High-order Polynomial Expansions Research Articles

First-principles study of large-amplitude dynamic Jahn–Teller effects in vanadium tetrafluoride

Transition metal tetrahalides are a class of highly symmetric molecules for which very few spectroscopic data exist. Exploratory ab initio calculations of electronic potential energy functions indicate that the equilibrium molecular geometries of the vanadium, niobium, and tantalum tetrafluorides (i.e., VF4, NbF4, and TaF4) exhibit strong distortions from the tetrahedral configuration in their electronic ground state (2E) and first excited state (2T2) along the nuclear displacement coordinates of e symmetry. The distortions result from the E × e and T2 × e Jahn-Teller (JT) effects, respectively. In addition, there are weaker distortions in the 2T2 state along the coordinates of t2 symmetry due to the T2 × t2 JT effect. The description of the large-amplitude dynamics induced by these JT effects requires the construction of JT Hamiltonians beyond the standard model of JT theory, which is based on Taylor expansions up to second order in normal-mode displacements. These higher-order JT Hamiltonians were constructed in this work by expansions of the electronic potentials of the title molecule in terms of symmetry invariant polynomials in symmetry-adapted nuclear displacement coordinates for the bending modes of VF4. A multi-configuration electronic structure method was employed to determine the coefficients of these high-order polynomial expansions from first principles. Using these large-amplitude Jahn-Teller Hamiltonians, the vibronic spectra of VF4 were computed. The spectra illustrate the effects of large-amplitude fluxional nonadiabatic dynamics due to exceptionally strong E × e and T2 × e JT couplings. In addition, the vibronic spectrum of the T2 × (e + t2) JT effect, including the bending mode of t2 symmetry, was computed. The spectrum displays strong inter-mode coupling effects exhibiting a vibronic structure, which is substantially different from that predicted by independent-mode approximation. These results represent the first ab initio study of dynamical Jahn-Teller effects in VF4.

NEMHEX solution of benchmark problems on thermal and fast breeder reactor cores with hexagonal lattice

This paper gives the solution of some benchmark problems for reactor core calculations in hexagonal geometry using a computer code NEMHEX (Nodal Expansion Method for HEXagonal geometry). The problems cover a wide range of thermal reactor cores including large heavy water, fast breeder and VVER reactors in both two and three dimensional. The code is based on higher order nodal expansion method (NEM) for the solution of the neutron diffusion equation in 3-D hexagonal geometry. In this method, flux in each mesh/node and energy group is expanded in terms of polynomial up to third order in X–Y plane and fourth order in axial direction (Z). Weighted residual method is used to calculate the higher order expansion coefficients of the polynomials. Both the partially integrated radial and axial leakages are approximated by the quadratic polynomial. Higher order polynomial expansion [with respect to the work by Lawrence (DIF3D nodal neutronics option for two-and three-dimensional diffusion theory calculations in hexagonal geometry, 1983)] of flux as well as leakage in each node improves the accuracy in power distribution of VVER-1000 type reactor (which is having large size lattice) in terms of maximum reduction in error from 8 to 2%. The final inhomogeneous response matrix equation involves spatial moments of nodal flux and surface averaged partial currents across the surfaces of node. These equations are solved using conventional fission source iteration. This code is used for calculation of effective multiplication factor, neutron flux and fission power distribution. The results of the analysis establish the adequacy of NEMHEX for thermal as well as fast reactor core calculations in hexagonal geometry.

Analysis of spacecraft disposal solutions from LPO to the Moon with high order polynomial expansions

This paper presents the analysis of disposal trajectories from libration point orbits to the Moon under uncertainty. The paper proposes the use of polynomial chaos expansions to quantify the uncertainty in the final conditions given an uncertainty in initial conditions and disposal manoeuvre. The paper will compare the use of polynomial chaos expansions against high order Taylor expansions computed with point-wise integration of the partials of the dynamics, the use of the covariance matrix propagated using a unscented transformation and a standard Monte Carlo simulation. It will be shown that the use of the ellipsoid of uncertainty, that corresponds to the propagation of the covariance matrix with a first order Taylor expansions, is not adequate to correctly capture the dispersion of the trajectories that can intersect the Moon. Furthermore, it will be shown that polynomial chaos expansions better represent the distribution of the final states compared to Taylor expansions of equal order and are comparable to a full scale Monte Carlo simulations but at a fraction of the computational cost.

Computation of maximal local (un)stable manifold patches by the parameterization method

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh–Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition.

Density functionals for surface science: Exchange-correlation model development with Bayesian error estimation

A methodology for semiempirical density functional optimization, using regularization and cross-validation methods from machine learning, is developed. We demonstrate that such methods enable well-behaved exchange-correlation approximations in very flexible model spaces, thus avoiding the overfitting found when standard least-squares methods are applied to high-order polynomial expansions. A general-purpose density functional for surface science and catalysis studies should accurately describe bond breaking and formation in chemistry, solid state physics, and surface chemistry, and should preferably also include van der Waals dispersion interactions. Such a functional necessarily compromises between describing fundamentally different types of interactions, making transferability of the density functional approximation a key issue. We investigate this trade-off between describing the energetics of intramolecular and intermolecular, bulk solid, and surface chemical bonding, and the developed optimization method explicitly handles making the compromise based on the directions in model space favored by different materials properties. The approach is applied to designing the Bayesian error estimation functional with van der Waals correlation (BEEF--vdW), a semilocal approximation with an additional nonlocal correlation term. Furthermore, an ensemble of functionals around BEEF--vdW comes out naturally, offering an estimate of the computational error. An extensive assessment on a range of data sets validates the applicability of BEEF--vdW to studies in chemistry and condensed matter physics. Applications of the approximation and its Bayesian ensemble error estimate to two intricate surface science problems support this.

Validation of the Rayleigh–Ritz method for the postbuckling analysis of rectangular plates with application to delamination growth

Postbuckling solutions of laminated rectangular plates are obtained by the Rayleigh–Ritz method using von Karman’s nonlinear strain displacement relations and high-order polynomial expansions of the displacements. The potential energy function and the nonlinear algebraic equations governing the undetermined coefficients are obtained by Mathematica. Reasonably accurate solutions for the membrane forces, the bending and twisting moments and the pointwise energy-release rates generally require more undeterminated coefficients. Such refined postbuckling solutions show significant non-uniformity of the in-plane forces and strains and certain boundary effects characterized by concentration of the curvatures and the bending moment.

Adaptive Meshing for Deep-drawing Process

This paper incorporates the concept of adaptive meshing for finite-element analysis of the deep-drawing process. In adaptive meshing, the mesh is automatically refined both in the areas of insufficient accuracy and sharp stress gradients. The Zienkiewicz-Zhu error estimator based upon the difference between the finite-element solution and the corresponding smoothened solution is used to judge the accuracy both at the element and the global levels. The post-processing for determining more accurate solutions is done by fitting a higher order polynomial expansion to the fmite-element solution in nodal patches. An illustrative problem is solved and the adaptive refmement at different stages of deformation is presented.