4. Element type selection

Element type selection is the single most consequential meshing decision in industrial FEA. It determines the polynomial order of the displacement interpolation, the number of integration points, the locking and shear behavior, the mesh density required for a target accuracy, and ultimately whether a stress, modal, or thermal result can be trusted. This chapter builds a decision framework around four families: linear (first-order) tetrahedra, quadratic (second-order) tetrahedra, hexahedra, and structural elements (shells and beams).

Interpolation order and what it buys you

A linear tetrahedron (Tet4) interpolates displacement linearly across four nodes. Its strain field is constant within each element, so bending is represented only by a staircase of constant-strain blocks. This produces severe artificial stiffness in bending-dominated problems — a phenomenon called shear locking and, more generally, poor bending response. A quadratic tetrahedron (Tet10) adds midside nodes and a quadratic displacement field, so strain varies linearly and bending is captured with far fewer elements. The practical consequence is that Tet10 typically needs roughly one-tenth the element count of Tet4 for the same displacement accuracy in a bending problem, at the cost of more DOFs per element and higher solver memory.

Hexahedra behave differently. A linear hex (Hex8) with full integration is prone to shear locking, but with reduced integration plus hourglass control it performs well in bending and is extremely efficient. A quadratic hex (Hex20) is the gold standard for stress accuracy per DOF in solid mechanics, but it is expensive to generate on complex geometry. The trade is not "which is best" but "which fits the geometry, the physics, and the budget."

Choosing by analysis type

The thermal row deserves emphasis. Heat conduction is governed by a second-order (Laplacian) operator, and the temperature field is typically much smoother than a stress field near a notch. A linear Tet4 mesh that would be unacceptable for stress may give a perfectly usable temperature distribution. However, if you then feed that temperature field into a thermal-stress analysis, the stress result inherits the poor bending behavior of Tet4 — so the element choice must be made for the most demanding physics in a coupled workflow.

Shells and beams: when to reduce dimensionality

A shell element is valid when one dimension (thickness) is small relative to the other two and the through-thickness stress is negligible. The standard validity window is a span-to-thickness ratio greater than about 10 for Kirchhoff-type shells and greater than about 5 for Mindlin-Reissner shells that include transverse shear. Below that, use solids. Shells carry membrane, bending, and transverse shear; the analyst must define thickness, offset (mid-surface vs. reference surface), and section integration points. Getting the offset wrong is one of the most common sources of a factor-of-two error in bending stress.

Beam elements are valid when length-to-cross-section ratio exceeds roughly 10 and the cross-section does not distort. They recover axial, bending, torsion, and shear resultants, and stresses are computed from section properties rather than from a 3D stress field. For modal analysis, beams are extremely efficient for frame structures, but their mass and stiffness must match the real section — a hollow section modeled as solid will over-predict frequency.

Worked example: bracket under bending

Consider an L-shaped steel bracket, 120 mm × 80 mm × 6 mm, fixed at one leg and loaded with 500 N at the other. The dominant physics is bending at the fillet. A reasonable workflow:

1. Model the bracket as a solid and mesh with Tet10, target element size 2 mm at the fillet, 5 mm elsewhere. This yields roughly 40,000 elements and ~200,000 DOFs.
2. Run a linear static analysis. Extract von Mises stress at the fillet surface. Suppose the peak reads 210 MPa.
3. Refine the fillet to 1 mm and re-run. If the peak moves to 225 MPa (+7%), the original mesh was under-resolved. Continue refining until the change is below 5%, or until you recognize a stress singularity (a sharp re-entrant corner with no fillet radius will not converge — the stress rises without bound).
4. Cross-check with a shell model using S8 elements and a mid-surface offset of t/2. If the shell result is 150 MPa, the offset or the section definition is wrong; a correct shell model should land within ~10% of the converged solid result away from the singularity.

Modal and thermal specifics

In modal analysis, mass representation matters as much as stiffness. Tet4 with lumped mass over-stiffens the structure and shifts natural frequencies high; the error can exceed 20% for the first bending mode. Tet10 or Hex20 with consistent mass is the safe choice. For a frame structure, beam elements with correct section properties reproduce the first ten modes within a few percent of a solid model at a fraction of the cost.

In thermal analysis, the temperature field is diffusive and smooth. A Tet4 mesh with 3 mm elements often resolves a steady-state temperature distribution to within 1–2% of a much finer mesh. The element choice becomes critical only when you couple thermal results into a stress analysis, at which point the stress mesh requirements dominate. A practical strategy is to use a coarse Tet4 mesh for the thermal solve, then map temperatures onto a finer Tet10 stress mesh — but verify that the mapping does not smooth out sharp thermal gradients at surfaces, which would under-predict thermal stress.

Selection checklist

• Identify the dominant physics first: bending, modal, or thermal.
• For bending stress, prefer Hex20 or Tet10; never accept Tet4 without a convergence study.
• For modal, avoid Tet4 and lumped mass on bending-dominated parts.
• For thermal-only, Tet4 or Hex8 is usually sufficient; re-evaluate if coupled to stress.
• Use shells when span/thickness > 10 and through-thickness stress is negligible; verify offset and section integration.
• Use beams when length/section > 10 and the section does not distort; verify section properties against the real geometry.
• Always run at least one refinement pass and compare against an independent element family before reporting a peak value.

The element type is not a cosmetic choice. It sets the mathematical ceiling on what your mesh can represent, and no amount of mesh refinement can overcome a formulation that cannot bend. Choose the element family from the physics, then use convergence to confirm the mesh density — in that order.