Calculations

Calculations

• Overall dimensions

  The width and height on the front view and the depth on the top view.

  width = u₁ − u₀; height = v₁ − v₀

  Drawing sheet views, dimensions and PDF drawing.js

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• Numbers in the PDF

  Writes one coordinate or size into the PDF content.

  Drawing sheet views, dimensions and PDF drawing.js

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• PDF page content

  Writes the sheet's lines, fills and texts in millimeters under a millimeter to point transform.

  1 mm = 72 ÷ 25.4 pt

  Drawing sheet views, dimensions and PDF drawing.js

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• Section of a faceted mesh

  The section view's visible lines, cut outline, extents and the cutting plane's trace.

  Drawing section view of a faceted part drawing.js

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• Binary STL

  Writes each triangle's unit normal and three corners.

  n = (v₁ − v₀) × (v₂ − v₀) ÷ |(v₁ − v₀) × (v₂ − v₀)|

  Part export to STL, 3MF, DXF and STEP export.js

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• Numbers in a 3MF file

  Writes one vertex coordinate as text.

  Part export to STL, 3MF, DXF and STEP export.js

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• 3MF model

  Writes the mesh's vertices and triangles as a 3MF package in millimeters.

  Part export to STL, 3MF, DXF and STEP export.js

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• Numbers in a DXF file

  Writes one coordinate, radius or angle as text.

  Part export to STL, 3MF, DXF and STEP export.js

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• Sketch to DXF

  Writes a sketch's lines, circles, arcs, splines and points as DXF R12 entities.

  angle in degrees = radians × 180 ÷ π, reduced to [0, 360)

  Part export to STL, 3MF, DXF and STEP export.js

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• Flat pattern to DXF

  Writes a sheet metal flat pattern's outline, bend lines and bend notes.

  Part export to STL, 3MF, DXF and STEP export.js

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• Numbers in a STEP file

  Writes one real number in STEP's exponent form.

  Part export to STL, 3MF, DXF and STEP export.js

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• Outer loop of a planar STEP face

  Picks the outer boundary of a planar face as the loop with the largest signed area.

  area = ½ n · Σ pᵢ × pᵢ₊₁

  Part export to STL, 3MF, DXF and STEP export.js

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• Plane placement of a STEP face

  The x direction of the plane a planar face lies on.

  x = (e − (e · n) n) ÷ |e − (e · n) n|, e = (1,0,0) unless |n_x| ≥ 0.9, then (0,1,0)

  Part export to STL, 3MF, DXF and STEP export.js

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• Triangle normal

  The unit normal of one triangle and twice its area.

  n = (b − a) × (c − a); unit normal = n ÷ |n|

  Planar faces of a mesh faces.js

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• Coplanar triangle groups

  Joins triangles that share an edge and lie in one plane.

  n₁ · n₂ > 1 − 1e-5 and |d₁ − d₂| < 1e-5 × max(1, max |coordinate|)

  Planar faces of a mesh faces.js

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• Groups from the kernel's faces

  Uses the kernel's own face of each triangle as the group.

  Planar faces of a mesh faces.js

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• Face outline and area in its plane

  The outline loops of a planar face in two dimensions and the face's area.

  area = ½ Σ (x_j y_i − x_i y_j)

  Planar faces of a mesh faces.js

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• Dropping collinear outline vertices

  Removes outline vertices that lie on a straight run.

  |u × v| ÷ (|u| |v|) < 1e-6 and u · v > 0

  Planar faces of a mesh faces.js

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• Elastic matrix

  The 6 × 6 isotropic stress-strain matrix for a unit modulus.

  λ = ν ÷ ((1 + ν)(1 − 2ν)); μ = 1 ÷ (2(1 + ν)); D_ii = λ + 2μ, D_ij = λ, shear = μ

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

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• Hexahedron stiffness and stress operators

  The 24 × 24 stiffness of a unit cube and the operators that give stress at its corners and center.

  K = Σ_gauss Bᵀ D B × ⅛; K_e = K_cc − K_ci K_ii⁻¹ K_ic; σ = (E ÷ h) S u

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

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• Small dense solve

  Solves a small system by Gaussian elimination with row exchanges.

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

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• Stiffness times displacement

  Applies the whole model's stiffness to a vector, element by element.

  y = Σ_e s_e × K_e x_e, s_e = E h × scale_e

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

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• Node-block preconditioner

  Inverts each node's 3 × 3 diagonal block of the stiffness.

  A⁻¹ = adj(A) ÷ det(A)

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

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• Constraint projections

  Turns each node's held directions into a projection that removes them.

  P = I − Σ q qᵀ over the orthonormal held directions q

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

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• Eigenvalues of a small symmetric matrix

  Eigenvalues and vectors by cyclic Jacobi rotations.

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

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

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• Is the body held

  Checks that the supports stop all six rigid motions.

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

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• Gravity load

  Shares each element's weight equally to its eight corners.

  corner force = scale × ρ g h³ ÷ 8 × unit direction

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

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• Stress at element corners and center

  The six stress components and von Mises stress at each element's eight corners and center, and the largest center value.

  σ = scale × (E ÷ h) × S u_e − scale × E α ΔT ÷ (1 − 2ν) on the direct components

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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Sphere surface

  A triangle surface of a sphere from rings and segments.

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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Lattice Boltzmann fields for a script

  Velocity and density of every cell as tensors.

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

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• 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

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• 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

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• 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

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• 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

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