Finite Gradient Research Articles

Variational gradient plasticity at finite strains. Part I: Mixed potentials for the evolution and update problems of gradient-extended dissipative solids

This work outlines a theoretical and computational framework of finite inelasticity with length scales based on a rigorous exploitation of mixed variational principles. In contrast to classical local approaches to inelasticity based on locally evolving internal variables, order parameter fields are taken into account governed by additional balance-type partial differential equations including micro-structural boundary conditions. This incorporates non-local dissipative effects based on length scales, which reflect properties of the material micro-structure. Typical examples are phase field theories, gradient damage and strain gradient plasticity. We outline unified minimization and saddle point principles for the evolution problem of first-order gradient-type standard dissipative solids. Particular emphasis is put on mixed multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These settings are needed for models with threshold functions formulated in the space of the driving forces, in particular for finite gradient plasticity. The central aim is to define constitutive rate-type and algorithmic incremental potentials, whose variational derivatives govern the coupled macro- and micro-balances in both the continuous as well as the time-discrete setting. Their existence underlines the inherent symmetry of standard dissipative solids. We demonstrate geometrically consistent constructions of these potentials for important classes of finite inelasticity, providing a fresh look on models of multiplicative and additive gradient plasticity of single crystals and amorphous materials. The potentials provide maximum compact representations of those complex material models, are the cornerstones of a subsequent mixed finite element design and should be considered as the primary objects for the theoretical and computational modeling.

Convective cells of internal gravity waves in the earth's atmosphere with finite temperature gradient

Abstract. In this paper, we have investigated vortex structures (e.g. convective cells) of internal gravity waves (IGWs) in the earth's atmosphere with a finite vertical temperature gradient. A closed system of nonlinear equations for these waves and the condition for existence of solitary convective cells are obtained. In the atmosphere layers where the temperature decreases with height, the presence of IGW convective cells is shown. The typical parameters of such structures in the earth's atmosphere are discussed.

Assembly of complex cell microenvironments using geometrically docked hydrogel shapes

Cellular communities in living tissues act in concert to establish intricate microenvironments, with complexity difficult to recapitulate in vitro. We report a method for docking numerous cellularized hydrogel shapes (100-1,000 µm in size) into hydrogel templates to construct 3D cellular microenvironments. Each shape can be uniquely designed to contain customizable concentrations of cells and molecular species, and can be placed into any spatial configuration, providing extensive compositional and geometric tunability of shape-coded patterns using a highly biocompatible hydrogel material. Using precisely arranged hydrogel shapes, we investigated migratory patterns of human mesenchymal stem cells and endothelial cells. We then developed a finite element gradient model predicting chemotactic directions of cell migration in micropatterned cocultures that were validated by tracking ∼2,500 individual cell trajectories. This simple yet robust hydrogel platform provides a comprehensive approach to the assembly of 3D cell environments.

Intrinsic spin noise in MgO magnetic tunnel junctions

We consider two intrinsic sources of noise in ultra-sensitive magnetic field sensors based on MgO magnetic tunnel junctions, coming both from 25Mg nuclear spins (I = 5/2, 10% natural abundance) and S = 1 Mg-vacancies. While nuclear spins induce noise peaked in the MHz frequency range, the vacancies noise peaks in the GHz range. We find that the nuclear noise in submicron devices has a similar magnitude than the 1/f noise, while the vacancy-induced noise dominates in the GHz range. Interestingly, the noise spectrum under a finite magnetic field gradient may provide spatial information about the spins in the MgO layer.

A Formulation of Finite Gradient Crystal Plasticity with Systematic Separation of Long‐ and Short‐Range States

AbstractWith the ongoing trend of miniaturization and nanotechnology, the predictive modeling of size effects plays an increasingly important role in metal plasticity. The description of such size effects requires gradient‐extended theories of crystal plasticity. We outline a new viscous regularized formulation of rate‐independent gradient crystal plasticity for full multislip scenarios. To this end, we exploit in a systematic manner the long‐ and short‐range nature of the involved variables. It is shown that the evolution of the short‐range state is fully determined by the evolution of the long‐range state. This separation into long‐ and short‐range states is systematically exploited in the algorithmic treatment by a new update structure, where the short‐range variables play the role of a local history base. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Invariance of Poisson measures under random transformations

Nous montrons que les mesures de Poisson sont invariantes par les transformations aléatoires qui préservent les mesures d’intensité, et dont le gradient aux différences finies satisfait une condition d’annulation cyclique. La preuve de ce résultat repose sur des identités de moments d’intérêt indépendant pour les intégrales stochastiques de Poisson adaptées et anticipantes, et est inspirée par la méthode de Üstünel et Zakai (Probab. Theory Related Fields 103 (1995) 409–429) sur l’espace de Wiener, bien que l’algèbre correspondante soit plus compliquée que dans le cas Wiener. Les exemples d’application incluent des transformations conditionnées par des ensembles aléatoires tels que l’enveloppe convexe d’une mesure aléatoire de Poisson.

Parameter identification of systems with preload nonlinearities based on the finite impulse response model and negative gradient search

This paper considers identification problems of a class of nonlinear systems. By introducing a switching function and the finite impulse response model, we obtain the identification model of a preload nonlinear system and develop a stochastic gradient algorithm to the parameters of the system. Furthermore, a stochastic gradient algorithm with a forgetting factor is given to improve the identification accuracy. A simulation results indicate that the forgetting factor can improve the parameter estimation accuracy.

Non-local gyrokinetic model of linear ion-temperature-gradient modes

The non-local properties of anomalous transport in fusion plasmas are still an elusive topic. In this work, a theory of non-local linear ion-temperature-gradient (ITG) drift modes while retaining non-adiabatic electrons and finite temperature gradients is presented, extending the previous work [S. Moradi et al., Phys. Plasmas 18, 062106 (2011)]. A dispersion relation is derived to quantify the effects on the eigenvalues of the unstable ion temperature gradient modes and non-adiabatic electrons on the order of the fractional velocity operator in the Fokker-Planck equation. By solving this relation for a given eigenvalue, it is shown that as the linear eigenvalues of the modes increase, the order of the fractional velocity derivative deviates from two and the resulting equilibrium probability density distribution of the plasma, i.e., the solution of the Fokker-Planck equation, deviates from a Maxwellian and becomes Lévy distributed. The relative effect of the real frequency of the ITG mode on the deviation of the plasma from Maxwellian is larger than from the growth rate. As was shown previously the resulting Lévy distribution of the plasma may in turn significantly alter the transport as well.

On molecular diffusion in nanostructured porous media: interfacial exchange kinetics and surface diffusion

Water-vapour transport in nanostructured composite materials is poorly understood because diffusion and interfacial exchange kinetics are coupled. We formulate an interfacial balance that couples diffusion in dispersed and continuous phases to adsorption, absorption and interfacial surface diffusion. This work is motivated by water-vapour transport in cellulose fibre-based barriers, but the model applies to nanostructured porous media such as catalysts, chromatography columns, nanocomposites, cementitious structures and biomaterials. The interfacial balance can be applied in an analytical or a computational framework to porous media with any microstructural geometry. Here, we explore its capabilities in a model porous medium: randomly dispersed solid spheres in a continuous (humid) gas. We elucidate the roles of equilibrium moisture uptake, solid, gas and surface diffusion coefficients, inclusion size and interfacial exchange kinetics on the effective diffusivity. We then apply the local model to predict water-vapour transport rates under conditions in which the effective diffusivity varies through the cross section of a dense, homogeneous membrane that is subjected to a finite moisture-concentration gradient. As the microstructural length scale decreases from micrometres to nanometres, interfacial exchange kinetics and surface diffusion produce a maximum in the tracer flux. This optimal flux is flanked, respectively, by interfacial-kinetic- and diffusion-limited transport at smaller and larger microscales.

Paramecium swimming in capillary tube

Swimming organisms in their natural habitat need to navigate through a wide range of geometries and chemical environments. Interaction with boundaries in such situations is ubiquitous and can significantly modify the swimming characteristics of the organism when compared to ideal laboratory conditions. We study the different patterns of ciliary locomotion in glass capillaries of varying diameter and characterize the effect of the solid boundaries on the velocities of the organism. Experimental observations show that Paramecium executes helical trajectories that slowly transition to straight lines as the diameter of the capillary tubes decreases. We predict the swimming velocity in capillaries by modeling the system as a confined cylinder propagating longitudinal metachronal waves that create a finite pressure gradient. Comparing with experiments, we find that such pressure gradient considerations are necessary for modeling finite sized ciliary organisms in restrictive geometries.

Thermal artifact on the spin Seebeck effect in metallic thin films deposited on MgO substrates

Spin Seebeck effect (SSE) was investigated in Heusler compound Co2MnSi (CMS) and ferromagnetic binary alloy Ni80Fe20 (Py) thin films deposited on MgO substrates at different ambient temperatures, T = 30 K to 295 K, by maintaining a finite temperature gradient along the film plane. Low temperature enhancement of voltage signal and a peak around T = 70 K were detected in a Pt wire placed on the higher temperature end of CMS (or Py) film. It has been found that the low temperature enhancement was dominated by a thermal artifact, which was generated by the anomalous Nernst effect with an additional perpendicular temperature gradient across the film plane. The remarkable mismatch between thermal conductivities of MgO and CMS (or Py) at low temperatures is considered to give rise to the perpendicular temperature gradient causing the thermal artifact. We showed that consideration of thermal conductivity matching between thin film and substrate is essential for performing experiments of SSE correctly; moreover, the observation of SSE could be realized only when minimization of the thermal artifact is possible.

Variational Model for Optimization of Finite-Burn Escape Trajectories Using a Direct Method

A variational model is used to obtain the necessary gradient information for a multiple finite-burn trajectory problem. The augmented state vector includes position, velocity, mass, and control variables such as thrust magnitude, and the variables that describe the time evolution of the thrust direction unit vector. The augmented state vector and its associated state equation are used in the variational model to formulate analytical expressions for the gradients of the problem functions with respect to the problem variables. A general finite-burn trajectory problem is formulated for an inertially fixed thrust vector steering model and a fixed-plane linearly varying thrust vector steering model that constrains the thrust vector to be normal to a rotation axis, which itself is part of the control variable set. Both steering models are used in the optimization of a three-finite-burn lunar-escape-trajectory example, using a direct optimization method with explicit numerical integration. The performance of the optimization procedure is compared for both thrust vector steeringmodels, using variational gradients and standard numerical finite difference gradients. The ability to obtain the most accurate gradient information for the direct optimization of simple finite-burn thrust steering models is a motivation of the current study.

Asymmetric tearing mode in the presence of viscosity

The linear stability of the tearing mode (TM) in a plasma column is investigated in the presence of viscosity and finite equilibrium current density gradients (i.e., asymmetries). It is shown that for low β, both effects are essential in order to properly describe the mode behaviour close to marginality. In particular, the theory introduces a critical threshold for the destabilization, such that the perturbation grows only if Δ′>Δ′cr. The value of Δ′cr depends on the equilibrium configuration and on the plasma parameters. Most importantly, Δ′cr can take negative values, thus allowing unstable tearing modes for Δ′ < 0 (even in the absence of bootstrap current).

Estimation of Magnetization Transition Width and Shape by Inverse Reciprocity

Very narrow transition width must be attained to achieve areal densities greater than 1 Tbit/inch <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Not only the grain size, but also the grain switching behavior of the recording medium should be considered. In this paper, the magnetization distribution is calculated by an inverse reciprocity operation on the measured readback waveform obtained from a spin-stand. The calculated magnetization distribution at the transitions between bits was less steep than the error function, which suggested the presence of island-like switched grains away from the transition line due to the finite head field gradient and switching field distribution.

Magnetic-field-dependent pinning potential in LiFeAs superconductor from its Campbell penetration depth

A true critical current density, $j_{c}$, as opposite to commonly measured relaxed persistent (Bean) current, $j_{B}$, was extracted from the Campbell penetration depth, $\lambda_{C}(T,H)$ measured in single crystals of LiFeAs. The effective pinning potential is non-parabolic, which follows from the magnetic field - dependent Labusch parameter $\alpha$. At the equilibrium (upon field - cooling), $\alpha(H)$ is non-monotonic, but it is monotonic at a finite gradient of the vortex density. This behavior leads to a faster magnetic relaxation at the lower fields and provides a natural \emph{dynamic} explanation for the fishtail (second peak) effect. We also find the evidence for strong pinning at the lower fields. The inferred field dependence of the pinning potential is consistent with the evolution from strong pinning, through collective pinning and, eventually, to a disordered vortex lattice. The values of $j_{c}(2\text{K}) \simeq 2\times10^{6}$ A/cm$^{2}$ provide an upper estimate of the current carrying capability of LiFeAs. Overall, vortex behavior of almost isotropic, fully-gapped LiFeAs is very similar to highly anisotropic d-wave cuprate superconductors, the similarity that requires further studies in order to understand unconventional superconductivity in cuprates and pnictides.

Interaction of turbulence with magnetic islands: effect on bootstrap current

A finite radial temperature gradient has been observed to be maintained within magnetic islands in gyro-kinetic turbulence simulations despite the fast motion along the field, and is related to the trapped particles which do not move along the field around the island. Recent calculations of the interaction of drift-wave turbulence with magnetic islands have shown that turbulence can exist within the separatrix, which in turn allows only the partial flattening of electron temperature profiles. Here we calculate, using a minimal drift-kinetic model, the effect on the bootstrap current in a tokamak. Consequences for the stability of the neoclassical tearing mode are discussed.

Shape optimization of shell structures based on NURBS description using automatic differentiation

AbstractThe consolidation of the link among four fields in computational mathematics and mechanics is the main objective of this work. Surfaces based on NURBS (non‐uniform rational B‐spline), mathematical optimization, the finite element method (FEM) in structural analysis, and automatic differentiation (AD) are applied in shape optimization of shells. This problem is performed taking into account the fact that material and mechanical characteristics influence both, the structural shape and the thickness variation, in order to obtain the best performance with respect to a specific criterium. Some techniques were implemented to modify the shell geometry conserving the same parameterization without a new finite element mesh generation. The shape modification is carried out using an optimization code based on the data obtained by a finite element analysis and gradients evaluation. In this work the optimization procedure is performed using an SQP (Sequential Quadratic Programming) algorithm, where the variables are the control points in homogeneous coordinates, the knot vectors, and the thickness. The functional value is determined by the FEM and gradients are evaluated using AD. Some examples are analyzed and discussed. As a consequence of the shape optimization, shells with high structural performance and aesthetically beautiful shapes can be obtained. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Conditions and growth rate of Rayleigh instability in a Hall thruster under the effect of ion temperature

Rayleigh instability is investigated in a Hall thruster under the effect of finite temperature and density gradient of the plasma species. The instability occurs only when the frequency of the oscillations ω falls within a frequency band described by k{y}u₀+1/k_{y}∂²u_{0}/∂x²+Ω/k_{y}n_{0}∂n₀/∂x≪ω<sqrt[Y{i}T{i}k{y}²/M+ω{p}{i}²(Ω²+Y{e}T{e}k{y}²/m)/(ω{p}{e}²+Ω²+Y{e}T{e}k{y}²/m], where u₀ is the drift velocity of the electrons, Ω is their gyration frequency under the effect of the magnetic field, k{y} is the wave propagation constant, n₀ is the plasma density together with ∂n₀/∂x as the density gradient, and T{i}(T{e}), M(m), Y{i}(Y{e}), and ω{p}{i}(ω{p}{e}) are the temperature, mass, specific heat ratio, and plasma frequency of the ions (electrons), respectively. A relevant Rayleigh equation is derived and solved numerically using the fourth-order Runge-Kutta method for investigating the perturbed potential under the effect of electron drift velocity, channel length, magnetic field, ion temperature, and electron temperature. The instability grows faster because of the magnetic field, ion temperature, and drift velocity of the electrons but its growth rate is reduced because of the electron temperature, channel length, and also its far distances from the anode.

Metamaterial tunnel barrier gives broadband microwave transmission

A simple structure comprising a metal mesh, symmetrically surrounded by subwavelength thickness dielectric layers, is shown to give near total microwave transmission over a broad frequency range. The mesh may be considered to be a tunnel barrier since it behaves as an ideal plasmonic metamaterial with a negative effective permittivity and no loss. The introduction of the mesh into the dielectric cavity imposes a finite gradient on the electromagnetic fields at the two mesh-dielectric interfaces. This defines a finite wavelength of the zeroth order Fabry-Pérot-type mode, which would otherwise be infinite. Suitable choice of the mesh parameters yields a broad band of near total transmission associated with the overlap of this zeroth order mode with that of the first order half-wavelength Fabry-Pérot-type resonance.

Effects of finite gradient B drift on collisional boundary layer analysis for neoclassical toroidal plasma viscosity in tokamaks

The effects of finite gradient B drift are included in the collisional boundary layer analysis to improve the accuracy of the neoclassical toroidal plasma viscosity in tokamaks that have error fields or magnetohydrodynamic activities present. Depending on the sign of the electric charge of the species and that of the radial electric field, the effects of finite gradient B drift can either reduce, if the E × B drift is in the same direction of the gradient B drift, or enhance, if these two drifts are in the opposite direction, the magnitude of the neoclassical toroidal plasma viscosity. Here, E is the electric field and B is the magnetic field. However, because the gradient B drift depends on the effective pitch angle, the net effects have to be properly weighted by integrating over the particle energy.