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DOE OSTI2025

Development of a Digital Twin for Hydrogen Dispersion and Safety Assessment in an Electrolyzer Based Hydrogen Production Facility

Digital twin models are virtual representations of physical systems that use real-time data to simulate and optimize performance. This study presents the development and initial implementation of a digital twin (DT) for the electrolyzer-based hydrogen production facility at NREL's Advanced Research on Integrated Energy Systems (ARIES), focused on enhancing safety and optimizing sensor placement through physics-based simulations and metadata integration. The DT incorporates detailed facility-specific information, including component layout, leak locations, and controlled release parameters, to model hydrogen dispersion under varying environmental conditions. Using steady-state computational fluid dynamics (CFD) simulations informed by real meteorological data, such as wind speed, direction, and vertical wind profiles, the DT enables visualization of hydrogen plume behavior and spatial concentration distributions. Comparative analysis between high and low wind speed scenarios illustrates the significant influence of wind dynamics on plume shape and extent, with horizontal momentum dominating dispersion at higher speeds, while buoyancy effects become more prominent under low wind conditions. These simulations generate a rich dataset embedded within the DT, allowing users to assess potential leak outcomes and identify optimal sensor locations based on concentration thresholds. The model supports scenario-based analysis to guide safety strategies and equipment deployment for open-area hydrogen infrastructure. The digital twin thus serves as a dynamic platform for virtual prototyping, providing predictive insight into hydrogen behavior and enhancing risk-informed decision-making. This initial phase establishes a validated foundation for future integration of transient, uncontrolled leak scenarios and real-time sensor feedback, positioning the DT as a critical tool for safety design, operational planning, and adaptive monitoring in hydrogen systems. Overall, the approach demonstrates the value of combining environmental data with digital simulations to inform safer and more efficient deployment of hydrogen technologies.

08 HYDROGEN↗

DOE OSTI2026

Development of a Digital Twin for Hydrogen Dispersion and Safety Assessment in an Electrolyzer-Based Hydrogen Production Facility: Preprint

Digital twin models are virtual representations of physical systems that use real-time data to simulate and optimize performance. This study presents the development and initial implementation of a digital twin (DT) for the electrolyzer-based hydrogen production facility at the National Renewable Energy Laboratory (NREL)'s Advanced Research on Integrated Energy Systems (ARIES), focused on enhancing safety and optimizing sensor placement through physics-based simulations and metadata integration. The DT incorporates detailed facility-specific information, including component layout, leak locations, and controlled release parameters, to model hydrogen dispersion under varying environmental conditions. Using steady-state computational fluid dynamics (CFD) simulations informed by real meteorological data, such as wind speed, direction, and vertical wind profiles, the DT enables visualization of hydrogen plume behavior and spatial concentration distributions. Comparative analysis between high and low wind speed scenarios illustrates the significant influence of wind dynamics on plume shape and extent, with horizontal momentum dominating dispersion at higher speeds, while buoyancy effects become more prominent under low wind conditions. These simulations generate a rich dataset embedded within the DT, allowing users to assess potential leak outcomes and identify optimal sensor locations based on concentration thresholds. The model supports scenario-based analysis to guide safety strategies and equipment deployment for open-area hydrogen infrastructure. The digital twin thus serves as a dynamic platform for virtual prototyping, providing predictive insight into hydrogen behavior and enhancing risk-informed decision-making. This initial phase establishes a validated foundation for future integration of transient, uncontrolled leak scenarios and real-time sensor feedback, positioning the DT as a critical tool for safety design, operational planning, and adaptive monitoring in hydrogen systems. Overall, the approach demonstrates the value of combining environmental data with digital simulations to inform safer and more efficient deployment of hydrogen technologies.

08 HYDROGEN↗

DOE OSTI2025

Net Present Value Optimization of a Natural Gas Combined Cycle Plant with CO 2 Capture using a Water-Lean Solvent Considering Transient Electricity Price for Multiple Regions

Global CO 2 emissions are increasing at about a 1.5% rate per year. Fossil fuel-based plants are one of the main contributors to this rise. In the power generation industry, fossil fuel plants are dominant, and many plants are under development. In this study, a natural gas combined cycle (NGCC) power plant with postcombustion capture using a leading water-lean solvent is considered. For optimal design and operating schedule, large-scale dynamic optimization is undertaken for net present value (NPV) optimization. The first principle dynamic model of NGCC is developed, including a model of the highly efficient H-class gas turbines. For computational tractability of the dynamic optimization problem, a reduced-order model is developed by using the Hankel singular value decomposition. A waterlean solvent, N-(2-ethoxyethyl)-3-morpholinopropan-1-amine, is used for carbon capture. A model of the capture system is developed in Aspen Plus, which is used to develop a reduced-order model by using ALAMO, a machine learning software. In addition, a reduced model of the CO 2 compression system with a dehydration unit is also considered. The integrated system is used for NPV optimization by using the Python-based PYOMO platform. The PCC process is analyzed for three configurations-conventional packed bed, rotating packed bed (RPB), and a combination of RPB and direct contact cooler. The NPV optimization is performed for 14 regional markets by considering year-long clustered and continuous locational marginal price data with a 1 h interval. Optimization results show that the PCC can achieve 90% CO 2 capture with a positive NPV for six regions. Sensitivity studies conducted by using the PCC configurations indicate that the process is economically feasible for 9 regions out of 14 regional electricity markets with NPV values in the range of 33−540 $MM.

cabon capture↗

NASA NTRS1990

Tele-Autonomous control involving contact

Object localization and its application in tele-autonomous systems are studied. Two object localization algorithms are presented together with the methods of extracting several important types of object features. The first algorithm is based on line-segment to line-segment matching. Line range sensors are used to extract line-segment features from an object. The extracted features are matched to corresponding model features to compute the location of the object. The inputs of the second algorithm are not limited only to the line features. Featured points (point to point matching) and featured unit direction vectors (vector to vector matching) can also be used as the inputs of the algorithm, and there is no upper limit on the number of the features inputed. The algorithm will allow the use of redundant features to find a better solution. The algorithm uses dual number quaternions to represent the position and orientation of an object and uses the least squares optimization method to find an optimal solution for the object's location. The advantage of using this representation is that the method solves for the location estimation by minimizing a single cost function associated with the sum of the orientation and position errors and thus has a better performance on the estimation, both in accuracy and speed, than that of other similar algorithms. The difficulties when the operator is controlling a remote robot to perform manipulation tasks are also discussed. The main problems facing the operator are time delays on the signal transmission and the uncertainties of the remote environment. How object localization techniques can be used together with other techniques such as predictor display and time desynchronization to help to overcome these difficulties are then discussed.

Shao, Lejun↗

DOE OSTI2023

Grid-Integrated Production of Fischer-Tropsch Synfuels from Nuclear Power

Idaho National Laboratory (INL) investigates the relative economic profitability of an integrated energy system (IES) coupling an NPP with a synfuel production process at selected case study locations across the United States. In the synfuel IES, a high-temperature steam electrolysis (HTSE) plant is thermally and electrically coupled with an NPP to produce zero-carbon hydrogen. The synthetic fuel is produced from this H 2 combined with a CO 2 supply using the reverse water gas shift process followed by the Fischer-Tropsch (FT) reaction. This analysis considers a system in which the CO 2 is sourced from regional CO 2 emitters via the construction and operation of pipeline-based CO 2 supply networks. Locating the FT plant at the same site as the NPP and HTSE plants enables the NPP to provide zero-carbon heat and power to the HTSE plant and zero-carbon power to the FT plant as well as avoid the requirement for long-distance H 2 product transport from the HTSE plant to the FT plant. Hydrogen storage is used to enable the NPP to dispatch power to the electrical grid (instead of the HTSE plant) when grid demand increases, thus enabling the FT plant to continue to operate in a steady-state production mode. The ability to cease hydrogen production for several hours within each day enables the NPP to provide power to the grid to balance the electricity market during peak periods and maximize revenues for the nuclear synfuel IES. The FT process design considered has a 99% carbon conversion efficiency. The use of nuclear energy and nuclear energy-derived hydrogen enables synfuel production to achieve this high level of carbon utilization. Additionally, the life-cycle carbon emissions of the nuclear-based synfuel production process are very low, with WTW emissions of approximately 25 gCO 2 e/MJ, including steam credits (generated from FT process excess heat), and approximately 7 gCO 2 e/MJ, if steam credits are excluded. This compares favorably with the WTW emissions of 90.5 gCO 2 e/MJ for a compression-ignition, direct injection (CIDI) vehicle with a fuel economy of 31.6 miles per gallon gasoline equivalent (MPGGE), using low-sulfur diesel produced using conventional petroleum production and refining processes. Several NPPs in various regions of the U.S. are considered as case study analyses. Supply locations and transportation via pipeline of the CO 2 feedstock to the NPP site are analyzed through the National Energy Technology Laboratory (NETL) CO 2 Transport Cost model. The team finds that the amount of CO 2 generated by different sectors is sufficient for the synfuel production process at all locations considered. The CO 2 transportation costs are functions of the distance of the source to the NPP location, the CO 2 capture cost at the source, and the quantity of CO 2 transported. Historical electricity prices for the NPP case study locations are collected and analyzed. Monthly average prices, price range, and duration of negative-price periods vary among these locations. For each location, an auto-regressive moving average (ARMA) model is trained on historical electricity price data. ARMA validation is done to ensure the synthetic price distributions represent one of historical prices with high fidelity. Synthetic time series from these ARMA models are used in a coupled dispatch and system optimization in the Holistic Energy Resource Optimization Network (HERON) to compute the differential net present value (NPV) of the IES. The team finds that this econometric is positive, ranging from $14M–1.3bn (2020) depending on the location. The optimal synfuel IES configuration to obtain this increase in NPV often maximizes the size of the synfuel production process with regards to the size of the NPP. However, the team shows that the NPP still plays a stabilizing role for the grid: In periods of high prices and high loads, more electricity from the NPP is sent to the grid. A high variability of electricity prices and extreme maximum prices tend to drive up electricity production. While it requires significant investment, the synfuel IES could increase the economic profitability for the existing fleet of LWRs across the country while still maintaining the grid stabilizer role of NPPs. During its lifetime, the main costs for the nuclear synfuel IES are the carbon feedstock transportation costs, followed by the capital expenses (CAPEX) and operation and maintenance (O&M) costs while the revenue comes first from the IRA H 2 production tax credit (PTC) and then from the sales of synfuel products. The profitability of the synfuel IES is most sensitive to the value of the hydrogen PTC and the synfuel products as well as the cost of the carbon feedstock, highlighting the importance of governmental incentives regarding hydrogen, carbon emissions, and synfuel in driving the deployment of future nuclear synfuel IESs.

08 HYDROGEN↗

NASA NTRS2018

Broadband Inlet Liner Design for the DGEN Aero-Propulsion Research Turbofan

The utilization of advanced fan designs (including higher bypass ratios) and shorter engine nacelles has highlighted a need for increased fan noise reduction over a broad frequency range. With these observations in mind, an acoustic liner optimization process has been developed to achieve improved broadband liner designs. A series of advanced liner design studies at increasing technology readiness levels has been conducted as the overall optimization methodology has been enhanced. In this work, further validation was pursued through the fabrication and testing of two liner designs for the NASA Glenn DGEN Aero-propulsion Research Turbofan. Specifically, single and three layer liners were designed, fabricated, and tested. Analysis of the predicted attenuation results showed them to generally match expected trends for the conditions considered. The effects of source distribution on predicted directivity were also illustrated through the flexibility of a statistical source model. As part of the analysis, the overall design and evaluation capability was extended such that external observer locations may be included in the optimization process. This enhancement will provide a much wider design space for the development of advanced broadband liners. These preliminary results provide further confidence in the design tools and enhancements made to the overall liner design process.

Nark, Douglas M.↗

DOE OSTI2024

LogPath: Log data based energy consumption analysis enabling electric vehicle path optimization

Vehicle navigation and path optimization require a more meticulous approach when it deals with EVs (electric vehicles) and SDVs (software-defined vehicles), due to lengthy charging times and the lack of charging infrastructure. Long-distance freight EV trucking needs path guidance with accurate energy consumption estimates to prevent charging-related failures. We developed a novel energy consumption estimation approach that only uses battery log data to extract major vehicle parameters to increase EV navigation accuracy without additional sensors. This is enabled by extracting multiple drive modes from the log data for analysis. The system provides 1) routes, 2) charge locations, 3) charging times, and 4) optimal vehicle speeds that guarantee the shortest travel time. Here we successfully validated the system using log data collected from an EV and Tesla's Supercharging map in the US and compared it with the commercially available navigation system, Tesla's trip planner, whose capabilities solely include charging time and routing.

EV (Electric vehicles) navigation↗

DOE OSTI2020

Distributionally robust facility location problem under decision-dependent stochastic demand

While the traditional facility location problem considers exogenous demand, in some applications, locations of facilities could affect the willingness of customers to use certain types of services, e.g., carsharing, and therefore they also affect realizations of random demand. Moreover, a decision maker may not know the exact distribution of such endogenous demand and how it is affected by location choices. In this paper, we consider a distributionally robust facility location problem, in which we interpret the moments of stochastic demand as functions of facility-location decisions. We reformulate a two-stage decision-dependent distributionally robust optimization model as a monolithic formulation, and then derive exact mixed-integer linear programming reformulation as well as valid inequalities when the means and variances of demand are piecewise linear functions of location solutions. We conduct extensive computational studies, in which we compare our model with a decision-dependent deterministic model, as well as stochastic programming and distributionally robust models without the decision-dependent assumption. Here, the results show superior performance of our approach with remarkable improvement in profit and quality of service under various settings, in addition to computational speed-ups given by formulation enhancements. These results draw attention to the need of considering the impact of location decisions on customer demand within this strategic-level planning problem.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Optimal design of microseismic monitoring network: Synthetic study for the Kimberlina CO2 storage demonstration site

Microseismic monitoring can play a crucial role to ensure safe long-term geological carbon storage. For reliable long-term monitoring for CO2-injection-induced microseismic events, a surface seismic array is desirable in addition to borehole geophone sensors. Optimal design of surface seismic network is of great interest to achieve cost-effective monitoring. We develop a methodology to determine the optimal number of surface seismic stations with a geometrically satisfactory distribution for given monitoring regions. We design an optimal microseismic monitoring network based on widely-accepted guiding principles, and the relationship between the location accuracy of microseismic events and the total number of seismic stations. We determine the optimal number of seismic stations based on the trade-off curve of the event location accuracy vs. the total number of seismic stations. We apply our optimal design method to the Kimberlina carbon storage site in California. We use a synthetic Kimberlina model to show that approximately 20 surface seismic stations with a geometrically satisfactory distribution are preferred for the best trade-off between the cost and the event location accuracy.

58 GEOSCIENCES↗

NASA NTRS1991

SALSA - A Synthesis Array for Lunar Submillimeter Astronomy

A conceptual design is described for a lunar submillimeter wavelength interferometer called SALSA, a Synthesis Array for Lunar Submillimeter Astronomy. Its design is greatly simplified over conventional submillimeter wavelength arrays because it takes advantage of a beam waveguide to relay signals to a centrally-located receiver system. The array is optimized to synthesize images at wavelengths between 60 and 300 microns, and it has an angular resolution of 10 milliarcsecs at 60 microns. Observations in this region provide unique insights into fundamental issues in astrophysics such as star formation, but are impossible from the earth's surface because of atmospheric absorption. The baseline design for SALSA consists of twelve, 3.5-meter diameter antennas arranged in a Y-shaped configuration consisting of three 0.5-km long arms, each with four antennas distributed according to a power-law function of distance. Such a beam-waveguide approach has significant advantages, since all the high-power-consumption items (active cryogenics, correlator) are at a central location and could share a single power source. In addition, the antennas can be quite simple and maintenance is greatly reduced. The primary technology challenges for developing such an array are the coherent detectors (mixers and local oscillator sources), and ensuring the dimensional stability of the optical elements under the extreme lunar thermal environment (85-385 K).

Mahoney, M. J.↗

DOE OSTI2026

Extreme-scale EV charging infrastructure planning for last-mile delivery using high-performance parallel computing

Here, this paper addresses stochastic charger location and allocation problems under queue congestion for last-mile delivery using electric vehicles (EVs). The objective is to decide where to open charging stations and how many chargers of each type to install, subject to budgetary and waiting-time constraints. We formulate the problem as a mixed-integer non-linear program, where each station-charger pair is modeled as a multiserver queue with stochastic arrivals and service times to capture the notion of waiting in fleet operations. The model is extremely large, with billions of variables and constraints for a typical metropolitan area; even loading the model in solver memory is difficult, let alone solving it. To address this challenge, we develop a Lagrangian-based dual decomposition framework that decomposes the problem by station and leverages parallelization on high-performance computing systems, where the subproblems are solved by using a cutting plane method and their solutions are collected at the master level. We also develop a three-step rounding heuristic to transform the fractional subproblem solutions into feasible integral solutions. Computational experiments on data from the Chicago metropolitan area with hundreds of thousands of households and thousands of candidate stations show that our approach produces high-quality solutions in cases where existing exact methods cannot even load the model in memory. We also analyze various policy scenarios, demonstrating that combining existing depots with newly built stations under multiagency collaboration substantially reduces costs and congestion. These findings offer a scalable and efficient framework for developing sustainable large-scale EV charging networks.

Capacity allocation↗

DOE OSTI2020

Regional Energy Deployment System (ReEDS) Model Documentation: Version 2019

The Regional Energy Deployment System (ReEDS) model is a capacity expansion and dispatch model that is primarily used for the contiguous U.S. electric power sector. The model relies on system-wide least cost optimization to estimate the type and location of future generation and transmission capacity. This document describes details of how the model is formulated, how it functions, and many of the key inputs.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2021

Regional Energy Deployment System (ReEDS) Model Documentation (Version 2020)

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS2022

Exploring Applications of Machine Learning for Wildfire Monitoring and Detection using Unmanned Aerial Vehicles

Wildfires are increasing in frequency and severity around the world, including the United States. The losses caused by wildfires could be mitigated if high-risk areas, hotspots, and flare-ups could be monitored continuously, such as through the use of Unmanned Aerial Vehicles (UAVs). This paper documents exploratory efforts using machine learning to determine efficient flight paths for UAVs and to detect wildfires using image classification. On path planning, three machine learning techniques—Genetic Algorithm, Simulated Annealing, and Dynamic Programming—were explored. Genetic Algorithm was found to be an effective approach for path planning for wildfire monitoring and surveillance by UAVs. For a scenario of 25 locations in a circular arrangement, the algorithm was able to return the optimal path. The accuracy and execution time was found to be sensitive to the algorithm hyperparameters selected, which was especially evident in scenarios with hundreds or thousands of locations. Simulated Annealing was also found to be an effective approach for UAV path planning, with a major benefit of avoiding getting trapped in local minima and being straightforward to implement. Like Genetic Algorithm, the performance of Simulated Annealing was also found to be sensitive to the algorithm hyperparameters selected. By comparison, Dynamic Programming guarantees optimality for any number of locations, but it was found to be less practical in terms of execution time for scenarios with more than about a couple dozen locations. On wildfire detection, image classification using deep learning with a convolutional neural network was explored. Transfer learning was found to be a useful technique to efficiently train deep learning models. Also, it was determined that GPU processing can increase training speed by an order of magnitude, which enables significantly faster development. For a validation test set of 500 images, there were only two false negatives and zero false positives. These results demonstrate that detecting wildfires in static cameras using machine learning is feasible and establish a baseline for using images captured by UAVs in flight for wildfire detection.

Wildfire management↗

DOE OSTI2025

Meshfree simulation and prediction of recrystallized grain size in friction stir processed 316L stainless steel

Friction stir processing (FSP) is a promising solid-phase microstructural modification technique that can repair and enhance damaged stainless steel surfaces exposed to harsh environments. The quality of the repaired material is closely correlated to the recrystallized grain size in the stir zone (SZ), which is influenced by the thermomechanical conditions dictated by FSP process parameters. Thus, establishing a reliable relationship between these parameters and recrystallized grain size in the SZ is crucial for optimizing repair quality. However, existing experimental approaches often rely on indirect temperatures measured far from the SZ, along with rough strain rate estimations, which are imprecise and time-consuming. Meanwhile, existing mesh-based modeling methods usually face numerical challenges when dealing with the large material deformations inherent in FSP. Here, to address these issues, this study introduces a meshfree process model for FSP based on the smoothed particle hydrodynamics (SPH) method, aimed at predicting process conditions under different parameters. The model is validated using experimental data from 11 combinations of tool traverse and rotation speeds on 316 L stainless steel. Correlations between process parameters, material flow, temperature, strain, strain rate, and recrystallized grain size are revealed through SPH simulations and electron backscatter diffraction (EBSD) imaging. The results show that in situ SZ temperatures range from 1071 to 1322°C, which exceed the tool temperature by over 300°C. Furthermore, SZ temperature, strain rate, and grain size increase monotonically with higher tool temperature and faster traverse speed. A relationship is then established between the model-predicted Zener-Hollomon parameter and the recrystallized grain size based on EBSD data, expressed as ln(d) = -0.364 ln(Z) + 14.673. Finally, this relationship exhibits satisfactory accuracy with errors of less than 26.9% in predicting grain sizes at various SZ locations, which offers valuable insights for optimizing FSP repair processes for 316 L stainless steel.

316L stainless steel↗

DOE OSTI2023

Investigating a Renewable-Resource-Targeting Mobile Aquaculture System Using Route Optimization Based on Optimal Foraging Theory

Aquaculture systems require careful consideration of location, which determines water conditions, pollution impacts, and hazardous conditions. Mobility may be able to address these factors while also supporting the targeting of renewable energy sources such as wind, wave, and solar power throughout the year. In this paper, a purpose-built mobile aquaculture ship is identified and modeled with a combination of renewable energy harvesting capabilities as a case study with the objective of assessing the potential benefits of targeting high renewable energy potentials to power aquaculture operations. A route optimization algorithm is created and tuned to simulate the mobility of the aquaculture platform and cost-basis comparisons are made to a stationary system. The small spatial variability in renewable energy potential when combining multiple resources significantly limits the benefits of a mobile, renewable-targeting aquaculture system. On the other hand, the consistent energy harvest from a blend of renewable energy types (13 kW installed wind capacity, 661 m 2 installed solar, and 1 m characteristic width wave-energy converter) suggests that the potential benefits of a mobile platform for offshore aquaculture (mitigation of environmental and social concerns, any potential positive impact on yields, hazard avoidance, etc.) can likely be pursued without significant increases in energy harvester costs.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

NASA NTRS2014

Spatio-Temporal Video Segmentation with Shape Growth or Shrinkage Constraint

We propose a new method for joint segmentation of monotonously growing or shrinking shapes in a time sequence of noisy images. The task of segmenting the image time series is expressed as an optimization problem using the spatio-temporal graph of pixels, in which we are able to impose the constraint of shape growth or of shrinkage by introducing monodirectional infinite links connecting pixels at the same spatial locations in successive image frames. The globally optimal solution is computed with a graph cut. The performance of the proposed method is validated on three applications: segmentation of melting sea ice floes and of growing burned areas from time series of 2D satellite images, and segmentation of a growing brain tumor from sequences of 3D medical scans. In the latter application, we impose an additional intersequences inclusion constraint by adding directed infinite links between pixels of dependent image structures.

Segmentation↗

DOE OSTI2023

Optimal design of chemoepitaxial guideposts for the directed self-assembly of block copolymer systems using an inexact Newton algorithm

Directed self-assembly (DSA) of block copolymers (BCPs) is one of the most promising developments in the cost-effective production of nanoscale devices. The process makes use of the natural tendency for BCP melts to form nanoscale structures upon phase separation. The phase separation can be directed through the use of chemically patterned substrates to promote the formation of morphologies that are essential to the production of semiconductor devices. Moreover, the design of substrate pattern can be formulated as an optimization problem for which we seek optimal substrate designs that effectively produce given target morphologies. In this paper, we adopt a phase field model given by a nonlocal Cahn–Hilliard partial differential equation (PDE) based on the minimization of the Ohta–Kawasaki free energy, and present an efficient PDE-constrained optimization framework for the optimal design problem. The design variables are the locations of circular- or strip-shaped guiding posts that are used to model the substrate chemical pattern. To solve the ensuing optimization problem, we propose a variant of an inexact Newton conjugate gradient algorithm tailored to this problem. Additionally, we demonstrate the effectiveness of our computational strategy on numerical examples that span a range of target morphologies. Owing to our second-order optimizer and fast state solver, the numerical results demonstrate five orders of magnitude reduction in computational cost over previous work. The efficiency of our framework and the fast convergence of our optimization algorithm enable us to rapidly solve the optimal design problem in not only two, but also three spatial dimensions.

97 MATHEMATICS AND COMPUTING↗