While a wealth of experience in the development of uncertainty quantification methods and software tools exists at present, a cohesive software package utilizing massively parallel computing resources does not. The thrust of the work to be discussed herein is the development of such a toolkit, which has leveraged existing software frameworks (e.g., DAKOTA (Design Analysis Kit for OpTimizAtion)) where possible, and has undertaken additional development efforts when necessary. The contributions of this paper are two-fold. One, the design and structure of the toolkit from a software perspective will be discussed, detailing some of its distinguishing features. Second, the toolkit's capabilities will be demonstrated by applying a subset of its available uncertainty quantification techniques to an example problem involving multiple engineering disciplines, nonlinear solid mechanics and soil mechanics. This example problem will demonstrate the toolkit's suitability in quantifying uncertainty in engineering applications of interest modeled using very large computational system models.

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By using zeros of elliptic integrals we establish an upper bound for the number of limit cycles that emerge from the period annulus of the Hamiltonian XH in the system X, = XH + e{P,Q), where H -y 7 + x* and P, Q are polynomials in x, y, ( N . \ as a function of the degrees of P and Q. In particular, if (P,Q) = I ^a.ix ',0 I \i = 2 ) with N = 2k + 1 or 2* + 2, this upper bound is Jf c -1.