Calculations

Calculations

• Solve with a Cholesky factorization

  Solves RᵀR X = B by two triangular substitutions.

  CraftScript dense linear algebra cs-lib-linalg.js

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• QR factorization

  Factors a matrix as an orthogonal matrix times an upper triangular one.

  H = I − β v vᵀ, β = 2 ÷ (vᵀv)

  CraftScript dense linear algebra cs-lib-linalg.js

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• Symmetric eigenvalues and vectors

  All eigenvalues, and optionally eigenvectors, of a symmetric matrix, in ascending order.

  CraftScript dense linear algebra cs-lib-linalg.js

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• Generalized symmetric eigenproblem

  Eigenvalues and vectors of K v = λ M v for symmetric K and positive definite M.

  M = RᵀR; C = R⁻ᵀ K R⁻¹; C y = λ y; v = R⁻¹ y

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• Choice of solver for A x = b

  Picks the method for the backslash operator and linsolve, for dense and sparse matrices.

  CraftScript dense linear algebra cs-lib-linalg.js

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• Matrix inverse

  The inverse of a square matrix.

  CraftScript dense linear algebra cs-lib-linalg.js

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• Determinant

  The determinant of a square matrix.

  det A = sign × Π U_ii

  CraftScript dense linear algebra cs-lib-linalg.js

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• Vector and matrix norms

  The 1, 2, infinity, p and Frobenius norms.

  ‖x‖₂ = √(Σ xᵢ²); ‖A‖₂ = √(λ_max(AᵀA))

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• Condition number

  The 2-norm condition number of a matrix.

  cond A = √(λ_max(AᵀA) ÷ λ_min(AᵀA))

  CraftScript dense linear algebra cs-lib-linalg.js

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• Sparse matrix from triplets

  Builds a sparse matrix from row, column and value lists, adding values that share a place.

  CraftScript sparse matrices and direct factorization cs-lib-linalg.js

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• Weighted sum of two sparse matrices

  a·A + b·B for matrices of the same size.

  C_ij = a × A_ij + b × B_ij

  CraftScript sparse matrices and direct factorization cs-lib-linalg.js

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• Sparse matrix times vector

  The product of a sparse matrix and a vector.

  y_r = Σ_k val_k × x_col(k)

  CraftScript sparse matrices and direct factorization cs-lib-linalg.js

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• Sparse matrix times sparse matrix

  The product of two sparse matrices.

  CraftScript sparse matrices and direct factorization cs-lib-linalg.js

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• Symmetry test

  Decides whether a sparse matrix is symmetric, which selects the solver.

  max |S − Sᵀ| ≤ 1e-10 × max |S|

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• Envelope size after reordering

  Counts the stored entries and the work of the skyline factorization for an ordering.

  entries = Σ wᵢ; work = Σ wᵢ² ÷ 2, wᵢ = i − firstᵢ + 1

  CraftScript sparse matrices and direct factorization cs-lib-linalg.js

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• Envelope Cholesky factorization and solve

  Factors a symmetric positive definite sparse matrix and solves with the factor.

  L_ij = (A_ij − Σ_k L_ik L_jk) ÷ L_jj; L_ii = √(A_ii − Σ_k L_ik²)

  CraftScript sparse matrices and direct factorization cs-lib-linalg.js

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• Assembly of element matrices

  Adds weighted element matrices into one sparse matrix.

  K = Σ_e w_e × P_eᵀ Ke P_e

  CraftScript sparse matrices and direct factorization cs-lib-linalg.js

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• Preconditioned conjugate gradients

  Solves a symmetric positive definite sparse system with the pcg builtin.

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Jacobi preconditioner

  Scales a residual by the inverse of the matrix diagonal.

  zᵢ = rᵢ ÷ A_ii

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Incomplete LU factorization

  The ILU(0) preconditioner used by BiCGSTAB.

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• BiCGSTAB

  Solves a general sparse system by iteration.

  stop when ‖r‖ ÷ ‖b‖ < tol

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• Rigid-body vectors for AMG

  Builds the translation and rotation vectors the multigrid preconditioner uses.

  rotation about the centroid: (−(y − c_y), x − c_x)

  CraftScript iterative solvers and sparse eigenvalues cs-lib-linalg.js

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• Smallest eigenvalues of a sparse problem

  The k smallest eigenvalues and vectors of K v = λ M v.

  stop when |λⱼ − λⱼ(previous)| ≤ tol × |λⱼ| for every j ≤ k

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• One operator on two numbers

  Adds, subtracts, multiplies, divides, compares or takes the remainder of two numbers.

  mod(x, y) = x − ⌊x ÷ y⌋ × y; rem(x, y) = x − trunc(x ÷ y) × y

  CraftScript array arithmetic cs-tensor.js

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• Hypotenuse

  √(x² + y²) for two numbers.

  m × √((x ÷ m)² + (y ÷ m)²), m = max(|x|, |y|)

  CraftScript array arithmetic cs-tensor.js

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• Power

  x raised to the power y.

  CraftScript array arithmetic cs-tensor.js

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• Elementwise loops

  Applies an operator to every element of one or two arrays.

  CraftScript array arithmetic cs-tensor.js

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• Matrix product

  The product of two dense matrices in a script expression.

  C_ij = Σ_p A_ip × B_pj

  CraftScript array arithmetic cs-tensor.js

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• Backslash for dense matrices

  Solves A x = b, or the least-squares problem when A is not square.

  CraftScript array arithmetic cs-tensor.js

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• Number of elements in a range

  How many values a:s:b has.

  n = ⌊(b − a) ÷ s + max(3ε × max(|a|, |b|) ÷ |s|, 1e-10)⌋ + 1

  CraftScript array arithmetic cs-tensor.js

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• Values of a range

  The values a, a + s, a + 2s … up to b.

  x_k = a + k × s

  CraftScript array arithmetic cs-tensor.js

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• Sum, product, mean, maximum and minimum

  Reduces an array along one dimension.

  mean = (Σ xₖ) ÷ n

  CraftScript array arithmetic cs-tensor.js

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• Unit factors

  The factors that turn the given force, length and torque units into newtons, millimeters and newton millimeters.

  Design-it sizing and hand checks design-it.js

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• Arguments to working units

  Converts the given force, lengths and torque to N, mm and N·mm.

  value × unit factor

  Design-it sizing and hand checks design-it.js

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• Rounding to stock sizes and to reported decimals

  Rounds a thickness up to a stock plate or a 0.5 mm step, and rounds reported values.

  r2(v) = round(100 v) ÷ 100; r3(v) = round(1000 v) ÷ 1000

  Design-it sizing and hand checks design-it.js

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• Bolt positions in the flange

  The heights of the two bolts and the flange height.

  y_b = t + 1.5 × hole; spacing = clamp(round(a ÷ 10) × 5, 30, 100); H = y_t + 1.5 × hole

  Design-it sizing and hand checks design-it.js

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• Bracket checks

  Bending and shear stress, tip deflection and bolt loads of the bracket, each with its limit and safety factor.

  σ = 6 F a ÷ (b t²); τ = 1.5 F ÷ (b t); δ = F a³ ÷ (3 E I), I = b t³ ÷ 12; T = M y₂ ÷ (y₁² + y₂²); (T ÷ T_all)² + (V ÷ V_all)² ≤ 1

  Design-it sizing and hand checks design-it.js

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• Bracket sizing, volume and mass

  The thickness for strength and for the deflection limit, the bolt size, and the bracket's volume and mass.

  t = √(6 M ÷ (b σ_allow)); t = ∛(4 F a³ ÷ (E b δ)); V = b (L t + (H − t) t) − 2 (π ÷ 4) hole² t; mass = V × density × 1e-6

  Design-it sizing and hand checks design-it.js

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• Shaft checks

  Von Mises stress, torsional shear and twist of a round shaft.

  σ' = 32 ÷ (π d³) × √(M² + ¾T²); τ = 16 T ÷ (π d³); θ = T L ÷ (G J), G = E ÷ (2(1 + ν)), J = π d⁴ ÷ 32

  Design-it sizing and hand checks design-it.js

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• Shaft sizing, volume and mass

  The required diameter, the standard size above it, and the shaft's volume and mass.

  d = ∛(32 SF √(M² + ¾T²) ÷ (π S_y)); V = (π ÷ 4) d² L

  Design-it sizing and hand checks design-it.js

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• The built part

  Sketches and extrudes the sized bracket or shaft as part features.

  density (g/cm³) = density (kg/m³) ÷ 1000

  Design-it sizing and hand checks design-it.js

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• Edges and facet normals

  Finds the mesh's edges, each facet's unit normal, and which edges are sharp.

  sharp when n₁ · n₂ < cos 20°

  Drawing sheet views, dimensions and PDF drawing.js

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• One projected view with hidden lines

  Projects the mesh onto a view plane, finds the view's extents, and splits each drawn edge into visible and hidden parts.

  (u, v, w) = (p · right, p · up, p · dir)

  Drawing sheet views, dimensions and PDF drawing.js

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• Merging collinear segments

  Joins touching or overlapping segments on the same line and removes hidden lines under visible ones.

  Drawing sheet views, dimensions and PDF drawing.js

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• Section hatching

  Hatch lines at an angle inside the cut outline.

  Drawing sheet views, dimensions and PDF drawing.js

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• Clip to the detail circle

  Keeps the parts of segments inside the detail view's circle.

  A t² + B t + C = 0; t = (−B ± √(B² − 4AC)) ÷ 2A

  Drawing sheet views, dimensions and PDF drawing.js

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• Holes for the hole table

  Collects each hole's axis, diameter and callout from the kernel's cylindrical faces.

  diameter = 2 × smallest radius on the axis

  Drawing sheet views, dimensions and PDF drawing.js

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• Hole table rows

  The X and Y of each hole measured from the view's lower-left corner, and its tag.

  x = o · right − u₀; y = o · up − v₀

  Drawing sheet views, dimensions and PDF drawing.js

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• Sheet scale

  Picks the drawing scale for a paper size.

  first s in the ISO 5455 series with width × s ≤ 0.82 × area width and height × s ≤ 0.82 × area height

  Drawing sheet views, dimensions and PDF drawing.js

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• Dimension text

  The text of an overall dimension or a hole table position.

  text = value rounded to 2 decimals

  Drawing sheet views, dimensions and PDF drawing.js

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