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Journal articles on the topic 'Surface integrals'

Author: Grafiati

Published: 5 June 2025

Last updated: 16 July 2025

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1

Shomurodova, Dilafroz. "SURFACE INTEGRALS IN MATHEMATICAL ANALYSIS." International journal of advanced research in education, technology and management 2, no. 2 (2023): 68–71. https://doi.org/10.5281/zenodo.7677840.

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In this article, methods of working examples related to the calculation of surface integrals of the first type and some applications of surface integrals, that is, finding the mass of several bodies, are considered, and a convenient method of teaching them to students of higher education institutions is analyzed.       

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2

Xu, Xia, and Li Na Wang. "Research on Integrals of the Second Category Curved Surface." Applied Mechanics and Materials 672-674 (October 2014): 2017–20. http://dx.doi.org/10.4028/www.scientific.net/amm.672-674.2017.

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In this thesis,surface integrals and the relationship between characteristics of the second class and second class double integral surface integrals, analyzes the application of the law of symmetry integral calculation, even if the integration region derived symmetric integrand is an odd function integral value characteristic is not necessarily zero. And pointed out that the feasibility of the method is definitely second class surface integrals.

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3

Gordon, W. B., and H. J. Bilow. "Reduction of surface integrals to contour integrals." IEEE Transactions on Antennas and Propagation 50, no. 3 (2002): 308–11. http://dx.doi.org/10.1109/8.999621.

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4

Karaiev, Artem, and Elena Strelnikova. "Singular integrals in axisymmetric problems of elastostatics." International Journal of Modeling, Simulation, and Scientific Computing 11, no. 01 (2020): 2050003. http://dx.doi.org/10.1142/s1793962320500038.

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Singular integral equations arisen in axisymmetric problems of elastostatics are under consideration in this paper. These equations are received after applying the integral transformation and Gauss–Ostrogradsky’s theorem to the Green tensor for equilibrium equations of the infinite isotropic medium. Initially, three-dimensional problems expressed in Cartesian coordinates are transformed to cylindrical ones and integrated with respect to the circumference coordinate. So, the three-dimensional axisymmetric problems are reduced to systems of one-dimensional singular integral equations requiring the evaluation of linear integrals only. The thorough analysis of both displacement and traction kernels is accomplished, and similarity in behavior of both kernels is established. The kernels are expressed in terms of complete elliptic integrals of first and second kinds. The second kind elliptic integrals are nonsingular, and standard Gaussian quadratures are applied for their numerical evaluation. Analysis of external integrals proved the existence of logarithmic and Cauchy’s singularities. The numerical treatment of these integrals takes into account the presence of this integrable singularity. The numerical examples are provided to testify accuracy and efficiency of the proposed method including integrals with logarithmic singularity, Catalan’s constant, the Gaussian surface integral. The comparison between analytical and numerical data has proved high precision and availability of the proposed method.

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5

Bleszynski, Elizabeth H., Marek K. Bleszynski, and Thomas Jaroszewicz. "Reduction of Singular Surface Integrals to Nonsingular Line Integrals in Integral Equations for Planar Geometries." IEEE Transactions on Antennas and Propagation 64, no. 11 (2016): 4760–69. http://dx.doi.org/10.1109/tap.2016.2602356.

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6

Rathod, H. T., and H. S. Govinda Rao. "Integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 39, no. 3 (1998): 355–85. http://dx.doi.org/10.1017/s0334270000009450.

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AbstractThis paper concerns with analytical integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space. The volume integration of trivariate polynomials over linear polyhedra is computed as sum of surface integrals in R3 on application of the well known Gauss's divergence theorem and by using triangulation of the linear polyhedral boundary. The surface integrals in R3 over an arbitrary triangle are connected to surface integrals of bivariate polynomials in R2. The surface integrals in R2 over a simple polygon or over an arbitrary triangle are computed by two different approaches. The first algorithm is obtained by transforming the surface integrals in R2 into a sum of line integrals in a one-parameter space, while the second algorithm is obtained by transforming the surface integrals in R2 over an arbitrary triangle into a parametric double integral over a unit triangle. It is shown that the volume integration of trivariate polynomials over linear polyhedra can be obtained as a sum of surface integrals of bivariate polynomials in R2. The computation of surface integrals is proposed in the beginning of this paper and these are contained in Lemmas 1–6. These algorithms (Lemmas 1–6) and the theorem on volume integration are then followed by an example for which the detailed computational scheme has been explained. The symbolic integration formulas presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, for example, the volume, centre of mass, moments of inertia etc., required in engineering design processes.

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7

Tzoulis, A., and T. F. Eibert. "Review of singular potential integrals for method of moments solutions of surface integral equations." Advances in Radio Science 2 (May 27, 2005): 93–99. http://dx.doi.org/10.5194/ars-2-93-2004.

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Abstract. Accurate evaluation of singular potential integrals is essential for successful method of moments (MoM) solutions of surface integral equations. In mixed potential formulations for metallic and dielectric scatterers, kernels with 1/R and r1/R singularities must be considered. Several techniques for the treatment of these singularities will be reviewed. The most common approach solves the MoM source integrals analytically for specific observation points, thus regularizing the integral. However, in the case of r1/R a logarithmic singularity remains for which numerical evaluation of the testing integral is still difficult. A recently by Yl¨a-Oijala and Taskinen proposed remedy to this issue is discussed and evaluated within a hybrid finite element – boundary integral technique. Convergence results for the MoM coupling integrals are presented where also higher-order singularity extraction is considered.

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8

Dawass, Noura, Peter Krüger, Sondre K. Schnell, et al. "Kirkwood-Buff Integrals Using Molecular Simulation: Estimation of Surface Effects." Nanomaterials 10, no. 4 (2020): 771. http://dx.doi.org/10.3390/nano10040771.

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Kirkwood-Buff (KB) integrals provide a connection between microscopic properties and thermodynamic properties of multicomponent fluids. The estimation of KB integrals using molecular simulations of finite systems requires accounting for finite size effects. In the small system method, properties of finite subvolumes with different sizes embedded in a larger volume can be used to extrapolate to macroscopic thermodynamic properties. KB integrals computed from small subvolumes scale with the inverse size of the system. This scaling was used to find KB integrals in the thermodynamic limit. To reduce numerical inaccuracies that arise from this extrapolation, alternative approaches were considered in this work. Three methods for computing KB integrals in the thermodynamic limit from information of radial distribution functions (RDFs) of finite systems were compared. These methods allowed for the computation of surface effects. KB integrals and surface terms in the thermodynamic limit were computed for Lennard–Jones (LJ) and Weeks–Chandler–Andersen (WCA) fluids. It was found that all three methods converge to the same value. The main differentiating factor was the speed of convergence with system size L. The method that required the smallest size was the one which exploited the scaling of the finite volume KB integral multiplied by L. The relationship between KB integrals and surface effects was studied for a range of densities.

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9

Beale, J. Thomas, Wenjun Ying, and Jason R. Wilson. "A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces." Communications in Computational Physics 20, no. 3 (2016): 733–53. http://dx.doi.org/10.4208/cicp.030815.240216a.

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AbstractWe present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about O(h3), where h is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.

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10

Movshovich, Yevgenya. "Surface integrals and harmonic functions." Journal of Inequalities and Applications 2005, no. 4 (2005): 913098. http://dx.doi.org/10.1155/jia.2005.443.

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11

Uglanov, A. V. "Surface integrals in Frechet spaces." Sbornik: Mathematics 189, no. 11 (1998): 1719–37. http://dx.doi.org/10.1070/sm1998v189n11abeh000374.

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12

Benn, I. M., and W. P. Wood. "Surface integrals for electromagnetic energy." Journal of Mathematical Physics 34, no. 7 (1993): 2936–49. http://dx.doi.org/10.1063/1.530106.

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13

Jiménez, Raúl, and J. E. Yukich. "Nonparametric estimation of surface integrals." Annals of Statistics 39, no. 1 (2011): 232–60. http://dx.doi.org/10.1214/10-aos837.

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14

Agafonov, Sergey I., and Thaís G. P. Alves. "Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs." Advances in Geometry 24, no. 2 (2024): 263–73. http://dx.doi.org/10.1515/advgeom-2024-0008.

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Abstract We prove that if the geodesic flow on a surface has an integral which is fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for the existence of fractional-linear integrals: such an integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves.

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15

Peters, James, Roberto Alfano, Peter Smith, Arturo Tozzi, and Tane Vergilie. "Geometric realizations of homotopic paths over curved surfaces." Filomat 38, no. 3 (2024): 793–802. http://dx.doi.org/10.2298/fil2403793p.

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This paper introduces geometric realizations of homotopic paths over simply-connected surfaces with non-zero curvature as a means of comparing and measuring paths between antipodes with either a Feynman path integral or Woodhouse contour integral, resulting in a number of extensions of the Borsuk Ulam Theorem. All realizations of homotopic paths reside on a Riemannian surface S, which is simplyconnected and has non-zero curvature at every point in S. A fundamental result in this paper is that for any pair of antipodal surface points, a path can be found that begins and ends at the antipodal points. The realization of homotopic paths as arcs on a Riemannian surface leads to applications in Mathematical Physics in terms of Feynman path integrals on trajectory-of-particle curves and Woodhouse countour integrals for antipodal vectors on twistor curves. Another fundamental result in this paper is that the Feynman trajectory of a particle is a homotopic path geometrically realizable as a Lefschetz arc.

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16

T. Shivaram, K., and S. Kiran. "A simple and efficient wavelet approach for evaluating surface integral over curved domain." International Journal of Engineering & Technology 7, no. 4.5 (2018): 511. http://dx.doi.org/10.14419/ijet.v7i4.5.21145.

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This paper presents, a simple and efficient wavelet approach for computing the surface integrals over irregular or curved dom ain, the limit of the integrals are nonlinear function are transformed to standard two square by using finite element basis function, Haar wavelet based integration technique is applied to evaluation of surface integral over curved domain, the computational efficiency of the method is illustrated with several numerical examples.

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17

Jacobs, Ralf T., Thomas Wondrak, and Frank Stefani. "Singularity consideration in the integral equations for contactless inductive flow tomography." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 37, no. 4 (2018): 1366–75. http://dx.doi.org/10.1108/compel-08-2017-0361.

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Purpose The contactless inductive flow tomography is a procedure that enables the reconstruction of the global three-dimensional flow structure of an electrically conducting fluid by measuring the flow-induced magnetic flux density outside the melt and by subsequently solving the associated linear inverse problem. The purpose of this study is to improve the accuracy of the computation of the forward problem, since the forward solution primarily determines the accuracy of the inversion. Design/methodology/approach The tomography procedure is described by a system of coupled integral equations where the integrals contain a singularity when a source point coincides with a field point. The integrals need to be evaluated to a high degree of precision to establish an accurate foundation for the inversion. The contribution of a singular point to the value of the surface and volume integrals in the system is determined by analysing the behaviour of the fields and integrals in the close proximity of the singularity. Findings A significant improvement of the accuracy is achieved by applying higher order elements and by attributing special attention to the singularities inherent in the integral equations. Originality/value The contribution of a singular point to the value of the surface integrals in the system is dependent upon the geometry of the boundary at the singular point. The computation of the integrals is described in detail and the improper surface and volume integrals are shown to exist. The treatment of the singularities represents a novelty in the contactless inductive flow tomography and is the focal point of this investigation.

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18

KISHIDA, Michiya, Kazuaki SASAKI, and Shiroh MACHINO. "Accuracy of numerical surface integrals in the indirect boundary integral method." Transactions of the Japan Society of Mechanical Engineers Series A 57, no. 533 (1991): 181–87. http://dx.doi.org/10.1299/kikaia.57.181.

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19

Wang, Zhongxiang, Cai Chang, and Jian Lu. "Direct Calculating Method for Integral of Multivariate Functions Based on Mathematica." Journal of Education and Culture Studies 9, no. 1 (2024): p16. https://doi.org/10.22158/jecs.v9n1p16.

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Mathematica is a comprehensive and efficient general mathematical software that integrates numerical and symbolic calculation, graphics and animation, programming and interactive demonstrations. This paper presents the methodology and operational procedures for the direct computation of multiple integrals, curve integrals and surface integrals utilizing Mathematica software. Through the application examples, it can be intuitively seen that this way of calculating multiple integrals with Mathematical software provides a very convenient, fast and effective way to verify the correctness of the integral calculation methods and results.

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20

Guiggiani, M., G. Krishnasamy, T. J. Rudolphi, and F. J. Rizzo. "A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations." Journal of Applied Mechanics 59, no. 3 (1992): 604–14. http://dx.doi.org/10.1115/1.2893766.

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The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

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21

Guo, Quanxin, Jian-Juei Wang, and R. J. Clifton. "Three-Dimensional Analysis of Surface Cracks in an Elastic Half-Space." Journal of Applied Mechanics 63, no. 2 (1996): 287–94. http://dx.doi.org/10.1115/1.2788862.

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A numerical method is presented for analyzing arbitrary planar cracks in a half-space. The method is based on the fundamental solution for a dislocation loop in a half-space, which is derived from the Mindlin solution (Mindlin, Physics, Vol. 7, 1936) for a point force in a half-space. By appropriate replacement of the Burgers vectors of the dislocation by the differential crack-opening displacement, a singular integral equation is obtained in terms of the gradient of the crack opening. A numerical method is developed by covering the crack with triangular elements and by minimizing the total potential energy. The singularity of the kernel, when the integral equation is expressed in terms of the gradient of the crack opening, is sufficiently weak that all integrals exist in the regular sense and no special numerical procedures are required to evaluate the contributions to the stiffness matrix. The integrals over the source elements are converted into line integrals along the perimeter of the element and evaluated analytically. Numerical results are presented and compared with known results for both surface breaking cracks and near surface cracks.

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22

Lyrintzis, Anastasios S. "Surface Integral Methods in Computational Aeroacoustics—From the (CFD) Near-Field to the (Acoustic) Far-Field." International Journal of Aeroacoustics 2, no. 2 (2003): 95–128. http://dx.doi.org/10.1260/147547203322775498.

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A review of recent advances in the use of surface integral methods in Computational AeroAcoustics (CAA) for the extension of near-field CFD results to the acoustic far-field is given. These integral formulations (i.e. Kirchhoff's method, permeable (porous) surface Ffowcs-Williams Hawkings (FW-H) equation) allow the radiating sound to be evaluated based on quantities on an arbitrary control surface if the wave equation is assumed outside. Thus only surface integrals are needed for the calculation of the far-field sound, instead of the volume integrals required by the traditional acoustic analogy method (i.e. Lighthill, rigid body FW-H equation). A numerical CFD method is used for the evaluation of the flow-field solution in the near field and thus on the control surface. Diffusion and dispersion errors associated with wave propagation in the far-field are avoided. The surface integrals and the first derivatives needed can be easily evaluated from the near-field CFD data. Both methods can be extended in order to include refraction effects outside the control surface. The methods have been applied to helicopter noise, jet noise, propeller noise, ducted fan noise, etc. A simple set of portable Kirchhoff/FW-H subroutines can be developed to calculate the far-field noise from inputs supplied by any aerodynamic near/mid-field CFD code.

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23

De Cicco, Virginia. "Lower semicontinuity for nonautonomous surface integrals." Rendiconti Lincei - Matematica e Applicazioni 26, no. 1 (2015): 1–21. http://dx.doi.org/10.4171/rlm/688.

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24

Ren, Zhengyong, Huang Chen, Jingtian Tang, and Feng Zhou. "A volume-surface integral approach for direct current resistivity problems with topography." GEOPHYSICS 83, no. 5 (2018): E293—E302. http://dx.doi.org/10.1190/geo2017-0577.1.

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We have developed an accurate volume-surface integral formula for 3D direct current (DC) resistivity forward modeling with heterogeneous conductivities and arbitrary homogeneous topography. First, a volume-surface integral formula is derived from its elliptic boundary value problem in terms of an artificial analytical function defined over the full space. That leads to a volume integral accounting for underground anomalous regions and a surface integral over the surface topography. Then, tetrahedral grids are used to discretize the volume anomalous bodies and triangular grids are adopted to approximate the complicated surface topography. The use of unstructured grids enables our volume-surface integral formula to deal with realistic earth models with complex geometries and conductivity distributions. Furthermore, linear shape functions are assumed in the tetrahedral and triangular elements to obtain the final system of linear equations. In the final system matrix, singularity-free analytical expressions are developed for entries arising from volume integrals over tetrahedral elements and Gaussian quadrature formulas are used to calculate surface integrals over triangular elements. To guarantee the accuracy of the final numerical solutions, direct solvers are used. At the end, three synthetic models are used to verify our newly developed volume-surface integral formula by comparison with published analytical solutions and finite-element solutions. Due to its high accuracy, solutions of our volume-surface integral approach can act as an efficient benchmark tool for other numerical solutions for complicated DC models with arbitrary homogeneous topographies.

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HAKIMELAHI, B., and N. SOLTANI. "3D J-integral evaluation using the computation of line and surface integrals." Fatigue & Fracture of Engineering Materials & Structures 33, no. 10 (2010): 661–72. http://dx.doi.org/10.1111/j.1460-2695.2010.01478.x.

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Sheng, Wei Tian, Zhen Ying Zhu, Kuo Yang, and Mei Song Tong. "Efficient Evaluation of Weakly Singular Integrals Arising From Electromagnetic Surface Integral Equations." IEEE Transactions on Antennas and Propagation 61, no. 6 (2013): 3377–81. http://dx.doi.org/10.1109/tap.2013.2253296.

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27

Zhu, Ming-Da, Xi-Lang Zhou, and Wen-Yan Yin. "Efficient Evaluation of Double Surface Integrals in Time-Domain Integral Equation Formulations." IEEE Transactions on Antennas and Propagation 61, no. 9 (2013): 4653–64. http://dx.doi.org/10.1109/tap.2013.2266313.

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Beuc, Robert, Mladen Movre, and Berislav Horvatić. "On the Approximate Evaluation of Some Oscillatory Integrals." Atoms 7, no. 2 (2019): 47. http://dx.doi.org/10.3390/atoms7020047.

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To determine the photon emission or absorption probability for a diatomic system in the context of the semiclassical approximation it is necessary to calculate the characteristic canonical oscillatory integral which has one or more saddle points. Integrals like that appear in a whole range of physical problems, e.g., the atom–atom and atom–surface scattering and various optical phenomena. A uniform approximation of the integral, based on the stationary phase method is proposed, where the integral with several saddle points is replaced by a sum of integrals each having only one or at most two real saddle points and is easily soluble. In this way we formally reduce the codimension in canonical integrals of “elementary catastrophes” with codimensions greater than 1. The validity of the proposed method was tested on examples of integrals with three saddle points (“cusp” catastrophe) and four saddle points (“swallow-tail” catastrophe).

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Kim, Jeongmin, and Sunghwan Moon. "Isometry property and inversion of the Radon transform over a family of paraboloids." Filomat 38, no. 23 (2024): 8047–52. https://doi.org/10.2298/fil2423047k.

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Integral geometry problems involve finding a desired function from its integrals on a surface. These problems are closely intertwined with the generalized Radon transform, and obtaining an inversion formula for it is pivotal in solving integral geometry problems. The applications of integral geometry span various fields, including tomography, radar, and radiology. Particularly noteworthy is the recovery of a function from integrals over a parabola, which holds significance in reflection seismology. In our study, we concentrate on the transform that maps a real-valued smooth function with compact support to integrals over the paraboloid. This transform, along with its dual, can be expressed as convolutions of kernels and given functions, and we have derived inversion formulas based on their isometric properties.

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Cheshkova, M. A. "Examples of surfaces of constant mean curvature." Differential Geometry of Manifolds of Figures, no. 50 (2019): 148–54. http://dx.doi.org/10.5922/0321-4796-2019-50-17.

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A surface in E3 is called parallel to the surface M if it consists of the ends of constant length segments, laid on the normals to the surfaces M at points of this surface. The tangent planes at the corresponding points will be parallel. For surfaces in E3 the theorem of Bonnet holds: for any surface M that has constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature. Using Bonnet's theorem for a surfaces of revolution of constant positive Gaussian curvature, surfaces of constant mean curvature are constructed. It is proved that they are also surfaces of revolution. A family of plane curvature lines (meridians) is described by means of elliptic integrals. The surfaces of constant Gaussian curvature are also described by means of elliptic integrals. Using the mathematical software package, the surfaces under consideration are constructed.

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Ren, Zhengyong, Huang Chen, Chaojian Chen, Yiyuan Zhong, and Jingtian Tang. "New analytical expression of the magnetic gradient tensor for homogeneous polyhedrons." GEOPHYSICS 84, no. 3 (2019): A31—A35. http://dx.doi.org/10.1190/geo2018-0741.1.

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We have developed a new analytical expression for the magnetic-gradient tensor for polyhedrons with homogeneous magnetization vectors. Instead of performing the direct derivative on the closed-form solutions of the magnetic field, it is obtained by first transforming the volume integrals of the magnetic-field tensor into surface integrals over polyhedral facets, in terms of the gradient theorem. Second, the surface divergence theorem transforms the surface integrals over polyhedral facets into edge integrals and structure-simplified surface integrals. Third, we develop analytical expressions for these edge integrals and simplified surface integrals. We use a synthetic prismatic target to verify the accuracies of the new analytical expression. Excellent agreements are obtained between our results and those calculated by other published formulas. The new analytical expression of the magnetic-gradient tensor can play a fundamental role in advancing magnetic mineral explorations, environmental surveys, unexploded ordnance and submarine detection, aeromagnetic and marine magnetic surveys because more and more magnetic tensor data have been collected by magnetic-tensor gradiometry instruments.

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NWOGU, OKEY G. "Interaction of finite-amplitude waves with vertically sheared current fields." Journal of Fluid Mechanics 627 (May 25, 2009): 179–213. http://dx.doi.org/10.1017/s0022112009005850.

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A computationally efficient numerical method is developed to investigate nonlinear interactions between steep surface gravity waves and depth-varying ocean currents. The free-surface boundary conditions are used to derive a coupled set of equations that are integrated in time for the evolution of the free-surface elevation and tangential component of the fluid velocity at the free surface. The vector form of Green's second identity is used to close the system of equations. The closure relationship is consistent with Helmholtz's decomposition of the velocity field into rotational and irrotational components. The rotational component of the flow field is given by the Biot–Savart integral, while the irrotational component is obtained from an integral of a mixed distribution of sources and vortices over the free surface. Wave-induced changes to the vorticity field are modelled using the vorticity transport equation. For weak currents, an explicit expression is derived for the wave-induced vorticity field in Fourier space that negates the need to numerically solve the vorticity transport equation. The computational efficiency of the numerical scheme is further improved by expanding the kernels of the boundary and volume integrals in the closure relationship as a power series in a wave steepness parameter and using the fast Fourier transform method to evaluate the leading-order contribution to the convolution integrals. This reduces the number of operations at each time step from O(N2) to O(NlogN) for the boundary integrals and O[(NM)2] to O(NlogN) for the volume integrals, where N is the number of horizontal grid points and M is the number of vertical layers, making the model an order of magnitude faster than traditional boundary/volume integral methods. The numerical model is used to investigate nonlinear wave–current interaction in depth-uniform current fields and the modulational instability of gravity waves in an exponentially sheared current in deep water. The numerical results demonstrate that the mean flow vorticity can significantly affect the growth rate of extreme waves in narrowband sea states.

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Zhu, J., A. H. Shah, and S. K. Datta. "Transient Response of a Composite Plate With Delamination." Journal of Applied Mechanics 65, no. 3 (1998): 664–70. http://dx.doi.org/10.1115/1.2789109.

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Transient response of a composite plate with a near-surface delamination has been studied in this paper. A new technique developed by the authors to evaluate the Cauchy Principal Value integrals and the weakly singular integrals involved in the boundary integral equations has been employed and modifided to treat the corner points on the boundary. The time harmonic Green’s functions appearing in the boundary integral equation are evaluated by combining a stiffness method and the modal summation technique. To circumvent the difficulties associated with the evaluation of hypersingular integrals for cracks, the multidomain technique is employed. The accuracy and efficiency of the method are checked by comparing the displacements in a uniaxial graphite-epoxy plate containing a delamination with results obtained by a hybrid method. It is shown that the presence of the delamination significantly alters the surface response spectra of the plate. Results are presented in both time and frequency domains. The results show that the technique would be useful for ultrasonic nondestructive evaluation of defects in composite and anisotropic plates, and for studying dynamic response of such plates to impact.

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Shpakivskyi, V. S., and T. S. Kuzmenko. "Integral theorems for the quaternionic G-monogenic mappings." Analele Universitatii "Ovidius" Constanta - Seria Matematica 24, no. 2 (2016): 271–81. http://dx.doi.org/10.1515/auom-2016-0042.

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Abstract In the paper [1] considered a new class of quaternionic mappings, so- called G-monogenic mappings. In this paper we prove analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, and the Cauchy integral formula for G-monogenic mappings.

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35

Feldmann, P., and S. W. Director. "Integrated circuit quality optimization using surface integrals." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 12, no. 12 (1993): 1868–79. http://dx.doi.org/10.1109/43.251150.

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Umul, Yusuf Z. "Rubinowicz transform of the MTPO surface integrals." Optics Communications 281, no. 23 (2008): 5641–46. http://dx.doi.org/10.1016/j.optcom.2008.08.008.

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37

Klees, Roland. "Numerical calculation of weakly singular surface integrals." Journal of Geodesy 70, no. 11 (1996): 781–97. http://dx.doi.org/10.1007/bf00867156.

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Matho, K. "Determining the Fermi Surface from Arpes Integrals." Acta Physica Polonica A 97, no. 1 (2000): 201–4. http://dx.doi.org/10.12693/aphyspola.97.201.

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Gray, L. J., J. M. Glaeser, and T. Kaplan. "Direct Evaluation of Hypersingular Galerkin Surface Integrals." SIAM Journal on Scientific Computing 25, no. 5 (2004): 1534–56. http://dx.doi.org/10.1137/s1064827502405999.

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40

Murphy, William C., and Thomas F. George. "Overlap integrals for atom-metal surface interactions." International Journal of Quantum Chemistry 28, no. 5 (1985): 631–39. http://dx.doi.org/10.1002/qua.560280509.

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41

Stankiewicz, Barbara. "Surface states – an effect of overlap integrals." physica status solidi (b) 134, no. 2 (1986): 691–97. http://dx.doi.org/10.1002/pssb.2221340228.

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Maas, Leo R. M. "On the surface area of an ellipsoid and related integrals of elliptic integrals." Journal of Computational and Applied Mathematics 51, no. 2 (1994): 237–49. http://dx.doi.org/10.1016/0377-0427(92)00009-x.

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43

Shamshiev, Fazliddin Tulaevich. "INTEGRABILITY OF THE EQUATIONS OF MOTION USING LOCAL INTEGRALS." Physical and mathematical science 3, no. 1 (2022): 5. https://doi.org/10.5281/zenodo.7198943.

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In our previous paper [1], potential is obtained that admit local integrals of motion for steadystate stellar systems. The concept of a local integral, in contrast to D.Lynden-Bell [2], as a separate isolated surface, was introduced by V.A.Antonov [3] for an arbitrarily given field. Here we have the possibility of finding a trajectory in the presence of a local integral, in contrast to the true integral of motion with a local integral it is not always possible to completely find trajectories. As an example, using [1] we present the result of one of the cases of integrability of the equations of motion for a space model. Key words: stellar dynamics, gravitational potential, integrals of motion.  

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44

Kholshevnikov, Konstantin V., and Vladimir B. Titov. "Minimal velocity surface in the restricted circular Three-Body-Problem." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 65, no. 4 (2020): 734–42. http://dx.doi.org/10.21638/spbu01.2020.413.

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In the framework of the restricted circular Three-Body-Problem, the concept of the minimum velocity surface S is introduced, which is a modification of the zero-velocity surface (Hill surface). The existence of Hill surface requires occurrence of the Jacobi integral. The minimum velocity surface, other than the Jacobi integral, requires conservation of the sector velocity of a zero-mass body in the projection on the plane of the main bodies motion. In other words, there must exist one of the three angular momentum integrals. It is shown that this integral exists for a dynamic system obtained after a single averaging of the original system by longitude of the main bodies. Properties of S are investigated. Here is the most significant. The set of possible motions of the zero-mass body bounded by the surface S is compact. As an example the surfaces S for four small moons of Pluto are considered in the framework of the averaged problem Pluto — Charon — small satellite. In all four cases, S represents a topological torus with small cross section, having a circumference in the plane of motion of the main bodies as the center line.

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45

Ren, Zhengyong, Yiyuan Zhong, Chaojian Chen, Jingtian Tang, and Kejia Pan. "Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order." GEOPHYSICS 83, no. 1 (2018): G1—G13. http://dx.doi.org/10.1190/geo2017-0219.1.

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A new singularity-free analytical formula has been developed for the gravity field of arbitrary 3D polyhedral mass bodies with horizontally and vertically varying density contrast using third-order polynomial functions. First, the observation sites are moved to the origin of the coordinate system. Then, the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies. Furthermore, singularity-free closed-form solutions are derived for these line integrals over the edges. Thus, the observation sites can be located inside, on, or outside the 3D distributions. A synthetic prismatic mass body is adopted to verify the accuracy and singularity-free property of our newly developed analytical expressions. Excellent agreements are obtained between our solutions and other published closed-form solutions with relative errors in the order of [Formula: see text] to [Formula: see text]. In addition, an octahedral model and a near-Earth asteroid model are used to verify the accuracy of the presented method for complicated target structures by comparing the results with those from a high-order Gaussian quadrature approach.

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Singh, Bijendra. "Simultaneous computation of gravity and magnetic anomalies resulting from a 2‐D object." GEOPHYSICS 67, no. 3 (2002): 801–6. http://dx.doi.org/10.1190/1.1484524.

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This paper presents a new algorithm for the simultaneous computation of gravity and magnetic anomalies resulting from an infinitely long (2‐D) body with an arbitrary polygonal cross‐section. With the assumption of uniform volume density and magnetization, the gravity or magnetic field may be expressed as the field resulting from an equivalent distribution of surface mass density or surface pole density, respectively, over the surface of the source body. The resulting surface integrals are reduced to new line integrals using Stokes' theorem. The components of the fields for each bounding surface are expressed in terms of a new line integral and the solid angle subtended by the surface at the point of observation. Since these analytical solutions are similar in form, a direct relation is derived between gravity and magnetic fields, which allows their simultaneous computation. Hence, the same computer program can be used to compute the gravity field, the magnetic field, or both fields simultaneously. This new approach will find wide applications in the joint inversion of potential field data, as it will make the numerical computations much faster.

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NATSIOPOULOS, GEORGIOS. "ALTERNATIVE TIME DOMAIN BOUNDARY INTEGRAL EQUATIONS FOR THE SCALAR WAVE EQUATION USING DIVERGENCE-FREE REGULARIZATION TERMS." Journal of Computational Acoustics 17, no. 02 (2009): 211–18. http://dx.doi.org/10.1142/s0218396x09003938.

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In this short note alternative time domain boundary integral equations (TDBIE) for the scalar wave equation are formulated on a surface enclosing a volume. The technique used follows the traditional approach of subtracting and adding back relevant Taylor expansion terms of the field variable, but does not restrict this to the surface patches that contain the singularity only. From the divergence-free property of the added-back integrands, together with an application of Stokes' theorem, it follows that the added-back terms can be evaluated using line integrals defined on a cut between the surface and a sphere whose radius increases with time. Moreover, after a certain time, the line integrals may be evaluated directly. The results provide additional insight into the theoretical formulations, and might be used to improve numerical implementations in terms of stability and accuracy.

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Latipova, Shahnoza Salim qizi. "POSITIVE AND NEGATIVE DIRECTIONS OF THE SURFACE INTEGRAL OF THE SECOND TYPE." Multidisciplinary Journal of Science and Technology 4, no. 3 (2024): 382–89. https://doi.org/10.5281/zenodo.10870014.

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Lin, Guobin. "The Dynamical and Kinetic Equations of Four-Five-Six-Wave Resonance for Ocean Surface Gravity Waves in Water with a Finite Depth." Symmetry 16, no. 5 (2024): 618. http://dx.doi.org/10.3390/sym16050618.

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Based on the Hamilton canonical equations for ocean surface waves with four-five-six-wave resonance conditions , the determinate dynamical equation of four-five-six-wave resonances for ocean surface gravity waves in water with a finite depth is established, thus leading to the elimination of the nonresonant second-, third-, fourth-, and fifth-order nonlinear terms though a suitable canonical transformation. The four kernels of the equation and the 18 coefficients of the transformation are expressed in explicit form in terms of the expansion coefficients of the gravity wave Hamiltonian in integral-power series in normal variables. The possibilities of the existence of integrals of motion for the wave momentum and the wave action are discussed, particularly the special integrals for the latter. For ocean surface capillary–gravity waves on a fluid with a finite depth, the sixth-order expansion coefficients of the Hamiltonian in integral-power series in normal variables are concretely provided, thus naturally including the classical fifth-order kinetic energy expansion coefficients given by Krasitskii.

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50

Marchand, Richard. "Surface charge and surface current densities at material boundaries." American Journal of Physics 92, no. 2 (2024): 158–60. http://dx.doi.org/10.1119/5.0164442.

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In electromagnetism, materials with a polarization density P→ or a magnetization density M→ are known to exhibit a bound surface charge density σb=P→·n̂ or a surface current density κ→b=M→×n̂, respectively, where n̂ is the unit vector perpendicular to the material boundary surface, directed outward. These expressions can be obtained from volume integrations for the electric potential V, or the magnetic vector potential A→, in which the integrals are restricted to the material volumes delimited by their respective boundaries. In that case, applying the divergence theorem leads to surface integrals on material boundaries and to the above-mentioned surface quantities. In this paper, a simple derivation is presented, which shows that both σb and κ→b are included in the expressions for the volume charge or current densities, provided that the divergence and curl operators are evaluated at the boundary so as to account for discontinuities at interfaces.

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