Calculations

Calculations

• Element heat capacity

  The heat capacity matrix of one element, or its lumped diagonal for transient runs.

  M = Σₚ wₚ det Jₚ ρc NₐN_b; lumped: Mₐₐ × (ΣM ÷ Σ diagonal)

  Quadratic tetrahedron element matrices tet-elements.js

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• Element heat generation load

  The nodal heat input of a uniform internal heat source.

  fₐ = Σₚ wₚ det Jₚ Q Nₐ

  Quadratic tetrahedron element matrices tet-elements.js

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• Temperature gradient at the element center

  The temperature gradient used for the heat flux of an element.

  ∇T = Σ ∇Nₐ Tₐ at L = (¼, ¼, ¼, ¼)

  Quadratic tetrahedron element matrices tet-elements.js

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• Element thermal load

  The nodal forces that a temperature rise produces in one element.

  f = Σₚ wₚ det Jₚ E α ΔTₚ (D₀₀ + D₀₁ + D₀₂) ∇N

  Quadratic tetrahedron element matrices tet-elements.js

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• Stresses at the four Gauss points

  The stresses used to build the geometric stiffness of a buckling study.

  σ = E · D (B u − α ΔT [1 1 1 0 0 0])

  Quadratic tetrahedron element matrices tet-elements.js

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• Element geometric stiffness

  The stress stiffness of one element for linear buckling.

  Gₐ_b = Σₚ wₚ det Jₚ ∇Nₐᵀ σₚ ∇N_b

  Quadratic tetrahedron element matrices tet-elements.js

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• Face areas and volume of the tetrahedral mesh

  The area of every boundary face and the mesh volume handed to the heat and buckling routines.

  area = Σ ∫Nₐ dA; volume = Σₑ Σₚ wₚ det Jₚ

  Quadratic tetrahedron element matrices tet-elements.js

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• Stiffness and mass assembly

  Adds every element's stiffness (and mass) into the model's sparse matrices and totals the volume.

  K = Σₑ Kₑ; volume = Σₑ volumeₑ

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Matrix times vector

  Multiplies the symmetric stiffness, stored as its lower triangle, by a vector.

  y = K x

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Constrained stiffness

  Builds the stiffness with supports applied.

  K̃ = P K P + α (I − P)

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Scale of the supported directions

  The value α put on the diagonal for supported directions.

  α = Σ Kᵢᵢ ÷ N

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Incomplete Cholesky preconditioner

  Factors the constrained stiffness on its own sparsity pattern to speed up the iteration.

  L Lᵀ ≈ K̃ on the pattern of K̃

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Conjugate-gradient solve for displacements

  Solves the constrained system for the displacement of every node.

  stop when ‖r‖ ÷ ‖P f‖ < tol

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Loads on faces and gravity

  Turns forces, pressures and gravity into nodal loads and totals them.

  force: t = total ÷ A; pressure: t = −p n; f = Σ N t dA; gravity: f = ρ g ∫N dV

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Support directions on faces

  Finds which nodes of a supported face are held and the normal direction used for a sliding support.

  held when distance to the exact face ≤ 0.002 × mean edge length

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Is the part held

  Checks that the supports stop all six rigid-body motions.

  free when an eigenvalue of Σ rᵀr is ≤ 10⁻⁹ × the largest, or ≤ 10⁻¹²

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Natural frequencies

  The lowest natural frequencies and mode shapes of the supported part.

  K̃ x = λ M x; f = √max(λ, 0) ÷ 2π

  Tetrahedral stress, vibration and buckling solve tetfea.js

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• Temperature rise for thermal stress

  The temperature rise at every node over the stress-free temperature, and the load it adds.

  ΔT = T − T_ref (20 when not given); f += thermal load

  Tetrahedral stress, vibration and buckling solve tet-elements.js

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• Buckling pre-stress and geometric stiffness

  Solves the pre-stress and assembles the stress stiffness that the buckling eigen-solve uses.

  K u = f (tol 10⁻¹⁰); G = −Σₑ Gₑ; each inner solve tol 10⁻⁸

  Tetrahedral stress, vibration and buckling solve tet-elements.js

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• Principal stresses

  The largest, middle and smallest principal stress of a stress state.

  s₁ = m + 2q cos φ, s₃ = m + 2q cos(φ + 2π ÷ 3), s₂ = 3m − s₁ − s₃; φ = acos(det ÷ 2q³) ÷ 3

  Stress, displacement and error results on tetrahedra tetfea.js

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• Equivalent strain

  The von Mises equivalent strain from the six engineering strains.

  ε_eq = √(⅔ (a² + b² + c² + (γ_xy² + γ_yz² + γ_zx²) ÷ 2)), a, b, c = normal strains − mean

  Stress, displacement and error results on tetrahedra tetfea.js

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• Gauss points to corners

  The value at an element corner of the linear field that passes through the four Gauss-point values.

  cᵢ = (gᵢ − 0.1381966… × Σg) ÷ (0.5854101… − 0.1381966…)

  Stress, displacement and error results on tetrahedra tetfea.js

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• Largest von Mises stress in an element

  The unaveraged maximum von Mises stress of each element and the largest over the mesh.

  max over the 4 Gauss points and the 4 extrapolated corners

  Stress, displacement and error results on tetrahedra tetfea.js

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• Nodal stress, strain and displacement, and their maxima

  The values the plot colors by and the maxima the result reports.

  σ at a node = mean over its elements; |u| = √(u_x² + u_y² + u_z²); maximum over the nodes

  Stress, displacement and error results on tetrahedra tetfea.js

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• Error indicator per element

  How far the averaged stress field is from each element's own stresses, in the energy norm.

  e_K² = ∫ (σ* − σₕ)ᵀ C⁻¹ (σ* − σₕ) dV; U = ∫ σₕᵀ C⁻¹ σₕ dV

  Stress, displacement and error results on tetrahedra tetfea.js

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• Support reactions

  The total reaction force and the reaction of each support group.

  r = K u − f at the supported nodes, kept along the supported directions; R = Σ r

  Stress, displacement and error results on tetrahedra tetfea.js

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• Reported maxima, safety factor and error percentage

  The numbers the study result states: error percentage, maximum stress and displacement, principal stress range and minimum safety factor.

  η = 100 √(Σe² ÷ (U + Σe²)); safety = yield ÷ maximum von Mises

  Stress, displacement and error results on tetrahedra tetfea.js

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• Error estimate of several fields

  Combines the error indicators of mode shapes, or of a buckling pre-stress and its modes, into one estimate.

  e_K² = Σ_f e_K,f² ÷ U_f; η = 100 √(Σe² ÷ (n + Σe²))

  Stress, displacement and error results on tetrahedra tetfea.js

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• Mesh volume and mass

  The volume and mass of the meshed part shown with the study.

  volume = Σₑ Σₚ wₚ det Jₚ; mass in kg = volume × ρ (t/mm³) × 1000

  Stress, displacement and error results on tetrahedra tetfea.js

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• Error estimate of a heat solution

  The estimated error of a steady temperature solution, per element and as a percentage.

  e_K² = ∫ (q* − qₕ)·(q* − qₕ) ÷ k dV; η = 100 √(Σe² ÷ (U + Σe²))

  Stress, displacement and error results on tetrahedra tet-elements.js

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• Nodal values onto plot vertices

  Copies a solved nodal field (stress, displacement, temperature, strain) to the vertices of the plot mesh.

  Result fields sent to the plot, legend and probe tet-worker.js

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• Heat flux size per vertex

  The heat flux magnitude in W/m² at each plot vertex.

  q = flux × 10⁶; |q| = √(q_x² + q_y² + q_z²)

  Result fields sent to the plot, legend and probe tet-worker.js

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• Mode shape per vertex

  A vibration or buckling mode shape scaled so its largest displacement is 1.

  shape = u ÷ max |u|

  Result fields sent to the plot, legend and probe tet-worker.js

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• Plot mesh of the tetrahedral model

  The surface of the tetrahedral mesh as triangles for the result view, each boundary face split in four through its midside nodes.

  Result fields sent to the plot, legend and probe tet-worker.js

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• Grid field sampled at plot vertices

  Samples a voxel-grid result (temperature, flux, temperature rise, mode shape) at each vertex of the plot mesh.

  Result fields sent to the plot, legend and probe thermal-worker.js

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• Heat flux size per vertex on a voxel mesh

  The heat flux magnitude in W/m² at each plot vertex of a voxel study.

  q = flux × 10⁶; |q| = √(q_x² + q_y² + q_z²)

  Result fields sent to the plot, legend and probe thermal-worker.js

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• Buckling mode shape on a voxel mesh

  A buckling mode shape sampled on the plot mesh and scaled to a largest displacement of 1.

  shape = u ÷ max |u|

  Result fields sent to the plot, legend and probe thermal-worker.js

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• Fields of each time step on tetrahedra

  Copies the stress, displacement and temperature-rise fields of every stored time step to the plot vertices.

  Result fields sent to the plot, legend and probe thermal-worker.js

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• Longest edge of a tetrahedron

  The length of the longest of an element's six edges.

  L = √max((Δx)² + (Δy)² + (Δz)²)

  Automatic mesh refinement of tetrahedral studies tetadapt.js

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• Which elements to refine and to what size

  Marks the elements with the largest error and gives each a new target edge length.

  ē = (target ÷ 100) √((U + Σe²) ÷ m); h_new = h × clamp((e_K ÷ ē)^(−1/2), 0.5, 0.7)

  Automatic mesh refinement of tetrahedral studies tetadapt.js

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• New boundary vertices onto the exact faces

  Moves a vertex created on the boundary onto the true surface of its faces, unless that would damage the elements around it.

  accept when every tet keeps volume > 0 and min dihedral ≥ min(10°, before)

  Automatic mesh refinement of tetrahedral studies tetadapt.js

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• Volume of the refined mesh

  The volume of the refined linear mesh, kept in its statistics.

  V = Σₑ volume of tet e

  Automatic mesh refinement of tetrahedral studies tetadapt.js

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• When the refinement stops

  Compares each pass's estimated error with the target and decides whether to refine again.

  stop when η ≤ target, or passes used, or elements > limit

  Automatic mesh refinement of tetrahedral studies tetadapt.js

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• Closest point on a triangle

  The nearest point of a surface triangle to a given point, and the squared distance.

  Tetrahedral mesh of a part tetmesh.js

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• Ray against a triangle

  Where a ray crosses a surface triangle, used for inside tests, wall thickness and cut points.

  t = (e₂·q) ÷ det, det = e₁·(d × e₂)

  Tetrahedral mesh of a part tetmesh.js

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• Is a point inside the part

  Decides whether a point lies inside the closed surface.

  inside when most clean rays cross the surface an odd number of times

  Tetrahedral mesh of a part tetmesh.js

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• Projection onto a face

  Moves a point onto a plane, a cylinder, a recorded surface or, failing those, the face's triangles.

  plane: p − ((p − o)·n) n; cylinder: axis point + r × radial ÷ |radial|

  Tetrahedral mesh of a part tetmesh.js

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• Projection onto the meeting of several faces

  Moves a point onto the edge or corner where two or three faces meet.

  repeat: project onto each face in turn, until the move < 10⁻¹² × diagonal

  Tetrahedral mesh of a part tetmesh.js

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• Signed volume of a tetrahedron

  The volume of one element, positive for the expected vertex order.

  V = (u · (v × w)) ÷ 6

  Tetrahedral mesh of a part tetmesh.js

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• Dihedral angles and aspect ratio of a tetrahedron

  The smallest and largest dihedral angle and the aspect ratio of one element.

  θ = atan2(1.5 V |e|, −n₁·n₂ ÷ 4); aspect = L_max ÷ (2√6 × r_in), r_in = 3V ÷ ΣA

  Tetrahedral mesh of a part tetmesh.js

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