Calculations

Calculations

• Body mesh read back for named faces and edges

  Copies the current body's vertices and triangles so its faces and edges can be named and measured by later features.

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

  • PartCraft
  32-bit in part

• Fillet tangent points and arc

  Where a round of radius r touches the two faces of an edge, its center, and the points of its arc.

  s_A = ±r × (1 − n_A·n_B) ÷ (u_A·n_B); center = O + u_A × s_A ± n_A × r; angle = acos(v₀·v₁ ÷ r²)

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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• Chamfer distances

  The two setback distances of a chamfer, for equal, two-distance and distance-and-angle modes.

  θ = acos(u_a·u_b); angle mode: d_b = d × sin α ÷ sin(π − α − θ)

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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• Section and prism of a fillet or chamfer

  The cross-section that is extruded along the edge and cut from (or added to) the body.

  section pushed past both faces by 0.01 × size; prism length = edge length

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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• Shell by offsetting the body

  Hollows the body to a wall thickness, or thickens it outward, and opens the picked flat faces.

  inward: body − (body ⊖ ball(t)); outward: (body ⊕ ball(t)) − body

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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• Draft: new normal and hinge line

  The plane a drafted face turns to, and the line where the face meets the neutral plane.

  n₂ = unit(n_perp × cos a + pull × sin a); q = n × (d₁ − d₂ n·m) ÷ (1 − (n·m)²) + m × (d₂ − d₁ n·m) ÷ (1 − (n·m)²)

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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• Draft: outline of the face

  The outline of the drafted face, extruded into the prism that is trimmed by the old and new planes.

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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• Thread: extent of the round face

  Where a picked cylinder face starts and ends along its axis and whether it is a shaft or a bore, which set the thread's start, length and callout.

  t₀, t₁ = min and max of (p − o)·d over the face's vertices; full length = t₁ − t₀

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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  32-bit in part

• Mesh body from an imported file

  Scales an imported mesh to millimeters, turns it Y up and makes it a solid body.

  (x, y, z) → (x × s, z × s, −y × s) for a Z-up file; s = unit factor

  Fillet, chamfer, draft, thread and mesh bodies on the mesh kernel kernel.js

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  32-bit in part

• Matching tolerance and chord sag

  The distance within which a triangle's corners count as lying on a surface.

  sag = 1.5 × (1 − cos(π ÷ max(8, segments))); tol = 2e-6 × diagonal + 1e-7

  Named faces and edges from the body mesh meshfaces.js

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• Triangle on a surface

  Decides whether a triangle lies on a plane, cylinder, cone, sphere, torus or thread zone.

  plane: |(p − o)·n| ≤ tol; cylinder: −(tol + sag × r) ≤ ρ − r ≤ tol + 1e-6 × r

  Named faces and edges from the body mesh meshfaces.js

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• Face area and center

  The area of each named face and its area-weighted center.

  area = Σ |(b − a) × (c − a)| ÷ 2; center = Σ (triangle center × area) ÷ area

  Named faces and edges from the body mesh meshfaces.js

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• Edge length, ends and middle

  The length of an edge between two faces, its two ends and its middle point.

  len = Σ |pᵢ − pᵢ₋₁|; middle = the point at len ÷ 2 along the polyline

  Named faces and edges from the body mesh meshfaces.js

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• Straight edge or circle

  Classifies an edge as a line, a circle or a curve and gives a circle its center, axis and radius.

  line: every point within 10 × tol of the chord; circle: every point at one radius and one height about a round face's axis

  Named faces and edges from the body mesh meshfaces.js

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  32-bit in part

• Circle edge points put on the circle

  Moves a circular edge's ends and middle exactly onto its circle.

  p′ = c + r × w ÷ |w|, w = (p − c) − n × ((p − c)·n); closed circle: middle = 2c − a

  Named faces and edges from the body mesh meshfaces.js

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• Surfaces of pattern and mirror copies

  Moves a tool's named planes, axes and centers with the matrix of a pattern or mirror copy.

  point′ = M × point; direction′ = unit(linear part of M × direction)

  Named faces and edges from the body mesh meshfaces.js

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• Circle of an edge

  Center, radius and axis of a circular edge from its two ends and its middle point.

  c = A + ((n × ab) |ac|² + (ac × n) |ab|²) ÷ (2 |n|²), n = ab × ac; full circle: c = (a + middle) ÷ 2, r = |a − middle| ÷ 2

  Measuring and mass properties in 3D measure.js

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• Area of a picked group of facets

  The area and normal of a picked face when no face table is available.

  area = Σ |(b − a) × (c − a)| ÷ 2; plane if |Σ cross| > 0.999 × 2 × area

  Measuring and mass properties in 3D measure.js

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• Closest point on a triangle

  The point of a facet nearest to a given point, for the fallback distance.

  Measuring and mass properties in 3D measure.js

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• Closest points of two segments

  The nearest pair of points on two mesh edges, for the fallback distance.

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

  Measuring and mass properties in 3D measure.js

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• Minimum distance over the facets

  The smallest distance between two picked entities when the exact kernel's distance is not available.

  min over point–point, point–triangle, point–segment and segment–segment pairs

  Measuring and mass properties in 3D measure.js

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• Distance, angle and gaps between two picks

  Point-to-point distance, the angle between planes, lines or axes, the gap between parallel planes and the distance between centers or parallel axes.

  angle = acos(clamp(d₁·d₂)) in degrees; plane gap = |(o₂ − o₁)·n|; axis gap = |w − d (w·d)|

  Measuring and mass properties in 3D measure.js

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• Volume, center of mass and inertia from the mesh

  Mass properties of a closed triangle mesh at density 1.

  V = Σ a·(b × c) ÷ 6; ∫xxᵀ dV = Σ V_t ÷ 20 × (Σ pᵢpᵢᵀ + (Σpᵢ)(Σpᵢ)ᵀ); about the center: C − V g gᵀ

  Measuring and mass properties in 3D measure.js

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• Mass properties of a placed part

  Mass in grams and the inertia tensor in world axes for one placed part.

  I_world = (density ÷ 1000) × R I Rᵀ; center_world = M × center

  Measuring and mass properties in 3D measure.js

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• Mass properties of several parts together

  Total mass, combined center of mass and inertia tensor of an assembly.

  center = Σ mᵢ cᵢ ÷ Σ mᵢ; I = Σ (Iᵢ + mᵢ (|dᵢ|² E − dᵢ dᵢᵀ)), dᵢ = cᵢ − center

  Measuring and mass properties in 3D measure.js

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• Principal moments and axes

  The eigenvalues and eigenvectors of the inertia tensor.

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

  Measuring and mass properties in 3D measure.js

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• Exact minimum distance

  The minimum distance between two picked faces, edges, points or bodies and the two closest points.

  Measuring and mass properties in 3D measure-brep.js

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• Exact volume, area, center and inertia

  Mass properties of the part's exact body at density 1.

  I_xy = I((x + y) ÷ √2) − (I_xx + I_yy) ÷ 2, and the same for yz and xz

  Measuring and mass properties in 3D measure-brep.js

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• Curved edges of an assembly drawing view

  Turns each visible or hidden curve of a hidden-line view into the line segments the drawing keeps.

  n = clamp(⌈span ÷ size × 160⌉, 8, 240) equal parameter steps

  Measuring and mass properties in 3D measure-brep.js

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• Binary STL coordinates

  Reads the corner coordinates of each triangle from a binary STL file.

  Mesh import, repair and storage meshimport.js

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• 3MF vertex placement

  Applies the unit of each model file and the chain of build and component transforms to every vertex.

  p′ = [x s, y s, z s, 1] × M (row vectors, 4 × 3); s = unit of the part file ÷ unit of the root file

  Mesh import, repair and storage meshimport.js

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• Welding coincident points

  Merges points that lie within a tolerance of each other and maps every index to the point that is kept.

  merge when (Δx² + Δy² + Δz²) ≤ tol²; the earliest point is kept

  Mesh import, repair and storage meshimport.js

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• Repair tolerances from the mesh size

  Sets the weld distance, the gap-closing distance and the smallest triangle area kept, then welds and cleans.

  exact = 1e-7 × size; gap = 1e-3 × size; area limit = 1e-14 × size²; size = bounding-box diagonal

  Mesh import, repair and storage meshimport.js

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• Closing small gaps

  Merges boundary points that lie within the gap distance of another boundary point.

  Mesh import, repair and storage meshimport.js

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• Outward orientation by signed volume

  Turns each closed shell so that it encloses positive volume.

  6 V = Σ (a − o)·((b − o) × (c − o)), o = the shell's first point

  Mesh import, repair and storage meshimport.js

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• Filling a hole

  Closes a boundary loop with triangles.

  Newell normal n = Σ ((a_y − b_y)(a_z + b_z), …); ear when cross(a, b, c) > 0 and no other point is inside

  Mesh import, repair and storage meshimport.js

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• Growing flat faces

  Joins neighboring triangles into one flat face when they share a plane.

  join when n_s·n_t ≥ cos 0.05° and every corner of t is within tol of s's plane; tol = 2e-6 × size + 1e-7

  Mesh import, repair and storage meshimport.js

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• Plane of a flat face

  The normal and origin of each flat face a sketch can sit on.

  n = unit(Σ n_t A_t); origin = Σ (triangle center × A_t) ÷ Σ A_t

  Mesh import, repair and storage meshimport.js

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  32-bit in part

• Mesh kept in the part file

  Packs the repaired mesh into the text the part file stores.

  vertices → Float32Array → base64

  Mesh import, repair and storage meshimport.js

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  32-bit in part

• Mesh read back from the part file

  Unpacks and checks the stored mesh when a part is opened or rebuilt.

  Mesh import, repair and storage meshimport.js

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  32-bit in part

• Motion laws

  How far along its travel the joint is at each fraction of the time.

  linear: s; ease: s²(3 − 2s); sine: ½ − ½ cos(πs); round trip: 2s or 2 − 2s; value = from + (to − from) × law(k ÷ steps)

  Motion study of an assembly motion.js

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• Default tracked point

  The middle of a part's bounding box, tracked when no point is given.

  center = (min + max) ÷ 2 per axis

  Motion study of an assembly motion.js

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  32-bit in part

• Solids for the clearance check

  Rebuilds each part as a solid so the gap between parts can be measured at every step.

  Motion study of an assembly motion.js

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• Velocity and acceleration

  The rate of the joint value and each tracked point's velocity, speed and acceleration.

  v_k = (p_(k+1) − p_(k−1)) ÷ (t_(k+1) − t_(k−1)); a_k = (p_(k−1) − 2p_k + p_(k+1)) ÷ Δt²

  Motion study of an assembly motion.js

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• Peak speed, peak acceleration and travel

  The largest speed and acceleration of each tracked point and the length of its path.

  travel = Σ |p_k − p_(k−1)|

  Motion study of an assembly motion.js

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• Numbers in the CSV export

  Formats each time, value, position, speed and gap written to the CSV file.

  text = value.toFixed(5), trailing zeros removed

  Motion study of an assembly motion.js

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• Bilinear material constants

  Shear modulus, bulk modulus and hardening modulus from E, ν, yield stress and tangent modulus.

  G = E ÷ (2(1 + ν)); K = E ÷ (3(1 − 2ν)); H = E × E_t ÷ (E − E_t)

  Nonlinear and contact stress study nonlinear.js

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• Stress update at a Gauss point

  The stress, the new plastic strain and the algorithmic tangent for a given total strain.

  q = √(1.5 s:s); f = q − (σ_y + H α); Δγ = f ÷ (3G + H); σ = p 1 + (1 − 3GΔγ ÷ q) s

  Nonlinear and contact stress study nonlinear.js

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• Element tangent and internal force

  Integrates each element's tangent stiffness and internal force over its four Gauss points.

  K_e = Σ w Bᵀ D_ep B; f_e = Σ w Bᵀ σ; w = weight × det J

  Nonlinear and contact stress study nonlinear.js

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• Load-stepped Newton solve

  Raises the load in steps and iterates each step to equilibrium.

  r = λ f − f_int(u) (− contact); converged when |r| ÷ (λ |f|) < tol

  Nonlinear and contact stress study nonlinear.js

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