Calculations

Calculations

• Residence time distribution

  The distribution of times a pulse of tracer takes to leave a flow-through box, with its mean, variance and the nominal time.

  E = flux ÷ Σ(flux × Δt); mean = Σ t E Δt; variance = Σ t² E Δt − mean²; nominal = V ÷ Q

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• VTK export

  Writes the velocity and tracer fields as a legacy ASCII VTK file.

  2D mixing solver (lattice Boltzmann) cfd/lbm.js

  • CadCraft
  64-bit

• Guard on every written measure

  Decides what is written when a value is missing or not a number.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Arc angles brought into one turn

  Brings an angle into the range from 0 up to 2π before it is written.

  r = a mod 2π; r + 2π when r < 0

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Arc

  Writes an arc's center, radius and two angles.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Line

  Writes a line's two end points.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Polyline vertices

  Writes the vertices of a lightweight polyline.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Vertex widths

  Writes the start and end width at a polyline vertex.

  equal widths: one value, −|width|

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Linetype pattern length

  The total length of a linetype's dash pattern.

  length = Σ |element length|

  DWG object values written and read dwg/objects-write.js

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• Text direction

  The x-axis direction written for a block of text.

  x axis = (cos θ, sin θ, 0)

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Block scale

  Writes the three scale factors of a block reference in the format's compact form.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Single-line text

  Writes a text's insertion point, height, rotation, width factor and oblique angle.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Extension record values

  Writes the numbers kept in an extension record.

  DWG object values written and read dwg/objects-write.js

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  64-bit

• Block content matrix of a multileader

  The 4 × 4 matrix written for a multileader's block content when the model gives none.

  M = translate(l) · rotate_z(θ) · scale(s)

  DWG object values written and read dwg/mleader-write.js

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  64-bit

• Line read

  Reads a line's two end points.

  DWG object values written and read dwg/objects-read.js

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  64-bit

• Polyline vertices read

  Reads the vertices of a lightweight polyline.

  DWG object values written and read dwg/objects-read.js

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  64-bit

• North direction of older geolocation data

  Turns the north angle of a version 1 geolocation object into a direction vector.

  north = (sin a, cos a)

  DWG object values written and read dwg/objects-read.js

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  64-bit

• Fit point parameters and knots

  The parameter of each fit point along the spline, and the knot vector built from them.

  u₀ = 0; uᵢ = uᵢ₋₁ + |Qᵢ − Qᵢ₋₁|; knots = [0,0,0,0, u₁ … uₙ₋₂, L,L,L,L]

  Spline through fit points dwg/spline-fit.js

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  64-bit

• Cubic basis functions at a parameter

  The four B-spline basis values that weigh the control points at one fit point.

  Cox–de Boor recurrence, degree 3

  Spline through fit points dwg/spline-fit.js

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  64-bit

• Start condition

  The rows that pin the first control point and set the spline's start tangent or a natural start.

  P₀ = Q₀; P₁ − P₀ = T₀ × u₄ ÷ 3 or P₀ ÷ u₄ − P₁ (1 ÷ u₄ + 1 ÷ u₅) + P₂ ÷ u₅ = 0

  Spline through fit points dwg/spline-fit.js

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  64-bit

• End condition

  The rows that pin the last control point and set the spline's end tangent or a natural end.

  Pₘ₋₁ = Qₙ₋₁; Pₘ₋₁ − Pₘ₋₂ = T₁ × (L − uₘ₋₁) ÷ 3 or the zero second derivative row

  Spline through fit points dwg/spline-fit.js

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  64-bit

• Solve for the control points

  Solves the banded system for the x, y and z of every control point.

  A · P = B

  Spline through fit points dwg/spline-fit.js

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  64-bit

• Lineweight to the file's lineweight index

  Picks the stored lineweight that is nearest to the one the layer or object has.

  index = the i with the smallest |table[i] − lineweight|

  Values changed when a DWG is written dwg/write.js

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  Exact

• Ellipse with a ratio above 1

  Rewrites an ellipse whose minor axis is longer than its major axis so the file holds a ratio of 1 or less.

  major′ = (−m_y, m_x) × ratio; ratio′ = 1 ÷ ratio; start and end − π ÷ 2

  Values changed when a DWG is written dwg/write.js

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• Ray and construction line direction

  Writes the direction of a ray or construction line as a unit vector.

  d′ = d ÷ |d|

  Values changed when a DWG is written dwg/write.js

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• Drawing extents

  The lower left and upper right corner of model space written to the file header.

  min and max of x, y, z over the defining points

  Values changed when a DWG is written dwg/write.js

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• Default angles of a light

  The hotspot and falloff angles written for a light that has none of its own.

  hotspot = π ÷ 4; falloff = π ÷ 4 + 0.0872664626

  Values changed when a DWG is written dwg/solid-write.js

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  64-bit

• Standard gravity

  The acceleration that turns a member's or plate's mass into its own weight.

  g = 9.80665 m/s²

  Member axes and section properties frame/frame-model.js

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  Exact

• Member length and local axes

  The length of a member and the three unit vectors its forces and section are measured in.

  L = |P_j − P_i|; x = (P_j − P_i) ÷ L; z = unit(Z − (Z·x) x); y = z × x; then y and z turned by the roll

  Member axes and section properties frame/frame-model.js

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• Zero-length member test

  Removes a member whose two ends are at the same place.

  |P_j − P_i| < 1e-6 mm

  Member axes and section properties frame/frame-model.js

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• Area and second moments of a polygon section

  Area, centroid, second moments about the centroid, principal moments and their angle for a section outline with holes.

  A = ½ Σ cᵢ; Iyy = (1 ÷ 12) Σ (v₀² + v₀v₁ + v₁²) cᵢ − A c_v²; cᵢ = u₀v₁ − u₁v₀; I₁,₂ = (Iyy + Izz) ÷ 2 ± √(((Iyy − Izz) ÷ 2)² + Iyz²)

  Member axes and section properties frame/frame-model.js

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• Torsion constant

  The Saint-Venant torsion constant J of a catalog section or of an outline.

  I with fillets: J = ⅔ B T³ + ⅓ (D − 2T) t³ + 2 α D₁⁴ − 0.420 T⁴; open: Σ b t³ ÷ 3; tube: 4 Am² t ÷ p; round tube: π (D⁴ − (D − 2t)⁴) ÷ 32

  Member axes and section properties frame/frame-model.js

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• Section properties for the analysis

  Chooses the area, the two second moments, J and the roll a drawn section is analyzed with.

  A = the profile's exact area if given, else the polygon's; Iy, Iz = Iyy, Izz, or I₁, I₂ with roll = the principal angle

  Member axes and section properties frame/frame-model.js

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• A load split onto the member's axes

  Turns one point or distributed load in a global or local direction into pieces along the member's own three axes.

  piece = load × (direction · local axis); a, b clamped to [0, L]

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Inverse of the released block

  Inverts the small stiffness block of a member's released end actions.

  Gauss–Jordan with partial pivoting

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Member stiffness with released ends

  Removes pinned or sliding end actions from a member's 12 × 12 stiffness.

  k_c = k_kk − k_kr · k_rr⁻¹ · k_rk

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Fixed-end loads with released ends

  Moves the load a released end cannot carry onto the member's other end actions.

  p_c = p_k − k_kr · k_rr⁻¹ · p_r

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Member stiffness in global axes

  Turns a member's stiffness from its own axes to the model's.

  k_g = Tᵀ · k · T, by 3 × 3 blocks

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Directions nothing stiffens

  Finds translations and rotation directions at a node that no member resists, so they can be held instead of stopping the run.

  translation: diagonal + spring ≤ 1e-14 × largest diagonal; rotation: eigenvalue ≤ 1e-10 × largest rotational diagonal

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Eigenvalues of a symmetric 3 × 3 matrix

  The principal stiffnesses and directions of the rotation block at a node.

  cyclic Jacobi rotations; t = sign(θ) ÷ (|θ| + √(θ² + 1))

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Loads of a case or combination

  Adds up node forces and member loads of every case in a set, each times its factor, including the members' own weight.

  own weight w = −factor × f × ρ × 1e-9 × A × g (N/mm along −Z)

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Member end forces and reactions

  The six forces at each end of every member and the forces at the supports, from the solved displacements.

  f = k_c · u_local − p_c; reaction = Σ Tᵀ f − loads applied at the node

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• One step of iterative refinement

  Solves the leftover out-of-balance once more and adds the correction to the displacements.

  r = loads − member forces; x₂ = x + K⁻¹ r, kept if max |r₂| < max |r|

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Sum of the reactions

  The total force and moment of all support reactions, for the equilibrium check printed in the report.

  ΣF = Σ R; ΣM = Σ (M + r × R) about the global origin

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Sum of the applied loads

  The total force and moment of every load of a set about the global origin.

  ΣF = Σ F; ΣM = Σ (M + r × F); distributed loads by 4-point Gauss

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Linear combination of results

  Adds computed load sets, each times a factor, into one result.

  u = Σ fᵢ uᵢ, the same for end forces, reactions, loads and totals

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Internal forces at a cut

  N, Vy, Vz, T, My and Mz at a distance x from the member's start.

  S = −(F_i + ∫₀ˣ q); M = −(M_i + r × F_i + Σ r × q)

  Forces and deflection along a member frame/frame-solve.js

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• Deflection along the member

  The member's displacement between its ends, from its curvature.

  v″ = M_z ÷ (E I_z); w″ = −M_y ÷ (E I_y); u′ = N ÷ (E A); trapezoids, twice

  Forces and deflection along a member frame/frame-solve.js

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• Station values and deflection off the chord

  The values kept at each station, and how far the member bows from the straight line between its ends.

  rel = √(v_rel² + w_rel²); v_rel = v − v_end × x ÷ L

  Forces and deflection along a member frame/frame-solve.js

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• Largest and smallest values along a member

  The maximum and minimum of each internal force, where they occur, and the largest deflection with its span ratio.

  max and min over the stations; ratio = L ÷ deflection

  Forces and deflection along a member frame/frame-solve.js

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