A computational scheme is developed for sampling-based evaluation of a function whose inputs are statistically variable. After a general abstract framework is developed, it is applied to initialize and evolve the size and orientation of cracks within a finite domain, such as a finite element or similar subdomain. The finite element is presumed to be too large to explicitly track each of the potentially thousands (or even millions) of individual cracks in the domain. Accordingly, a novel binning scheme is developed that maps the crack data to nodes on a reference grid in probability space. The scheme, which is clearly generalizable to applications involving arbitrary numbers of random variables, is illustrated in the scope of planar deformations of a brittle material containing straight cracks. Assuming two random variables describe each crack, the cracks are assigned uniformly random orientations and non-uniformly random sizes. Their data are mapped to a computationally tractable number of nodes on a grid laid out in the unit square of probability space so that Gauss points on the grid may be used to define an equivalent subpopulation of the cracks. This significantly reduces the computational cost of evaluating ensemble effects of large evolving populations of random variables. ‡ In biology, for example, this binning framework could be used to identify a tractably small subset of millions of children (having known initial descriptors like height, weight, geographic location, socio-economic status, etc. from hospital birth records if in developed nations); these representative (binned) children could then be tracked over their lifetimes to develop a growth model predictive of the larger 7 billion world population. As a different application, this binning scheme could decimate a gigapixel image down to megapixels or smaller. § For an integral over a cell of length L cell , the conventional scaled Gauss weight is k g D Lcell Lgauss K g , where K g is the g th standard reference Gauss weight and L gauss is the length of the standard Gauss reference domain (usually L gauss D 2 corresponding to a Gauss domain from -1 to 1).
INPUTSnumber of Gauss points per cell = 2 point locations, r p D ¹0:11; 0:22; 0:37; 0:56; 0:92º zoom exponent = 1 (no zooming and hence equal-sized grid cells) node locations, r i D ¹0; 0:5; 1º OUTPUTS (and intermediate results to help with debugging) Node volumes, V i D ¹0:25; 0:5; 0:25º Number of points per node, N i D ¹1:6; 2:44; 0:96º Node weights, w i D ¹0:32; 0:488; 0:192º Gauss-point locations, r g D ¹0:105662; 0:394338; 0:605662; 0:894338º Standard Gauss weights on range OE 1; 1 = OEK 1 ; K 2 = ¹1; 1º Scaled standard Gauss weights: k g D len cell len gauss K g =¹0:25; 0:25; 0:25; 0:25º Shape functions at Gauss points: S i .r g /= ¹.0:788675; 0Sum these pairs: q.r g / = ¹1:21576; 1:04024; 0:932044; 0:811956º Multiply these by the k g values to obtain the W g occupancies. Alternatively, if intermediate results (like k g ) are not sought, the bin occupancies may be found by directly applying t...