Calculations

Calculations

• Mesh quality summary

  The smallest and largest dihedral angle, the worst and mean aspect ratio and a histogram, as shown with the study.

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Element size from surface curvature

  Estimates the radius of curvature between neighboring surface triangles of one face and the element size it allows.

  angle = acos(n₁·n₂); r ≈ |c₁ − c₂| ÷ angle; size = r × curv

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Element size from wall thickness and gaps

  Measures the wall behind and the gap in front of each surface triangle and the element size they allow.

  wall: size = distance ÷ thick; gap: size = distance ÷ 1.5

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Background lattice spacing

  Chooses the cube size of the background lattice so the part's box is a whole number of cubes where possible.

  s = L_min ÷ ceil(L_min ÷ h − 10⁻⁹); per axis: L ÷ n when within 8 % of s

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Longest edge chosen for bisection

  Picks the edge of an element that is split, the same way in every element that shares it.

  compare round(length² ÷ unit × 10⁷); ties by vertex numbers

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Where a lattice edge crosses the surface

  The point on a lattice edge where it leaves the part.

  p = a + t (b − a), t from the first ray hit; else 40 bisections of the inside test

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Lattice vertices warped onto the surface

  Moves a lattice vertex onto a nearby cut point so no very short edge is created.

  warp when fraction of the edge < α (0.24999, 0.41189 or 0.33)

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Closest point on the part's edges

  Finds the nearest point on the part's edge polylines, used to snap boundary vertices onto sharp edges.

  t = clamp(((p − a)·d) ÷ |d|², 0, 1); distance = √min d²

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Boundary vertices onto exact faces

  Moves each boundary vertex onto the true surface of its faces when all of them have an exact record.

  move when 10⁻¹² × diagonal ≤ distance ≤ 0.25 h and the tets around keep min dihedral ≥ min(10°, before)

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Mesh volume

  The volume of the linear tetrahedral mesh, kept in its statistics.

  V = Σₑ volume of tet e

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Interior vertices moved to improve angles

  Moves interior vertices of poor elements when that raises the worst dihedral angle around them.

  accept a trial position when the worst angle rises by more than 10⁻⁹ degrees

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Jacobian check of a quadratic element

  The Jacobian determinant at the four corners and four Gauss points of a 10-node tetrahedron.

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Midside nodes onto the exact surface

  Places the middle node of each boundary edge on the true surface so curved faces stay curved.

  start at (a + b) ÷ 2; move onto the faces when every Jacobian stays > 0.2 × its straight value

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Stefan–Boltzmann constant

  The radiation constant in the solver's units, W/(mm²·K⁴).

  σ = 5.670374419 × 10⁻¹⁴

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Unit factors from SI

  The factors that convert conductivity, flux, film coefficient, heat generation and heat capacity from SI to the solver's millimeter units.

  k × 10⁻³; flux and h × 10⁻⁶; generation and ρc × 10⁻⁹

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Unit-cube element matrices

  The conduction, capacity and thermal-load matrices of a unit cube, scaled for every voxel element.

  K = Σ_g ∇Nₐ·∇N_b ÷ 8; M = Σ_g NₐN_b ÷ 8

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Thermal load of the incompatible-mode element

  The share of a thermal load carried by the element's internal bubble modes, and their thermal stress.

  f = E α h² (TH ÷ (1 − 2ν) + Aᵀ Fᵢ) ΔT, A = −Kᵢᵢ⁻¹ Kᵢc

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Voxel element routines

  Conduction, capacity, heat load, gradient, thermal load, stresses and geometric stiffness of one voxel element.

  K = k h K_unit; C = ρc h³ M_unit (lumped: ρc h³ ÷ 8); ∇T = Σ G₀ Tₐ ÷ h

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Area of a boundary quad

  The area of a four-node boundary facet.

  A = |d₁ × d₂| ÷ 2 (d₁, d₂ the diagonals)

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Volume and facet areas of the voxel mesh

  The mesh volume and the area of every exposed voxel face.

  volume = elements × h³; facet area = h²

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Incomplete Cholesky preconditioner

  Factors the conduction matrix on its own pattern to speed up the temperature solve.

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Conjugate-gradient solve for temperatures

  Solves the heat system for the temperature of every free node.

  stop when ‖r‖ ÷ ‖b‖ < 10⁻¹⁰

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Heat flux from a power

  Turns a total power on picked faces into a flux per unit area.

  q = P ÷ A (meshed area); flux input: q × scale

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Radiation as a film coefficient

  The equivalent film coefficient of a radiating facet at its current temperature.

  h_r = ε σ (T² + T∞²)(T + T∞) × scale, T in kelvin (°C + 273.15)

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Steady solve with radiation

  Repeats the temperature solve with updated radiation coefficients until the temperatures settle.

  stop when max |T_new − T_old| ≤ 10⁻⁶ K, at most 60 passes

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Transient time stepping

  Advances the temperature field in time and stores the frames shown on the time slider.

  (C ÷ Δt + θK) T⁺ = (C ÷ Δt − (1 − θ)K) T + f, Δt = t ÷ steps, time of step s = s × Δt

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Heat balance

  The heat entering through each boundary condition, generated, stored, and what is left over.

  imbalance = inflow − outflow − stored; fixed temperature: Q = (K T)ₙ − other heat into node n

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• Heat flux

  The heat flux of every element, its average at the nodes and the largest element value.

  q = −k ∇T; |q| = √(q_x² + q_y² + q_z²)

  Heat conduction, steady and transient thermal.js

  • PartCraft
  64-bit

• 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

  • PartCraft
  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

  • PartCraft
  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

  • PartCraft
  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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  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

  • PartCraft
  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

  • PartCraft
  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

  • PartCraft
  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

  • PartCraft
  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

  • PartCraft
  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

  • PartCraft
  64-bit