Computational Physics

Conventional Monte Carlo simulations are stochastic in the sense that the acceptance of a trial move is decided by comparing a computed acceptance probability with a random number, uniformly distributed between 0 and 1. Here we consider the case that the weight determining the acceptance probability itself is fluctuating. This situation is common in many numerical studies. We show that it is possible to construct a rigorous Monte Carlo algorithm that visits points in state space with a probability proportional to their average weight. The same approach has the potential to transform the methodology of a certain class of high-throughput experiments or the analysis of noisy datasets.

The straightforward automatic-differentiation and the hand-differentiated incremental iterative methods are interwoven to produce a hybrid scheme that captures some of the strengths of each strategy. With this compromise, discrete aerodynamic sensitivity derivatives are calculated with the efficient incremental iterative solution algorithm of the original flow code. Moreover, the principal advantage of automatic differentiation is retained (i.e., all complicated source code for the derivative calculations is constructed quickly with accuracy). The basic equations for second-order sensitivity derivatives are presented, which results in a comparison of four different methods. Each of these four schemes for second-order derivatives requires that large systems are solved first for the first-order derivatives and, in all but one method, for the first-order adjoint variables. Of these latter three schemes, two require no solutions of large systems thereafter. For the other two for which additional systems are solved, the equations and solution procedures are analogous to those for the first-order derivatives. From a practical viewpoint, implementation of the second-order methods is feasible only with software tools such as automatic differentiation, because of the extreme complexity and large number of terms. First- and second-order sensitivities are calculated accurately for two airfoil problems, including a turbulent-flow example. In each of these two sample problems, three dependent variables (coefficients of lift, drag, and pitching-moment) and six independent variables (three geometric-shape and three flow-condition design variables) are considered. Several different procedures are tested, and results are compared on the basis of accuracy, computational time, and computer memory. For first-order derivatives, the hybrid incremental iterative scheme obtained with automatic differentiation is competitive with the best hand-differentiated method. Furthermore, it is at least two to four times faster than central finite differences, without an overwhelming penalty in computer memory.

In particle transport applications there are numerous physical constructs in which heterogeneities are randomly distributed. The quantity of interest in these problems is the ensemble average of the flux, or the average of the flux over all possible material ‘realizations.’ The Levermore-Pomraning closure assumes Markovian mixing statistics and allows a closed, coupled system of equations to be written for the ensemble averages of the flux in each material. Generally, binary statistical mixtures are considered in which there are two (homogeneous) materials and corresponding coupled equations. The solution process is iterative, but convergence may be slow as either or both materials approach the diffusion and/or atomic mix limits. A three-part acceleration scheme is devised to expedite convergence, particularly in the atomic mix-diffusion limit where computation is extremely slow. The iteration is first divided into a series of ‘inner’ material and source iterations to attenuate the diffusion and atomic mix error modes separately. Secondly, atomic mix synthetic acceleration is applied to the inner material iteration and S2 synthetic acceleration to the inner source iterations to offset the cost of doing several inner iterations per outer iteration. Finally, a Krylov iterative solver is wrapped around each iteration, inner and outer, to further expedite convergence. A spectral analysis is conducted and iteration counts and computing cost for the new two-step scheme are compared against those for a simple one-step iteration, to which a Krylov iterative method can also be applied.

Using a mathematical framework originally developed for the development of PML schemes in computational electromagnetics, we develop a set of strongly well-posed PML equations for the absorption of acoustic and vorticity waves in two-dimensional convective acoustics under the assumption of a spatially constant mean flow. A central piece in this development is the development of a variable transformation that conserves the dispersion relation of the physical space equations. The PML equations are given for layers being perpendicular to the direction of the mean flow as well as for layers being parallel to the mean flow. The efficacy of the PML scheme is illustrated by solving the equations of acoustics using a 4th order scheme, confirming the accuracy as well as stability of the proposed scheme.

The effect of local preconditioning on boundary conditions is analyzed for the subsonic, one-dimensional Euler equations. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions and different preconditioners whose intent is to accelerate low Mach number computations. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective when used with preconditioning, and asymptotically, at low Mach numbers, initial disturbances do not decay. Other boundary conditions are shown to be perfectly nonreflective in conjunction with preconditioning. Two-dimensional numerical results confirm the trends predicted by the one-dimensional analysis.

The equation for electrostatic potential which arises from the implicit moment method in plasma simulation is a nonsymmetric elliptic equation. We present results using a simple multigrid method as a preconditioner to General Minimum RESidual (GMRES) to iteratively solve this nonsymmetric elliptic equation in two dimensions. It is demonstrated that a simple multigrid method produces an effective preconditioner. It is also demonstrated that under some conditions the required number of linear iterations is independent of grid dimension. Results are presented for both uniform and nonuniform grid problems.

Keywords: 07.05.Tp: computer modeling and simulations 62.20.M: structural failure of materials 83.10.Mj: molecular dynamics a b s t r a c t Using large scale molecular dynamics simulation, we investigate the deleterious effect of hydrogen in Zr. We consider both dilute and concentrated limit of H. In the dilute and concentrated H limits, we study the effect of 1e5 atomic percentage of hydrogen, and that of ε-ZrH 2 precipitate having 5e10 nm diameters , respectively. From the stressestrain curves and micro-structure analysis at different strain values, we characterize the deformation behavior and correlate our result with previously reported mechanisms. We show hydrogen atoms in dilute limit help in dislocation multiplication, following the hydrogen-enhanced localized plasticity mechanism. In the concentrated limit, on the other hand, dis-locations and cracks nucleate from precipitateematrix interface, indicating the decohesion mechanism as primary method for Zr degradation. These findings are corroborated with a nucleation and growth model as expressed in KolmogoroveJohnsoneMehleAvrami equation.