Follower Force
Theory and Physics
What is a Follower Force?
Professor, what is a "follower force"?
A load whose direction changes following deformation. A typical example is internal pressure: when a container deforms, the orientation (normal direction) of the pressure-acting surface changes, and thus the direction of the pressure also changes.
Examples of Follower Forces
| Load | Follower? | Reason |
|---|---|---|
| Internal/External Pressure | Yes | Follows the surface normal direction |
| Gravity | No | Always downward |
| Rocket Thrust | Yes | Follows the vehicle's orientation |
| Wind Load | No (usually) | Fixed direction |
| Contact Force | Yes | Follows the contact surface normal |
Internal pressure is a follower force... When it expands, the area changes and the magnitude of the force changes too.
Both area change and direction change contribute to nonlinear effects. Follower forces are automatically considered when NLGEOM=YES.
Summary
Ziegler's Paradox and the Instability of Follower Forces
The stability of follower forces (forces where the external load follows deformation) is famous for the paradox shown by Wilhelm Ziegler (ETH Zurich) in 1952. There exists "Ziegler's Paradox" where a continuum that should become stable with added viscous damping instead becomes unstable. This demonstrates the peculiarity of follower force systems where static and dynamic stability do not coincide, and is deeply involved in the analysis of flutter-type instability (Beck's column problem).
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Haven't you experienced your body 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, which is the assumption that "acceleration can be ignored because the force is applied slowly". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That is 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". 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 contents" (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 common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—it 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, right? 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 continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
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 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
Follower Forces in FEM
In nonlinear analysis with NLGEOM=YES, for each iteration:
1. Recalculate the surface normal direction based on the deformed shape
2. Pressure × deformed area = Follower force
3. Reflect in the right-hand side of the global equation
Abaqus: *DLOAD, P (pressure) automatically becomes follower with NLGEOM=YES.
Comparison with Non-Follower Forces
Results can differ by 10-20% depending on whether internal pressure is treated as follower in large deformation (e.g., balloon expansion). With NLGEOM=NO, internal pressure acts fixed to the initial surface.
Summary
Finite Element Formulation of Follower Forces and Load Stiffness Matrix
To incorporate follower forces into FEM, it is necessary to add a "load stiffness matrix" (Kσ_load) that depends on deformation to the tangent stiffness. For uniform pressure (normal direction follower force), the load stiffness matrix becomes anti-symmetric. While the standard full Newton method (Newton-Raphson) can automatically consider follower forces by updating the load stiffness each analysis, note that errors may arise depending on the update frequency in the Modified Newton method.
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Computational cost is low, but stress accuracy is low. 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 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 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
Instead of applying the full load at once, apply in small increments. 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"—the initial answer is rough, but accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it 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 with the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Follower Forces in Practice
Problems where follower forces are important:
- Large deformation (expansion) of pressure vessels
- Balloon/airbag inflation
- Rocket thrust direction
- Brake friction force (direction changes with deformation)
Practical Checklist
Thrust Follower Effect in Solid Rockets
The thrust nozzle of a solid rocket is a typical example of a follower force that always generates a rearward resultant force relative to the deformed vehicle body. The Japan Aerospace Exploration Agency (JAXA) applied nonlinear dynamic analysis including follower forces to the flight load analysis of the Epsilon rocket (first launched in 2013) and incorporated into the design standard the conservative evaluation of structural loads during roll programs as 3-5% higher.
Analogy of Analysis Flow
The analysis flow is actually very similar to cooking. First, 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 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 across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the answer must be correct because the computer produced it".
How to Think 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 real-world constraint conditions is actually the most important step in the entire analysis.
Software Comparison
Tools
All solvers support follower forces with NLGEOM=YES. No difference.
Discovery of Follower Forces: Euler's Compression Buckling Research
The problem of follower forces (follower loads) emerged in nascent form when Leonhard Euler studied rod compression buckling in 1744. Since ABAQUS 6.14, the `FOLLOWER FORCE` option allows explicit specification of follower loads; there is a case where buckling load decreased by up to 23% when the follower nature of nozzle combustion gas pressure was not considered in rocket engine thrust analysis.
The Three Most Important Questions for Selection
- "What to solve?": Does the required physical model/element type for follower forces (follower loads) have support? For example, in fluids, the presence of LES support; in structures, the capability to handle contact/large deformation makes a difference.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic transmission (GUI) and manual transmission (script) in cars.
- "How far to extend?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced Research
Variational Principle for Non-Conservative Force Systems: Extension of Hamilton's Principle
Since follower forces are non-conservative, the usual potential energy minimization principle cannot be used. It is necessary to use an extended version of Lagrange's equations, Hamilton's variational principle (principle of virtual work), to explicitly incorporate the work of non-conservative forces. Research by Bolotin (Moscow) in the 1970s introduced probability theory into the stability analysis of non-conservative forces, laying the foundation for Dynamic stability theory.
Troubleshooting
Troubles
When Stiffness Appears to Decrease with Load Increase in Follower Force Analysis
If Newton-Raphson method convergence worsens in follower force analysis and apparent stiffness decreases as load increments accumulate, the sign or magnitude of the load stiffness matrix may be inappropriate. When using Abaqus pressure follower force (FOLLOWER FORCE TYPE=PRESSURE), set the iterative convergence criterion strictly to 1e-6 or lower and check the residual at each load step. It is good to verify stability by making load steps finer (1/5 to 1/10).
If You Think "The Analysis Doesn't Match"
- First, take a deep breath—Randomly changing settings in a panic will only complicate the problem further
- Create a minimal reproducible case—Reproduce the follower force (follower load) problem in its simplest form. "Subtractive debugging" is the most efficient
- Change only one thing and re-run—Making multiple changes simultaneously makes it unclear what worked. The principle of "controlled experiment" same as in scientific experiments
- Return to physics—If the calculation result is non-physical, like "an object floating against gravity", suspect a fundamental mistake in the input data
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