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A Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Reacting Hypersonic Flows

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced, is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this talk. Moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) [1],[2] are then presented for solving compressible flow problems with discontinuities based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. In the MDG-ICE formulation, both conservative quantities and grid geometry are considered as independent variables. A space-time DG formulation is used to solve the multi-material compressible Euler equations in the standard discontinuous solution space and the discrete grid geometry is solved using a variational formulation in a continuous space. A self-adaptive Levenberg-Marquardt method is utilized to solve the resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation. A number of numerical experiments for a variety of flow problems are conducted to assess the accuracy and performance of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to deliver the designed order of both h- and p-convergence even for discontinuous solutions, and detect all types of interfaces, via interface condition enforcement and satisfy, via grid movement, the compressible Euler equations and the associated interface condition.

Discontinuous Galekin↗

NASA NTRS2002

CAPRI: Using a Geometric Foundation for Computational Analysis and Design

CAPRI (Computational Analysis Programming Interface) is a software development tool intended to make computerized design, simulation and analysis faster and more efficient. The computational steps traditionally taken for most engineering analysis (Computational Fluid Dynamics (CFD), structural analysis, etc.) are: Surface Generation, usually by employing a Computer Aided Design (CAD) system; Grid Generation, preparing the volume for the simulation; Flow Solver, producing the results at the specified operational point; Post-processing Visualization, interactively attempting to understand the results. It should be noted that the structures problem is more tractable than CFD; there are fewer mesh topologies used and the grids are not as fine (this problem space does not have the length scaling issues of fluids). For CFD, these steps have worked well in the past for simple steady-state simulations at the expense of much user interaction. The data was transmitted between phases via files. In most cases, the output from a CAD system could go IGES files. The output from Grid Generators and Solvers do not really have standards though there are a couple of file formats that can be used for a subset of the gridding (i.e. PLOT3D) data formats and the upcoming CGNS). The user would have to patch up the data or translate from one format to another to move to the next step. Sometimes this could take days. Instead of the serial approach to analysis, CAPRI takes a geometry centric approach. CAPRI is a software building tool-kit that refers to two ideas: (1) A simplified, object-oriented, hierarchical view of a solid part integrating both geometry and topology definitions, and (2) programming access to this part or assembly and any attached data. The connection to the geometry is made through an Application Programming Interface (API) and not a file system.

Haimes, Robert↗

NASA NTRS2017

Development and Validation of The SMAP Enhanced Passive Soil Moisture Product

Since the beginning of its routine science operation in March 2015, the NASA SMAP observatory has been returning interference-mitigated brightness temperature observations at L-band (1.41 GHz) frequency from space. The resulting data enable frequent global mapping of soil moisture with a retrieval uncertainty below 0.040 cu m/cu m at a 36 km spatial scale. This paper describes the development and validation of an enhanced version of the current standard soil moisture product. Compared with the standard product that is posted on a 36 km grid, the new enhanced product is posted on a 9 km grid. Derived from the same time-ordered brightness temperature observations that feed the current standard passive soil moisture product, the enhanced passive soil moisture product leverages on the Backus-Gilbert optimal interpolation technique that more fully utilizes the additional information from the original radiometer observations to achieve global mapping of soil moisture with enhanced clarity. The resulting enhanced soil moisture product was assessed using long-term in situ soil moisture observations from core validation sites located in diverse biomes and was found to exhibit an average retrieval uncertainty below 0.040 cu m/cu m. As of December 2016, the enhanced soil moisture product has been made available to the public from the NASA Distributed Active Archive Center at the National Snow and Ice Data Center.

SMAP Enhanced↗

NASA NTRS2002

Multi-Zone Liquid Thrust Chamber Performance Code with Domain Decomposition for Parallel Processing

Computational Fluid Dynamics (CFD) has considerably evolved in the last decade. There are many computer programs that can perform computations on viscous internal or external flows with chemical reactions. CFD has become a commonly used tool in the design and analysis of gas turbines, ramjet combustors, turbo-machinery, inlet ducts, rocket engines, jet interaction, missile, and ramjet nozzles. One of the problems of interest to NASA has always been the performance prediction for rocket and air-breathing engines. Due to the complexity of flow in these engines it is necessary to resolve the flowfield into a fine mesh to capture quantities like turbulence and heat transfer. However, calculation on a high-resolution grid is associated with a prohibitively increasing computational time that can downgrade the value of the CFD for practical engineering calculations. The Liquid Thrust Chamber Performance (LTCP) code was developed for NASA/MSFC (Marshall Space Flight Center) to perform liquid rocket engine performance calculations. This code is a 2D/axisymmetric full Navier-Stokes (NS) solver with fully coupled finite rate chemistry and Eulerian treatment of liquid fuel and/or oxidizer droplets. One of the advantages of this code has been the resemblance of its input file to the JANNAF (Joint Army Navy NASA Air Force Interagency Propulsion Committee) standard TDK code, and its automatic grid generation for JANNAF defined combustion chamber wall geometry. These options minimize the learning effort for TDK users, and make the code a good candidate for performing engineering calculations. Although the LTCP code was developed for liquid rocket engines, it is a general-purpose code and has been used for solving many engineering problems. However, the single zone formulation of the LTCP has limited the code to be applicable to problems with complex geometry. Furthermore, the computational time becomes prohibitively large for high-resolution problems with chemistry, two-equation turbulence model, and two-phase flow. To overcome these limitations, the LTCP code is rewritten to include the multi-zone capability with domain decomposition that makes it suitable for parallel processing, i.e., enabling the code to run every zone or sub-domain on a separate processor. This can reduce the run time by a factor of 6 to 8, depending on the problem.

Homayun K. Navaz↗

NASA NTRS1998

Low Dissipative High Order Shock-Capturing Methods using Characteristic-Based Filters

An approach which closely maintains the non-dissipative nature of classical fourth or higher- order spatial differencing away from shock waves and steep gradient regions while being capable of accurately capturing discontinuities, steep gradient and fine scale turbulent structures in a stable and efficient manner is described. The approach is a generalization of the method of Gustafsson and Olsson and the artificial compression method (ACM) of Harten. Spatially non-dissipative fourth or higher-order compact and non-compact spatial differencings are used as the base schemes. Instead of applying a scalar filter as in Gustafsson and Olsson, an ACM like term is used to signal the appropriate amount of second or third-order TVD or ENO types of characteristic based numerical dissipation. This term acts as a characteristic filter to minimize numerical dissipation for the overall scheme. For time-accurate computations, time discretizations with low dissipation are used. Numerical experiments on 2-D vortical flows, vortex-shock interactions and compressible spatially and temporally evolving mixing layers showed that the proposed schemes have the desired property with only a 10% increase in operations count over standard second-order TVD schemes. Aside from the ability to accurately capture shock-turbulence interaction flows, this approach is also capable of accurately preserving vortex convection. Higher accuracy is achieved with fewer grid points when compared to that of standard second-order TVD or ENO schemes. To demonstrate the applicability of these schemes in sustaining turbulence where shock waves are absent, a simulation of 3-D compressible turbulent channel flow in a small domain is conducted.

Yee, H. C.↗

NASA NTRS1998

Low Dissipative High Order Shock-Capturing Methods Using Characteristic-Based Filters

An approach which closely maintains the non-dissipative nature of classical fourth or higher- order spatial differencing away from shock waves and steep gradient regions while being capable of accurately capturing discontinuities, steep gradient and fine scale turbulent structures in a stable and efficient manner is described. The approach is a generalization of the method of Gustafsson and Oisson and the artificial compression method (ACM) of Harten. Spatially non-dissipative fourth or higher-order compact and non-compact spatial differencings are used as the base schemes. Instead of applying a scalar filter as in Gustafsson and Olsson, an ACM like term is used to signal the appropriate amount of second or third-order TVD or ENO types of characteristic based numerical dissipation. This term acts as a characteristic filter to minimize numerical dissipation for the overall scheme. For time-accurate computations, time discretizations with low dissipation are used. Numerical experiments on 2-D vortical flows, vortex-shock interactions and compressible spatially and temporally evolving mixing layers showed that the proposed schemes have the desired property with only a 10% increase in operations count over standard second-order TVD schemes. Aside from the ability to accurately capture shock-turbulence interaction flows, this approach is also capable of accurately preserving vortex convection. Higher accuracy is achieved with fewer grid points when compared to that of standard second-order TVD or ENO schemes. To demonstrate the applicability of these schemes in sustaining turbulence where shock waves are absent, a simulation of 3-D compressible turbulent channel flow in a small domain is conducted.

Yee, H. C.↗

NASA NTRS2019

Utilization of Airborne Observations to Assess Model Parametrizations of Critical RH Profiles in the Arctic Ocean

Observational paucity is a reality in the Arctic Ocean. This is especially true for near-surface variables such as temperature, moisture, heat fluxes and BL clouds. As a result, modeling has become one of the major avenues for understanding current and future Arctic trends. Reanalyses are frequently used to force global ocean circulation and sea-ice models. But in northern high latitudes, model integrations and reanalyses are known to have large uncertainties in temperature and humidity profiles, and in boundary layer cloudiness. These are common sources of error in the surface radiative budget terms.An important way to diagnose these biases spatially and temporally is by using satellite remote sensing data. However, remotely-sensed observations also have large uncertainties, especially in near-surface temperature and relative humidity profiles.In situ observational studies are important in bridging our knowledge gap in regions such as the Arctic Ocean. Here, we utilize airborne and ship observations from the ARISE, ACME-IV, and ASCOS campaigns to construct critical relative humidity (RH) profiles over the Beaufort Sea. Such profiles are used as parameterization inputs in the NASA GOES global model to derive the total water condensate in a model grid-box, which determines the cloud fraction. Currently, the critical RH profiles are derived by global AIRS data, relaying mostly on mid-latitude regions, which are not necessarily relevant to the Arctic.We derive campaign-wide mean, standard deviation, and critical RH values, for grid size of 50x50 km and altitude bins between 50 to 400 m, covering both open-ocean and sea-ice covered regions. We compare profiles over open ocean and sea-ice, and look at correlations between the observed critical RH values and water condensate (by cloud number concentration) from observations versus the modeled ones. We then input our calculated values of minimal critical RH values into a set of GEOS single column model (SCM) simulations over the ARISE and ASCOS regions and compare the differences between the predicted values of cloud liquid water path (LWP), ice water path (IWP) and surface fluxes with the observed ones under the range of input parameterizations. Finally, we discuss the implications on surface radiative budget predictions in this region.

Segal-Rozenhaimer, Michal↗

DOE OSTI2024

On Compatible Transfer Operators in Nonsymmetric Algebraic Multigrid

The standard goal for an effective algebraic multigrid (AMG) algorithm is to develop relaxation and coarse-grid correction schemes that attenuate complementary error modes. In the nonsymmetric setting, coarse-grid correction Π will almost certainly be nonorthogonal (and divergent) in any known standard product, meaning ∥Π∥ > 1. This introduces a new consideration, that one wants coarse-grid correction to be as close to orthogonal as possible, in an appropriate norm. In addition, due to nonorthogonality, Π may actually amplify certain error modes that are in the range of interpolation. Relaxation must then not only be complementary to interpolation, but also rapidly eliminate any error amplified by the nonorthogonal correction, or the algorithm may diverge. Here this paper develops analytic formulae on how to construct “compatible” transfer operators in nonsymmetric AMG such that ∥Π∥ = 1 in some standard matrix-induced norm. Discussion is provided on different options for the norm in the nonsymmetric setting, the relation between “ideal” transfer operators in different norms, and insight into the convergence of nonsymmetric reduction-based AMG.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Low-Cost, Screen-Printed Silver Metal Complex Inks for Silicon Heterojunction Solar Cells

Screen printing using metal particle pastes, the current photovoltaic industry metallization standard, provides fast and reliable metal grids for silicon solar cells. Recently, metal complex or reactive metal inks are attracting research interest due to their significantly low cost and higher performance compared to traditional nanoparticle silver pastes. In this work, we demonstrate, for the first time, screen-printed high-efficiency silicon heterojunction solar cells metallized by silver metal complex inks on industrial G1-size (158.75 x 158.75 mm2) wafers. We demonstrate screen-printed Ag metal complex ink grid patterns with continuous fingers ~100-120 ..mu..m wide. The printed Ag grid is very thin (~1 ..mu..m), which is an order of magnitude thinner than the current ~20-30 ..mu..m fingers printed with low-temperature nanoparticle-based pastes. Double printing allows silicon heterojunction devices with efficiencies >20%. This is the highest efficiency so far, to our knowledge, of industrial solar cell precursors using this metallization technology. Simulation results suggested that increasing the thickness of the metal film does not significantly improve efficiency due to the dense, highly conductive films. So, a single print of ~1 ..mu..m finger would be enough to produce cells that perform similarly to a ~20 ..mu..m thick nanoparticle paste printed cells. Additionally, solar cells printed on G1 wafers with silver metal complex ink required more than 10 times less silver (~0.03 g) compared to those using silver/copper nanoparticle paste (~0.4 g of Ag). These results indicate that metal complex inks are a very promising replacement for silver nanoparticle pastes for industrial-scale metallization in an age of resource scarcity and high costs of noble metals.

14 SOLAR ENERGY↗

DOE OSTI2024

Impact of Electric Vehicle Charging Station Reliability, Resilience, and Location on Electric Vehicle Adoption

While the majority of electric vehicle (EV) charging events in the United States occur at home, issues with public charging stations are consistently found to be a top reason that potential EV buyers do not purchase an EV, demonstrating that both EVSE reliability and availability impacts EV adoption. This report explores multiple parameters that impact EVSE reliability and deployment, which in turn impact EV sales. These include extreme weather, codes and standards, region (urban vs. rural), and grid network type. Grid reliability was not found to impact EV adoption. The relationships between EV station reliability, station resilience, grid resilience, and EV adoption are largely outside the scope of the National Renewable Energy Laboratory's (NREL's) Automotive Deployment Options Projection Tool (ADOPT) and other vehicle adoption models, so the methodology of this report is varied. Section 2 sets the baseline for infrastructure reliability, user satisfaction, and maintenance practices. Section 3 explores the ways that electric vehicle supply equipment (EVSE) reliability impacts the relationship between EVSE and EV adoption. Section 4 shows how geographical categories such as urban, rural, large grid, off-grid, or microgrid can be helpful in EVSE deployment strategies, as well as how the relationship between EVSE and EV adoption differs among these categories. Section 5 investigates the impacts of grid reliability and infrastructure resilience on EV adoption. Finally, Section 6 reverses the perspective to examine the impact that EVs and EVSE have on grid resilience and reliability. As recent funding initiatives result in an expansion of public chargers across the United States, as well as an increase in the uptime of existing chargers, EV adoption will likely grow.

33 ADVANCED PROPULSION SYSTEMS↗

DOE OSTI2025

Bridging the Gap on Data and Analysis for Distribution System Planning: Information That Utilities Can Provide Regulators, State Energy Offices and Other Stakeholders

Electric utilities conduct planning annually to ensure their distribution system meets technical standards, policies, and regulations; addresses forecasted grid conditions; satisfies customer needs; and advances utility priorities. The plan identifies grid deficiencies, analyzes potential solutions, and prioritizes capital investments and other expenditures. About 20 U.S. states and jurisdictions require regulated utilities to file some type of distribution system plan with the public utility commission for review. Requirements for sharing distribution system data and analyses vary widely, from few specific requirements to a detailed list of information that must be provided. While utilities conduct extensive analysis to develop distribution system plans, in most jurisdictions regulators and stakeholders do not know what data are available and how the utility uses the data in planning and investing. This report aims to bridge the gap by increasing understanding of the types of data and analyses utilities employ to develop distribution system plans and how the information affects their decision-making. The report describes information that states and stakeholders can ask for related to 11 data categories: -Forecasting loads and distributed energy resources (DERs) -Scenario analysis -Worst-performing circuits -Asset management strategy -Hosting capacity analysis -Value of DERs -Grid needs assessment -Cost-effectiveness framework for investments -Distribution system investment strategy and implementation -Geotargeted programs -Non-wires alternatives procurements.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2025

Virtual Power Plants and Distributed Energy Resource Management Systems

Virtual Power Plants (VPPs) are aggregations of DERs that can balance electrical loads and provide utility-scale and utility-grade grid services like a traditional power plant. This presentation covers VPP definition, State-of-the-Art, Grid Architectures, Example VPP studies, VPP Standards, and VPP Roadmap.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1986

Prediction of monthly 700 mb heights using simulated medium-range numerical forecasts

Standardized 700 mb heights at 126 grid points over the Northern Hemisphere during 1948-80 were used to quantify the persistence of the heights from one month to the next (using various antecedent period lengths) and to determine predictive skill for calendar-month 700 mb heights using antecedent and simulated prognostic height information. The calculations showed that optimum period length and its corresponding average pattern correlation vary widely over the course of the year, being largest in summer and winter. The skill of models using simulated prog heights was found to be much superior to those using antecedent heights.

Harnack, R.↗

NASA NTRS2011

Transverse Pupil Shifts for Adaptive Optics Non-Common Path Calibration

A simple new way of obtaining absolute wavefront measurements with a laboratory Fizeau interferometer was recently devised. In that case, the observed wavefront map is the difference of two cavity surfaces, those of the mirror under test and of an unknown reference surface on the Fizeau s transmission flat. The absolute surface of each can be determined by applying standard wavefront reconstruction techniques to two grids of absolute surface height differences of the mirror under test, obtained from pairs of measurements made with slight transverse shifts in X and Y. Adaptive optics systems typically provide an actuated periscope between wavefront sensor (WFS) and commonmode optics, used for lateral registration of deformable mirror (DM) to WFS. This periscope permits independent adjustment of either pupil or focal spot incident on the WFS. It would be used to give the required lateral pupil motion between common and non-common segments, analogous to the lateral shifts of the two phase contributions in the lab Fizeau. The technique is based on a completely new approach to calibration of phase. It offers unusual flexibility with regard to the transverse spatial frequency scales probed, and will give results quite quickly, making use of no auxiliary equipment other than that built into the adaptive optics system. The new technique may be applied to provide novel calibration information about other optical systems in which the beam may be shifted transversely in a controlled way.

Bloemhof, Eric E.↗

NASA NTRS2018

File Specification for M2AMIP Products: Version Number - 1.0

The Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2) is an atmospheric reanalysis computed with the Goddard Earth Observing (EOS) System, Version 5.12.4 (GEOS) data assimilation system (Gelaro et al., 2017). To supplement the reanalysis, the GEOS General Circulation Model (GCM) used in MERRA-2 has been used to generate a 10-member ensemble of simulations, configured following the convention of the Atmospheric Model Intercomparison Project (AMIP; Gates et al., 1992). Each ensemble member was initialized using meteorological fields from a different date in November 1979. The AMIP simulations used the sea-surface temperature (SST) and sea-ice boundary conditions that were used in MERRA-2 (Bosilovich et al., 2016). This 10-member ensemble of AMIP simulations, denoted M2AMIP, is available for download in a group of self-describing files, which are documented in this office note. All data collections are provided on the same horizontal grid as MERRA-2. This grid has 576 points in the longitudinal direction and 361 points in the latitudinal direction, corresponding to a resolution of 0.625 degrees by 0.5 degrees. Although data collections are available at this grid, all fields are computed on a cubed-sphere grid with an approximate resolution of 50 km by 50 km and are then spatially interpolated to the latitude-longitude grid. There are no changes in the vertical grids used: variables are provided on either the native vertical grid of 72 model layers, or interpolated to 42 standard pressure levels. Unlike MERRA, no data collections are available at the vertical layer edges. More details on the grid are provided in Section 4. MERRA-2 introduced observation-based precipitation forcing for the land surface parameterization and the corresponding variable PRECTOTCORR in the MERRA-2 FLX (surface turbulent fluxes and related quantities) and LFO (land-surface forcing) collections (see Section 6; Reichle et al., 2017). While this variable is still available for M2AMIP, there was no observation-based forcing, making the value identical to the model derived precipitation, PRECTOT. Similarly, without data assimilation, the values for the analysis increments, D*DTANA, in the tendency and vertically integrated file collections are zero. The M2AMIP data are available for download online through the NASA Center for Climate Simulation (NCCS) DataPortal (https://portal.nccs.nasa.gov/datashare/gmao_m2amip/). Data are arranged in subdirectories based on ensemble member, followed by year and month. Control files that are compatible with the Grid Analysis and Display System (GrADS) are available in the ctl_daily and ctl_monthly directories for the hourly, three hourly, and monthly mean data. Control files for the monthly mean diurnal cycle can be found in the ctl_diurnal subdirectory within the directory for each individual ensemble member.

M2AMIP↗

DOE OSTI2025

Thermal and electric multidomain dynamic model for integration of power grid distribution with behind-the-meter devices

As renewable energy sources like solar and wind power become more integrated into the grid, coordinated control of behind-the-meter devices is crucial for enhancing grid flexibility and reliability and for meeting cost targets, with standardized models being developed to support this transition. The increasing flexibility and uncertainty of integrated renewable energy grids, along with interactions between various subsystems, make traditional steady-state modeling insufficient to capture transient and dynamic behaviors. Current models (e.g., composite load and battery equivalent models) focus on thermodynamic or electrical characteristics but overlook critical electromechanical interactions. This limits the ability to share performance information for grid services and hampers fast dynamic simulations. In addition, motor stalling is usually triggered by a fault event and attributed to the characteristics of the mechanical torque of the motor, resulting in absorption of a large amount of reactive power during the stalling period. Further, this significant withdrawal of reactive power will deteriorate the dynamic voltage stability of power grids and cause delayed voltage recovery. Therefore, an in-depth modeling of the thermodynamics or mechanical torque is essential to study the impacts of the realistic torque characteristics of those behind-the-meter devices on power system voltage stability. This study developed a dynamic multidomain model for building HVAC systems, such as air-source heat pumps, to simulate their thermal and electrical responses to grid transients. The model can accurately predict power metrics with a mean absolute percentage error of 10%, by validating against with power system computer-aided design performance data. Case studies demonstrate the model capability of capturing the transient response to sudden voltage changes, rapid load fluctuations, and system shutdowns respectively. During a sudden voltage drop (30% for 0.1s), a fully loaded heat pump’s motor speed dropped, continued declining, and shut down after 3.6s, with severe power oscillations and a torque spike. A partially loaded unit experienced temporary oscillations but stabilized. Under higher building loads, compressor speed increased from 64% to 100%, with power and torque rising before stabilizing. In safety-triggered shutdowns, power decreased after minor fluctuations, and torque briefly spiked before dropping to zero.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1984

Synoptic scale wind field properties from the SEASAT SASS

Dealiased SEASAT SEASAT A Scatterometer System SASS vector winds obtained during the Gulf Of Alaska SEASAT Experiment GOASEX program are processed to obtain superobservations centered on a one degree by one degree grid. The grid. The results provide values for the combined effects of mesoscale variability and communication noise on the individual SASS winds. These superobservations winds are then processed further to obtain estimates of synoptic scale vector winds stress fields, the horizontal divergence of the wind, the curl of the wind stress and the vertical velocity at 200 m above the sea surface, each with appropriate standard deviations of the estimates for each grid point value. They also explain the concentration of water vapor, liquid water and precipitation found by means of the SMMR Scanning Multichannel Microwave Radiometer at fronts and occlusions in terms of strong warm, moist air advection in the warm air sector accompanied by convergence in the friction layer. Their quality is far superior to that of analyses based on conventional data, which are shown to yield many inconsistencies.

Pierson, W. J., Jr.↗

NASA NTRS1986

Multigrid methods in structural mechanics

Although the application of multigrid methods to the equations of elasticity has been suggested, few such applications have been reported in the literature. In the present work, multigrid techniques are applied to the finite element analysis of a simply supported Bernoulli-Euler beam, and various aspects of the multigrid algorithm are studied and explained in detail. In this study, six grid levels were used to model half the beam. With linear prolongation and sequential ordering, the multigrid algorithm yielded results which were of machine accuracy with work equivalent to 200 standard Gauss-Seidel iterations on the fine grid. Also with linear prolongation and sequential ordering, the V(1,n) cycle with n greater than 2 yielded better convergence rates than the V(n,1) cycle. The restriction and prolongation operators were derived based on energy principles. Conserving energy during the inter-grid transfers required that the prolongation operator be the transpose of the restriction operator, and led to improved convergence rates. With energy-conserving prolongation and sequential ordering, the multigrid algorithm yielded results of machine accuracy with a work equivalent to 45 Gauss-Seidel iterations on the fine grid. The red-black ordering of relaxations yielded solutions of machine accuracy in a single V(1,1) cycle, which required work equivalent to about 4 iterations on the finest grid level.

Raju, I. S.↗