Final report: Stochastic partial differential equations applied to the predictability of complex multiscale phenomena

The objectives of this research remain as stated in our proposal of November 1997. We report on progress in the quantification of uncertainty and prediction, with applications to flow in porous media and to shock wave physics. The main strength of this work is an innovative theory for the quantification of uncertainty based on models for solution errors in numerical simulations. We also emphasize a deep connection to application communities, including those in DOE Laboratories

Selfsimilarity and growth in Birkhoff sums for the golden rotation

We study Birkhoff sums S(k,a) = g(a)+g(2a)+...+g(ka) at the golden mean rotation number a with periodic continued fraction approximations p(n)/q(n), where g(x) = log(2-2 cos(2 pi x). The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena. We relate the boundedness of log- averaged Birkhoff sums S(k,a)/log(k) and the convergence of S(q(n),a) with the existence of an experimentally established limit function f(x) = lim S([x q(n)])(p(n+1)/q(n+1))-S([x q(n)])(p(n)/q(n)) for n to infinity on the interval [0,1]. The function f satisfies a functional equation f(ax) + (1-a) f(x)= b(x) with a monotone function b. The limit lim S(q(n),a) for n going to infinity can be expressed in terms of the function f.Comment: 14 pages, 8 figure

Applications Of Front Tracking To The Simulation Of Resin Transfer Molding

Resin Transfer Molding, as a method for the manufacture of advanced fiber reinforced composite materials, is attractive because it offers the possibility of lower manufacturing costs and more complex shapes than the traditional manufacturing methods. A major issue in this new manufacturing process is the elimination of void spaces in the resin fill operation, so that products with high quality are manufactured. In this paper we present a two phase, two component air solubility model to study the formation and migration of the macro and micro voids. The numerical solutions of the model are obtained through a front tracking code. The front tracking method has the distinguishing feature of preserving sharp interfaces throughout the simulation. The results demonstrate that the model proposed here has desirable qualitative agreement with experimental results. Based on these results, we make numerical predictions to show how modeling could improve the manufacturing process and hence enhance ..