Calculations

Calculations

• 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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• 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

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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

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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)

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• Corner where two fitted lines meet

  Places a corner of the sketch at the intersection of the two fitted lines on either side of it.

  den = u_d × v_d; t = ((v_p − u_p) × v_d) ÷ den; corner = u_p + t·u_d

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• Arc ends and refit

  Extends each arc along the outline while the points stay on its circle, fits the circle again and puts the two end points on it.

  grow while | |P − c| − r | ≤ min(0.75, tol ÷ 2); end = c + r·(P − c) ÷ |P − c|

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• Squaring of near-horizontal and near-vertical lines

  Makes a line that is within the squaring angle of horizontal or vertical exactly so, by giving both ends the same coordinate.

  angle = |atan2(B_y − A_y, B_x − A_x)| in degrees; shared y = (A_y + B_y) ÷ 2 or shared x = (A_x + B_x) ÷ 2

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• Scale of a silhouette

  Millimeters per pixel from one stated length, and the mapping of pixel coordinates to sketch coordinates.

  mm = stated length ÷ pixel extent; X = (x − x₀) × mm; Y = (y₁ − y) × mm

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• Scale of a pen drawing

  Millimeters per pixel for traced pen lines, from one stated length and the extent of the traced lines and arcs.

  extent from line ends, arc ends and arc centers; mm = stated length ÷ extent

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• Dot product in chunks

  Each thread sums its share of a dot product in chunks of 4096 entries.

  P_c = Σ aᵢ·bᵢ over chunk c

  Threaded solver kernels and conjugate gradients pool.js

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• Sum of the chunk partials

  Adds the chunk sums of a dot product in chunk order on the coordinating thread.

  a·b = Σ P_c

  Threaded solver kernels and conjugate gradients pool.js

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• Conjugate-gradient update

  Advances the solution and the residual by one step and sums the new residual's square.

  x += α·p; r −= α·q; P_c = Σ rᵢ²

  Threaded solver kernels and conjugate gradients pool.js

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• Sparse matrix times vector

  y = A·x for a matrix stored row by row, with the rows split between the threads.

  y_r = Σ_k v_k · x[col_k]

  Threaded solver kernels and conjugate gradients pool.js

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• Voxel stiffness times vector

  y = K·x for a voxel model without assembling K: each node gathers the contributions of its elements.

  y_n = Σ over elements at n of s_e · (K_e · x_e) rows of n, plus the 3 × 3 penalty block of a constrained node

  Threaded solver kernels and conjugate gradients pool.js

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• Node-block preconditioner and smoother

  Multiplies each node's three values by its 3 × 3 block, or adds a damped correction ω·M·(b − r).

  dst = M·src or dst += ω · M · (b − src)

  Threaded solver kernels and conjugate gradients pool.js

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• Constraint projection

  Sets the values of inactive nodes to zero and applies each constrained node's 3 × 3 projector.

  v_n = 0 (inactive); v_n ← P_n · v_n (constrained)

  Threaded solver kernels and conjugate gradients pool.js

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• Multigrid restriction

  Carries a residual from a fine grid to the grid of half the resolution.

  r_c(N) = Σ over the 27 fine neighbors of w · r_f; w = product per axis of 1 (same node) or ½ (between nodes)

  Threaded solver kernels and conjugate gradients pool.js

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• Multigrid prolongation

  Adds a coarse-grid correction to the fine grid by linear interpolation.

  x_f(n) += w · Σ x_c over the 1, 2, 4 or 8 coarse nodes around n; w = ½ per axis in which n lies between coarse nodes

  Threaded solver kernels and conjugate gradients pool.js

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• Geometric stiffness times vector

  y = s·(G ⊗ I₃)·x from per-element blocks, used by the buckling study.

  y_n = s · Σ over elements at n of Σ_b G_e[a, b] · x_b

  Threaded solver kernels and conjugate gradients pool.js

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• Conjugate gradients for a voxel model

  Solves K·u = f for a voxel stress model on the thread pool.

  res = ‖r‖ ÷ ‖b‖; stop when res < tol; α = (r·z) ÷ (p·Kp); β = (r·z)_new ÷ (r·z)_old

  Threaded solver kernels and conjugate gradients pool.js

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• Conjugate gradients for a sparse matrix

  Solves A·x = b for an assembled sparse matrix on the thread pool.

  res = ‖r‖ ÷ ‖b‖; stop when res < tol; α = (r·z) ÷ (p·Ap)

  Threaded solver kernels and conjugate gradients pool.js

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• Bend allowance

  The length of the neutral layer through a bend, which the flat pattern gives to that bend.

  BA = θ × (r + k × t)

  Sheet metal bends and flat pattern sheetmetal.js

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• Chain of profile lines

  Orders the lines of an open profile end to end, or reports that they are not one chain.

  two ends are the same point when their distance ≤ 1e-6 × max(1, largest coordinate size)

  Sheet metal bends and flat pattern sheetmetal.js

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• Bend angle, center and tangent length

  For every corner of the profile: the signed bend angle, the radius on the path side and the offset side, the distance the bend takes from each leg, and the bend's center.

  θ = atan2(u × w, u · w); d = r_path × tan(|θ| ÷ 2); c = P + left(u) × sign(θ) × r_path − u × d

  Sheet metal bends and flat pattern sheetmetal.js

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• Arc of a bend in the profile

  The arc of a bend between its two tangent points, with its middle point and 17 points along it.

  sweep = a₁ − a₀ brought to (0, 2π] or [−2π, 0); point = c + r × (cos t, sin t)

  Sheet metal bends and flat pattern sheetmetal.js

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• Straight runs between bends

  The flat length of each leg: its drawn length less what the bends at its ends take.

  straightⱼ = |Pⱼ₊₁ − Pⱼ| − dⱼ − dⱼ₊₁

  Sheet metal bends and flat pattern sheetmetal.js

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