Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “basis functions”

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.

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

At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS2006

A Nonlinear Reduced Order Method for Prediction of Acoustic Fatigue

The goal of this investigation is to assess the quality of high-cycle-fatigue life estimation via a reduced order method, for structures undergoing geometrically nonlinear random vibrations. Modal reduction is performed with several different suites of basis functions. After numerically solving the reduced order system equations of motion, the physical displacement time history is obtained by an inverse transformation and stresses are recovered. Stress ranges obtained through the rainflow counting procedure are used in a linear damage accumulation method to yield fatigue estimates. Fatigue life estimates obtained using various basis functions in the reduced order method are compared with those obtained from numerical simulation in physical degrees-of-freedom.

Przekop, Adam↗
NASA NTRS2011

Evaluation of the Reflection Coefficient of Microstrip Elements for Reflectarray Antennas

Basis functions were studied and identified that provide efficient and accurate solutions for the induced patch currents and the reflection phase in microstrip reflect arrays. The integral equation of an infinite array of microstrip elements in the form of patches or crossed dipoles excited by a uniform plane wave is solved by the method-of-moments. Efficient choices of entire domain basis functions that yield accurate results have been described.

Rengarajan, Sembiam↗
NASA NTRS1992

Negative-energy states in the Dirac-Hartree-Fock problem - The effect of omission of two-electron integrals involving the small component

The effect of omission of two-electron integrals involving basis functions for the small component of the wavefunction on the eigenvalue spectrum in the Dirac-Hartree-Fock problem is studied. From an analysis of the Fock matrix it is shown that omission of these integrals moves the negative-energy states down, not up. Their complete omission does not give rise to intruder states. The appearance of intruder states occurs when only some of the core integrals are omitted, due to the nature of particular contraction schemes used for the core basis functions. Use of radially localized functions rather than atomic functions alleviates the intruder state problem.

Dyall, Kenneth G.↗
NASA NTRS1995

Two-dimensional mesh embedding for Galerkin B-spline methods

A number of advantages result from using B-splines as basis functions in a Galerkin method for solving partial differential equations. Among them are arbitrary order of accuracy and high resolution similar to that of compact schemes but without the aliasing error. This work develops another property, namely, the ability to treat semi-structured embedded or zonal meshes for two-dimensional geometries. This can drastically reduce the number of grid points in many applications. Both integer and non-integer refinement ratios are allowed. The report begins by developing an algorithm for choosing basis functions that yield the desired mesh resolution. These functions are suitable products of one-dimensional B-splines. Finally, test cases for linear scalar equations such as the Poisson and advection equation are presented. The scheme is conservative and has uniformly high order of accuracy throughout the domain.

Shariff, Karim↗
NASA NTRS1996

Refining Linear Fuzzy Rules by Reinforcement Learning

Linear fuzzy rules are increasingly being used in the development of fuzzy logic systems. Radial basis functions have also been used in the antecedents of the rules for clustering in product space which can automatically generate a set of linear fuzzy rules from an input/output data set. Manual methods are usually used in refining these rules. This paper presents a method for refining the parameters of these rules using reinforcement learning which can be applied in domains where supervised input-output data is not available and reinforcements are received only after a long sequence of actions. This is shown for a generalization of radial basis functions. The formation of fuzzy rules from data and their automatic refinement is an important step in closing the gap between the application of reinforcement learning methods in the domains where only some limited input-output data is available.

Berenji, Hamid R.↗
NASA NTRS2007

Simple and Efficient Numerical Evaluation of Near-Hypersingular Integrals

Recently, significant progress has been made in the handling of singular and nearly-singular potential integrals that commonly arise in the Boundary Element Method (BEM). To facilitate object-oriented programming and handling of higher order basis functions, cancellation techniques are favored over techniques involving singularity subtraction. However, gradients of the Newton-type potentials, which produce hypersingular kernels, are also frequently required in BEM formulations. As is the case with the potentials, treatment of the near-hypersingular integrals has proven more challenging than treating the limiting case in which the observation point approaches the surface. Historically, numerical evaluation of these near-hypersingularities has often involved a two-step procedure: a singularity subtraction to reduce the order of the singularity, followed by a boundary contour integral evaluation of the extracted part. Since this evaluation necessarily links basis function, Green s function, and the integration domain (element shape), the approach ill fits object-oriented programming concepts. Thus, there is a need for cancellation-type techniques for efficient numerical evaluation of the gradient of the potential. Progress in the development of efficient cancellation-type procedures for the gradient potentials was recently presented. To the extent possible, a change of variables is chosen such that the Jacobian of the transformation cancels the singularity. However, since the gradient kernel involves singularities of different orders, we also require that the transformation leaves remaining terms that are analytic. The terms "normal" and "tangential" are used herein with reference to the source element. Also, since computational formulations often involve the numerical evaluation of both potentials and their gradients, it is highly desirable that a single integration procedure efficiently handles both.

Fink, Patrick W.↗
NASA NTRS2011

Assessment of CFD-based Response Surface Model for Ares I Supersonic Ascent Aerodynamics

The Ascent Force and Moment Aerodynamic (AFMA) Databases (DBs) for the Ares I Crew Launch Vehicle (CLV) were typically based on wind tunnel (WT) data, with increments provided by computational fluid dynamics (CFD) simulations for aspects of the vehicle that could not be tested in the WT tests. During the Design Analysis Cycle 3 analysis for the outer mold line (OML) geometry designated A106, a major tunnel mishap delayed the WT test for supersonic Mach numbers (M) greater than 1.6 in the Unitary Plan Wind Tunnel at NASA Langley Research Center, and the test delay pushed the final delivery of the A106 AFMA DB back by several months. The aero team developed an interim database based entirely on the already completed CFD simulations to mitigate the impact of the delay. This CFD-based database used a response surface methodology based on radial basis functions to predict the aerodynamic coefficients for M > 1.6 based on only the CFD data from both WT and flight Reynolds number conditions. The aero team used extensive knowledge of the previous AFMA DB for the A103 OML to guide the development of the CFD-based A106 AFMA DB. This report details the development of the CFD-based A106 Supersonic AFMA DB, constructs a prediction of the database uncertainty using data available at the time of development, and assesses the overall quality of the CFD-based DB both qualitatively and quantitatively. This assessment confirms that a reasonable aerodynamic database can be constructed for launch vehicles at supersonic conditions using only CFD data if sufficient knowledge of the physics and expected behavior is available. This report also demonstrates the applicability of non-parametric response surface modeling using radial basis functions for development of aerodynamic databases that exhibit both linear and non-linear behavior throughout a large data space.

Hanke, Jeremy L.↗
NASA NTRS2020

INVESTIGATING THE FEASIBILITY AND STABILITY FOR MODELING ACOUSTIC WAVE SCATTERING USING A TIME-DOMAIN BOUNDARY INTEGRAL EQUATION WITH IMPEDANCE BOUNDARY CONDITION

Reducing aircraft noise is a major objective in the field of computational aeroacoustics. When designing next generation quiet and environmentally friendly aircraft, it is important to be able to accurately and efficiently predict the acoustic scattering by an aircraft body from a given noise source. Acoustic liners are an effective tool for aircraft noise reduction and are characterized by a frequency-dependent impedance. Converted into the time-domain using Fourier transforms, an impedance boundary condition can be used to simulate the acoustic wave scattering by geometric bodies treated with acoustic liners. This work considers using either an impedance or an admittance (inverse of impedance) boundary condition to allow for acoustic scattering problems to be modeled with geometries consisting of both un-lined and lined surfaces. Three acoustic liner models are discussed: the Extended Helmholtz Resonator Model, the Three-Parameter Impedance Model, and the Broadband Impedance Model. In both the Helmholtz and Three-Parameter models, liner impedance is specified at a given frequency, whereas the Broadband model allows for the investigation of multiple frequencies simultaneously. The impedance and admittance boundary conditions for acoustic liners are derived for each model and coupled with a time-domain boundary integral equation. The scattering solution is obtained iteratively using a boundary element method with constant spatial and third-order temporal basis functions. Time-domain boundary integral equations are unfortunately prone to numerical instabilities due to resonant frequencies resulting from non-trivial solutions in the interior domain. When reformulated with the Burton-Miller method, the instabilities are eliminated. Using a Burton-Miller reformulation, the stability of the boundary element method assuming a liner boundary condition is assessed using eigenvalue analysis. The stability of each liner model is discussed, and it is shown that the Three-Parameter and Broadband models are sufficient for modeling an acoustic liner on the surface of scattering bodies. The Helmholtz model demonstrates strict limitations for stability, whereas the Three-Parameter and Broadband models are stable for most cases. Also included in this work is an assessment of the spatial accuracy of the time-domain boundary element method with respect to the surface element basis functions, as well as a performance study of the numerical algorithm.

computational mathematics↗
NASA NTRS1995

Exponentially accurate approximations to piece-wise smooth periodic functions

A family of simple, periodic basis functions with 'built-in' discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representations of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.

Greer, James↗
NASA NTRS1993

Analytical evaluation of a surface integral expressing the coupling between interior and exterior volumes in a FE-IE approach

Recently an approach which combines the finite element technique and an integral equation to determine the fields scattered by inhomogeneous bodies of complicated shape has been proposed. Basically, a mathematical surface which encloses the scatterers is introduced, thus dividing the space into an interior and an exterior volume, in which the finite element technique and an integral equation for EM scattering, respectively, are applied. The integral equation is set up for the tangential components of the fields at the surface, while the interior volume the unknowns are the total fields. Continuity of the tangential fields at the boundary, as required by Maxwell's equations, is imposed, thus coupling the two methods to obtain a consistent solution. The coupling term is expressed by a surface integral formed by the dot product of a FE basis function and an IE testing function, or viceversa. By choosing the boundary to be a surface of revolution and by making a convenient selection of IE basis (testing) functions, it is possible to evaluate the integrals analytically on surfaces such as curved triangles, curved quadrilaterals and curved pentagons. We will illustrate the salient steps involved in setting up and carrying out these integrals and discuss what class of basis (testing) functions and analytic surfaces of revolution they are applicable to. Analytic calculations offer the advantage of better accuracy than purely numerical ones, and, when combined with them, often shed light on issues of numerical convergence and limiting values. Furthermore, they may reduce computation time and storage requirements.

Zuffada, C.↗
NASA NTRS2003

Further Developments in the MLPG Method for Beam Problems

An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions as in the conventional MLPG method. Linear test functions are also chosen, leading to a variation of the MLPG method that is computationally efficient compared to the conventional implementation. The MLPG method is evaluated by applying the formulation to a variety of patch tests, thin beam problems, and problems with load discontinuities. The formulation successfully reproduces exact solutions to machine accuracy when higher order power and spline functions are chosen as test functions or when the linear test function is used, and when constructing the trial functions, the order of the basis function is properly balanced by the order of the weight function. For mixed boundary value problems, deflections, slopes, moments, and shear forces are calculated to the same accuracy by the MLPG method without the use of elaborate post-processing techniques. Problems with load discontinuities require special care - when a reasonable number of nodes are used, the method yields very accurate results.

Raju, I. S.↗
NASA NTRS2009

A Finite Element Theory for Predicting the Attenuation of Extended-Reacting Liners

A non-modal finite element theory for predicting the attenuation of an extended-reacting liner containing a porous facesheet and located in a no-flow duct is presented. The mathematical approach is to solve separate wave equations in the liner and duct airway and to couple these two solutions by invoking kinematic constraints at the facesheet that are consistent with a continuum theory of fluid motion. Given the liner intrinsic properties, a weak Galerkin finite element formulation with cubic polynomial basis functions is used as the basis for generating a discrete system of acoustic equations that are solved to obtain the coupled acoustic field. A state-of-the-art, asymmetric, parallel, sparse equation solver is implemented that allows tens of thousands of grid points to be analyzed. A grid refinement study is presented to show that the predicted attenuation converges. Excellent comparison of the numerically predicted attenuation to that of a mode theory (using a Haynes 25 metal foam liner) is used to validate the computational approach. Simulations are also presented for fifteen porous plate, extended-reacting liners. The construction of some of the porous plate liners suggest that they should behave as resonant liners while the construction of others suggest that they should behave as broadband attenuators. In each case the finite element theory is observed to predict the proper attenuation trend.

Watson, W. R.↗
NASA NTRS2011

Experiments in Reconstructing Twentieth-Century Sea Levels

One approach to reconstructing historical sea level from the relatively sparse tide-gauge network is to employ Empirical Orthogonal Functions (EOFs) as interpolatory spatial basis functions. The EOFs are determined from independent global data, generally sea-surface heights from either satellite altimetry or a numerical ocean model. The problem is revisited here for sea level since 1900. A new approach to handling the tide-gauge datum problem by direct solution offers possible advantages over the method of integrating sea-level differences, with the potential of eventually adjusting datums into the global terrestrial reference frame. The resulting time series of global mean sea levels appears fairly insensitive to the adopted set of EOFs. In contrast, charts of regional sea level anomalies and trends are very sensitive to the adopted set of EOFs, especially for the sparser network of gauges in the early 20th century. The reconstructions appear especially suspect before 1950 in the tropical Pacific. While this limits some applications of the sea-level reconstructions, the sensitivity does appear adequately captured by formal uncertainties. All our solutions show regional trends over the past five decades to be fairly uniform throughout the global ocean, in contrast to trends observed over the shorter altimeter era. Consistent with several previous estimates, the global sea-level rise since 1900 is 1.70 +/- 0.26 mm/yr. The global trend since 1995 exceeds 3 mm/yr which is consistent with altimeter measurements, but this large trend was possibly also reached between 1935 and 1950.

Ray, Richard D.↗
NASA NTRS1995

The Schwinger Variational Method

Variational methods have proven invaluable in theoretical physics and chemistry, both for bound state problems and for the study of collision phenomena. The application of the Schwinger variational (SV) method to e-molecule collisions and molecular photoionization has been reviewed previously. The present chapter discusses the implementation of the SV method as applied to e-molecule collisions. Since this is not a review of cross section data, cross sections are presented only to server as illustrative examples. In the SV method, the correct boundary condition is automatically incorporated through the use of Green's function. Thus SV calculations can employ basis functions with arbitrary boundary conditions. The iterative Schwinger method has been used extensively to study molecular photoionization. For e-molecule collisions, it is used at the static exchange level to study elastic scattering and coupled with the distorted wave approximation to study electronically inelastic scattering.

Huo, Winifred M.↗
NASA NTRS2003

Electron Collisional Excitation Rates for OI USING the B-Spline R-Matrix Approach

The B-spline R-matrix approach has been used to calculate electron collisional excitation strengths and rates for transitions between the 3P, 1D, and 1S states of ground configuration and from these states to the states of the excited 2s(sup 2)2p(sup 3)ns (n = 3-5), 2s(sup 2)2p(sup 3)np (n = 3-4), 2s(sup 2)2p(sup 3)nd (n = 3-4), 2s(sup 2)2p(sup 3)4f, and 2s2p(sup 5) configurations. The nonorthogonal orbitals are used for an accurate description of both the target wave functions and the R-matrix basis functions. The thermally averaged collision strengths are obtained from the collision strengths by integrating over a Maxwellian velocity distribution of electron energies, and these are tabulated over a temperature range from 1000 to 60,000 K. The parametric functions of scaled energy have also been obtained to represent collision strengths over a wide energy range or thermally averaged collision strengths at any desired temperature.

Zatsarinny, O.↗
NASA NTRS2006

New Accurate Oscillator Strengths and Electron Excitation Collision Strengths for N1

The nonorthogonal orbitals technique in a multiconfiguration Hartree-Fock approach is used to calculate oscillator strengths and transition probabilities of N(I) lines. The relativistic effects are allowed by means of Breit-Pauli operators. The length and velocity forms of oscillator strengths show good agreement for most transitions. The B-spline R-matrix with pseudostates approach has been used to calculate electron excitation collision strengths and rates. The nonorthogonal orbitals are used for an accurate description of both target wave functions and the R-matrix basis functions. The 24 spectroscopic bound and autoionizing states together with 15 pseudostates are included in the close-coupling expansion. The collision strengths for transitions between fine-structure levels are calculated by transforming the LS-coupled K-matrices to K-matrices in an intermediate coupling scheme. Thermally averaged collision strengths have been determined by integrating collision strength over a Maxwellian distribution of electron energies over a temperature range suitable for the modeling of astrophysical plasmas. The oscillator strengths and thermally averaged collision strengths are presented for transitions between the fine-structure levels of the 2s(sup 2)p(sup 3) (sup 4)S(sup 0), (sup 2)D(sup 0), (sup 2)P(sup 0), 2s2p(sup 4) (sup 4)P, 2s(sup 2)2p(sup 2)3s (sup 4)P, and (sup 2)P terms and from these levels to the levels of the 2s(sup 2)2p(sup 2)3p (sup 2)S(sup 0), (sup 4)D(sup 0), (sup 4)P(sup 0), (sup 4)S(sup 0), (sup 2)D(sup 0), (sup 2)P(sup 0),2s(sup 2)2p(sup 2)3s(sup 2)D, 2s(sup 2)2p(sup 2)4s(sup 4)P, (sup 2)P, 2s(sup 2)2p(sup 2)3d(sup 2)P, (sup 4)F,(sup 2)F,(sup 4)P, (sup 4)D, and (sup 2)D terms. Thermally averaged collision strengths are tabulated over a temperature range from 500 to 50,000 K.

Tayal, S. S.↗
NASA NTRS1994

Scattering from perfectly conducting and resistive strips on a grounded dielectric slab

The scattering properties of perfectly conducting and resistive strips are predicted for strips which are located on a dielectric slab backed by a perfectly conducting ground plane. The spectral domain Green's function is used to relate the currents and fields on the strip, and the resulting integral equation is solved using the method of moments. Both TE and TM strips are examined using piecewise linear and pulse subdomain basis functions, respectively, to model the current on the strip. Calculated results are compared with results measured at the NASA Langley Research Center.

PULSE SUBDOMAIN BASIS FUNCTION↗