xfem-extended-fem

name	xfem-extended-fem
description	Extended finite element method (XFEM) — enrichment functions, level set method, crack growth simulation, stress intensity factor extraction, cohesive zone, Abaqus/ANSYS XFEM.
metadata	{"priority":7,"promptSignals":{"phrases":["XFEM","extended finite element","crack growth simulation","crack propagation FEM","level set crack","XFEM Abaqus"],"minScore":3}}

Extended Finite Element Method (XFEM) — Complete Skill

Motivation

Standard FEM requires mesh to conform to crack geometry; crack growth → remeshing XFEM enriches displacement field to represent discontinuities without mesh modification

XFEM Enrichment

Enhanced displacement field: u_h(x) = Σ_I N_I(x) u_I [standard FEM] + Σ_J N_J(x) H(x) a_J [Heaviside enrichment — for cut elements] + Σ_K N_K(x) Σ_α F_α(x) b_K^α [near-tip enrichment — for crack tip elements]

Heaviside function H(x): H(x) = +1 if x above crack; -1 if below crack Represents displacement jump across crack faces

Near-tip enrichment functions F_α(x): F_1 = √r sin(θ/2) F_2 = √r cos(θ/2) F_3 = √r sin(θ/2) sin(θ) F_4 = √r cos(θ/2) sin(θ) These are Williams asymptotic fields (singular at crack tip)

Level Set Method for Crack Tracking

Two level set functions: φ(x): signed distance from crack surface (φ=0 on crack; + ahead, – behind in plane) ψ(x): signed distance from crack tip extension line

Crack tip location: φ=0 and ψ<0 Element categorization: cut (|φ_min| varies sign), tip (contains tip), standard (outside)

Stress Intensity Factor Extraction

Domain integral method (J-integral → K_I, K_II): J = ∫_Ω (σ_ij × ∂u_i/∂x₁ - W δ₁ⱼ) × ∂q/∂x_j dΩ W = strain energy density; q = weight function (smooth, 1 at tip → 0 at boundary)

Interaction integral for mixed mode: J_int = ∫ interaction terms with auxiliary Mode I/II fields K_I = √(J E* / (1 + K_II²/K_I²)) for plane strain E* = E/(1-ν²)

Accuracy: 1–3% error with proper enrichment and integration

Crack Growth Criterion

Maximum hoop stress criterion (Erdogan-Sih): Propagation direction θ_0: K_I sin θ_0 + K_II(3cos θ_0 - 1) = 0 Propagation when: K_eff = K_I cos²(θ_0/2) - 3/2 K_II sin(θ_0) ≥ K_IC

Fatigue crack growth: Paris law per cycle da/dN = C (ΔK_eff)^m Level sets updated; mesh remains fixed; crack grows by Δa each load cycle step

Cohesive Zone Model (CZM)

Often combined with XFEM for ductile fracture: Traction-separation law in crack-plane elements: t_n = K_nn δ_n (before damage initiation) After damage: t_n = (1-D) K_nn δ_n (D = damage variable 0→1)

Damage initiation: max traction or max displacement criterion Damage evolution: linear or exponential softening to G_IC fracture energy

XFEM in Abaqus

Setup:

• Define crack geometry using Assembly → Crack (Seam crack or XFEM crack)
• Material: assign fracture energy G_IC, K_IC (from material or CZM traction-separation law)
• Step: Static; activate CONTOURS output for SIF (SIFS, KINOS)
• Request PHILSM, PSILSM fields for level set visualization
• Request STATUSXFEM for element damage state

Output: J-integral over contours; K_I, K_II, K_III per contour ring; crack mouth opening displacement (CMOD)

XFEM in ANSYS Mechanical

• Insert: Fracture → Crack (Arbitrary crack, Semi-elliptical, or Smart Crack Growth)
• Smart Crack Growth: automated propagation with remeshing or XFEM
• Fatigue: combine with nCode or Workbench Fatigue module for da/dN integration
• Output: SIFS (KI, KII, KIII), J-integral, T-stress

Validation

Compare K_I to analytical solutions (edge crack: K_I = F σ √(πa); embedded: K_I = σ √(πa)) J = K_I² / E* (plane strain) → check consistency Mesh refinement at crack tip: 5–10 elements within 0.1a radius for accuracy

Limitations

XFEM not well-suited for:

• Large-scale plasticity (requires other models)
• Very curved cracks in 3D (level set accuracy degrades)
• Intersecting cracks (topology complex)
• Manufacturing processes (use Lagrangian / adaptive remeshing instead)

Output

Provide: K_I and K_II [MPa√m] vs. K_IC, crack propagation angle θ_0 [deg], N cycles to critical crack size (Paris integration), critical crack length a_c [mm], XFEM convergence check (J contour values), margin of safety on fracture.