Self-Contact
Theory and Physics
What is Self-Contact?
Professor, is self-contact a problem where a structure contacts "itself"?
Yes. It's a problem where a part of a structure contacts another part due to large deformation. Examples include rubber O-ring compression, sheet metal bending, tire deformation, balloon inflation, etc.
Normal contact is between "two separate bodies," but self-contact is between "different surfaces of the same body." Contact detection is complex.
Settings in FEM
Summary
Key Points:
- A structure contacts itself — Occurs in large deformation
- Contact detection is complex — Monitoring distances between surfaces within the same body
- General Contact (automatic contact) is convenient — Self-contact is also automatically detected
Mathematical Definition of Self-Contact
Self-contact is a phenomenon where different parts of a single object come into contact with each other. It has a higher degree of freedom in deformation compared to normal contact problems and is characterized by the inability to identify contact pairs in advance. Mathematically, it is formulated as a condition where the mapping φ on the initial configuration Ω ceases to be injective. In 1987, Ciarlet & Nečas proved the existence theorem for "deformations without self-intersection" in Sobolev spaces.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward during sudden braking? That "feeling of being pulled" 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 cannot be omitted in 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 — an iron rod and a rubber band, which stretches more under the same force? Obviously, the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong." Stiffness is "resistance to deformation," strength is "resistance to failure" — different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) 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 tire pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but ending up with "compression" — sounds like a joke, but it actually happens when coordinate systems rotate 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. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibrational energy for a smoother ride. What if damping were zero? Buildings would keep 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, and the stress-strain relationship is linear.
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity or creep, constitutive law extensions are needed.
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 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
Implementation of Self-Contact
```
*CONTACT, TYPE=GENERAL CONTACT
*CONTACT INCLUSIONS, ALL EXTERIOR
```
General Contact + ALL EXTERIOR automatically detects self-contact on all exterior surfaces.
```
*CONTACT_AUTOMATIC_SINGLE_SURFACE
1 $ Part set ID
```
Summary
High-Speed Search via Bucket Sort
The bottleneck in self-contact detection is the O(N²) exhaustive pair search. The bucket sort (spatial hashing) method implemented in LS-DYNA Version 930 (around 1993) divides the computational domain into cells and only considers nodes within the same or adjacent cells as pair candidates, reducing computational complexity to O(N logN). Even today, adaptive bucket sizes combined with GPU parallelization form the backbone of mainstream algorithms.
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)
Capable of representing curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2-3 times. Recommended when stress evaluation is important.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass mode (zero-energy mode). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). 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. Provides quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial values or every few iterations. Lower cost per iteration but linear convergence speed.
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 in small increments rather than all at once. The arc-length method (Riks method) can trace beyond extremum 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) than to search sequentially from the first page (direct).
Relationship Between Mesh Order and Accuracy
1st order elements are like "approximating a curve with a ruler" — represented by straight polylines, so accuracy is limited. 2nd order elements are like "flexible curves" — can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Self-Contact in Practice
Practical Checklist
Automotive Door Crash Analysis
In the late 1990s, during the crash test reproduction analysis for the Ford Explorer door, self-contact occurred when the inner trim panel folded. Analyses that did not consider this showed intrusion amounts more than double the measured values. Models applying LS-DYNA's SINGLE_SURFACE contact reported intrusion errors within ±8%, enabling door beam shape optimization without physical experiments, as documented in SAE Paper 1999-01-3155.
Analogy of the Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy ingredients (prepare CAD model), do the prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question — which step in cooking is most prone to failure? Actually, it's the "prep work." If mesh quality is poor, the results will be a mess no matter how good 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 is far from reality. Confirm that results stabilize across 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 like "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 real-world constraints is actually the most crucial step in the entire analysis.
Software Comparison
Self-Contact Tools
Selection Guide
Evolution of LS-DYNA Self-Contact
LS-DYNA3D's self-contact feature was first published by John Hallquist in a 1987 LLNL internal report. Initially called SINGLE_SURFACE contact, it specialized in folding thin-shell structures. Version 940 (1994) added segment-based self-contact, enabling compression of thick objects. The current R14 version implements asynchronous bucket reconstruction for MPPP (Massively Parallel Processing), making self-contact analysis on a scale of 1 billion elements practical.
Three Most Important Questions for Selection
- "What are you solving?": Does the required physical model/element type for self-contact (self-contact) have support? For example, fluid
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