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NASA NTRS1999

Space-Time Conservation Element and Solution Element Method Being Developed

The engineering research and design requirements of today pose great computer-simulation challenges to engineers and scientists who are called on to analyze phenomena in continuum mechanics. The future will bring even more daunting challenges, when increasingly complex phenomena must be analyzed with increased accuracy. Traditionally used numerical simulation methods have evolved to their present state by repeated incremental extensions to broaden their scope. They are reaching the limits of their applicability and will need to be radically revised, at the very least, to meet future simulation challenges. At the NASA Lewis Research Center, researchers have been developing a new numerical framework for solving conservation laws in continuum mechanics, namely, the Space-Time Conservation Element and Solution Element Method, or the CE/SE method. This method has been built from fundamentals and is not a modification of any previously existing method. It has been designed with generality, simplicity, robustness, and accuracy as cornerstones. The CE/SE method has thus far been applied in the fields of computational fluid dynamics, computational aeroacoustics, and computational electromagnetics. Computer programs based on the CE/SE method have been developed for calculating flows in one, two, and three spatial dimensions. Results have been obtained for numerous problems and phenomena, including various shock-tube problems, ZND detonation waves, an implosion and explosion problem, shocks over a forward-facing step, a blast wave discharging from a nozzle, various acoustic waves, and shock/acoustic-wave interactions. The method can clearly resolve shock/acoustic-wave interactions, wherein the difference of the magnitude between the acoustic wave and shock could be up to six orders. In two-dimensional flows, the reflected shock is as crisp as the leading shock. CE/SE schemes are currently being used for advanced applications to jet and fan noise prediction and to chemically reacting flows.

Chang, Sin-Chung↗

NASA NTRS2007

A Comparison of Two Balance Calibration Model Building Methods

Simulated strain-gage balance calibration data is used to compare the accuracy of two balance calibration model building methods for different noise environments and calibration experiment designs. The first building method obtains a math model for the analysis of balance calibration data after applying a candidate math model search algorithm to the calibration data set. The second building method uses stepwise regression analysis in order to construct a model for the analysis. Four balance calibration data sets were simulated in order to compare the accuracy of the two math model building methods. The simulated data sets were prepared using the traditional One Factor At a Time (OFAT) technique and the Modern Design of Experiments (MDOE) approach. Random and systematic errors were introduced in the simulated calibration data sets in order to study their influence on the math model building methods. Residuals of the fitted calibration responses and other statistical metrics were compared in order to evaluate the calibration models developed with different combinations of noise environment, experiment design, and model building method. Overall, predicted math models and residuals of both math model building methods show very good agreement. Significant differences in model quality were attributable to noise environment, experiment design, and their interaction. Generally, the addition of systematic error significantly degraded the quality of calibration models developed from OFAT data by either method, but MDOE experiment designs were more robust with respect to the introduction of a systematic component of the unexplained variance.

DeLoach, Richard↗

NASA NTRS2011

A Method of Strengthening Composite/Metal Joints

The term tape setback method denotes a method of designing and fabricating bonded joints between (1) box beams or other structural members made of laminated composite (matrix/ fiber) materials and (2) metal end fittings used to fasten these structural members to other structural members. The basic idea of the tape setback method is to mask the bonded interface between the metallic end fitting and composite member such that the bond does not extend out to the free edges of the composite member. The purpose served by the tape setback method is to strengthen the joints by decoupling stress concentrations from edge defects, which can cause premature failures. A related prior method that serves a similar purpose, involving the use of tapered adherends at the joints, can be too difficult and costly to be acceptable in some applications. The tape setback method offers an easier, less costly alternative. The structural members to which the method was originally applied were box beams in the form of composite tubes having flat faces with rounded corners. The end fittings were plugs made of a low-thermal- expansion nickel/iron alloy (see figure). In computational-simulation studies of tensile and compressive loading of members without tape setback, stresses were found to be concentrated at the free end edges of the composite tubes, and inspection of members that had been subjected to real tension and compression tests showed that cracks started at the free end edges. As applied to these members, the tape setback method makes them less vulnerable to initiation of failure at edge defects produced during fabrication. In real tension tests of comparable members without and with tape setback, the average mean tensile strength of the members with tape setback was found to be 1.9 times that of the members without tape setback.

Polis, Daniel L.↗

NASA NTRS2012

Method for Pre-Conditioning a Measured Surface Height Map for Model Validation

This software allows one to up-sample or down-sample a measured surface map for model validation, not only without introducing any re-sampling errors, but also eliminating the existing measurement noise and measurement errors. Because the re-sampling of a surface map is accomplished based on the analytical expressions of Zernike-polynomials and a power spectral density model, such re-sampling does not introduce any aliasing and interpolation errors as is done by the conventional interpolation and FFT-based (fast-Fourier-transform-based) spatial-filtering method. Also, this new method automatically eliminates the measurement noise and other measurement errors such as artificial discontinuity. The developmental cycle of an optical system, such as a space telescope, includes, but is not limited to, the following two steps: (1) deriving requirements or specs on the optical quality of individual optics before they are fabricated through optical modeling and simulations, and (2) validating the optical model using the measured surface height maps after all optics are fabricated. There are a number of computational issues related to model validation, one of which is the "pre-conditioning" or pre-processing of the measured surface maps before using them in a model validation software tool. This software addresses the following issues: (1) up- or down-sampling a measured surface map to match it with the gridded data format of a model validation tool, and (2) eliminating the surface measurement noise or measurement errors such that the resulted surface height map is continuous or smoothly-varying. So far, the preferred method used for re-sampling a surface map is two-dimensional interpolation. The main problem of this method is that the same pixel can take different values when the method of interpolation is changed among the different methods such as the "nearest," "linear," "cubic," and "spline" fitting in Matlab. The conventional, FFT-based spatial filtering method used to eliminate the surface measurement noise or measurement errors can also suffer from aliasing effects. During re-sampling of a surface map, this software preserves the low spatial-frequency characteristic of a given surface map through the use of Zernike-polynomial fit coefficients, and maintains mid- and high-spatial-frequency characteristics of the given surface map by the use of a PSD model derived from the two-dimensional PSD data of the mid- and high-spatial-frequency components of the original surface map. Because this new method creates the new surface map in the desired sampling format from analytical expressions only, it does not encounter any aliasing effects and does not cause any discontinuity in the resultant surface map.

Sidick, Erkin↗

NASA NTRS2012

CREME96 and Related Error Rate Prediction Methods

Predicting the rate of occurrence of single event effects (SEEs) in space requires knowledge of the radiation environment and the response of electronic devices to that environment. Several analytical models have been developed over the past 36 years to predict SEE rates. The first error rate calculations were performed by Binder, Smith and Holman. Bradford and Pickel and Blandford, in their CRIER (Cosmic-Ray-Induced-Error-Rate) analysis code introduced the basic Rectangular ParallelePiped (RPP) method for error rate calculations. For the radiation environment at the part, both made use of the Cosmic Ray LET (Linear Energy Transfer) spectra calculated by Heinrich for various absorber Depths. A more detailed model for the space radiation environment within spacecraft was developed by Adams and co-workers. This model, together with a reformulation of the RPP method published by Pickel and Blandford, was used to create the CR ME (Cosmic Ray Effects on Micro-Electronics) code. About the same time Shapiro wrote the CRUP (Cosmic Ray Upset Program) based on the RPP method published by Bradford. It was the first code to specifically take into account charge collection from outside the depletion region due to deformation of the electric field caused by the incident cosmic ray. Other early rate prediction methods and codes include the Single Event Figure of Merit, NOVICE, the Space Radiation code and the effective flux method of Binder which is the basis of the SEFA (Scott Effective Flux Approximation) model. By the early 1990s it was becoming clear that CREME and the other early models needed Revision. This revision, CREME96, was completed and released as a WWW-based tool, one of the first of its kind. The revisions in CREME96 included improved environmental models and improved models for calculating single event effects. The need for a revision of CREME also stimulated the development of the CHIME (CRRES/SPACERAD Heavy Ion Model of the Environment) and MACREE (Modeling and Analysis of Cosmic Ray Effects in Electronics). The Single Event Figure of Merit method was also revised to use the solar minimum galactic cosmic ray spectrum and extended to circular orbits down to 200 km at any inclination. More recently a series of commercial codes was developed by TRAD (Test & Radiations) which includes the OMERE code which calculates single event effects. There are other error rate prediction methods which use Monte Carlo techniques. In this chapter the analytic methods for estimating the environment within spacecraft will be discussed.

Adams, James H., Jr.↗

NASA NTRS2013

Method for Reduction of Silver Biocide Plating on Metal Surfaces

Silver ions in aqueous solutions (0.05 to 1 ppm) are used for microbial control in water systems. The silver ions remain in solution when stored in plastic containers, but the concentration rapidly decreases to non-biocidal levels when stored in metal containers. The silver deposits onto the surface and is reduced to non-biocidal silver metal when it contacts less noble metal surfaces, including stainless steel, titanium, and nickel-based alloys. Five methods of treatment of contact metal surfaces to deter silver deposition and reduction are proposed: (1) High-temperature oxidation of the metal surface; (2) High-concentration silver solution pre-treatment; (3) Silver plating; (4) Teflon coat by vapor deposition (titanium only); and (5) A combination of methods (1) and (2), which proved to be the best method for the nickel-based alloy application. The mechanism associated with surface treatments (1), (2), and (5) is thought to be the development of a less active oxide layer that deters ionic silver deposition. Mechanism (3) is an attempt to develop an equilibrium ionic silver concentration via dissolution of metallic silver. Mechanism (4) provides a non-reactive barrier to deter ionic silver plating. Development testing has shown that ionic silver in aqueous solution was maintained at essentially the same level of addition (0.4 ppm) for up to 15 months with method (5) (a combination of methods (1) and (2)), before the test was discontinued for nickel-based alloys. Method (1) resulted in the maintenance of a biocidal level (approximately 0.05 ppm) for up to 10 months before that test was discontinued for nickel-based alloys. Methods (1) and (2) used separately were able to maintain ionic silver in aqueous solution at essentially the same level of addition (0.4 ppm) for up to 10 months before the test was discontinued for stainless steel alloys. Method (3) was only utilized for titanium alloys, and was successful at maintaining ionic silver in aqueous solution at essentially the same level of addition (0.4 ppm) for up to 10 months before the test was discontinued for simple flat geometries, but not for geometries that are difficult to Teflon coat.

Steele, John↗

NASA NTRS2012

Test of Shi et al. Method to Infer the Magnetic Reconnection Geometry from Spacecraft Data: MHD Simulation with Guide Field and Antiparallel Kinetic Simulation

When analyzing data from an array of spacecraft (such as Cluster or MMS) crossing a site of magnetic reconnection, it is desirable to be able to accurately determine the orientation of the reconnection site. If the reconnection is quasi-two dimensional, there are three key directions, the direction of maximum inhomogeneity (the direction across the reconnection site), the direction of the reconnecting component of the magnetic field, and the direction of rough invariance (the "out of plane" direction). Using simulated spacecraft observations of magnetic reconnection in the geomagnetic tail, we extend our previous tests of the direction-finding method developed by Shi et al. (2005) and the method to determine the structure velocity relative to the spacecraft Vstr. These methods require data from four proximate spacecraft. We add artificial noise and calibration errors to the simulation fields, and then use the perturbed gradient of the magnetic field B and perturbed time derivative dB/dt, as described by Denton et al. (2010). Three new simulations are examined: a weakly three-dimensional, i.e., quasi-two-dimensional, MHD simulation without a guide field, a quasi-two-dimensional MHD simulation with a guide field, and a two-dimensional full dynamics kinetic simulation with inherent noise so that the apparent minimum gradient was not exactly zero, even without added artificial errors. We also examined variations of the spacecraft trajectory for the kinetic simulation. The accuracy of the directions found varied depending on the simulation and spacecraft trajectory, but all the directions could be found within about 10 for all cases. Various aspects of the method were examined, including how to choose averaging intervals and the best intervals for determining the directions and velocity. For the kinetic simulation, we also investigated in detail how the errors in the inferred gradient directions from the unmodified Shi et al. method (using the unperturbed gradient) depended on the amplitude of the calibration errors. For an accuracy of 3 for the maximum gradient direction, the calibration errors could be as large as 3% of reconnection magnetic field, while for the same accuracy for the minimum gradient direction, the calibration errors could only be as large as 0.03% of the reconnection magnetic field. These results suggest that the maximum gradient direction can normally be determined by the unmodified Shi et al. method, while the modified method or some other method must be used to accurately determine the minimum gradient direction. The structure velocity was found with magnitude accurate to 2% and direction accurate to within 5%.

Denton, R.↗

NASA NTRS2015

Methods for Stem Cell Production and Therapy

The present invention relates to methods for rapidly expanding a stem cell population with or without culture supplements in simulated microgravity conditions. The present invention relates to methods for rapidly increasing the life span of stem cell populations without culture supplements in simulated microgravity conditions. The present invention also relates to methods for increasing the sensitivity of cancer stem cells to chemotherapeutic agents by culturing the cancer stem cells under microgravity conditions and in the presence of omega-3 fatty acids. The methods of the present invention can also be used to proliferate cancer cells by culturing them in the presence of omega-3 fatty acids. The present invention also relates to methods for testing the sensitivity of cancer cells and cancer stem cells to chemotherapeutic agents by culturing the cancer cells and cancer stem cells under microgravity conditions. The methods of the present invention can also be used to produce tissue for use in transplantation by culturing stem cells or cancer stem cells under microgravity conditions. The methods of the present invention can also be used to produce cellular factors and growth factors by culturing stem cells or cancer stem cells under microgravity conditions. The methods of the present invention can also be used to produce cellular factors and growth factors to promote differentiation of cancer stem cells under microgravity conditions.

Claudio, Pier Paolo↗

NASA NTRS2015

De-Aliasing Through Over-Integration Applied to the Flux Reconstruction and Discontinuous Galerkin Methods

High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

High-Order↗

NASA NTRS2016

Modified Spectral Fatigue Methods for S-N Curves With MIL-HDBK-5J Coefficients

The rainflow method is used for counting fatigue cycles from a stress response time history, where the fatigue cycles are stress-reversals. The rainflow method allows the application of Palmgren-Miner's rule in order to assess the fatigue life of a structure subject to complex loading. The fatigue damage may also be calculated from a stress response power spectral density (PSD) using the semi-empirical Dirlik, Single Moment, Zhao-Baker and other spectral methods. These methods effectively assume that the PSD has a corresponding time history which is stationary with a normal distribution. This paper shows how the probability density function for rainflow stress cycles can be extracted from each of the spectral methods. This extraction allows for the application of the MIL-HDBK-5J fatigue coefficients in the cumulative damage summation. A numerical example is given in this paper for the stress response of a beam undergoing random base excitation, where the excitation is applied separately by a time history and by its corresponding PSD. The fatigue calculation is performed in the time domain, as well as in the frequency domain via the modified spectral methods. The result comparison shows that the modified spectral methods give comparable results to the time domain rainflow counting method.

Irvine, Tom↗

NASA NTRS2017

Formal Methods Tool Qualification

Formal methods tools have been shown to be effective at finding defects in safety-critical digital systems including avionics systems. The publication of DO-178C and the accompanying formal methods supplement DO-333 allows applicants to obtain certification credit for the use of formal methods without providing justification for them as an alternative method. This project conducted an extensive study of existing formal methods tools, identifying obstacles to their qualification and proposing mitigations for those obstacles. Further, it interprets the qualification guidance for existing formal methods tools and provides case study examples for open source tools. This project also investigates the feasibility of verifying formal methods tools by generating proof certificates which capture proof of the formal methods tool's claim, which can be checked by an independent, proof certificate checking tool. Finally, the project investigates the feasibility of qualifying this proof certificate checker, in the DO-330 framework, in lieu of qualifying the model checker itself.

Wagner, Lucas G.↗

NASA NTRS2018

Evaluation of Multiple Flow Constrained Area Capacity Setting Methods for Collaborative Trajectory Options Program

The purpose of this study was to compare flow constrained area (FCA) capacity setting methods for Collaborative Trajectory Options Program (CTOP) as they pertain to the Integrated Demand Management (IDM) concept. IDM uses flow balancing to manage air traffic across multiple FCAs with a common downstream constraint, as well as constraints at the respective FCA locations. FCA capacity rates can be set manually, but generating capacities for multiple, interdependent FCAs could potentially over-burden a user. A new enhancement to CTOP called the FCA Balance Algorithm (FBA) was developed at NASA Ames Research Center to improve the process of allocating capacity across multiple flow constrained segments in the airspace. The FBA evaluates the predicted demand and capacity across multiple FCAs and dynamically generates capacity settings for the FCAs that best meet capacity limits for all identified constraints. In a human-in-the-loop simulation study, both manual and automated capacity setting methods were evaluated in terms of their overall feasibility using measures of system performance, human performance, and qualitative feedback. Subject matter experts were asked to use three different methods to allocate capacity to three FCAs, either (1) by manually setting capacity for every 60-minute time window, (2) by manually setting capacity for every 15-minute time window, or (3) by using the FBA capability to automatically generate capacity settings. Results showed no significant differences in terms of overall system performance, indicated by similar ground delay and airport throughput numbers between methods. However, differences in individual strategies afforded by the manual methods allowed some participants to achieve system-wide delay that was much lower than the average. The FBA was the fastest method of capacity setting, and it received the lowest subjective rating scores on physical task load, mental task load, task difficulty and task complexity out of the three methods. Finally, participants explained through qualitative feedback that there were many benefits to using the FBA, such as ease of use, accuracy, and low risk of human input error. Participants did not experience the same limitations with the FBA that they did with the manual methods, such as reduced accuracy in the 60-minute manual condition, or high complexity in the 15-minute/manual condition. These results suggest that the FBA automation enhancement to CTOP maintains system performance while improving human performance. Therefore, the FBA could be introduced as a way to mitigate operator workload while planning a CTOP.

NextGen↗

NASA NTRS2015

Survey and Assessment of Crew Performance Evaluation Methods Applicable to Human Spacecraft Design

Space is an unforgiving environment where the actions of the crew play a critical role in their health and safety. Given the limited number of crewmembers typically onboard a spacecraft and the multitude of complex systems they must operate, the performance of each individual is of paramount importance. Spacecraft habitat layout and operations are two main drivers affecting crew performance efficiency. Having the capability to analyze and compare crew performance across various spacecraft configurations can help identify improvements early in the conceptual design process where changes are less costly to implement, ultimately reducing overall project costs and improving long-term operations of the system. Currently, there are few comprehensive methods readily available for evaluating crew integration within a spacecraft in the conceptual design phase. In order to address this shortcoming, the goal of this work was to analyze various specialized evaluation methods found in analogous industries that have potential application to human spacecraft design. A survey of more than 400 human performance evaluation methods was completed. Over twenty different attributes were identified for each method and a variety of analyses were conducted to characterize and evaluate their potential use for assessing human spacecraft design options. The analysis revealed a particular deficiency of quantitative evaluation methods that are applicable early in the systems engineering design phase. It also identified five existing methods that could be supplemented to achieve the needs of an early design evaluation method. Additional discussion describes potential issues that must be overcome when developing a method specific for use in human spacecraft evaluations.

crew performance↗

NASA NTRS2018

Evaluation of Multiple Flow Constrained Area Capacity Setting Methods for Collaborative Trajectory Options Program

NextGen↗

NASA NTRS2018

High-precision Method for Cyclic Loading of Small-Animal Vertebrae to Assess Bone Quality

One potentially important bone quality characteristic is the response of bone to cyclic (repetitive) mechanical loading. In small animals, such as in rats and mice, cyclic loading experiments are particularly challenging to perform in a precise manner due to the small size of the bones and difficult-to-eliminate machine compliance. Addressing this issue, we developed a precise method for ex vivo cyclic compressive loading of isolated mouse vertebral bodies. The method has three key characteristics: 3D-printed support jigs for machining plano-parallel surfaces of the tiny vertebrae; pivotable loading platens to ensure uniform contact and loading of specimen surfaces; and specimen-specific micro-CT-based finite element analysis to measure stiffness to prescribe force levels that produce the same specified level of strain for all test specimens. To demonstrate utility, we measured fatigue life for three groups (n = 5-6 per group) of L5 vertebrae of C57BL/6J male mice, comparing our new method against two methods commonly used in the literature. We found reduced scatter of the mechanical behavior for this new method compared to the literature methods. In particular, for a controlled level of strain, the standard deviation of the measured fatigue life was up to 5-fold lower for the new method (F-ratio = 4.9; p < 0.01). The improved precision for this new method for biomechanical testing of small-animal vertebrae may help elucidate aspects of bone quality.

Bone quality↗

NASA NTRS2019

Image Processing Methods for Coronal Hole Segmentation, Matching, and Map Classification

The paper presents the results from a multi-year effort to develop and validate image processing methods forselecting the best physical models based on solar image observations. The approach consists of selecting the physical models based on their agreement with coronal holes extracted from the images. Ultimately, the goal is to use physical models to predict geomagnetic storms. We decompose the problem into three subproblems: (i) coronal hole segmentation based on physical constraints, (ii) matching clusters of coronal holes between different maps, and (iii) physical map classification. For segmenting coronal holes, we develop a multi-modal method that uses segmentation maps from three different methods to initialize a level-set method that evolves the initial coronal hole segmentation to the magnetic boundary. Then, we introducea new method based on Linear Programming for matchingclusters of coronal holes. The final matching is then performedusing Random Forests. The methods were carefully validatedusing consensus maps derived from multiple readers, manualclustering, manual map classification, and method validation for50 maps. The proposed multi-modal segmentation method significantly outperformed SegNet, U-net, Henney-Harvey, and FCNby providing accurate boundary detection. Overall, the methodgave a 95.5% map classification accuracy.

Jatla, Venkatesh↗

NASA NTRS2020

Evaluation of Tablet-Based Methods for Assessment of Contrast Sensitivity

Some astronauts have suffered degradation of vision during long-duration space flight, suffering from a condition that has come to be known as Spaceflight Associated Neuro-ocular Syndrome (SANS). While related morphological changes can be observed with imaging technologies such as optical coherence tomography (OCT), it may be useful to have a rapid method for functional vision assessment. In this paper, we compare three tablet-based methods for rapid assessment of contrast sensitivity. First, a relatively novel method developed expressly for touch screens, in which the subject ”swipes” a frequency/contrast sweep grating to indicate the boundary between visible and invisible patterns; second, a method-of-adjustment task in which the subject adjusts the contrast of a grating patch up and down to bracket the visual threshold; and third, a traditional temporal two-alternative forced choice (2AFC) task, in which the subject is presented with a near-threshold stimulus in one of two intervals, and must report the interval containing the stimulus. The swipe method shows variability comparable to the 2AFC method, and shows good agreement in estimates of the spatial frequency of peak sensitivity. The absolute sensitivity estimated with the swipe method is higher than that of the other methods, perhaps because subjects are biased to trace outside of the visible pattern region, or perhaps due to stimulus differences.

Mulligan, Jeffrey B.↗

NASA NTRS2019

Meshless Local Petrov-Galerkin (MLPG) Method with Orthogonal Polynomials for Euler-Bernoulli Beam Problems

In this paper, the feasibility of orthogonal polynomials in the meshless local Petrov Galerkin method (MLPG) method is studied. The orthogonal polynomials, Chebyshev and Legendre polynomials, are used in this MLPG method as trial functions. The test functions used were power functions with smooth derivatives at their ends. The performance of these methods is studied by applying these methods to Euler-Bernoulli beam problems. The MLPG-Galerkin and Legendre methods passed all the patch tests for simple beam problems. Next the formulations are tested on complex beam problems such as beams with partial loadings and continuous beam problems. Problems with load discontinuities and additional supports require special attention. Near discontinuities, judicious choice of number of nodes and nodal placements are needed to obtain accurate deflections, slopes, moments and shear forces. As polynomial functions are used, the large number of nodes can create a transformation matrix that is ill-conditioned, resulting in problems with the inversion of the matrix. The conditioning worsens as the number of nodes are increased beyond 20. Quadruple precision was needed for models to obtain accurate solutions. Even with quadruple precision the accuracy of the method suffers as the number of nodes is increased beyond 20. This appears to be a drawback of the MLPG-Chebyshev and MLPG-Legendre methods.

Raju, Ivatury S.↗