Search NASA⌕ Search

SEARCH · Search NASA

Results for “Surface integration”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Finite deformation implementation of a mixed-mode single-integral type cohesive zone with reorienting surfaces of separation

To model material ductile failure and crack propagation, cohesive zone elements can be embedded along potential fracture paths in a finite element simulation. When damage criteria are met, elements in the mesh decohere, simulating the formation and propagation of a crack. In this paper, we present a novel computational algorithm based on finite deformation theory, essential to modeling crack initiation and growth in solids undergoing large deformations. This new algorithm was formulated within a Lagrangian frame of reference to extend previous cohesive zone algorithms to include modeling crack growth in finite deformation contexts. The local coordinate system, necessary for defining an embedded cohesive zone, is constructed based upon the current configuration and is updated within the nonlinear iteration process, thereby resulting in the convergence of the solution for a growing crack in a large deformation quasi-static setting. The model’s accuracy was demonstrated by comparing finite element model simulation results with the analytic case of a constant surface separation, as shown in the verification examples. The power and efficacy of the algorithm to capture large deformations during crack growth were then demonstrated with a double cantilever beam example case. It indicates that the model can be applied to a variety of physical circumstances for predicting crack initiation and growth with delamination and fracture.

42 ENGINEERING↗
DOE OSTI2020

An arbitrarily high-order three-dimensional Cartesian-grid method for reconstructing interfaces from volume fraction fields

Here Tthis work describes a newly developed, arbitrarily high-order Cartesian-grid method for reconstructing material interfaces from a volume fraction field. The method begins by identifying all of the grid cells in the volume fraction field that are intersected by the interface and need to be approximated by the reconstruction scheme. Finite-differences are used to calculate the gradient of the volume fraction field and provide an estimate of the surface normal in all of the interfacial grid cells. Groups of connected grid cells are then identified which all have the same dominant component of the normal vector. This grouping by orientation determines the proper dependent variable to use in the surface reconstruction (e.g. for a 2D curve, this step determines if the surface will be approximated by a function of x or y). A cumulative integral over the surface is constructed and fit using b-splines for two-dimensional problems or tensor-product b-splines for three-dimensional problems. This construction allows for the interface to be recovered through application of the second fundamental theorem of calculus. Fitting the cumulative integral with $\mathscr{N}$ th-order b-splines (or tensor-product b-splines) yields an ($\mathscr{N}$-1) th-order convergence rate of the interface shape. Differentiation of the b-spline interface function(s) allows for the high-order approximation of the normal vector and curvature to be obtained directly anywhere along b-spline. Together, the proposed reconstruction technique can achieve arbitrarily high mesh convergence rates. Validation tests are presented with mesh convergence rates ranging from fourth- to tenth-order.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Chip-integrated metasurface full-Stokes polarimetric imaging sensor

Abstract Polarimetric imaging has a wide range of applications for uncovering features invisible to human eyes and conventional imaging sensors. Chip-integrated, fast, cost-effective, and accurate full-Stokes polarimetric imaging sensors are highly desirable in many applications, which, however, remain elusive due to fundamental material limitations. Here we present a chip-integrated Meta surface-based Full-Stokes Polar imetric Im aging sensor (MetaPolarIm) realized by integrating an ultrathin (~600 nm) metasurface polarization filter array (MPFA) onto a visible imaging sensor with CMOS compatible fabrication processes. The MPFA is featured with broadband dielectric-metal hybrid chiral metasurfaces and double-layer nanograting polarizers. This chip-integrated polarimetric imaging sensor enables single-shot full-Stokes imaging (speed limited by the CMOS imager) with the most compact form factor, records high measurement accuracy, dual-color operation (green and red) and a field of view up to 40 degrees. MetaPolarIm holds great promise to enable transformative applications in autonomous vision, industry inspection, space exploration, medical imaging and diagnosis.

47 OTHER INSTRUMENTATION↗
DOE OSTI2021

Hierarchical Tactile Sensation Integration from Prosthetic Fingertips Enables Multi-Texture Surface Recognition

Multifunctional flexible tactile sensors could be useful to improve the control of prosthetic hands. To that end, highly stretchable liquid metal tactile sensors (LMS) were designed, manufactured via photolithography, and incorporated into the fingertips of a prosthetic hand. Three novel contributions were made with the LMS. First, individual fingertips were used to distinguish between different speeds of sliding contact with different surfaces. Second, differences in surface textures were reliably detected during sliding contact. Third, the capacity for hierarchical tactile sensor integration was demonstrated by using four LMS signals simultaneously to distinguish between ten complex multi-textured surfaces. Four different machine learning algorithms were compared for their successful classification capabilities: K-nearest neighbor (KNN), support vector machine (SVM), random forest (RF), and neural network (NN). The time-frequency features of the LMSs were extracted to train and test the machine learning algorithms. The NN generally performed the best at the speed and texture detection with a single finger and had a 99.2 ± 0.8% accuracy to distinguish between ten different multi-textured surfaces using four LMSs from four fingers simultaneously. The capability for hierarchical multi-finger tactile sensation integration could be useful to provide a higher level of intelligence for artificial hands.

Abd, Moaed A. (ORCID:0000000284954244)↗
DOE OSTI2025

A double copy from twisted (co)homology at genus g

We study a family of generalized hypergeometric integrals defined on punctured Riemann surfaces of genus g. These integrals are closely related to g-loop string amplitudes in chiral splitting, where one leaves the loop-momenta, moduli and all but one puncture un-integrated. We study the twisted homology groups associated to these integrals, and determine their intersection numbers. We make use of these homology intersection numbers to write a double-copy formula for the "complex" version of these integrals -- their closed-string analogues. To verify our findings, we develop numerical tools for the evaluation of the integrals in this work. This includes the recently introduced Enriquez kernels -- integration kernels for higher-genus polylogarithms.

Pokraka, Andrejz [Brown University; Amsterdam Univ↗
DOE OSTI2022

Solving Coupled Surface and Subsurface Flow with Multirate Time Integration [Slides]

This report details an end of summer internship. The report lists the objective, "Add more multirate time integration methods to Amanzi" and concludes that, "Multirate methods can be used to speed up simulations and to get higher orders of accuracy", and "Coupled surface and subsurface simulations could benefit from using multirate schemes".

47 OTHER INSTRUMENTATION↗
DOE OSTI2022

Surface analysis insight note: Synthetic line shapes, integration regions and relative sensitivity factors

Here, methods for estimating photoemission intensity from X-ray photoelectron spectroscopy data are examined. The role played by “synthetic” bell-shaped curves, integration intervals, background curves, and the use of relative sensitivity factors (RSFs) in reporting percentage atomic concentration for a sample is presented. In particular, photoemission lines with differing energy distributions obtained from the NaCl sample surface are used to demonstrate how a comparison of photoemission intensities is dependent on the line shapes, background curves, and appropriate use of RSFs.

42 ENGINEERING↗
DOE OSTI2025

Evaluating post-fire watershed response to varying burn severity and precipitation regimes using fully-distributed and integrated hydrologic models

Wildfires can cause significant changes in vegetation and soil, which may lead to increased surface runoff and soil erosion, thereby affecting water cycling within ecosystems. This study uses the Advanced Terrestrial Simulator (ATS), an integrated and fully distributed hydrologic model at the watershed scale, to examine post-fire hydrologic responses in selected watersheds with varying burn severities in the Pacific Northwest region of the United States. The model integrates surface overland flow, subsurface flow, and canopy biophysical processes. We have developed a new fire module in ATS to account for changes in soil hydraulic properties caused by fire in the topsoil layer. Modeling results indicate that, in the year following a high-severity burn, watershed-averaged evapotranspiration decreases by 25%. Additionally, post-fire peak flows increase by 18-29% in watersheds burned with medium to high severity due to changes in soil properties. Conversely, a low-severity burn results in less than a 1% increase in post-fire peak flow. Furthermore, a high-severity fire causes a 38% reduction in the infiltration rate within the affected watershed during the first post-fire wet season. Hypothetical numerical experiments with varying precipitation regimes after a high-severity fire show that post-fire peak flows can increase by 1-29% due to fire-induced changes in soil hydraulic properties. This study highlights the importance of using fully distributed hydrologic models to quantify disturbance-feedback loops, which are essential for understanding the complexities brought about by spatial heterogeneity in post-fire landscapes.

ATS↗
DOE OSTI2026

Unlocking the Future of Aircraft Manufacturing: The Environmental Benefits of Laser Patterning for Surface Enhancement of Aircraft-Certified Alloys

Surface protection and functional modification of aircraft-certified aluminum alloys are essential for corrosion resistance, durability, and long-term airworthiness. At the same time, increasingly restrictive environmental regulations motivate the development of alternatives to legacy wet-chemical surface treatments. This study presents an integrated assessment of ultrafast femtosecond laser surface texturing as a surface functionalization approach for Aluminum 6061 alloys within an aerospace manufacturing and sustainability context. Ultrashort-pulse laser processing enables controlled micro- and nano-scale surface topographical modification with limited thermal impact, allowing adjustment of wettability and surface functionality while preserving bulk material integrity. As a dry and contactless process, femtosecond laser treatment eliminates the use of hazardous chemicals, reduces consumable inputs, and generates minimal secondary waste. A streamlined cradle-to-gate life cycle assessment conducted in accordance with ISO 14040/14044 indicates a lower global-warming potential per functional unit compared with conventional surface treatments, including anodization, plasma-assisted coatings, and organic coating systems. Complementary qualitative analyses addressing environmental health and safety, supply-chain risk, and ESG alignment indicate potential advantages related to occupational safety, regulatory compliance, waste management, and end-of-life recyclability. The investigation is performed on planar Aluminum 6061 reference surfaces with a treated area of 25 mm 2 , providing a controlled laboratory-scale basis for analyzing process behavior, functional surface modification, and associated environmental metrics. Within this defined scope, the results support further evaluation of femtosecond laser surface texturing as a surface engineering option for future aerospace manufacturing.

corrosion resistance↗
DOE OSTI2023

Kinetically versus thermodynamically controlled factors governing elementary pathways of GaP(111) surface oxidation

GaP and related III-V semiconductors have attracted interest as high-efficiency photoelectrochemical electrodes for solar water splitting. However, their efficacy is linked to the presence, identity, and integrity of native surface oxides, which are structurally and chemically complex and evolve during operation. Using ambient pressure X-ray photoelectron spectroscopy (APXPS) coupled with ab initio simulations, we track key chemical motifs expressed during evolution of GaP(111) surface oxides, their associated reaction kinetics, and their correlations with electronic properties. We identify two distinct thermal regimes corresponding to kinetically and thermodynamically controlled oxidation. Below 600 K, exposure to O2 generates kinetically facile Ga-O-Ga configurations, whereas higher temperatures cause activated oxygen to insert into Ga–P bonds as part of a thermodynamically driven transformation into a complex, heterogeneous 3D network of surface POx (1≤x ≤ 4) groups and Ga2O3 species, the latter of which eventually dominates upon depletion of surface phosphorus. Here our study highlights the critical competition between kinetic and thermodynamic factors during GaP oxidation, yielding insights for fabricating stable III-P-based photoelectrodes with precisely engineered surface properties.

08 HYDROGEN↗
DOE OSTI2025

Integrating Characteristic Arctic Vegetation in a Land Surface Model Improves Representation of Carbon Dynamics Across a Tundra Landscape

Arctic warming is altering vegetation and carbon dynamics with global implications, yet Earth System Model (ESM) predictions in the Arctic remain highly uncertain, in part due to historically limited data for model parameterization and validation. As such, ESMs typically represent Arctic ecosystems in an oversimplified manner. Recently, nine plant functional types (PFTs) designed to realistically represent tundra vegetation were integrated into the Energy Exascale Earth System Model (E3SM) Land Model (ELM) and parameterized using plot-scale observations from a single site. Additional evaluation was needed to determine their transferability across the Arctic. Here, in this study, we evaluated whether refined representation of tundra vegetation improved model accuracy by conducting spatially explicit 100 × 100 m resolution ELM simulations on Alaska's Seward Peninsula. Simulations with the default two-PFT configuration and with the nine Arctic-specific PFTs were benchmarked against observations of net ecosystem exchange, gross primary production, and aboveground biomass from multiple data streams including an eddy covariance flux tower, flux chambers, and aircraft and unoccupied aerial system hyperspectral remote sensing. Evaluation revealed that Arctic-specific PFT simulations produced more realistic landscape-level carbon exchanges, and better captured observed heterogeneity in biomass and productivity, explaining 60%–70% of spatial variance (R 2 = 0.6–0.7) compared to just 12%–18% (R 2 = 0.12–0.18) with the default configuration. However, the refined model failed to reproduce observed aboveground biomass for highly productive alder-willow communities, requiring further evaluation of carbon allocation parameterizations for tall shrubs that are increasingly expanding across tundra landscapes. Our results demonstrate that enhanced representation of vegetation heterogeneity boosts predictive understanding of tundra carbon dynamics, facilitating regional to pan-Arctic model and remote-sensing scaling.

Murphy, Bailey A. [Oak Ridge National Laboratory (↗
DOE OSTI2023

Preliminary verification of the MCNP perturbation and fixed-source tally sensitivity tools

Integral benchmark experiments are vital in the adjustment and validation of the nuclear data that govern predictive simulations across the nuclear community. The nuclear data sensitivity capabilities of the Monte Carlo N-Particle (MCNP ®) transport code are currently limited; however, expanding sensitivity capabilities will allow benchmark experiments to be designed to resolve compensating errors and adjust nuclear data where previously prohibitively difficult. This paper provides details of a preliminary verification for the use of (i.) the recently revised perturbation and (ii.) developmental fixed-source sensitivity tools within MCNP to calculate sensitivities of tallied responses (such as current integrated over a surface, F1, and flux averaged over a cell, F4) to nuclear data in fixed-source simulations. Energy-binned and energy-integrated sensitivities calculated with these tools are compared against sensitivities calculated using a central-difference approximation. Here, the verification is completed for four configurations of a benchmarked system using a 4.5-kg plutonium sphere surrounded by varying amounts of copper and/or polyethylene. The results show that sensitivities calculated with the perturbation and fixed-source sensitivity tools agree with the central-difference-based approach.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Integrating New Arctic Plant Functional Types in a Land Surface Model Using Above- and Belowground Field Observations: Modeling Archive

This Modeling Archive is in support of an NGEE Arctic publication "Integrating New Arctic Plant Functional Types in a Land Surface Model Using Above- and Belowground Field Observations". We used field observations of biomass and leaf traits (Salmon et al., 2019b) across a gradient of plant communities on the Seward Peninsula in northwest Alaska (Breen et al., 2020; Iversen et al., 2017) to replace the original ELM configuration with nine, arctic-specific PFTs. Original arctic PFTs in the model were (1) broadleaf deciduous boreal shrub and (2) C3 arctic grass. New PFTs were (1) lichen, (2) bryophyte, (3) evergreen dwarf shrub, (4) deciduous dwarf shrub, (5) deciduous low shrub, (6) deciduous low to tall shrub, (7) alder shrub, (8) forb, and (9) graminoid. New PFTs were parameterized and evaluated using site-level trait and biomass measurements (Salmon et al., 2019b), root C:N ratios and rooting depth patterns (Salmon et al., 2019c), and root biomass (Salmon et al., 2020). This archive contains forcing data and model output from ELM simulations conducted at the Kougarok Hillslope site (Kougarok Road Mile Marker 64). Simulations were conducted for the Kougarok Hillslope site using meteorological driving data from the Scenarios Network for Alaska and Arctic Planning (SNAP) downscaled climate projection dataset using NCAR-CCSM forcing (Bieniek et al., 2020 dataset; Walsh et al., 2018). The archive includes three versions of model simulations representing a progression from the original model configuration to an Arctic-specific configuration (see Output section below). Simulation 1 used grid cell data from a global E3SM configuration, including the fractional area of E3SM PFTs assigned to the grid cell containing the Kougarok Hillslope site in global simulations. This simulation did not distinguish between different plant communities on the Kougarok Hillslope, but instead used a single point simulation to represent the entire area. Simulation 2 used default E3SM PFT definitions combined with adjusted depth to bedrock for plant communities with shallow rocky layers to represent the role of abiotic soil factors in driving site differences (based on depth measurements of Iversen et al., 2019a), and adjusted the relative areas of the broadleaf deciduous boreal shrub and C3 arctic grass PFTs to reflect the observed spatial coverage of shrub and non-shrub PFTs across Kougarok Hillslope plant communities. Simulation 3 used the new arctic PFT definitions and parameterizations based on the vegetation types present at the site, along with the adjusted soil depths used in Simulation 2. Included are *.pdf, *.nc (NetCDF), *tar.gz, *.py, and *.bash files. We would like to thank Mary's Igloo Native Corporation for allowing us to perform this research on their land and giving us the opportunity to share this research with their community. Thanks to Shawn Serbin and Alistair Rogers for helpful discussions. Holly Vander Stel, Joanne Childs, and Stan Wullschleger helped with collection of field data. The Next-Generation Ecosystem Experiments: Arctic (NGEE Arctic), was a research effort to reduce uncertainty in Earth System Models by developing a predictive understanding of carbon-rich Arctic ecosystems and feedbacks to climate. NGEE Arctic was supported by the Department of Energy's Office of Biological and Environmental Research. The NGEE Arctic project had two field research sites: 1) located within the Arctic polygonal tundra coastal region on the Barrow Environmental Observatory (BEO) and the North Slope near Utqiagvik (Barrow), Alaska and 2) multiple areas on the discontinuous permafrost region of the Seward Peninsula north of Nome, Alaska. Through observations, experiments, and synthesis with existing datasets, NGEE Arctic provided an enhanced knowledge base for multi-scale modeling and contributed to improved process representation at global pan-Arctic scales within the Department of Energy's Earth system Model (the Energy Exascale Earth System Model, or E3SM), and specifically within the E3SM Land Model component (ELM).

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

In-Situ Nanoscale Focusing of Extreme Ultraviolet Solid-State High Harmonics

Extreme ultraviolet light delivering radiation with a wavelength shorter than approximately 100 nm is now available from solid-state sources. However, despite exceptional progress, efficient focusing of extreme ultraviolet photons to their ultimate diffraction limit remains a formidable challenge because of the precision of the focusing by curved, optical surfaces. Here we integrate coherent short-wavelength high-order harmonics from a MgO crystal, with a high-numerical-aperture, nanostructured, focusing element etched onto the surface of the crystal itself. We focus extreme ultraviolet light, seventh harmonic of an 800-nm laser, with a zone plate of numerical aperture 0.35, down to a waist radius of 150 nm, with 18% focusing efficiency. The estimated intensity approaches 10 7 W/cm 2 . Future developments may demonstrate nanoscale laser ablation and miniaturization of extreme ultraviolet coherent sources on a chip.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Scalable fabrication of Chip-integrated 3D-nanostructured electronic devices via DNA-programmable assembly

DNA-based self-assembly methods have demonstrated powerful and unique capabilities to encode nanomaterial structures through the prescribed placement of inorganic and biological nanocomponents. However, the challenge of selectively growing DNA superlattices on specific locations of surfaces and their integration with conventional nanofabrication has hindered the fabrication of three-dimensional (3D) DNA-assembled functional devices. Here, we present a scalable nanofabrication technique that combines bottom-up and top-down approaches for selective growth of 3D DNA superlattices on gold microarrays. This approach allows for the fabrication of self-assembled 3D-nanostructured electronic devices. DNA strands are bound onto the gold arrays, which anchor DNA origami frames and promote ordered framework growth on the specific areas of the surface, enabling control of the lateral placement and orientation of superlattices. DNA frameworks selectively grown on the pads are subsequently templated to nanoscale silica and tin oxide (SnO x ) that follow the architecture, as confirmed by structural and chemical characterizations. The fabricated SnO x superlattices are integrated into devices that demonstrate photocurrent response.

36 MATERIALS SCIENCE↗
DOE OSTI2022

A coupled ground heat flux–surface energy balance model of evaporation using thermal remote sensing observations

Abstract. One of the major undetermined problems in evaporation (ET) retrieval using thermal infrared remote sensing is the lack of a physically based ground heat flux (G) model and its integration within the surface energy balance (SEB) equation. Here, we present a novel approach based on coupling a thermal inertia (TI)-based mechanistic G model with an analytical surface energy balance model, Surface Temperature Initiated Closure (STIC, version STIC1.2). The coupled model is named STIC-TI. The model is driven by noon–night (13:30 and 01:30 local time) land surface temperature, surface albedo, and a vegetation index from MODIS Aqua in conjunction with a clear-sky net radiation sub-model and ancillary meteorological information. SEB flux estimates from STIC-TI were evaluated with respect to the in situ fluxes from eddy covariance measurements in diverse ecosystems of contrasting aridity in both the Northern Hemisphere and Southern Hemisphere. Sensitivity analysis revealed substantial sensitivity of STIC-TI-derived fluxes due to the land surface temperature uncertainty. An evaluation of noontime G (Gi) estimates showed 12 %–21 % error across six flux tower sites, and a comparison between STIC-TI versus empirical G models also revealed the substantially better performance of the former. While the instantaneous noontime net radiation (RNi) and latent heat flux (LEi) were overestimated (15 % and 25 %), sensible heat flux (Hi) was underestimated (22 %). Overestimation (underestimation) of LEi (Hi) was associated with the overestimation of net available energy (RNi−Gi) and use of unclosed surface energy balance flux measurements in LEi (Hi) validation. The mean percent deviations in Gi and Hi estimates were found to be strongly correlated with satellite day–night view angle difference in parabolic and linear pattern, and a relatively weak correlation was found between day–night view angle difference versus LEi deviation. Findings from this parameter-sparse coupled G–ET model can make a valuable contribution to mapping and monitoring the spatiotemporal variability of ecosystem water stress and evaporation using noon–night thermal infrared observations from future Earth observation satellite missions such as TRISHNA, LSTM, and SBG.

Bhattacharya, Bimal K.↗
DOE OSTI2022

Fiber-optic sensor packaging assessment and benchmarking for embedded sensing applications

Fiber optic-based sensing for non-destructive evaluation (NDE) and structural health monitoring of various infrastructure and energy assets is an increasingly important sensing scheme. Because of their immunity to electromagnetic interferences, versatility in sensing mechanisms and designs as well as capability for distributed sensing, the fiber optic sensing is of great interest for embedding within infrastructures using advanced manufacturing methods. However, the protection of fiber optic sensors during advanced manufacturing-based embedding is an important aspect for infrastructure monitoring in practice. While successful installation of fiber optic sensors has been accomplished through application of epoxy or adhesives on infrastructure surfaces, there has been only limited investigation of fiber protection packaging schemes which are compatible with in-situ repair techniques such as the metallic cold-spray. With increasing interest and importance of cold spray-based repair processes, the integration of fiber optic sensors simultaneously during the repair is equally important. In this work, we focus our efforts on investigating various types of packaging schemes for their compatibility and integrability on the surfaces of metallic structures such as oil and gas pipelines during an in-situ repair coating. First, we consider sandblasting using silica sand particles of various size for optical fibers on the surface of a metallic coupon as a useful proxy test of sensor packaging stability. Structural damage of packaging is characterized by means of optical microscopy and the optical integrity of embedded fiber optic cables is examined using optical backscatter reflectometry (OBR). In addition, initial investigation of metallic cold-spray embedded fiber optic cables under various packaging on metallic substrates is also considered with early conclusions and future directions.

Fiber-optic sensors, Sensors packaging, Metallic C↗
DOE OSTI2020

Higher-dimensional generalizations of Berry curvature

A family of finite-dimensional quantum systems with a nondegenerate ground state gives rise to a closed two-form on the parameter space, the curvature of the Berry connection. Its integral over a surface detects the presence of degeneracy points inside the volume enclosed by the surface. In this work, we seek generalizations of the Berry curvature to gapped many-body systems in $\textit{D}$ spatial dimensions which can detect gapless or degenerate points in the phase diagram of a system. Field theory predicts that in spatial dimension $\textit{D}$ the analog of the Berry curvature is a closed ($\textit{D}$ + 2) -form on the parameter space (the Wess-Zumino-Witten form). We construct such closed forms for arbitrary families of gapped interacting lattice systems in all dimensions. We show that whenever the integral of the Wess-Zumino-Witten form over a ($\textit{D}$ + 2)-dimensional surface in the parameter space is nonzero, there must be gapless edge modes for at least one value of the parameters. These edge modes arise even when the bulk system is in a trivial phase for all values of the parameters and are protected by the nontrivial topology of the phase diagram.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗