Calculations

Calculations

• Reactions and compliance

  The support reactions and the work of the loads.

  R = (K u − f) projected on the held directions; compliance = Σ fᵢ uᵢ

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

  • PartCraft
  64-bit

• Field value at a point

  Interpolates a nodal result at a point inside the grid.

  value = Σ wᵢ vᵢ ÷ Σ wᵢ over the active corners of the cell

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

  • PartCraft
  64-bit

• Lumped mass

  The mass at each node for the modal analysis.

  node mass = Σ over its elements of ρ h³ ÷ 8

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

  • PartCraft
  64-bit

• Natural frequencies

  The lowest natural frequencies and mode shapes.

  f = √λ ÷ (2π); stop when |λ − λ(previous)| ≤ 1e-6 × λ for every wanted mode

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

  • PartCraft
  64-bit

• Mesh volume and mass

  The volume and mass reported with a study.

  volume = elements × h³; mass (kg) = volume × ρ (t/mm³) × 1000

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

  • PartCraft
  64-bit

• Minimum safety factor

  The yield stress divided by the largest nodal von Mises stress.

  safety = S_y ÷ σ_max

  Voxel stress analysis: 32-bit steps in loads and results fea.js

  • PartCraft
  32-bit in part

• Values on the plot mesh

  Stress, displacement size and displacement vector at each vertex of the plot mesh, read by the hover probe and the legend.

  Voxel stress analysis: 32-bit steps in loads and results fea.js

  • PartCraft
  32-bit in part

• Force and pressure on faces

  Shares a force or a pressure over the nodes of the picked faces.

  node force = magnitude × weightᵢ ÷ Σ weight × direction; pressure magnitude = p × area

  Voxel stress analysis: 32-bit steps in loads and results fea.js

  • PartCraft
  32-bit in part

• Cylinder surface

  A triangle surface of a cylinder with n sides.

  (r cos(2π i ÷ n), r sin(2π i ÷ n))

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Sphere surface

  A triangle surface of a sphere from rings and segments.

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Structured hexahedron grid

  Node coordinates and elements of a box divided into equal cells.

  x = o + l × i ÷ n

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Voxel mesh nodes

  Node coordinates of a voxel grid as a script mesh.

  x = origin + i × h

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Hexahedron stiffness for a script

  The 24 × 24 stiffness of a cube of side h.

  K = E × h × K_unit

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Hexahedron stress operator for a script

  The matrix that turns a cube's nodal displacements into stresses at its corners and center.

  σ = (E ÷ h) × S u

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Mean stress of quadratic tetrahedra

  Each element's stress as the average of its four Gauss points.

  σ_e = ¼ Σ σ_gauss

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Voxel element stress for a script

  Stress components and von Mises stress at the center of each voxel element.

  σ = (E ÷ h) × S_center u_e

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Voxel solve for a script

  Solves a voxel model for displacements with the multigrid conjugate gradient solver.

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Lattice Boltzmann fields for a script

  Velocity and density of every cell as tensors.

  CraftScript meshes and element kernels cs-lib-mesh.js

  • PartCraft
  64-bit

• Fluid properties

  Density and viscosity of air and water at 20 °C.

  ρ_air = p ÷ (R T); μ_air = μ₀ (T ÷ T₀)^1.5 (T₀ + S) ÷ (T + S)

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Lattice links and boundary terms

  Connects each fluid cell to its neighbors and sets the term of every boundary link.

  moving wall: a = 6 wᵢ (cᵢ · u); interpolated wall when |q − ½| > 1e-6, q clamped to [0.001, 1]

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Population arriving from a boundary

  The value a boundary link returns to its cell.

  q < ½: 2q f_o + (1 − 2q) f_behind; q ≥ ½: (f_o + (2q − 1) f_q) ÷ (2q); pressure: −f_o + 2 wᵢ ρ_w (1 + 4.5 (c·u)² − 1.5 u²)

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Initial state and relaxation times

  Fills every cell with the equilibrium for the starting velocity and sets the two relaxation rates.

  f_eq = wᵢ (1 + 3 c·u + 4.5 (c·u)² − 1.5 u²); τ⁻ = ½ + Λ ÷ (τ − ½)

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Boundary links as linear terms

  Writes every wall, moving-wall and slip link as two coefficients and a constant, for the time step's inner loop.

  f = c₁ f₁ + c₂ f₂ + a; q < ½: c₁ = 2q, c₂ = 1 − 2q; q ≥ ½: c₁ = 1 ÷ (2q), c₂ = (2q − 1) ÷ (2q)

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• One time step: stream and collide

  Pulls the populations into each cell, takes density and velocity, and relaxes toward equilibrium.

  ρ = Σ fᵢ; u = (Σ cᵢ fᵢ + F ÷ 2) ÷ ρ

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Collision with an eddy viscosity

  The collision used in turbulent runs, with a per-cell viscosity and a stabilizer.

  τ = ½ + 3(ν + ν_t); τ_s = ¼ + √(1/16 + 4.5 √2 C² |Π| ÷ ρ); τ = max(τ, τ_s)

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Mass through boundaries and force on walls

  Sums the net mass rate per boundary group and the momentum-exchange force on the part.

  mass = Σ (f_in − f_out); F = −Σ cᵢ (f_in + f_out)

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Stepping to a steady state

  Repeats time steps until the velocity field stops changing.

  res = √(Σ |u − u_prev|² ÷ Σ |u|²) ÷ steps between checks

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Lattice units from physical ones

  Chooses the lattice speed, viscosity, relaxation time and time step for a cell size, speed and fluid.

  Re_cell = U dx ÷ ν; u = min(0.05, √(0.03 ÷ k), 0.5 Re_cell); τ = 3 u ÷ Re_cell + ½; dt = u dx ÷ U

  Flow study: lattice Boltzmann solver flow.js

  • PartCraft
  64-bit

• Face normal and area

  The area-weighted outward normal of a face, its area and how flat it is.

  area = Σ |nₜ| ÷ 2; flatness = |Σ nₜ| ÷ (2 × area)

  Flow study: fluid cells, openings and wall distances flow-domain.js

  • PartCraft
  64-bit

• Area of an opening's holes

  The area of the holes in a flat face, which is the opening's true area.

  loop area = ½ Σ (pₐ × p_b) · n; holes = Σ −area over clockwise loops

  Flow study: fluid cells, openings and wall distances flow-domain.js

  • PartCraft
  64-bit

• Lid cells over an opening

  Finds the cells just behind a flat face that cover its holes.

  raster area = hole pixels × (h ÷ 2)²

  Flow study: fluid cells, openings and wall distances flow-domain.js

  • PartCraft
  64-bit

• Wall distance along a link

  Where the link from a fluid cell to a solid cell crosses the part's surface.

  t = (e₂ · (tv × e₁)) ÷ (e₁ · (d × e₂))

  Flow study: fluid cells, openings and wall distances flow-domain.js

  • PartCraft
  64-bit

• Frontal area

  The area of the part seen along the flow, used as the reference area for drag and lift coefficients.

  area = covered pixels × pixel size², pixel size = larger extent ÷ 1024

  Flow study: fluid cells, openings and wall distances flow-domain.js

  • PartCraft
  64-bit

• Automatic cell size

  Picks the cell size that gives about a target number of cells in the grid box.

  h = ∛(volume ÷ target), rounded up to 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 or 10 times a power of ten

  Flow study: fluid cells, openings and wall distances flow-study.js

  • PartCraft
  64-bit

• Cell count estimate

  The number of grid cells a study will use.

  n = ⌈extent ÷ h − 1e-6⌉ + margin cells, per axis

  Flow study: fluid cells, openings and wall distances flow-study.js

  • PartCraft
  64-bit

• Lift direction

  The direction lift is measured along, made square to the flow.

  l = l₀ − (l₀ · d) d, then divided by its length

  Flow study: fluid cells, openings and wall distances flow-study.js

  • PartCraft
  64-bit

• Inlet flow and speed

  Each inlet's flow rate, mean speed and the speed set on its lattice links.

  Q = v × A (or the given Q); U = Q ÷ A; link speed = Q × (A_face ÷ A) ÷ (lattice link area × dx²); D = √(4A ÷ π)

  Flow study: results in physical units flow-worker.js

  • PartCraft
  64-bit

• Speed and pressure scales

  The factors between lattice and physical speed and pressure, and the density set at each outlet.

  c_u = dx ÷ dt; c_p = ρ c_u²; ρ_outlet = 1 + 3 (p − p_ref) ÷ c_p

  Flow study: results in physical units flow-worker.js

  • PartCraft
  64-bit

• Second steady-state test

  Watches the pressure drop or the drag force between checks before the run is accepted as steady.

  |q − q_prev| ÷ max(|q|, 1e-30) < 1e-4 on two checks in a row

  Flow study: results in physical units flow-worker.js

  • PartCraft
  64-bit

• Mass flow, mean pressure and pressure drop

  The mass and volume flow through each opening, its mean pressure, and the pressure drop between inlets and outlets.

  ṁ = lattice mass × ρ dx³ ÷ dt; p = p_ref + (mean ρ − 1) ÷ 3 × c_p; Δp = flow-weighted mean inlet p − outlet p

  Flow study: results in physical units flow-worker.js

  • PartCraft
  64-bit

• Force, drag and lift coefficients

  The force on the part in newtons and, for external flow, drag, lift and their coefficients.

  F = lattice force × ρ dx⁴ ÷ dt²; C_d = F · d ÷ (½ ρ U² A)

  Flow study: results in physical units flow-worker.js

  • PartCraft
  64-bit

• Reynolds number

  The Reynolds number of the study.

  Re = U √(4A ÷ π) ÷ ν (internal); Re = U L ÷ ν (external)

  Flow study: results in physical units flow-worker.js

  • PartCraft
  64-bit

• Inflow eddy viscosity

  The turbulence level given to inflows in a turbulent run.

  ν_t = max(3ν, √1.5 × 0.09^0.25 × I × u × ℓ), I = 5 % (internal) or 1 % (external)

  Flow study: results in physical units flow-worker.js

  • PartCraft
  64-bit

• Mass balance

  The share of the inflow that does not leave through the outlets.

  balance = (ṁ_in − ṁ_out) ÷ ṁ_in

  Flow study: results in physical units flow-ui.js

  • PartCraft
  64-bit

• The mesh handed to the flow solver

  Builds the job for a flow study from the part and the study's settings.

  Flow study: 32-bit steps in the mesh and the result fields flow-study.js

  • PartCraft
  32-bit in part

• Cell fields, maximum speed and pressure range

  Velocity and pressure of every cell in physical units, and the largest speed and lowest and highest pressure the results state.

  v = u × c_u; p = p_ref + (ρ − 1) ÷ 3 × c_p; v_max = max |v|

  Flow study: 32-bit steps in the mesh and the result fields flow-worker.js

  • PartCraft
  32-bit in part

• Pressure on the part's surface

  The static pressure at each vertex of the plot mesh, sampled a little way into the fluid.

  p = Σ (wᵢ + 1e-6) pᵢ ÷ Σ (wᵢ + 1e-6) over the fluid cells around the sample point

  Flow study: 32-bit steps in the mesh and the result fields flow-worker.js

  • PartCraft
  32-bit in part

• Result probe

  Shows the speed, velocity and pressure of the cell under the pointer, or the surface pressure.

  speed = √(v_x² + v_y² + v_z²)

  Flow study: 32-bit steps in the mesh and the result fields flow-ui.js

  • PartCraft
  32-bit in part

• Lattice Mach number warning

  The fastest cell's speed as a share of the lattice speed of sound.

  Ma = v_max × dt ÷ dx × √3

  Flow study: 32-bit steps in the mesh and the result fields flow-ui.js

  • PartCraft
  32-bit in part

• Model constants

  The constants of the turbulence model.

  c_w1 = c_b1 ÷ κ² + (1 + c_b2) ÷ σ

  Flow study: turbulence model and wall functions flow-turb.js

  • PartCraft
  64-bit