Calculations

Calculations

• Structural load of a temperature rise

  Adds every element's thermal load into the nodal force vector of a thermal-stress study.

  f = Σₑ ∫ Bᵀ D ε_th dV

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Boundary conditions and area scale

  Converts the study's heat inputs to solver units and scales them for the difference between true and meshed face area.

  scale = true face area ÷ meshed area (1 when either is 0)

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Minimum and maximum temperature, maximum flux

  The lowest and highest temperature with their locations, and the largest heat flux in W/m².

  max flux in W/m² = max |q| ÷ 10⁻⁶

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Material values in solver units

  Conductivity and volumetric heat capacity of the material in the solver's units.

  k × 10⁻³; ρc = density × cp × 10⁻⁹

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Volume and mass of a voxel thermal study

  The meshed volume and the mass shown in the study summary.

  mass in kg = volume (mm³) × density (kg/m³) × 10⁻⁹

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Thread diameters

  The major, minor and pitch diameter and the thread depth of a size.

  ISO: depth = 5√3 ÷ 16 × P, d₁ = d − 2 × depth, d₂ = d − 0.649519 P; trapezoidal: depth = P ÷ 2, d₂ = d − P ÷ 2

  Thread dimensions and modeled threads threads.js

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  64-bit

• Size that fits a round face

  Chooses the thread size nearest a shaft's or a bore's diameter.

  shaft: nearest d; bore: nearest (d₁ + d₂) ÷ 2; accepted when the difference ≤ 0.12 × the face diameter

  Thread dimensions and modeled threads threads.js

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  64-bit

• Profile corners over one pitch

  The crest, flank and root corners of the thread profile as axial position and radius.

  ISO: crest P ÷ 8, root P ÷ 4; trapezoidal: crest = root = P (0.5 − tan(π ÷ 12) ÷ 2); flank = (P − crest − root) ÷ 2

  Thread dimensions and modeled threads threads.js

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  64-bit

• Profile radius at an axial position

  The radius of the thread profile at any position along the axis.

  r = r₀ + (r₁ − r₀)(w − u₀) ÷ (u₁ − u₀), w = u mod P

  Thread dimensions and modeled threads threads.js

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• Cross-section of a modeled thread

  The polygon cut square to the axis whose twisted extrusion is the thread.

  point = (r cos θ, r sin θ), r = profile(∓P θ ÷ 2π)

  Thread dimensions and modeled threads threads.js

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• Helical solid

  The thread's solid shape from the cross-section.

  turns = h ÷ P; slices = ceil(turns × max(24, segments)); twist = 360° × turns

  Thread dimensions and modeled threads threads.js

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  64-bit

• Cutting tool of a modeled thread

  The solid that is cut from a shaft or a bore to leave the thread, with its clearance.

  profile shifted by ∓clearance; start moved back by ceil(ext₀ ÷ P) × P

  Thread dimensions and modeled threads threads.js

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  64-bit

• Numbers in the thread callout

  Formats the pitch and the thread length written in a drawing's thread note.

  text = value rounded to 3 decimals, trailing zeros dropped

  Thread dimensions and modeled threads threads.js

  • PartCraft
  64-bit

• Density filter

  Averages each element's density with its neighbors within the minimum member size.

  x̃ₘ = Σ w xₙ ÷ Σ w, w = r − distance (in elements)

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Element strain energy

  uᵀ Kₑ u for every element, the quantity the sensitivities are built from.

  s = uₑᵀ Kₑ uₑ

  Generative design by topology optimization topopt.js

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  64-bit

• Stiffness scale, compliance and sensitivities

  The stiffness of each element from its density, the compliance of the design and how the compliance changes with each density.

  E = E_min + xᵖ (1 − E_min); c = Σ E k uᵀKₑu; ∂c/∂x = −p xᵖ⁻¹ (1 − E_min) k uᵀKₑu

  Generative design by topology optimization topopt.js

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  64-bit

• Threshold projection

  Pushes filtered densities toward 0 or 1 when the projection option is on.

  x = (tanh βη + tanh β(x̃ − η)) ÷ (tanh βη + tanh β(1 − η)), η = 0.5

  Generative design by topology optimization topopt.js

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  64-bit

• Density update at the target volume

  Updates every density and finds the multiplier at which the design's volume meets the target.

  x_new = clamp(x √(−∂c ÷ (∂v λ)), x ± move, 0, 1); bisect λ until (l₂ − l₁) ÷ (l₁ + l₂) ≤ 10⁻⁵

  Generative design by topology optimization topopt.js

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  64-bit

• Final compliance and returned densities

  The compliance of the final design and the density field handed to the next steps.

  c = Σ f·u

  Generative design by topology optimization topopt.js

  • PartCraft
  32-bit in part

• How many elements the solid design keeps

  The number of densest elements kept so the design holds its optimized volume, and the density of the last one kept.

  n = round(Σx), at least the preserved elements; level = density of the n-th densest

  Generative design by topology optimization topopt.js

  • PartCraft
  32-bit in part

• Surface level that holds the optimized volume

  Finds the density level whose surface encloses the optimized material volume.

  target = Σx × h³; 10 bisections of the level between 0.05 and 0.95

  Generative design by topology optimization topopt.js

  • PartCraft
  32-bit in part

• Surface vertex in a cell

  Places one surface vertex in each grid cell the surface passes through.

  t = (vₐ − iso) ÷ (vₐ − v_c), kept in [0.1, 0.9]; vertex = mean of the edge crossings

  Generative design by topology optimization topopt.js

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  64-bit

• Smoothed surface mesh and its volume

  Smooths the surface, restores its volume, and returns the mesh with its volume.

  Generative design by topology optimization topopt.js

  • PartCraft
  32-bit in part

• Surface smoothing

  Smooths the surface mesh without shrinking it as much as plain averaging would.

  p ← p + f (mean of neighbors − p), f alternating 0.5 and −0.53

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Volume restored after smoothing

  Moves every vertex along its normal by one distance so the mesh volume is the voxel volume again.

  d = (V₀ − V) ÷ area, three steps

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Volume of a closed mesh

  The volume enclosed by a triangle mesh.

  V = Σ a · (b × c) ÷ 6

  Generative design by topology optimization topopt.js

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  64-bit

• Check of the solid design

  The mass, volume, largest stress and displacement and the safety factor of the kept elements, solved again with the real material.

  volume = elements × h³; mass in kg = volume × ρ (t/mm³) × 1000; safety = yield ÷ max von Mises

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Compliance for another material

  Rescales the compliance of a shape when an outcome uses a different material.

  c = c₀ × E₀ ÷ E

  Generative design by topology optimization topopt.js

  • PartCraft
  64-bit

• Unit vector

  Scales a vector to length 1.

  a ÷ |a|

  Vectors, plane frames and transforms vec.js

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  64-bit

• Frame of a plane

  The origin and the x, y and normal axes given to a sketch plane.

  o = (p·n) n; x = Y × n (or world X made square to n when |n_y| > 0.9); y = n × x

  Vectors, plane frames and transforms vec.js

  • PartCraft
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• Sketch point to model space

  The 3D position of a sketch point (u, v) on its plane.

  p = o + u x + v y

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Model point to sketch coordinates

  The sketch coordinates of a 3D point on a plane.

  u = (p − o)·x; v = (p − o)·y

  Vectors, plane frames and transforms vec.js

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  64-bit

• Rotation about an axis

  The matrix that turns geometry by an angle about an axis through a point.

  R = c I + s [d]× + (1 − c) d dᵀ, c = cos a, s = sin a, a = deg × π ÷ 180; translation = p − R p

  Vectors, plane frames and transforms vec.js

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  64-bit

• Reflection in a plane

  The matrix that mirrors geometry in a plane.

  M = I − 2 n nᵀ; translation = 2 (o·n) n

  Vectors, plane frames and transforms vec.js

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  64-bit

• Matrix applied to a point

  Transforms a point by a 4 × 4 matrix.

  p′ = M₃ₓ₃ p + t

  Vectors, plane frames and transforms vec.js

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  64-bit

• Matrix product

  Multiplies two 4 × 4 matrices.

  C = A B

  Vectors, plane frames and transforms vec.js

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• Small linear system

  Solves A x = b for a small dense system.

  Gaussian elimination with partial pivoting, then back substitution

  Vectors, plane frames and transforms vec.js

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• Number shown with up to four decimals

  Formats a length or number for display.

  text = value rounded to dp decimals (4 by default); |value| < 10⁻⁹ shown as 0

  Vectors, plane frames and transforms vec.js

  • PartCraft
  64-bit

• Triangle buffer for drawing

  Copies the part's mesh into the vertex buffer the GPU draws.

  3D view of the part view3d.js

  • PartCraft
  32-bit, display only

• Result colors on the part

  Spreads per-vertex or per-corner result colors onto the drawn triangles.

  3D view of the part view3d.js

  • PartCraft
  32-bit, display only

• What is under the pointer

  Finds the face, edge or plane under the pointer.

  3D view of the part view3d.js

  • PartCraft
  32-bit, display only

• Grid size and origin

  The number of cubes along each axis and where the grid starts.

  n = max(1, ceil(size ÷ h − 10⁻⁶)); origin = box center − n h ÷ 2

  Voxel grid of a part voxel.js

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• Bounding box of the mesh

  The smallest box around the part's surface triangles.

  Voxel grid of a part voxel.js

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• Study size estimate

  The rough element count, node count and memory shown before a voxel study is meshed.

  elements ≈ round(volume ÷ h³)

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Where a row's ray crosses the surface

  The x positions at which the ray through a row of cube centers enters and leaves the part.

  x = (w₀ xₐ + w₁ x_b + w₂ x_c) ÷ (w₀ + w₁ + w₂), w from the y/z projection

  Voxel grid of a part voxel.js

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  64-bit

• Cubes between two crossings

  Marks the cubes of a row whose centers lie between an entry and an exit crossing.

  i from ceil((x_in − origin) ÷ h − 0.5) to floor((x_out − origin) ÷ h − 0.5)

  Voxel grid of a part voxel.js

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  64-bit

• Position of a grid node

  The coordinates of a grid node from its index.

  p = origin + (i, j, k) × h

  Voxel grid of a part voxel.js

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  64-bit

• Distance from a point to a triangle

  The squared distance from a grid node or cube center to a surface triangle.

  Voxel grid of a part voxel.js

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  64-bit

• Faces nearest a point

  Which of the part's faces a grid node lies on, with each face's normal there.

  faces with d² ≤ max(d_min² + (10⁻³ h)², (d_min + 10⁻³ h)²)

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit

• Tributary area of surface nodes

  The share of a picked face's area carried by each grid node, used to spread loads.

  each exposed cube face with all four corners on the picked faces gives h² ÷ 4 to each corner; area = Σ h²

  Voxel grid of a part voxel.js

  • PartCraft
  64-bit