Car Crashworthiness: Are There Physical Limits to Improvement?

Computer simulation of car crash is certainly among the most computationally demanding tasks that engineers in the automotive industry perform. The size and cost of computer models for car crash is steadily growing and models with millions of finite elements are not uncommon. Huge compute power is needed to run such a model in acceptable time. But does this growth of modelsize and bandwidth really pay off? Is it really possible to build safer cars by building larger and more sophisticated computer models? Are there any intrinsic limitations to crash-worthiness of automobiles dictated by the physics of crash, which make it impossible to improve performance beyond the already high standards of today? The answer may be articulated along the following lines.

Models are only models. A model, no matter how refined, tells only part of the truth. Using a model that misses some physics introduces a limitation as to how much the real product can be improved.
Signals recorded by accelerometers during car crash tests contain a non-negligble amount of chaos. Chaotic systems (such as the atmosphere) are not fully predictable. While certain patterns may be observed (e.g. the seasons) predictions are essentially impossible.
While crash is an exquisitely stochastic (uncertain) phenomenon, the car industry ignores this fundamental fact of physics, and insists on building deterministic models. Lab tests are not fully representatiove of real crash conditions. Moreover, imperfections in the manufacturing process make it practically impossible to build two identical cars. Therefore, a single crash test may be very misleading.

Numerical Models. Many people overlook the fact that models are already fruit of sometimes quite drastic simplifications of the actual physics. For example, the familiar Euler-Bernoulli beam differential equation is result of the following assumptions:

The material is continuum.
The beam is slender.
The constraints are perfect.
The material is linear and elastic.
The effects of shear are neglected.
Rotational inertia effects are neglected.
The displacements are small.

While the above assumptions might at first glance seem reasonable, their combined effect may in many cases be responsible for a substantial “loss of physics”. Models never represent reality completely, and this incompleteness is always source of unexpected errors. Experience shows that models which capture correctly more than 90% of reality are quite rare. Car crash is a highly non-linear phenomenon, which has very little in common with bending of long and slender beams. The amount of phenomena involved in crash (material and weld failure, buckling, crushing, etc.) is so broad that a fidelity of 90% is, most probably, unreachable with contemporary simulation technology. But the question remains. Even if models were “perfect”, would it be possible to build safer cars than those on the roads today?

Chaos and crash. Tests for chaotic content in a time-history can be performed with a variety of mathematical techniques. The basic tests for chaos are:

Poincaré sections (or return maps)
log-linear power spectrum
Hausdorff dimension
Correlation dimension
Lyapunov Characteristic Exponents

For the purpose of discussion it is not necessary to go into the details of these tests. An example of a typical crash pulse is illustrated in the figure below.

Twenty years ago, while working at the BMW R&D Center in Munich, I have tested a real crash signal (i.e. not one generated in a computer simulation) for chaotic content. The above tests revealed indeed that the signal contained a fair amount of chaos. In fact, the fractal dimension of the crash pulse of 1.8 – non-integer dimension points to a fractal – confirmed that there is hidden chaos in the crash response. But what does this mean? Why is the presence of chaos a nuisance? Contrary to popular belief, chaos does not imply randomness. Chaos is a deterministic phenomenon which may be described by closed-form equations. From a purely aesthetical point of view chaos may look like randomness but it is, essentially, a deterministic phenomenon. Chaotic phenomena and systems have the nasty characteristic of being extremely sensitive to initial conditions. Ultimately, this means one thing: since in car crash it is practically impossible to repeat the same initial conditions, even in a lab, every crash test is a unique event. Even if it were possible to manufacture two identical cars, the chaotic nature of crash would still make each crash an unrepeatable event. In a ground-braking stochastic crash experiment performed by the author and colleagues in 1997 at the University of Stuttgart’s Compute Center (RUS) on a 512 CPU Cray T3E, it was discovered that the angle of impact in car crash is a key variable which determines how the structure will deform. This apparently innocent variable – what difference can 1 or 2 degrees make? – was found to be more important than design details or material properties. It was in fact found that the response bifurcated around the nominal impact angle of 90 degrees – in other words, hitting the barrier at 89 or 91 degrees made a huge difference in the response. One or two degrees can make a huge difference. In order for a Finite Element (FE) model to show this, the model clearly has to embrace enough physics. The bottom line, however, is that the angle of impact, which cannot be determined a-priori, not even in a lab test, happens to be an initial condition that influences strongly crash behavior. If that were not enough, crash is a chaotic phenomenon that depends strongly on the angle of impact. Each crash is a unique and unrepeatable event.

Uncertainty modeling. Because of manufacturing imperfections, assembly tolerances, material property scatter, etc., it is impossible to manufacture two identical cars. But this is only one side of the medal. Once you take a new car on the road, it starts to age. Each time a bump is hit, a few weld points are lost. Due to weld-point failure and corrosion, the structure becomes softer. This not only modifies the response in the case of crash, it also reduces ride comfort and handling. After three or four years you are essentially driving quite a different car than the one you purchased. The process of aging is irreversible as much for cars as it is for humans. The bottom line? First of all, certification crash tests are performed on a brand new car (and in clinical conditions) while in reality it is an aged car that suffers a crash (and not precisely in a lab!). While one could accept this because of cost constraints, what is not easy to accept is that with very rare exceptions cars are designed neglecting the above mentioned uncertainties. A deterministic model is built in which everything is nominal. All welds are in place, no manufacturing tolerances, perfect materials. This model is then used in computer simulations to come up with a design. If that were not enough, these deterministic (optimistic) models are used to optimize the design, i.e. to deliver the best possible crashworthiness.

The answer to our question, at this point, is evident. The underlying buckling-dominated nature of car crash makes it impossible to predict behavior even in a lab. The chaotic component present in crash adds extreme sensitivity to intitial conditions, such as angle of impact. Manufacturing tolerances make it impossible to manufacture two identical cars. Because of the existence of chaos and manufacturing uncertainties it is impossibe to repeat the same car crash, even in a lab. Just like there are no two identical stock-market crashes, or two identical earthquakes or tornados, there are no two equal car crashes. The physics of crash make optimization of crashworthiness a futile exercise. You cannot optimize a car for an a-priori unknown crash scenario, just like you cannot design an optimal asset portfolio for an unknown future stock-market crash. There is no optimal building design for an unknown earthquake of which you only know that one day it will come.

Should you ever experience a crash in your own car, you can be sure that it will not be against a standard barrier and with a speed of 50 kph. And at exactly ninety degrees.