XFEM（拡張有限要素法）
Theory and Physics
What is XFEM?
Professor, what's so great about XFEM?
Principle of XFEM
Adds enrichment functions to the FEM displacement field:
- First term: Standard FEM
- Second term: Heaviside function $H$ — Discontinuity (jump) at the crack surface
- Third term: Crack tip enrichment $F_\alpha$ — Singular field of $\sqrt{r}$
You can add cracks without changing the mesh. No remeshing even during crack propagation!
This is the revolutionary advantage of XFEM. The crack position is described using the Level Set method.
Summary
The Birth of XFEM: The 1999 Revolution
XFEM (eXtended FEM) was proposed in 1999 by Belytschko and Black (Northwestern University). In conventional FEM, remeshing was required every time a crack propagated, but XFEM "adds" Heaviside and crack tip enrichment functions to the existing mesh, allowing crack representation without changing the mesh. Practical application accelerated with improvements by Moës and Dolbow in 2004.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness = strong" is incorrect. Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External force term (load term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A typical pitfall here: getting the load direction wrong. Intending "tension" but ending up with "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because vibration energy is converted to heat through air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibration energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, stress-strain relationship is linear
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definition)
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considers only equilibrium between external and internal forces
- Non-applicable cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior such as plasticity and creep requires constitutive law extension
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
XFEM FEM Settings
```
*ENRICHMENT, NAME=crack, TYPE=STATIONARY CRACK
element_set
*CONTOUR INTEGRAL, XFEM, CONTOURS=5, TYPE=J
```
STATIONARY CRACK (J/K evaluation for static cracks) or PROPAGATION CRACK (crack propagation).
Crack Propagation Criteria
Summary
Tracking Cracks by Combining with Level Set Method
XFEM is typically combined with the Level Set Method (LSM) to track crack geometry. Two Level Set functions, ψ (normal direction) and φ (tangential direction), describe the crack surface and tip, and the Level Sets are updated according to the crack growth direction. Both ANSYS SMART and Abaqus XFEM modules internally integrate LSM and XFEM, requiring users to only set crack growth criteria (maximum principal stress, SIF comparison, etc.).
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with mid-side nodes)
Can represent curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended: when stress evaluation is critical.
Full integration vs Reduced integration
Full integration: Risk of over-constraint (locking). Reduced integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (ZZ estimator, etc.). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Achieves quadratic convergence within convergence radius, but computational cost is high.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$
Load Increment Method
Applies total load not all at once, but in small increments. The arc-length method (Riks method) can track beyond limit points on the load-displacement curve.
Analogy: Direct Method vs Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to open it at an estimated location and adjust forward/backward (iterative method) than to search sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
XFEM in Practice
Simulation of crack nucleation and propagation. Cracks in welded structures, fatigue cracks in piping.
Practical Checklist
Predicting Crack Growth in Welded Structures Using XFEM
EPRI (Electric Power Research Institute, USA) adopted XFEM for crack growth evaluation in nuclear power plant piping welds. Compared to the conventional manual crack shape update method, automatic tracking with XFEM reduced analysis time by 80%. The superiority of XFEM, which can track crack propagation even when cracks bend under thermal stress without remeshing, was demonstrated.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (post-processing for visualization). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If the mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the result is correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer will be far from reality. Confirm that results stabilize with at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is the same as "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling the real-world constraint conditions is actually the most important step in the entire analysis.
Software Comparison
XFEM Tools
Example of Utilizing Abaqus XFEM Module
XFEM is implemented in Abaqus/Standard via the *ENRICHMENT function, controlling crack propagation with SIGEPS (maximum principal strain criterion) or KCRIT (critical SIF criterion). EDF (Électricité de France) uses Abaqus XFEM for SCC (stress corrosion cracking) propagation analysis in nuclear primary system piping, achieving a more physically detailed evaluation than the conventional Engineering Assessment (FA-3) method with only a 10% increase in time.
The Three Most Important Questions for Selection
- "What are you solving?"
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