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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

An eigenvalue analysis of finite-difference approximations for hyperbolic IBVPs

The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L (sub 2) stability on a finite domain.

Warming, Robert F.↗
NASA NTRS1996

First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H(exp 1) product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity with estimates that are uniform in the Lame constants.

Cai, Z.↗
DOE OSTI2014

Solutions to higher-order anisotropic parabolic equations in unbounded domains

The paper is devoted to a certain class of doubly nonlinear higher-order anisotropic parabolic equations. Using Galerkin approximations it is proved that the first mixed problem with homogeneous Dirichlet boundary condition has a strong solution in the cylinder D=(0,∞)×Ω, where Ω⊂R{sup n}, n≥3, is an unbounded domain. When the initial function has compact support the highest possible rate of decay of this solution as t→∞ is found. An upper estimate characterizing the decay of the solution is established, which is close to the lower estimate if the domain is sufficiently 'narrow'. The same authors have previously obtained results of this type for second order anisotropic parabolic equations. Bibliography: 29 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Asymptotics for inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation

We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Quantum systems with position-dependent mass and spin-orbit interaction via Rashba and Dresselhaus terms

We consider a particle with spin 1/2 with position-dependent mass moving in a plane. Considering separately Rashba and Dresselhaus spin-orbit interactions, we write down the Hamiltonian for this problem and solve it for Dirichlet boundary conditions. Our radial wavefunctions have two contributions: homogeneous ones which are written as Bessel functions of non-integer orders—that depend on angular momentum m—and particular solutions which are obtained after decoupling the non-homogeneous system. In this process, we find non-homogeneous Bessel equation, Laguerre, as well as biconfluent Heun equation. We also present the probability densities for m = 0, 1, 2 in an annular quantum well. Our results indicate that the background as well as the spin-orbit interaction naturally splits the spinor components.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1974

Analysis of twisting stiffness for a multifiber composite layer

The twisting stiffness of a rectangular cross section consisting of a single row of solid circular cross-section fibers embedded in a matrix is analyzed. The problem is formulated as a Dirichlet torsion problem of a multielement region and solved by the boundary-point least-squares method. Numerical results for a single-fiber square cross section compare favorably with previous relaxation-method results. New numerical results for three and five-fiber composites suggest that the torsional rigidity of a multifiber composite can be approximated from the torsional rigidities of single and three-fiber models.

Bert, C. W.↗
DOE OSTI2026

Quantum properties of non-Dirichlet boundary conditions in gravity

The Euclidean path integral for gravity is enriched by the addition of boundaries, which provide useful probes of thermodynamic properties. Common boundary conditions include Dirichlet conditions on the boundary induced metric; microcanonical conditions, which refers to fixing some components of the Brown-York boundary stress tensor; and conformal conditions, in which the conformal structure of the induced metric and the trace of the extrinsic curvature are fixed. Boundaries also present interesting problems of consistency. The Dirichlet problem is known, under various (and generally different) conditions, to be inconsistent with perturbative quantization of graviton fluctuations, to exhibit thermodynamic instability, or to require infinite fine-tuning in the presence of matter fluctuations. We extend some of these results to other boundary conditions. We find that similarly to the Dirichlet problem, the graviton fluctuation operator is not elliptic with microcanonical boundaries, and the nonelliptic modes correspond to “boundary-moving diffeomorphisms.” However, we argue that microcanonical factorization of path integrals—essentially, the insertion of microcanonical constraints on two-sided surfaces in the bulk—is not affected by the same issues of ellipticity. We also show that for a variety of matter field boundary conditions, matter fluctuations renormalize the gravitational bulk and boundary terms differently, so that the classical microcanonical or conformal variational problems are not preserved unless an infinite fine-tuning is performed.

Draper, Patrick [Univ. of Illinois at Urbana-Champ↗
DOE OSTI2015

Scattering anomalies in a resonator above the thresholds of the continuous spectrum

We consider the Dirichlet spectral problem for the Laplace operator in a multi-dimensional domain with a cylindrical outlet to infinity, a Helmholtz resonator. Using asymptotic analysis of the scattering matrix we demonstrate different types of reflection of high-amplitude near-threshold waves. One scattering type or another, unstable or stable with respect to variations of the resonator shapes, is determined by the presence or absence of stabilizing solutions at the threshold frequency, respectively. In a waveguide with two cylindrical outlets to infinity, we discover the effect of almost complete passage of the wave under 'fine tuning' of the resonator. Bibliography: 26 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems

In this paper, we propose nonoverlapping localized exponential time differencing (ETD) methods for diffusion problems. The model time-dependent diffusion equation is first reformulated on subdomains based on the nonoverlapping domain decomposition, in which Neumann boundary conditions are imposed on the interfaces for the subdomain problems and Dirichlet type conditions are enforced to form a space-time interface problem. After spatial discretization by standard central finite differences and temporal integration with the first or second order ETD methods, the fully discrete interface problem is obtained. Such an interface problem is then solved iteratively either at each time step or over the whole time interval: the former involves the solution of stationary problems in each subdomain at each iteration while the latter involves the solution of time-dependent subdomain problems at each iteration. For both approaches, we prove that localized ETD solutions conserve mass exactly and converge in time to the exact space semidiscrete solution. Numerical experiments in two dimensions are also presented to illustrate the performance of the proposed methods.

Mass conservation↗
DOE OSTI2018

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

In axisymmetric fusion reactors, the equilibrium magnetic configuration can be expressed in terms of the solution to a semi-linear elliptic equation known as the Grad–Shafranov equation, the solution of which determines the poloidal component of the magnetic field. When the geometry of the confinement region is known, the problem becomes an interior Dirichlet boundary value problem. We propose a high order solver based on the Hybridizable Discontinuous Galerkin method. The resulting algorithm (1) provides high order of convergence for the flux function and its gradient, (2) incorporates a novel method for handling piecewise smooth geometries by extension from polygonal meshes, (3) can handle geometries with non-smooth boundaries and x-points, and (4) deals with the semi-linearity through an accelerated two-grid fixed-point iteration. The effectiveness of the algorithm is verified with computations for cases where analytic solutions are known on configurations similar to those of actual devices (ITER with single null and double null divertor, NSTX, ASDEX upgrade, and Field Reversed Configurations).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

A cut-cell finite volume – finite element coupling approach for fluid–structure interaction in compressible flow

We present a loosely coupled approach for the solution of fluid–structure interaction problems between a compressible flow and a deformable structure. The method is based on staggered Dirichlet–Neumann partitioning. The interface motion in the Eulerian frame is accounted for by a conservative cut-cell Immersed Boundary method. The present approach enables sub-cell resolution by considering individual cut-elements within a single fluid cell, which guarantees an accurate representation of the time-varying solid interface. The cut-cell procedure inevitably leads to non-matching interfaces, demanding for a special treatment. A Mortar method is chosen in order to obtain a conservative and consistent load transfer. We validate our method by investigating two-dimensional test cases comprising a shock-loaded rigid cylinder and a deformable panel. Moreover, the aeroelastic instability of a thin plate structure is studied with a focus on the prediction of flutter onset. Finally, we propose a three-dimensional fluid–structure interaction test case of a flexible inflated thin shell interacting with a shock wave involving large and complex structural deformations.

42 ENGINEERING↗
DOE OSTI2017

Functional level-set derivative for a polymer self consistent field theory Hamiltonian

We derive functional level-set derivatives for the Hamiltonian arising in self-consistent field theory, which are required to solve free boundary problems in the self-assembly of polymeric systems such as block copolymer melts. In particular, we consider Dirichlet, Neumann and Robin boundary conditions. We provide numerical examples that illustrate how these shape derivatives can be used to find equilibrium and metastable structures of block copolymer melts with a free surface in both two and three spatial dimensions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Vlasov tokamak equilibria with sheared toroidal flow and anisotropic pressure

By choosing appropriate deformed Maxwellian ion and electron distribution functions depending on the two particle constants of motion, i.e., the energy and toroidal angular momentum, we reduce the Vlasov axisymmetric equilibrium problem for quasineutral plasmas to a transcendental Grad-Shafranov-like equation. This equation is then solved numerically under the Dirichlet boundary condition for an analytically prescribed boundary possessing a lower X-point to construct tokamak equilibria with toroidal sheared ion flow and anisotropic pressure. Depending on the deformation of the distribution functions, these steady states can have toroidal current densities either peaked on the magnetic axis or hollow. These two kinds of equilibria may be regarded as a bifurcation in connection with symmetry properties of the distribution functions on the magnetic axis.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Lifshits Tails for Randomly Twisted Quantum Waveguides

We consider the Dirichlet Laplacian H{sub γ} on a 3D twisted waveguide with random Anderson-type twisting γ. We introduce the integrated density of states N{sub γ} for the operator H{sub γ}, and investigate the Lifshits tails of N{sub γ}, i.e. the asymptotic behavior of N{sub γ}(E) as E↓infsupp dN{sub γ}. In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

Asymptotics of the Resonant Tunneling of High-Energy Electrons in Two-Dimensional Quantum Waveguides of Variable Cross-Section

A waveguide occupies a strip in ℝ{sup 2} having two identical narrows of small diameter ε. An electron wave function satisfies the Helmholtz equation with the homogeneous Dirichlet boundary condition. The energy of electrons may be rather high, i.e., any (fixed) number of waves can propagate in the strip far from the narrows. As ε → 0, a neighborhood of a narrow is assumed to transform into a neighborhood of the common vertex of two vertical angles. The part of the waveguide between the narrows as ε = 0 is called the resonator. An asymptotics of the transmission coefficient is obtained in the waveguide as ε → 0. Near a degenerate eigenvalue of the resonator, the leading term of the asymptotics has two sharp peaks. Positions and shapes of the resonant peaks are described.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied — in particular, the number of nodal domains, ν of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrödinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, ν satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of mmodkn, given a particular k, for a set of quantum numbers, m,n. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations. - Highlights: • We find that the number of nodal domains of eigenfunctions of integrable, planar billiards satisfy a class of difference equations. • The eigenfunctions labelled by quantum numbers (m,n) can be classified in terms of mmodkn. • A theorem is presented, realising algebraic representations of geometrical patterns exhibited by the domains. • This work presents a connection between integrable systems and difference equations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗