Automobile Collision Simulation
Theory and Physics
Collision Simulation Overview
Professor, could you explain the overall picture of automotive collision simulation?
Automotive crash safety simulation is one of the largest application areas of FEM. For a single vehicle, 20 to 50 collision scenarios are evaluated using FEM, with only a few final confirmation tests on actual vehicles.
Collision Physics
The fundamentals of collision are energy absorption and occupant protection:
1. Kinetic Energy — $E = mv^2/2$. 1500 kg × (56 km/h)² / 2 ≈ 180 kJ
2. Energy Absorption — The front crash zone absorbs 180 kJ through plastic deformation
3. Deceleration — The average deceleration during the collision determines the impact on occupants
4. Restraint System — Seatbelts + airbags decelerate the occupants
So the structure absorbs as much as 180 kJ of energy?
Energy is dissipated through plastic deformation of steel sheets (bending, buckling, tearing). This "controlled destruction" is the design philosophy of the crash zone.
Numerical Methods
The numerical method for collision simulation is explicit FEM (Central Difference Method). LS-DYNA is the world standard.
Features:
- Large Deformation, Plasticity, Contact, and Fracture all occur simultaneously
- Millions of elements in a full vehicle model
- 50 to 200 ms of analysis time
- Strain rate dependent material models are essential
Summary
Key Points:
- 180 kJ Energy Absorption — Plastic deformation of the crash zone
- Explicit FEM (LS-DYNA) is the world standard
- 20 to 50 collision scenarios evaluated by FEM — Actual vehicle tests are for final confirmation only
- Material strain rate dependency is important — Cowper-Symonds law
The Foundation of Crash Safety is Thin-Walled Crushing Theory
The axial crushing theory proposed by Alexander (1960) expresses the energy absorption of corrugated deformation of a thin-walled cylinder as the product of plate thickness, diameter, and yield stress. This theory was utilized by Ford and GM for cross-section optimization of front rails, becoming the analytical foundation for maintaining crash performance while downsizing during the 1980s fuel efficiency regulations. In current vehicle body CAE, a full frontal collision of 10 milliseconds is calculated within 100 ms using the finite element method.
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 assumes "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted in 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's Hooke's law $F=kx$, the essence of the stiffness term. So here's 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 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"—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 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—intentionally absorbing 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, and 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 equilibrium between external and internal forces
- Non-applicable cases: Geometric nonlinearity is required for large deformation/large rotation problems. Constitutive law extension is needed for nonlinear material behavior like plasticity and creep
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load 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 dependency |
| 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
Collision Simulation Setup
Basic LS-DYNA settings:
```
*CONTROL_TERMINATION
0.120 $ 120 ms
*CONTROL_TIMESTEP
0.0, 0.9, 0, 0.0, 0.0, 0, 0, 0
*INITIAL_VELOCITY_GENERATION
1, 0., 0., 0., 0., -15556., 0. $ 56 km/h (in mm/ms units)
*CONTACT_AUTOMATIC_GENERAL
0 $ Full automatic contact
```
So *CONTACT_AUTOMATIC_GENERAL automatically detects contact between all parts.
In collisions, it's impossible to predict what will contact beforehand. Full automatic contact comprehensively detects contact between all parts. LS-DYNA's automatic contact operates stably even with models of millions of elements.
Summary
Explicit Crash Analysis Time Step is 1μs
Crash simulation uses a stable time step determined by element size/sound speed from the CFL stability condition. For typical vehicle body analysis (minimum element 5mm, steel plate C=5000m/s), the time step is about 1μs, calculating a 100ms collision in 100,000 steps. Parallel computing (256 cores) analyzing a full vehicle model of about 7 million elements in 2-4 hours is the industry standard cycle time in the 2020s.
Linear Elements (1st Order)
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-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 mode (zero-energy mode). 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. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix with 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 extremum points on load-displacement relationships.
Analogy of 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: opening to an estimated page and adjusting forward/backward (iterative method) is more efficient than searching 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
Collision Simulation Practice
Achieving Euro NCAP 5 stars is the development target. All patterns are pre-verified with FEM.
Practice Checklist
NCAP Test is 64km/h Full Overlap Collision
The Euro NCAP full frontal test is a 64 km/h full overlap collision with a fixed barrier, evaluating occupant dummy (Hybrid III 50th percentile) head HIC, chest 3ms acceleration, etc. The 2022 version added the MPDB (Moving Progressive Deformable Barrier) test, making aggressivity evaluation towards the other vehicle in partial overlap collisions also mandatory. Automakers simultaneously calculate all these test modes in LS-DYNA, with some cases exceeding 3000 CPU hours per test cycle.
Analysis Flow Analogy
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—in cooking, which step is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, no matter how excellent the solver, the results will be a mess.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "calculation ran = results are 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. Verify 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".
Approach to 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 truly fully fixed?" "Is this load truly uniformly distributed?"—Correctly modeling real-world constraint conditions is actually the most important step in the entire analysis.
Software Comparison
Collision Simulation Tools
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