Calculations

Calculations

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  32-bit in part

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• Scale matrix about a fixed point

  The 4 × 4 matrix that scales a body about the center, minimum corner, base center or origin of its box.

  a = (min + max) ÷ 2 (center); translation = a × (1 − f)

  Scale feature for printing print-features.js

  • PartCraft
  64-bit

• Scaled triangles

  Applies the scale to every corner of a body's triangles, with the box found from the triangles themselves.

  x′ = f_x·x + t_x; y′ = f_y·y + t_y; z′ = f_z·z + t_z

  Scale feature for printing print-features.js

  • PartCraft
  64-bit

• Rotation about an axis

  Turns a vector about a unit axis by an angle; used for a plane at an angle to another.

  v′ = v·cos a + (u × v)·sin a + u·(u · v)·(1 − cos a)

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Point where three planes meet

  The single point that lies on three planes.

  solve [nₐ; n_b; n_c] · p = (nₐ·oₐ, n_b·o_b, n_c·o_c)

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Axis where two planes meet

  The line of intersection of two planes: its direction and a point on it.

  d = nₐ × n_b; solve nₐ·x = cₐ, n_b·x = c_b, d·x = d·(oₐ + o_b) ÷ 2

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Offset plane and plane at an angle

  A plane parallel to another at a distance, and a plane turned about an axis by an angle in degrees.

  o′ = o + n·d; n′ = rotate(n ⊥ axis, axis, angle × π ÷ 180)

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Plane through three points

  The plane through three points, with its center at their mean.

  n = (b − a) × (c − a); center = (a + b + c) ÷ 3

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Mid plane between two planes

  The plane half way between two planes, or the one that bisects them when they are not parallel.

  n = (nₐ − n_b) ÷ |nₐ − n_b|; c = (oₐ·nₐ − o_b·n_b) ÷ |nₐ − n_b|; parallel and same facing: o = oₐ + n·((o_b − oₐ)·n) ÷ 2

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Plane tangent to a cylinder

  The plane that touches a cylindrical face on the side of a chosen point.

  radial = w − d·(w · d); n = radial ÷ |radial|; o = foot on the axis + n·r

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Plane through a line and a point

  The plane that contains a straight edge or axis and one more point.

  n = d × (p − o)

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Where a line meets a plane or a cylinder

  The point where a straight edge or axis crosses a plane, a flat face or a cylindrical face.

  plane: t = ((o_s − o) · n) ÷ (d · n); cylinder: A·t² + B·t + C = 0, t = (−B ± √(B² − 4AC)) ÷ 2A

  Reference planes, axes and points refgeom.js

  • PartCraft
  64-bit

• Slider range of the section

  The lowest and highest value of n·p over the model's box, the two ends of the section slider.

  lo, hi = min, max of n · p over the 8 box corners

  Section view plane section.js

  • PartCraft
  32-bit, display only

• Section through a picked face

  The cutting plane through a flat face, placed by the instance matrix of an assembly.

  n = unit(M · n_face); c = n · (M · o_face)

  Section view plane section.js

  • PartCraft
  32-bit, display only

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• 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

  • PartCraft
  64-bit

• Developed length and bend lines

  The total flat length of a bent profile and where each bend line lies on the flat.

  length = Σ straights + Σ BA; bend line at x + BA ÷ 2; angle° = θ × 180 ÷ π

  Sheet metal bends and flat pattern sheetmetal.js

  • PartCraft
  64-bit

• Joining straight segments

  Joins two straight segments of the flat outline that continue each other through a point no third segment touches.

  collinear when |d₁ × d₂| ≤ 1e-9 × |d₁| × |d₂| and d₁ · d₂ > 0

  Sheet metal bends and flat pattern sheet-flat.js

  • PartCraft
  64-bit

• Points along an arc for the extent

  Sampled points on an arc or circle of the flat, used to find its box.

  point = c + r × (cos a, sin a), a = (a₀ + sweep × k ÷ n) × π ÷ 180

  Sheet metal bends and flat pattern sheet-flat.js

  • PartCraft
  64-bit

• Flat of a bent base

  The developed rectangle of an open bent profile: flat length by depth, with its bend lines and area.

  box = L × D; area = L × D

  Sheet metal bends and flat pattern sheet-flat.js

  • PartCraft
  64-bit