Jacobian Of Transformation Research Articles

Analytical study of magnetic properties of quantum dot/ring system: Rashba, magnetic field, and electron-electron effects

Abstract Magnetic behavior of a two-electron quantum dot/ring system are analytically studied with electron-electron (e-e) interaction taking into account the Rashba spin-orbit interaction (SOI) and magnetic field. The Jacobi transformation has been employed to separate the Hamiltonian of the system to the center of mass and relative terms. The Schrödinger equation is analytically solved, and energy spectra are obtained. Then, the magnetization and susceptibility are calculated. The magnetization decreases by rising the magnetic field without and with SOI and also without e-e interaction. Also, SOI slightly modifies the magnetization of the system without e-e interaction. The susceptibility displays a peak structure as the magnetic field changes from low values to high values. The susceptibility by considering e-e interaction and without SOI is always negative and its value decreases by rising the magnetic field. The susceptibility displays a transition from diamagnetic to paramagnetic with e-e interaction and SOI.

Logarithmic terms in discrete heat kernel expansions in the quadrant

In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set, and given length. After renormalisation, this number may be interpreted as a discrete heat kernel in the quadrant. We propose a new method to compute complete asymptotic expansions of these numbers of walks as their length tends to infinity, based on two main ingredients: explicit expressions for the underlying generating functions in terms of elliptic Jacobi theta functions along with a duality known as Jacobi transformation. This duality allows us to pass from a classical Taylor expansion of the series to an expansion at the critical point of the model. We work through two examples. First, we present our approach on the well-known Kreweras model, which is algebraic, and show how to obtain a complete asymptotic expansion in this case. We then consider a more generic (so-called infinite group) model and find the associated complete asymptotic expansion. In this second case, we prove the existence of logarithmic terms in the asymptotic expansion, and we relate the coefficients appearing in the expansion to polyharmonic functions. To our knowledge, this is the first time that logarithmic terms have been observed in the asymptotics of a class of lattice walks confined to a quadrant.

A generalized curvilinear solver for spherical shell Rayleigh-Bénard convection

Summary A three-dimensional finite-difference solver has been developed and implemented for Boussinesq convection in a spherical shell. The solver transforms any complex curvilinear domain into an equivalent Cartesian domain using Jacobi transformation and solves the governing equations in the latter. This feature enables the solver to account for the effects of the non-spherical shape of the convective regions of planets and stars. Apart from parallelization using MPI, implicit treatment of the viscous terms using a pipeline alternating direction implicit scheme and HYPRE multigrid accelerator for pressure correction makes the solver efficient for high-fidelity direct numerical simulations. We have performed simulations of Rayleigh-Bénard convection at two Rayleigh numbers Ra = 105 and 107 while keeping the Prandtl number fixed at unity (Pr = 1). The average radial temperature profile and the Nusselt number match very well, both qualitatively and quantitatively, with the existing literature. Closure of the turbulent kinetic energy budget, apart from the relative magnitude of the grid spacing compared to the local Kolmogorov scales, ensures sufficient spatial resolution.

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.

A Hermite-type collocation mesh-free approach for simulating incompressible viscous fluid flows

In this work, a novel Hermite-type mesh-free model for incompressible viscous fluid flows is proposed, as a way, to overcome the numerical difficulties experienced by solving under weak or strong formulation the pure stream-function Navier–Stokes equation. In this model a dimensionless virtual domain is utilized, in order to make an accurate dimensionless shape-functions of the recently developed Hermite-type collocation Weighted Least Square approximation, by using dimensionless scale parameters. Then, a Jacobian transformation matrix is built to express the shape-functions spatial derivatives from the virtual to the considered domain. The resolution of the obtained nonlinear algebraic system is performed by the High Order Continuation Method. Numerical tests were investigated on incompressible viscous isothermal fluid flows in lid-driven cavity, backward-facing step, sudden expansions and around various obstacle shapes. They show the advantages of the proposed model compared to the reference Hermite-type collocation models used for pure stream-function equation and the weak Galerkin procedures as Finite Element Method and Element Free Galerkin method, in term of computation accuracy, number of degree of freedom, implementation simplicity and computation time. The numerical and experimental results of literature confirm the validity and capability of the proposed dimensionless model to simulate fluid flows in a expended-lengths inflow–outflow geometry and around obstacles of circular, inclined elliptical and Joukowski airfoil shapes.

Concept of Bilinear Transformation, Jacobian, and Conformal Mapping with Applications

This article explores the concepts of bilinear transformation, Jacobian transformation, and conformal mapping, focusing on their essential properties and presenting key results. The discussion revolves around isogonal transformation, conformal transformation, Jacobian transformation, and bilinear transformation, as well as critical points and fixed points.

Random vibration analysis of nonlinear structure with pounding tuned mass damper

To mitigate the undesired vibrations of the structure, a cost-effective approach is to set up the nonlinear dynamic vibration absorber (NLDVA) on the structure. As an alternative NLDVA, the Pounding Tuned Mass Damper (PTMD) gathered extensively attention in recent years. However, the random vibration analysis of structure coupled with PTMD still poses a challenge. Besides, there remains an urgent need to construct a more comprehensive recovery coefficient (RC) for the two-stage damping mechanism of PTMD. To fill in this gap, this work involves the nonlinear random vibration of a type of two degree-of-freedom system composing of a main structure (MS) coupled to a PTMD device, in which the collision process of the system is portrayed by a segmented recovery coefficient (RC) model. Specifically, an equivalent system of structure-PTMD is firstly constructed by coordinate transformation. Then, the approximate probability density function (PDF) of equivalent system is performed by combining non-smooth conversion and stochastic averaging method (SAM). Further, the approximate PDF solution of the structure-PTMD can be determined by resorting to Jacobi transformation and verified numerically. Remarkably, the approximate PDF solutions of the system can also be applied to estimate the power spectral density of the system response. Being analytical, the analysis in this work yields results in the closed-form expressions that enable one to acquire new insight into the dynamic features of the structure-PTMD system. Moreover, the proposed scheme will promote the design and application of PTMD in practical projects from a theoretical point of view.

Identifying the factors influencing the voltage quality of 6kV grids when using electric excavators in surface mining

Purpose. The research purpose is to study the relationship between the number of electric excavators and their impact on reducing the voltage losses. As a result of the research, it becomes possible to obtain factors that can help the manager to correctly understand the effect of power compensation caused by the over-excitation mode of electric excavators. Methods. The paper uses the Jacobian matrix transformation to simulate the power flows of electric excavators, the driving mechanisms of which are mainly synchronous motors. The input data for the simulation is the in-situ measurement data representing the inverse power flow. A diagram and a software to determine the factors corresponding to the number of electric excavators are also provided. Findings. A cross-reference table has been compiled showing the ratio of factors corresponding to the number of electric excavators in a 6kV grid. An appropriate software has also been developed, including a table for correcting typical equations for calculating voltage losses. Originality. The proposed factor is conditioned by over-excitation mode of excavators operating as compensation machines in a 6kV grid. Practical implications. When calculating the voltage loss in a 6kV grid of surface mines, if the design feeder contains electric excavators, a modified factor should be added to give a correct idea of the voltage quality.

Detection Tolerant Black-Box Adversarial Attack Against Automatic Modulation Classification With Deep Learning

Advances in adversarial attack and defense technologies will enhance the reliability of deep learning (DL) systems spirally. Most existing adversarial attack methods make overly ideal assumptions, which creates the illusion that the DL system can be attacked simply and has restricted the further improvement on DL systems. To perform practical adversarial attacks, a detection tolerant black-box adversarial-attack (DTBA) method against DL-based automatic modulation classification (AMC) is presented in this article. In the DTBA method, the local DL model as a substitution of the remote target DL model is trained first. The training dataset is generated by an attacker, labeled by the target model, and augmented by Jacobian transformation. Then, the conventional gradient attack method is utilized to generate adversarial attack examples toward the local DL model. Moreover, before launching attack to the target model, the local model estimates the misclassification probability of the perturbed examples in advance and deletes those invalid adversarial examples. Compared with related attack methods of different criteria on public datasets, the DTBA method can reduce the attack cost while increasing the rate of successful attack. Adversarial attack transferability of the proposed method on the target model has increased by more than 20%. The DTBA method will be suitable for launching flexible and effective black-box adversarial attacks against DL-based AMC systems.

Boundary element analysis of thin structures using a dual transformation method for weakly singular boundary integrals

This paper focuses on the weakly singular integrals in thin structures of 3D boundary element method. In the pioneering woks for the weakly singular integrals, the singularities are eliminated through introducing transformations whose Jacobian at the singular point is zero. When using these methods for weakly singular boundary integrals on elements with a large-angle and large side length ratio, near singularity will appear. In this paper, a dual transformation method is proposed to accurately calculate weakly singular integrals in which the near singularity is also considered. The numerical implementation is divided into several steps. Firstly, the (α, β) coordinate transformation method is used to eliminate the weak singularity by introducing the α factors in the transformation Jacobian. Then extract the integral form with near singularities about β. Finally, a corresponding distance transformation is constructed to eliminate its near singularity. The results show that the proposed method can accurately integrate for arbitrary element shape and source point position. The dual transformation method may be a good choice to evaluate weakly singular integral for thin structural problem.

Equation of state based analytical formulation for optimization of sCO2 Brayton cycle

Real gas thermodynamic cycles employ state-of-the-art property routines to compute the state properties as ideal gas relations are inaccurate. Multiparameter equation of state (EOS), though accurate, is computationally time consuming due to iterative solutions needed for complex cycles. Alternatively, look-up tables alleviate the computational time but compromise the accuracy due to interpolation and extrapolation schemes. This paper proposes a novel analytical formulation to overcome the tradeoff between accuracy and computation cost. Helmholtz energy-based EOS in combination with Jacobian transformations and real gas thermodynamics is used to arrive at a single equation linking the state points of a thermodynamic cycle. The efficacy of the formulation is demonstrated to deduce the optimum pressure ratio while maximizing specific work or efficiency of a simple recuperated supercritical CO2 Brayton cycle. The difference in optimum pressure ratio predicted by the analytical formulation and conventional property-routine is inconsequential compared to an order of gain in the computational time.

Joint simulation through orthogonal factors generated by the L-SHADE optimization method

Due to its better results over the traditional co-simulation methods, joint simulation of variables through orthogonal factors has gained popularity. In this approach, practitioners transform variables into orthogonal factors, simulate them independently, and back-transform the results into the initial space. It is unlikely to generate spatially uncorrelated factors at more than two lags. Therefore, the idea of simultaneous diagonalization has become the topic of so many studies. Some approaches have used the Jacobi transformation to replace a multivariate minimization problem with a sequence of univariate problems. However, these studies do not pay attention to the interruption of the previously minimized univariate problems while solving the new one. Therefore, this study aimed to minimize the objective function by considering all the univariate problems at once using the L-SHADE optimization method. The proposed method was applied to the Meiduk porphyry copper deposit to generate the L-SHADE factors. L-SHADE was efficient due to high speed of convergence and giving logical answer. Comparison among the L-SHADE factors and those of principal component analysis and Min/max autocorrelation factors showed better performance of the L-SHADE method such that the cross-variograms of the L-SHADE factors did not show noticeable spatial pattern and generally had smaller values. Sequential simulation was used to produce fifty equiprobable realizations from the L-SHADE factors. The proposed approach could reproduce the means, variances, correlation coefficients, cumulative distributions, and auto/cross-variograms of the variables in the simulations. Therefore, the simulations were reliable to be used in long-term production planning. A weighted multi-objective model using binary integer variables was developed to find the best possible production plan among different production scheduling alternatives developed based on the simulations. Different weighting scenarios were considered to investigate the impacts of economic (net present value), resources (ore extraction), and environmental (waste removal) objectives on stakeholder interests.

Direct dynamic-simulation approach to trajectory optimization

This paper proposes a new direct method for an efficient trajectory optimization using the point that the dynamics of a deterministic system are uniquely determined by initial states and controls imposed over the time horizon of interest. To effectively implement this concept, the Hermite spline is adopted to interpolate the continuous controls and the system dynamics are integrated with corresponding control parameters in prior. As a result, the optimal control problem can be transcribed into a nonlinear programming problem which has no dynamic equality constraints and no intermediate states in its design variables. In addition, the paper proposes an efficient recursive Jacobian estimation technique and introduces a Jacobian transformation matrix to straightforwardly handle the general state constraints. Important properties of the present method are thoroughly investigated through its applications to the trajectory optimization for a soft lunar landing from a parking orbit, including the detailed analyses for the de-orbiting phase. The computed results are compared with those using the pseudo-spectral method to demonstrate an extreme outperformance of the proposed method in the aerospace applications over the traditional direct method.

Conservational integrals of the fourth-order phase field model for brittle fracture via Noether theorem

As a regularization method for modeling fracture phenomenon, the phase field method cannot accurately capture the location of crack tip. J-integral is an important physical parameter to quantify the stress state of crack tip. In this study, the J-, M- and L-integrals of the fourth-order phase field model for brittle fracture are proposed based on the Noether theorem, and the corresponding infinitesimal generator of Lie group are given. The J- and L-integral models in the phase field fracture system are proved to be path-independent. Besides, the numerical implementation of the J-integral is conducted by using the domain integral method. In order to calculate the third derivative term of phase field in the J-integral model, a 9×9 Jacobian transformation matrix is constructed. Moreover, the J-integrals with and without effect of damage phase field are compared with analytic solution by numerical examples. The path-independence and Γ-convergence of the J-integral of phase field model are numerically verified. The proposed integral models can overcome the drawback of the fracture phase field method and provide the characteristic parameters of fracture mechanics to evaluate fracture behavior of brittle material.

Some Estimates to Best Approximation of Functions by Fourier-Jacobi Differential Operators

In this paper, an extension of the idea of the best approximation in the Hölder spaces with respect to Fourier-Jacobi operators by moduli of smoothness is studied. A special form of the moduli of smoothness is considered to get a strong convergence. Further, advanced approaches of approximation and some direct and inverse results are proved. Moreover, the Jackson-type estimate of functions in Hölder spaces by Jacobi transformations to algebraic polynomials with generalized de la Vallée Poussin mean are established.

New and Efficient Implementation of CC3.

We present a new and efficient implementation of the closed shell coupled cluster singles and doubles with perturbative triples method (CC3) in the electronic structure program eT. Asymptotically, a ground state calculation has an iterative cost of 4nV4nO3 floating point operations (FLOP), where nV and nO are the number of virtual and occupied orbitals, respectively. The Jacobian and transpose Jacobian transformations, required to iteratively solve for excitation energies and transition moments, both require 8nV4nO3 FLOP. We have also implemented equation of motion (EOM) transition moments for CC3. The EOM transition densities require recalculation of triples amplitudes, as nV3nO3 tensors are not stored in memory. This results in a noniterative computational cost of 10nV4nO3 FLOP for the ground state density and 26nV4nO3 FLOP per state for the transition densities. The code is compared to the CC3 implementations in CFOUR, DALTON, and PSI4. We demonstrate the capabilities of our implementation by calculating valence and core excited states of l-proline.

Spectral wave modeling of tsunami waves in Pohang New Harbor (South Korea) and Paradip Port (India)

A coupled numerical model is developed based on the spectral element method (SEM) and boundary element method (BEM) to predict the characteristics of tsunami wave response on Pohang New Harbor (PNH), South Korea, and Paradip Port, India. The current numerical model is developed to analyze the impacts on the coastal region with slowly varying bathymetry under the resonance conditions. In this method, the boundary integral equation of each boundary segment is discretized into the spectral elements using the Chebyshev’s polynomial, and the corresponding boundary integrals are transformed using the Jacobian of transformation. This leads to a highly accurate depiction of the boundary edges or corners of the complex domain. Convergence and error analyses are conducted on the rectangular port using the BEM and the present method, which shows that the later approach enhances the overall numerical accuracy of the model. The simulation results for normal day wave (ordinary wave propagating in the ocean under normal conditions) and tsunami waves are validated with previous studies and measurement data at Pohang New Harbor (PNH), South Korea. In addition, the calculated spectral density for tsunami waves at PNH is also compared with the Hokkaido tsunami (reported on July 12, 1993) data at tide gauge record station in Pohang. The amplification factor is determined for normal day wave and tsunami waves at key locations inside the PNH and Paradip Port for different directional incident waves. Further, the spectral density is also estimated at the same locations with respect to the wave period to understand the consequences of tsunami waves on the coastal ports. Therefore, the coupled numerical model is an efficient tool to predict tsunami wave impact on a realistic port. The causes and countermeasure are suggested to reduce the risk of normal day waves and tsunami waves.

Efficient modelling of beam-like structures with general non-prismatic, curved geometry

The analysis of three-dimensional (3D) stress states can be complex and computationally expensive, especially when large deflections cause a nonlinear structural response. Slender structures are conventionally modelled as one-dimensional beams but even these rather simpler analyses can become complicated, e.g. for variable cross-sections and planforms (i.e. non-prismatic curved beams). In this paper, we present an alternative procedure based on the recently developed Unified Formulation in which the kinematic description of a beam builds upon two shape functions, one for the beam’s axis, the other for its cross-section. This approach predicts 3D displacement and stress fields accurately and is computationally efficient in comparison with 3D finite elements. However, current modelling capabilities are limited to the use of prismatic elements. As a means for further applicability, we propose a method to create beam elements with variable planform and variable cross-section, i.e. of general shape. This method employs an additional set of shape functions which describes the geometry of the structure exactly. These functions are different from those used for describing the kinematics and provide local curvilinear basis vectors upon which 3D Jacobian transformation matrices are produced to define non-prismatic elements. The model proposed is benchmarked against 3D finite element analyses, as well as analytical and experimental results available in the literature. Significant computational efficiency gains over 3D finite elements are observed for similar levels of accuracy, for both linear and geometrically nonlinear analyses.

Accelerated multimodel Newton-type algorithms for faster convergence of ground and excited state coupled cluster equations.

We introduce a multimodel approach to solve coupled cluster equations, employing a quasi-Newton algorithm for the ground state and an Olsen algorithm for the excited states. In these algorithms, both of which can be viewed as Newton algorithms, the Jacobian matrix of a lower level coupled cluster model is used in Newton equations associated with the target model. Improvements in convergence then imply savings for sufficiently large molecular systems, since the computational cost of macroiterations scales more steeply with system size than the cost of microiterations. The multimodel approach is suitable when there is a lower level Jacobian matrix that is much more accurate than the zeroth order approximation. Applying the approach to the CC3 equations, using the CCSD approximation of the Jacobian, we show that the time spent to determine the ground and valence excited states can be significantly reduced. We also find improved convergence for core excited states, indicating that similar savings will be obtained with an explicit implementation of the core-valence separated CCSD Jacobian transformation.

Forward and inverse problems of Type A uncertainty evaluation

• The Type A uncertainty analysis can be divided into forward and inverse problems. • Forward problem gives estimator’s PDFs when population parameters are known. • Inverse problem gives posteriori PDFs when estimates are determined by experiment. • A posteriori PDFs are result of the proper transformation of estimators’ PDFs. • Jacobian of transformation is the ratio of estimate to population scale parameter. In the article method of Type A uncertainty evaluation of the location and scale parameters of the population based on solution of the forward and inverse problems is proposed. The essence of forward problem is determination of the distributions of estimates m of location and s of scale parameters. The inverse problem is solved when values of the estimates are determined, and its essence is determination of the a posteriori distributions of location and scale parameters and also their standard and expanded uncertainties. A general method of the determining the distributions in inverse problem bases on the transformation of distributions obtained in the forward problem by the Jacobian which is the ratio of estimate ( s ) to the scale parameter ( σ ). Independently on population distribution the standard uncertainties of the location and scale parameters are depend on value of 1/sqrt( n − 3), where n is a number of observations.