fea-fundamentals

name	fea-fundamentals
description	Finite Element Analysis fundamentals — element types, stiffness matrix, shape functions, assembly, boundary conditions, solver types, convergence, error estimation, verification & validation.
metadata	{"priority":8,"promptSignals":{"phrases":["FEA","finite element","stiffness matrix","element type","FEA setup","boundary condition","FEA convergence","ANSYS","Abaqus","simulation"],"minScore":3}}

FEA Fundamentals — Complete Skill

The FEA Process

1. Pre-processing: geometry → mesh → material → BCs → loads
2. Solving: assemble [K]{d} = {F}, solve for displacements {d}
3. Post-processing: compute stress/strain from {d}, assess results

Stiffness Matrix (Global Equation)

[K]{d} = {F} [K] = Σ[kₑ] (assembled from element stiffness matrices) {d} = nodal displacement vector [DOF × 1] {F} = nodal force vector [DOF × 1]

Element stiffness (general): [kₑ] = ∫_Vₑ [B]ᵀ[C][B] dV [B] = strain-displacement matrix (derivatives of shape functions) [C] = material constitutive matrix (elastic moduli)

Shape Functions and Interpolation

Within element: u(x) = [N]{dₑ} [N] = shape function matrix (interpolate between nodes) ε = [B]{dₑ} = ∂[N]/∂x × {dₑ} σ = [C]{ε}

Linear element (1D bar, 2-node): N₁ = (1 - ξ)/2, N₂ = (1 + ξ)/2 where ξ ∈ [-1, +1] [B] = [-1/L, 1/L] [kₑ] = AE/L × [1 -1; -1 1] ← exact for uniform bar

Quadratic element (1D, 3-node): N₁ = ξ(ξ-1)/2, N₂ = 1-ξ², N₃ = ξ(ξ+1)/2 More accurate — captures quadratic displacement fields

Element Types

1D Elements

Type	DOF/Node	Use
Bar (truss)	1 (u)	Axial only, pin joints
Beam (Euler-B)	2 (v, θ)	Bending, no shear deformation
Timoshenko beam	2 (v, θ) + shear correction	Thick beams, L/h < 10
Pipe element	6 (u,v,w,θx,θy,θz)	Piping systems

2D Elements

Type	Nodes	Integration	Use
CST (3-node tri)	3	1 pt	Simple, poor performance, avoid
LST (6-node tri)	6	3 pt	Better, midside nodes
Q4 (4-node quad)	4	2×2	Standard, shear locking risk
Q8 (8-node quad)	8	3×3	Best 2D, preferred
CPS4R (reduced)	4	1 pt	Abaqus; add hourglass control

Plane stress: σ_z = τ_xz = τ_yz = 0 (thin plates, in-plane loads) Plane strain: ε_z = γ_xz = γ_yz = 0 (long prismatic bodies, tunnels, pipes) Axisymmetric: 2D elements, revolve around axis (rings, pressure vessels)

3D Elements

Type	Nodes	Integration	Recommend
TET4 (linear tet)	4	1 pt	Avoid — constant stress, poor
TET10 (quadratic tet)	10	4 pt	Good for complex geometry, auto-mesh
HEX8 (linear brick)	8	2×2×2	Good but shear locking risk
HEX20 (quadratic brick)	20	3×3×3	Best 3D, highest accuracy
PENTA15 (wedge)	15		Transition zones

Golden rule: Use HEX20 or TET10 over TET4. Never use TET4 for stress analysis.

Shell Elements

SHELL181 (ANSYS), S4R (Abaqus): 4-node, reduced integration, hourglass control S8R: 8-node quadratic shell (thick shells, large rotations) Use for: thin structures t/L < 0.1, pressure vessels, sheet metal

Boundary Conditions

Displacement BCs (Essential)

Fix DOFs to prevent rigid body motion:

• 3D solid: fix 6 DOFs minimum (3T + 3R or 3 non-collinear points)
• Symmetry plane XZ: fix UY, ROTX, ROTZ at symmetry plane
• Antisymmetry plane: fix UX, UZ, ROTY

Symmetry conditions: Symmetry about X-plane: U_x = 0, θ_y = 0, θ_z = 0 (all nodes on plane) Half-model: apply symmetry BC, apply half the total load

Natural BCs (Traction/Force)

Surface tractions, pressure loads, point forces Distributed load → equivalent nodal forces via shape functions Pressure on face: p → F_nodal = ∫[N]ᵀ p dA (computed by FEA software)

Solver Types

Linear Static

[K]{d} = {F} — single solve Assumptions: small displacements, linear elastic, static equilibrium Used for: most structural, pressure vessel checks

Nonlinear Static (iterative — Newton-Raphson)

[K_T(d)]{Δd} = {F} - {F_int(d)} until residual < tolerance K_T = tangent stiffness (updated each iteration) Types: geometric NL (large displacement), material NL (plasticity), contact NL

Modal Analysis

([K] - ω²[M]){φ} = {0} → eigenvalue problem Extract natural frequencies ω_i and mode shapes φ_i No loads applied; use consistent or lumped mass matrix [M]

Transient Dynamic

[M]{ẍ} + [C]{ẋ} + [K]{x} = {F(t)} Newmark-β method (implicit, unconditionally stable: β=0.25, γ=0.5) Time step: Δt ≤ T_min/20 (highest mode of interest)

Harmonic (Frequency Response)

(-ω²[M] + iω[C] + [K]){X} = {F₀} Frequency sweep → transfer function H(ω)

Convergence

h-refinement: reduce element size, check stress convergence p-refinement: increase polynomial order (same mesh) Convergence criterion: result changes < 5% with mesh refinement

Convergence study: Coarse → Medium → Fine mesh → plot stress vs. mesh size If converging: use Fine mesh result (or Richardson extrapolation) Richardson: σ_exact ≈ σ_fine + (σ_fine - σ_med)/(r^p - 1) where r = mesh refinement ratio, p = convergence order

Stress singularities: sharp re-entrant corners → σ → ∞ as mesh refines Solution: add fillet radius; or use K_I fracture mechanics approach

Error Indicators

Nodal stress averaging error: σ_error = (σ_avg - σ_elem)/σ_avg × 100% Target: < 5-10% error across transition elements

Energy norm error: e_h = √(∫(σ - σ_ref)ᵀ[C]⁻¹(σ - σ_ref)dV) Most robust convergence metric

V&V (Verification and Validation)

Verification: "solving the equations right" — compare to analytical, benchmark Validation: "solving the right equations" — compare to physical test data

NAFEMS benchmark problems: standard test cases for FEA software verification Hand calculation check: simple cases should match within 5%

Software Guide

Software	Primary Use	Strength
ANSYS Mechanical	General structural, multiphysics	Industry standard, APDL scripting
Abaqus	Nonlinear, contact, composites	Best NL capabilities, Explicit (impact)
NASTRAN (MSC/NX)	Aerospace structures	Certification-grade, SOL 101/103/105
LS-DYNA	Crash, impact, explicit	Fastest explicit, automotive
COMSOL	Multiphysics (thermal+structural+fluid)	Coupled physics
FreeCAD FEM	Open source, basic	Learning/prototyping

Output

Provide: element type recommendation, mesh density guidance (elements across thickness), BCs for symmetry, solver type, convergence criterion, expected accuracy, key post-processing quantities.