Gap Element
Theory and Physics
What is a Gap Element?
Professor, what is a "gap element"?
A gap element is an element that has a gap (clearance) between two points and transmits force only when they are in contact. It is used as a simplified model for contact problems.
Physical Image
A gap element is a "one-dimensional contact element":
- Gap is open ($\delta < g$) → Force is zero
- Gap is closed ($\delta \geq g$) → Transmits force as a compression spring (stiffness $k$)
Here, $\delta$ is the relative displacement between the two nodes, and $g$ is the initial gap.
It's a simple nonlinearity: "when the gap closes, it becomes a spring."
Correct. Force-displacement relationship:
Applications
| Application | Explanation |
|---|---|
| Bolt hole clearance | Load transfer when a bolt contacts the hole |
| Bearing stopper | Contact beyond a certain displacement |
| Pipe support | Unidirectional support only (lift-off) |
| Contact due to thermal expansion | Gap closes with temperature rise |
What do you mean by pipe support being unidirectional only?
When a pipe deflects downward, it rests on the support; when it deflects upward, it lifts off from the support. It transmits force only in the downward direction and is free in the upward direction. This is a typical application of gap elements.
Element Names by Solver
| Solver | Element Name | Notes |
|---|---|---|
| Nastran | CGAP | Specify direction, gap amount, closure stiffness |
| Abaqus | *GAP / GAPUNI | 1D gap. ITT element |
| Ansys | CONTA178 | Node-to-node contact element |
Does Abaqus have ITT (Interface) elements separate from GAP elements?
In Abaqus, the general contact definition (*CONTACT PAIR / *GENERAL CONTACT) is more flexible and recommended than gap elements (*GAP). Gap elements are used only for simplified 1D contact.
Gap Element vs. Contact Definition
| Comparison | Gap Element | Surface Contact Definition |
|---|---|---|
| Degrees of Freedom | One direction only | Entire surface |
| Setup Effort | Low | High |
| Accuracy | 1D approximation | Accurate contact pressure distribution |
| Nonlinearity | Weak | Strong |
| Friction | Supported only by Nastran CGAP | Fully supported |
So, use gap elements for simplicity and full contact definition for precision.
Exactly. Gap elements are used when a simple binary judgment of "contact/no contact" is sufficient. If contact pressure distribution or sliding is important, a general contact definition is necessary.
Summary
Let me organize the theory of gap elements.
Key points:
- Transmits force only when the gap is closed — A one-dimensional contact element
- Force = 0 (open) or $k(\delta - g)$ (closed) — Simple nonlinearity
- Modeling of pipe supports, stoppers, clearances — Widely used in practice
- Nastran CGAP is the most widely used — The standard for simplified contact
- Use full contact definition for precise contact — Gap elements are a simplified model
Theoretical Origin of Gap Elements
The first attempt to handle contact problems with FEM was in 1963, extending Hertz's contact theory with the matrix method. The formulation as a gap element was pioneered by Wilson & Parkes in their 1972 paper, establishing the binary contact method that treats the opening/closing between two nodes as an "ON/OFF switch." This method became the prototype for the current ANSYS CONTA171 element.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, i.e., "mass × acceleration." Have you ever experienced being thrown forward when slamming on the brakes? 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 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"—they are different concepts.
- External force term (load term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (e.g., pressure, contact force). 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 is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common 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. That's because the vibration energy is converted to heat by 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 keep swaying 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 otherwise specified): 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
Nonlinear Solution Method for Gap Elements
Gap elements are nonlinear, right? Can't they be used in linear analysis?
The opening/closing of a gap is a state change, so it is inherently nonlinear. However, many solvers process it pseudo-linearly using iterative methods.
Procedure:
1. Perform linear analysis assuming all gap elements are "open."
2. Check the relative displacement of each gap. Change elements that should close to "closed."
3. Re-analyze with the updated stiffness.
4. Iterate until the state of all gaps stabilizes.
Is it thanks to this iteration that CGAP can be used even in Nastran's SOL 101 (linear static analysis)?
In Nastran, using CGAP in SOL 106 (nonlinear static analysis) is the formal way, but SOL 101 also has a "linear contact" feature that performs gap opening/closing iterations. It's faster than full nonlinear analysis but cannot handle complex contact.
Closure Stiffness Setting
How do you set the stiffness $k$ when the gap closes?
Ideally, it would be "infinite" (perfectly rigid contact), but numerically, a finite large stiffness is used via the penalty method.
Guidelines:
- $k \approx 10 \sim 100$ × the stiffness of the contact surface (equivalent to $EA/L$)
- Too large → condition number worsens, convergence becomes difficult
- Too small → excessive penetration
Finding the "just right $k$" seems difficult.
Abaqus's *CONTACT definition automatically calculates the penalty stiffness, but with gap elements, manual setting is required. Start with $k$ = 10 times the structural stiffness and adjust so that penetration is less than 1% of the plate thickness.
Summary
Let me organize the numerical methods for gap elements.
Key points:
- Converge gap opening/closing with iterative methods — Treated as iterations of linear analysis
- Closure stiffness is 10~100 times the structural stiffness — Neither too large nor too small
- Simplified contact possible even in Nastran's SOL 101 — But SOL 106 is the formal method
- Full contact definition recommended for precise contact — Gap elements are a simplified model
Penalty Method vs. Lagrange Method
There are two main approaches to the numerical implementation of gap elements: the penalty method and the Lagrange multiplier method. The penalty method can be implemented without changing the dimension of the stiffness matrix and is easier to implement, but the choice of penalty coefficient directly affects accuracy. In 1974, Bathe and Wilson showed that the Lagrange method provides more stable condition numbers, which became a guideline for subsequent high-precision contact solver development.
Linear Elements (1st-order elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with 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 important.
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 (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 the tangent stiffness matrix each iteration. Provides quadratic convergence within the convergence radius but has high computational cost.
Modified New
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