Order Approximation Research Articles

Accurate Electromagnetic Force Analysis of Offshore Wind Power Transformer Windings Based on Fractional Order Lumped Parameter Model

Accurate calculation of the electromagnetic force distribution of transformer windings under different loads and fault conditions is of great significance for transformer maintenance, condition evaluation and life prediction. Due to the influence of offshore wind power systems, offshore wind power transformers have high harmonic content and large changes in load rates, which can easily cause the coil destabilization, winding deformation or even damage because of the uneven distribution of the electromagnetic force. To improve the accuracy of electromagnetic force calculation, this paper proposes a fractional order numerical method. First, a three-dimensional axisymmetric transformer model and a symmetrical lumped parameter equivalent circuit model are established, respectively, based on field-circuit coupling. Second, the fractional order approximation of circuit components is realized by using the improved Oustaloup filter. In the fractional order model, the transformer is replaced by the lumped parameter equivalent circuit model. Third, as in the calculation process for integer order electromagnetic force, the integer order current has a large error, and the current waveform does not match the actual power frequency. The fractional order current and electromagnetic force at the 0.9 order are closer to the rated value. Finally, the effects of different load rates, three-phase short circuits and harmonic conditions are studied with the fractional order model. Compared with the traditional integer order finite element electromagnetic model, the fractional order equivalent circuit model established in this paper is more accurate and suitable for electromagnetic force calculation. The proposed method is significant for the structural design and state detection of transformers and also could be applied in the analysis of other dry-type transformers.

On the asymptotic stability of N-soliton solution for the short pulse equation with weighted Sobolev initial data

In this work, we are devoted to studying the Cauchy problem of the short pulse (SP) equation with weighted Sobolev initial data. By developing the ∂¯-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution q(x,t) in solitonic region of space-time, (yt)=ξ for any fixed ξ=ξ0∈(0,+∞), and give bounds for the error decaying as |t|→∞. Based on the resulting asymptotic behavior, the asymptotic approximation of the SP equation is characterized with the soliton term confirmed by N(I)-soliton on discrete spectrum with residual error up to O(t−1). Combining with previous results presented by Yang and Fan, that is the long time asymptotic expansion of the solution q(x,t) in solitonic region ξ=yt∈(−∞,0), our results show that the soliton resolution conjecture of the SP equation is fully solved in all space-time solitonic regions. Moreover, considering properties of the scattering map of SP equation, we derive an asymptotic stability result for a class of initial values.

Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves

Abstract We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Padé approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e., away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.

A sediment transport theory based on distortion-free-boundary nonhomogeneous fluid flows

Over the last 50 years several sediment transport models in coastal environments based on Shallow Water(SW) type models have been developed in the literature. The water flow over an abrupt moving topography quickly spatially variable becomes accelerated and strongly varied arising the turbulence (distortion). The acceleration and strong variation of the flow facilitate the transport of a large quantity of sediments present at the bottom while modifying it. The mathematical models based on SW type models widely used to describe the sediment transport phenomena do not account the distortion effects. Indeed, it is well-known that the SW models are derived from first order approximation of long wave theory. The acceleration and strong variation of the water flow near the bottom is due to the distortion of the horizontal velocity profile along the vertical direction. One can regard distortion as a combination of strain and rotation. The effect of the rotational component is to weaken the effect of the strain somewhat. In this work, we put in place a king theory of sediment transport derived from the second order approximation of long wave theory that can describe sediment transport processes in distortion-free-boundary nonhomogeneous fluid flows. The derived model accounts the distortion (fluctuation with great correlation lengths) that creates the turbulence. Moreover, the model differentiates the fluid velocity from sediment velocity (phase-lag) near the sediment bed. The proposed theory significantly reduces the modeling errors observed in several sediment transport models based on nonhomogeneous shallow water equations and has a great potential to increase the predictive power of sediment transport models in rivers, lakes, coastal flows, ocean basins and so on. The proposed theory improves several existing sediment transport theories recently developed in the literature and can be apply with some degree confidence.

Gravity inversion by second order approximation applied to the Styrian Basin (Austria)

AbstractAn approximative second order gravity inversion scheme by a truncated power series expansion is applied to derive the thickness of the Neogene, mostly clastic, sedimentary section of the Styrian Basins in South–East Austria, which are sub‐basins of the Pannonian Basin System. These sub‐basins with a derived thickness of up to 4 km are of interest for geothermal exploitation because of the increased geothermal gradient and heat flow observed in the Pannonian Basin in general and a geothermal gradient of 4°–5°/100 m measured in some wells in the Styrian Basin. The Styrian Basin also has been an area for hydrocarbon exploration in the past 50 years, with oil and gas show encountered in several exploration wells and one sub‐commercial gas discovery. The Miocene and Plio–Pleistocene volcanism in the Styrian Basin caused by Miocene crustal thinning is discussed in terms of the influence to the gravity inversion taking the aeromagnetic field into account. The volcanism is of relevance for the geothermal prospectivity but poses problems for the single layer‐based gravity inversion scheme. Results are discussed from a computational side comparing observed and calculated gravity fields but also the match with well data is discussed. In terms of gravity inversion methodology, the presented can be viewed as an approximative fast‐track approach.

Pairing EU directives and their national implementing measures: A dataset for semantic search

European Directives (EUDs) are binding upon Member States as to the results to be achieved, but leave to national authorities the choice of form and methods. Therefore, Member States adopt ad hoc National Implementing Measures (NIMs) that mostly reproduce the contents of EUDs and transpose them into domestic legislation. This well-known process is defined as “legal harmonization” and consists of the gradual, although ambiguous, approximation of national legal orders as a result of the adoption of European legislation. In order to contribute to the analysis of this phenomenon, we collect a large and unique dataset composed of European and domestic legislative sources, which is an essential requirement to automatically pair EUDs and the corresponding NIMs, in light of their semantic similarity. The first results show the feasibility of the proposed task to discern NIMs from national legislation that does not contribute to implementing EUDs, thus constituting the foundation for a semantic search engine. We believe that our effort can promote future applications and research directions, with the ultimate aim to support traditional legal methodology, facilitate citizens’ access to rights, support public administrations, and, more in general, promote democracy and the rule of law in the European Union. Data and source code are available at https://doi.org/10.17632/mkx5sb3mnw.

Simulations in Science and Engineering for University Education

In this paper, the strategies and tests that guarantee a reliable solution of partial differential equations (PDE) solved with the finite difference method are proposed. The elements to be considered are presented so that the solution of equations in partial derivatives is the closest to reality. The experience of the proponents and previous work play an important role in validating simulations in science and engineering problems. A series of tests are proposed that must be carried out to validate and make the simulation results reliable, due to the fact that the selection of the finite difference scheme, the initial data, the boundary conditions and the restrictions in the time step, the stability and convergence of the solution cannot be guaranteed a priori. There are situations in which the solution is stable but converges to values ??that have no physical meaning. To illustrate the strategies and tests inherent in a simulation, such that the solutions can be trusted, the 3D wave equation (three spatial dimensions and one temporal dimension) is solved numerically, by implementing a code in FORTRAN 90, implementing a scheme finite difference to second order approximation. The stability and convergence of the solution are studied according to the schemes proposed in this research, in such a way that the results are reliable, and guarantee the possible practical implementation of the solutions. Finally, the evolution of the scalar field for different times is shown, validated with the principle of conservation of energy. The strategies implemented in this work to guarantee the reliability of the simulations are also valid when solving EDP problems with neural networks, genetic algorithms and other intelligent computing techniques.

Study of performance of low-rank nonnegative tensor factorization methods

Abstract In the present paper we compare two different iterative approaches to constructing nonnegative tensor train and Tucker decompositions. The first approach is based on idea of alternating projections and randomized sketching for factorization of tensors with nonnegative elements. This approach can be useful for both TT and Tucker formats. The second approach consists of two stages. At the first stage we find the unconstrained tensor train decomposition for the target array. At the second stage we use this initial approximation in order to fix it within moderate number of operations and obtain the factorization with nonnegative factors either in tensor train or Tucker model. We study the performance of these methods for both synthetic data and hyper-spectral image and demonstrate the clear advantage of the latter technique in terms of computational time and wider range of possible applications.

On the efficiency of modified regression-type mean imputation scheme under two-phase sampling

Human-based surveys such as medical and social science surveys are often characterized by non-response or missing observations. In this study, a new class of regression-type mean imputation method that uses X̄n as an estimate of X̄ was suggested. Using partial derivative approach, the MSEs of the class of estimators presented were derived up to first order approximation under two cases. Case I: when the secondary sample S2 of size n(n<n1) is a subset of preliminary sample S1[S2 ⸦ S1] , and Case II: is when secondary sample S2 is a subset of universal set ΩN . Conditions for which the new estimator was more efficient than the other estimators studied were derived. The results of numerical examples through simulations revealed that the suggested class of estimators is more efficient.

Improved inference for a boundary parameter

AbstractThe limiting distributions of statistics used to test hypotheses about parameters on the boundary of their domains may provide very poor approximations to the finite‐sample behaviour of these statistics, even for very large samples. We review theoretical work on this problem, describe hard and soft boundaries and iceberg estimators, and give examples highlighting how the limiting results greatly underestimate the probability that the parameter lies on its boundary even in very large samples. We propose and evaluate some simple remedies for this difficulty based on normal approximation for the profile score function, and then outline how higher order approximations yield excellent results in a range of hard and soft boundary examples. We use the approach to develop an accurate test for the need for a spline component in a linear mixed model.

Higher order Morita approximation and its validity for random copolymer adsorption onto homogeneous and periodic heterogeneous surfaces

Adsorption of a single AB random copolymer (RC) chain onto homogeneous and inhomogeneous ab surfaces with a regular periodic pattern is studied theoretically. For the averaging over disorder in the RC sequence, the constrained annealed approximation, known as the Morita approximation is employed. A general scheme for constructing the Morita approximation of an arbitrary order m is proposed; it is based on representation of the RC monomer sequence as a Markov chain of overlapping –1)-ads. The problem is solved within the framework of the generating functions approach for the two-dimensional partially directed walk model of the polymer on a square lattice. Temperature dependences of various adsorption characteristics are obtained. The accuracy of the Morita approximation is assessed by comparison with the numerical results for RC with quenched sequences obtained by the transfer matrix approach. It is shown that for RC adsorption on a homogeneous surface for AB-copolymers with adsorbing A and neutral B blocks, it is sufficient to use the Morita approximation of second or third order. If the non-adsorbing B block is repelled from the surface, then a higher order of the approximation (5th–6th) is required. An important indicator of validity of the Morita approximation is the entropy: if the order of the approximation is not high enough, the entropy becomes negative in the low-temperature regime which means that the Morita approximation is wrong in this order. In the case when the surface is periodically inhomogeneous, the Morita approximation works worse and very high orders are required, which can be beyond the computational capabilities. Moreover, the Morita approximations become degenerate: approximations of two or more consecutive orders give the same result.

High order approximations of solutions to initial value problems for linear fractional integro-differential equations

High order approximations of solutions to initial value problems for linear fractional integro-differential equations

Heat Transfer in a Silo Containing Rice: Solution by the Explicit Method of the Finite Differences with Approximations of Order 2 and 4 for a One-Dimensional Transient Model

Rice is one of the most consumed cereals in the world, and the cultivation of this crop has significant relevance in the southern region of Brazil. When subjected to inadequate conditions of temperature and humidity, rice becomes susceptible to attacks from pests and fungi and, therefore, care in the storage process is of paramount importance, since this is largely responsible for the quality of the harvest. Such care allows the food to arrive without harm to the consumer. In this sense, the mathematical modeling, among numerous possibilities, allows for evaluating the internal temperature of a silo and, through this, taking preventive measures so that the grains maintain their quality. The objective of this work is to model the heat transfer process in a silo prototype containing rice in husk through the explicit finite difference method for a one-dimensional and transient model considering two approaches centered on the spatial derivative: error of order 2 and 4. In addition, the thermal diffusivity of the grain with average value was analyzed. The results obtained by the solutions were analyzed through graphs and statistical indexes comparing with the experimental data of the literature, and the computer simulation was performed through the Google Colab platform. The chosen methodology proved effective for the work, and the predicted temperatures for the approximations of order 2 and 4 denote similarities both graphically and in the precision of the statistical indexes.

The validity of the [formula omitted] lattice calculations for the finite asymmetric [formula omitted] matter using the central potentials

The lattice version of the lowest order constrained variational (LOCV) approach at the three-body cluster approximation is applied to the finite asymmetric nucleonic matter (ASM) in the periodic boxes (PBLOCV). The PBLOCVASM three- to two-body cluster energy ratios (R), using the central Bethe homework and (AV′8)c potentials, are obtained for small cubic PBs. The smallness of the ratios R could be considered evidence for the validity of the lowest order approximation. It is demonstrated that the truncation of the energy cluster expansion at the lowest order is confirmed for the Bethe homework interaction case. Also, the finite size effects and their impact on the symmetry energy coefficient are investigated. It is observed that using simple potentials and small finite nucleonic systems, one cannot reproduce the experimental prediction of the symmetry energy coefficient, even by including the three-body cluster energy contribution.

Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps

We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.

A Second-Order Difference Scheme for Solving a Class of Fractional Differential Equations

Introduction. Increasing accuracy in the approximation of fractional integrals, as is known, is one of the urgent tasks of computational mathematics. The purpose of this study is to create and apply a second-order difference analog to approximate the fractional Riemann-Liouville integral. Its application is investigated in solving some classes of fractional differential equations. The difference analog is designed to approximate the fractional integral with high accuracy.Materials and Methods. The paper considers a second-order difference analogue for approximating the fractional Riemann-Liouville integral, as well as a class of fractional differential equations, which contains a fractional Caputo derivative in time of the order belonging to the interval (1, 2).Results. To solve the above equations, the original fractional differential equations have been transformed into a new model that includes the Riemann-Liouville fractional integral. This transformation makes it possible to solve problems efficiently using appropriate numerical methods. Then the proposed difference analogue of the second order approximation is applied to solve the transformed model problem.Discussion and Conclusions. The stability of the proposed difference scheme is proved. An a priori estimate is obtained for the problem under consideration, which establishes the uniqueness and continuous dependence of the solution on the input data. To evaluate the accuracy of the scheme and verify the experimental order of convergence, calculations for the test problem were carried out.

New second order approximation method for posterior precision evaluation of weighted total least squares estimates with analytical expression

Several first order approximate methods to evaluate the posterior precision of weighted total least squares estimates have been developed but there is still some space for the methods of precision estimation. Therefore, a new second order approximate estimation method (SOA) is proposed which can consider the biases of the estimated parameters. The formulae of the new SOA are derived from the partial differential equation of objective function and the definition of variance covariance. And then, with the help of the second order Taylor expansion, the bias from the nonlinearity of EIV can be modified to improve the precision representation ability. Two simulated experiments are designed to verify the proposed SOA, and numerical results show that the proposed SOA is better and closer to the reference value than four selected competitors.

Mathematical modeling of magnetized nanofluid flow over extending and contracting pipe with heat source and chemical reaction: Error estimation and stability analysis.

The aim of the current study is to examine the magnetohydrodynamic (MHD) flow over the permeable pipe containing the nanoparticles with a heat transport mechanism. The leading equations of flow are obtained in terms of partial differential equations (PDEs). The suitable transformation is applied to alter PDEs to ordinary differential equations (ODEs) and solved numerically via the RK4 method. The novelty of the current work is to investigate the MHD nanofluid overextending and shrinking pipe enclosing the chemical reaction with a heat reservoir. The outcomes of different factors are depicted through graphs and tables on the flow phenomena. It is observed that for both situations of wall extension or contraction with injection, the temperature is the increasing function of the thermophoresis and Brownian motion factors. It is explored that the heat is upraised when the heat generation is augmented but decays for heat absorption. The point to be noted here is that the Hartmann number and Prandtl number enhance the heat curves in the presence of a heat source. It is also observed that when the thermophoresis factor is increased the nanoparticles concentration is also enhanced via heat source/sink. The error estimation is computed for different order approximations. For the confirmation of the mathematical modeling, the current study is validated with the previous.

Stochastic viscosity approximations of Hamilton–Jacobi equations and variance reduction

We consider the computation of free energy-like quantities for diffusions when resorting to Monte Carlo simulation is necessary, for instance in high dimension. Such stochastic computations typically suffer from high variance, in particular in a low noise regime, because the expectation is dominated by rare trajectories for which the observable reaches large values. Although importance sampling, or tilting of trajectories, is now a standard technique for reducing the variance of such estimators, quantitative criteria for proving that a given control reduces variance are scarce, and often do not apply to practical situations. The goal of this work is to provide a quantitative criterion for assessing whether a given bias reduces variance, and at which scale. We rely for this on a recently introduced notion of stochastic solution for Hamilton–Jacobi–Bellman (HJB) equations. Based on this tool, we introduce the notion of k-stochastic viscosity approximation (SVA) of a HJB equation. We next prove that such approximate solutions are associated with estimators having a relative variance of order k − 1 at log-scale. In particular, a sampling scheme built from a 1-SVA has bounded variance as noise goes to zero. Finally, in order to show that our definition is relevant, we provide examples of stochastic viscosity approximations of order one and two, with a numerical illustration confirming our theoretical findings.

Application of Splines of the Seventh Order Approximation to the Solution of the Fredholm Integral Equations with Weekly Singularity

We consider the construction of a numerical solution to the Fredholm integral equation of the second kind with weekly singularity using polynomial spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. In the beginning, in the middle, and at the end of the integration interval, we apply various modifications of the basis splines of the seventh order of approximation. We use the Gaussian-type quadrature formulas to calculate the integrals with a weakly singularity. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order to solve integral equations are given.