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Journal articles on the topic 'Volume Surface Integral Equation'

Author: Grafiati

Published: 6 September 2023

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1

Gomez, Luis J., Abdulkadir C. Yucel, and Eric Michielssen. "Volume-Surface Combined Field Integral Equation for Plasma Scatterers." IEEE Antennas and Wireless Propagation Letters 14 (December 2015): 1064–67. http://dx.doi.org/10.1109/lawp.2015.2390533.

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2

Kaplan, Meydan, and Yaniv Brick. "A fast solver framework for acoustic hybrid integral equations." Journal of the Acoustical Society of America 152, no. 4 (October 2022): A119. http://dx.doi.org/10.1121/10.0015743.

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Reliable modeling of the scattering by acoustically large and geometrically complex objects can be achieved by means of subdomain-dependent problem formulation and a numerically rigorous solution. While the objects’ inhomogeneity has driven the development of differential equation formulations and solvers, integral equation formulations, where the object’s background is modeled via a Green’s function, are advantageous for unbounded domains. In the hybrid integral equations approach (Usner et al., 2006), the interaction of separate subdomains with external fields is described by pertinent integral equations. Their Galerkin discretization leads to a dense blocked stiffness matrix. The development of compressed representations of the matrix, which are necessary for the treatment of large systems, becomes non-trivial due to the multitude of integral equation kernels and the different geometrical and physical characteristic of the subdomains. As part of the development of a fast hybrid integral equation solver framework, we consider the case of objects composed of large inhomogeneous volumes, modeled as incompressible fluids, and of simplified solids, modeled via surface integral equations. A hybrid integral equation formulation is derived and solved numerically. The iterative solution is accelerated by employing the butterfly-compressed hierarchical representation of the stiffness matrix, recently used for acoustic volume integral equations (Kaplan, 2022).

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3

Remis, R., and E. Charbon. "An Electric Field Volume Integral Equation Approach to Simulate Surface Plasmon Polaritons." Advanced Electromagnetics 2, no. 1 (February 16, 2013): 15. http://dx.doi.org/10.7716/aem.v2i1.23.

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In this paper we present an electric field volume integral equation approach to simulate surface plasmon propagation along metal/dielectric interfaces. Metallic objects embedded in homogeneous dielectric media are considered. Starting point is a so-called weak-form of the electric field integral equation. This form is discretized on a uniform tensor-product grid resulting in a system matrix whose action on a vector can be computed via the fast Fourier transform. The GMRES iterative solver is used to solve the discretized set of equations and numerical examples, illustrating surface plasmon propagation, are presented. The convergence rate of GMRES is discussed in terms of the spectrum of the system matrix and through numerical experiments we show how the eigenvalues of the discretized volume scattering operator are related to plasmon propagation and the medium parameters of a metallic object.

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4

Usner, B. C., K. Sertel, and J. L. Volakis. "Doubly periodic volume–surface integral equation formulation for modelling metamaterials." IET Microwaves, Antennas & Propagation 1, no. 1 (2007): 150. http://dx.doi.org/10.1049/iet-map:20050344.

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5

Ewe, Wei-Bin, Hong-Son Chu, and Er-Ping Li. "Volume integral equation analysis of surface plasmon resonance of nanoparticles." Optics Express 15, no. 26 (2007): 18200. http://dx.doi.org/10.1364/oe.15.018200.

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6

Amundsen, Lasse. "The propagator matrix related to the Kirchhoff‐Helmholtz integral in inverse wavefield extrapolation." GEOPHYSICS 59, no. 12 (December 1994): 1902–10. http://dx.doi.org/10.1190/1.1443577.

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The Kirchhoff‐Helmholtz formula for the wavefield inside a closed surface surrounding a volume is most commonly given as a surface integral over the field and its normal derivative, given the Green’s function of the problem. In this case the source point of the Green’s function, or the observation point, is located inside the volume enclosed by the surface. However, when locating the observation point outside the closed surface, the Kirchhoff‐Helmholtz formula constitutes a functional relationship between the field and its normal derivative on the surface, and thereby defines an integral equation for the fields. By dividing the closed surface into two parts, one being identical to the (infinite) data measurement surface and the other identical to the (infinite) surface onto which we want to extrapolate the data, the solution of the Kirchhoff‐Helmholtz integral equation mathematically gives exact inverse extrapolation of the field when constructing a Green’s function that generates either a null pressure field or a null normal gradient of the pressure field on the latter surface. In the case when the surfaces are plane and horizontal and the medium parameters are constant between the surfaces, analysis in the wavenumber domain shows that the Kirchhoff‐Helmholtz integral equation is equivalent to the Thomson‐Haskell acoustic propagator matrix method. When the medium parameters have smooth vertical gradients, the Kirchhoff‐Helmholtz integral equation in the high‐frequency approximation is equivalent to the WKBJ propagator matrix method, which also is equivalent to the extrapolation method denoted by extrapolation by analytic continuation.

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7

Roco, M. C., and S. Mahadevan. "Scale-up Technique of Slurry Pipelines—Part 2: Numerical Integration." Journal of Energy Resources Technology 108, no. 4 (December 1, 1986): 278–85. http://dx.doi.org/10.1115/1.3231277.

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A kinetic energy turbulence model has been proposed for the computer flow simulation and scale-up of slurry pipelines (in Part 1 [1]). The numerical integration is performed by using a modified finite volume technique, with application to high-convective two-phase flows in two and three dimensions (in Part 2). The mixture kinetic energy and eddy viscosity turbulence models are compared. The one-equation eddy-viscosity turbulence model (εt - model) is formulated in Part 2 and applied for the multi-species particle slurry flow in cylindrical pipes. A modified finite volume technique is proposed for high convective transport equations, for one and two-phase flows. The integral formulation per volume yields surface and volume integrals, that are stored and counted only by interfaces using a multidimensional approach. The nonlinear distributions in volumes and on interfaces are approximated employing the derivatives in the normal and tangent directions to the bounding surfaces. Linear, analytical (upwind) and logarithmic laws of interpolations are considered for internal flows. The numerical approach was tested with good results for transport equations of momentum and various contaminants (solid particles, temperature, eddy-viscosity) in pipes. Experimental data for one and two-phase flows are compared to the integral finite volume predictions. The proposed finite volume technique can economically simulate complex flow situations encountered in the slurry pipeline scale-up applications.

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8

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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9

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 (June 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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10

Jin, J. M., V. V. Liepa, and C. T. Tai. "A Volume-Surface Integral Equation for Electromagnetic Scattering by Inhomogeneous Cylinders." Journal of Electromagnetic Waves and Applications 2, no. 5-6 (January 1988): 573–88. http://dx.doi.org/10.1163/156939388x00170.

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11

Branda˜o, Mauri´cio P. "Mixed Volume Boundary Element Approach for Aerodynamics." Applied Mechanics Reviews 44, no. 11S (November 1, 1991): S36—S45. http://dx.doi.org/10.1115/1.3121370.

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A review is presented of theoretical methods in aerodynamics and aeroacoustics which lead to the present approach. A formulation is developed for the analysis of three-dimensional, unsteady, and viscous flows. The integral solution to the Ffowcs Williams and Hawkings equation mixes surface and volume contributions. The surface terms are treated following the traditional boundary element technique. Special care is taken in revealing the hidden singularities of curved surfaces. The volume terms are treated following a cell-based approach. The concept of finite-part of singular integrals is used to interpret these terms. A computational procedure is proposed to determine simultaneously the pressure and velocity fields around bodies moving in incompressible and compressible fluids. Finally, current and future directions of research regarding the present method are pointed out and discussed.

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12

Čiupaila, Regimantas, Mifodijus Sapagovas, and Olga Štikonienė. "Numerical solution of nonlinear elliptic equation with nonlocal condition." Nonlinear Analysis: Modelling and Control 18, no. 4 (October 25, 2013): 412–26. http://dx.doi.org/10.15388/na.18.4.13970.

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Two iterative methods are considered for the system of difference equations approximating two-dimensional nonlinear elliptic equation with the nonlocal integral condition. Motivation and possible applications of the problem present in the paper coincide with the small volume problems in hydrodynamics. The differential problem considered in the article is some generalization of the boundary value problem for minimal surface equation.

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13

Wang, Zhi Li, and Yan Fen Geng. "FVCOM2D: A Two Dimensional Semi-Implicit Finite Volume Free-Surface Ocean Model." Applied Mechanics and Materials 170-173 (May 2012): 2248–55. http://dx.doi.org/10.4028/www.scientific.net/amm.170-173.2248.

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A two dimensional semi-implicit finite volume free-surface ocean model(FVFOM2D) based on unstructured C-grid is built, in which the momentum equation is discretized on the faces of each cell, the continuity equation is discretized on the cell. The model is discretized by semi-implicit finite volume method, in that the free-surface is semi-implicit and the bottom friction is implicit, thereby removing stability limitations associated with the surface gravity wave and friction. The remaining terms in the momentum equations are discretized explicitly by integral finite volume method and second order Adams-Bashforth method. The performance of the present model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in Pear River Estuary.

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14

Zhong, Siyang, and Xin Zhang. "A sound extrapolation method for aeroacoustics far-field prediction in presence of vortical waves." Journal of Fluid Mechanics 820 (May 8, 2017): 424–50. http://dx.doi.org/10.1017/jfm.2017.219.

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Off-surface integral solutions to an inhomogeneous wave equation based on acoustic analogy could suffer from spurious wave contamination when volume integrals are ignored for computation efficiency and vortical/turbulent gusts are convected across the integration surfaces, leading to erroneous far-field directivity predictions. Vortical gusts often exist in aerodynamic flows and it is inevitable their effects are present on the integration surface. In this work, we propose a new sound extrapolation method for acoustic far-field directivity prediction in the presence of vortical gusts, which overcomes the deficiencies in the existing methods. The Euler equations are rearranged to an alternative form in terms of fluctuation variables that contains the possible acoustical and vortical waves. Then the equations are manipulated to an inhomogeneous wave equation with source terms corresponding to surface and volume integrals. With the new formulation, spurious monopole and dipole noise produced by vortical gusts can be suppressed on account of the solenoidal property of the vortical waves and a simple convection process. It is therefore valid to ignore the volume integrals and preserve the sound properties. The resulting new acoustic inhomogeneous convected wave equations could be solved by means of the Green’s function method. Validation and verification cases are investigated, and the proposed method shows a capacity of accurate sound prediction for these cases. The new method is also applied to the challenging airfoil leading edge noise problems by injecting vortical waves into the computational domain and performing aeroacoustic studies at both subsonic and transonic speeds. In the case of a transonic airfoil leading edge noise problem, shocks are present on the airfoil surface. Good agreements of the directivity patterns are obtained compared with direct computation results.

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15

Maulana, Ahmat Rif’an, Mahmud Yunus, and Dwi Ratna Sulistyaningrum. "The Constructions of Egg-Shaped Surface Equations using Hugelschaffer’s Egg-Shaped Curve." Indonesian Journal of Physics 26, no. 2 (December 29, 2015): 26–30. http://dx.doi.org/10.5614/itb.ijp.2015.26.2.2.

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Hugelschaffer’s egg-shaped curve is egg-shaped curve that is constructed by two non-concentric circles using Newton’s transformation known as hyperbolism. This study has goals to construct the egg-shaped surface equations using Hugelschafer’s egg-shaped curve that is rotated on x-axis, y-axis and z-axis; to get the volume formula of the egg-shaped solid and the egg-shaped surface area and also to visualize the egg-shaped surface equations using GeoGebra. Hugelschaffer’s egg-shaped curve is selected because its equation is simple. The procedures of the construction of the egg-shaped surface equations are done by drawing the curve on xy-plane and xz-plane, then it is rotated on axes of the coordinate. Whereas, the volume formula of the egg-shaped solid is gotten by using the disk method of the volume integral. The egg-shaped surface area is attained by using the integral of surface area. Visualisation of the egg-shaped surface equations are done by choosing vary of parameter values of the equations that aims to know the effect of the parameter values with the shaped surface.

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16

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 (September 1, 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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17

Yla-Oijala, Pasi, Johannes Markkanen, Seppo Jarvenpaa, and Sami P. Kiminki. "SURFACE AND VOLUME INTEGRAL EQUATION METHODS FOR TIME-HARMONIC SOLUTIONS OF MAXWELL'S EQUATIONS (Invited Paper)." Progress In Electromagnetics Research 149 (2014): 15–44. http://dx.doi.org/10.2528/pier14070105.

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18

Ye, Yuhuang, Jiade Yuan, and Kaixiong Su. "A Volume-Surface Integral Equation Solver for Radiation from Microstrip Antenna on Anisotropic Substrate." International Journal of Antennas and Propagation 2012 (2012): 1–4. http://dx.doi.org/10.1155/2012/120208.

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A volume-surface integral equation (VSIE) solver is presented for the calculation of electromagnetic radiation from arbitrary shaped microstrip antenna on anisotropic substrate. The method of moments (MoM) is used to convert the integral equation into a matrix equation, where the equivalent volume current and surface current are expanded into a finite series of SWG and RWG basis function, respectively. A simple strip model is incorporated in the VSIE to simplify the analysis of the probe-fed microstrip antenna. The present approach is sufficiently versatile in handling microstrip antenna with arbitrary shaped anisotropic dielectric substrate. Numerical results indicate the reliability and accuracy of the proposed method.

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19

Liu, Jinbo, Hongyang Chen, Hui Zhang, Jin Yuan, and Zengrui Li. "A Modified Hybrid Integral Equation to Electromagnetic Scattering from Composite PEC-Dielectric Objects Containing Closed-Open PEC Junctions." Applied Computational Electromagnetics Society 36, no. 6 (August 6, 2021): 642–49. http://dx.doi.org/10.47037/2020.aces.j.360604.

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To efficiently analyze the electromagnetic scattering from composite perfect electric conductor (PEC)-dielectric objects with coexisting closed-open PEC junctions, a modified hybrid integral equation (HIE) is established as the surface integral equation (SIE) part of the volume surface integral equation (VSIE), which employs the combined field integral equation (CFIE) and the electric field integral equation (EFIE) on the closed and open PEC surfaces, respectively. Different from the traditional HIE modeled for the objects whose closed and open PEC surfaces are strictly separate, the modified HIE can be applied to the objects containing closed-open junctions. A matrix equation is obtained by using the Galerkin’s method of moments (MoM), which is augmented with the spherical harmonics expansion-based multilevel fast multipole algorithm (SE-MLFMA), improved by the mixed-potential representation and the triangle/tetrahedron-based grouping scheme. Because in the improved SE-MLFMA, the memory usage for storing the radiation patterns of basis functions is independent of the SIE type in the VSIE, it is highly appropriate for the fast solution of the VSIE that contains the HIE. Various numerical experiments demonstrate that during the calculation of composite objects containing closed-open PEC junctions, the application of the modified HIE in the VSIE can give reliable results with fast convergence speed.

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20

Li, Ya-Nan, Hai-Ying Yao, and Le-Wei Li. "A Fast Volume-surface Integral Equation Solver for Scattering Properties of NIMs." PIERS Online 3, no. 3 (2007): 273–77. http://dx.doi.org/10.2529/piers061003050010.

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21

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 (April 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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22

Deng, Xiaoqiao, Chang Qing Gu, Bingzheng Xu, and Zhuo Li. "A FAST VOLUME-SURFACE INTEGRAL EQUATION SOLVER FOR SCATTERING FROM HIGH-CONTRAST MATERIALS." Progress In Electromagnetics Research M 27 (2012): 83–95. http://dx.doi.org/10.2528/pierm12092902.

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23

Usner, B. C., K. Sertel, M. A. Carr, and J. L. Volakis. "Generalized Volume-Surface Integral Equation for Modeling Inhomogeneities Within High Contrast Composite Structures." IEEE Transactions on Antennas and Propagation 54, no. 1 (January 2006): 68–75. http://dx.doi.org/10.1109/tap.2005.861579.

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24

Gomez, Luis J., Abdulkadir C. Yucel, and Eric Michielssen. "Low-Frequency Stable Internally Combined Volume-Surface Integral Equation for High-Contrast Scatterers." IEEE Antennas and Wireless Propagation Letters 14 (2015): 1423–26. http://dx.doi.org/10.1109/lawp.2015.2410290.

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25

Hu, Jun, Yin Li, Xiang Feng, and Zaiping Nie. "Non-conformal geometry discretization scheme for hybrid volume and surface integral equation method." International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 25, no. 5-6 (June 27, 2012): 573–86. http://dx.doi.org/10.1002/jnm.1852.

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26

Liu, Jinbo, Jiming Song, Hui Zhang, and Zengrui Li. "Solving the Surface Current Distribution for Open PEC-Dielectric Objects Using the Volume Surface Integral Equation." IEEE Antennas and Wireless Propagation Letters 21, no. 1 (January 2022): 89–93. http://dx.doi.org/10.1109/lawp.2021.3118676.

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27

Yu, Chun, and Cai-Cheng Lu. "Analysis of finite and curved frequency-selective surfaces using the hybrid volume-surface integral equation approach." Microwave and Optical Technology Letters 45, no. 2 (2005): 107–12. http://dx.doi.org/10.1002/mop.20738.

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28

Nadobny, J., P. Wust, M. Seebass, P. Deuflhard, and R. Felix. "A volume-surface integral equation method for solving Maxwell's equations in electrically inhomogeneous media using tetrahedral grids." IEEE Transactions on Microwave Theory and Techniques 44, no. 4 (April 1996): 543–54. http://dx.doi.org/10.1109/22.491022.

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29

Chen, Jia Qi, and Yue Yuan Zhang. "Hybrid Accelerated Method via Volume-Surface Integral Equation for Efficient Calculation of Monostatic RCS." Applied Mechanics and Materials 719-720 (January 2015): 881–85. http://dx.doi.org/10.4028/www.scientific.net/amm.719-720.881.

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A novel efficient hybrid accelerated method is proposed for the fast analysis of the monostatic electromagnetic scattering problems arising from volume-surface integral equations (VSIE) formulation. In the first step, by utilizing the low rank property, several largest eigenvalues and corresponding eigenvectors of the multiple right hand sides can be computed and saved efficiently by adaptive cross approximation (ACA) algorithm. The iterative solution of linear equations is required at these principle eigenvectors. Compared with solving linear equations at each angle repeatedly, the proposed method is able to greatly reduce the number of equations. In the second step, a disturbed symmetric successive over-relaxation (D-SSOR) preconditioner is constructed to speed up the convergence rate of iterative methods. Numerical results demonstrate that the present method can reduce the computational time significantly for monostatic VSIE calculation with high accuracy.

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30

Hodges, Ben R. "Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage." Hydrology and Earth System Sciences 23, no. 3 (March 7, 2019): 1281–304. http://dx.doi.org/10.5194/hess-23-1281-2019.

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Abstract. New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

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31

Ding, Da-zhi, E. K. N. Yung, Daoxiang Wang, and Rushan Chen. "Efficient Analysis of Periodic Structures with Arbitrary Shape Using Volume-surface Integral Equation Method." PIERS Online 1, no. 6 (2005): 677–80. http://dx.doi.org/10.2529/piers050119222455.

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32

Xiao-Chun Nie, Ning Yuan, L. W. Li, Y. B. Gan, and Tat Soon Yeo. "A fast volume-surface integral equation solver for scattering from composite conducting-dielectric objects." IEEE Transactions on Antennas and Propagation 53, no. 2 (February 2005): 818–24. http://dx.doi.org/10.1109/tap.2004.841323.

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33

Monin, Maxime Y., Lyes Rahmouni, Adrien Merlini, and Francesco P. Andriulli. "A Hybrid Volume-Surface-Wire Integral Equation for the Anisotropic Forward Problem in Electroencephalography." IEEE Journal of Electromagnetics, RF and Microwaves in Medicine and Biology 4, no. 4 (December 2020): 286–93. http://dx.doi.org/10.1109/jerm.2020.2966121.

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34

Sarkar, T. K., and E. Arvas. "An integral equation approach to the analysis of finite microstrip antennas: volume/surface formulation." IEEE Transactions on Antennas and Propagation 38, no. 3 (March 1990): 305–12. http://dx.doi.org/10.1109/8.52238.

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35

Li, Mengmeng, Tao Zhuang, and Rushan Chen. "Volume integral equation equivalence principle algorithm domain decomposition with body of revolution equivalence surface." IET Microwaves, Antennas & Propagation 12, no. 3 (January 9, 2018): 375–79. http://dx.doi.org/10.1049/iet-map.2017.0717.

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36

Menshov, Anton, and Vladimir I. Okhmatovski. "Surface–Volume–Surface Electric Field Integral Equation for Magneto-Quasi-Static Analysis of Complex 3-D Interconnects." IEEE Transactions on Microwave Theory and Techniques 62, no. 11 (November 2014): 2563–73. http://dx.doi.org/10.1109/tmtt.2014.2360838.

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37

Svanadze, Merab. "Steady vibration problems in the coupled linear theory of porous elastic solids." Mathematics and Mechanics of Solids 25, no. 3 (November 28, 2019): 768–90. http://dx.doi.org/10.1177/1081286519888970.

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This paper concerns the coupled linear theory of elasticity for isotropic porous materials. In this theory the coupled phenomena of the concepts of Darcy’s law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions, and its basic properties are presented. The radiation conditions are established and Green’s identities are obtained. The uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single layer and double layer) and volume potentials are constructed and the basic properties of these potentials are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

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38

Bhattacharya, A. K., and N. L. Arora. "A hybrid integral equation finite volume scheme for transonic potential flow about complex configurations." Aeronautical Journal 98, no. 972 (February 1994): 35–48. http://dx.doi.org/10.1017/s0001924000050193.

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AbstractA hybrid integral equation finite volume scheme has been developed for the calculation of transonic potential flow about complex configurations. A new technique has been used for evaluating the potential values in the field cells intersecting the body surface panels. These potential values then serve as Dirichlet boundary conditions for computing the potentials in the field by a finite volume Successive Line Over Relaxation (SLOR) scheme. In this approach there is no need to evaluate the potentials anywhere in the field by direct application of Green's third identity, thus significantly reducing computer processing time and storage requirement, while improving accuracy of surface pressure prediction and shock capture, as results indicate. The capability of tackling additional complex geometry with ease, the primary advantage of the integral equation approach, is demonstrated by using the same field grid for wing-alone and wing-body combination cases, while maintaining the solution accuracy.

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39

He, Yuan, Jian Feng Li, Xiao Jun Jing, and Mei Song Tong. "Fast Solution of Volume–Surface Integral Equations for Multiscale Structures." IEEE Transactions on Antennas and Propagation 67, no. 12 (December 2019): 7649–54. http://dx.doi.org/10.1109/tap.2019.2943321.

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40

Vavrukh, M. V., and D. V. Dzikovskyi. "Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n=0 and n=1." Mathematical Modeling and Computing 8, no. 2 (2021): 338–58. http://dx.doi.org/10.23939/mmc2021.02.338.

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Calculations of characteristics of stars with axial rotation in the frame of polytropic model are based on the solution of mechanical equilibrium equation – differential equation of second order in partial derivatives. Different variants of approximate determinations of integration constants are based on traditional in the theory of stellar surface approximation, namely continuity of gravitational potential in the surface vicinity. We proposed a new approach, in which we used simultaneously differential and integral forms of equilibrium equations. This is a closed system and allows us to define in self-consistent way integration constants, the polytrope surface shape and distribution of matter over volume of a star. With the examples of polytropes n=0 and n=1, we established the existence of two rotation modes (with small and large eccentricities). It is proved that the polytrope surface is the surface of homogeneous rotational ellipsoid for the case n=0. The polytrope characteristics with n=1 in different approximations were calculated as the functions of angular velocity. For the first time it has been calculated the deviation of polytrope surface at fixed value of angular velocity from the surface of associated rotational ellipsoid.

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41

He, Mang, Jinbo Liu, Binbin Wang, Chuanfang Zhang, and Houjun Sun. "On the Use of Continuity Condition in the Fast Solution of Volume-Surface Integral Equation." IEEE Antennas and Wireless Propagation Letters 16 (2017): 625–28. http://dx.doi.org/10.1109/lawp.2016.2594206.

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42

Yucel, Abdulkadir C., Luis J. Gomez, and Eric Michielssen. "Internally Combined Volume-Surface Integral Equation for EM Analysis of Inhomogeneous Negative Permittivity Plasma Scatterers." IEEE Transactions on Antennas and Propagation 66, no. 4 (April 2018): 1903–13. http://dx.doi.org/10.1109/tap.2018.2800638.

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43

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 (July 2, 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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44

Pukalskyi, I., and I. Luste. "OPTIMAL CONTROL IN THE MULTIPOINT BOUNDARY VALUE PROBLEM FOR 2B-PARABOLIC EQUATIONS." Bukovinian Mathematical Journal 10, no. 1 (2022): 110–19. http://dx.doi.org/10.31861/bmj2022.01.10.

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The potential theory method was used to study the existence of a solution of a multi- point boundary value problem for a 2b-parabolic equation. Using the Green’s function of a homogeneous boundary value problem for a 2b-parabolic equation, the integral Fredholm equation of the second kind is placed in accordance with the multipoint boundary value problem. Taking into account the constraints on the coeﬃcients of the nonlocal condition and using the sequential approximation method, an integrated image of the solution of the nonlocal problem at the initial moment of time and its estimation in the Holder spaces are found. Estimates of the solution of a nonlocal multipoint boundary value problem at ﬁxed moments of time given in a nonlocal condition are found by means of estimates of the components of the Green’s function of the general boundary value problem for the 2b-parabolic equation. Taking into account the obtained estimates and constraints on coeﬃcients in multipoint problem, estimates of the solution of the multipoint problem for the 2b-parabolic equations and its derivatives in Holder spaces are established. In addition, the uniqueness and integral image of the solution of the general multipoint problem for 2b-parabolic equations is justiﬁed. The obtained result is applied to the study of the optimal system control problem described by the general multipoint boundary value problem for 2b-parabolic equations. The case of simultaneous internal, initial and boundary value control of solutions to a multipoint parabolic boundary value problem is considered. The quality criterion is deﬁned by the sum of volume and surface integrals. The necessary and suﬃcient conditions for the existence of an optimal solution of the system described by the general multipoint boundary value problem for 2b-parabolic equations with limited internal, initial and boundary value control are established.

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Wünsche, M., Jan Sladek, Vladimir Sladek, and S. Hrcek. "Computation of Effective Material Properties in Smart Composite Materials by a Symmetric Galerkin BEM." Key Engineering Materials 665 (September 2015): 9–12. http://dx.doi.org/10.4028/www.scientific.net/kem.665.9.

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In this paper, the symmetric Galerkin boundary element method (SGBEM) will be developed and applied for boundary value problems with layered and fiber reinforced piezoelectric representative volume elements (RVE) and real macroscopic structures. Mechanical and electric loadings are considered to determine the effective material properties. For this purpose, the resulting boundary value problem is formulated as boundary integral equations (BIEs). The Galerkin method is applied for the spatial discretization of the boundary to solve the BIEs numerically. The required surface derivatives of the generalized displacements are computed directly with a boundary integral equation. Numerical examples will be presented and discussed to show the efficiency of the present SGBEM and the influence of the fiber variation on the effective material properties.

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46

Krstajic, B., Z. Andelic, S. Milojkovic, S. Babic, and S. Salon. "Nonlinear 3D magnetostatic field calculation by the integral equation method with surface and volume magnetic charges." IEEE Transactions on Magnetics 28, no. 2 (March 1992): 1088–91. http://dx.doi.org/10.1109/20.123871.

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47

Yuan, N., T. S. Yeo, X. C. Nie, and L. W. Li. "RCS Computation of Composite Conducting-Dielectric Objects with Junctions using the Hybrid Volume-Surface Integral Equation." Journal of Electromagnetic Waves and Applications 19, no. 1 (January 2005): 19–36. http://dx.doi.org/10.1163/1569393052955107.

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48

Hu, Y. L., and R. S. Chen. "Analysis of Scattering From Composite Conducting Dispersive Dielectric Objects by Time-Domain Volume-Surface Integral Equation." IEEE Transactions on Antennas and Propagation 64, no. 5 (May 2016): 1984–89. http://dx.doi.org/10.1109/tap.2016.2535107.

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49

BUELER, ED. "Stable finite volume element schemes for the shallow-ice approximation." Journal of Glaciology 62, no. 232 (March 18, 2016): 230–42. http://dx.doi.org/10.1017/jog.2015.3.

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ABSTRACTThe isothermal, non-sliding shallow-ice approximation, combined with mass conservation, is a fundamental model for ice-sheet and glacier flow. It determines the ice extent, geometry and velocity by the solution of a free-boundary problem. In this paper, the steady-state form of this problem is solved directly, without time-stepping, thereby demonstrating a fully implicit scheme with no stability restrictions. The classical Mahaffy (1976) finite difference method is first reinterpreted as a ‘finite volume element’ scheme that has both an everywhere-defined approximate thickness function and an approximation of the conservation equation in flux integral form. From this reinterpretation an improved scheme is built by using better quadrature in the integral and upwinding on that part of the flux which is proportional to the bed gradient. The discrete equations are then solved by a parallel Newton scheme which respects the constraint that ice thickness is non-negative. The results show good accuracy on both flat-bed and bedrock-step exact solutions. The method is then applied at high resolution to model the steady-state geometry of the Greenland ice sheet, using only bedrock elevation and present-day surface mass balance as input data.

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50

Tuan, N. A., and Y. C. Shiah. "Bem Study of 3D Heat Conduction in Multiply Adjoined Anisotropic Media with Quadratic Domain Heat Generation." Journal of Mechanics 35, no. 02 (January 3, 2019): 225–31. http://dx.doi.org/10.1017/jmech.2018.47.

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ABSTRACTIn engineering, it is quite often to have applications of the heat transfer of conduction having domain heat generation present inside. The paper aims to present boundary element formulations for analyzing the three-dimensional heat-conduction in dissimilarly bonded anisotropic media involving quadratic volume heat sources. In this paper, the additional volume integral present in the boundary integral equation is analytically transformed to the boundary surface for the volume heat sources modeled by quadratic functions. The technique of domain-mapping is employed to treat the three-dimensional anisotropic heat conduction in multiply adjoined media with proper interfacial conditions provided. For showing our successful implementation, a few example cases are analyzed with verification of independent analyses by the finite element method.

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