Divergence Theorem Research Articles

Extended Theory and Analysis of Potential Applications of Gauss's Law

This article analyzes and discusses the extended theory of Gauss's law and its potential applications in different fields. In the theoretical foundation section, the article elaborates on the definition, theory, and formulas of the traditional Gauss's law, providing mathematical derivations. Additionally, it reviews the applications of Gauss's law in electrostatics and gravitational fields, including aspects such as electric field flux, electric potential displacement, gravitational field, and mass distribution. In the potential application analysis section, the article explores the potential applications of Gauss's law in fields such as communications, acoustics, and medicine The expanded theories and potential applications of Gauss's theorem provide us with a unified mathematical framework at the macroscopic level for describing and explaining physical phenomena in different fields. By applying Gauss's theorem and its extended forms, we can better understand and explore various phenomena in the natural world, applying them to acoustics, medicine, and scientific research, thus advancing technological development and addressing practical problems.

Applying an Oriented Divergence Theorem to Swept Face Remap

Applying an Oriented Divergence Theorem to Swept Face Remap

On the creation of a photon by an electromagnetic wave ball

Recent literature has shown, theoretically as well as experimentally, that while a beam splitter does not split a single photon, it nonetheless divides the electromagnetic wave into transmitted and reflected, with both containing energies. This implies the existence of a spacetime of pure electromagnetic waves of energies but without particles; also, it prompts the question on how much energy a photon loses after coming out of a beam splitter, which impacts on interferometry and hence quantum communication. This paper shows that, by Gauss divergence theorem, the gravitational force inside an electromagnetic wave ball results in a point energy that is three times as the wave energy; thus, a 50/50 beam splitter is to cause a photon to lose half of a quarter, or 1/8, of its initial wave energy.

Volume integral model for algebraic image reconstruction and computed tomography

Industrial computed tomography (CT) has seen widespread adoption as an inspection technique due to its ability to resolve small defects and perform high-resolution measurements on complex structures. The reconstruction of CT data is usually performed using filtered back-projection (FBP) methods, such as the Feldkamp-Davis-Kress (FDK) method, and are selected as they offer a good compromise between reconstruction time and quality. More recently, iterative reconstruction algorithms have seen a resurgence in research interest as computing power has increased. Iterative reconstruction algorithms, such as the algebraic reconstruction technique (ART), use a reconstruction approach based on linear algebra to determine voxel attenuation coefficients based on the measured attenuation of the sample at the detector and calculation of the ray paths traversing the voxel grid. This offers a more precise model for CT reconstruction but at the cost of computational complexity and reconstruction time. Existing ART implementations are based on the 2D weighting models of the binary integral method (BIM), line integral method (LIM) and area integral method (AIM). For full 3D reconstruction, BIM and LIM only offer approximations leading to numerical inaccuracies. AIM for 2D reconstruction is mathematically exact but considers the divergent nature of a fan beam for 2D only. For a full 3D volumetric reconstruction, the X-ray cone beam is divergent in all directions and therefore AIM cannot be applied in its current form. A novel voxel weighting method for 3D volumetric image reconstruction using ART and providing a mathematically exact fractional volume weighting is introduced in this paper and referred to as the volume integral method (VIM). A set of algorithms is provided based on computer graphics techniques to determine ray/voxel intersections with volume reconstruction computed based on the divergence theorem. A set of experimental configurations is developed to provide a comparison against existing methods and conclusions are provided. Optimisation is achieved through graphic acceleration.

Efficient sequential PLIC interface positioning for enhanced performance of the three-phase VoF method

This paper presents an efficient algorithm for the sequential positioning, also called nested dissection, of two planes in an arbitrary polyhedron. Two planar interfaces are positioned such that the first plane truncates a given volume from this arbitrary polyhedron and the next plane truncates a second given volume from the residual polyhedron. This is a relevant task in the numerical simulation of three-phase flows when resorting to the geometric Volume-of-Fluid (VoF) method (Hirt and Nichols, 1981) with a Piecewise Linear Interface Calculation (PLIC) (Youngs, 1982). An efficient algorithm for this task significantly speeds up the three-phase PLIC algorithm. The present study describes a method based on a recursive application of the Gaussian divergence theorem, where the fact that the truncated polyhedron shares multiple faces with the original polyhedron can be exploited to reduce the computational effort. A careful choice of the coordinate system origin for the volume computation allows for successive positioning of two planes without reestablishing polyhedron connectivity. Combined with a highly efficient root finding, this results in a significant performance gain in the reconstruction of the three-phase interface configurations.The performance of the new method is assessed in a series of carefully designed numerical experiments. Compared to a conventional decomposition-based approach, the number of iterations and, thus, of the required truncations was reduced by up to an order of magnitude. The PLIC positioning run-time was reduced by about 90% in our reference implementation. Integrated into the multi-phase flow solver Free Surface 3D (FS3D), an overall performance gain of about 20% was achieved. The reference implementation of the efficient sequential PLIC positioning is available as a Fortran module at https://doi.org/10.18419/darus-2488. Allowing for simple integration into existing numerical schemes, the proposed algorithm is self-contained, requiring no external decomposition libraries.

On subspace concentration for dual curvature measures

We study subspace concentration of dual curvature measures of convex bodies K satisfying γ(−K)⊆K for some γ∈(0,1]. We present upper bounds on the subspace concentration depending on γ, which, in particular, retrieves the known results in the symmetric setting. The proof is based on a unified approach to prove necessary subspace concentration conditions via the divergence theorem.

Mimetic finite difference operators and higher order quadratures

Mimetic finite difference operators textbf{D}, textbf{G} are discrete analogs of the continuous divergence (div) and gradient (grad) operators. In the discrete sense, these discrete operators satisfy the same properties as those of their continuum counterparts. In particular, they satisfy a discrete extended Gauss’ divergence theorem. This paper investigates the higher-order quadratures associated with the fourth- and sixth- order mimetic finite difference operators, and show that they are indeed numerical quadratures and satisfy the divergence theorem. In addition, extensions to curvilinear coordinates are treated. Examples in one and two dimensions to illustrate numerical results are presented that confirm the validity of the theoretical findings.

Second-order accurate normal reconstruction from volume fractions on unstructured meshes with arbitrary polyhedral cells

This paper introduces a novel method for the computation of accurate normal fields from volume fractions on unstructured polyhedral meshes. For surfaces of considerable curvature variation, the method is second-order accurate with respect to symmetric volume differences and first-order accurate with respect to normal field angular deviation. For this purpose, an averaged normal is locally computed, i.e. in each mesh cell, by fitting a plane in a least squares sense with appropriate weights to the volume fraction data of neighbouring cells while explicitly accounting for volume conservation in the cell at hand. The resulting minimization problem is solved approximately by means of a Newton-type method. Moreover, employing the Reynolds transport theorem allows for both assessing the regularity of the derivatives of the least squares error functional with respect to the free parameters and to exactly compute these derivatives at small additional costs. Since the divergence theorem implies that the volume fraction can be written as a sum of face-based quantities, our method considerably simplifies the numerical procedure for applications in three spatial dimensions while demonstrating an inherent ability to robustly deal with unstructured meshes.We discuss the theoretical foundations, regularity and appropriate error measures, along with the details of the numerical algorithm. Finally, numerical results for convex and non-convex hypersurfaces embedded in cuboidal and tetrahedral meshes are presented together with a convergence study considering two different error measures. Moreover, our findings are complemented by profound insights into the minimization procedure.

Free in-plane vibration of irregular laminated plate with curved edges based on boundary-type Chebyshev–Ritz method

The free in-plane vibration characteristics of laminated plate with arbitrarily geometrical shape and generally restrained edges, including curved and inclined edges, are investigated based on the boundary-type Chebyshev–Ritz method (BCRM). By adopting the divergence theorem, the double integral encountered in solving the energy equation of irregular laminates is converted to the integral on the boundary edge, which effectively reduces the computational complexity of the involved domain. The Chebyshev polynomials are utilized as test functions and the primitives of domain integrand functions are deduced according to the properties of Chebyshev polynomials. General boundary conditions on straight or even curved edges are achieved by introducing normal and tangential springs and assigning reasonable stiffness values. Natural frequencies and modes of laminated composite plates with straight edges (triangle, square, trapezoid, and hexagon), laminated composite plates with curved edge (ellipse), and irregular laminated composite plates are obtained. The method validation is carried out by comparison of the obtained solutions with literature and finite element results. The effects of variations in layering mode, number of layers, boundary conditions, and geometry on the eigenpairs of composite plate are extensively discussed.

Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non‐smooth coefficients

AbstractWe prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C1‐boundary, the other stated for sets with finite perimeter.

Failure probability estimation and detection of failure surfaces via adaptive sequential decomposition of the design domain

We propose an algorithm for selection of points from the design domain of small to moderate dimension and for failure probability estimation. The proposed active learning detects failure events and progressively refines the boundary between safe and failure domains thereby improving the failure probability estimation. The method is particularly useful when each evaluation of the performance function g(x) is very expensive and the function can be characterized as either highly nonlinear, noisy, or even discrete-state (e.g., binary). In such cases, only a limited number of calls is feasible, and gradients of g(x) cannot be used. The input design domain is progressively segmented by expanding and adaptively refining a mesh-like lock-free geometrical structure. The proposed triangulation-based approach effectively combines the features of simulation and approximation methods. The algorithm performs two independent tasks: (i) the estimation of probabilities through an ingenious combination of deterministic cubature rules and the application of the divergence theorem and (ii) the sequential extension of the experimental design with new points. The sequential selection of points from the design domain for future evaluation of g(x) is carried out through a new decision approach, which maximizes instantaneous information gain in terms of the probability classification that corresponds to the local region. The extension may be halted at any time, e.g., when sufficiently accurate estimations are obtained. Due to the use of the exact geometric representation in the input domain, the algorithm is most effective for problems of a low dimension, not exceeding eight. The method can handle random vectors with correlated non-Gaussian marginals. When the values of the performance function are valid and credible, the estimation accuracy can be improved by employing a smooth surrogate model based on the evaluated set of points. Finally, we define new factors of global sensitivity to failure based on the entire failure surface weighted by the density of the input random vector.

Rayleigh‐Taylor Instability Observed at the Dayside Magnetopause Under Northward Interplanetary Magnetic Field

AbstractUnder northward interplanetary magnetic field, periodical fluctuations with the period of 85 s have been observed by one of the Time History of Events and Macroscale Interactions during Substorms spacecraft at the dayside magnetopause when the solar wind dynamic pressure suddenly dropped. The observed magnetic field distortions, characterized by the compression of the main component as well as the flapping of the perpendicular components, are in line with the theoretical predictions of the Rayleigh‐Taylor (R‐T) instability excited at the magnetopause. With the convective electric field and the mean electric field removed, the perturbations in the electric field are identified as typical two sinusoidal signals with a 90° phase difference, consistent with previous prediction of the attendant electrostatic field arising from the R‐T instability. The transverse motions of the plasmas resulting from the electric drifting driven by the electrostatic field are observed both in pitch‐angle distributions and in distribution functions. The growth rate of the R‐T instability is about 43 s, implying that the instability has time to grow during the disturbances. Using the Gauss theorem, the calculated net charge density based on the electric field observation is one part of 27 compared to the totally observed plasma density, indicating that only a tiny fraction of the charge separates to set up the electrostatic field, which is due to a collective effect in the plasma.

Conservative semi-Lagrangian method for continuity equation with various time steps in different parts of computational domain

The system of Navier Stokes differential equations describes gas flow near rocket nozzle. To find solution of this system in general case, scientists and engineers use numerical methods. Modern numerical methods do not allow engineers to model all features of gas flow. It happened because of sophisticated physical processes and limitations of computational hardware. So, at least where are two ways to improve it: to enlarge hardware or reduce computational complicacy. The global aim of many science investigations is to develop numerical approach with reduced computational complicacy and at the same time without loss of computational accuracy. In 1959, Aksel C. Wiin-Nielsen proposed a new and effective trajectories method for problem of numerical forecasting. In 1966, K. M. Magomedov developed similar approach (method of characteristics) for numerical modelling of space (three dimensional on space) gas flow. In 1982, O. Pironneau showed a new and effective approach for two dimensional approximation of the Navier Stokes problem. It was based on method of characteristics also. Nowadays, these methods are called semi-Lagrangian or Eulerian Lagrangian methods. They use Lagrangian nature of the transport process. To apply this advantage, scientists decompose each equation of Navier Stokes system into three parts: convective part (hyperbolic type), elliptic part and part of right-hand side. Scientists use semi-Lagrangian approach to approximate the convective parts of equations. To develop and test modern algorithms from family of semi-Lagrangian methods, we use continuity equation from the system of Navier Stokes equations. Conservative versions of semi-Lagrangian approach are based on Gauss Ostrogradsky (divergence) theorem. It allows scientists to get conservation low (balance equation) for numerical solution in norm of L1 space. Aim of our investigation is to use different time steps in different parts of computation domain. It enables us to obtain at the same time three advantages: convergence of numerical solution to exact solution, reduction of computational complicacy, implementation of conservation low (balance equation) without weight coefficients. For this purpose we decompose computational domain into two parts (subdomains) and use different time steps in them. Main complication is design algorithm in the boundaries of computational subdomains. We use one dimensional (on space) problem to demonstrate ability of developing numerical method with described advantages. Generalization of considered approach for two- or even three-dimensional cases allows engineers to model gas flow more accurately and without artificial viscosity. It is essentially important in the parts of computational domain with high level of solution gradient.

A hybrid direct FE2 method for modeling of multiscale materials and structures with strain localization

Strain localization is a common phenomenon existing in various multiscale materials or structures, e.g., the bulking bands of thin-walled structures, the local collapse of porous materials, and the crack in solid materials, etc. However, this phenomenon cannot be captured by the conventional homogenization methods, such as mean-field homogenization (e.g., Mori–Tanaka method), first-order or high-order computational homogenization (e.g., the Finite Element Square method), etc., due to the high strain gradient associated with strain localization. Aiming at this, a hybrid Direct FE2 method is proposed by combining the D-FE2 (Direct FE2) and the traditional FE methods, while the multiscale structure is modeled using the D-FE2 method in the region exhibiting low deformation gradient, and the other region displaying high deformation gradient is modeled using the traditional FE method. Moreover, a node displacement constraint and an overall node displacement constraint derived from the multilevel equilibrium equations using the Gauss–Ostrogradsky theorem are respectively prescribed to the interface between the D-FE2 model and the FE model of the multiscale structure, to enforce the energy equilibrium and deformation continuity. The proposed hybrid D-FE2 method is then applied to predict the strain localization behavior of multiscale materials or structures, including local bulking of honeycomb structures, in-situ crack propagation, and localized plastic deformation in fiber reinforced composites, etc. Comparison of the simulation results obtained from the hybrid D-FE2 method and the traditional FE method validates the accuracy, efficiency and ease of numerical implementation of the proposed hybrid D-FE2 method.

The ℓ-adic hypergeometric function and associators

We introduce an $\ell$-adic analogue of Gauss's hypergeometric function arising from the Galois action on the fundamental torsor of the projective line minus three points. Its definition is motivated by a relation between the KZ-equation and the hypergeometric differential equation in the complex case. We show two basic properties, analogues of Gauss's hypergeometric theorem and of Euler's transformation formula for our $\ell$-adic function. We prove them by detecting a connection of a certain two-by-two matrix specialization of even unitary associators with the associated gamma function, which extends the result of Ohno and Zagier.

Gauss Theorem and Pointlike Charges: When Infinitesimals Make the Difference

A classic application of the Gauss theorem is examined, showing how the choice of the Gaussian surface and the pointlike nature of the charge can determine results somehow unexpected for the student. Particular care must be paid not only to points where the field becomes infinite, but also to infinitesimal displacements of the charge and their direction. Pedagogical implications are outlined.

The force density in electrical machines modeled as tension and pressure gradients of magnetic field lines

This paper shows how to model the force density in electrical machines based on the field lines of the magnetic flux density. The force density is written as two vector components: the magnetic tension force and the magnetic pressure gradient force. This approach has been applied in physics but never to forces in engineering problems. The magnetic tension force acts to straighten bent field lines, based on the curvature of the flux density. The magnetic pressure gradient force acts from regions of high flux density to regions of low flux density. Both force densities are derived from the Lorentz force using the tnb-frame of Frenet–Serret formulas and shown to be equivalent to the divergence of the Maxwell stress tensor. It is shown how the force density could describe the forces in a synchronous machine, including both the angular torque of the load and the radial forces between the rotor and the stator. It could also be linked to the power flow and thereby to the energy flux of Poynting’s vector. The force densities could be used to improve the understanding of the Maxwell stress tensor, since they are easier to illustrate as vectors compared to the matrix form of the Maxwell stress tensor. It also shows the location of the force density, which could improve the use of enclosing volumes when calculating the force based on the divergence theorem with the Maxwell stress tensor.

The Application of Calculus to Gauss's Theorem for Electrostatic Fields

The Application of Calculus to Gauss's Theorem for Electrostatic Fields

Investigation of the value and mechanism of atmospheric plasmoid charge formation

The study of a long-lived atmospheric plasmoid has been continued. (Gatchina plasmoid). The electric charge of the plasmoid was detected earlier by us using a plasma probe. Here the measuring equipment was improved, which allowed to take into account the influence of bias current. The measured charge of the negative plasmoid was about -35 nC. The cause of the electric charge is investigated. It consists in the inhomogeneity of the plasma and, as a consequence, a special configuration of the electric field during discharge. In the column of emission plasma above the central electrode, the field is directed into the plasmoid (for positive central electrode). In streamers coming out of the plasmoid and in space above the plasmoid, the field is directed outward. To find out the reason for the appearance of the charge, the Gauss theorem is applied. The column of emission plasma gives a small contribution to the integral. The integral will be determined by the second section having a different sign. Thus, the plasmoid has a charge during discharge. After the current is turned off, the charge is retained due to the rapid cooling of the plasma. It is noted that when obtaining other long-lived plasmoids, there is a similar configuration of the field during discharge. A similar discharge structure occurs when a linear lightning strikes the electric network and a ball lightning appears from an electrical outlet, a speaker, etc. This work introduces a new object of plasma physics a charged atmospheric plasmoid. Keywords: atmospheric plasmoid, high-voltage discharge, plasma probe, ball lightning, electric charge.

Real-Time Surface-Based Volume Constraints on Mass-Spring Model in Unity3D

This paper describes a parallel method to simulate real-time 3D deformable objects using volume preservation constraints on a mass-spring model (MSM) to achieve plausible results in real-time performance. Instead of considering a volumetric mesh which is mostly used to simulate deformable objects, we purely take the surface of the 3D object into account to reduce the time complexity and obtain a high-quality deformable. In the conventional MSM, we can simply control the shape of the deformable object through the stiffness and damping coefficients which is beneficial for our volume constraint. We use the divergence theorem and implicit constraint enforcement scheme to maintain the volume of the object and deform it freely. The surface-based volume constraint is applied to correct the force of the spring network. The proposed algorithm was designed on compute shader in Unity3D and runs outside the normal rendering pipeline in the graphics processing unit (GPU) in order to utilize the massive parallel process to accelerate the performance of the simulation. The performance of the simulation can be accelerated by using the parallel processing method on the GPU with an average speedup factor of 4.27 using the conventional mass-spring method, and an average speedup factor of 2.54 for the volume preservation constraint method. We present several scenes which demonstrate the volume-preserving deformations using the conventional mass-spring method and volume-preservation constraint method and the volume loss of deformable objects for all 3D models is compared. The volume preservation method obtains a volume loss significantly lower than the conventional MSM for all experimental tests.