Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Level set method”

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 1,207 records · Page 67
All sourcesNASA NTRSDOE OSTI
NASA NTRS1999

Choosing Meteorological Input for the Global Modeling Initiative Assessment of High Speed Aircraft

The Global Modeling Initiative (GMI) science team is developing a three dimensional chemistry and transport model (CTM) to be used in assessment of the atmospheric effects of aviation. Requirements are that this model be documented be validated against observations, use a realistic atmospheric circulation, and contain numerical transport and photochemical modules representing atmospheric processes. The model must also retain computational efficiency to be tractable to use for multiple scenarios and sensitivity studies. To meet these requirements, a facility model concept was developed in which the different components of the CTM are evaluated separately. The first use of the GMI model will be to evaluate the impact of the exhaust of supersonic aircraft on the stratosphere. The assessment calculations will depend strongly on the wind and temperature fields used by the CTM. Three meteorological data sets for the stratosphere are available to GMI: the National Center for Atmospheric Research Community Climate Model (CCM2), the Goddard Earth Observing System Data Assimilation System (GEOS DAS), and the Goddard Institute for Space Studies general circulation model (GISS). Objective criteria were established by the GMI team to identify the data set which provides the best representation of the stratosphere. Simulations of gases with simple chemical control were chosen to test various aspects of model transport. The three meteorological data sets were evaluated and graded based on their ability to simulate these aspects of stratospheric measurements. This paper describes the criteria used in grading the meteorological fields. The meteorological data set which has the highest score and therefore was selected for GMI is CCM2. This type of objective model evaluation establishes a physical basis for interpretation of differences between models and observations. Further, the method provides a quantitative basis for defining model errors, for discriminating between different models, and for ready re-evaluation of improved models. These in turn will lead to a higher level of confidence in assessment calculations.

Douglass, A. R.↗
NASA NTRS2016

Confidence Intervals for Laboratory Sonic Boom Annoyance Tests

Commercial supersonic flight is currently forbidden over land because sonic booms have historically caused unacceptable annoyance levels in overflown communities. NASA is providing data and expertise to noise regulators as they consider relaxing the ban for future quiet supersonic aircraft. One deliverable NASA will provide is a predictive model for indoor annoyance to aid in setting an acceptable quiet sonic boom threshold. A laboratory study was conducted to determine how indoor vibrations caused by sonic booms affect annoyance judgments. The test method required finding the point of subjective equality (PSE) between sonic boom signals that cause vibrations and signals not causing vibrations played at various amplitudes. This presentation focuses on a few statistical techniques for estimating the interval around the PSE. The techniques examined are the Delta Method, Parametric and Nonparametric Bootstrapping, and Bayesian Posterior Estimation.

Rathsam, Jonathan↗
DOE OSTI2024

Data-Driven Computation of Probabilistic Marching Cubes for Efficient Visualization of Level-Set Uncertainty

Uncertainty visualization is an important emerging research area. Being able to visualize data uncertainty can help scientists improve trust in analysis and decision-making. However, visualizing uncertainty can add computational overhead, which can hinder the efficiency of analysis. In this paper, we propose novel data-driven techniques to reduce the computational requirements of the probabilistic marching cubes (PMC) algorithm. PMC is an uncertainty visualization technique that studies how uncertainty in data affects level-set positions. However, the algorithm relies on expensive Monte Carlo (MC) sampling for the multivariate Gaussian uncertainty model because no closed-form solution exists for the integration of multivariate Gaussian. In this work, we propose the eigenvalue decomposition and adaptive probability model techniques that reduce the amount of MC sampling in the original PMC algorithm and hence speed up the computations. Our proposed methods produce results that show negligible differences compared with the original PMC algorithm demonstrated through metrics, including root mean squared error, maximum error, and difference images. We demonstrate the performance and accuracy evaluations of our data-driven methods through experiments on synthetic and real datasets.

Athawale, Tushar↗
DOE OSTI2019

Thermonuclear fusion rates for tritium + deuterium using Bayesian methods

The 3 H(d, n) 4 He reaction has a large low-energy cross section and will likely be utilized in future commercial fusion reactors. This reaction also takes place during Big Bang nucleosynthesis. Studies of both scenarios require accurate and precise fusion rates. To this end, we implement a one-level, two-channel R-matrix approximation into a Bayesian model. Our main goals are to predict reliable astrophysical S-factors and to estimate R-matrix parameters using the Bayesian approach. All relevant parameters are sampled in our study, including the channel radii, boundary condition parameters, and data set normalization factors. In addition, we take uncertainties in both measured bombarding energies and S-factors rigorously into account. Thermonuclear rates and reactivities of the 3 H(d, n) 4 He reaction are derived by numerically integrating the Bayesian S-factor samples. The present reaction rate uncertainties at temperatures between 1.0 MK and 1.0 GK are in the range of 0.2% to 0.6%. Our reaction rates differ from previous results by 2.9% near 1.0 GK.Our reactivities are smaller than previous results, with a maximum deviation of 2.9% near a thermal energy of 4 keV. The present rate or reactivity uncertainties are more reliable compared to previous studies that did not include the channel radii, boundary condition parameters, and data set normalization factors in the fitting. Finally, we investigate previous claims of electron screening effects in the published 3 H(d, n) 4 He data. No such effects are evident and only an upper limit for the electron screening potential can be obtained.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1996

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.

Vanek, Petr↗
DOE OSTI2023

Complete collision data set for electrons scattering on molecular hydrogen and its isotopologues: IV. Vibrationally-resolved ionization of the ground and excited electronic states

Here, we present a comprehensive set of vibrationally-resolved cross sections for electron-impact ionization of molecular hydrogen and its isotopologues (H 2 , D 2 , T 2 , HD, HT, and DT) in both the ground and excited electronic states. We apply the adiabatic-nuclei molecular convergent close-coupling (MCCC) method to calculate cross sections from threshold to 1000 eV for ionization of the ground and excited vibrational levels of the X 1 Σ$^{+}_{g}$, B 1 Σ$^{+}_{u}$, C 1 π u , EF 1 Σ$^{+}_{g}$, a 3 Σ$^{+}_{g}$, and c 3 π u electronic states, representing all states with united-atoms-limit principle quantum number n=1–2. The cross sections are presented in graphical form and provided as both numerical values and analytic fit functions in supplementary data files. The data can also be downloaded from the MCCC database at mccc-db.org.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2018

Modelling wind power spatial-temporal correlation in multi-interval optimal power flow: A sparse correlation matrix approach

The significantly increasing deployment of wind power necessitates that system operation considers the spatial-temporal correlation of power forecast from different wind power plants. How to model this spatial-temporal correlation in the actual system dispatch is challenging. In this paper, a sparse correlation matrix is utilized to efficiently model the spatial-temporal correlation of wind power forecast in the generation dispatch model. A novel, adjustable, and distributionally-robust chance-constrained multi-interval optimal power flow (ADRCC-MIOPF) model is proposed to obtain reliable economic dispatch (ED) solutions. The spatial-temporal correlation of wind power plants power forecasts is represented by the sparse correlation covariance matrix obtained from historical, time series wind power forecast error data. The chance constraints in the ADRCC-MIOPF model are transformed into a set of second-order-cone (SOC) constraints in which an adjustable coefficient in the chance constraints controls the robustness of the ADRCC-MIOPF model to the wind power forecast errors distribution. Case studies performed on the PJM 5-bus system and IEEE 118-bus system verify the proposed method to improve the system security and reduce the cost especially under the high wind penetration levels. All the cases can be solved within several minutes for both the small and large cases which validates the efficiency of the proposed sparse matrix model. In addition, considering the spatial-temporal correlation of wind power forecast and the distributional robustness of wind power forecast error leads to a more reliable economic dispatch with lower system violations.

17 WIND ENERGY↗
DOE OSTI2018

Two-dimensional simulations of mixing in classical novae: The effect of white dwarf composition and mass

Context. Classical novae are explosive phenomena that take place in stellar binary systems. They are powered by mass transfer from a low-mass main sequence star onto either a CO or ONe white dwarf. The material accumulates for 104–105 yr until ignition under degenerate conditions, resulting in a thermonuclear runaway. The nuclear energy released produces peak temperatures of ~0.1–0.4 GK. During these events, 10 –7 –10 –3 M ⊙ enriched in intermediate-mass elements, with respect to solar abundances, are ejected into the interstellar medium. Yet, the origin of the large metallicity enhancements and the inhomogeneous distribution of chemical species observed in high-resolution spectra of ejected nova shells is not fully understood. Aims. Recent multidimensional simulations have demonstrated that Kelvin-Helmholtz instabilities that operate at the core-envelope interface can naturally produce self-enrichment of the accreted envelope with material from the underlying white dwarf at levels that agree with observations. Yet, such multidimensional simulations have been performed for a small number of cases and much of the parameter space remains unexplored. Methods. We explored the dredge-up, driven by Kelvin-Helmholtz instabilities, for white dwarf masses in the range 0.8–1.25 M ⊙ and different core compositions, that is, CO-rich and ONe-rich substrates. We present a set of five numerical simulations performed in two dimensions aimed at analyzing the possible impact of the white dwarf mass, and composition, on the metallicity enhancement and explosion characteristics. Results. At the time we stop the simulations, we observe greater mixing (~30% higher when measured in the same conditions) and more energetic outbursts for ONe-rich substrates than for CO-rich substrates and more massive white dwarfs.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Electronic spectrum of non-tetrahedral acceptors in CdTe:Cl and CdTe:Bi,Cl single crystals

The electronic spectra of complex acceptors in compensated CdTe:Cl, CdTe:Ag,Cl, and CdTe:Bi,Cl single crystals are studied using low-temperature photoluminescence (PL) measurements under both nonresonant and resonant excitation of distant donor–acceptor pairs (DAP). The wavelength modulation of the excitation source combined with the analysis of the differential PL signal is used to enhance narrow spectral features obscured because of inhomogeneous line broadening and/or excitation transfer for selectively excited DAPs. For the well-known tetrahedral (T{sub D}) Ag{sub Cd} acceptor, the energies of four excited states are measured, and the values obtained are shown to be in perfect agreement with the previous data. Moreover, splitting between the 2P{sub 3/2} (Γ{sub 8}) and 2S{sub 3/2} (Γ{sub 8}) states is clearly observed for Ag{sub Cd} centers located at a short distance (5–7 nm) from a hydrogen-like donor (Cl{sub Te}). This splitting results from the reduction of the T{sub D} symmetry taking place when the acceptor is a member of a donor–acceptor pair. For the Cl-related complex acceptor with an activation energy of ∼121 meV (A-center), the energies of eight excited states are measured. It is shown that this defect produces low-symmetry central-cell correction responsible for the strong splitting of S-like T{sub D} shells. The energy spectrum of the Bi-related shallow acceptor with an activation energy of ∼36 meV is measured as well. The spectrum obtained differs drastically from the hydrogen-like set of levels, which indicates the existence of repulsive low-symmetry perturbation of the hydrogen-like Coulomb potential. It is also shown that the spectra of selectively excited PL recorded for a macroscopic ensemble of distant donor–acceptor pairs allow one to detect the low symmetry of acceptors of a given type caused by their complex nature or by the Jahn–Teller distortion. This method does not require any additional (external) field and is applicable to acceptors in diverse zinc-blende compound semiconductors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

A New Method for Generating Probability Tables in the Unresolved Resonance Region

One new method for constructing probability tables in the unresolved resonance region (URR) has been developed. This new methodology is an extensive modification of the single-level Breit-Wigner (SLBW) pseudo-resonance pair sequence method commonly used to generate probability tables in the URR. The new method uses a Monte Carlo process to generate many pseudo-resonance sequences by first sampling the average resonance parameter data in the URR and then converting the sampled resonance parameters to the more robust R-matrix limited (RML) format. Furthermore, for each sampled set of pseudo-resonance sequences, the temperature-dependent cross sections are reconstructed on a small grid around the energy of reference using the Reich-Moore formalism and the Leal-Hwang Doppler broadening methodology. We then use the effective cross sections calculated at the energies of reference to construct probability tables in the URR. The RML cross-section reconstruction algorithm has been rigorously tested for a variety of isotopes, including 16 O, 19 F, 35 Cl, 56 Fe, 63 Cu, and 65 Cu. The new URR method also produced normalized cross-section factor probability tables for 238 U that were found to be in agreement with current standards. The modified 238 U probability tables were shown to produce results in excellent agreement with several standard benchmarks, including the IEU-MET-FAST-007 (BIG TEN), IEU-MET-FAST-003, and IEU-COMP-FAST-004 benchmarks.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2017

Erratum: First-Principles Approach to Calculating Energy Level Alignment at Aqueous Semiconductor Interfaces [Phys. Rev. Lett. 113, 176802 (2014)]

We note, while extending our approach in our Letter to calculate energy level alignment at aqueous TiO 2 interfaces, we discovered two problems with our application of the GW methodology. Briefly, in our approach, we used a DFT + U approach for the underlying structure determination and as an input to the GW calculations. Then, GW calculations were carried out at the full-frequency spectrum-self-consistent level. All calculations were performed with the projector augmented wave (PAW) method as it is implemented in VASP. The specific semiconductors considered were GaN and ZnO. The first problem was caused by improper handling of the U parameter when initializing the follow-up GW calculations, leading to a final calculated band gap for GaN and ZnO that was too large. Second, we used an older set of pseudopotentials for the PAW method. In a paper that was published while our original manuscript was under review, the impact of the completeness of the partial waves representing empty states in the PAW method was analyzed, showing significant issues with the convergence of the GW results for certain materials, particularly ZnO. Specifically, those issues included use of PAW pseudopotentials that deviate too much from norm conservation, an unduly restricted basis set size that constrains the number of empty states available, and use of a small cutoff for the dielectric matrix size that can result in a false convergence with respect to the number of empty states included. Our GW calculations for GaN and ZnO were affected by these problems.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Transmission Interface Limits for High-Spatial Resolution Capacity Expansion Modeling

Large-scale capacity expansion models typically rely on estimates of the power transfer limits between modeled zones. Accurate estimation of these interface transfer limits (ITLs) requires modeling the underlying transmission network. Here we expand on a maximum flow optimization method that uses linearized power flow to estimate transfer limits. We apply this method to a data set of the U.S. transmission network to estimate ITLs between U.S. counties. By calculating ITLs using different subsets of the network, we evaluate how the size of the network used in the estimation affects the results. The results show diminishing returns to ITL accuracy after six hops, suggesting that a network subset can reasonably be used to approximate ITLs. The county-level estimates produced in this study will support more spatially resolved capacity expansion modeling and will help inform policy making at local and national levels.

capacity planning↗
DOE OSTI2021

A Bilevel Voltage Regulation Operation for Distribution Systems With Self-Operated Microgrids

The emerging of microgrids in distribution systems has significantly enhanced the resilience of power grids. However, the operators of a distribution system and microgrids therein can be different and have accessibility to different devices. To model the operation of such a grid, this work proposes a bilevel formulation and probes into the voltage regulation operation, considering the interaction between different systems. The proposed bilevel formulation considers the cooperation of active energy resources (AER), transformer tap-changers, and capacitor banks that are controlled by different operators. To facilitate the solution time of the target bilevel optimization, the lower-level problems with different objectives are modeled using deep neural networks (DNNs) which are then converted into a set of constraints. Hence, the bilevel problem can be reformed to a single-level problem. Lastly, the proposed solution procedures are validated using a customized joint system constructed by the IEEE 123-bus system and a real distribution system in Iowa. According to the numerical validation results, the solution time of the proposed nonlinear activation function based DNN model is 69 times faster than other methods in solving voltage regulation with a bilevel structure.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Data‐driven identification of environmental variables influencing phenotypic plasticity to facilitate breeding for future climates

Summary Phenotypic plasticity describes a genotype's ability to produce different phenotypes in response to different environments. Breeding crops that exhibit appropriate levels of plasticity for future climates will be crucial to meeting global demand, but knowledge of the critical environmental factors is limited to a handful of well‐studied major crops. Using 727 maize ( Zea mays L.) hybrids phenotyped for grain yield in 45 environments, we investigated the ability of a genetic algorithm and two other methods to identify environmental determinants of grain yield from a large set of candidate environmental variables constructed using minimal assumptions. The genetic algorithm identified pre‐ and postanthesis maximum temperature, mid‐season solar radiation, and whole season net evapotranspiration as the four most important variables from a candidate set of 9150. Importantly, these four variables are supported by previous literature. After calculating reaction norms for each environmental variable, candidate genes were identified and gene annotations investigated to demonstrate how this method can generate insights into phenotypic plasticity. The genetic algorithm successfully identified known environmental determinants of hybrid maize grain yield. This demonstrates that the methodology could be applied to other less well‐studied phenotypes and crops to improve understanding of phenotypic plasticity and facilitate breeding crops for future climates.

Kusmec, Aaron↗
DOE OSTI2015

Algorithms for propagating uncertainty across heterogeneous domains

We address an important research area in stochastic multi-scale modeling, namely the propagation of uncertainty across heterogeneous domains characterized by partially correlated processes with vastly different correlation lengths. This class of problems arise very often when computing stochastic PDEs and particle models with stochastic/stochastic domain interaction but also with stochastic/deterministic coupling. The domains may be fully embedded, adjacent or partially overlapping. The fundamental open question we address is the construction of proper transmission boundary conditions that preserve global statistical properties of the solution across different subdomains. Often, the codes that model different parts of the domains are black-box and hence a domain decomposition technique is required. No rigorous theory or even effective empirical algorithms have yet been developed for this purpose, although interfaces defined in terms of functionals of random fields (e.g., multi-point cumulants) can overcome the computationally prohibitive problem of preserving sample-path continuity across domains. The key idea of the different methods we propose relies on combining local reduced-order representations of random fields with multi-level domain decomposition. Specifically, we propose two new algorithms: The first one enforces the continuity of the conditional mean and variance of the solution across adjacent subdomains by using Schwarz iterations. The second algorithm is based on PDE-constrained multi-objective optimization, and it allows us to set more general interface conditions. The effectiveness of these new algorithms is demonstrated in numerical examples involving elliptic problems with random diffusion coefficients, stochastically advected scalar fields, and nonlinear advection-reaction problems with random reaction rates.

Cho, Heyrim↗
DOE OSTI2023

Study on Electrostatic Separation of Quinoline Insolubles from Coal Tar Pitch

The feasibility of electrical separation in the removal of quinoline insoluble (QI) particles from coal tar pitch (CTP) was experimentally investigated. QI particle involvement prohibits the effective fabrication of high-quality value-added products, such as carbon fibers, from CTP. A substantial and sustainable CTP market exists around the world; therefore, a strong incentive exists to develop a viable technical approach to effectively and economically remove QI particles from CTP. The electrical separation method shows promise to achieve this technical goal (Cao et al., 2012). Even with the given setup (wire- cylinder adapted), critical issues remain to be addressed for this method to be applicable to the CTP: (1) identifying the wash oil used in the original QI separation, (2) understanding the mechanism of QI separation, (3) characterizing the deposit, and (4) identifying the QI. This study integrated a set of experimental and analysis techniques into the electrical separation tests to address the aforementioned issues. Key findings are as follows: 1. Electric current responses: • The electric current level of the CTP–wash oil solution reported by Cao et al. (2012) can be attained by using a mixture of 25% quinoline and 75% toluene for the CTPs examined in this study under the same electrical load condition. • The electric current tends to decrease during the test period because of the decreasing number of charged particles. • The reversed field corresponded to the configuration of electrostatic precipitation for positive corona discharge. The high electric field can result in dielectric breakdown and lead to an abrupt current surge. 2 Deposit response and solvent candidates: • Deposit of particles in CTP mixture can be effectively implemented in a wire-cylinder configuration as proposed and examined in this study. • Deposit depends on the solvents. Among the solvents tested, two- and three-part solvents that included quinoline and ethanol (i.e., 25% quinoline and 75% toluene; 50% quinoline and 50% toluene; and 33% wash oil, 33% BTX, and 34% ethanol) produced the highest deposit weight. • The deposit process examined in this study is derived from the charged particles. An appreciable relation exists between deposit weight and electric charge. 3. QI removal efficiency (RE): • The RE of the electrostatic separation can reach as high as 76.2% for Carbores, and 61.6% for Koppers. • The RE can be further enhanced if the field level increases from 0.16 kV/mm that was used in the current study to 0.23 kV/mm, according to the relation established between the RE and the electric field. 4. Testing of the modeling system: • The motion of particles is originally driven by the charge-based electric force in the cases tested. The solid particle separation mechanism is similar to that of QI deposition in CTP. • The mechanical movement of solid particles can be strongly affected by the gravitational and viscous forces in the electric field. 5. EDS analysis and chemical compositions: • The main chemical composition of deposit QI matches that of as-prepared CTP QI. The deposit QI is derived from the same group of the CTP QI. • In addition to carbon, the QI contains oxygen, sodium, aluminum, silicon, iron, and sulfur. The work for the near future is also discussed.

01 COAL, LIGNITE, AND PEAT↗
NASA NTRS1982

On the linearity of the SWP camera of the international ultraviolet explorer /IUE/ - A correction algorithm

An algorithm is presented for correcting IUE low resolution spectral images obtained with the SWP camera for some of the non-linearity effects reported by Bohlin et al. (1980). The non-linearity problem, which affects SWP images processed at Goddard Space Flight Center in the period May 22, 1978 to July 7, 1979 and at VILSPA in the period June 14, 1978 to August 6, 1979, was essentially due to the use of an Intensity Transfer Function (ITF) that erroneously included a blank image in the 20 percent exposure level. The correction algorithm described here was adopted by the three IUE Agencies in November 1979 as being suitable for most IUE users. It has the advantages to be applicable to any kind of low resolution SWP spectra, to introduce errors which are usually less than the intrinsic photometric errors, and to be of simple application. The results obtained by applying the method to a representative set of spectra of both point and extended sources are reported. In addition, a new evaluation of linearity and reproducibility of the SWP spectral data is provided, based on the improved ITF.

Holm, A.↗
NASA NTRS2011

Technique for Solving Electrically Small to Large Structures for Broadband Applications

Fast iterative algorithms are often used for solving Method of Moments (MoM) systems, having a large number of unknowns, to determine current distribution and other parameters. The most commonly used fast methods include the fast multipole method (FMM), the precorrected fast Fourier transform (PFFT), and low-rank QR compression methods. These methods reduce the O(N) memory and time requirements to O(N log N) by compressing the dense MoM system so as to exploit the physics of Green s Function interactions. FFT-based techniques for solving such problems are efficient for spacefilling and uniform structures, but their performance substantially degrades for non-uniformly distributed structures due to the inherent need to employ a uniform global grid. FMM or QR techniques are better suited than FFT techniques; however, neither the FMM nor the QR technique can be used at all frequencies. This method has been developed to efficiently solve for a desired parameter of a system or device that can include both electrically large FMM elements, and electrically small QR elements. The system or device is set up as an oct-tree structure that can include regions of both the FMM type and the QR type. The system is enclosed with a cube at a 0- th level, splitting the cube at the 0-th level into eight child cubes. This forms cubes at a 1st level, recursively repeating the splitting process for cubes at successive levels until a desired number of levels is created. For each cube that is thus formed, neighbor lists and interaction lists are maintained. An iterative solver is then used to determine a first matrix vector product for any electrically large elements as well as a second matrix vector product for any electrically small elements that are included in the structure. These matrix vector products for the electrically large and small elements are combined, and a net delta for a combination of the matrix vector products is determined. The iteration continues until a net delta is obtained that is within the predefined limits. The matrix vector products that were last obtained are used to solve for the desired parameter. The solution for the desired parameter is then presented to a user in a tangible form; for example, on a display.

Jandhyala, Vikram↗