Calculations

Calculations

• Support directions on faces

  Finds which nodes of a supported face are held and the normal direction used for a sliding support.

  held when distance to the exact face ≤ 0.002 × mean edge length

  Tetrahedral stress, vibration and buckling solve tetfea.js

  • PartCraft
  64-bit

• Is the part held

  Checks that the supports stop all six rigid-body motions.

  free when an eigenvalue of Σ rᵀr is ≤ 10⁻⁹ × the largest, or ≤ 10⁻¹²

  Tetrahedral stress, vibration and buckling solve tetfea.js

  • PartCraft
  64-bit

• Natural frequencies

  The lowest natural frequencies and mode shapes of the supported part.

  K̃ x = λ M x; f = √max(λ, 0) ÷ 2π

  Tetrahedral stress, vibration and buckling solve tetfea.js

  • PartCraft
  64-bit

• Temperature rise for thermal stress

  The temperature rise at every node over the stress-free temperature, and the load it adds.

  ΔT = T − T_ref (20 when not given); f += thermal load

  Tetrahedral stress, vibration and buckling solve tet-elements.js

  • PartCraft
  64-bit

• Buckling pre-stress and geometric stiffness

  Solves the pre-stress and assembles the stress stiffness that the buckling eigen-solve uses.

  K u = f (tol 10⁻¹⁰); G = −Σₑ Gₑ; each inner solve tol 10⁻⁸

  Tetrahedral stress, vibration and buckling solve tet-elements.js

  • PartCraft
  64-bit

• Longest edge of a tetrahedron

  The length of the longest of an element's six edges.

  L = √max((Δx)² + (Δy)² + (Δz)²)

  Automatic mesh refinement of tetrahedral studies tetadapt.js

  • PartCraft
  64-bit

• Which elements to refine and to what size

  Marks the elements with the largest error and gives each a new target edge length.

  ē = (target ÷ 100) √((U + Σe²) ÷ m); h_new = h × clamp((e_K ÷ ē)^(−1/2), 0.5, 0.7)

  Automatic mesh refinement of tetrahedral studies tetadapt.js

  • PartCraft
  64-bit

• New boundary vertices onto the exact faces

  Moves a vertex created on the boundary onto the true surface of its faces, unless that would damage the elements around it.

  accept when every tet keeps volume > 0 and min dihedral ≥ min(10°, before)

  Automatic mesh refinement of tetrahedral studies tetadapt.js

  • PartCraft
  64-bit

• Volume of the refined mesh

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

  V = Σₑ volume of tet e

  Automatic mesh refinement of tetrahedral studies tetadapt.js

  • PartCraft
  64-bit

• When the refinement stops

  Compares each pass's estimated error with the target and decides whether to refine again.

  stop when η ≤ target, or passes used, or elements > limit

  Automatic mesh refinement of tetrahedral studies tetadapt.js

  • PartCraft
  64-bit

• Closest point on a triangle

  The nearest point of a surface triangle to a given point, and the squared distance.

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Ray against a triangle

  Where a ray crosses a surface triangle, used for inside tests, wall thickness and cut points.

  t = (e₂·q) ÷ det, det = e₁·(d × e₂)

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Is a point inside the part

  Decides whether a point lies inside the closed surface.

  inside when most clean rays cross the surface an odd number of times

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Projection onto a face

  Moves a point onto a plane, a cylinder, a recorded surface or, failing those, the face's triangles.

  plane: p − ((p − o)·n) n; cylinder: axis point + r × radial ÷ |radial|

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Projection onto the meeting of several faces

  Moves a point onto the edge or corner where two or three faces meet.

  repeat: project onto each face in turn, until the move < 10⁻¹² × diagonal

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Signed volume of a tetrahedron

  The volume of one element, positive for the expected vertex order.

  V = (u · (v × w)) ÷ 6

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• Dihedral angles and aspect ratio of a tetrahedron

  The smallest and largest dihedral angle and the aspect ratio of one element.

  θ = atan2(1.5 V |e|, −n₁·n₂ ÷ 4); aspect = L_max ÷ (2√6 × r_in), r_in = 3V ÷ ΣA

  Tetrahedral mesh of a part tetmesh.js

  • PartCraft
  64-bit

• 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