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DOE OSTI2024

Managing Subsurface Pressure Buildup and Interference in Commercial-Scale CO 2 Storage Project with Proximal Injection Wells

Large-scale decarbonization using carbon capture and storage (CCS) is likely to involve many commercial-scale CO 2 storage projects located in close proximity to each other. This close proximity raises concerns over pressure interference among the storage projects. Pressure interference between injection and storage efforts can reduce the practicable CO 2 storage resource and force wells to inject CO 2 at a lower rate to avoid the fracture pressure thresholds per United States Environmental Protection Agency (EPA) Class VI well regulations to preserve injection and confining zone integrity and potentially mitigate against inducing seismic activity. These analyses employ numerical full-physics reservoir modeling to evaluate how pressure buildup fronts and CO 2 plumes evolve under commercial-scale injection volumes of CO 2 in which multiple storage sites located in close proximity occur in tandem. The simulation models mimic injection at pseudo basin-scale and assume homogeneous saline formation(s) as storage targets with a pair of upper and lower homogeneous seal layer/s. These analyses specifically investigate the efficacy of two basin-wide reservoir pressure management strategies in addressing the technical challenges associated with pressure buildup and CO 2 plume commingling. The strategies explored include: 1) enlarging the area of injection well spacing (WS) and 2) storing CO 2 in a stacked sequence (SSS) of saline formations compared to a single formation. The storage and confining zones properties assumed were held common across the scenarios, unless specified otherwise. Analyses results show that after injecting 4 million tons per year for 30 years using 4 separate wells (each injecting 1 million metric tons per year), the radius of CO 2 plume extends to a mere 3 km or less from injection wells. Meanwhile, the radius of pressure buildup ranges on the order of tens to a few hundreds of kilometers, depending on the magnitude of pressure buildup threshold that one would use to define the front. CO 2 plume commingling from different injection wells appears to occur 50 years post-injection, especially under scenarios with narrowly spaced (i.e., < 5 km apart) injection well locations. Findings from sensitivity cases on the well spacing suggest that storage formations modeled would require different well spacing to avoid fracture pressure thresholds. For instance, modeled storage formations with high fracture gradients (i.e., 0.8 psi/ft) would need less than 5–km well spacing, whereas those with lower fracture gradients (i.e., 0.7 psi/ft) would need approximately 20–km well spacing, based on assumed modeling parameters. Under stacked injection, the pressure challenges (described above) still exist but are more alleviated due to distributing the same injection volume across more available reservoir volume. These analyses demonstrate that stacked-sequence storage can effectively address the challenges, while still providing the same target CO 2 storage volumes and allowing a large number of storage projects to be deployed in the same basin by better utilizing the available storage resource across different reservoir depths. Among cases modeled, the resulting pressure buildup front is most suppressed when each storage project distributes injection volumes over several wells, each of which injects a portion of the total CO 2 across the stacked sequence. This strategy results in the smallest CO 2 aerial footprint amongst scenarios evaluated but also shows the largest reduction in the pressure buildup at the top of perforation at the injection wells (upwards of approximately 42 percent compared to the commercial-scale single-formation storage), the result of which is crucial to maintain caprock integrity. The findings presented by this research draw attention to the importance of greater coordination among storage operators and regulatory stakeholders to foster the upscaling and deployment of CCS. These analyses provide insights into required decision-making when considering multi-project deployment in a shared basin. Because these analyses evaluate a very specific geologic situation, they bear further investigations across other geologic situations.

42 ENGINEERING↗

DOE OSTI2026

Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study

Neural scaling laws play a pivotal role in the performance of deep neural networks and have been observed in a wide range of tasks. However, a complete theoretical framework for understanding these scaling laws remains underdeveloped. In this paper, we explore the neural scaling laws for deep operator networks, which involve learning mappings between function spaces, with a focus on the Chen and Chen style architecture. These approaches, which include the popular Deep Operator Network (DeepONet), approximate the output functions using a linear combination of learnable basis functions and coefficients that depend on the input functions. We establish a theoretical framework to quantify the neural scaling laws by analyzing its approximation and generalization errors. We articulate the relationship between the approximation and generalization errors of deep operator networks and key factors such as network model size and training data size. Moreover, we address cases where input functions exhibit low-dimensional structures, allowing us to derive tighter error bounds. These results also hold for deep ReLU networks and other similar structures. Our results offer a partial explanation of the neural scaling laws in operator learning and provide a theoretical foundation for their applications.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2013

The NASA-Goddard Multi-Scale Modeling Framework - Land Information System: Global Land/atmosphere Interaction with Resolved Convection

The present generation of general circulation models (GCM) use parameterized cumulus schemes and run at hydrostatic grid resolutions. To improve the representation of cloud-scale moist processes and landeatmosphere interactions, a global, Multi-scale Modeling Framework (MMF) coupled to the Land Information System (LIS) has been developed at NASA-Goddard Space Flight Center. The MMFeLIS has three components, a finite-volume (fv) GCM (Goddard Earth Observing System Ver. 4, GEOS-4), a 2D cloud-resolving model (Goddard Cumulus Ensemble, GCE), and the LIS, representing the large-scale atmospheric circulation, cloud processes, and land surface processes, respectively. The non-hydrostatic GCE model replaces the single-column cumulus parameterization of fvGCM. The model grid is composed of an array of fvGCM gridcells each with a series of embedded GCE models. A horizontal coupling strategy, GCE4fvGCM4Coupler4LIS, offered significant computational efficiency, with the scalability and I/O capabilities of LIS permitting landeatmosphere interactions at cloud-scale. Global simulations of 2007e2008 and comparisons to observations and reanalysis products were conducted. Using two different versions of the same land surface model but the same initial conditions, divergence in regional, synoptic-scale surface pressure patterns emerged within two weeks. The sensitivity of largescale circulations to land surface model physics revealed significant functional value to using a scalable, multi-model land surface modeling system in global weather and climate prediction.

cumulus schemes↗

NASA NTRSDate not supplied

Scaling Methodology for Buckling of Composite Conical Shells in Axial Compression

Conical shells are commonly used as structural components for launch vehicles. The axial compression experienced during launch is one of the sizing load cases, because it can lead to loss of stability. As Because experimentally testing these large full-scale structures is cumbersome and expensive, it is studied how reduced-scale shells can be designed such that their buckling behavior is representative of the full-scale scale shell behavior. An analytical, sequential scaling procedure methodology is developed based on the nondimensional governing equations for composite conical shells with a symmetric, balanced layup and negligible flexural anisotropy. The buckling behavior of the shells of different size is compared using linear and nonlinear finite element analyses, and good comparisons . Accurate results are obtained for the considered shells in terms of buckling load, displacement, and mode. The inclusion of a geometric imperfections affects reduces the prediction accuracy, but it does not to the extent that the methodology is no longer valid cause the methodology to fail.

Buckling↗

DOE OSTI2020

Application of Dynamical System Scaling to Bubble Dynamics

Dynamical System Scaling (DSS) provides a useful method for analyzing, categorizing and scaling time-dependent processes. A key feature of DSS is the temporal displacement rate, D, which relates the natural process time to the reference clock time. Because of its property of being invariant under a two-parameter affine transformation, it provides an underlying basis for process scaling. Application of DSS to bubble dynamics serves to elucidate the physical significance of the temporal displacement rate and its role in scaling a variety of bubble dynamics processes. This paper shows that the temporal displacement rate consists of the sum of dimensionless groups that govern the bubble dynamics. Preserving the temporal displacement rate for a prototype and a scaled model results in similitude of the time-dependent normalized bubble size, growth rates and interfacial accelerations for different fluid conditions. For vapor bubble growth in a superheated liquid, it is shown that inertial controlled bubble growth occurs when D=0 and thermally controlled bubble growth occurs when D=1 .

dynamical system scaling, SMR, bubble dynamics, DS↗

DOE OSTI2021

Why Plan for Utility-Scale Solar?

The Solar@Scale project is a DOE Solar Energy Technologies Office-funded collaboration between the American Planning Association and International City/County Management Association (ICMA) to provide best practices and guidance for local communities in planning for large-scale PV development. As a member of the Solar@Scale advisory committee, I have been asked to present on trends and land use projections for large-scale solar to demonstrate why local governments will likely see growing demand for solar energy development. These slides represent an introduction to projections from NREL's Standard Scenarios, Cambium, LBNL's Utility-Scale Solar report, and other data sources on the level of solar energy development that can be expected across the country.

48 EE - Weatherization and Intergovernmental Progr↗

DOE OSTI2022

Application of Dynamical System Scaling for Accelerated Fuel Qualification Efforts

There are considerable ongoing research and testing campaigns to qualify new fuel designs such as metallic fuels for advanced reactor designs and the Accident Tolerant Fuel (ATF) campaign for light water reactors (LWR). The typical research and development lifecycle needed to qualify a nuclear fuel design, under ideal conditions, can take up to 20-25 years which limits the ability for new fuels to make fast deployments into commercial, test, and research reactors. While there are several efforts to accelerate nuclear fuel qualification through advanced modeling coupled with state-of-the-art experiments, there is not yet a framework to methodically quantify and rank distortions occurring in experimental test specimens needed to validate nuclear fuel performance codes. This study uses the Dynamical System Scaling (DSS) methodology to quantify transient scaling distortions that occur over experiment and simulated transients. DSS analysis metrics provide a researcher with several tools and information embedded within test data to identify dominant phenomena, associated timescales, and quantify a transient?s overall dynamic distortion. This is demonstrated with separate analyses of the SETH-C and CHF-SERTTA thermal-hydraulic experiments in the TREAT facility at Idaho National Laboratory. The outcome of this work is a scaling and data analysis approach to account for the influence a scaled fuel test specimen?s geometric and temporal distortions have on its ability to be representative of the full-scale design.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2023

EVs@Scale Lab Consortium Semi-Annual Stakeholder Meeting

The U.S. Department of Energy (DOE) Electric Vehicles at Scale Lab Consortium (EVs@Scale Lab Consortium) is accelerating research to support the establishment of a secure and scalable national network of charging infrastructure. This network will be critical to support tens of millions of light-, medium-, and heavy-duty EVs on American roads by 2030. The EVs@Scale Lab Consortium brings together national laboratories and key stakeholders to conduct infrastructure research and development (R&D) that advances innovations in, and sets unified standards for, high-power and wireless charging. The effort will also develop technologies to integrate vehicle charging with the power grid, and develop cybersecurity measures to protect drivers, vehicles, equipment, and the grid. The first hybrid EVs@Scale Lab Consortium Semiannual Stakeholder Meeting was held at ANL on September 27-28, 2023, to identify research, development, and deployment needs to accelerate technology development for electric vehicles at scale and explore opportunities for collaboration across government, academia, and industry.

advanced charging and grid interface technologies↗

DOE OSTI2024

2024 Electric Vehicles at Scale Semiannual Stakeholder Meeting

advanced charging and grid interface technologies↗

DOE OSTI2022

Experimental validation of multiphysics model simulations of the thermal response of a cement clinker rotary kiln at laboratory scale

Abstract An increasing demand for buildings, transportation systems and civil infrastructure development has driven expansion of cement consumption world‐wide, producing a significant increase in related global energy demand. With approximately 7% of the world‐wide industrial energy consumption (10.7 exajoules [EJ]), the cement industry is the third most energy intensive industrial processes and a key component for concrete, the most consumed composite material in the global construction industry. In cement manufacturing, the cement kiln accounts for most of the energy consumption in the production process. As the heart of a cement plant, the cement kiln is where the kiln feed primarily containing calcium oxide (CaO), silica (SiO 2 ), alumina (Al 2 O 3 ), and iron (Fe 2 O 3 ) are thermally and chemically transformed into clinker minerals. The presented work developed a multiphysics model, designed and built a laboratory‐scale rotary cement clinker kiln, and produced cement clinker at laboratory‐scale. The model was developed to study the interaction between the various thermal, fluid dynamic and chemical interactions involved in the sintering process used to form Portland cement clinker in an effort to reduce energy use. The analytical model was validated through experimental testing using a unique laboratory‐scale rotary cement kiln developed during the investigation. Also demonstrated was the feasibility of producing clinker at laboratory scale. This modeling and lab scale tests were designed to better understand the clinker sintering process so that operational and quality decisions can be made to optimize energy consumption without compromising cement clinker quality. The computational fluid dynamics modeling was developed in COMSOL Multiphysics 6.0. The characteristics of the combustion fluid flow, concentration of species, temperature and heat transfer were studied for a turbulent flow of methane (CH 4 ) gas and oxygen (O 2 ). Theory suggests that heat transfer impacts the cement production process but the multiphysics model more accurately describes the convection, conduction, and radiant heat transfer in the kilning process and thus allows for a better understanding of the energy exchange driving the chemical reactions that produce Portland cement. Clinker minerals were formed because of appropriate burning conditions implemented during experimental model validation.

Tabares, Juan David↗

DOE OSTI2020

Applying cumulative effects to strategically advance large-scale ecosystem restoration

Programs restoring large terrestrial-aquatic ecosystems are expected to increase in area and number over the next decade. Factors ensuring long-term effectiveness at the program scale differ from those at the individual restoration project scale, necessitating a new synthesis. Here we contextualize hydrological, geomorphological, and biological effects of large-scale ecosystem restoration activities within a cumulative-effects paradigm. Evidence of eight modes of cumulative effects benefiting species and ecosystems was elucidated by reviewing restoration efforts in the greater Florida Everglades, Gulf Coast, lower Columbia River and estuary, Puget Sound estuary, Missouri River, northeastern coastal states, San Francisco Bay and Sacramento Delta. Additive and nonlinear cumulative effects of interacting restoration projects propagated across landscape and regional geographic scales, and throughout trophic levels, affecting population structures. Applying a cumulative-effects paradigm to planning, implementing, and evaluating large-scale restoration will avoid countervailing effects and capitalize on positive feedback, advancing evidence-based programs to recover priority species and ecosystems.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Implications from Monitoring Gopher Tortoises at Two Spatial Scales

A problem that conservation biologists face is how to monitor species, given both resource limitations and the inherent challenges of assessing long-term demographic processes. We assessed gopher tortoise (Gopherus polyphemus) abundance at a landscape scale and at the scale of 3 local populations within the Conecuh National Forest (CNF).USA between 1991 and 2017. Landscape-level data were collected 26 from line transect distance sampling arranged uniformly across the CNF and collected during a single season (2011); data for local populations were generated from long-term mark-recapture of individuals at three sites selected based on prior knowledge of high density at each. At a landscape scale, we estimated 5,242 (95% CI = 3,538-7,768) tortoises occurred across the approximately 34,000-ha forest, yielding a density of 0.14-0.32 tortoises/ ha. These low densities across the landscape suggest that, on average, management activities across the property have not allowed tortoise populations to retain social structure needed for long-term persistence. The three local populations, however, contained 25-60 individuals and densities of 1.9–6.9 tortoises/ha. Over the study period, populations at two sites were stable and the third experienced significant population growth. Mean annual survival of individuals was 0.89 and invariant across size classes. Altogether, line transect distance sampling is important for assessing landscape-scale abundance of tortoises but may fail to detect local clusters of high-density sites important for population persistence. Our mark-recapture efforts at the local scale revealed that small populations on these high-density sites can exhibit long-term stability or growth even though they do not meet current established criteria for viability. Improved models that incorporate immigration and emigration and better reflect dynamics of peripheral populations would assist in determining how such populations best contribute to species recovery and regional conservation targets.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2022

A micromorphic filter for determining stress and deformation measures from direct numerical simulations of lower length scale behavior

A micromorphic filter is presented for the extraction of stress and deformation measures from underlying direct numerical simulation (DNS) of microstructural mechanical response. The filter is consistent with the micromorphic continuum theory of Eringen and Suhubi (1964), and allows the interrogation of a DNS to generate higher order constitutive models. The filter adopts aspects of the overlap coupling techniques for atomistic-continuum scale-bridging in order to update the nodal degrees of freedom of the finite-element-based filter. Here, two methods for the extraction of quantities of interest from the underlying DNS are presented. The first is a least squares fitting technique which is appropriate for finite-element-style macro-scale domains, and the second is a variationally based approach which generalizes the method to any macro-scale domain which has a variational form. No restrictions on the form of the micro-scale simulation are inherent to the approach, provided that the required quantities (stress, position, etc.) can be provided. The filter is applied to a simple homogeneous medium undergoing homogeneous deformation to demonstrate the recovery of a classical continuum response, what we call the null case. We then apply the framework to the same homogeneous medium with a central and edge penny-shaped crack to show micromorphic effects. We demonstrate that varying the size of the filter causes the micromorphic behaviors to arise and then dissipate as the filter becomes smaller than the crack's length scale. Additionally we show the effect of filter size on a chiral element network along with a plate perforated with evenly spaced and perturbed holes through the thickness. The micromorphic effects are shown to be present for the chiral elements under any loading but become most pronounced when a single element is located within each averaging domain. Micromorphic effects become pronounced for the plate upon the onset of localization.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Higgs coupling measurements and the scale of new physics

A primary goal of present and future colliders is measuring the Higgs couplings to Standard Model (SM) particles. Any observed deviation from the SM predictions for these couplings is a sign of new physics whose energy scale can be bounded from above by requiring tree-level unitarity. In this paper, we extend previous work on unitarity bounds from the Higgs cubic coupling to Higgs couplings to vector bosons and top quarks. We find that HL-LHC measurements of these couplings compatible with current experimental bounds may point to a scale that can be explored at the HL-LHC or a next-generation collider. Our approach is completely model-independent: we assume only that there are no light degrees of freedom below the scale of new physics, and allow arbitrary values for the infinitely many couplings beyond the SM as long as they are in agreement with current measurements. We also extend and clarify the methodology of this analysis, and show that if the scale of new physics is above the TeV scale, then the deviations can be described by the leading higher-dimension gauge invariant operator, as in the SM effective field theory.

43 PARTICLE ACCELERATORS↗

DOE OSTI2026

Spaced out: An economic framework to explore the impacts of PV panel spacing on large-scale farming in Colorado

CONTEXT Agrivoltaic systems co-locate solar technologies with agricultural operations on an integrated plot of land and potentially provide benefits to both energy and agricultural systems. To date, large-scale (>5-MW) agrivoltaic projects in the United States have been limited to grazing and ecovoltaic applications, raising questions about the impact and scalability of agrivoltaic crop systems. Many agrivoltaic designs raise the height of the solar panels to accommodate agricultural practices while keeping energy density high. However, raising the panels results in increased photovoltaic (PV) development costs, which often are higher than the economic returns of crop production underneath the panels. This leads to unfavorable project economics and the need for other agrivoltaic solutions than raising panels. OBJECTIVE To explore other solutions, we perform an initial feasibility analysis for an agrivoltaic solution that can integrate with large-scale farming practices by increasing the row spacing in between panels. Increased PV row spacing is a low-cost approach for scaling agrivoltaics to accommodate crop production and this spacing can be tailored to required crop equipment for different regions. Increasing row spacing will reduce the power density (PV installed per acre), but in areas that are not land limited, these agrivoltaic designs could be economically feasible. Our analysis establishes a framework for a feasibility analysis for where and with what crops spaced out panel agrivoltaic solutions might be economical. METHODS Using a case study for large-scale agriculture crops in Colorado, we establish a framework for wide-row agrivoltaic economic feasibility analysis. We utilized the System Advisor Model to calculate technoeconomic metrics to compare different row spacing solutions and capture tradeoffs of these system designs. RESULTS AND CONCLUSIONS We find that, in some circumstances, wider row agrivoltaic solutions that allow for continued mechanized crop production can provide economic benefits over a traditional utility-scale PV system. For most crops examined in this analysis, roughly $\$$200/acre in agricultural profit justified spacing out the panels to at least 31.7 ft. to accommodate agrivoltaic configurations versus PV only configurations. Additionally, opportunities for increased agricultural revenue with agrivoltaic systems allow PV project economics to tolerate a larger range of CAPEX variability while remaining economically viable relative to the PV only configurations. SIGNIFICANCE This framework can be adapted for a wide variety of crops and regions and allows for examination of economically favorable sites for future agrivoltaic systems that utilize different configuration and expand opportunities for agrivoltaics.

14 SOLAR ENERGY↗

DOE OSTI2022

Scale-dependent enhancement of productivity and stability in xenic Nannochloropsis cultures

Although bacterial amendments can enhance algal productivity and stability, the development of algal-bacteria consortia for commercial-scale utilization is limited. Here, for this work, we used an established high-throughput approach to generate algal-bacteria consortia, and tested consortia performance at spatial scales from microplates to 320 L raceway ponds. We used both lab and field-reared strains of Nannochloropsis oceanica to build consortia. In some experiments, we imposed environmental perturbations to test the ability of bacteria to enhance algal culture stability. In repeated assays at the scale of well plates, flasks, and bioreactors, strong effects of bacterial amendments on N. oceanica were observed. These effects were most dramatic when cultures experienced stressors such as temperature perturbations or removal of CO 2 augmentation. Isolates that were advanced for field testing included species in the genera Algoriphagus, Oceanicaulis, and Marinobacter. When consortia were generated in the field, positive effects of bacterial amendments were not observed. The amended bacteria were outcompeted, and bacterial community composition across treatments converged after the first grow out. These results highlight the complexity of using consortia in open systems, where interactions between the existing bacterial community, inoculated bacteria, and changing environmental conditions are layered upon other differences in scale and cultivation regimes. Moreover, functionally redundant bacteria are likely present in the field. Following this work, we hypothesize that tight interactions (e.g., obligate relationships between partners) will scale more predictably to outdoor systems. We suggest relying on true synthetic ecological approaches in which the relationships between bacteria and algae partners are well understood, or synthetic ecological approaches coupled with high throughput approaches to design and test consortia. We also recommend future work to examine the effect of algae-bacteria inoculation ratios on productivity and stability, track dynamics of partners through time, and manage ponds to retain beneficial symbioses.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2022

Comparing and contrasting In-Vial and full-scale systems for sparging volatile analytes

Here, in-vial sparging was demonstrated as an effective, practical alternative to a full-scale sparging system for supporting the analysis of volatile constituents. Using elemental mercury (Hg 0 ) and toluene as representative purgeable analytes, the mass removal for various sparge configurations was measured and a reduced order model was developed and validated. In the primary experiments, Hg 0 in the sparge gas was trapped on activated carbon or gold, thermally desorbed, and quantified using atomic absorption or atomic fluorescence spectroscopy. Toluene experiments using the same in-vial sparge apparatus and sparge parameters were performed to demonstrate the applicability of the reduced order model to a broad range of compounds. Toluene removal was tracked by measuring the remaining toluene in sparged aliquots using Ultraviolet-visible (UV–Vis) spectroscopy. For the sparging, flow rates varied from 25 to 75 mL/min for periods from 0 to 30 min. Sparge performance, mass removal as a function of time, and sparge gas volume were measured for both in-vial and full-scale systems. A model based on dimensionless Henry's Law coefficient, normalized sparge gas volume, and fractional extent of equilibrium matched the experimental data for both compounds and provides a practical tool for future applications. For the conditions tested in this study, the calibrated model indicated that the sparge gas in the in-vial system reached approximately 33% of its equilibrium value before exiting the water surface, while a full-scale system reached approximately 100%. The tests validated the quality, reproducibility, and predictability of sparging performance for both full scale and in-vial sparge systems. Related factors such as waste generation, worker risk, and labor were also assessed. Full scale sparge systems provide the advantage of lower detection levels due to larger sample volume, while the in-vial sparge systems provide advantages for most other factors; including automatability, reducing secondary wastes, lessening the need to clean and check the sparge apparatus, and lowering labor and costs. The data and associated reduced order model support continued development and deployment of in-vial sparge platforms as a practical option for analysis of purgeable analytes such as volatile organic compounds and volatile metals/organometallics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

A multiscale design method using interpretable machine learning for phononic materials with closely interacting scales

Manipulating the dispersive characteristics of vibrational waves is beneficial for many applications, e.g., high-precision instruments. architected hierarchical phononic materials have sparked promise tunability of elastodynamic waves and vibrations over multiple frequency ranges. In this article, hierarchical unit-cells are obtained, where features at each length scale result in a band gap within a targeted frequency range. Our novel approach, the ‘‘hierarchical unit-cell template method,’’ is an interpretable machine-learning approach that uncovers global unit-cell shape/topology patterns corresponding to predefined band-gap objectives. A scale-separation effect is observed where the coarse-scale band-gap objective is mostly unaffected by the fine-scale features despite the closeness of their length scales, thus enabling an efficient hierarchical algorithm. Moreover, the hierarchical patterns revealed are not predefined or self-similar hierarchies as common in current hierarchical phononic materials. Furthermore, our approach offers a flexible and efficient method for the exploration of new regions in the hierarchical design space, extracting minimal effective patterns for inverse design in applications targeting multiple frequency ranges.

Architected materials↗