Second-order Partial Derivatives Research Articles

Correlations between Fukui Indices and Reactivity Descriptors Based on Sanderson's Principle.

A previous study presented a very simple model mainly based on Sanderson's principle for estimating important reactivity descriptors, such as the local hardness and the local electrophilicity. In another study, a new way of calculating the Fukui function was obtained that resulted in a new operative formula for the function. We also obtained the second-order partial derivative of the electron density with regard to the number of electrons. The current study analyzes the relationships between the two models and justifies the origin of these relationships.

EL: Local Image Descriptor Based on Extreme Responses to Partial Derivatives of 2D Gaussian Function

We propose a two-part local image descriptor EL (Edges and Lines), based on the strongest image responses to the first- and second-order partial derivatives of the two-dimensional Gaussian function. Using the steering theorems, the proposed method finds the filter orientations giving the strongest image responses. The orientations are quantized, and the magnitudes of the image responses are histogrammed. Iterative adaptive thresholding of histogram values is then applied to normalize the histogram, thereby making the descriptor robust to nonlinear illumination changes. The two-part descriptor is empirically evaluated on the HPatches benchmark for three different tasks, namely, patch verification, image matching, and patch retrieval. The proposed EL descriptor outperforms the traditional descriptors such as SIFT and RootSIFT on all three evaluation tasks and the deep-learning-based descriptors DeepCompare, DeepDesc, and TFeat on the tasks of image matching and patch retrieval.

Force Reconstruction for Uncertain Structure Based on Interval Model and Second-Order Perturbation Theory

In order to reconstruct the upper and lower bounds of dynamic excitations applied on the uncertain structure, an algorithm based on interval model and second-order perturbation theory is presented in this paper. First, interval model is built up by expressing the uncertain parameters of structure in interval form. Next according to second-order perturbation theory, structure characteristic matrices and input force vector are approximated as second-order Taylor polynomial expansion at the midpoint of uncertain parameters. After that the input force’s midpoint, first-order and second-order partial derivatives are respectively calculated by existing step-by-step integration method. Then addition and subtraction of the three components obtained in previous step are operated. Ultimately the upper and lower bounds of dynamic load can be identified. Numerical simulation results demonstrate this method is with the characteristic of high efficiency and precision. In addition, it is able to remain a relatively strong robustness under noise turbulence.

A Novel Framework to Harmonise Satellite Data Series for Climate Applications

Fundamental and thematic climate data records derived from satellite observations provide unique information for climate monitoring and research. Since any satellite only operates over a relatively short period of time, creating a climate data record also requires the combination of space-borne measurements from a series of several (often similar) satellite sensors. Simply combining calibrated measurements from several sensors can, however, produce an inconsistent climate data record. This is particularly true of older, historic sensors whose behaviour in space was often different from their behaviour during pre-launch calibration and more scientific value can be derived from considering the series of historical and present satellites as a whole. Here, we consider harmonisation as a process that obtains new calibration coefficients for revised sensor calibration models by comparing calibrated measurements over appropriate satellite-to-satellite matchups, such as simultaneous nadir overpasses and which reconciles the calibration of different sensors given their estimated spectral response function differences. We present the concept of a framework that establishes calibration coefficients and their uncertainty and error covariance for an arbitrary number of sensors in a metrologically-rigorous manner. We describe harmonisation and its mathematical formulation as an inverse problem that is extremely challenging when some hundreds of millions of matchups are involved and the errors of fundamental sensor measurements are correlated. We solve the harmonisation problem as marginalised errors in variables regression. The algorithm involves computation of first and second-order partial derivatives using Algorithmic Differentiation. Finally, we present re-calibrated radiances from a series of nine Advanced Very High Resolution Radiometer sensors showing that the new time series has smaller matchup differences compared to the unharmonised case while being consistent with uncertainty statistics.

Quasistatic Stability of Frictionless Planar Enveloping Grasps

This paper discusses stability of frictionless planar enveloping grasps. Linear spring stiffness is included in the joints and the finger surface of a multifingered hand. The grasp stability is analyzed using potential energy stored in the grasp. The first- and the second-order partial derivatives of the energy are derived analytically. Effect of surface curvature deviation is also analyzed.

Parameter estimation of 2D polynomial phase signals using NU sampling and 2D CPF

The two-dimensional (2D) cubic phase function (CPF) is known as a highly accurate 2D polynomial phase signal estimator, but it has limited applicability due to the requirement for the 3D search for second-order partial phase derivatives. The authors propose an interpolation-based approach simulating non-uniform (NU) signal sampling in order to reduce the 2D CPF calculation complexity. The NU resampling enables the 2D CPF evaluation using the 2D fast Fourier transform and searches over mixed-phase parameter. The computational complexity is reduced from O ( N 5 ) to O ( N 3 log 2 N ) . The additional stage with dechirping, filtering and phase unwrapping is introduced to refine parameter estimates.

Numerical simulations of multilingual competition dynamics with nonlocal derivative

The dynamics of the language competition model is considered in this paper. The classical system is converted to non-integer order case by replacing the second-order partial derivative with the Riesz fractional derivative. A well-known numerical approximation methods based on the Fourier spectral algorithm in space and the third-order exponential time-differencing scheme are formulated to numerically simulate the three component fractional-in-space reaction-diffusion system in one and high dimensions for different values of α. Numerical results indicate α ∈ (1, 1.5] as the key control parameter that can influence the coexistence of various speakers over a period of time.

Local averaged path integration method approach for nonlinear dynamic systems

Traditional path integration method always has a limited time step length, whose minimum is based on the subdivision degree of integral space. In order to numerically analyze a smaller time step probability evolution without significant increase in computing time, a new local averaged path integration (LAPI) method is investigated. The basic idea of this method is giving a local averaged value of transition probability density function (TPDF). Then the numerical integration of Chapman–Kolmogorov (CK) equation to the local averaged probability density function (PDF) will be calculated in lower space subdivision degree. Second-order partial derivative of PDF can be used to correct the deviation between local averaged PDF and the real one. It is demonstrated that the proposed method can give a more efficient calculation compared with traditional Gauss–Legendre path integration (GLPI). In the case of our method, smaller time step value is permitted with the same space subdivision degree, and smaller space subdivision degree is needed to keep the accuracy of stationary PDF with the same time step value. This merits also exist when analyzing transient PDF, which leads to a better capability of capturing the evolution of the PDF.

Second-Order Integral-Form Gauss’s Variational Equations Under Impulsive Control

Second-order integral-form Gauss’s variational equations (GVEs) under impulsive control are derived for both nonsingular orbital elements and classical orbital elements. These equations are used to estimate the changes of the orbital elements under assigned impulsive vector. These GVEs are obtained by solving the second-order partial derivatives of orbital elements with respect to velocity vector. In addition, using analytical series approximation in the second-order GVEs, the required impulsive control is obtained for specified orbital-element variations. Based on the proposed second-order GVEs, coplanar two-impulse maneuver problems are solved to perform corrections of three and four orbital elements, respectively. Compared with the classical linear GVEs, the proposed second-order GVEs are more accurate to estimate the orbital-element variations and to solve for the required impulsive control. Several numerical examples are provided to validate the accuracy gain of the second-order GVEs.

Numerical patterns in system of integer and non-integer order derivatives

This research work contributes to the formation of spatial patterns in fractional-order reaction-diffusion systems. The classical second-order partial derivatives in such systems are replaced with the Riemann–Liouville fractional derivative of order α ∈ (1, 2]. We equally propose a novel numerical scheme for the approximation in space, and the resulting system of equations is advance in time with the improved fourth-order exponential time differencing method. Mathematical analysis of general two-component integer and non-integer order derivatives are provided. To guarantee the correct choice of the parameters in the main dynamics, we carry-out their linear stability analysis. Theorems regarding the local-stability and the conditions for a Hopf-bifurcation to occur are also provided. The proposed numerical method is applied to solve two non-integer-order models, namely the biological (predator-prey) and chemical (activator-inhibitor) systems. We observed some amazing patterns that are completely missing in the classical case at different values of fractional power α in high dimensions that evolve in fractional reaction-diffusion equations.

A novel stability-based EMG-assisted optimization method for the spine

Traditional electromyography-assisted optimization (TEMG) models are commonly employed to compute trunk muscle forces and spinal loads for the design of clinical/treatment and ergonomics/prevention programs. These models calculate muscle forces solely based on moment equilibrium requirements at spinal joints. Due to simplifications/assumptions in the measurement/processing of surface EMG activities and in the presumed muscle EMG-force relationship, these models fail to satisfy stability requirements. Hence, the present study aimed to develop a novel stability-based EMG-assisted optimization (SEMG) method applied to a musculoskeletal spine model in which trunk muscle forces were estimated by enforcing equilibrium conditions constrained to stability requirements. That is, second-order partial derivatives of the potential energy of the musculoskeletal model with respect to its generalized coordinates were enforced to be positive semi-definite. Fifteen static tasks in upright and flexed postures with and without a hand load at different heights were simulated. The SEMG model predicted different muscle recruitments/forces (generally larger global and local muscle forces) and spinal loads (slightly larger) compared to the TEMG model. Such task-specific differences were dependant on the assumed magnitude of the muscle stiffness coefficient in the SEMG model. The SEMG model-predicted and measured L4-L5 intradiscal pressures were in satisfactory agreement during simulated activities.

Performance analysis and improvement of direct position determination based on Doppler frequency shifts in presence of model errors: case of known waveforms

The direct position determination (DPD) method based on Doppler frequency shifts for signals with known waveforms is first proposed by Amar and Weiss. Although this method exhibits excellent asymptotic performance, but does not account for the effects of uncertainties in the receiver positions and velocities. These uncertainties may result in a considerable reduction of localization accuracy. In this paper, the statistical performance of the DPD estimator is investigated under receiver position and velocity errors (also called model errors). We derive an analytical expression for the mean square error in the estimated source location in the case where the estimator assumes that the receiver positions and velocities are accurate, but they in fact contain errors. The main difficulty in the mathematical analysis is that the DPD cost function is not explicit with respect to the emitter position. Consequently, some algebraic manipulation is required to derive closed-form expressions for the first- and second-order partial derivatives of the cost function. The Cramer–Rao bounds (CRBs) for the target position estimation are also deduced in the presence and absence of model errors. These CRBs provide insights into the effects of model errors on the localization accuracy. Two improved DPD methods are developed based on our analysis results. The first has lower complexity than the common grid search, and the second exhibits increased robustness to model errors. Simulation results support and corroborate the theoretical developments in this paper.

Modeling of thin-walled structures interacting with acoustic media as constrained two-dimensional continua

The transient interaction of acoustic media and elastic shells is considered on the basis of the transition function approach. The three-dimensional hyperbolic initial boundary-value problem is reduced to a two-dimensional problem of shell theory with integral operators approximating the acoustic medium effect on the shell dynamics. The kernels of these integral operators are determined by the elementary solution of the problem of acoustic waves diffraction at a rigid obstacle with the same boundary shape as the wetted shell surface. The closed-form elementary solution for arbitrary convex obstacles can be obtained at the initial interaction stages on the background of the so-called “thin layer hypothesis”. Thus, the shell–wave interaction model defined by integro-differential dynamic equations with analytically determined kernels of integral operators becomes hence two-dimensional but nonlocal in time. On the other hand, the initial interaction stage results in localized dynamic loadings and consequently in complex strain and stress states that require higher-order shell theories. Here the modified theory of I.N.Vekua–A.A.Amosov-type is formulated in terms of analytical continuum dynamics. The shell model is constructed on a two-dimensional manifold within a set of field variables, Lagrangian density, and constraint equations following from the boundary conditions “shifted” from the shell faces to its base surface. Such an approach allows one to construct consistent low-order shell models within a unified formal hierarchy. The equations of the N th-order shell theory are singularly perturbed and contain second-order partial derivatives with respect to time and surface coordinates whereas the numerical integration of systems of first-order equations is more efficient. Such systems can be obtained as Hamilton–de Donder–Weyl-type equations for the Lagrangian dynamical system. The Hamiltonian formulation of the elementary N th-order shell theory is here briefly described.

A Hydraulic Friction Model for One-Dimensional Unsteady Channel Flows with Experimental Demonstration

As a critical parameter of the steady uniform friction model, the roughness coefficient changes with flow unsteadiness in flood events; i.e., the flow conditions of the stream segment significantly affect the flow resistance. In this study, a modified formula was established to improve the unsteady friction simulation; ten terms relating to the first- and second-order time and space partial derivatives of hydraulic parameters were selected as additional terms. The results of a hydraulic experiment show that the hysteresis between flow depth and mean cross-sectional velocity cannot be neglected in unsteady flows that disturb the performance of a steady uniform friction model. Six terms have a strong correlation with objective friction. Further, three of them have a small variance in correlation coefficient. Then, the composition of the proposed formula was determined. The results show that adding too many additional terms provides better performance in the calibration phase, yet reduces the accuracy of the validation phase because of an overfitting phenomenon. The optimal number of additional terms is three, and the established formula can improve the unsteady friction simulation.

Investigation of Numerical Dissipation in Classical and Implicit Large Eddy Simulations

The quantitative measure of dissipative properties of different numerical schemes is crucial to computational methods in the field of aerospace applications. Therefore, the objective of the present study is to examine the resolving power of Monotonic Upwind Scheme for Conservation Laws (MUSCL) scheme with three different slope limiters: one second-order and two third-order used within the framework of Implicit Large Eddy Simulations (ILES). The performance of the dynamic Smagorinsky subgrid-scale model used in the classical Large Eddy Simulation (LES) approach is examined. The assessment of these schemes is of significant importance to understand the numerical dissipation that could affect the accuracy of the numerical solution. A modified equation analysis has been employed to the convective term of the fully-compressible Navier–Stokes equations to formulate an analytical expression of truncation error for the second-order upwind scheme. The contribution of second-order partial derivatives in the expression of truncation error showed that the effect of this numerical error could not be neglected compared to the total kinetic energy dissipation rate. Transitions from laminar to turbulent flow are visualized considering the inviscid Taylor–Green Vortex (TGV) test-case. The evolution in time of volumetrically-averaged kinetic energy and kinetic energy dissipation rate have been monitored for all numerical schemes and all grid levels. The dissipation mechanism has been compared to Direct Numerical Simulation (DNS) data found in the literature at different Reynolds numbers. We found that the resolving power and the symmetry breaking property are enhanced with finer grid resolutions. The production of vorticity has been observed in terms of enstrophy and effective viscosity. The instantaneous kinetic energy spectrum has been computed using a three-dimensional Fast Fourier Transform (FFT). All combinations of numerical methods produce a k − 4 spectrum at t * = 4 , and near the dissipation peak, all methods were capable of predicting the k − 5 / 3 slope accurately when refining the mesh.

The lumped-parameter model on two-phase and superheated geothermal reservoir

Tank models (also known as “lumped parameter” models) may be used to describe the dynamical response of a geothermal reservoir. In this paper, a generalized combined lumped-parameter model consisting of both two-phase and superheated geothermal fluid is developed. The model is inherently nonlinear and multivariate and therefore it is difficult to compute Hessian matrix (second-order partial derivatives) or Jacobian matrix. We show how the BFGS algorithm, a kind of quasi-Newton method, based on Armijo line search can be used for parameter estimation. Using this algorithm, two different scenarios are matched with data generated by tank model itself.

Second-order moments of Schell-model beams with various correlation functions in atmospheric turbulence.

The general formulae for second-order moments of Schell-model beams with various correlation functions in atmospheric turbulence are derived and validated by the Bessel-Gaussian Schell-model beams and cosine-Gaussian-correlated Schell-model beams. Our finding shows that the second-order moments of partially coherent Schell-model beams are related to the second-order partial derivatives of source spectral degree of coherence at the origin. The formulae we provide are much more convenient to analyze and research propagation problems in turbulence.

Non-stationary random response analysis of structures with uncertain parameters

This paper proposes a non-stationary random response analysis method of structures with uncertain parameters. The structural physical parameters and the input parameters are considered as random variables or interval variables. By using the pseudo-excitation method and the direct differentiation method (DDM), the analytical expression of the time-varying power spectrum and the time-varying variance of the structure response can be obtained in the framework of first order perturbation approaches. In addition, the analytical expression of the first-order and second-order partial derivative (e.g., time-varying sensitivity coefficient) for the time-varying power spectrum and the time-varying variance of the structure response expressed via the uncertainty parameters can also be determined. Based on this and the perturbation technique, the probabilistic and non-probabilistic analysis methods to calculate the upper and lower bounds of the time-varying variance of the structure response are proposed. Finally the effectiveness of the proposed method is demonstrated by numerical examples compared with the Monte Carlo solutions and the vertex solutions.

Second‐order sensitivity analysis of electromagnetic performance with respect to surface nodal displacements for reflector antennas and its benefits in integrated structural–electromagnetic optimisation

Integrated structural–electromagnetic optimisation design is an interesting area in the field of antenna design. Due to the urgent requirements on second-order sensitivity analysis to benefit the integrated iterative procedure, based on the authors’ previous work, an analytic second-order partial derivative formula of electromagnetic performance – directivity with respect to surface nodal displacements for reflector antennas is deduced. In this formula, the electromagnetic performance is obtained by superposition of radiation integrals on each small triangular element. Then, a perturbation of surface nodal displacement is introduced to re-express the distorted radiation integral. By partially differentiating the distorted phase error in radiation integral with respect to nodal perturbation over the corresponding triangular element twice, the second-order partial derivative is derived. Two reflectors – an axis-symmetrical one and an offset one, are illustrated in simulation to show the second-order partial derivatives in contour plots. A comparative simulation is performed to illustrate its benefits in integrated structural–electromagnetic optimisation.

Correcting errors from spatial upscaling of nonlinear greenhouse gas flux models

Ecological models are used to quantify processes over large regions. When the model is nonlinear and input variables are spatially averaged, the regional mean will be in error.A formula for estimating the upscaling error can be derived from Taylor expansion of the model (Bresler and Dagan 1988). We test this for simple models under three different input distributions (Gaussian, exponential, lognormal). In several cases the formula is exact, in others it provides a reasonable approximation.We then study models for emissions of methane, ammonia, and nitrous oxide across the UK. We scale from 1 × 1 km to 32 × 32 km. The UK-average upscaling errors are −12%, −48% and −3%, well estimated using the formula.The formula is a useful tool for modellers desiring to correct upscaling error for their application. Calculation of second-order partial derivatives of model output is required, for which we provide R-code.