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Although applications such as those described by Schweiger et al. demonstrate the power of the deterministic nite element method combined with reliability methods, they do not take advantage of the nite element method s power to model stochastic spatial variation. This requires modifying the nite element method to incorporate random elds and, in particular, the spatial correlation of the material properties. Most studies that model random elds are Monte Carlo simulations. That is, each analysis is a representation of the random eld. The simplest approach is simply to generate the material properties for each element randomly and independently. While we have seen this done in some quick studies, we do not recommend it because it ignores the correlation structure of the material properties. The deterministic approach of the previous section implicitly assumes that, within each material area, the properties are perfectly correlated, and this approach assumes they are perfectly uncorrelated. As will be pointed out below, the critical condition often lies between these extremes.

for (int i=1; i < 100; i++)

you would have:

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Based on Matheron s (1973) original insight, Mantoglu and Wilson (1982) described the turning bands method as a way to generate correlated random elds. Matheron originated the name turning bands. It is assumed that the eld is de ned by some trend and by some correlated, random, Normally distributed departure from that trend. First, the trend is removed; it can be added back in later. This leaves the problem of generating a random eld with zero mean and correlated, Normally distributed, random error. The method is illustrated in Figure 19.3 for the case of a two-dimensional eld. The line labeled x is a coordinate axis. Line i is one of a set of turning bands lines, oriented by an angle i with respect to the x axis. The parameter i describes distances along this line, and short increments along the line are . A unit vector oriented along line i is ui . A one-dimensional, correlated random process is generated along each of the turning bands lines and de ned in increments . N is a point in the two-dimensional or threedimensional region where the random eld is to be generated; xN is the vector from the origin to point N . The projection of N onto the line i is the point Ni , and the distance from the origin to Ni is the vector dot product (xN ui ). The value of the one-dimensional random process at Ni is zi ( Ni ) = zi (xN ui ). Then, the value of the correlated random eld at N is found by summing the contributions of all L lines: 1 zs (xN ) = L

5

for (int i=2; i < 200; i += 2)

which is of course functionally equivalent. This example is trivial and would do very little to deter reversers, but you could create far more complex encodings that would cause significant confusion with regards to the variable s meaning and purpose. It should be noted that this type of transformation is better applied at the binary level, because it might actually be eliminated (or somewhat modified) by a compiler during the optimization process.

zi (xN ui )

Restructuring arrays means that you modify the layout of some arrays in a way that preserves their original functionality but confuses reversers with regard to their purpose. There are many different forms to this transformation, such as merging more than one array into one large array (by either interleaving the elements from the arrays into one long array or by sequentially connecting the two arrays). It is also possible to break one array down into several smaller arrays or to change the number of dimensions in an array. These transformations are not incredibly potent, but could somewhat increase the confusion factor experienced by reversers. Keep in mind that it would usually be possible for an automated deobfuscator to reconstruct the original layout of the array.

(19.5)

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