大変形（幾何学的非線形）解析
Theory and Physics
What is Large Deformation?
Professor, how is "Large Deformation" analysis different from regular FEM?
Regular linear FEM assumes infinitesimal deformation: the shape before and after deformation are "almost the same". Large deformation analysis handles problems where the shape changes significantly due to deformation. Equilibrium is evaluated on the deformed shape.
Sources of Geometric Nonlinearity
Three nonlinear effects:
1. Large strain — Strain is not infinitesimal ($\varepsilon << 1$). Rubber, etc.
2. Large rotation — Element rotation is not small. Large deformation of beams or shells.
3. Follower force — Load direction follows the deformation. Pressure loads, etc.
All of these are ignored in linear analysis, right?
Linear analysis assumptions: $\varepsilon << 1$, rotation $\theta << 1$, loads act on the initial shape. If any of these do not hold, NLGEOM=YES (large deformation option) is required.
When is NLGEOM Needed?
| Condition | NLGEOM Required? |
|---|---|
| Strain > 5% | Mandatory |
| Displacement/Dimension Ratio > 10% | Mandatory |
| Rotation Angle > 10° | Mandatory |
| Pressure Load (large area change) | Required |
| Post-buckling behavior | Mandatory |
| Rubber/Hyperelasticity | Mandatory |
So if displacement is more than 10% of the dimension, large deformation is needed.
For a 1 mm thick plate, if it deflects more than 0.1 mm, large deformation. It's needed more often than you might think.
NLGEOM Settings
Summary
Key Points:
- Evaluate equilibrium on the deformed shape — Linear analysis stays on the initial shape.
- Large strain + Large rotation + Follower force — Three nonlinear effects.
- Displacement/Dimension Ratio > 10% is mandatory — Needed more often than you think.
- NLGEOM=YES (Abaqus), SOL 106/400 (Nastran), NLGEOM ON (Ansys)
Green and Almansi Finite Strain
Finite deformation theory requires two configurations: "current configuration" and "reference configuration". Green-Lagrange strain (reference configuration basis) and Almansi strain (current configuration basis) coincide under infinitesimal deformation, but when the stretch ratio exceeds 1.2, a difference of over 10% arises. The distinction between these two types of strain, independently proposed by Green and Almansi in the 1900s, is directly linked to the difference between Total Lagrangian (reference configuration) and Updated Lagrangian (current configuration) FEM formulations.
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 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—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"—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), 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"—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—deliberately absorbing vibrational 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.
- Infinitesimal deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and 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: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
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 to N in mm system, N in m system. |
Numerical Methods and Implementation
Newton-Raphson Method
What is the basic algorithm for large deformation analysis?
Newton-Raphson Method: Apply load in increments, iteratively satisfying equilibrium at each increment.
What is the basic algorithm for large deformation analysis?
Newton-Raphson Method: Apply load in increments, iteratively satisfying equilibrium at each increment.
1. Load increment — Apply total load divided into $n$ steps.
2. Equilibrium iteration — Newton-Raphson iteration at each increment until internal and external forces match.
3. Tangent stiffness matrix update — Recalculate stiffness based on deformed shape.
So it solves simultaneous equations repeatedly at each increment. Much heavier than linear analysis.
Linear analysis solves simultaneous equations once. Large deformation analysis solves them $n$ increments × $m$ iterations. Computational cost is 10 to 100 times higher.
Total Lagrangian Method vs. Updated Lagrangian Method
Abaqus's NLGEOM=YES uses UL. Nastran's SOL 106 is TL-based.
Summary
Arc-Length Method and Snap-Through Tracking in Large Deformation Analysis
When the load-displacement curve shows "snap-back", it cannot be tracked with normal load control. The Riks method (arc-length method), proposed by Kemper and Riks in 1972, simultaneously increments load and displacement, enabling tracking up to unstable equilibrium paths. Applications to industrial analysis, such as shell snap-through and buckling deformation of rubber seals, have been standardized since the 1980s as the RIKS step in Abaqus.
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 Midside 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.
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. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix at 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
Apply total load not all at once, but in small increments. The arc-length method (Riks method) can track beyond extremum points in the load-displacement relationship.
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 judge based on total cost-effectiveness.
Practical Guide
Large Deformation in Practice
Typical problems requiring large deformation analysis:
| Problem | Reason for Large Deformation |
|---|---|
| Rubber components | Strain > 100% |
| Sheet metal forming | Large strain + large rotation |
| Cables/Ropes | Geometric stiffness change |
| Membrane structures | Initial shape is "flat", undergoes large deformation in use |
| Post-buckling | Deformed shape is important |
| Medical devices (stents) | Large deformation during expansion |
Practical Checklist
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