Calculations

Calculations

• Sketch curves as kernel edges

  Turns the lines, arcs, circles and splines of a sketch into the kernel's edges for a loft.

  arc: start, middle and end points c + r × (cos, sin) at a₀, a₀ + sweep ÷ 2, a₀ + sweep, then an arc through the three points

  Surface features built with the kernel surface.js

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• Network surface as a kernel face

  Hands the surface fitted through a network of curves to the kernel and sets the tolerance its seams are sewn with.

  face tolerance 1e-9; sewing tolerance = max(1e-7, 1e-7 × size)

  Surface features built with the kernel surface.js

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• Area of a surface body

  The area of each surface body left after a rebuild.

  area = Σ over the faces of OpenCascade's surface area

  Surface features built with the kernel surface.js

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• Basis functions and derivatives

  The B-spline basis functions and their derivatives at a parameter.

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Point and derivatives of a NURBS curve

  A rational curve's point and first two derivatives at a parameter.

  C = A ÷ w; C′ = (A′ − w′·C) ÷ w; C″ = (A″ − 2w′·C′ − w″·C) ÷ w

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Point and derivatives of a line, circle or ellipse

  A profile curve's point and first two derivatives at a parameter.

  circle and ellipse: C = c + a·cos t·x + b·sin t·y

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Point and derivatives of a B-spline surface

  A B-spline surface's point and all derivatives up to second order at (u, v).

  homogeneous sums A[k][l] = Σ Nᵤ⁽ᵏ⁾·Nᵥ⁽ˡ⁾·w·P; S[k][l] by the quotient rule with binomial factors 1, 2, 1

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Closest point on a sphere, torus or cone

  The nearest point, the normal there and the distance, in closed form.

  sphere: c + r·n̂, d = | |p − c| − r |; torus: tube center m = c + R·ê, then m + r·n̂; cone: τ = ((ρ − r₀)·k + t) ÷ (1 + k²)

  Exact surfaces for the tetrahedral mesher surfaces.js

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• One-dimensional Newton minimizer

  Refines the parameter of the nearest point on a profile curve.

  Δt = −f′ ÷ f″ (or one sample spacing downhill when f″ ≤ 0); stop when |Δt| ≤ 1e-15 × (|b − a| + |t|)

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Start points for the minimizer

  Samples the distance along a curve and runs Newton from the two best samples.

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Closest point on a surface of revolution

  The nearest point on a profile turned about an axis.

  minimize ½((ρ(v) − ρ_p)² + (t(v) − t_p)²) over the profile parameter v

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Closest point on an extruded surface

  The nearest point on a curve swept along a direction.

  minimize ½ |across(p − C(u))|²; P = C(u) + ((p − C(u)) · d)·d

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Closest point on a B-spline surface

  The nearest point on a B-spline face, kept inside the face's parameter box.

  Newton on ∇½|S(u, v) − p|² = 0 with the Hessian [Sᵤ·Sᵤ + Sᵤᵤ·r, Sᵤ·Sᵥ + Sᵤᵥ·r; …]

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Smallest radius of curvature near a point

  The radius of curvature the mesher uses to size elements on a curved face.

  sphere, torus: r; cone: ρ × √(1 + k²); else 1 ÷ max |Δn| ÷ moved over three directions 60° apart

  Exact surfaces for the tetrahedral mesher surfaces.js

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• Number formatter

  Writes one number as macro text.

  text = toPrecision(v, 15); |v| < 1e-14 → 0

  Numbers written into the VBA macro export swmacro.js

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• Millimeters to meters

  Converts a length or a coordinate from millimeters to the meters the macro uses.

  m = round(mm × 10⁶) ÷ 10⁹

  Numbers written into the VBA macro export swmacro.js

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• Degrees to radians

  Converts an angle in degrees to the radians the macro uses.

  rad = deg × π ÷ 180

  Numbers written into the VBA macro export swmacro.js

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• Unit-vector component

  Writes a component of a sketch plane's normal or x direction.

  u = round(v × 10¹²) ÷ 10¹²

  Numbers written into the VBA macro export swmacro.js

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• Sketch dimension value and text position

  Works out the value of each sketch dimension in meters or radians and where its text sits.

  length: value ÷ 1000; angle: value × π ÷ 180; arc radius = |start − center|

  Numbers written into the VBA macro export swmacro.js

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• Pick point on a face

  Finds a point inside a face, used by the macro to select that face.

  centroid of the largest triangle = (a + b + c) ÷ 3, moved onto the plane: p − ((p − o)·n) n

  Numbers written into the VBA macro export swmacro.js

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• Countersink depth and taper

  The depth and half angle of the tapered cut that stands in for a countersink.

  depth = (D − d) ÷ 2 ÷ tan(angle × π ÷ 360); taper = angle ÷ 2 × π ÷ 180

  Numbers written into the VBA macro export swmacro.js

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• Expected volume and box after each step

  Writes the part's volume and bounding box after a feature, for the macro to compare with what it built.

  volume in m³ = volume in mm³ × 10⁻⁹; box corners through the millimeter-to-meter conversion

  Numbers written into the VBA macro export swmacro.js

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• Body check tolerances in the macro

  The comparison the macro makes between the body it built and the expected volume and box.

  |v − vol| ≤ 10⁻⁴ × |vol| + 10⁻¹⁵; |box − expected| ≤ 10⁻⁵ m + 0.02 × diagonal

  Numbers written into the VBA macro export swmacro.js

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• Final volume comparison

  Writes the finished part's volume into the macro and the test that reports whether the rebuilt part matches.

  |v − vol| ≤ 10⁻⁴ × vol, v = reported volume × 10⁹ (mm³)

  Numbers written into the VBA macro export swmacro.js

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• Numbers in equations

  Writes a parameter expression as equation text for the macro, including its numeric constants.

  constant → formatNumber(value); pow(a, b) → (a) ^ (b); log(x) → log(x) ÷ log(10)

  Numbers written into the VBA macro export swmacro.js

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• 64-point rule on the tetrahedron

  Builds the integration points and weights used for mass and body loads.

  x = u, y = v(1 − u), z = w(1 − u)(1 − v); weight = wₐ w_b w_c ÷ 8 × (1 − u)² (1 − v)

  Quadratic tetrahedron element matrices tetfea.js

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• Shape functions of the 10-node tetrahedron

  The ten shape functions and their derivatives at a point of the element.

  corner: N = L(2L − 1); midside: N = 4 LᵢLⱼ

  Quadratic tetrahedron element matrices tetfea.js

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• Jacobian and physical derivatives

  The element's Jacobian determinant at a point and the shape-function derivatives in x, y and z.

  J = Σ xₐ ∂Nₐ/∂ξ; det J by cofactors; ∂N/∂x = J⁻ᵀ ∂N/∂ξ

  Quadratic tetrahedron element matrices tetfea.js

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• Element stiffness

  The 30 × 30 stiffness matrix of one element and its volume.

  Kₑ = Σₚ wₚ det Jₚ E · Bᵀ D B; volume = Σₚ wₚ det Jₚ

  Quadratic tetrahedron element matrices tetfea.js

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• Element mass and shape-function integrals

  The consistent mass matrix of one element and ∫N dV for each node, used for gravity.

  Mₑ = Σₚ ρ wₚ det Jₚ NₐN_b; ∫Nₐ dV = Σₚ wₚ det Jₚ Nₐ

  Quadratic tetrahedron element matrices tetfea.js

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• Stress and strain at a Gauss point

  The six strains and six stresses at one Gauss point from the element's displacements.

  ε = B u; σ = E · D (ε − α ΔT [1 1 1 0 0 0])

  Quadratic tetrahedron element matrices tetfea.js

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• 6-point rule on a triangle face

  The points and weights used to integrate loads, areas and heat exchange over boundary faces.

  Quadratic tetrahedron element matrices tetfea.js

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• Shape functions of the 6-node face

  The six shape functions of a boundary face and their derivatives.

  corner: N = L(2L − 1); midside: N = 4 LᵢLⱼ

  Quadratic tetrahedron element matrices tetfea.js

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• Integration over a boundary face

  The area-weighted normal and position at each rule point of a curved boundary face.

  n dA = (∂x/∂L₁ × ∂x/∂L₂) × w ÷ 2

  Quadratic tetrahedron element matrices tetfea.js

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• Area element on a face for heat terms

  The area weight at each rule point of a 6-node face, for heat exchange and heat loads.

  dA = |∂x/∂L₁ × ∂x/∂L₂| × w ÷ 2

  Quadratic tetrahedron element matrices tet-elements.js

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• Element conductivity matrix

  The 10 × 10 heat conduction matrix of one element.

  Kₑ = Σₚ wₚ det Jₚ k ∇Nₐ·∇N_b

  Quadratic tetrahedron element matrices tet-elements.js

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• Element heat capacity

  The heat capacity matrix of one element, or its lumped diagonal for transient runs.

  M = Σₚ wₚ det Jₚ ρc NₐN_b; lumped: Mₐₐ × (ΣM ÷ Σ diagonal)

  Quadratic tetrahedron element matrices tet-elements.js

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• Element heat generation load

  The nodal heat input of a uniform internal heat source.

  fₐ = Σₚ wₚ det Jₚ Q Nₐ

  Quadratic tetrahedron element matrices tet-elements.js

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• Temperature gradient at the element center

  The temperature gradient used for the heat flux of an element.

  ∇T = Σ ∇Nₐ Tₐ at L = (¼, ¼, ¼, ¼)

  Quadratic tetrahedron element matrices tet-elements.js

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• Element thermal load

  The nodal forces that a temperature rise produces in one element.

  f = Σₚ wₚ det Jₚ E α ΔTₚ (D₀₀ + D₀₁ + D₀₂) ∇N

  Quadratic tetrahedron element matrices tet-elements.js

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• Stresses at the four Gauss points

  The stresses used to build the geometric stiffness of a buckling study.

  σ = E · D (B u − α ΔT [1 1 1 0 0 0])

  Quadratic tetrahedron element matrices tet-elements.js

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• Element geometric stiffness

  The stress stiffness of one element for linear buckling.

  Gₐ_b = Σₚ wₚ det Jₚ ∇Nₐᵀ σₚ ∇N_b

  Quadratic tetrahedron element matrices tet-elements.js

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• Face areas and volume of the tetrahedral mesh

  The area of every boundary face and the mesh volume handed to the heat and buckling routines.

  area = Σ ∫Nₐ dA; volume = Σₑ Σₚ wₚ det Jₚ

  Quadratic tetrahedron element matrices tet-elements.js

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• Stiffness and mass assembly

  Adds every element's stiffness (and mass) into the model's sparse matrices and totals the volume.

  K = Σₑ Kₑ; volume = Σₑ volumeₑ

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Matrix times vector

  Multiplies the symmetric stiffness, stored as its lower triangle, by a vector.

  y = K x

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Constrained stiffness

  Builds the stiffness with supports applied.

  K̃ = P K P + α (I − P)

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Scale of the supported directions

  The value α put on the diagonal for supported directions.

  α = Σ Kᵢᵢ ÷ N

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Incomplete Cholesky preconditioner

  Factors the constrained stiffness on its own sparsity pattern to speed up the iteration.

  L Lᵀ ≈ K̃ on the pattern of K̃

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Conjugate-gradient solve for displacements

  Solves the constrained system for the displacement of every node.

  stop when ‖r‖ ÷ ‖P f‖ < tol

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Loads on faces and gravity

  Turns forces, pressures and gravity into nodal loads and totals them.

  force: t = total ÷ A; pressure: t = −p n; f = Σ N t dA; gravity: f = ρ g ∫N dV

  Tetrahedral stress, vibration and buckling solve tetfea.js

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