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All sources NASA NTRS DOE OSTI

NASA NTRS2011

Broadband Via-Less Microwave Crossover Using Microstrip-CPW Transitions

The front-to-back interface between microstrip and CPW (coplanar waveguide) typically requires complex fabrication or has high radiation loss. The microwave crossover typically requires a complex fabrication step. The prior art in microstrip-CPW transition requires a physical vias connection between the microstrip and CPW line on a separate layer. The via-less version of this transition was designed empirically and does not have a close form solution. The prior art of the micro wave crossover requires either additional substrate or wire bond as an air bridge to isolate two microwave lines at the crossing junction. The disadvantages are high radiation loss, no analytical solution to the problem, lengthy simulation time, and complex fabrication procedures to generate air bridges or via. The disadvantage of the prior crossover is a complex fabrication procedure, which also affects the device reliability and yield. This microstrip-CPW transition is visualized as two microstrip-slotline transitions combined in a way that the radiation from two slotlines cancels each other out. The invention is designed based on analytical methods; thus, it significantly reduces the development time. The crossover requires no extra layer to cross two microwave signals and has low radiation loss. The invention is simple to fabricate and design. It produces low radiation loss and can be designed with low insertion loss, with some tradeoff with signal isolation. The microstrip-CPW transition is used as an interface to connect between the device and the circuit outside the package. The via-less microwave crossover is used to allow two signals to cross without using an extra layer or fabrication processing step to enable this function. This design allows the solution to be determined entirely though analytical techniques. In addition, a planar via-less microwave crossover using this technique was proposed. The experimental results show that the proposed crossover at 5 GHz has a minimum isolation of 32 dB. It also has low in-band insertion loss and return loss of 1.2 dB and 18 dB, respectively, over more than 44 percent of bandwidth at room temperature. This microstrip-CPW transition requires the microstrip line to be split into two sections. Each section is connected to a microstrip quarter-wavelength openended stub. A slotline is also placed perpendicular to the microstrip section. The slot is connected to a grounded-end quarter-wavelength slotline and generates a microstrip-slotline transition. When two of these sections are placed in parallel and with the microstrip section combined at transition, a microstrip- CPW transition is formed. The slotline radiation is suppressed as two slots are excited with the electric field in an opposite direction, which cancels the radiation in far field. The invention on the crossover consists of the invented microstrip-CPW transitions combined back-to-back and a microstrip low-pass filter. One signal is crossed through to the microstrip layer, while the other signal is crossed through the CPW line located on the ground plane of the microstrip line. The microstrip low-pass filter produces a narrow line at the crossing point to enhance the system isolation. It also produces broadband response in the operating frequency band. The microstrip-CPW transition allows a microwave signal to travel from microstrip line to CPW line with low radiation loss. The crossover allows two microwave signals to cross with minimal parasitic coupling.

Stevenson, Thomas↗

DOE OSTI2023

Geometric Optimization of an Electrochemical Purification Cell to Prevent Corrosion in CSP Plants During Operation

When exposed to moisture or oxygen, molten chloride salts produce corrosive impurities which degrade containment alloys. This can significantly decrease the lifetime and increase costs of molten-salt-based systems. To overcome this barrier, we designed and modeled an electrochemical purification cell to remove the corrosive impurity MgOH+. Various reactor architectures, including continuous stirred tank reactors (CSTRs) and plug flow reactors (PFRs) were investigated. Steady-state thermoelectric properties were evaluated using analytical methods, allowing assessment of the effects of structure and design parameters such as flow rate, cell length, and cross-sectional area of molten salt. The results suggest that our design could most effectively increase reliability and decrease costs of molten-chloride-salt-based systems by protecting them during continuous operation using an annular plug flow reactor.

analytical modeling↗

NASA NTRS1972

Atmospheric pressure, density, temperature and wind variations between 50 and 200 km

Data on atmospheric pressure, density, temperature and winds between 50 and 200 km were collected from sources including Meteorological Rocket Network data, ROBIN falling sphere data, grenade release and pitot tube data, meteor winds, chemical release winds, satellite data, and others. These data were analyzed by a daily difference method and results on the distribution statistics, magnitude, and spatial structure of the irregular atmospheric variations are presented. Time structures of the irregular variations were determined by the analysis of residuals from harmonic analysis of time series data. The observed height variations of irregular winds and densities are found to be in accord with a theoretical relation between these two quantities. The latitude variations (at 50 - 60 km height) show an increasing trend with latitude. A possible explanation of the unusually large irregular wind magnitudes of the White Sands MRN data is given in terms of mountain wave generation by the Sierra Nevada range about 1000 km west of White Sands. An analytical method is developed which, based on an analogy of the irregular motion field with axisymmetric turbulence, allows measured or model correlation or structure functions to be used to evaluate the effective frequency spectra of scalar and vector quantities of a spacecraft moving at any speed and at any trajectory elevation angle.

Justus, C. G.↗

NASA NTRS1992

Flow analysis and design optimization methods for nozzle-afterbody of a hypersonic vehicle

This report summarizes the methods developed for the aerodynamic analysis and the shape optimization of the nozzle-afterbody section of a hypersonic vehicle. Initially, exhaust gases were assumed to be air. Internal-external flows around a single scramjet module were analyzed by solving the 3D Navier-Stokes equations. Then, exhaust gases were simulated by a cold mixture of Freon and Ar. Two different models were used to compute these multispecies flows as they mixed with the hypersonic airflow. Surface and off-surface properties were successfully compared with the experimental data. The Aerodynamic Design Optimization with Sensitivity analysis was then developed. Pre- and postoptimization sensitivity coefficients were derived and used in this quasi-analytical method. These coefficients were also used to predict inexpensively the flow field around a changed shape when the flow field of an unchanged shape was given. Starting with totally arbitrary initial afterbody shapes, independent computations were converged to the same optimum shape, which rendered the maximum axial thrust.

Baysal, O.↗

NASA NTRS1994

Computational predictive methods for fracture and fatigue

The damage-tolerant design philosophy as used by aircraft industries enables aircraft components and aircraft structures to operate safely with minor damage, small cracks, and flaws. Maintenance and inspection procedures insure that damages developed during service remain below design values. When damage is found, repairs or design modifications are implemented and flight is resumed. Design and redesign guidelines, such as military specifications MIL-A-83444, have successfully reduced the incidence of damage and cracks. However, fatigue cracks continue to appear in aircraft well before the design life has expired. The F16 airplane, for instance, developed small cracks in the engine mount, wing support, bulk heads, the fuselage upper skin, the fuel shelf joints, and along the upper wings. Some cracks were found after 600 hours of the 8000 hour design service life and design modifications were required. Tests on the F16 plane showed that the design loading conditions were close to the predicted loading conditions. Improvements to analytic methods for predicting fatigue crack growth adjacent to holes, when multiple damage sites are present, and in corrosive environments would result in more cost-effective designs, fewer repairs, and fewer redesigns. The overall objective of the research described in this paper is to develop, verify, and extend the computational efficiency of analysis procedures necessary for damage tolerant design. This paper describes an elastic/plastic fracture method and an associated fatigue analysis method for damage tolerant design. Both methods are unique in that material parameters such as fracture toughness, R-curve data, and fatigue constants are not required. The methods are implemented with a general-purpose finite element package. Several proof-of-concept examples are given. With further development, the methods could be extended for analysis of multi-site damage, creep-fatigue, and corrosion fatigue problems.

Cordes, J.↗

DOE OSTI2025

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part I: Model Formulation

Here, this paper formulates a new particle-in-cell method for the Vlasov–Maxwell system. Under the Lorenz gauge condition, Maxwell’s equations for the electromagnetic fields can be written as a collection of scalar and vector wave equations. The use of potentials for the fields motivates the adoption of a Hamiltonian formulation for particles that employs the generalized (conjugate) momentum. A notable advantage offered by the Hamiltonian formulation is the elimination of time derivatives in the Lorenz gauge formulation that are required by the standard Newton–Lorentz treatment of the particles. This allows the fields to retain the full time-accuracy guaranteed by the field solver. The resulting updates for particles require only knowledge of the fields and their spatial derivatives. An analytical method for constructing these spatial derivatives is presented that exploits the underlying integral solution used in the field solver for the wave equations. Moreover, these derivatives are demonstrated to converge at the same rate as the fields in both time and space. The Method of Lines Transpose field solver we consider in this work is globally first-order accurate in time and high-order accurate in space (e.g., fourth- and fifth-order) and belongs to a larger class of methods which are unconditionally stable, can address geometry, and leverage $\mathcal {O}(N)$ fast summation methods for efficiency. We demonstrate the method on several well-established benchmark problems on bounded domains, including a plasma sheath as well as a relativistic particle beam. The efficacy of the proposed formulation is established by comparing with a second-order accurate finite-difference time-domain method that employs a leapfrog time advance for particles and a charge conserving map suitable for bounded domains. The new method shows mesh-independent numerical heating properties even in cases where the plasma Debye length is smaller than the grid spacing. This is an important feature of the new method for problems defined on bounded domains, because it permits the use of coarser grids in space in the representation of the fields. Such a capability has significant implications for the simulation of plasmas in bounded domains with complex geometry, where the ratio between the largest and smallest cells can vary significantly. The use of high-order spatial approximations in the new method also means that fewer grid points are required in order to achieve a fixed accuracy. Our results also suggest that the new method can be used with fewer simulation particles per cell compared to the benchmark explicit method, which permits further computational savings.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

A Critical Review of the Circular Economy for Lithium-Ion Batteries and Photovoltaic Modules: Status, Challenges, and Opportunities

To meet net-zero emissions and cost targets for power production, recent analysis indicates that photovoltaic (PV) capacity in the United States could exceed 1 TW by 2050 alongside comparable levels of energy storage capacity, mostly from batteries. For comparison, the total U.S. utility-scale power capacity from all energy sources in 2020 was 1.2 TW (EIA 2022), of which solar satisfied approximately 3% (DOE 2021). With such massive scales of deployment, questions have arisen regarding issues of material supply for manufacturing, end-of-life management of technologies, environmental impacts across the life cycle, and economic costs to both individual consumers and society at large. A set of solutions to address these issues center on the development of a circular economy - shifting from a take-make-waste linear economic model to one that retains the value of materials and products as long as possible, recovering materials at end of life to recirculate back into the economy. With limited global experience, scholars and practitioners have begun to investigate circular economy pathways, focusing on applying novel technologies and analytical methods to fast-growing sectors like renewable energy. This critical review aims to synthesize the growing literature to identify key insights, gaps, and opportunities for research and implementation of a circular economy for two of the leading technologies that enable the transition to a renewable energy economy: solar PV and lithium-ion batteries (LIBs). We apply state-of-the-science systematic literature review procedures to critically analyze over 3,000 publications on the circular economy of solar PV and LIBs, categorizing those that pass a series of objective screens in ways that can illuminate the current state of the art, highlight existing impediments to a circular economy, and recommend future technological and analytical research. We conclude that while neither PV nor LIB industries have reached a circular economy, they are both on a path towards increased circularity. Based on our assessment of the state of current literature and scientific understanding, we recommend research move beyond its prior emphasis on recycling technology development to more comprehensively investigate other CE strategies, more holistically consider economic, environmental and policy aspects of CE strategies, increase leveraging of digital information systems that can support acceleration towards a CE, and to continue to study CE-related aspects of LIB and PV markets.

circular economy↗

NASA NTRS2003

A new method to evaluate human-robot system performance

One of the key issues in space exploration is that of deciding what space tasks are best done with humans, with robots, or a suitable combination of each. In general, human and robot skills are complementary. Humans provide as yet unmatched capabilities to perceive, think, and act when faced with anomalies and unforeseen events, but there can be huge potential risks to human safety in getting these benefits. Robots provide complementary skills in being able to work in extremely risky environments, but their ability to perceive, think, and act by themselves is currently not error-free, although these capabilities are continually improving with the emergence of new technologies. Substantial past experience validates these generally qualitative notions. However, there is a need for more rigorously systematic evaluation of human and robot roles, in order to optimize the design and performance of human-robot system architectures using well-defined performance evaluation metrics. This article summarizes a new analytical method to conduct such quantitative evaluations. While the article focuses on evaluating human-robot systems, the method is generally applicable to a much broader class of systems whose performance needs to be evaluated.

Space Flight/instrumentation↗

NASA NTRS2001

Advanced Methods for Aircraft Engine Thrust and Noise Benefits: Nozzle-Inlet Flow Analysis

Three connected sub-projects were conducted under reported project. Partially, these sub-projects are directed to solving the problems conducted by the HU/FM&AL under two other NASA grants. The fundamental idea uniting these projects is to use untraditional 3D corrugated nozzle designs and additional methods for exhaust jet noise reduction without essential thrust lost and even with thrust augmentation. Such additional approaches are: (1) to add some solid, fluid, or gas mass at discrete locations to the main supersonic gas stream to minimize the negative influence of strong shock waves forming in propulsion systems; this mass addition may be accompanied by heat addition to the main stream as a result of the fuel combustion or by cooling of this stream as a result of the liquid mass evaporation and boiling; (2) to use porous or permeable nozzles and additional shells at the nozzle exit for preliminary cooling of exhaust hot jet and pressure compensation for non-design conditions (so-called continuous ejector with small mass flow rate; and (3) to propose and analyze new effective methods fuel injection into flow stream in air-breathing engines. Note that all these problems were formulated based on detailed descriptions of the main experimental facts observed at NASA Glenn Research Center. Basically, the HU/FM&AL Team has been involved in joint research with the purpose of finding theoretical explanations for experimental facts and the creation of the accurate numerical simulation technique and prediction theory for solutions for current problems in propulsion systems solved by NASA and Navy agencies. The research is focused on a wide regime of problems in the propulsion field as well as in experimental testing and theoretical and numerical simulation analysis for advanced aircraft and rocket engines. The F&AL Team uses analytical methods, numerical simulations, and possible experimental tests at the Hampton University campus. We will present some management activity and theoretical numerical simulation results obtained by the FM&AL Team in the reporting period in accordance with the schedule of the work.

Morgan, Morris H.↗

NASA NTRS2007

Aerospace Toxicology and Microbiology

Toxicology dates to the very earliest history of humanity with various poisons and venom being recognized as a method of hunting or waging war with the earliest documentation in the Evers papyrus (circa 1500 BCE). The Greeks identified specific poisons such as hemlock, a method of state execution, and the Greek word toxos (arrow) became the root of our modern science. The first scientific approach to the understanding of poisons and toxicology was the work during the late middle ages of Paracelsus. He formulated what were then revolutionary views that a specific toxic agent or "toxicon" caused specific dose-related effects. His principles have established the basis of modern pharmacology and toxicology. In 1700, Bernardo Ramazzini published the book De Morbis Artificum Diatriba (The Diseases of Workers) describing specific illnesses associated with certain labor, particularly metal workers exposed to mercury, lead, arsenic, and rock dust. Modern toxicology dates from development of the modern industrial chemical processes, the earliest involving an analytical method for arsenic by Marsh in 1836. Industrial organic chemicals were synthesized in the late 1800 s along with anesthetics and disinfectants. In 1908, Hamilton began the long study of occupational toxicology issues, and by WW I the scientific use of toxicants saw Haber creating war gases and defining time-dosage relationships that are used even today.

James, John T.↗

DOE OSTI2020

Analytical Study to Use the Excess Digester Gas of Wastewater Treatment Plants

This study presents an analytical method that can be used to enhance the power production rate and the energy-saving at wastewater treatment plants. The digester used at wastewater treatment plants produces digester gas by anaerobic digestion, with which biofuel production can be achieved. Biofuels can be used to meet some of the energy requirements of the wastewater treatment facility through combined heat and power (CHP) gas engines (cogeneration). Using micro gas turbine (MGT), a CHP technology can be introduced in wastewater treatment plants (WWTPs). The combination of MGTs and absorption chillers is a promising technology as it produces electricity, heating, and cooling simultaneously. The study demonstrated how the waste heat of MGTs could be used to drive absorption chillers. In this analytical study, a detailed technical and economic analysis is provided on the trigeneration system, i.e., the integration of MGTs and absorption chillers driven by waste digester gas of the wastewater treatment plants. It can meet the heating and cooling demands of the plants, which promote the reduction of utility costs. The technology presented is also useful for other thermal energy users.

09 BIOMASS FUELS↗

NASA NTRS1998

Inverse Compton Scattering in Mildly Relativistic Plasma

We investigated the effect of inverse Compton scattering in mildly relativistic static and moving plasmas with low optical depth using Monte Carlo simulations, and calculated the Sunyaev-Zel'dovich effect in the cosmic background radiation. Our semi-analytic method is based on a separation of photon diffusion in frequency and real space. We use Monte Carlo simulation to derive the intensity and frequency of the scattered photons for a monochromatic incoming radiation. The outgoing spectrum is determined by integrating over the spectrum of the incoming radiation using the intensity to determine the correct weight. This method makes it possible to study the emerging radiation as a function of frequency and direction. As a first application we have studied the effects of finite optical depth and gas infall on the Sunyaev-Zel'dovich effect (not possible with the extended Kompaneets equation) and discuss the parameter range in which the Boltzmann equation and its expansions can be used. For high temperature clusters (k(sub B)T(sub e) greater than or approximately equal to 15 keV) relativistic corrections based on a fifth order expansion of the extended Kompaneets equation seriously underestimate the Sunyaev-Zel'dovich effect at high frequencies. The contribution from plasma infall is less important for reasonable velocities. We give a convenient analytical expression for the dependence of the cross-over frequency on temperature, optical depth, and gas infall speed. Optical depth effects are often more important than relativistic corrections, and should be taken into account for high-precision work, but are smaller than the typical kinematic effect from cluster radial velocities.

Molnar, S. M.↗

NASA NTRS1991

Flow analysis and design optimization methods for nozzle afterbody of a hypersonic vehicle

This report summarizes the methods developed for the aerodynamic analysis and the shape optimization of the nozzle-afterbody section of a hypersonic vehicle. Initially, exhaust gases were assumed to be air. Internal-external flows around a single scramjet module were analyzed by solving the three dimensional Navier-Stokes equations. Then, exhaust gases were simulated by a cold mixture of Freon and Argon. Two different models were used to compute these multispecies flows as they mixed with the hypersonic airflow. Surface and off-surface properties were successfully compared with the experimental data. In the second phase of this project, the Aerodynamic Design Optimization with Sensitivity analysis (ADOS) was developed. Pre and post optimization sensitivity coefficients were derived and used in this quasi-analytical method. These coefficients were also used to predict inexpensively the flow field around a changed shape when the flow field of an unchanged shape was given. Starting with totally arbitrary initial afterbody shapes, independent computations were converged to the same optimum shape, which rendered the maximum axial thrust.

Baysal, Oktay↗

DOE OSTI2021

A Review of Nuclear Forensics: 2016-2020

The study of nuclear forensics harkens back to the Manhattan Project-era, when scientists first started to analyze the debris from the 1945 Trinity test. Political turmoil stemming from the Cold War and the rehabilitation of Germany following WWII has led to new challenges in international security involving nuclear proliferation. Nuclear materials have, on occasion, been lost, misplaced, or stolen from former Soviet countries, and illicit materials have been interdicted all over the world. The National Technical Nuclear Forensics Center (NTNFC) was established in 2006, and has been at the forefront of drive to advance nuclear forensic capabilities in the United States. The ultimate goal of nuclear forensics is to examine nuclear and other radioactive materials using analytical techniques to determine origin and history of the material, particularly in the context of law enforcement investigations. Nuclear forensics can be divided into two parts: predetonation and post-detonation. Pre-detonation forensics, as the name implies, is the investigation of a nuclear material or weapon that has not been detonated or involved in an explosion, whereas postdetonation forensics is the study of activation or fission products in debris or the environment following the use of a nuclear or radiological dispersal device (RDD). Both parts require a number of analytical chemical and radiochemical techniques to determine identification of the material. Many advancements in analytical techniques, including rapidity, sample size, and forensic signatures have been made in recent years. The analytical methods that can be used in a nuclear forensic investigation, such as mass spectrometry and gamma spectroscopy, have been described in detail in previous reviews, including Straub et.al, and will not be explained here. This review will discuss recent publications (from 2016 to present) describing advancements of techniques such as radiochronometry, morphology, development of novel reference materials, and inter-laboratory collaborations for both pre- and post-detonation nuclear forensics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Mesoscale modeling and semi-analytical approach for the microstructure-aware effective thermal conductivity of porous polygranular materials

Here we established a comprehensive modeling approach for investigating the microstructure-aware effective thermal conductivity ($κ_{eff}$) for porous microstructures containing solid particles and gaseous pores. Our approach combines the mesoscale computational modeling framework and the semi-analytical method, allowing for efficient prediction of $κ_{eff}$ for realistic porous microstructures, while considering complicated microstructural thermal conduction pathways effectively in the prediction. We used the diffuse-interface mesoscale computational model to generate extensive simulated $κ_{eff}$ data for realistic digital representations of microstructures with wide ranges of porosity ($f_p$), thermal conductivity of the gas phase ($κ_g$), and thermal conductivity of the solid phase ($κ_s$). From the simulated data, we identified two property variation regimes for $κ_{eff}$: (1) a slow $κ_{eff}$ increase for $κ_s ~ κ_g$; and (2) a faster $κ_{eff}$ increase for $κ_s \gg κ_g$. To capture the key features of the relationship between the microstructure and $κ_{eff}$, we derived a semi-analytical model by introducing structure and intensification factors. The two new factors incorporate the calibrated effective contribution of the solid volume with $κ_s$ and additional interfacial effects into the prediction of $κ_{eff}$, respectively, allowing for consideration of parallel, serial, and interfacial conduction mechanisms effectively. Using the selected simulation data, we quantified key model parameters within the semi-analytical model and verified that the parameterized model exhibits excellent agreement with simulated $κ_{eff}$ for the entire range of the parameter space.

36 MATERIALS SCIENCE↗

NASA NTRS2021

Experimental and Computational Investigation of Flow Boiling in Microgravity

This study explores use of Computational Fluid Dynamics (CFD) to predict near-saturated flow boiling of FC-72 in microgravity. The computational method employs transient analysis to predict detailed interfacial behavior and heat transfer characteristics along a rectangular channel heated along two opposite walls. Predicted results are validated against experimental temperature measurements and high-speed video images captured during a series of parabolic aircraft maneuvers for three sets of operating conditions which include variations of both mass velocity and wall heat flux. The computational method is based on the multi-phase volume of fluid (VOF) model, which is combined with appropriate phase change and turbulence models, and accounts for both shear-lift force on bubbles and conjugate heat transfer along the heating walls. A key advantage of the CFD method is ability to capture details that are very difficult to measure experimentally, including detailed spatial variations of bubble shape, void fraction, mixture fluid temperature, liquid velocity, and vapor velocity, results for which are presented for each of the three test cases. Different flow regimes predicted along the heated length exhibit a number of dominant mechanisms including bubble nucleation, bubble growth, coalescence, vapor blankets, interfacial waviness, and residual liquid sub-layer, all of which agree well with experiment. Vapor velocity is shown to increase appreciably along the heated length because of increased void fraction, while liquid velocity experiences large fluctuations. Non-equilibrium effects are accentuated with increasing mass velocity, contributing minor deviations of fluid temperature from simulations compared to those predicted by the analytical method. Predicted wall temperature is fairly uniform in the middle of the heated length but increases in the entrance region, due to sensible heat transfer in the subcooled liquid, and decreases toward the exit, mostly because of flow acceleration resulting from increased void fraction.

two-phase cooling; flow boiling; CFD; microgravity↗

NASA NTRS2016

Monte Carlo Simulation of a Knudsen Effusion Mass Spectrometer Sampling System

Knudsen flow is easily simulated with a Monte Carlo method. In this study we develop a Visual Basic for Excel (VBA) code to simulate the molecular beam from a vaporizing solid. The system at NASA Glenn uses the restricted collimation method of Chatillon and colleagues, which consists of two apertures between the effusion cell and the ionizer. The diameter of the first aperture is smaller than the diameter of the effusion cell orifice, so the ionizer effectively only sees inside the effusion cell. The code is able to calculate the transmission coefficient through the cell orifice, through the cell orifice and the first aperture, and through the cell orifice and first and second aperatures. Calculated transmission coefficients through the cell orifice are compared to tabulated values to validate the code. Then transmission coefficients are calculated through the cell orifice and both apertures to the ionizer. This allows the geometry (aperture spacing and diameters) of the sampling system the sampling system to be optimized. Calculated transmission factors are also compared to literature values calculated via an analytic method.

Radke, Michael J.↗

NASA NTRS2020

An Overview of the Null-Field Method. II: Convergence and Numerical Stability

In this paper we provide an analysis of the convergence and numerical stability of the null-field method with discrete sources. We show that (i) if the null-field scheme is numerically stable then we can decide whether or not convergence can be achieved; (ii) if the null-field scheme is numerically unstable then we cannot draw any conclusion about the convergence issue; and (iii) the numerical stability is closely related to the property of a tangential system of radiating discrete sources to form a Riesz basis. Our numerical analysis indicates that for prolate spheroids and localized vector spherical wave functions, the null-field scheme is numerically unstable (this system of vector functions does not form a Riesz basis), while for distributed vector spherical wave functions, the numerical instability is not so pronounced (this system of discrete sources almost possesses the property of being a Riesz basis). We also describe an analytical method for computing the surface integrals in the framework of the conventional null-field method with localized vector spherical wave functions which increases the stability of the numerical scheme.

Null-field method↗