Calculations

Calculations

• 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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• 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

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

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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|

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

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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

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

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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

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

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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

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

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• Density unit change

  Converts a material's density from kg/m³ to tonnes/mm³, the unit the solver uses.

  ρ (t/mm³) = ρ (kg/m³) × 1e-12

  Contact and magnet study results phys-worker.js

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• Element values averaged to nodes

  Gives each node the mean of the values of the elements that share it.

  value at a node = Σ element values ÷ number of elements at the node

  Contact and magnet study results phys-worker.js

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• Maximum stress, plastic strain and displacement

  The maxima the study reports, and where they occur.

  |u| = √(u_x² + u_y² + u_z²); maximum over the elements (stress, plastic strain) and over the nodes (displacement)

  Contact and magnet study results phys-worker.js

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• Per-vertex result arrays

  Copies the nodal stress, plastic strain, displacement size and displacement vector to the vertices of the plot mesh.

  Contact and magnet study results phys-worker.js

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  32-bit in part

• Placing a part's mesh in the assembly

  Moves each mesh node of a placed part to the assembly frame with the instance's 4 × 4 matrix.

  x′ = M · (x, y, z, 1)

  Contact and magnet study results phys-worker.js

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• Contact penalty stiffness

  Chooses the stiffness of the contact springs for each pair of touching faces when none is given.

  k_n = autoPenalty(min(E_slave, E_master), h_min, factor); factor = 100 (frictionless) or 1000 (bonded)

  Contact and magnet study results phys-worker.js

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• Total contact force and peak pressure

  Adds the force at every contact point and finds the highest contact pressure.

  F = Σ forceᵢ; p_max = max pressureᵢ

  Contact and magnet study results phys-worker.js

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• Maximum flux density, forces and region areas

  The largest flux density of a magnet study and where it is, the force on each requested body, and the area of each region.

  B_max = max over the elements of |B|; region area = Σ element areas

  Contact and magnet study results phys-worker.js

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• Flux density and potential fields

  The flux density of every element and the magnetic vector potential of every node, as returned to the page.

  Contact and magnet study results phys-worker.js

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• Maxima from a results file read back

  The largest displacement and von Mises stress in a CalculiX results file read back onto the study's mesh.

  σ_vm = √(½·((a − b)² + (b − c)² + (c − a)²) + 3·(d² + e² + f²)); |u| = √(u_x² + u_y² + u_z²)

  Contact and magnet study results phys-worker.js

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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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• Scale factors

  The three factors a scale feature multiplies by, from the feature or from its shrinkage allowance.

  Scale feature for printing print-features.js

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