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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.

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At least 199 records · Page 11
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Test-Based Modeling of Photovoltaic Inverter Impact on Distribution Systems

Distributed PV has important effects on overcurrent, overvoltage and islanding protection. These effects are more difficult to analyze because the faulted behavior of PV inverters is less well understood than for rotating machines. Full transient models of the inverter can be accurate, but that approach requires inverter design details and simulation software that doesn’t easily interoperate with utility protection system software. Further, anomalous model behaviors often emerge at different grid conditions, and these are difficult to fix in a proprietary model. Testing provides another approach to characterize inverters, validate models or answer questions of application. This earlier work found major impacts from the inverter design, firmware, operating point and balance-of-system components. Our objective is to characterize PV inverters from black-box testing, leading to standard test reports that could provide parameter sets for widely-implemented models in software tools. The tool vendors can then develop standard inverter models, as they did for rotating machines and exciters. Besides fully detailed transient models, inverters can be represented with steady-state current sources, dynamically controlled current sources or non-linear frequency-dependent black boxes. Type-test data could support the last three kinds of model. Short circuit and sag tests would encompass the basic fault types, reporting the maximum transient and rms currents over all power levels and points-on-wave within four time ranges : 1) 0 to 1 cycles, for close and latch consideration 2) 1 to 4 cycles, for equipment withstand 3) 4 to 30 cycles, for interrupt ratings 4) 30 cycles and beyond, for backup protection analysis Open circuit tests would vary output power and power factor, and report the maximum rms voltage from 1 to 10 cycles after the event. Without models, the type-test data could also provide upper limits on the expected overvoltages and overcurrents.

Wieserman, Laura M.↗
NASA NTRS1989

Effects of cumulus entrainment and multiple cloud types on a January global climate model simulation

An improved version of the GISS Model II cumulus parameterization designed for long-term climate integrations is used to study the effects of entrainment and multiple cloud types on the January climate simulation. Instead of prescribing convective mass as a fixed fraction of the cloud base grid-box mass, it is calculated based on the closure assumption that the cumulus convection restores the atmosphere to a neutral moist convective state at cloud base. This change alone significantly improves the distribution of precipitation, convective mass exchanges, and frequencies in the January climate. The vertical structure of the tropical atmosphere exhibits quasi-equilibrium behavior when this closure is used, even though there is no explicit constraint applied above cloud base.

Yao, Mao-Sung↗
NASA NTRS1990

User's guide: Nimbus-7 Earth radiation budget narrow-field-of-view products. Scene radiance tape products, sorting into angular bins products, and maximum likelihood cloud estimation products

The archived Earth radiation budget (ERB) products produced from the Nimbus-7 ERB narrow field-of-view scanner are described. The principal products are broadband outgoing longwave radiation (4.5 to 50 microns), reflected solar radiation (0.2 to 4.8 microns), and the net radiation. Daily and monthly averages are presented on a fixed global equal area (500 sq km), grid for the period May 1979 to May 1980. Two independent algorithms are used to estimate the outgoing fluxes from the observed radiances. The algorithms are described and the results compared. The products are divided into three subsets: the Scene Radiance Tapes (SRT) contain the calibrated radiances; the Sorting into Angular Bins (SAB) tape contains the SAB produced shortwave, longwave, and net radiation products; and the Maximum Likelihood Cloud Estimation (MLCE) tapes contain the MLCE products. The tape formats are described in detail.

Kyle, H. Lee↗
DOE OSTI2017

Advanced Grid Support Functionality Testing for Florida Power and Light

This report describes the results of laboratory testing of advanced photovoltaic (PV) inverter testing undertaken by the National Renewable Energy Laboratory (NREL) on behalf of the Florida Power and Light Company (FPL). FPL recently commissioned a 1.1 MW-AC PV installation on a solar carport at the Daytona International Speedway in Daytona Beach, Florida. In addition to providing a source of clean energy production, the site serves as a live test bed with 36 different PV inverters from eight different manufacturers. Each inverter type has varied support for advanced grid support functions (GSFs) that are becoming increasingly commonplace, and are being required through revised interconnection standards such as UL1741, IEEE1547, and California (CA) Rule 21. FPL is interested in evaluating the trade-offs between different GSFs, their compliance to emerging standards, and their effects on efficiency and reliability. NREL has provided a controlled laboratory environment to undertake such a study. This work covered nine different classes of tests to compare inverter capabilities and performance for four different inverters that were selected by FPL. The test inverters were all three-phase models rated between 24-36 kW, and containing multiple PV input power point trackers. Advanced grid support functions were tested for functional behavior, and included fixed power factor operation, voltage-ride through, frequency ride-through, volt-var control, and frequency-Watt control. Response to abnormal grid conditions with GSFs enabled was studied through anti-islanding, fault, and load rejection overvoltage tests. Finally, efficiency was evaluated across a range of operating conditions that included power factor, output power, and input voltage variations. Test procedures were derived from requirements of a draft revision of UL741, CA Rule 21, and/or previous studies at NREL. This reports summarizes the results of each test case, providing a comparative performance analysis between the four test inverters. Inverters were mostly able to meet the requirements of their stated GSF capabilities, with deviations from expected results discussed throughout the report. There were mixed results across the range of abnormal tests, and results were often dependent on the capability of each test inverter to deploy the GSFs of interest. Detailed test data has been provided to FPL to support future decision making with respect to inverter selection and GSF deployment in the field.

24 POWER TRANSMISSION AND DISTRIBUTION↗
NASA NTRS2007

Mapping Nearby Terrain in 3D by Use of a Grid of Laser Spots

A proposed optoelectronic system, to be mounted aboard an exploratory robotic vehicle, would be used to generate a three-dimensional (3D) map of nearby terrain and obstacles for purposes of navigating the vehicle across the terrain and avoiding the obstacles. The difference between this system and the other systems would lie in the details of implementation. In this system, the illumination would be provided by a laser. The beam from the laser would pass through a two-dimensional diffraction grating, which would divide the beam into multiple beams propagating in different, fixed, known directions. These beams would form a grid of bright spots on the nearby terrain and obstacles. The centroid of each bright spot in the image would be computed. For each such spot, the combination of (1) the centroid, (2) the known direction of the light beam that produced the spot, and (3) the known baseline would constitute sufficient information for calculating the 3D position of the spot.

Padgett, Curtis↗
NASA NTRS2006

Apparent Transition Behavior of Widely-Used Turbulence Models

The Spalart-Allmaras and the Menter SST kappa-omega turbulence models are shown to have the undesirable characteristic that, for fully turbulent computations, a transition region can occur whose extent varies with grid density. Extremely fine two-dimensional grids over the front portion of an airfoil are used to demonstrate the effect. As the grid density is increased, the laminar region near the nose becomes larger. In the Spalart-Allmaras model this behavior is due to convergence to a laminar-behavior fixed point that occurs in practice when freestream turbulence is below some threshold. It is the result of a feature purposefully added to the original model in conjunction with a special trip function. This degenerate fixed point can also cause nonuniqueness regarding where transition initiates on a given grid. Consistent fully turbulent results can easily be achieved by either using a higher freestream turbulence level or by making a simple change to one of the model constants. Two-equation kappa-omega models, including the SST model, exhibit strong sensitivity to numerical resolution near the area where turbulence initiates. Thus, inconsistent apparent transition behavior with grid refinement in this case does not appear to stem from the presence of a degenerate fixed point. Rather, it is a fundamental property of the kappa-omega model itself, and is not easily remedied.

Rumsey, Christopher L.↗
DOE OSTI2022

Expert perspectives on the wind plant of the future

Abstract Wind power technology has changed rapidly in recent years. Technology innovation, evolving power markets, and competing land and ocean uses continue to influence the design and operation of wind turbines and plants. Anticipating these trends and their impact on future facilities can inform commercial strategies and research priorities. Drawing from a recent survey of 140 of the world's foremost wind experts, we identify expectations of future wind plant design in 2035, both for onshore and offshore wind. Experts anticipate continued growth in turbine size, to 5.5 (onshore) and 17 MW (offshore), with plants located in increasingly less favorable wind and siting regimes. They expect plant sizes of 1,100 MW for fixed‐bottom and 600 MW for floating offshore wind. Experts forecast enhanced grid‐system value from wind through significant to widespread use of larger rotors, hybrid projects with batteries and hydrogen production, and more. To explain experts' perspectives on future plant design and operation, we identify five mechanisms: economies of unit, plant, and resource scale; grid‐system value economies; and production efficiencies. We characterize learning effects as a moderating influence on the strength of these mechanisms. In combination, experts predict that these design choices support levelized cost of energy reductions of 27% (onshore) and 17%–35% (floating and fixed‐bottom offshore) by 2035 compared to today, while enhancing wind energy's grid service offerings. Our findings provide a much‐needed benchmark for representing future wind technologies in power sector models and address a critical research gap by explaining the economics behind wind energy design choices.

17 WIND ENERGY↗
DOE OSTI2025

Quantum computation of SU(2) lattice gauge theory with continuous variables

We present a quantum computational framework for pure SU(2) lattice gauge theory, using continuous variables instead of discrete qubits to represent the infinite-dimensional Hilbert space of the gauge fields. We consider a ladder as well as a two-dimensional grid of plaquettes, detailing the use of gauge fixing to reduce the degrees of freedom and simplify the Hamiltonian. We demonstrate how system dynamics, ground states, and energy gaps can be computed using the continuous-variable approach to quantum computing. Our results indicate that it is feasible to study non-Abelian gauge theories with continuous variables, providing new avenues for understanding the real-time dynamics of quantum field theories.

Gauge Symmetry↗
DOE OSTI2017

High Resolution Stereoscopic PIV Measurements in a 5 x 5 Rod Bundle with Mixing Vane

Spacer grids are used in the pressurized water reactor (PWR) to fix, clamp the fuel rods, and enhance the turbulence mixing within the spaces of the rods and overall heat transfer performance. Proper understanding on the thermal-hydraulic characteristics of the turbulent mixing and heat transfer created by the space grids is important. However, spacer grids having complex design structure with dimple, mixing vane and spring have been challenges for computational fluid dynamics (CFD) turbulent models. High fidelity experimental data with spatial-temporal resolutions is essential to validate and assess the performances of CFD models. Baglietto and Ninokata applied an improved realizable anisotropic turbulence models via direct numerical simulation (DNS)-based near-wall formulation data to study the wall shear stresses and flow structures in tight bundle subchannels. These authors have shown that their models are able to capture the stress distribution and velocity field at an acceptable accuracy. The authors numerically studied the flow field characteristics at the region downstream of the spacer grids, and validated their results to particle imagery velocimetry (PIV) measurements. Dominguez and Hassan performed two-dimensional time resolved PIV (TR-PIV) measurements within a 5 x 5 PWR rod bundle with spacer-grids that utilized a matching index of refraction (MIR) approach. Their averaged data showed a rather uniform direction velocity profile after the spacer grid while the downstream illustrated a no slip wall profile. In this paper, we presents flow measurements in the MIR 5 x 5 fuel rod bundle with a spacer grid. Time resolved stereoscopic PIV (TR-SPIV) measurements were performed to obtain high spatial-temporal resolution two-dimensional three-component (2D3C) velocity vector fields in the subchannels between the rods. Flow statistics such as mean velocity, and Reynolds stresses were computed. The obtained experimental data can be used to improve the computational modeling and validate the numerical models that are currently applied to light and pressurized water reactors.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
NASA NTRS1980

Grid evolution in time asymptotic problems

A technique for generating systems of coordinates for solving time asymptotic problems is described which provides a simple way of moving the mesh points in physical space and reduces the error in the solution relative to that obtained using a fixed mesh. First order partial differential equations are formulated for the grid point velocity in transient problems. Local flow information and boundary motion are used to determine the interior grid point motion.

Rai, M. M.↗
DOE OSTI2023

An Assessment of Deploying Advanced Pumped Storage Hydropower Technology in U.S. Electricity Markets

Pumped storage hydropower (PSH) is a type of hydropower technology where energy can be stored and generated by moving water between two reservoirs of differing elevations. In addition to providing 97% of the total utility-scale hydropower storage in the United States, PSH plants have operational characteristics - such as high ramp rates and the ability to provide reserves - that contribute to greater flexibility and reliability of the power grid. New PSH technologies can provide additional flexibility beyond existing, fixed-speed units. With the emergence of high levels of variable renewable energy resources (e.g., wind and solar), energy storage is expected to be crucial to the reliability and reliance of the power grid in a low-carbon future. At diurnal and longer durations, PSH plants have some of the lowest costs per unit of energy, have been proven to be reliable and efficient, are not cycle-limited, and typically have long lives, often exceeding 50 years. The work presented here focuses on a new generation of PSH: ternary PSH and quaternary PSH (together referred to as T/Q-PSH). Given recent experience in Europe, grid operators in the United States and elsewhere are increasingly focusing their attention on T/Q-PSH as a proven, financeable technology that can offer utility-scale, long-duration, fast-acting energy storage capabilities and grid services. T/Q-PSH differs from conventional PSH in that it can provide fast-response ancillary services during both generating and pumping operations. This capability together with fast mode switching times are key attributes that make T/Q-PSH technologies attractive for managing and stabilizing electricity systems with high amounts of variable renewable energy.

ENERGY PLANNING, POLICY, AND ECONOMY,HYDRO ENERGY↗
NASA NTRS2006

Validation of a Pressure-Based Combustion Simulation Tool Using a Single Element Injector Test Problem

The traditional design and analysis practice for advanced propulsion systems, particularly chemical rocket engines, relies heavily on expensive full-scale prototype development and testing. Over the past decade, use of high-fidelity analysis and design tools such as CFD early in the product development cycle has been identified as one way to alleviate testing costs and to develop these devices better, faster and cheaper. Increased emphasis is being placed on developing and applying CFD models to simulate the flow field environments and performance of advanced propulsion systems. This necessitates the development of next generation computational tools which can be used effectively and reliably in a design environment by non-CFD specialists. A computational tool, called Loci-STREAM is being developed for this purpose. It is a pressure-based, Reynolds-averaged Navier-Stokes (RANS) solver for generalized unstructured grids, which is designed to handle all-speed flows (incompressible to hypersonic) and is particularly suitable for solving multi-species flow in fixed-frame combustion devices. Loci-STREAM integrates proven numerical methods for generalized grids and state-of-the-art physical models in a novel rule-based programming framework called Loci which allows: (a) seamless integration of multidisciplinary physics in a unified manner, and (b) automatic handling of massively parallel computing. The objective of the ongoing work is to develop a robust simulation capability for combustion problems in rocket engines. As an initial step towards validating this capability, a model problem is investigated in the present study which involves a gaseous oxygen/gaseous hydrogen (GO2/GH2) shear coaxial single element injector, for which experimental data are available. The sensitivity of the computed solutions to grid density, grid distribution, different turbulence models, and different near-wall treatments is investigated. A refined grid, which is clustered in the vicinity of the solid walls as well as the flame, is used to obtain a steady state solution which may be considered as the best solution attainable with the steady-state RANS methodology. From a design point of view, quick turnaround times are desirable; with this in mind, coarser grids are also employed and the resulting solutions are evaluated with respect to the fine grid solution.

Thakur, Siddarth↗
NASA NTRS2007

Apparent Transition Behavior of Widely-Used Turbulence Models

The Spalart-Allmaras and the Menter SST k-omega turbulence models are shown to have the undesirable characteristic that, for fully turbulent computations, a transition region can occur whose extent varies with grid density. Extremely fine two-dimensional grids over the front portion of an airfoil are used to demonstrate the effect. As the grid density is increased, the laminar region near the nose becomes larger. In the Spalart-Allmaras model this behavior is due to convergence to a laminar-behavior fixed point that occurs in practice when freestream turbulence is below some threshold. It is the result of a feature purposefully added to the original model in conjunction with a special trip function. This degenerate fixed point can also cause non-uniqueness regarding where transition initiates on a given grid. Consistent fully turbulent results can easily be achieved by either using a higher freestream turbulence level or by making a simple change to one of the model constants. Two-equation k-omega models, including the SST model, exhibit strong sensitivity to numerical resolution near the area where turbulence initiates. Thus, inconsistent apparent transition behavior with grid refinement in this case does not appear to stem from the presence of a degenerate fixed point. Rather, it is a fundamental property of the k-omega model itself, and is not easily remedied.

Rumsey, Christopher L.↗
DOE OSTI2018

Case Study: Nuclear-Renewable-Water Integration in Arizona

This document reports the application of the N-R HES software framework to a case study for Arizona Public Service (APS); the manager and part owner of the Palo Verde (PV) nuclear power plant. The case study is a work in progress of which this report presents a detailed description of the current model input data, assumptions and the corresponding results produced by the developed software framework. The goal of the report is, that with this information APS (together with INL) should be able to identify where more detailed data and more accurate model assumptions are necessary to reproduce all driving physical and economic phenomena in order to capture enough complexity of the reality to allow APS to use the results for their strategic decisions. The N-R HES software framework started being developed at INL two years ago [ref]. The framework has reached some level of maturity so that it can be applied to more than simple demonstration cases, e.g. real industry problems. Nevertheless, more capabilities are constantly added to accommodate the special needs of these challenging real life problems. The N-R HES framework is built on top of the RAVEN code [ref] and uses it as driver and workflow manager for all calculations. The framework has specifically been developed for the economic assessment of N-R HES systems. There are four main cornerstones of the N-R HES simulation framework: 1) generation of stochastic time series, 2) a probabilistic analysis and optimization set of algorithms available in RAVEN, 3) a set of models for representation of the physical behavior of N-R HES, and 4) a RAVEN plug-in that maps physical performance into economic performance called CashFlow. Within this framework, a broad spectrum of questions related to N-R HES can be addressed. One of the challenges currently of high interest is that the increasing penetration of variable renewables is altering the profile of the net demand (demand after removing all non curtailable renewable energy sources), with which the other generators on the grid have to cope. The N-R HES software framework is capable to analyze the potential feasibility of mitigating the resultant volatility in the net electricity demand. One possible solution to that problem currently intensively studied by the energy industry is adding stabilizing loads to the grid. These loads can be external industrial processes that are able to ramp production up and down quickly or adding variable internal loads that will lift the base load of the power plant. The latter is a possible solution currently studied by APS. APS if currently anticipating several challenges that will likely arise in the near future. One of them is, as mentioned, coping with the rapid growth of variable renewable energy (VRE) sources put on the grid. Although APS can sell its electricity for a fixed retail price to cover the local demand first, it also sells electricity in excess of the local demand at the Palo Verde energy hub. Due to the growing VRE penetration, the Palo Verde hub electricity spot price is negative more and more frequently. To mitigate this price volatility, APS is seeking to add more base load, so that the quantity of excess energy to be sold for prices potentially less than the internal retail price or even negative prices at the hub is reduced. A second challenge APS is facing is that their cooling water acquisition contract with the Sub Regional Operating Group (SROG) will expire soon and a renewal can only be done for a significantly higher price of the water. Therefore, APS is also seeking for alternative sources for their cooling water. An opportunity currently under investigation is to pump a limited quantity of brackish water from the regional ground water. Although much cheaper than the water from the new SROG contract, the salinity of the brackish water is so high that a blend of brackish and SROG water would need additional treatment to improve its quality for use in the cooling towers. An on-site reverse osmosis (RO) desalination plant is envisaged to reduce the salinity to an acceptable level. This report investigates the economic impact of such an RO plant. As one can see, the RO plant would help with both problems APS is facing: First, it would rise the APS base load helping to mitigate VRE induced hub price volatility. From this perspective, the RO plant can be seen as a stabilizing load on the grid as would be any other industrial process. Second, the RO’s goods, e.g. the clean water produced can be used by APS itself for a (hopefully) lower cost than 100% SROG water would be. The report will first describe the APS water procurement strategy in detail and lay out the possible alternative scenarios studied in this document. This is covered in Chapter 2. Next, in Chapter 3, the APS case model developed within the N-R HES software framework is described. This chapter includes also a description of the needed framework developments to accommodate the APS study.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

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

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

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1977

NASTRAN hydroelastic modal studies. Volume 3: NASTRAN 3-D hydroelastic analysis and MESHGEN user's manuals

The method used to formulate the fluid/structure equations is described in the updates to the NASTRAN Theoretical Manual. The basis for defining the fluid are three dimensional finite elements connected to fluid grid points defining the Eulerian pressure at a point fixed in space. The use of a pressure single degree of freedom at each point rather than three displacements allows a finer mesh of elements with a reasonable matrix order. In the formulation of the fluid/structure system the interior fluid degrees of freedom are transformed and removed from the solution matrices. The eigenvalues of the combination are extracted from small, fully dense, symmetric mass and stiffness matrices, efficiently processed with the Givens method. The solution matrices are defined by the free surface displacements and the reduced structure coordinates.

Source record↗
NASA NTRS2001

Mars Constellation Design

In this study, several different Mars constellation designs were evaluated. Various figures of merit were used to examine constellation configurations in order to optimize the performance of Mars Network. Plots were also obtained in order to make a clear comparison of the various figures of merit. In the following analysis, six figures of merit were used in the comparison (average time to achieve a 1 meter fix, maximum time to achieve a 1 meter fix, maximum wait time between satellite sightings, percent of time a satellite is in view, percent of cases to achieve a 1 meter fix in 10 minutes, and percent of cases to achieve a 1 meter fix in 1 hour). For the communication figures of merit (maximum wait time between satellite sightings and percent of time a satellite is in view), an initial run over a period of 10 days was used to produce the data. For the remaining figures of merit, the simulation was first restarted at least 100 times over a period of 1000 Earth days. Each time the simulation was continued until a 1 meter fix was achieved at all stations. The ground station grid was spaced 20 deg in longitude and 5 deg in latitude. In order to obtain the following graphs the values were first averaged over longitude and then an average of the positive and negative latitude values was made (i.e., -90 deg lat and 90 deg lat were averaged). In each case, the resulting figures of merit are shown. For some cases, the difference between the given case and the 'Eagles 2 Satellites' case is also given.

Source record↗
NASA NTRS2006

Calculation of Thermally-Induced Displacements in Spherically Domed Ion Engine Grids

An analytical method for predicting the thermally-induced normal and tangential displacements of spherically domed ion optics grids under an axisymmetric thermal loading is presented. A fixed edge support that could be thermally expanded is used for this analysis. Equations for the displacements both normal and tangential to the surface of the spherical shell are derived. A simplified equation for the displacement at the center of the spherical dome is also derived. The effects of plate perforation on displacements and stresses are determined by modeling the perforated plate as an equivalent solid plate with modified, or effective, material properties. Analytical model results are compared to the results from a finite element model. For the solid shell, comparisons showed that the analytical model produces results that closely match the finite element model results. The simplified equation for the normal displacement of the spherical dome center is also found to accurately predict this displacement. For the perforated shells, the analytical solution and simplified equation produce accurate results for materials with low thermal expansion coefficients.

Soulas, George C.↗