SEARCH · Search NASA

Results for “DIRICHLET PROBLEM”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 73 records · Page 4

All sources NASA NTRS DOE OSTI

NASA NTRS1980

A general panel method for the analysis and design of arbitrary configurations in incompressible flows

A method for solving the linear integral equations of incompressible potential flow in three dimensions is presented. Both analysis (Neumann) and design (Dirichlet) boundary conditions are treated in a unified approach to the general flow problem. The method is an influence coefficient scheme which employs source and doublet panels as boundary surfaces. Curved panels possessing singularity strengths, which vary as polynomials are used, and all influence coefficients are derived in closed form. These and other features combine to produce an efficient scheme which is not only versatile but eminently suited to the practical realities of a user-oriented environment. A wide variety of numerical results demonstrating the method is presented.

Johnson, F. T.↗

DOE OSTI2023

Dirichlet-type absorbing boundary conditions for peridynamic scalar waves in two-dimensional viscous media

Construction of absorbing boundary conditions (ABCs) for nonlocal models is generally challenging, primarily due to the fact that nonlocal operators are commonly associated with volume constrained boundary conditions. Moreover, application of Fourier and Laplace transforms, which are essential for the majority of available methods for ABCs, to nonlocal models is complicated. In this paper, we propose a simple method to construct accurate ABCs for peridynamic scalar wave-type problems in viscous media. The proposed ABCs are constructed in the time and space domains and are of Dirichlet type. Consequently, their implementation is relatively simple, since no derivatives of the wave field are required. The proposed ABCs are derived at the continuum level, from a semi-analytical solution of the exterior domain using harmonic exponential basis functions in space and time (plane-wave modes). The numerical implementation is done using a meshfree collocation approach employed within a boundary layer adjacent to the interior domain boundary. The modes satisfy the peridynamic numerical dispersion relation, resulting in a compatible solution of the interior region (near-field) with that of the exterior region (far-field). The accuracy and stability of the proposed ABCs are demonstrated with several numerical examples in two-dimensional unbounded domains.

42 ENGINEERING↗

DOE OSTI2025

A Domain-Decomposed A-ϕ Formulation Based on Lagrange Multipliers for Low-Frequency Problems

A domain-decomposed A-ϕ formulation based on Lagrange multipliers is proposed to simulate low-frequency elec- tromagnetic problems. This method partitions the computational domain into smaller subdomains, allowing each subdomain to be independently formulated using Lagrange multipliers as Dirichlet boundary conditions, while ensuring continuity of the fields across the interfaces. A mixed finite element method, utilizing both vector and scalar basis functions, is employed to discretize the formulation, resulting in a global system to be solved. The proposed method is validated using TEAM Problem 7 at 50 Hz, demonstrating its effectiveness in handling complex geometries and addressing the low-frequency breakdown issues commonly encountered in traditional finite element methods.

Hossain, Amzad↗

DOE OSTI2024

Controlling radioisotope proportions when randomly sampling from Dirichlet distributions in PyRIID

As machine learning models for radioisotope quantification become more powerful, likewise the need for high-quality synthetic training data grows as well. For problem spaces that involve estimating the relative isotopic proportions of various sources in gamma spectra it is necessary to generate training data that accurately represents the variance of proportions encountered. In this report, we aim to provide guidance on how to target a desired variance of proportions which are randomly when using the PyRIID Seed Mixer, which samples from a Dirichlet distribution. We provide a method for properly parameterizing the Dirichlet distribution in order to maintain a constant variance across an arbitrary number of dimensions, where each dimension represents a distinct source template being mixed. We demonstrate that our method successfully parameterizes the Dirichlet distribution to target a specific variance of proportions, provided that several conditions are met. This allows us to follow a principled technique for controlling how random mixture proportions are generated which are then used downstream in the synthesis process to produce the final, noisy gamma spectra.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2022

A coupling approach for linear elasticity problems with spatially non-coincident discretized interfaces

Here we present a new method for coupled linear elasticity problems whose finite element discretization may lead to spatially non-coincident discretized interfaces. Our approach combines the classical Dirichlet–Neumann coupling formulation with a new set of discretized interface conditions obtained through Taylor series expansions. We show that these conditions ensure linear consistency of the coupled finite element solution. We then formulate an iterative solution method for the coupled discrete system and apply the new coupling approach to two representative settings for which we also provide several numerical illustrations. The first setting is a mesh-tying problem in which both coupled structures have the same Lamé parameters whereas the second setting is an interface problem for which the Lamé parameters in the two coupled structures are different.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Discrete Green’s functions and spectral graph theory for computationally efficient thermal modeling

Here, this work concerns solutions of the heat equation with the spectral graph method, for which the temperature is defined at discrete points in the domain and the spatial relationship among the points is described by a graph. The heat equation on the graph is solved using matrix techniques involving the eigenvectors and eigenvalues of the Laplacian matrix. The spectral graph approach precludes the computationally intensive meshing and numerous time-integration steps of the finite element method. In the present work, the spectral graph method is extended to include heat loss at the boundaries with a generalized boundary condition, and physics-based edge weights are introduced which simplify the calibration process. From this approach a discrete Green’s function is defined which allows for solutions under a variety of heating conditions including: space-varying initial conditions; time-and-space varying internal heating; and, time-and-space-varying heating at boundaries of type 1 (Dirichlet), type 2 (Neumann) and type 3 (Robin). Results are provided for benchmark heat transfer problems in one spatial dimension and in three spatial dimensions, and verification is provided by comparison with exact analytical solutions and finite difference solutions. The spectral graph method converges within 0.4% error of the analytical solution. The practical utility of the approach is demonstrated by thermal simulation of a multilayer additive manufacturing process. The spectral graph results are compared to experimentally-obtained temperature data for two metal parts, with error less than 5% of the experimental measurements, with computation time less than one minute on a desktop computer.

36 MATERIALS SCIENCE↗

NASA NTRS1984

Boundary conditions in Chebyshev and Legendre methods

Two different ways of treating non-Dirichlet boundary conditions in Chebyshev and Legendre collocation methods are discussed for second order differential problems. An error analysis is provided. The effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated.

Canuto, C.↗

NASA NTRS1986

Preconditioned conjugate residual methods for the solution of spectral equations

Conjugate residual methods for the solution of spectral equations are described. An inexact finite-difference operator is introduced as a preconditioner in the iterative procedures. Application of these techniques is limited to problems for which the symmetric part of the coefficient matrix is positive definite. Although the spectral equation is a very ill-conditioned and full matrix problem, the computational effort of the present iterative methods for solving such a system is comparable to that for the sparse matrix equations obtained from the application of either finite-difference or finite-element methods to the same problems. Numerical experiments are shown for a self-adjoint elliptic partial differential equation with Dirichlet boundary conditions, and comparison with other solution procedures for spectral equations is presented.

Wong, Y. S.↗

NASA NTRS1976

A higher order panel method for general analysis and design applications in subsonic flow

A higher-order panel method is described for numerical solution of boundary-value problems relating to steady inviscid irrotational incompressible subsonic fluid flow in a domain. Both Neumann and Dirichlet boundary conditions are treated; two types of auxiliary conditions are used to remove the degrees of freedom that arise from specifying only the derivative of the perturbation velocity potential. Four general network types and two expansions of the induced potential kernel are employed in the numerical solution. Some results are presented which illustrate the modeling options and numerical characteristics of the method.

Johnson, F. T.↗

DOE OSTI2022

Physics‐Informed Machine Learning Method for Large‐Scale Data Assimilation Problems

Abstract We develop a physics‐informed machine learning approach for large‐scale data assimilation and parameter estimation and apply it for estimating transmissivity and hydraulic head in the two‐dimensional steady‐state subsurface flow model of the Hanford Site given synthetic measurements of said variables. In our approach, we extend the physics‐informed conditional Karhunen‐Loéve expansion (PICKLE) method to modeling subsurface flow with unknown flux (Neumann) and varying head (time‐dependent Dirichlet) boundary conditions. We demonstrate that the PICKLE method is comparable in accuracy with the standard maximum a posteriori (MAP) method, but is significantly faster than MAP for large‐scale problems. Both methods use a mesh to discretize the computational domain. In MAP, the parameters and states are discretized on the mesh; therefore, the size of the MAP parameter estimation problem directly depends on the mesh size. In PICKLE, the mesh is used to evaluate the residuals of the governing equation, while the parameters and states are approximated by the truncated conditional Karhunen‐Loéve expansions with the number of parameters controlled by the smoothness of the parameter and state fields, and not by the mesh size. For a considered example, we demonstrate that the computational cost of PICKLE increases near linearly (as N 1.15 ) with the number of grid nodes N , while that of MAP increases much faster (as N 3.28 ). We also show that once trained for one set of Dirichlet boundary conditions (i.e., one river stage), the PICKLE method provides accurate estimates of the hydraulic head for any value of the Dirichlet boundary conditions (i.e., for any river stage).

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1994

LES, DNS and RANS for the analysis of high-speed turbulent reacting flows

The purpose of this research is to continue our efforts in advancing the state of knowledge in large eddy simulation (LES), direct numerical simulation (DNS), and Reynolds averaged Navier Stokes (RANS) methods for the computational analysis of high-speed reacting turbulent flows. In the second phase of this work, covering the period 1 Sep. 1993 - 1 Sep. 1994, we have focused our efforts on two research problems: (1) developments of 'algebraic' moment closures for statistical descriptions of nonpremixed reacting systems, and (2) assessments of the Dirichlet frequency in presumed scalar probability density function (PDF) methods in stochastic description of turbulent reacting flows. This report provides a complete description of our efforts during this past year as supported by the NASA Langley Research Center under Grant NAG1-1122.

Givi, Peyman↗

NASA NTRS2000

High-order Two-way Artificial Boundary Conditions for Nonlinear Wave Propagation with Backscattering

When solving linear scattering problems, one typically first solves for the impinging wave in the absence of obstacles. Then, by linear superposition, the original problem is reduced to one that involves only the scattered waves driven by the values of the impinging field at the surface of the obstacles. In addition, when the original domain is unbounded, special artificial boundary conditions (ABCs) that would guarantee the reflectionless propagation of waves have to be set at the outer boundary of the finite computational domain. The situation becomes conceptually different when the propagation equation is nonlinear. In this case the impinging and scattered waves can no longer be separated, and the problem has to be solved in its entirety. In particular, the boundary on which the incoming field values are prescribed, should transmit the given incoming waves in one direction and simultaneously be transparent to all the outgoing waves that travel in the opposite direction. We call this type of boundary conditions two-way ABCs. In the paper, we construct the two-way ABCs for the nonlinear Helmholtz equation that models the laser beam propagation in a medium with nonlinear index of refraction. In this case, the forward propagation is accompanied by backscattering, i.e., generation of waves in the direction opposite to that of the incoming signal. Our two-way ABCs generate no reflection of the backscattered waves and at the same time impose the correct values of the incoming wave. The ABCs are obtained for a fourth-order accurate discretization to the Helmholtz operator; the fourth-order grid convergence is corroborated experimentally by solving linear model problems. We also present solutions in the nonlinear case using the two-way ABC which, unlike the traditional Dirichlet boundary condition, allows for direct calculation of the magnitude of backscattering.

Fibich, Gadi↗

NASA NTRS2002

Discontinuous Galerkin Methods for Turbulence Simulation

A discontinuous Galerkin (DG) method is formulated, implemented, and tested for simulation of compressible turbulent flows. The method is applied to turbulent channel flow at low Reynolds number, where it is found to successfully predict low-order statistics with fewer degrees of freedom than traditional numerical methods. This reduction is achieved by utilizing local hp-refinement such that the computational grid is refined simultaneously in all three spatial coordinates with decreasing distance from the wall. Another advantage of DG is that Dirichlet boundary conditions can be enforced weakly through integrals of the numerical fluxes. Both for a model advection-diffusion problem and for turbulent channel flow, weak enforcement of wall boundaries is found to improve results at low resolution. Such weak boundary conditions may play a pivotal role in wall modeling for large-eddy simulation.

Collis, S. Scott↗

NASA NTRS1992

Approximate Solutions Of Equations Of Steady Diffusion

Rigorous analysis yields reliable criteria for "best-fit" functions. Improved "curve-fitting" method yields approximate solutions to differential equations of steady-state diffusion. Method applies to problems in which rates of diffusion depend linearly or nonlinearly on concentrations of diffusants, approximate solutions analytic or numerical, and boundary conditions of Dirichlet type, of Neumann type, or mixture of both types. Applied to equations for diffusion of charge carriers in semiconductors in which mobilities and lifetimes of charge carriers depend on concentrations.

Edmonds, Larry D.↗

NASA NTRS1974

Electromagnetic and scalar diffraction by a right-angled wedge with a uniform surface impedance

The diffraction of an electromagnetic wave by a perfectly-conducting right-angled wedge with one surface covered by a dielectric slab or absorber is considered. The effect of the coated surface is approximated by a uniform surface impedance. The solution of the normally incident electromagnetic problem is facilitated by introducing two scalar fields which satisfy a mixed boundary condition on one surface of the wedge and a Neumann of Dirichlet boundary condition on the other. A functional transformation is employed to simplify the boundary conditions so that eigenfunction expansions can be obtained for the resulting Green's functions. The eigenfunction expansions are transformed into the integral representations which then are evaluated asymptotically by the modified Pauli-Clemmow method of steepest descent. A far zone approximation is made to obtain the scattered field from which the diffraction coefficient is found for scalar plane, cylindrical or sperical wave incident on the edge. With the introduction of a ray-fixed coordinate system, the dyadic diffraction coefficient for plane or cylindrical EM waves normally indicent on the edge is reduced to the sum of two dyads which can be written alternatively as a 2 X 2 diagonal matrix.

Hwang, Y. M.↗

NASA NTRS1995

Quantum field between moving mirrors: A three dimensional example

The scalar quantum field uniformly moving plates in three dimensional space is studied. Field equations for Dirichlet boundary conditions are solved exactly. Comparison of the resulting wavefunctions with their instantaneous static counterpart is performed via Bogolubov coefficients. Unlike the one dimensional problem, 'particle' creation as well as squeezing may occur. The time dependent Casimir energy is also evaluated.

Hacyan, S.↗

NASA NTRS1979

The accurate solution of Poisson's equation by expansion in Chebyshev polynomials

A Chebyshev expansion technique is applied to Poisson's equation on a square with homogeneous Dirichlet boundary conditions. The spectral equations are solved in two ways - by alternating direction and by matrix diagonalization methods. Solutions are sought to both oscillatory and mildly singular problems. The accuracy and efficiency of the Chebyshev approach compare favorably with those of standard second- and fourth-order finite-difference methods.

Haidvogel, D. B.↗

NASA NTRS1997

A Smoothed Boundary Condition for Reducing Nonphysical Field Effects

In this paper, we examine the problem associated with abruptly mixing boundary conditions in the context of a two-dimensional semiconductor device simulator. Explicitly, this paper addresses the transition between an ohmic-type Dirichlet condition and a passivated Neumann boundary. In the traditional setting, the details or the transition between the two boundary types are not addressed and an abrupt transition is assumed. Subsequently, the calculated observables (most notably the potential) exhibit discontinuous derivatives near the surface at the point where the boundary type switches. This paper proposes an alternative condition which models the progression between the two boundary types through the use of a finite length, smoothed boundary whereby the numerical discontinuities are eliminated. The physical and mathematical basis for this smoothed boundary condition is discussed and examples of the technique's implementation given. It is found that the proposed boundary condition is numerically efficient and can be implemented in pre-existing device simulators with relative ease.

Smith, Arlynn W.↗