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At least 757 records · Page 42

Transform methods for precision continuum and control models of flexible space structures

An open loop optimal control algorithm is developed for general flexible structures, based on Laplace transform methods. A distributed parameter model of the structure is first presented, followed by a derivation of the optimal control algorithm. The control inputs are expressed in terms of their Fourier series expansions, so that a numerical solution can be easily obtained. The algorithm deals directly with the transcendental transfer functions from control inputs to outputs of interest, and structural deformation penalties, as well as penalties on control effort, are included in the formulation. The algorithm is applied to several structures of increasing complexity to show its generality.

Lupi, Victor D.↗

Growth and Characterization of Epitaxial FeWO 4 Thin Films with Controlled Oxygen Stoichiometry

Here, we report the growth of single-phase epitaxial FeWO 4 thin films, using plasma-assisted molecular beam epitaxy, and investigate structural, optical, and electronic properties. The FeWO 4 films grow in (100) orientation on c-plane sapphire (0001) substrates and exhibit 3 rotational twin variants where FeWO 4 [001] is aligned to sapphire [100] equivalent in-plane directions. X-ray diffraction measurements indicate that the epitaxial FeWO 4 (100) structure is optimized when 80-100 W of rf power is applied to an atomic oxygen source during growth, yielding films with minimal strain and impurity phases or other orientations. In films grown with 120 W of rf power, FeWO 4 crystallites develop inhomogeneous and homogeneous strains and are potentially contaminated with Fe 3+ oxide phase impurities. In films grown with 60 W of rf power, FeWO 4 crystallites do not form fully epitaxial layers. X-ray photoelectron spectroscopy indicates that the structural changes are correlated with the Fe 3+ /Fe 2+ oxidation state ratio increasing from 0.6-1.4 with rf power from 60-120 W. X-ray fluorescence spectroscopy indicates that the Fe/W composition ratio is also increasing from 1.1-1.8 with rf power from 60-120 W. Ultraviolet and visible optical absorption spectra indicate a 1.8 ± 0.1 eV band gap with an additional interband absorption feature at 3.1 ± 0.1 eV in the 80-100 W films, with similar onsets observed in the 60 W films. In the 120 W films, the higher lying transition is shifted to 2.7 ± 0.1 eV due to the Fe 3+ enrichment. Electrical resistivity decreases over 2 orders of magnitude with oxidation from 10 4 -10 5 Ω cm in 60 W films to 120 ± 10 Ω cm in 120 W films. Thermopower measurements show p-type to n-type conductivity conversion when oxidation states shift from Fe 2+ majority in the 100 W films to Fe 3+ majority in the 120 W films. We conclude that electron polaron hopping driven by Fe 3+ is a dominant transport mechanism and a source of n-type conductivity in overoxidized FeWO 4 films.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Mixed time integration methods for transient thermal analysis of structures

The computational methods used to predict and optimize the thermal structural behavior of aerospace vehicle structures are reviewed. In general, two classes of algorithms, implicit and explicit, are used in transient thermal analysis of structures. Each of these two methods has its own merits. Due to the different time scales of the mechanical and thermal responses, the selection of a time integration method can be a different yet critical factor in the efficient solution of such problems. Therefore mixed time integration methods for transient thermal analysis of structures are being developed. The computer implementation aspects and numerical evaluation of these mixed time implicit-explicit algorithms in thermal analysis of structures are presented. A computationally useful method of estimating the critical time step for linear quadrilateral element is also given. Numerical tests confirm the stability criterion and accuracy characteristics of the methods. The superiority of these mixed time methods to the fully implicit method or the fully explicit method is also demonstrated.

Liu, W. K.↗

Customizable wave tailoring nonlinear materials enabled by bilevel inverse design

Abstract Passive wave transformation via nonlinearity is ubiquitous in settings from acoustics to optics and electromagnetics. It is well known that different nonlinearities yield different effects on propagating signals, which raises the question of “what precise nonlinearity is the best for a given wave tailoring application?” In this work, considering a one-dimensional spring-mass chain connected by polynomial springs (a variant of the Fermi-Pasta-Ulam-Tsingou system), we introduce a bilevel inverse design method which couples the shape optimization of structures for tailored constitutive responses with reduced-order nonlinear dynamical inverse design. We apply it to two qualitatively distinct problems—minimization of peak transmitted kinetic energy from impact, and pulse shape transformation—demonstrating our method’s breadth of applicability. For the impact problem, we obtain two fundamental insights. First, small differences in nonlinearity can drastically change the dynamic response of the system, from severely under- to outperforming a comparative linear system. Second, the oft-used strategy of impact mitigation via “energy locking” bistability can be significantly outperformed by our optimal nonlinearity. We validate this case with impact experiments and find excellent agreement. This study establishes a framework for broader passive nonlinear mechanical wave tailoring material design, with applications to computing, signal processing, shock mitigation, and autonomous materials.

Science & Technology - Other Topics↗

VZr_BCC_SolidSolution_128atoms_VASP6

We performed density functional theory (DFT) calculations for body-centered-cubic (BCC) structures with 128 lattices sites of solid solution binary alloys vanadium-zirconium (V-Zr). The electronic structures of alloys have been calculated using Vienna Ab initio Simulation Package (VASP). Within this package the DFT approach is used to reduce many-body Schrodinger equation to set of single particle Kohn-Sham (KS) equations. The generalized electronic exchange-correlation functional is described by generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization. The electron-ion interactions is described by pseudopotentials developed within the plane-wave basis projector augmented-wave (PAW) approach \cite{PAW}. These pseudopotentials are available at the VASP portal (http://cms.mpi.univie.ac.at/vasp/). Our calculations have been run with the pseudopotentials treating s and p semi-core states as valence in case for the elements V and Zr. The electronic densities and potentials are expanded over plane-waves with energy cutoff of 350 eV. 2x2x2 k-mesh and normal precision were used. The alloys were modeled by supercell containing 128 randomly distributed atoms. At initial step the atoms occupy perfect bcc lattice cites. This initial structure was optimized until energy changes less than 1e-6 eV, while forces acting on atoms don't exceed 1e-2 eV/angstrom. The electron-ion interaction is described by PAW pseudopotentials. The calculations have been collected by sampling chemical compositions across the entire compositional range. The chemical compositions have been sampled by progressively changing the number of atoms per constituent by 4. For each chemical composition of binaries and ternaries, the first-principle calculations have been run for 100 randomized arrangements of the constituents on the BCC lattice sites. We collected data for a total of 3,100 randomized atomic structures over 31 chemical compositions.

36 MATERIALS SCIENCE↗

TaTi_BCC_SolidSolution_128atoms_VASP6

We performed density functional theory (DFT) calculations for body-centered-cubic (BCC) structures with 128 lattices sites of solid solution binary alloys tantalum-titanium (Ta-Ti). The electronic structures of alloys have been calculated using Vienna Ab initio Simulation Package (VASP). Within this package the DFT approach is used to reduce many-body Schrodinger equation to set of single particle Kohn-Sham (KS) equations. The generalized electronic exchange-correlation functional is described by generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization. The electron-ion interactions is described by pseudopotentials developed within the plane-wave basis projector augmented-wave (PAW) approach. These pseudopotentials are available at the VASP portal (http://cms.mpi.univie.ac.at/vasp/). Our calculations have been run with the pseudopotentials treating s and p semi-core states as valence in case for the elements Hf and Nb. The electronic densities and potentials are expanded over plane-waves with energy cutoff of 350 eV. 2x2x2 k-mesh and normal precision were used. The alloys were modeled by supercell containing 128 randomly distributed atoms. At initial step the atoms occupy perfect bcc lattice cites. This initial structure was optimized until energy changes less than 1e-6 eV, while forces acting on atoms don't exceed 1e-2 eV/angstrom. The electron-ion interaction is described by PAW pseudopotentials. The calculations have been collected by sampling chemical compositions across the entire compositional range. The chemical compositions have been sampled by progressively changing the number of atoms per constituent by 4. For each chemical composition of binaries and ternaries, the first-principle calculations have been run for 100 randomized arrangements of the constituents on the BCC lattice sites. We collected data for a total of 3,100 randomized atomic structures over 31 chemical compositions. The calculations have been collected on Air Force HPC11 cluster using the VASP 6.5.1. Additional methodology and file structure information is available in the dataset README.txt file.

36 MATERIALS SCIENCE↗

HfNb_BCC_SolidSolution_128atoms_VASP6

We performed density functional theory (DFT) calculations for body-centered-cubic (BCC) structures with 128 lattices sites of solid solution binary alloys hafnium-niobium (Hf-Nb). The electronic structures of alloys have been calculated using Vienna Ab initio Simulation Package (VASP). Within this package the DFT approach is used to reduce many-body Schrodinger equation to set of single particle Kohn-Sham (KS) equations. The generalized electronic exchange-correlation functional is described by generalized gradient approximation with the Perdew-Burke-Ernzerhof parametrization. The electron-ion interactions is described by pseudopotentials developed within the plane-wave basis projector augmented-wave (PAW) approach. These pseudopotentials are available at the VASP portal (http://cms.mpi.univie.ac.at/vasp/). Our calculations have been run with the pseudopotentials treating s and p semi-core states as valence in case for the elements Hf and Nb. The electronic densities and potentials are expanded over plane-waves with energy cutoff of 350 eV. 2x2x2 k-mesh and normal precision were used. The alloys were modeled by supercell containing 128 randomly distributed atoms. At initial step the atoms occupy perfect BCC lattice cites. This initial structure was optimized until energy changes less than 1e-6 eV, while forces acting on atoms don't exceed 1e-2 eV/angstrom. The electron-ion interaction is described by PAW pseudopotentials. The calculations have been collected by sampling chemical compositions across the entire compositional range. The chemical compositions have been sampled by progressively changing the number of atoms per constituent by 4. For each chemical composition of binaries and ternaries, the first-principle calculations have been run for 100 randomized arrangements of the constituents on the BCC lattice sites. We collected data for a total of 3,029 randomized atomic structures over 31 chemical compositions. Further methodological and structural information is contained in the dataset README.txt file.

36 MATERIALS SCIENCE↗

Design optimization of lightweight automotive seatback through additive manufacturing compression overmolding of metal polymer composites

With the growing demand for enhanced automotive fuel efficiency and environmental sustainability, there is a need for lightweighting automotive components through innovative design and manufacturing processes. Here, this study leverages a combination of numerical iterative design optimization and hybrid additive manufacturing–compression molding (AM-CM) technique for metal polymer composites to lightweight an automotive seatback. The AM-CM process enables robust mechanical interlocking between metals and composites, boasting high stiffness and strength with low overall density. Replacing metallic components with such metal polymer composites allows for comparable mechanical performance while significantly reducing the overall weight. First, the automotive seatback design space is reduced to critical load carrying regions using topology optimization and high stress concentration areas are identified using finite element analysis. Next, a lightweight metal polymer subcomponent is designed for a high stress concentration region. The full seatback frame with spatially heterogeneous material-specific design is then iteratively optimized to enable enhanced stiffness with minimal weight. Overall, the automotive seatback frame designed with location-specific metal, polymer, and metal polymer composite materials weighs 20% less than the metal-only design while exhibiting similar stiffness.

36 MATERIALS SCIENCE↗

Controller reduction for effective interdisciplinary design of active structures

Control problems of large aerospace structures are intrinsically interdisciplinary and require strategies which address the complete interaction between flexible structures, electromechanical actuators and sensors, and feedback control algorithms. Current research and future directions which will require an interdisciplinary team effort in dynamics, control and optimization of such structures are being surveyed. It is generally agreed that the dynamics of space structures require large scale discrete modeling, resulting in thousands of discrete unknowns. Proven control strategies, on the other hand, employ a low order controller that is based on a reduced order model of structures. Integration of such low order controllers and large scale dynamics models often leads to serious deterioration of the closed loop stability margin and even instability. To alleviate this stability deterioration while low order controllers remain effective, the following approach was investigated: (1) retain low order controllers based on reduced order models of structures as the basic control strategy; (2) introduce a compensator that will directly account for the deterioration of stability margin due to controller-structure integration; and (3) assess overall performance of the integrated control structure system by developing measures of suboptimality in the combination of (1) and (2). The benefits include: simplicity in the design of basic controllers, thus facilitating the optimization of structure control interactions; increased understanding of the roles of the compensator so as to modify the structure as well as the basic controller, if necessary, for improved performance; and adaptability to localize controllers by viewing the compensator as a systems integration filter.

Balas, Mark J.↗

Design of a Manufacturing Demonstration Unit for a Composite Lunar Landing Gear Strut

The goal of the current work was to develop an analytical framework for design of composite struts using various levels of model fidelity. Rapid optimization trade studies were performed using low fidelity two-dimensional (2D) axisymmetric models with smeared composite properties. The optimum 2D model was compared with three-dimensional (3D) models with higher levels of fidelity in material property representation. Good agreement was found with all models. The buckling performance of the highest fidelity 3D model was found to be satisfactory for the intended loading conditions. An additional goal of this work was to perform an initial assessment of using automated fiber placement (AFP) and other advanced manufacturing methods to explore their feasibility for the fabrication of struts for lunar landers, strut-braced wings, and other aerospace components. The designs produced during this study are intended to be used to develop manufacturing demonstration units (MDU) that can be fabricated at the Integrated Structural Assembly of Advanced Composites (ISAAC) facility at Langley Research Center (LaRC) and tested in lab facilities at LaRC.

carbon-epoxy↗

Design of a Manufacturing Demonstration Unit for a Composite Lunar Landing Gear Strut

The goal of the current work was to develop an analytical framework for design of composite struts using various levels of model fidelity. Rapid optimization trade studies were performed using low fidelity two-dimensional (2D) axisymmetric models with smeared composite properties. The optimum 2D model was compared with three-dimensional (3D) models with higher levels of fidelity in material property representation. Good agreement was found with all models. The buckling performance of the highest fidelity 3D model was found to be satisfactory for the intended loading conditions. An additional goal of this work was to perform an initial assessment of using automated fiber placement (AFP) and other advanced manufacturing methods to explore their feasibility for the fabrication of struts for lunar landers, strut-braced wings, and other aerospace components. The designs produced during this study are intended to be used to develop manufacturing demonstration units (MDU) that can be fabricated at the Integrated Structural Assembly of Advanced Composites (ISAAC) facility at Langley Research Center (LaRC) and tested in lab facilities at LaRC.

carbon-epoxy↗

Molecular level insight into non-bilayer structure formation in thylakoid membranes: a molecular dynamics study

In oxygenic photosynthetic organisms, the light reactions are performed by protein complexes embedded in the lipid bilayer of thylakoid membranes (TMs). The organization of the bulk lipid molecules into bilayer structures provide optimal conditions for the build-up of the proton motive force (pmf) and its utilization for ATP synthesis. However, the lipid composition of TMs is dominated by the non-bilayer lipid species monogalactosyl diacylglycerol (MGDG), and functional plant TMs, besides the bilayer, contain large amounts of non-bilayer lipid phases. Bulk lipids have been shown to be associated with lumenal, stromal-side and marginal-region proteins and proposed to play roles in the self-assembly and photoprotection of the photosynthetic machinery. Furthermore, it has recently been pointed out that the generation and utilization of pmf for ATP synthesis according to the ‘protet’ or protonic charge transfer model Kell (Biochim Biophys Acta Bioenerg 1865(4):149504, 2024), requires high MGDG content Garab (Physiol Plant 177(2):e70230, 2025). In this study, to gain better insight into the structural and functional roles of MGDG, we employed all atom and coarse-grained molecular dynamics simulations to explore how temperature, hydration levels and varying MGDG concentrations affect the structural and dynamic properties of bilayer membranes constituted of plant thylakoid lipids. Our findings reveal that MGDG promotes increased membrane fluidity and dynamic fluctuations in membrane thickness. MGDG-rich stacked bilayers spontaneously formed inverted hexagonal phases; these transitions were enhanced at low hydration levels and at elevated but physiologically relevant temperatures. It can thus be inferred that MGDG plays important roles in heat and drought stress mechanisms.

59 BASIC BIOLOGICAL SCIENCES↗

Mass Mitigation in Structural Designs Via Dynamic Properties

The efforts reported here are focused on mitigating unnecessary mass in aerospace hardware. The approach to remove this undesired mass from the design is to leverage both the dynamic strength of materials and the frequency dependency of strain. Analytically predicted dynamic responses of structures are often applied as static loads in stress analyses that ultimately dictate the weight of a structural design. Assuming a dynamic response is a static load and then comparing resulting stress predictions to a static strength property is a long-standing engineering practice. Doing so is known to be, or is assumed to be, conservative. However, little indication of the order of magnitude of embedded conservatism has been identified. NASA/MSFC efforts in 2011, 2019, 2020, and now in 2021 have begun to qualitatively show the order of magnitude of that conservatism. A quick turnaround engineering method is pursued to leverage the subject facets of physics for the purpose of decreasing the weight of flight hardware. Tests performed using simple beams and significant observations are described.

conservative design loads↗

Design and Analysis of Shape Memory Spring Tires for Martian and Lunar Rover Vehicles

Shape memory alloys (SMAs) have played an important role in various innovative engineering and medical applications, such as aerospace actuators, vibration damping devices, and coronary stents. In applications, shape memory alloys are commonly utilized in two fundamentally different ways: (i) making use of the superelasticity/ pseudoelasticity (SE/PE) phenomena, as in applications in biomedical engineering, and (ii) taking advantages of the shape memory effect (SME), as is used for actuators. Their ability to act in such vastly different capacities is mainly due to their unique capability to recover large amounts of deformation produced by either applied stresses or temperature changes. One recent emerging application in use of SMAs has been in the area of non-pneumatic tire designs for Martian or Lunar roving vehicles. These vehicles require tires that are capable of traversing rugged terrain while withstanding extreme temperatures and atmospheric conditions. Inspired by the flexible wire mesh tires used on three Lunar Roving Vehicle (LRV) missions to the Moon on Apollo 15, 16, and 17, a new compliant tire technology was developed by the NASA Glenn Research Center (GRC) and Goodyear Tire & Rubber, known as the Spring Tire. The Spring Tire consists of several hundred coiled springs woven into a flexible mesh and formed into the shape of a tire. Like the LRV wire mesh tires, the original Spring Tires were made from spring steel and were prone to permanent deformation when undergoing high localized loads. Later, a new iteration of this technology was invented, known as the ‘Superelastic Tire’. This new technology incorporated the use of superelastic SMA springs, which could effectively undergo approximately 30 times more reversible deformation than the steel spring. It also provided even greater durability and allowed for more flexibility in design, such as the use of other structural forms to reduce mass or increase load carrying capacity. Because of the unique nature of both the SMA material and the complex interactions between the springs, designing Spring Tires for a specific application requires extensive effort. Historically, design decisions have relied on full-scale empirical testing; however, this is very expensive and time consuming, especially when multiple iterations are needed. Therefore, developing a large-scale, robust, and predictive numerical model entailing complex spring interactions and the shape memory material behavior within a tire construct is the first essential step towards a successful design program. The current work focuses on implementation of the user-defined Shape Memory Alloy (SMA) model, otherwise known as SMA-GVIPs, in the Finite Element analysis (FEA) program ABAQUS for large-scale simulations of the GRC-developed Spring Tires made of SMA. The novelty of this work lies in the thorough, detail-oriented, and computationally efficient finite element analysis of full-scale SMA tires. A well-thought material characterization plan followed by model validation and a systemic sensitivity study on SMA tires has never been reported in the previous literature. The main objective of this study is to help the team improve and optimize the structural design of the SMA tires through in-depth numerical analysis and sensitivity studies. Various design variables (wire diameter, coil diameter, pitch, bead angle, and number of springs) were varied to study their influence on the global load-displacement response of the tire construct. A detailed investigation of the three-dimensional stress states was also carried out to enhance our understanding of the local changes as the tire goes through global deformation. It was concluded that a robust numerical model with a good predictive capability, together with a thoughtfully crafted sensitivity study can result in improved design iterations required to reach a desired tire performance while, significantly reducing manufacturing, labor and testing expenses. A summary of the Finite Element (FE) model construction will be presented together with a description of the user-defined SMA model, characterization process, experimental results, model validation, and numerical sensitivity study results.

shape memory alloys↗

Variable-complexity aerodynamic optimization of an HSCT wing using structural wing-weight equations

A new approach for combining conceptual and preliminary design techniques for wing optimization is presented for the high-speed civil transport (HSCT). A wing-shape parametrization procedure is developed which allows the linking of planform and airfoil design variables. Variable-complexity design strategies are used to combine conceptual and preliminary-design approaches, both to preserve interdisciplinary design influences and to reduce computational expense. In the study, conceptual-design-level algebraic equations are used to estimate aircraft weight, supersonic wave drag, friction drag and drag due to lift. The drag due to lift and wave drag are also evaluated using more detailed, preliminary-design-level techniques. The methodology is applied to the minimization of the gross weight of an HSCT that flies at Mach 3.0 with a range of 6500 miles.

Hutchison, M. G.↗

Recent Progress in Solid‐State Lithium Batteries through Cathode Microstructure Engineering

A high‐performance cathode is necessary to realize the great potentials of solid‐state batteries such as high energy density and long cycle life. It is also needed to validate electrolyte performance, which is lacking. Currently, cathodes for solid‐state batteries are thinner and have lower cathode active material content than their lithium‐ion battery counterpart, resulting from insufficient conductivity and limiting the battery energy density. This review article provides an overview of recent development in cathode microstructures and their impact on battery properties, including compatibility between cathode and electrolyte, cathode architecture design, interface engineering, correlation between material properties, cathode processing approaches, and performance, as well as the advanced characterization methods used to understand the above correlations . Some perspectives on future development are shared including utilizing in situ and operando characterization tools to better understand dynamic evolution of the cathode/electrolyte interface, adapting artificial intelligence and machine learning to design and optimize cathode structures. The article is aimed to promote research interest on cathode development and advance solid‐state battery technologies.

cathode engineering↗

Microstructural Underpinnings of Giant Intrinsic Exchange Bias in Epitaxial NiCo 2 O 4 Thin Films

Understanding intrinsic exchange bias in nominally single-component ferromagnetic or ferrimagnetic materials is crucial for simplifying related device architectures. However, the mechanisms behind this phenomenon and its tunability remain elusive, which hinders the efforts to achieve unidirectional magnetization for widespread applications. Inspired by the high tunability of ferrimagnetic inverse spinel NiCo 2 O 4 , the origin of intrinsic exchange bias in NiCo 2 O 4 (111) films deposited on Al 2 O 3 (0001) substrates are investigated. The comprehensive characterizations, including electron diffraction, X-ray reflectometry and spectroscopy, and polarized neutron reflectometry, reveal that intrinsic exchange bias in NiCo 2 O 4 (111)/Al 2 O 3 (0001) arises from a reconstructed antiferromagnetic rock-salt Ni x Co 1-x O layer at the interface between the film and the substrate due to a significant structural mismatch. Remarkably, by engineering the interfacial structure under optimal growth conditions, it can achieve exchange bias larger than coercivity, leading to unidirectional magnetization. Such giant intrinsic exchange bias can be utilized for realistic device applications. This work establishes a new material platform based on NiCo 2 O 4 , an emergent spintronics material, to study tunable interfacial magnetic and spintronic properties.

36 MATERIALS SCIENCE↗

Spinel high-entropy oxides (FeNiCrMnZnX) 3 O 4 (X = Al, mg) as anode materials for high-performance lithium-ion batteries

To address the high cost, cobalt dependency, and resource constraints typical of conventional high-entropy oxide (HEO) anodes, this study reports the successful synthesis of two Co-free, six-component spinel-type HEOs(FeNiCrMnZnAl) 3 O 4 (HEO-Al) and (FeNiCrMnZnMg) 3 O 4 (HEO-Mg), via a sol-gel method. The distinct effects of Al 3+ and Mg 2+ incorporation on the electrochemical performance and lithium storage kinetics were systematically investigated. XRD, Raman, and TEM characterizations confirm that both materials possess a pure spinel phase, uniform particle size, and homogeneous elemental distribution. Notably, electrochemical evaluations reveal that HEO-Al delivers a superior reversible capacity of 480.7 mAh g −1 after 100 cycles at 0.1 A g −1 , and maintains 354.5 mAh g −1 after 1000 long-term cycles at 1 A g −1 , significantly outperforming HEO-Mg. Kinetic analysis indicates that HEO-Al exhibits lower charge transfer resistance, a higher Li + diffusion coefficient, and a pseudocapacitive contribution of up to 82%. Furthermore, these findings demonstrate that Al substitution effectively optimizes the structural stability and lithium storage kinetics of Co-free HEOs, providing a viable strategy for designing low-cost, highly stable HEO anode systems.

Anode materials↗