| 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.
