file

  Writes one vertex coordinate as text.

  Part export to STL, 3MF, DXF and STEP export.js

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• 3MF model

  Writes the mesh's vertices and triangles as a 3MF package in millimeters.

  Part export to STL, 3MF, DXF and STEP export.js

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• Numbers in a DXF file

  Writes one coordinate, radius or angle as text.

  Part export to STL, 3MF, DXF and STEP export.js

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• Sketch to DXF

  Writes a sketch's lines, circles, arcs, splines and points as DXF R12 entities.

  angle in degrees = radians × 180 ÷ π, reduced to [0, 360)

  Part export to STL, 3MF, DXF and STEP export.js

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• Flat pattern to DXF

  Writes a sheet metal flat pattern's outline, bend lines and bend notes.

  Part export to STL, 3MF, DXF and STEP export.js

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• Numbers in a STEP file

  Writes one real number in STEP's exponent form.

  Part export to STL, 3MF, DXF and STEP export.js

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• Outer loop of a planar STEP face

  Picks the outer boundary of a planar face as the loop with the largest signed area.

  area = ½ n · Σ pᵢ × pᵢ₊₁

  Part export to STL, 3MF, DXF and STEP export.js

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• Plane placement of a STEP face

  The x direction of the plane a planar face lies on.

  x = (e − (e · n) n) ÷ |e − (e · n) n|, e = (1,0,0) unless |n_x| ≥ 0.9, then (0,1,0)

  Part export to STL, 3MF, DXF and STEP export.js

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• Triangle normal

  The unit normal of one triangle and twice its area.

  n = (b − a) × (c − a); unit normal = n ÷ |n|

  Planar faces of a mesh faces.js

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• Coplanar triangle groups

  Joins triangles that share an edge and lie in one plane.

  n₁ · n₂ > 1 − 1e-5 and |d₁ − d₂| < 1e-5 × max(1, max |coordinate|)

  Planar faces of a mesh faces.js

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• Groups from the kernel's faces

  Uses the kernel's own face of each triangle as the group.

  Planar faces of a mesh faces.js

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• Face outline and area in its plane

  The outline loops of a planar face in two dimensions and the face's area.

  area = ½ Σ (x_j y_i − x_i y_j)

  Planar faces of a mesh faces.js

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• Dropping collinear outline vertices

  Removes outline vertices that lie on a straight run.

  |u × v| ÷ (|u| |v|) < 1e-6 and u · v > 0

  Planar faces of a mesh faces.js

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• Elastic matrix

  The 6 × 6 isotropic stress-strain matrix for a unit modulus.

  λ = ν ÷ ((1 + ν)(1 − 2ν)); μ = 1 ÷ (2(1 + ν)); D_ii = λ + 2μ, D_ij = λ, shear = μ

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Hexahedron stiffness and stress operators

  The 24 × 24 stiffness of a unit cube and the operators that give stress at its corners and center.

  K = Σ_gauss Bᵀ D B × ⅛; K_e = K_cc − K_ci K_ii⁻¹ K_ic; σ = (E ÷ h) S u

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Small dense solve

  Solves a small system by Gaussian elimination with row exchanges.

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Stiffness times displacement

  Applies the whole model's stiffness to a vector, element by element.

  y = Σ_e s_e × K_e x_e, s_e = E h × scale_e

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Node-block preconditioner

  Inverts each node's 3 × 3 diagonal block of the stiffness.

  A⁻¹ = adj(A) ÷ det(A)

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Constraint projections

  Turns each node's held directions into a projection that removes them.

  P = I − Σ q qᵀ over the orthonormal held directions q

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Eigenvalues of a small symmetric matrix

  Eigenvalues and vectors by cyclic Jacobi rotations.

  t = sign(θ) ÷ (|θ| + √(θ² + 1)), θ = (a_qq − a_pp) ÷ (2 a_pq)

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Is the body held

  Checks that the supports stop all six rigid motions.

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Gravity load

  Shares each element's weight equally to its eight corners.

  corner force = scale × ρ g h³ ÷ 8 × unit direction

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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• Stress at element corners and center

  The six stress components and von Mises stress at each element's eight corners and center, and the largest center value.

  σ = scale × (E ÷ h) × S u_e − scale × E α ΔT ÷ (1 − 2ν) on the direct components

  Voxel stress analysis: elements, loads, reactions and modes fea.js

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