Calculations

Calculations

• Filling a hole

  Closes a boundary loop with triangles.

  Newell normal n = Σ ((a_y − b_y)(a_z + b_z), …); ear when cross(a, b, c) > 0 and no other point is inside

  Mesh import, repair and storage meshimport.js

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• Growing flat faces

  Joins neighboring triangles into one flat face when they share a plane.

  join when n_s·n_t ≥ cos 0.05° and every corner of t is within tol of s's plane; tol = 2e-6 × size + 1e-7

  Mesh import, repair and storage meshimport.js

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• Plane of a flat face

  The normal and origin of each flat face a sketch can sit on.

  n = unit(Σ n_t A_t); origin = Σ (triangle center × A_t) ÷ Σ A_t

  Mesh import, repair and storage meshimport.js

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• Mesh kept in the part file

  Packs the repaired mesh into the text the part file stores.

  vertices → Float32Array → base64

  Mesh import, repair and storage meshimport.js

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• Mesh read back from the part file

  Unpacks and checks the stored mesh when a part is opened or rebuilt.

  Mesh import, repair and storage meshimport.js

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• Permeability of free space

  The constant μ₀ used for air, linear materials and the Maxwell stress.

  μ₀ = 4 × 10⁻⁷ × π

  Magnet study: 2D field, force and torque magnet.js

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• Reluctivity from a measured B–H curve

  Turns a table of B and H points into ν(B²) and its derivative for the solver.

  H(B) by a monotone cubic through the points; beyond the last point H = H_s + (B − B_s) ÷ μ₀; ν = H ÷ B; dν/dB² = (H′ ÷ B − H ÷ B²) ÷ (2B)

  Magnet study: 2D field, force and torque magnet.js

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• Banded Cholesky solve

  Factors the tangent matrix and solves for the correction of the potential.

  K = L Lᵀ in band storage; forward then back substitution

  Magnet study: 2D field, force and torque magnet.js

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• Triangle areas and shape-function gradients

  The area, center and the three gradient vectors of every triangle.

  d = (x₁ − x₀)(y₂ − y₀) − (x₂ − x₀)(y₁ − y₀); area = |d| ÷ 2; ∇Nᵢ = (bᵢ, cᵢ) ÷ d

  Magnet study: 2D field, force and torque magnet.js

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• Residual and tangent matrix

  Assembles the out-of-balance load and, when asked, the tangent stiffness for the current potential.

  R = load − Σ ν A (∇Nᵢ·∇A); K_ik = Σ A (ν ∇Nᵢ·∇N_k + 2ν′ (∇Nᵢ·∇A)(∇N_k·∇A))

  Magnet study: 2D field, force and torque magnet.js

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• Newton iteration

  Repeats solve and update until the potential stops changing.

  A ← A + α ΔA, α halved until the residual norm falls; stop when |ΔA| ÷ |A| < tol or |R| < 1e-14 × |load|

  Magnet study: 2D field, force and torque magnet.js

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• Flux density and energy per element

  The flux density in each triangle and the stored energy per unit depth.

  B = (∂A/∂y, −∂A/∂x); |B| = hypot; energy = Σ ½ ν(|B|²) |B|² × area

  Magnet study: 2D field, force and torque magnet.js

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• Force and torque by the Maxwell stress

  The force per meter of depth and the torque about the origin on everything inside a set of elements.

  t = ((B_x² − ½B²) n_x + B_x B_y n_y, B_x B_y n_x + (B_y² − ½B²) n_y) ÷ μ₀; F = Σ t × dl; τ = Σ (x t_y − y t_x) dl

  Magnet study: 2D field, force and torque magnet.js

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• Sketch regions in SI units

  Converts each profile to meters and sets its material, magnet direction or coil current density.

  x_m = x_mm × 1e-3; area_m² = area_mm² × 1e-3 × 1e-3; B_r = B (cos θ, sin θ); J = I ÷ area

  Magnet study: 2D field, force and torque magnet-study.js

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• Mesh sizes for the study

  The triangle size asked for in each body, in each air gap and in the surrounding air.

  base = size × (1 + 2 pad) ÷ 14 ÷ q; body: narrowest side ÷ 5 ÷ q; gap: gap ÷ 4 ÷ q

  Magnet study: 2D field, force and torque magnet-study.js

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• Air gap between two bodies

  The narrowest distance between two profiles that stand apart.

  d = min over the corners of each profile of the distance to the other's edges

  Magnet study: 2D field, force and torque magnet-study.js

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• Force and torque on each body

  The force in newtons and the torque about the body's own center, for the depth given.

  F = F_per_m × depth; τ = (τ_origin − (c_x F_y − c_y F_x)) × depth

  Magnet study: 2D field, force and torque magnet-study.js

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• Results as CSV

  Writes the peak flux density, forces and torques to a CSV file.

  number → toPrecision(8)

  Magnet study: 2D field, force and torque magnet-study.js

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• Orientation test

  Which side of a directed line a point lies on, used to walk the triangulation.

  (b_x − a_x)(c_y − a_y) − (b_y − a_y)(c_x − a_x)

  Magnet study: 2D field, force and torque magnetmesh.js

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• In-circle test

  Whether a new point lies inside the circle through a triangle's corners, which decides the Delaunay cavity.

  3 × 3 determinant of (a − d, b − d, c − d) with their squared lengths

  Magnet study: 2D field, force and torque magnetmesh.js

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• Circumcenter and quality of a triangle

  A triangle's circumcenter, circumradius and shortest edge, used to decide whether to refine it.

  u = (c_y |b|² − b_y |c|², b_x |c|² − c_x |b|²) ÷ (2 (b_x c_y − b_y c_x)), b and c relative to a

  Magnet study: 2D field, force and torque magnetmesh.js

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• Encroached boundary segment

  Whether a point lies inside the circle whose diameter is a boundary segment, in which case the segment is split at its midpoint.

  (p − mid)² < (|a − b|² ÷ 4) × (1 − 1e-12)

  Magnet study: 2D field, force and torque magnetmesh.js

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• Smallest and largest mesh angle

  The mesh quality figures shown with the result.

  angle = atan2(|u × v|, u·v) in degrees

  Magnet study: 2D field, force and torque magnetmesh.js

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• Coarse element matrix

  Forms Rᵀ K R for one fine element inside a coarse element.

  K_coarse += s × Rᵀ K R

  Geometric multigrid for the voxel stress model mg.js

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• Inverse of each node's 3 × 3 block

  Inverts the diagonal block of every node for the block Jacobi smoother.

  D⁻¹ = adj(D) ÷ det(D)

  Geometric multigrid for the voxel stress model mg.js

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• Damped block Jacobi sweep

  One smoothing step on a level.

  x ← x + ω D⁻¹ (b − A x), ω = 0.6

  Geometric multigrid for the voxel stress model mg.js

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• Restriction and prolongation

  Moves residuals to the coarser grid and corrections back to the finer one.

  weight = ½ for each direction in which the fine node lies between two coarse nodes

  Geometric multigrid for the voxel stress model mg.js

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• Coarsest-level factorization

  Factors the coarsest level's matrix once so each cycle can solve it directly.

  A = L Lᵀ in band storage, nodes renumbered with the longest axis slowest

  Geometric multigrid for the voxel stress model mg.js

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• Coarsest-level solve

  Forward and back substitution with the stored factor.

  Geometric multigrid for the voxel stress model mg.js

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• Motion laws

  How far along its travel the joint is at each fraction of the time.

  linear: s; ease: s²(3 − 2s); sine: ½ − ½ cos(πs); round trip: 2s or 2 − 2s; value = from + (to − from) × law(k ÷ steps)

  Motion study of an assembly motion.js

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• Default tracked point

  The middle of a part's bounding box, tracked when no point is given.

  center = (min + max) ÷ 2 per axis

  Motion study of an assembly motion.js

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• Solids for the clearance check

  Rebuilds each part as a solid so the gap between parts can be measured at every step.

  Motion study of an assembly motion.js

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• Velocity and acceleration

  The rate of the joint value and each tracked point's velocity, speed and acceleration.

  v_k = (p_(k+1) − p_(k−1)) ÷ (t_(k+1) − t_(k−1)); a_k = (p_(k−1) − 2p_k + p_(k+1)) ÷ Δt²

  Motion study of an assembly motion.js

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• Peak speed, peak acceleration and travel

  The largest speed and acceleration of each tracked point and the length of its path.

  travel = Σ |p_k − p_(k−1)|

  Motion study of an assembly motion.js

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• Numbers in the CSV export

  Formats each time, value, position, speed and gap written to the CSV file.

  text = value.toFixed(5), trailing zeros removed

  Motion study of an assembly motion.js

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• Bilinear material constants

  Shear modulus, bulk modulus and hardening modulus from E, ν, yield stress and tangent modulus.

  G = E ÷ (2(1 + ν)); K = E ÷ (3(1 − 2ν)); H = E × E_t ÷ (E − E_t)

  Nonlinear and contact stress study nonlinear.js

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• Stress update at a Gauss point

  The stress, the new plastic strain and the algorithmic tangent for a given total strain.

  q = √(1.5 s:s); f = q − (σ_y + H α); Δγ = f ÷ (3G + H); σ = p 1 + (1 − 3GΔγ ÷ q) s

  Nonlinear and contact stress study nonlinear.js

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• Element tangent and internal force

  Integrates each element's tangent stiffness and internal force over its four Gauss points.

  K_e = Σ w Bᵀ D_ep B; f_e = Σ w Bᵀ σ; w = weight × det J

  Nonlinear and contact stress study nonlinear.js

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• Load-stepped Newton solve

  Raises the load in steps and iterates each step to equilibrium.

  r = λ f − f_int(u) (− contact); converged when |r| ÷ (λ |f|) < tol

  Nonlinear and contact stress study nonlinear.js

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• Von Mises stress and plastic strain per element

  The largest von Mises stress and the largest equivalent plastic strain among each element's Gauss points.

  vm_e = max over 4 points of vonMises(σ); α_e = max over 4 points of α

  Nonlinear and contact stress study nonlinear.js

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• Reported maximum stress, displacement and safety factor

  The peak von Mises stress and where it is, the peak displacement, and the smallest safety factor against yield.

  max σ_vm = largest element value; max |u| = largest nodal √(u_x² + u_y² + u_z²); safety = yield ÷ max σ_vm

  Nonlinear and contact stress study nl-worker.js

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• Support reactions

  The total force the supports carry, overall and per support group.

  R = Σ over constrained nodes of (f_int − f_applied)

  Nonlinear and contact stress study nl-worker.js

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• Element values averaged to nodes

  The nodal stress and plastic strain fields the plot colors by.

  node value = mean of the values of the elements that share the node

  Nonlinear and contact stress study nl-worker.js

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• Separating the pieces of a part

  Splits a part's surface mesh into its connected pieces so each can be meshed on its own for a contact study.

  vertices are the same when round(x × 10⁴), round(y × 10⁴), round(z × 10⁴) agree

  Nonlinear and contact stress study nl-worker.js

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• Density unit for the solver

  Converts the material's density from kg/m³ to the solver's tonnes per mm³.

  ρ_t/mm³ = ρ_kg/m³ × 1e-12

  Nonlinear and contact stress study nl-worker.js

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• Von Mises stress from CalculiX results

  The von Mises stress at each node from the six stress components of a results file.

  σ_vm = √(½((a − b)² + (b − c)² + (c − a)²) + 3(d² + e² + f²))

  Nonlinear and contact stress study nl-worker.js

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• Signed area of a pixel outline

  Tells an outer outline from a hole and gives the hole size used to drop specks.

  a = ½ × Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ); hole kept when 2 × √(|a| ÷ π) ≥ minHole

  Photo to sketch: outline fitting and scale photo-sketch.js

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• Outline simplification

  Keeps only the outline points that lie farther than the tolerance from the chord between their neighbors.

  d = |dx·(Qy − Ay) − dy·(Qx − Ax)| ÷ L; keep the farthest point when d > tol

  Photo to sketch: outline fitting and scale photo-sketch.js

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• Circle through outline points

  The center and radius of the circle that best fits a run of outline points, with its largest and root-mean-square deviation.

  u = x − x̄, v = y − ȳ; solve [Σu² Σuv; Σuv Σv²]·(u_c, v_c) = ½·(Σu³ + Σuv², Σv³ + Σvu²); r = mean distance to the center

  Photo to sketch: outline fitting and scale photo-sketch.js

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• Line through outline points

  The direction of the straight line that best fits a run of outline points, through their centroid.

  θ = ½ × atan2(2·Σuv, Σu² − Σv²); d = (cos θ, sin θ)

  Photo to sketch: outline fitting and scale photo-sketch.js

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