Zeroth-order Approximation Research Articles

Kinetic theory models for granular mixtures with unequal granular temperature: Hydrodynamic velocity

Kinetic theory for granular mixtures with a separate granular temperature for each solid phase is considered. The particle phases are assumed to follow a Maxwellian distribution with zero-order approximation of the Boltzmann equation. A solution strategy for solving the integrals of the collisional closure relations is presented. The present analysis enables the solution of these integrals without neglecting the solid particle hydrodynamic velocities in the distribution function. However, the solution strategy limits the validity of the closure relations to relatively small and moderate hydrodynamic velocity differences. Dependent on the magnitude of the hydrodynamic velocity difference, the present closure relations can differ significantly from the simplified models where such velocity difference is traditionally neglected.

Scaling law for the slow flow of an unstable mechanical system coupled to a nonlinear energy sink

In this paper one first shows that the slow flow of a mechanical system with one unstable mode coupled to a Nonlinear Energy Sink (NES) can be reduced, in the neighborhood of a fold point of its critical manifold, to a normal form of the dynamic saddle-node bifurcation. This allows us to then obtain a scaling law for the slow flow dynamics and to improve the accuracy of the theoretical prediction of the mitigation limit of the NES previously obtained as part of a zeroth-order approximation. For that purpose, the governing equations of the coupled system are first simplified using a reduced-order model for the primary structure by keeping only its unstable modal coordinates. The slow flow is then derived by means of the complexification-averaging method and, by the presence of a small perturbation parameter related to the mass ratio between the NES and the primary structure, it appears as a fast-slow system. The center manifold theorem is finally used to obtain the reduced form of the slow flow which is solved analytically leading to the scaling law. The latter reveals a nontrivial dependence with respect to the small perturbation parameter of the slow flow dynamics near the fold point, involving the fractional exponents 1/3 and 2/3. Finally, a new theoretical prediction of the mitigation limit is deduced from the scaling law. In the end, the proposed methodology is exemplified and validated numerically using an aeroelastic aircraft wing model coupled to one NES.

Zeroth-Order Nonconvex Stochastic Optimization: Handling Constraints, High Dimensionality, and Saddle Points

In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting, and saddle point avoiding. To handle constrained optimization, we first propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. To facilitate zeroth-order optimization in high dimensions, we explore the advantages of structural sparsity assumptions. Specifically, (i) we highlight an implicit regularization phenomenon where the standard stochastic gradient algorithm with zeroth-order information adapts to the sparsity of the problem at hand by just varying the step size and (ii) propose a truncated stochastic gradient algorithm with zeroth-order information, whose rate of convergence depends only poly-logarithmically on the dimensionality. We next focus on avoiding saddle points in nonconvex setting. Toward that, we interpret the Gaussian smoothing technique for estimating gradient based on zeroth-order information as an instantiation of first-order Stein’s identity. Based on this, we provide a novel linear-(in dimension) time estimator of the Hessian matrix of a function using only zeroth-order information, which is based on second-order Stein’s identity. We then provide a zeroth-order variant of cubic regularized Newton method for avoiding saddle points and discuss its rate of convergence to local minima.

The problem of a hydrogen atom in a cavity: Oscillator representation solution versus analytic solution

Abstract A solution to the problem of a hydrogenic atom in a homogeneous dielectric medium with a concentric spherical cavity using the oscillator representation method (ORM) is presented. The results obtained by the ORM are compared with a known exact analytic solution. The energy levels of the hydrogenic atom in a spherical cavity exhibit a shallow-deep instability as a function of the cavity radius. The sharpness of the transition depends on the value of the dielectric constant of the medium. The results of the ORM agree well with the results obtained by the analytic solution when the shallow-deep transition is not too sharp (i.e., when the dielectric constant is not too large) for all values of the cavity radius. The ORM results in the zeroth order approximation diverge significantly in the region of the shallow-deep transition (i.e., for the values of the radius where the shallow-deep transition occurs) when the dielectric constant is high and as a result the transition is sharp. Even for the sharp transition, the ORM results again agree very well with the analytic results at least for the ground state when a commonly used approximation in the ORM is removed. The ORM methodology for the cavity model presented in this article can potentially be used for two-electron systems in a quantum dot.

Kinetic theory models for granular mixtures with unequal granular temperature. Derivation of analytical constitutive equations

We consider kinetic theory of granular mixtures in which each solid phase is described by its own granular temperature. The particle velocities are assumed to follow a Mawellian distribution with zero-order approximation of the Boltzmann equation. The literature proposes a number of approximate solutions to the constitutive equations resulting from particle collisions. Here, we derive and show that analytical solutions exist, and reveal that some of the proposed approximate solutions are inaccurate whereas others might give good approximations.We also review the balance laws and constitutive relations resulting from the mono- and poly-disperse kinetic granular flow model frameworks. Such a review has been done here because the model framework on kinetic theory of mono-disperse granular flow systems has sometimes been adopted in models for granular mixtures. The consequence of such a simplification is that the kinetic energy loss in a collision between two particles of different species is equally shared among them.

Component replacement TPA: A transmissibility-based structural modification method for in-situ transfer path analysis

In-situ transfer path analysis is a diagnostic method used to analyse the propagation of noise and vibration through complex built-up structures. Its defining feature is the independent characterisation of an assembly’s active components (i.e. vibration sources) by their blocked forces. This independent characterisation enables the downstream structural modification of an assembly without affecting the sources’ operational characteristics. In practical engineering structures, however, there is often a need to alter or replace components that reside within a vibration source, for example resilient mounts. An upstream structural modification of this sort would alter the blocked force and thus invalidate any response predictions made thereafter. Hence, an alternative approach is required. In the present paper a transmissibility-based structural modification method is introduced. We derive a set of equations that relate the blocked force and forward transfer functions obtained from an initial assembly, to those of an upstream modified assembly. Exact formulations are provided, together with first and zeroth order approximations for resiliently coupled structures. These component replacement expressions are verified by numerical examples.

Modeling and Decoupling Control of a Linear Permanent Magnet Actuator Considering Fringing Effect for Precision Engineering

In this article, accurate modeling and decoupling control of a linear ironless PM actuator are implemented for precision engineering. Given the fringing effect due to the finite length, the magnetic field of the PMs and the three-phase winding are both calculated by applying an improved Fourier series expansion. Subsequently, based on a distributed model, the associated flux linkage and winding inductances are obtained and validated. Compared with rotary motors of physical symmetry, the winding inductance matrix makes the electrical model of the finite-long linear actuator present a new non-uniform form, to which the classic decoupling technique is inapplicable directly. Therefore, improved decoupling strategies are established utilizing the zeroth-order approximation of the inductances based on the $dq0$ transformation, as well as a real-time updating scheme. Physical experiments show that high-order harmonics are eliminated by applying the non-uniform model, which makes it effective and applicable.

Estimation of C. elegans cell- and tissue volumes.

Although C. elegans is one of the best-studied model organisms, an estimate of its cell sizes and tissues is missing. Here we used the Virtual Worm that is based on electron microscopy images to calculate a zeroth-order approximation of cell and tissue sizes of C. elegans. We conclude that the intestine is the largest tissue, followed by the hypodermis, gonads, body wall muscles, pharynx, and neurons. Thus, we provide an approximation of tissue volumes of young adult C. elegans.

Organometallic ruthenium (η6-p-cymene) complexes interfering with quorum sensing and biofilm formation: an anti-infective approach to combat multidrug-resistance in bacteria

Organometallic ruthenium complexes of flavonoids as antiquorum sensing agents against pathogens likeChromobacterium violaceumATCC 12472,Pseudomonas aeruginosaPAO1 and methicillin-resistantS. aureus(MRSA).

Analysis of Lowest-Order Characteristics-Mixed FEMs for Incompressible Miscible Flow in Porous Media

The time discrete scheme of characteristics type is especially effective for convection-dominated diffusion problems. The scheme has been used in various engineering areas with different approximations in spatial direction. The lowest-order mixed method is the most popular one for miscible flow in porous media. The method is based on a linear Lagrange approximation to the concentration and the zero-order Raviart-Thomas approximation to the pressure/velocity. However, the optimal error estimate for the lowest-order characteristics-mixed FEM has not been presented although numerous effort has been made in last several decades. In all previous works, only first-order accuracy in spatial direction was proved under certain time-step and mesh size restrictions. The main purpose of this paper is to establish optimal error estimates, $i.e.$, the second-order in $L^2$-norm for the concentration and the first-order for the pressure/velocity, while the concentration is more important physical component for the underlying model. For this purpose, an elliptic quasi-projection is introduced in our analysis to clean up the pollution of the numerical velocity through the nonlinear dispersion-diffusion tensor and the concentration-dependent viscosity. Moreover, the numerical pressure/velocity of the second-order accuracy can be obtained by re-solving the (elliptic) pressure equation at a given time level with a higher-order approximation. Numerical results are presented to confirm our theoretical analysis.

Effects of Small Permanent Charge on PNP Models

In this paper, a stationary one-dimensional Poisson-Nernst-Planck model with permanent charge is studied under the assumption that n - 1 positively charged ion species have the same valence and the permanent charge is small. By expanding the singular solutions of Poisson-Nernst-Planck model with respect to small permanent charge, the explicit formulae for the zeroth order approximation and the first order approximation of individual flux can be obtained. Based on these explicit formulae, the effects of small permanent charges on individual flux are investigated.

Scalable Adaptive Batch Sampling in Simulation-Based Design With Heteroscedastic Noise

Abstract In this study, we propose a scalable batch sampling scheme for optimization of simulation models with spatially varying noise. The proposed scheme has two primary advantages: (i) reduced simulation cost by recommending batches of samples at carefully selected spatial locations and (ii) improved scalability by actively considering replicating at previously observed sampling locations. Replication improves the scalability of the proposed sampling scheme as the computational cost of adaptive sampling schemes grow cubicly with the number of unique sampling locations. Our main consideration for the allocation of computational resources is the minimization of the uncertainty in the optimal design. We analytically derive the relationship between the “exploration versus replication decision” and the posterior variance of the spatial random process used to approximate the simulation model’s mean response. Leveraging this reformulation in a novel objective-driven adaptive sampling scheme, we show that we can identify batches of samples that minimize the prediction uncertainty only in the regions of the design space expected to contain the global optimum. Finally, the proposed sampling scheme adopts a modified preposterior analysis that uses a zeroth-order interpolation of the spatially varying simulation noise to identify sampling batches. Through the optimization of three numerical test functions and one engineering problem, we demonstrate (i) the efficacy and of the proposed sampling scheme to deal with a wide array of stochastic functions, (ii) the superior performance of the proposed method on all test functions compared to existing methods, (iii) the empirical validity of using a zeroth-order approximation for the allocation of sampling batches, and (iv) its applicability to molecular dynamics simulations by optimizing the performance of an organic photovoltaic cell as a function of its processing settings.

Approximate analytical solution for ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas: isothermal flow

The propagation of an ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas under isothermal flow condition with an azimuthal magnetic field is investigated. The electrical conductivity is assumed to be negligible in the medium ahead of the shock wave, which after the passage of the shock wave becomes infinitely large. The magnetic pressure, azimuthal fluid velocity, and axial fluid velocity are assumed to be varying according to the power law with distance from the axis of symmetry in the undisturbed medium. The zeroth and first-order approximations are discussed by the aid of the power series method. Solutions for the zeroth-order approximation are constructed in analytical form. Distributions of hydrodynamical quantities are discussed. The effect of flow parameters, namely, shock wave Cowling number c∗, adiabatic exponent γ, rotational parameter L, and gas non-idealness parameter [Formula: see text] are studied on the flow variables. Due to the consideration of a rotating medium or due to the presence of magnetic field, the total energy of the disturbance increases, while with an increase in adiabatic exponent γ the total energy of the disturbance decreases. Density and pressure vanish near the axis of symmetry, thus forming a vacuum there.

Load Transfer from an Infinite Inhomogeneous Stringer to a Prestressed Elastic Strip Clamped at One Edge*

The plane contact problem of the transfer of a horizontal load from a weakly inhomogeneous infinite stringer to a prestressed elastic strip clamped at one edge is solved using the linearized theory of elasticity. The solution has a general form for the theory of large initial deformations and different theories of small initial deformations for an arbitrary elastic potential. The problem for the normal and tangential contact stresses is reduced to a governing system of recurrent systems of integro-differential equations, which is solved in powers of a small parameter. The zero-order approximation of the inhomogeneous problem is derived using the Fourier transform. The contact stresses are represented by Fourier integrals.

Explicit Multipole Formula for the Local Thermal Resistance in an Energy Pile—The Line-Source Approximation

This paper presents a closed-form quite handy formula for the local thermal resistance Rb between the temperature of the bulk heat-carrier fluid in the pipes, equally spaced on a concentric circle inside a circular energy pile, and the mean temperature at the periphery of the pile. The so-called multipole method is used to calculate the temperature field. An important improvement of the multipole method is presented, where Cauchy’s mean value theorem of analytical functions is used. The formula for thermal resistance Rb0 for the zero-order approximation (J = 0), where only line heat sources at the pipes are used, is presented. The errors using zeroth-order approximation (J = 0) are shown to be quite small by comparisons with eight-order approximation (J = 8) with its accuracy of more than eight digits. The relative error for the local thermal resistance Rb0 for the zero-order approximation (J = 0) lies below 5% for a wide range of input parameter values. These ranges are judged to cover most practical cases of application. The smallest local thermal resistance Rbmin is, with some exceptions, obtained when the pipes lie directly in contact with the pile periphery. A neat formula for this minimum is presented.

On the mechanics of composite sandwich plates with three-dimensional stress recovery

By using the variational-asymptotic method, a universal asymptotic model for composite sandwich plates is established to achieve a great compromise between efficiency and accuracy through the synthesis of two competing theories: equivalent single-layer theories and layer-wise theories. When each layer of a sandwich structure can be individually considered as an elastic plate, and all material constitutive constants of such plates can be assumed to be of the same order, an equivalent plate model can be constructed from the equivalent single-layer perspective. It has an asymptotically correct energy functional, capable of capturing the transverse deformations, but still limited to the zeroth-order approximation. Then, by taking into account mismatched constituent material properties, a universal model for predicting mechanical behavior of composite sandwich plates is then systematically derived from the layer-wise perspective. It has another asymptotically correct energy functional with respect to the core-layer’s plate variables and implements into a single unified representation for subsequent application to sandwich plate problems with any face-to-core-stiffness and length-to-thickness ratios. In particular, to resolve theoretical shortcomings found in published works, complementary theoretical procedures are incorporated and used into the present approach by considering the interlaminar transverse stress continuity conditions as essential conditions in conventional layer-wise theories and by providing three-dimensional recovery relations as necessary ingredients for theories based on various dimensional reduction processes. Finally, as a preliminary validation, several examples available in the literature are presented and investigated. Together with critical comparisons of the three-dimensional exact solutions, the close agreement demonstrates the capability and accuracy of this present approach.

Rotating ion beam effects on temperature gradient instability in completely ionized plasmas.

The aim of this paper is to investigate the effects of a rotating ion beam on the temperature gradient instability (TGI) in completely ionized plasmas. The interplay of the temperature and density gradients provides the basis for experiencing an unstable inhomogeneous plasma medium due to TGI taken under consideration. The density and temperature gradients are considered perpendicular to the magnetic field where a nonrelativistic rotating ion beam such as O^{+} is present. By implementing the kinetic theory together with a zeroth-order approximation of geometrical optics, the dielectric permittivity tensor of the inhomogeneous plasma is obtained where by a suitable linear eikonal equation, the growth rate of the TGI in the collisional regime is calculated in the presence of a rotating ion beam. In such a configuration an unstable condition is experienced in regions with opposite electron density and temperature gradients, where it is destabilized by the temperature and plasma density gradients and the frequent electron collisions. As a consequence, the results reveal that the TGI can be damped or modified through interaction with the rotating ion beam depending on the characteristics of the ion beam, namely, velocity and density.

Blast waves in a non-ideal self-gravitating gas with magnetic field

The propagation of the blast wave in self-gravitating non-ideal gas under the influence of axial magnetic field is investigated. The magnetic pressure and density are considered to be varying according to the power law. The approximate analytical solutions are obtained by expanding the flow variables in the form of power series using Sakurai’s technique. The zeroth- and first-order approximate solutions of self-gravitating non-ideal gas with magnetic field are discussed in detail. Solutions for zeroth-order approximation are constructed in analytical form. The effect of the flow parameters, namely parameter of non-idealness, adiabatic exponent, ambient density variation exponent and gravitational parameter on the flow variables, is discussed in detail. The results obtained in this work are compared with the results obtained by Sakurai and Nath, which are in good agreement

Response of Duffing's Oscillator to Harmonic Base Excitation and Significance of First Order Term

Responses of systems with nonlinear stiffness subjected to base harmonic excitation are determined. An expression to estimate the amplitude in the fundamental frequency of oscillation is derived from first principles using Lindstedt's method. It is observed that the amplitude determined using the zero order approximation is in error at low frequencies. Therefore, an expression for the first order approximation of the amplitude of response at the fundamental frequency is derived. Zero order and first order approximation terms together form the response. Characteristics showing the variation of the amplitude with the excitation frequency for various nonlinear spring parameters are presented. The issue at low frequencies is resolved by the incorporation of the first order term. An expression for the phase difference and the expression of the asymptote where the responses converge are also derived.

Introducing an Improved GRACE Global Point-Mass Solution—A Case Study in Antarctica

In the so-called point-mass modeling, surface densities are represented by point masses, providing only an approximated solution of the surface integral for the gravitational potential. Here, we propose a refinement for the point-mass modeling based on Taylor series expansion in which the zeroth-order approximation is equivalent to the point-mass solution. Simulations show that adding higher-order terms neglected in the point-mass modeling reduces the error of inverted mass changes of up to 90% on global and Antarctica scales. The method provides an alternative to the processing of the Level-2 data from the Gravity Recovery and Climate Experiment (GRACE) mission. While the evaluation of the surface densities based on improved point-mass modeling using ITSG-Grace2018 Level-2 data as observations reveals noise level of approximately 5.77 mm, this figure is 5.02, 6.05, and 5.81 mm for Center for Space Research (CSR), Goddard Space Flight Center (GSFC), and Jet Propulsion Laboratory (JPL) mascon solutions, respectively. Statistical tests demonstrate that the four solutions are not significant different (95% confidence) over Antarctica Ice Sheet (AIS), despite the slight differences seen in the noises. Therefore, the estimated noise level for the four solutions indicates the quality of GRACE mass changes over AIS. Overall, AIS shows a mass loss of −7.58 mm/year during 2003–2015 based on the improved point-mass solution, which agrees with the values derived from mascon solutions.