Equations Of Form Research Articles

Modeling and investigation of crack evaluation in metal tube based on magnetic field analysis method

In this paper, an analytical model of eddy current testing of a metal tube has been proposed to evaluate circumferential cracks from the signals of the magnetic field. First, based on Maxwell’s equations in time-harmonic form, magnetic vector potential equations for each region of space are built under two different configurations of the bobbin probe and the encircling probe respectively. Next, the magnetic vector potentials are solved by imposing conditions at the interface. Then, series expressions of the radical and axial magnetic induction intensities have been obtained. In addition, based on the proposed modeling, the influences of cracks at different locations on the surface magnetic field have been analyzed. The results have shown that the radial magnetic induction intensity increases while the axial magnetic induction intensity decreases for the internal and external cracks with the same shape and size on the surface of the metal tube. Moreover, the magnetic induction intensity has shown much more sensitive property to external cracks than it has done to internal cracks for the encircling probe. However, for the bobbin probe, the magnetic induction intensity has demonstrated almost the same sensitivity in detection for both internal and external cracks in the metal tube. The researching results have important theoretical and practical significance for understanding the physical process of eddy current testing, optimizing parameters of the experimental system and building the inversion model.

A comparative analysis of fractional model of second grade fluid subject to exponential heating: application of novel hybrid fractional derivative operator

In this article, a new approach to study the fractionalized second grade fluid flow is described by the different fractional derivative operators near an exponentially accelerated vertical plate together with exponentially variable velocity, energy and mass diffusion through a porous media is critically examined. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimentional form. For the sake of better rheology of second grade fluid, developed a fractional model by applying the new definition of Constant Proportional-Caputo hybrid derivative (CPC), Atangana Baleanu in Caputo sense (ABC) and Caputo Fabrizio (CF) fractional derivative operators that describe the generalized memory effects. For seeking exact solutions in terms of Mittag-Leffler and G-functions for velocity, temperature and concentration equations, Laplace integral transformation technique is applied. For physical significance of various system parameters on fluid velocity, concentration and temperature distributions are demonstrated through various graphs by using graphical software. Furthermore, for being validated the acquired solutions, accomplished a comparative analysis with some published work. It is also analyzed that for exponential heating and non-uniform velocity conditions, the CPC fractional operator is the finest fractional model to describe the memory effect of velocity, energy and concentration profile. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, in the literature, it is observed that to derived analytical results from fractional fluid models developed by the various fractional operators, is difficult and this article contributing to answer the open problem of obtaining analytical solutions the fractionalized fluid models.

Anomaly cancellation in string theory

The anomaly cancellation in superstring theory is known to hold at leading order in the curvature for the gauge groups [Formula: see text] and [Formula: see text]. The coefficients of the next-to-leading order terms may be evaluated, and a mechanism for cancellation is described, which would remain valid at higher genus when there exists a global splitting of the coordinates of supermoduli space. Since the spin-[Formula: see text] fields transform under the adjoint representation in these models, compactification of the heterotic string over [Formula: see text] or a fundamental 12-dimensional theory, from which superstrings are produced through an elliptic fibration, over [Formula: see text], provides another phenomenologically viable theory at lower energies compatible with the standard description of the elementary particle interactions. The 96 spin-[Formula: see text] fields now would transform under the fundamental representation of [Formula: see text] and the spin-one gauge fields would belong to the adjoint representation. The sum of the anomaly polynomials for the particle content of the [Formula: see text] model vanishes at [Formula: see text]. The contributions to the gravitational anomaly from the particles and antiparticles cancel by the CPT theorem and the duality transformations of polynomials of degree 6 in the curvature and the field strength. The existence of the interaction of a spin-2 charge, which is conserved only over a finite time interval, can be traced to nonlocal terms in the reduction of the string field theory to the gravitational sector. The source of the global gravitational anomaly cancellation in the modular form equations derived from an elliptic fibration of a 12-dimensional theory would restrict the compactifications and provide a method for preserving the absence of anomalies in four dimensions.

High Density Ratio Multi-Fluid Simulation with Peridynamics

Multiple fluid simulation has raised wide research interest in recent years. Despite the impressive successes of current works, simulation of scenes containing mixing or unmixing of high-density-ratio phases using particle-based discretizations still remains a challenging task. In this paper, we propose a peridynamic mixture-model theory that stably handles high-density-ratio multi-fluid simulations. With assistance of novel scalar-valued volume flow states, a particle based discretization scheme is proposed to calculate all the terms in the multi-phase Navier-Stokes equations in an integral form, We also design a novel mass updating strategy for enhancing phase mass conservation and reducing particle volume variations under high density ratio settings in multi-fluid simulations. As a result, we achieve significantly stabler simulations in mixture-model multi-fluid simulations involving mixing and unmixing of high density ratio phases. Various experiments and comparisons demonstrate the effectiveness of our approach.

Mathematical Model of a Robot-spider for Group Control Synthesis: Derivation and Validation

A six-legged spider robot is a complex object from the point of view of the problem of synthesizing a system for controlling its movement. To synthesize an advanced control system for such a robot, which must solve non-trivial problems of overcoming obstacles, functioning under conditions of external disturbances, etc., we first solve the problem of synthesizing an information model of the object, on the basis of which its control system will subsequently be built.The paper compares two methods for synthesizing the information model of a six-legged spider-robot. In the first method, an information model is automatically synthesized from a CAD model of a spider-robot in a MATLAB-based graphical programming environment Simulink. In the second method, the information model is synthesized in the environment of dynamic modeling of technical systems SimInTech on the basis of a system of differential equations in the Cauchy form. Control loops and external influences are added to the information models synthesized in each of the modeling environments. The study showed that each of the resulting models has both its own individual advantages and disadvantages. They are mainly related to taking into account the mutual influence of various blocks of models on each other. It is shown that, in the end, the two models complement each other and make it possible to obtain an advanced basis for further synthesis of the motion control system.The results obtained in this work make it possible to use information models as a basis for the development of a control system for a physical model of a six-legged spider-robot, printed on a 3D printer and assembled on the basis of the Arduino hardware platform.

A new equivalent acceleration-based non-uniform response spectrum method for efficient seismic SSI analysis

A complex multiple-support response spectrum (CMSRS) method with a concise and universal expression is required to estimate the seismic responses of soil–structure interaction (SSI) systems under non-uniform seismic excitations to replace existing complex methods. This paper introduces and defines a new concept called equivalent acceleration of boundary load (EABL) that converts widely used equivalent loads into equivalent accelerations, and further develops an efficient CMSRS-EABL method for the seismic analysis of the SSI system. First, EABL offers three main benefits: it simplifies the expressions without sacrificing precision, extends the applicable scope of viscous-spring artificial boundary (VSAB) to all stress-type artificial boundaries, and presents dynamic equations in a familiar form. Subsequently, the CMSRS–EABL formula was successfully deduced based on the equivalent simplified dynamic equations. This formula significantly simplifies the traditional method by reducing the effective modal participation factors, mean-square-roots, and cross-correlation coefficients by 4, 5 and 15 items, respectively. Finally, a rectangular construction surrounded by a VSAB–soil combination was used as an example to illustrate the convenience of the developed CMSRS–EABL method and verify its precision. The results indicate that the proposed method is characterized by greater simplicity and ingenuity, and it is capable of high precision similar to that of the conventional methods. We believe this method will be of significant benefit in evaluating the anti-seismic performance of deeply buried and long-standing structures.

A mathematical framework for evo-devo dynamics

Natural selection acts on phenotypes constructed over development, which raises the question of how development affects evolution. Classic evolutionary theory indicates that development affects evolution by modulating the genetic covariation upon which selection acts, thus affecting genetic constraints. However, whether genetic constraints are relative, thus diverting adaptation from the direction of steepest fitness ascent, or absolute, thus blocking adaptation in certain directions, remains uncertain. This limits understanding of long-term evolution of developmentally constructed phenotypes. Here we formulate a general, tractable mathematical framework that integrates age progression, explicit development (i.e., the construction of the phenotype across life subject to developmental constraints), and evolutionary dynamics, thus describing the evolutionary and developmental (evo-devo) dynamics. The framework yields simple equations that can be arranged in a layered structure that we call the evo-devo process, whereby five core elementary components generate all equations including those mechanistically describing genetic covariation and the evo-devo dynamics. The framework recovers evolutionary dynamic equations in gradient form and describes the evolution of genetic covariation from the evolution of genotype, phenotype, environment, and mutational covariation. This shows that genotypic and phenotypic evolution must be followed simultaneously to yield a dynamically sufficient description of long-term phenotypic evolution in gradient form, such that evolution described as the climbing of a fitness landscape occurs in “geno-phenotype” space. Genetic constraints in geno-phenotype space are necessarily absolute because the phenotype is related to the genotype by development. Thus, the long-term evolutionary dynamics of developed phenotypes is strongly non-standard: (1) evolutionary equilibria are either absent or infinite in number and depend on genetic covariation and hence on development; (2) developmental constraints determine the admissible evolutionary path and hence which evolutionary equilibria are admissible; and (3) evolutionary outcomes occur at admissible evolutionary equilibria, which do not generally occur at fitness landscape peaks in geno-phenotype space, but at peaks in the admissible evolutionary path where “total genotypic selection” vanishes if exogenous plastic response vanishes and mutational variation exists in all directions of genotype space. Hence, selection and development jointly define the evolutionary outcomes if absolute mutational constraints and exogenous plastic response are absent, rather than the outcomes being defined only by selection. Moreover, our framework provides formulas for the sensitivities of a recurrence and an alternative method to dynamic optimization (i.e., dynamic programming or optimal control) to identify evolutionary outcomes in models with developmentally dynamic traits. These results show that development has major evolutionary effects.

Pollutant dispersion model in landfills by GILTT technique

The expansion of urban centers leads to an increase in the generation of municipal solid waste (MSW). In this way, due to the exorbitant amount of MSW produced annually, studies on the form of disposal of this waste are necessary, so that they are properly disposed of, due to environmental safety and reduction of risks to human health. Therefore, mathematical modeling is an essential tool in the study about the dispersion of pollutants in the porous medium, in this case, in relation to how the water table can be reached by these pollutants. Thus, this work intends to carry out a study of a two-dimensional model of the dispersion of pollutants in sanitary landfills, considering the governing equation in the dimensionless form. In the problem under study, a continuous and uniform leakage from a MSW storage cell is considered. The adopted model is solved by the Generalized Integral Laplace Transform Technique (GILTT) method, whose solution is analytical, except for the truncation error in the infinite series. The results obtained show that the proposed method is effective with the physical characteristics of the problem.

Evaluation of Infiltration Modeling in the Cisadane Watershed in Indonesia: Existing and New Approach Equation

This study aims to evaluate the accuracy of semi-empirical models for determining infiltration based on the existing equations (Horton, Kostiakov and Kostiakov–Lewis) and a new form equation with optimized parameters which has good accuracy (lowest RMSE and MBE), sufficient linearity (highest R2), and a Willmott index close to 1. The location of this study was the Cisadane watershed, which is a priority watershed in Indonesia. The lowest values of average RMSE and MBE and highest values of R2 and Willmott index are given by Equation 1, which indicated that the model was closest to the field infiltration conditions. The distribution of the RMSE values for all survey points using Equation 1 resulted in small RMSE values ranging from 0.09 to 1.83. The distribution of R2 values for all survey points using Equation 1 yielded R2 values close to 1 (high linearity) with values ranging from 0.96 to 1. Based on these results, it can be concluded that Equation 1 is the most accurate infiltration model for the Cisadane watershed compared to the existing infiltration rate equation. This research can be used as an initial idea to create the infiltration rate model using other forms of equations with optimization parameters to produce a more representative model with field data.

Stability Analysis of Autonomous Microgrid Operated at Constant-Frequency with the Consideration of Reactive Power Balance

Large-scale autonomous microgrids have potential application values as they can increase renewable energy penetration level without compromising the stability of the existing large power systems. Before their widespread implementation, critical issues like stability analysis etc need to be solved. This paper analyses the stability in an autonomous microgrid operated at constant frequency with the consideration of reactive power balance. Difference equations of reactive power for the grid-forming generator are constructed separately from those for the grid-supporting and grid-feeding generators while the difference equations of the real power for all the generators are the same. For the voltage source inverter with its current controlled by proportional resonant controller, at the fundamental frequency, its output current is disentangled from its terminal voltage and is controlled to trace its reference accurately, namely . Therefore, each inverter can be modelled as an equivalent current source and the equivalent circuits for d-component and q-component can be separated from each other. Then, the nodal equations in matrix form for the microgrid system can be established readily. With these, the system level state-space equations are built to study the distribution of eigenvalues. By choosing proper coefficients for real power and reactive power reference generations and controller’s parameters, one can make all the eigenvalues falling in the left-hand-side of the complex plane. Therefore, the system is stable. Such a research paves the way for systematically searching good sets of coefficients and controller’s parameters which make system operate safely away from unstable region with necessary margin.

State-Spatial Viewpoint of Stress in Thermally Postbuckled and Nonlinearly Bent FG Saturated Poroelastic Circular Plates

A dimensionless discrete state-space mathematical model is proposed for stress analysis of an axially-symmetric saturated poroelastic circular plate being loaded by temperature rise of one of the flat faces. Computation of stress by means of this model is free from a need for differentiation. The curves of thermal postbuckling, thermal nonlinear bending, radial stress, circumferential stress and resultant radial force have been achieved by means of six state-space variables and differential quadrature procedure. The effects of loading temperature, boundary conditions, porosity, pore distribution and fluid pore pressure on the thermal post buckling and nonlinear bending phenomena have been investigated. The critical postbuckling thermal load has been determined by means of a numerical algorithm that does not converge to trivial solution because it starts from a relatively great numerical thermal load. Contrary to the governing equation in usual form, the governing equations in state space together with the state equations equalize the number of the governing assembled difference equations and that of the unknowns, and eventually do not require to apply the so-called auxiliary extra boundary conditions at the nodes adjacent to the boundary nodes. As another advantage, in state-spatial viewpoint, there is not any necessity for computing the coefficients of derivatives with order higher than unity.

Effect of Newtonian heating on double convection flow fractional differential type fluid

The influence of Newtonian heating (NH) is studied on heat-mass transfer problem of fractional differential type fluid. The proposed problem is modeled by using Caputo-Fabrizio fractional order derivative. A steady magnetic field of uniform strength is enforced in a perpendicular direction of flow. The set of governing equations in dimensional form accompanied by suitable initial and boundary conditions are changed into non-dimensional form by making use of some non-dimensional variables. The resulting problem is then converted into a model employing fractional differential operator by using the new Caputo-Fabrizio fractional differential operator. The analytical exact solution has been obtained by making use of Laplace transform technique. The numerical results are computed via MATHCAD software and displayed in various plots for different physical parameters and discussed.

Inverse scattering transform for the coupled modified complex short pulse equation: Riemann–Hilbert approach and soliton solutions

In this paper, we develop a Riemann–Hilbert (RH) approach to the inverse scattering transform for the coupled modified complex short pulse (cmcSP) equation, which appeared as a reduction of the four-component system of short pulse type introduced by Popowicz in 2017. This approach allows us to present the general soliton solutions of the cmcSP equation in parametric form. Compared with the early results for the scalar modified complex short pulse equation, the cmcSP equation possesses richer soliton solutions, which include smooth solitons, cuspons, breathers and their various interactions.

Terms of recurrence sequences in the solution sets of norm form equations

The structure as well as several arithmetic properties of the solution sets of norm form equations are of classical and recent interest. In this paper, we give a finiteness result for terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Our main theorem yields a common generalization of certain recent results from the literature.

Critical sets of solutions of elliptic equations in periodic homogenization

AbstractIn this paper we study critical sets of solutions of second‐order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first‐order correctors, we show that the ‐dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε, provided that doubling indices for solutions are bounded. The key step is an estimate of “turning” of an approximate tangent map, the projection of a non‐constant solution onto the subspace of spherical harmonics of order ℓ, when the doubling index for on a sphere is trapped between and , for r between 1 and a minimal radius . This estimate is proved by using harmonic approximation successively. With a suitable L2 renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets.

Uniform approximation of 2D Navier-Stokes equations with vorticity creation by stochastic interacting particle systems

We consider a stochastic interacting particle system in a bounded domain with reflecting boundary, including creation of new particles on the boundary prescribed by a given source term. We show that such particle system approximates 2D Navier–Stokes equations in vorticity form and impermeable boundary, the creation of particles modeling vorticity creation at the boundary. Kernel smoothing, more specifically smoothing by means of the Neumann heat semigroup on the space domain, allows to establish uniform convergence of regularized empirical measures to (weak solutions of) Navier–Stokes equations.

The Morley-type virtual element method for the Navier–Stokes equations in stream-function form

The nonconforming Morley-type virtual element method for the incompressible Navier–Stokes equations formulated in terms of the stream-function on simply connected polygonal domains (not necessarily convex) is designed. A rigorous analysis by using a new enriching operator is developed. More precisely, by employing such operator, we provide novel discrete Sobolev embeddings, which allow to establish the well-posedness of the discrete scheme and obtain optimal error estimates in broken H2-, H1- and L2-norms under minimal regularity condition on the weak solution. The velocity and vorticity fields are recovered via a postprocessing formulas. Furthermore, a new algorithm for pressure recovery based on a Stokes complex sequence is presented. Optimal error estimates are obtained for all the postprocessed variables. Finally, the theoretical error bounds and the good performance of the method are validated through several benchmark tests.

A cylindrical discontinuous Galerkin method for compressible flows in axisymmetric geometry

In this paper, we present a cylindrical discontinuous Galerkin method for compressible flows in axisymmetric geometry. The axisymmetric Euler equations in geometric flux form are discretized by the Runge–Kutta DG method. To ensure conservativeness when using limiters developed for the 1- and 2-dimensional planar problems, the basis is modified from the Legendre polynomials such that the non-zero order moments in each cell do not contribute to the cell average value. Several 1- and 2-dimensional cylindrical tests are implemented and compared with the reference results. The present method exhibits the expected order accuracy in smooth problems and has a good performance in some challenging tests such as Noh problem and Sedov problem.

Velocity Distribution Upstream of Circular and Square Intakes

In the present study, a practical method is developed to predict the radial velocity of the flow towards circular and square intakes. It is shown that one single spherical sink surface passing from the desired points that have radial velocity vectors of identical magnitude within the ambient fluid can be used for the prediction of the radial velocity. The specific imaginary center of the corresponding spherical sink surface should be located on the center-line axis downstream of the entrance of the intake. As opposed to the previous studies, the proposed method does not require the solution of equations in closed-integral form, and it can also be used for the orifice-intake not opening into the atmosphere (submerged outlet), and an intake with a pipe, or intake-pipe projecting into the ambient fluid. It is shown that velocity field can be predicted with the introduced method even in the case of circulation imposed or induced ambient fluid flow. The agreement between the theoretical results and the available experimental/numerical results was found to be good.

Local conservation laws of continuous Galerkin method for the incompressible Navier–Stokes equations in EMAC form

We consider local balances of momentum and angular momentum for the incompressible Navier–Stokes equations. First, we formulate new weak forms of the physical balances (conservation laws) of these quantities, and prove they are equivalent to the usual conservation law formulations. We then show that continuous Galerkin discretizations of the Navier–Stokes equations using the EMAC form of the nonlinearity preserve discrete analogues of the weak form conservation laws, both in the Eulerian formulation and the Lagrangian formulation (which are not equivalent after discretizations). Numerical tests illustrate the new theory.