Search NASA⌕ Search

SEARCH · Search NASA

Results for “convention equation”

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 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Foundations of magnetohydrodynamics

In this tutorial, a derivation of magnetohydrodynamics (MHD) valid beyond the usual ideal gas approximation is presented. Non-equilibrium thermodynamics is used to obtain conservation equations and linear constitutive relations. When coupled with Maxwell's equations, this provides closed fluid equations in terms of material properties of the plasma, described by the equation of state and transport coefficients. These properties are connected to microscopic dynamics using the Irving–Kirkwood procedure and Green–Kubo relations. Symmetry arguments and the Onsager–Casimir relations allow one to vastly simplify the number of independent coefficients. Importantly, expressions for current density, heat flux, and stress (conventionally Ohm's law, Fourier's law, and Newton's law) take different forms in systems with a non-ideal equation of state. The traditional form of the MHD equations, which is usually obtained from a Chapman–Enskog solution of the Boltzmann equation, corresponds to the ideal gas limit of the general equations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Equation of state of neutron-rich matter in d -dimensions

Nuclear systems under constraints, with high degrees of symmetries and/or collectivities may be considered as moving effectively in spaces with reduced spatial dimensions. We first derive analytical expressions for the nucleon specific energy E 0 (ρ), pressure P 0 (ρ), incompressibility coefficient K 0 (ρ) and skewness coefficient J 0 (ρ) of symmetric nucleonic matter (SNM), the quadratic symmetry energy E sym (ρ), its slope parameter L(ρ) and curvature coefficient K sym (ρ) as well as the fourth-order symmetry energy E sym,4 (ρ) of neutron-rich matter in general d spatial dimensions (abbreviated as "dD") in terms of the isoscalar and isovector parts of the isospin-dependent single-nucleon potential according to the generalized Hugenholtz-Van Hove (HVH) theorem. The equation of state (EOS) of nuclear matter in dD can be linked to that in the conventional 3-dimensional (3D) space by the ε-expansion which is a perturbative approach successfully used previously in treating second-order phase transitions and related critical phenomena and more recently in studying the EOS of cold atoms. The ε-expansion of nuclear EOS in dD based on a reference dimension d f = d – ε is shown to be effective with –1 ≲ ε ≲ 1 starting from 1 ≲ d f ≲ 3 in comparison with the exact expressions derived using the HVH theorem. Moreover, the EOS of SNM (with/without considering its potential part) is found to be reduced (enhanced) in lower (higher) dimensions, indicating in particular that the many-nucleon system tends to be deeper bounded but saturate at higher densities in spaces with lower dimensions. Here, the links between the EOSs in 3D and dD spaces from the ε-expansion provide new perspectives to the EOS of neutron-rich matter.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Decoupled Fréchet kernels based on a fractional viscoacoustic wave equation

We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion (Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. Here, we find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel.

58 GEOSCIENCES↗
DOE OSTI2020

Solving differential equations using deep neural networks

Recent work on solving partial differential equations (PDEs) with deep neural networks (DNNs) is presented. The paper reviews and extends some of these methods while carefully analyzing a fundamental feature in numerical PDEs and nonlinear analysis: irregular solutions. First, the Sod shock tube solution to the compressible Euler equations is discussed and analyzed. This analysis includes a comparison of a DNN-based approach with conventional finite element and finite volume methods, and demonstrates that the DNN is competitive in terms of degrees of freedom required for a given accuracy. Further, the DNN-based approach is extended to consider performance improvements and simultaneous parameter space exploration. Next, a shock solution to compressible magnetohydrodynamics (MHD) is solved for, and used in a scenario where experimental data is utilized to enhance a PDE system that is a priori insufficient to validate against the observed/experimental data. This is accomplished by enriching the model PDE system with source terms that are then inferred via supervised training with synthetic experimental data. The resulting DNN framework for PDEs enables straightforward system prototyping and natural integration of large data sets (be they synthetic or experimental), all while simultaneously enabling single-pass exploration of an entire parameter space.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Field–potential finite-difference time-domain (FiPo FDTD) technique for computational electromagnetics

Modeling light–matter interactions at the nanoscale requires accurate handling of coupled quantum and electromagnetic systems. This coupling requires information about the electric scalar potential Φ and the magnetic vector potential A, which are not typically calculated in standard computational electromagnetics implementations. To that end, we have developed a field–potential finite-difference time-domain (FiPo FDTD) algorithm, which solves a set of first-order equations for Φ and A alongside equations for the electric and magnetic fields E and H. The FiPo Basic code is essentially conventional FDTD, but with an added module that calculates the potentials. The FiPo Hybrid code self-consistently calculates both fields and potentials and is particularly suitable for coupling with quantum electronic transport solvers because it can be sourced by the potentials themselves. To terminate the domain and mimic infinite space, we have derived and implemented a convolutional perfectly matched layer (CPML) absorbing boundary condition for FiPo FDTD whose performance is on par with state-of-the-art CPMLs for standard FDTD. We present FiPo simulation results on several example systems.

Avazpour, L. [University of Wisconsin-Madison, WI ↗
DOE OSTI2020

An open source fast fluid dynamics model for data center thermal management

Although computational fluid dynamics (CFD) has been widely adopted to improve data center thermal management, the high computational demand limits its applications, such as multivariate optimal design and operation. Fast fluid dynamics (FFD), which has been applied for fast airflow simulation, shows great potential. However, few research applied FFD for optimal design and operation of data center thermal management. This research improves the FFD model for data centers and conducts a comprehensive evaluation and demonstration. First, the FFD model is improved by solving the advection and diffusion equations together using an upwind scheme instead of a semi-Lagrangian advection solver in the conventional FFD model. Second, new features for data centers are added, such as a pressure correction method to simulate plenum airflow and dynamic boundary conditions for IT racks. The new FFD model is first validated with two indoor environment cases and the results show that the new FFD model has slightly better overall prediction accuracy and faster speed compared to the conventional FFD model. It is also observed that both FFD models achieve acceptable accuracy, except for a few localized disparities with experimental data, which might be due to simplified handling of turbulence viscosity near the boundaries. Furthermore, validation with a real data center shows that the FFD model achieves a similar level of accuracy as CFD when compared to the experimental measurements with some level of uncertainties. It is then demonstrated for data center optimal design and operation, which saves 53.4–58.8% of annual energy while still meeting the thermal requirements. In conclusion, with a much faster speed and comparable accuracy compared to CFD, the FFD model parallelized on a graphics processing unit is promising for practical model-based data center early design and operation.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble

In this work we explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty lies in providing appropriate initial conditions or guesses, as widely used time-marching schemes and Newton’s method are highly sensitive to these choices when solving complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first demonstrate that PINNs, when combined with random initialization and deep ensemble method—originally developed for uncertainty quantification—can effectively uncover multiple solutions to nonlinear ordinary and partial DEs. Although training large ensembles of PINNs may appear computationally demanding, this can be done efficiently using vectorization techniques supported by modern deep learning frameworks, allowing many networks to be trained simultaneously. Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen–Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear DEs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An error-in-constitutive equations strategy for topology optimization for frequency-domain dynamics

This paper presents a topology optimization formulation for frequency-domain dynamics to reduce solution dependence upon initial guess and considered loading conditions. Due to resonance phenomena in undamped steady-state dynamics, objectives measuring dynamic response possess many local minima that may represent poor solutions to a design problem, an issue exacerbated for design with respect to multiple frequencies. In this work, we propose an extension of the modified error-in-constitutive-equations (MECE) method, used previously in material identification inverse problems, as a new approach for frequency-domain dynamics topology optimization to mitigate these issues. The main idea of the proposed framework is to incorporate an additional penalty-like term in the objective function that measures the discrepancy in the constitutive relations between stresses and strains and between inertial forces and displacements. Then, the design problem is cast within a PDE-constrained optimization formulation in which we seek displacements, stresses, inertial forces, and a density-field solution that minimize our new objective subject to conservation of linear momentum plus some additional constraints. We show that this approach yields superior designs to conventional gradient-based optimization approaches that solely use a functional of displacements as the objective, while strictly enforcing the constitutive equations. The MECE strategy integrates into a density-based topology optimization scheme for void–solid or two-phase material structural design. We highlight the merits of our approach in a variety of scenarios for direct frequency response design, considering multiple frequency load cases and structural objectives.

42 ENGINEERING↗
DOE OSTI2021

Theoretical Study of the Phenoxy Radical Recombination with the O( 3 P) Atom, Phenyl plus Molecular Oxygen Revisited

Quantum chemical calculations of the C 6 H 5 O 2 potential energy surface (PES) were carried out to study the mechanism of the phenoxy + O( 3 P) and phenyl + O 2 reactions. CASPT2(15e,13o)/CBS//CASSCF(15e,13o)/DZP multi-reference calculations were utilized to map out the minimum energy path for the entrance channels of the phenoxy + O( 3 P) reaction. Stationary points on the C 6 H 5 O 2 PES were explored at the CCSD(T)-F12/cc-pVTZ-f12//B3LYP/6-311++G** level for the species with a single-reference character of the wave function and at the CASPT2(15e,13o)/CBS//B3LYP/6-311++G** level of theory for the species with a multi-reference character of the wave function. Conventional, variational, and variable reaction coordinate transition state theories were employed in Rice-Ramsperger-Kassel-Marcus Master Equation calculations to assess temperature- and pressure-dependent phenomenological rate constants and product branching ratios. Here, the main bimolecular product channels of the phenoxy + O( 3 P) reaction are concluded to be para/ortho-benzoquinone + H, 2,4-cyclopentadienone + HCO, and, at high temperatures, also phenyl + O 2 . The main bimolecular product channels of the phenyl + O 2 reaction include 2,4-cyclopentadienone + HCO at lower temperatures, and phenoxy + O( 3 P) at higher temperatures. Both for the phenoxy + O( 3 P) and phenyl + O 2 reactions, the collisional stabilization of peroxybenzene at low temperatures and high pressures competes with the bimolecular product channels.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Developing Frequency Stability Constraint for Unit Commitment Problem Considering High Penetration of Renewables: Preprint

As zero-carbon electricity systems become the trend of future grid, the system inertia provided by conventional synchronous generators (SGs) keeps decreasing. The resultant lower system inertia will inevitably cause frequency stability problem, especially in the first few seconds after disturbance. To address this challenge, this paper proposes a frequency stability unit commitment constraint considering the fast frequency responses (FFRs) from inverter-based resources (IBRs). Thus, this paper first analyzes a frequency nadir differential equation that considers SG governor model and three kinds of FFR provided by IBRs. A system frequency nadir estimation framework is developed with consideration of various conventional SG and IBR models. The accuracy of this frequency estimation framework is validated by largest N-1 contingency simulation result of a real island system. Then, the adaptive inertia frequency stability unit commitment constraint is derived from sensitivity analysis method. Finally, we demonstrate the effectiveness of our developed constraints with one year day-ahead unit commitment results of the real island system.

fast frequency response↗
DOE OSTI2024

Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

DS-TIDE: Harnessing Dynamical Systems for Efficient Time-Independent Differential Equation Solving

Time-Independent Differential Equations (TIDEs) are central to modeling equilibrium behavior across a wide range of scientific and engineering domains, from electrostatics to porous media flow. Conventional numerical solvers offer reliable solutions but incur significant computational costs due to fine-grained discretization and iterative procedures. Machine learning-based approaches address this by replacing iterative solving processes with one-time inference; however, their sophisticated models require extensive training resources that often exceed those of traditional solvers. Consequently, designing a TIDE solver that achieves high accuracy, broad applicability, and exceptional computational efficiency remains a fundamental challenge. In this paper, we propose DS-TIDE, a novel hardware solver that is inspired by, and subsequently leverages, the intrinsic connection between Dynamical Systems (DS) and Differential Equations (DEs) to efficiently and accurately solve TIDEs. DS-TIDE employs a CMOS-compatible DS-based processor, whose physical states evolve under carefully designed DE-driven dynamics and naturally converge to equilibrium -- the solution of the target TIDE -- within ~1µs on a ~1-watt DS-TIDE processor. To enhance expressivity, DS-TIDE incorporates Heterogeneous Dynamics with Temporal Layering (HDTL), which solves TIDEs through a three-stage DS evolution -- conditioning, solving, and decoding -- each governed by specialized dynamics. The entire evolution process is analogous to an infinitely deep neural network temporally unrolled, offering the system the capability of representing complex equations. Furthermore, DS-TIDE is equipped with an on-device DS-DE Auto-Alignment mechanism that dynamically adapts intrinsic hardware dynamics within milliseconds, effectively aligning the system’s dynamics to diverse target DEs. Experimental results across TIDEs from a wide range of scientific and engineering domains demonstrate that DS-TIDE achieves ~10^3× speedup, ~10^5× energy savings, and competitive or superior accuracy compared to state-of-the-art numerical and ML-based solvers.

Liu, Chuan↗
DOE OSTI2020

Iron(III)–Oxo Cluster Chemistry with Dimethylarsinate Ligands: Structures, Magnetic Properties, and Computational Studies

A program has been initiated to develop Fe III /oxo cluster chemistry with the ‘pseudo-carboxylate’ ligand dimethylarsinate (Me 2 AsO 2 - ) for comparison with the well investigated Fe III /oxo/carboxylate cluster area. The synthesis and characterization of three polynuclear Fe III complexes are reported, [Fe 12 O 4 (O 2 C t Bu) 8 (O 2 AsMe 2 ) 17 (H 2 O) 3 ]Cl 3 (1), Na 2 [Fe 12 Na 2 O 4 (O 2 AsMe 2 ) 20 (NO 3 ) 6 (Me 2 AsO 2 H) 2 (H 2 O) 4 ](NO 3 ) 6 (2) and [Fe 3 (O 2 AsMe 2 ) 6 (Me 2 AsO 2 H) 2 (hqn) 2 ](NO 3 ) (3), where hqn is 8-hydroxyquinoline. The Fe 12 core of 1 is a type never previously encountered in Fe III carboxylate chemistry, consisting of two Fe 6 units each of which comprises two {Fe 3 (μ 3 -O 2- )} units bridged by three Me 2 AsO 2 - groups and linked into an Fe 12 loop structure by two anti-anti η 1 :η 1 :μ Me 2 AsO 2 - groups, a bridging mode extremely rare with carboxylates. 2 also consists of two Fe 6 units, differing in their ligation from those in 1, and this time linked together into a linear structure by a central {Na 2 (NO 3 ) 2 } bridging unit. 3 is a linear Fe 3 complex with no monoatomic bridges between Fe III ions, a very rare situation in Fe III chemistry with any ligands, and unprecedented in Fe carboxylate chemistry. The distinct differences observed in arsinate vs carboxylate ligation modes are rationalized largely based on the greater basicity of the former vs the latter. Variable-temperature dc and ac magnetic susceptibility data reveal all Fe 2 pairwise interactions to be antiferromagnetic. For 1 and 2, the different J ij couplings were estimated by use of a magnetostructural correlation for high nuclearity Fe III -oxo clusters and by density functional theory calculations using broken symmetry methods, allowing identification of their relative spin vector alignments and thus rationalization of their S = 0 ground states. The J ij values were then used as input values to give excellent fits of the experimental χM T vs T data. For 3, the fits of the experimental χM T vs T data to the Van Vleck equation or with PHI gave a very weak J 12 = -0.8(1) cm -1 (H = –2JŠ i ·Š j convention) between adjacent Fe III ions, and an S = 5/2 ground state. Furthermore, these initial Fe III arsinate complexes also provide structural parameters that help validate literature assignments of arsinate binding modes to iron oxide/hydroxide minerals as part of environmental concerns of using arsenic-containing herbicides in agriculture.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Extending the predictive power of perturbative QCD using the principle of maximum conformality and the Bayesian analysis

In addition to the evaluation of high-order loop contributions, the precision and predictive power of perturbative QCD (pQCD) predictions depends on two important issues: (1) how to achieve a reliable, convergent fixed-order series, and (2) how to reliably estimate the contributions of unknown higher-order terms. The recursive use of renormalization group equation, together with the Principle of Maximum Conformality (PMC), eliminates the renormalization scheme-and-scale ambiguities of the conventional pQCD series. The result is a conformal, scale-invariant series of finite order which also satisfies all of the principles of the renormalization group. In this paper we propose a novel Bayesian-based approach to estimate the size of the unknown higher order contributions based on an optimized analysis of probability distributions. We show that by using the PMC conformal series, in combination with the Bayesian analysis, one can consistently achieve high degree of reliability estimates for the unknown high order terms. Thus the predictive power of pQCD can be greatly improved. We illustrate this procedure for two pQCD observables: R e+e– and R τ , which are each known up to four loops in pQCD. Numerical analyses confirm that by using the scale-independent and more convergent PMC conformal series, one can achieve reliable Bayesian probability estimates for the unknown higher-order contributions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Data-Driven Learning for the Mori--Zwanzig Formalism: A Generalization of the Koopman Learning Framework

A theoretical framework which unifies the conventional Mori--Zwanzig formalism and the approximate Koopman learning of deterministic dynamical systems from noiseless observation is presented. In this framework, the Mori--Zwanzig formalism, developed in statistical mechanics to tackle the hard problem of construction of reduced-order dynamics for high-dimensional dynamical systems, can be considered as a natural generalization of the Koopman description of the dynamical system. We next show that, similar to the approximate Koopman learning methods, data-driven methods can be developed for the Mori--Zwanzig formalism with Mori's linear projection operator. We have developed two algorithms to extract the key operators, the Markov and the memory kernel, using time series of a reduced set of observables in a dynamical system. We have adopted the Lorenz `96 system as a test problem and solved for the above operators. These operators exhibit complex behaviors, which are unlikely to be captured by traditional modeling approaches in Mori--Zwanzig analysis. The nontrivial generalized fluctuation-dissipation relationship, which relates the memory kernel with the two-time correlation statistics of the orthogonal dynamics, was numerically verified as a validation of the solved operators. Here we present numerical evidence that the generalized Langevin equation, a key construct in the Mori--Zwanzig formalism, is more advantageous in predicting the evolution of the reduced set of observables than the conventional approximate Koopman operators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Machine Learning Moment Closure Models for the Radiative Transfer Equation III: Enforcing Hyperbolicity and Physical Characteristic Speeds

This is the third paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation. In our previous work (Huang et al. in J Comput Phys 453:110941, 2022), we proposed an approach to learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional $P_N$ closure. However, while the ML moment closure has better accuracy, it is not able to guarantee hyperbolicity and has issues with long time stability. In our second paper (Huang et al., in: Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures, 2021. arXiv:2105.14410), we identified a symmetrizer which leads to conditions that enforce that the gradient based ML closure is symmetrizable hyperbolic and stable over long time. The limitation of this approach is that in practice the highest moment can only be related to four, or fewer, lower moments. In this paper, we propose a new method to enforce the hyperbolicity of the ML closure model. Motivated by the observation that the coefficient matrix of the closure system is a lower Hessenberg matrix, we relate its eigenvalues to the roots of an associated polynomial. Here, we design two new neural network architectures based on this relation. The ML closure model resulting from the first neural network is weakly hyperbolic and guarantees the physical characteristic speeds, i.e., the eigenvalues are bounded by the speed of light. The second model is strictly hyperbolic and does not guarantee the boundedness of the eigenvalues. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, stability and generalizability of our hyperbolic ML closure model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Intrinsic Anomalous Hall Conductivity in a Nonuniform Electric Field

We study how the intrinsic anomalous Hall conductivity is modified in two-dimensional crystals with broken time-reversal symmetry due to weak inhomogeneity of the applied electric field. Focusing on a clean noninteracting two-band system without band crossings, we derive the general expression for the Hall conductivity at small finite wave vector q to order q 2 , which governs the Hall response to the second gradient of the electric field. Using the Kubo formula, we show that the answer can be expressed through the Berry curvature, Fubini-Study quantum metric, and the rank-3 symmetric tensor which is related to the quantum geometric connection and physically corresponds to the gauge-invariant part of the third cumulant of the position operator. We further compare our results with the predictions made within the semiclassical approach. By deriving the semiclassical equations of motion, we reproduce the result obtained from the Kubo formula in some limits. We also find, however, that the conventional semiclassical description in terms of the definite position and momentum of the electron is not fully consistent because of singular terms originating from the Heisenberg uncertainty principle. We thus present a clear example of a case when the semiclassical approach inherently suffers from the uncertainty principle, implying that it should be applied to systems in nonuniform fields with extra care.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗