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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 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
DOE OSTI2013

Resonant coupling applied to superconducting accelerator structures

The concept of resonant coupling and the benefits that accrue from its application is well known in the world of room temperature coupled cavity linacs. Design studies show that it can be applied successfully between sections of conventional elliptical superconducting coupled cavity accelerator structures and internally to structures with spoked cavity resonators. The coupling mechanisms can be designed without creating problems with high field regions or multipactoring. The application of resonant coupling to superconducting accelerators eliminates the need for complex cryogenic mechanical tuners and reduces the time needed to bring a superconducting accelerator into operation.

43 PARTICLE ACCELERATORS↗
DOE OSTI2025

An Adaptive Newton-Based Free-Boundary Grad–Shafranov Solver

Equilibria in magnetic confinement devices result from force balancing between the Lorentz force and the plasma pressure gradient. In an axisymmetric configuration like a tokamak, such an equilibrium is described by an elliptic equation for the poloidal magnetic flux, commonly known as the Grad–Shafranov equation. It is challenging to develop a scalable and accurate free-boundary Grad–Shafranov solver, since it is a fully nonlinear optimization problem that simultaneously solves for the magnetic field coil current outside the plasma to control the plasma shape. In this work, we develop a Newton-based free-boundary Grad–Shafranov solver using adaptive finite elements and preconditioning strategies. The free-boundary interaction leads to the evaluation of a domain-dependent nonlinear form of which its contribution to the Jacobian matrix is achieved through shape calculus. The optimization problem aims to minimize the distance between the plasma boundary and specified control points while satisfying two nontrivial constraints, which correspond to the nonlinear finite element discretization of the Grad–Shafranov equation and a constraint on the total plasma current involving a nonlocal coupling term. The linear system is solved by a block factorization, and AMG is called for subblock elliptic operators. The unique contributions of this work include the treatment of a global constraint, preconditioning strategies, nonlocal reformulation, and the implementation of adaptive finite elements. Furthermore, it is found that the resulting Newton solver is robust, successfully reducing the nonlinear residual to 1e-6 and lower in a small handful of iterations while addressing the challenging case to find a Taylor state equilibrium where conventional Picard-based solvers fail to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2015

A transmission line model for propagation in elliptical core optical fibers

The calculation of mode propagation constants of elliptical core fibers has been the purpose of extended research leading to many notable methods, with the classic step index solution based on Mathieu functions. This paper seeks to derive a new innovative method for the determination of mode propagation constants in single mode fibers with elliptic core by modeling the elliptical fiber as a series of connected coupled transmission line elements. We develop a matrix formulation of the transmission line and the resonance of the circuits is used to calculate the mode propagation constants. The technique, used with success in the case of cylindrical fibers, is now being extended for the case of fibers with elliptical cross section. The advantage of this approach is that it is very well suited to be able to calculate the mode dispersion of arbitrary refractive index profile elliptical waveguides. The analysis begins with the deployment Maxwell’s equations adjusted for elliptical coordinates. Further algebraic analysis leads to a set of equations where we are faced with the appearance of harmonics. Taking into consideration predefined fixed number of harmonics simplifies the problem and enables the use of the resonant circuits approach. According to each case, programs have been created in Matlab, providing with a series of results (mode propagation constants) that are further compared with corresponding results from the ready known Mathieu functions method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2013

Reweighted ℓ{sub 1} minimization method for stochastic elliptic differential equations

We consider elliptic stochastic partial differential equations (SPDEs) with random coefficients and solve them by expanding the solution using generalized polynomial chaos (gPC). Under some mild conditions on the coefficients, the solution is “sparse” in the random space, i.e., only a small number of gPC basis makes considerable contribution to the solution. To exploit this sparsity, we employ reweighted l{sub 1} minimization to recover the coefficients of the gPC expansion. We also combine this method with random sampling points based on the Chebyshev probability measure to further increase the accuracy of the recovery of the gPC coefficients. We first present a one-dimensional test to demonstrate the main idea, and then we consider 14 and 40 dimensional elliptic SPDEs to demonstrate the significant improvement of this method over the standard l{sub 1} minimization method. For moderately high dimensional (∼10) problems, the combination of Chebyshev measure with reweighted l{sub 1} minimization performs well while for higher dimensional problems, reweighted l{sub 1} only is sufficient. The proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Acoustic scattering by elastic cylinders of elliptical cross-section and splitting up of resonances

The scattering of a plane acoustic wave by an infinite elastic cylinder of elliptical cross section is studied from a modal formalism by emphasizing the role of the symmetries. More precisely, as the symmetry is broken in the transition from the infinite circular cylinder to the elliptical one, the splitting up of resonances is observed both theoretically and experimentally. This phenomenon can be interpreted using group theory. The main difficulty stands in the application of this theory within the framework of the vectorial formalism in elastodynamics. This method significantly simplifies the numerical treatment of the problem, provides a full classification of the resonances, and gives a physical interpretation of the splitting up in terms of symmetry breaking. An experimental part based on ultrasonic spectroscopy complements the theoretical study. A series of tank experiments is carried out in the case of aluminium elliptical cylinders immersed in water, in the frequency range 0 ≤ kr ≤ 50, where kr is the reduced wave number in the fluid. The symmetry is broken by selecting various cylinders of increasing eccentricity. More precisely, the greater the eccentricity, the higher the splitting up of resonances is accentuated. The experimental results provide a very good agreement with the theoretical ones, the splitting up is observed on experimental form functions, and the split resonant modes are identified on angular diagrams.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2013

Plasmoid solutions of the Hahm–Kulsrud–Taylor equilibrium model

The Hahm–Kulsrud (HK) [T. S. Hahm and R. M. Kulsrud, Phys. Fluids 28, 2412 (1985)] solutions for a magnetically sheared plasma slab driven by a resonant periodic boundary perturbation illustrate fully shielded (current sheet) and fully reconnected (magnetic island) responses. On the global scale, reconnection involves solving a magnetohydrodynamic (MHD) equilibrium problem. In systems with a continuous symmetry, such MHD equilibria are typically found by solving the Grad–Shafranov equation, and in slab geometry the elliptic operator in this equation is the 2-D Laplacian. Thus, assuming appropriate pressure and poloidal current profiles, a conformal mapping method can be used to transform one solution into another with different boundary conditions, giving a continuous sequence of solutions in the form of partially reconnected magnetic islands (plasmoids) separated by Syrovatsky current sheets. The two HK solutions appear as special cases.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Measurement Uncertainty of Highly Asymmetrically Curved Elliptical Mirrors Using Multi-Pitch Slope Stitching Technique

Soft X-ray off-axis elliptical mirrors bring new challenges for X-ray mirror metrology. These highly asymmetrically curved elliptical cylindrical mirrors with a total slope range >10 mrad are extremely challenging to measure. Their total slope range exceeds the measuring range of most angular sensors used for X-ray mirror inspection. To overcome this problem, it is possible to stitch partial slope data by measuring the mirror at different pitch angles (multi-pitch angles). By revisiting the theory of the multi-pitch Nano-accuracy Surface Profiler (NSP), we derive the sampling position error on the mirror surface as a function of the mirror height profile and the measurement error of the pitch rotation center. When measuring “extreme”, highly asymmetrically curved, elliptical mirrors, the calculation of the mirror height profile with iterative reconstruction outperforms the classical “flat assumption” (i.e., assuming that the mirror sag is negligible). As demonstrated by our simulations, a proper tolerance evaluation on the measurement of pitch rotation center is needed to assess the measurement accuracy (systematic error) for these strongly aspherical mirrors using the multi-pitch NSP technique. Taking a real design of an “extreme” elliptical mirror as a case study, we conduct a Monte Carlo simulation to mimic the measurement and characterization process to analyze the impact of several error sources. With the measurement uncertainty of the pitch rotation center, the multi-pitch NSP measurement can estimate the grazing angle θ and the chief ray location x o with their uncertainties, as well as the slope residuals.

36 MATERIALS SCIENCE↗
DOE OSTI2013

Lower bounds for sums of eigenvalues of elliptic operators and systems

Two-term lower bounds of Berzin-Li-Yau type are obtained for the sums of eigenvalues of elliptic operators and systems with constant coefficients and Dirichlet boundary conditions. The polyharmonic operator, the Stokes system and its generalizations, the two-dimensional buckling problem, and also the Klein-Gordon operator are considered. Bibliography: 32 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Model's sparse representation based on reduced mixed GMsFE basis methods

In this paper, we propose a model's sparse representation based on reduced mixed generalized multiscale finite element (GMsFE) basis methods for elliptic PDEs with random inputs. A typical application for the elliptic PDEs is the flow in heterogeneous random porous media. Mixed generalized multiscale finite element method (GMsFEM) is one of the accurate and efficient approaches to solve the flow problem in a coarse grid and obtain the velocity with local mass conservation. When the inputs of the PDEs are parameterized by the random variables, the GMsFE basis functions usually depend on the random parameters. This leads to a large number degree of freedoms for the mixed GMsFEM and substantially impacts on the computation efficiency. In order to overcome the difficulty, we develop reduced mixed GMsFE basis methods such that the multiscale basis functions are independent of the random parameters and span a low-dimensional space. To this end, a greedy algorithm is used to find a set of optimal samples from a training set scattered in the parameter space. Reduced mixed GMsFE basis functions are constructed based on the optimal samples using two optimal sampling strategies: basis-oriented cross-validation and proper orthogonal decomposition. Although the dimension of the space spanned by the reduced mixed GMsFE basis functions is much smaller than the dimension of the original full order model, the online computation still depends on the number of coarse degree of freedoms. To significantly improve the online computation, we integrate the reduced mixed GMsFE basis methods with sparse tensor approximation and obtain a sparse representation for the model's outputs. The sparse representation is very efficient for evaluating the model's outputs for many instances of parameters. To illustrate the efficacy of the proposed methods, we present a few numerical examples for elliptic PDEs with multiscale and random inputs. In particular, a two-phase flow model in random porous media is simulated by the proposed sparse representation method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

Divergence preserving reconstruction of the nodal components of a vector field from its normal components to edges

We have developed a new divergence preserving method for the reconstruction of the Cartesian components of a vector field from the orthogonal projection of a vector field to the normals to edges in two dimensional. In this method, discrete divergences computed from the nodal components and from the normal ones are exactly the same. Our new method consists of two stages. At the first stage, we use an extended version of the local procedure described in [J. Comput. Phys., 139:406–409, 1998] to obtain a ‘reference’ nodal vector. This local procedure is exact for linear vector fields; however, the discrete divergence is not preserved. Then, we formulate a constrained optimization problem, in which this reference vector plays the role of a target, and the divergence constraints are enforced by using Lagrange multipliers. It leads to the solution of ‘elliptic’ like discrete equations for the cell-centered Lagrange multipliers. The new global divergence preserving method is exact for linear vector fields. Here in this paper, we describe all details of our new method and present numerical results, which confirm our theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2019

A mixed $\ell_1$ regularization approach for sparse simultaneous approximation of parameterized PDEs

We introduce and assess a novel sparse polynomial technique for the simultaneous approximation of parameterized partial differential equations (PDEs) with deterministic and stochastic inputs. Our approach treats the numerical solution as a jointly sparse reconstruction problem through the reformulation of the standard basis pursuit denoising, where the set of jointly sparse vectors is infinite. To achieve global reconstruction of sparse solutions to parameterized elliptic PDEs over both physical and parametric domains, we combine the standard measurement scheme developed for compressed sensing in the context of bounded orthonormal systems with a novel mixed-norm based $\ell_1$ regularization method that exploits both energy and sparsity. Moreover, we are able to prove that, with minimal sample complexity, error estimates comparable to the best $s$-term and quasi-optimal approximations are achievable, while requiring only {\em a priori} bounds on polynomial truncation error with respect to the energy norm.Finally, we perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs

Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encountered in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in C 0 . Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes H 1 . Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Multi-stage preconditioners for thermal–compositional–reactive flow in porous media

We present a family of multi-stage preconditioners for coupled thermal-compositional-reactive reservoir simulation problems. The most common preconditioner used in industrial practice, the Constrained Pressure Residual (CPR) method, was designed for isothermal models and does not offer a specific strategy for the energy equation. For thermal simulations, inadequate treatment of the temperature unknown can cause severe conver-gence degradation. When strong thermal diffusion is present, the energy equation exhibits significant elliptic behavior that cannot be accurately corrected by CPR's second stage. In this work, we use Schur-complement decompositions to extract a temperature subsystem and apply an Algebraic MultiGrid (AMG) approximation as an additional preconditioning stage to improve the treatment of the energy equation. We present results for several two-dimensional hot air injection problems using an extra heavy oil, including challenging reactive In-Situ Combustion (ISC) cases. We show improved performance and robustness across different thermal regimes, from advection dominated (high Péclet number) to diffusion dominated (low Péclet number). The number of linear iterations is reduced by 40–85% compared to standard CPR for both homogeneous and heterogeneous media, and the new methods exhibit almost no sensitivity to the thermal regime.

42 ENGINEERING↗
DOE OSTI2012

Subsolutions That Are Close in the Uniform Norm Are Close in the Sobolev Norm as Well

A new type weighted reverse Poincare inequality is established for a difference of two continuous weak subsolutions of a linear second order uniformly elliptic partial differential equation in the ball.This result is the key to deriving the error estimate for the gradient of the analytically unknown value function of the optimal stochastic control problem from the uniform error of the value function itself in the related numerical approximation problems.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2025

Hardware acceleration for HPS algorithms in two and three dimensions

We provide a flexible, open-source framework for hardware acceleration, namely massively-parallel execution on general-purpose graphics processing units (GPUs), applied to the hierarchical Poincaré–Steklov (HPS) family of algorithms for building fast direct solvers for linear elliptic partial differential equations. To take full advantage of the power of hardware acceleration, we propose two variants of HPS algorithms to improve performance on two- and three-dimensional problems. In the two-dimensional setting, we introduce a novel recomputation strategy that minimizes costly data transfers to and from the GPU; in three dimensions, we modify and extend the adaptive discretization technique of Geldermans and Gillman [1] to greatly reduce peak memory usage. We provide an open-source implementation of these methods written in JAX, a high-level accelerated linear algebra package, which allows for the first integration of a high-order fast direct solver with automatic differentiation tools. We conclude with extensive numerical examples showing our methods are fast and accurate on two- and three-dimensional problems.

Fast direct solvers↗
DOE OSTI2015

On the statistical and transport properties of a non-dissipative Fermi-Ulam model

The transport and diffusion properties for the velocity of a Fermi-Ulam model were characterized using the decay rate of the survival probability. The system consists of an ensemble of non-interacting particles confined to move along and experience elastic collisions with two infinitely heavy walls. One is fixed, working as a returning mechanism of the colliding particles, while the other one moves periodically in time. The diffusion equation is solved, and the diffusion coefficient is numerically estimated by means of the averaged square velocity. Our results show remarkably good agreement of the theory and simulation for the chaotic sea below the first elliptic island in the phase space. From the decay rates of the survival probability, we obtained transport properties that can be extended to other nonlinear mappings, as well to billiard problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Final Report of the Project "From the finite element method to the virtual element method"

The Finite Element Method (FEM) is a powerful numerical tool that is being used in a large number of engineering applications. The FEM is constructed on triangular/tetrahedral and quadrilateral/hexahedral meshes. Extending the FEM to general polygonal/polyhedral meshes in straightforward way turns out to be extremely difficult and leads to very complex and computationally expensive schemes. The reason for this failure is that the construction of the basis functions on elements with a very general shape is a non-trivial and complex task. In this project we developed a new family of numerical methods, dubbed the Virtual Element Method (VEM) for the numerical approximation of partial differential equations (PDE) of elliptic type suitable to polygonal and polyhedral unstructured meshes. We successfully formulated, implemented and tested these methods and studied both theoretically and numerically their stability, robustness and accuracy for diffusion problems, convection-reaction-diffusion problems, the Stokes equations and the biharmonic equations.

97 MATHEMATICS AND COMPUTING↗