Volume Integral Method Research Articles

Construction and Verification of Two-dimensional Surface Hydraulic Calculation Model

Accurate calculation and prediction of the surface water flow process in the substation and surrounding area is of great significance for on-site flood control and disaster reduction of the substation. Aiming at the dynamic process of surface water flow, this paper simulated the spatial and temporal distribution characteristics of surface water flow by establishing a two-dimensional hydrodynamic calculation model. Based on the N-S equation, a finite volume integral method is used to discrete the governing equations. On this basis, a hydrodynamic model for two-dimensional numerical simulation of surface water flow is constructed, and the basic governing equations for two-dimensional surface shallow water movement are derived. Based on the Toce River physical model and water flow control experiment, the corresponding conceptual model is established. On this basis, the measured data are used to verify the constructed two-dimensional surface hydrodynamic calculation model. The results show that the model can better simulate the process of surface water flow and describe its temporal and spatial distribution characteristics. The research results will provide effective technical support for substation site flood warning and prevention.

The volatility of pollen extracts and their main constituents in aerosolized form via the integrated volume method (IVM) and the volatility basis set (VBS)

The volatility of pollen extracts and their main constituents in aerosolized form via the integrated volume method (IVM) and the volatility basis set (VBS)

Microseismic wavefield modelling in anisotropic elastic media using integral equation method

AbstractIn this paper, we present a frequency‐domain volume integral method to model the microseismic wavefield in heterogeneous anisotropic‐elastic media. The elastic wave equation is written as an integral equation of the Lippmann–Schwinger type, and the seismic source is represented as a general moment tensor. The actual medium is split into a background medium and a scattered medium. The background part of the displacement field is computed analytically, but the scattered part requires a numerical solution. The existing matrix‐based implementation of the integral equation is computationally inefficient to model the wavefield in three‐dimensional earth. An integral equation for the particle displacement is, hence, formulated in a matrix‐free manner through the application of the Fourier transform. The biconjugate gradient stabilized method is used to iteratively obtain the solution of this equation. The integral equation method is naturally target oriented, and it is not necessary to fully discretize the model. This is very helpful in the microseismic wavefield computation at receivers in the borehole in many cases; say, for example, we want to focus only on the fluid injection zone in the reservoir–overburden system and not on the whole subsurface region. Additionally, the integral equation system matrix has a low condition number. This provides us flexibility in the selection of the grid size, especially at low frequencies for given wave velocities. Considering all these factors, we apply the numerical scheme to three different models in order of increasing geological complexity. We obtain the elastic displacement fields corresponding to the different types of moment tensor sources, which prove the utility of this method in microseismic. The generated synthetic data are intended to be used in inversion for the microseismic source and model parameters.

Volume integral model for algebraic image reconstruction and computed tomography

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

An Air Gap Model for Magnetostatic Volume Integral Formulation

The presence of air gaps in magnetic modeling problems is well known. When the air gap is thin, its numerical treatment is complex because the integrals arising on the model lead to inaccuracy. This paper presents an air gap model for thin air gaps using the magnetostatic volume integral method for linear and nonlinear problems. The thin air gap is considered as a face region with a numerical treatment to improve the accuracy. An application to an academic case and a current transformer is reported to validate the model.

Low-rank compression techniques in integral methods for eddy currents problems

Volume integral methods for the solution of eddy current problems are very appealing in practice since only the conducting regions need to be meshed. However, they require the assembly and storage of a dense stiffness matrix. With the objective of cutting down assembly time and memory occupation, low-rank approximation techniques like the Adaptive Cross Approximation (ACA) have been considered a major breakthrough. Recently, the VINCO framework has been introduced to reduce significantly memory occupation and computational time thanks to a novel factorization of the dense stiffness matrix. The aim of this paper is introducing a new matrix compression technique enabled by the VINCO framework. We compare the performance of VINCO framework approaches with state-of-the-art alternatives in terms of memory occupation, computational time and accuracy by solving benchmark eddy current problems at increasing mesh sizes; the comparisons are carried out using both direct and iterative solvers. The results clearly indicate that the so-called VINCO-FAIME approach which exploits the Fast Multipole Method (FMM) has the best performance.

A computational package to compute the electrical resistivity tomography response for regular bodies immersed in a homogeneous half-space

A 3D forward algorithm is presented to model electrical resistivity tomography (ERT) measurements with the capability to use different electrode array configurations. We have implemented the algorithm as an open-source package with the assumption that the medium can be modeled as a homogeneous half-space with immersed bodies of different resistivities with regular geometry. This algorithm incorporates a novel numerical approximation by using the volume integral method, with the hypothesis of weak scattering, where the secondary currents generated in the bodies due to the main field are not considered. The algorithm also includes a new method called cubature to calculate volume integrals and the ERT response. In addition, the code has a module to display the results as color resistivity profiles. This method avoids the limitation of the complex geometry of the medium or even of the immersed bodies. The method is useful for planning a field survey, exploring the use of a specific electrode array, or implementing new arrays. The package, called the volume integral model of the electric response, is computationally efficient and can be easily ported to any operating system that supports MATLAB.

Dual Mesh Technique for Hydroacoustic Prediction of Cavitating Flows around a Three-Dimensional Bullet

Nowadays, hydroacoustics is of great importance. A major source of underwater noise is cavitation noise. One of the difficulties in cavitation noise prediction is its complex formation mechanism. In this paper, a cavitating bullet is taken as the research object, and detailed numerical simulation research of the local cavitation for this three-dimensional model is carried out. Ffowcs Williams and Hawkings equation (FW-H) is adopted. The quadrupole term is obtained by the direct volume integration method. Furthermore, dual mesh technique is adopted to achieve fast calculation. The results show that the dual mesh technology reduces the computational complexity of the nonlinear sound pressure significantly. With the appropriate acoustic grid size, the dual mesh method can ensure the same accuracy as the CFD grid. This provides a reliable way for the calculation of nonlinear sound pressure in the cavitation phenomenon.

Three-dimensional Analytical Modeling of Axial Flux Permanent Magnets Maglev Motor

The three-dimensional(3D) analytical model of the magnetic field in an Axial Flux Permanent Magnets Maglev Motor(AFPMMM) is proposed and investigated the influence of the structural parameters on electromagnetic characteristics. Firstly, the topology and working principle of the AFPMMM is introduced, and the model is transferred into a mathematical model in 3D cartesian coordinate. Then, the volume integral method and equivalent current sheets model is applied to find the 3D magnetic field distribution function of Halbach rotor. A unified form expression can be obtained by two dimensional discrete fourier transform (2-D DFT) is applied on the 3D magnetic field distribution function. Thirdly, the conductive and nonconductive regions of AFPMMM will be formulated by the second order vector potential (SOVP) to built the 3D analytic model. The expression of the lift force, torque and power losses was derived. Besides, the relationship between electromagnetic characteristics and structural parameters of the AFPMMM were analyzed based on 3D analytic model and validated using the 3D finite element analysis(FEA). Finally, the experiments based on a small scale prototype are carried out to verify the analytical results.

Near-field radiative heat transfer between irregularly shaped dielectric particles modeled with the discrete system Green's function method

Near-field radiative heat transfer (NFRHT) between irregularly shaped dielectric particles made of SiO2 and morphology characterized by Gaussian random spheres is studied. Particles are modeled using the discrete system Green's function (DSGF) approach, which is a volume integral numerical method based on fluctuational electrodynamics. This method is applicable to finite, three-dimensional objects, and all system interactions are defined independent of thermal excitation by a generalized system Green's function. The DSGF method is deemed suitable to model NFRHT between irregularly shaped particles after verification against the analytical solution for chains of two and three SiO2 spheres. The NFRHT results reveal that geometric irregularity in particles leads to a reduction of the total conductance from that of comparable perfect spheres at vacuum separation distances smaller than the particle size, a regime in which NFRHT is a surface phenomenon. At vacuum separation distances larger than the particle size, NFRHT becomes a volumetric process, and the total conductance between irregularly shaped particles converges to that of comparable perfect spheres. Spectral analysis reveals, however, that particle irregularity leads to damping and broadening of resonances at all separation distances, thereby highlighting the importance of the DSGF method for spectral engineering in the near field. The reduced spectral coherence when particle size is larger than the vacuum separation distance is attributed to coupling of surface phonon-polaritons within the randomly generated, distorted particle features. For particle size smaller than the vacuum separation distance, resonance broadening and damping is linked with the multiple localized surface phonon modes supported by the composite spherical harmonic morphologies of the Gaussian random spheres.

Totally volume integral of fluxes for discontinuous Galerkin method-two dimensional Euler equations

In this paper, the scheme of constructing high-order accurate totally volume discontinuous finite element method for the numerical solution of the 1D Euler equations is extended to 2D Euler equations on Cartesian meshes. In the present work, the boundary integral fluxes are transformed into volume integral by applying divergence theorem to the boundary integral of the Riemann fluxes. Therefore, the totally volume discontinuous finite element is independent on the boundary integral fluxes at the element boundaries as opposed to the classical discontinuous Galerkin method. The accuracy is obtained by applying high-order polynomial approximations within elements using the tensor product of Lagrange polynomial. For temporal integration, strong stability preserving Runge-Kutta method SSPRK (3, 3) is applied. The scheme is stabilized by using Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. For the spatial discretization, the polynomial of order 1and 2 are used, the shape function is constructed for the master (computational) element after applying the coordinate transformation for the physical domain, the transformation for the governing equations is performed to get it in the function of computational Cartesian, then the governing equations are put in conservative form. The numerical results of applying totally volume integral discontinuous Galerkin method for two-dimensional Euler equations presented in this paper show that the scheme is very accurate, fast, and effective even with shock appearance.

The Discrete Dipole Approximation: A Review

There are many methods for rigorously calculating electromagnetic diffraction by objects of arbitrary shape and permittivity. In this article, we will detail the discrete dipole approximation (DDA) which belongs to the class of volume integral methods. Starting from Maxwell’s equations, we will first present the principle of DDA as well as its theoretical and numerical aspects. Then, we will discuss the many developments that this method has undergone over time and the numerous applications that have been developed to transform DDA in a very versatile method. We conclude with a discussion of the strengths and weaknesses of the DDA and a description of the freely available DDA-based electromagnetic diffraction codes.

Electric Flux Density Learning Method for Solving 3-D Electromagnetic Scattering Problems

Inspired by a discretized formulation resulting from volume integral equation and method of moments, we propose an electric flux density learning method (EFDLM) using cascaded neural networks to solve 3-D electromagnetic (EM) scattering problems that involve lossless dielectric objects. The inputs of the proposed EFDLM consist of the contrast of the objects, the projections of incident field, and the first-order scattered field onto the testing functions, and the output is chosen as the normalized electric flux density. Analyses on the computational complexity, computation time, and memory usage of the EFDLM are conducted to fully understand its fundamental features. Numerical simulations clearly show that the proposed method outperforms black-box learning method, which chooses the contrast and incident field as its inputs and the total electric field as its output. It is also demonstrated that the EFDLM is able to solve the scattering problems of dielectric objects with higher contrasts by increasing the number of subnetworks. Further, the pros and cons of the proposed learning approach for solving EM scattering problems are discussed, where some caveats are provided to avoid using learning approaches in a black-box way.

Foundations of volume integral methods for eddy current problems

Integral methods for solving eddy current problems use Biot–Savart law to produce non-local constitutive relations that lead to fully populated generalized mass matrices, better known as inductance matrices. These formulations are appealing because—unlike standard Finite Element solutions—they avoid the generation of a mesh in the insulating regions.The aim of this paper is to alleviate the three main problems of volume integral methods. First, the computation of the inductance matrix elements is slow and also delicate because of the singularity in the integral equation. This paper introduces novel face basis functions that allow a much faster inductance matrix construction with respect to the standard one based on the Rao–Wilton–Glisson (RWG) or Raviart–Thomas (RT) basis functions. Second, our basis functions work for polyhedral elements formed by any number of faces (including prisms, hexahedra and pyramids), while producing the same results as RWG and RT basis functions for tetrahedral meshes.Third, the new basis functions allow to factorize the inductance matrix and to introduce a novel family of groundbreaking low-rank inductance matrix compression techniques that show several orders of magnitude improvement in memory occupation and computational effort than state-of-the-art alternatives, allowing to solve problems that otherwise cannot be faced.

A Quasi-Monte Carlo Volume Integration and Chebyshev Picard Iteration Method for Time-Parallel Nonlinear Seakeeping Computations

A Quasi-Monte Carlo Volume Integration and Chebyshev Picard Iteration Method for Time-Parallel Nonlinear Seakeeping Computations

A new benchmark problem for electromagnetic modelling of superconductors: the high-T c superconducting dynamo

The high-T c superconducting (HTS) dynamo is a promising device that can inject large DC supercurrents into a closed superconducting circuit. This is particularly attractive to energise HTS coils in NMR/MRI magnets and superconducting rotating machines without the need for connection to a power supply via current leads. It is only very recently that quantitatively accurate, predictive models have been developed which are capable of analysing HTS dynamos and explain their underlying physical mechanism. In this work, we propose to use the HTS dynamo as a new benchmark problem for the HTS modelling community. The benchmark geometry consists of a permanent magnet rotating past a stationary HTS coated-conductor wire in the open-circuit configuration, assuming for simplicity the 2D (infinitely long) case. Despite this geometric simplicity the solution is complex, comprising time-varying spatially-inhomogeneous currents and fields throughout the superconducting volume. In this work, this benchmark problem has been implemented using several different methods, including H-formulation-based methods, coupled H-A and T-A formulations, the Minimum Electromagnetic Entropy Production method, and integral equation and volume integral equation-based equivalent circuit methods. Each of these approaches show excellent qualitative and quantitative agreement for the open-circuit equivalent instantaneous voltage and the cumulative time-averaged equivalent voltage, as well as the current density and electric field distributions within the HTS wire at key positions during the magnet transit. Finally, a critical analysis and comparison of each of the modelling frameworks is presented, based on the following key metrics: number of mesh elements in the HTS wire, total number of mesh elements in the model, number of degrees of freedom, tolerance settings and the approximate time taken per cycle for each model. This benchmark and the results contained herein provide researchers with a suitable framework to validate, compare and optimise their own methods for modelling the HTS dynamo.

3D Semi-Analytical Modeling and Optimization of Fully HTS Ironless Axial Flux Electrical Machines

In this paper, a three-dimensional semi-analytical approach, based on the volume integral method, is developed to compute the forces and torque in fully high temperature superconducting (HTS) ironless axial flux machine (IAFM). The use of particular volumetric basis elements helps to speed up the computation of the magnetic field. The Maxwell stress tensor method is used for the force and torque calculations, where the electromagnetic state of the superconductor is taken into account. The developed model is then introduced into a multi objective optimization procedure, based on the genetic algorithms, to maximize the magnetic torque and minimize the superconducting wire length in the considered structure, while respecting the constraints linked to the physical and mechanical properties of the considered HTS tape. It is shown that the considered fully HTS machine has a higher torque density compared to conventional one.

Hydrodynamic Behavior in Solar Oil Heat Exchanger Ducts Fitted with Staggered Baffles and Fins

The attachment of turbulators, such as baffles, fins, ribs, bars, and blocks, inside the thermal solar receiver ducts, is among the most effective mechanisms for important thermal exchange by creating the turbulence, extending the trajectory of the flow, increasing the surface of heat exchange, forcing recycling cells, and hence a high thermal exchange. The solar finned and baffled heat exchangers are employed in a wide application interval, and it is important to examine the design of a duct for this configuration of the flow field and its effect on the heat transport phenomenon. In This study, dynamic field simulations are reported in horizontal rectangular form ducts, using three obstacles with oil HTF (heat transfer fluid). Two various finned and baffled duct configurations are treated, i.e., case (A) with one fin and two baffles, and case (B) with two fins and one baffle. Different hydrodynamic fields, i.e., X-velocity and Y-speed, as well as various X-velocity profiles in many flow stations, related to Re value, are analyzed. A computational approach is applied in order to simulate the oil flow, using finite volume (FV) integration method, SIMPLE discretization algorithm, QUICK interpolation scheme, Standard k-epsilon turbulence model, and ANSYS FLUENT 12.0 software. Simulation results reported an unstable flow structure, with powerful recycling cells, on the backsides of each fin and baffle, as a result of fluid detachment at their upper front sharp edges, in both studied models (A and B). As expected, the first duct model, i.e., Case A, has better X- and Y-velocity values, due to its large recirculation regions. In This paper, many physical phenomena, such as the turbulence, instability, flow separation, and the appearance of reverse secondary currents, are reported. As its data confirmed by many previous numerical and experimental results, the suggested new models of finned and baffled heat exchangers filled with high thermal conductivity oil, allow an improvement in the dynamic thermal-energy behavior of many thermal devices such as flat plate solar collectors.

Volatility parameters of secondary organic aerosol components determined using a thermal denuder

In this study, the characterization of a new thermal denuder and its application to determine the volatility parameters of important components of secondary organic aerosol (SOA) are presented in detail. The temperature ramping rate to achieve a set value was ~14.3 °C min−1 in the range of 25–150 °C. Particle loss during the penetration through the denuder was negligible in the temperature and effective residence time (RT) ranges of 25 °C–60 °C and 12.5 s–50 s, respectively. In addition, a vapor-phase aerosol reached equilibrium with the gas phase at an effective RT of approximately 24 s. Furthermore, volatility profile data generated using the thermal denuder with the help of a scanning particle sizer were used to determine the saturation pressure (P0sat) at 25 °C and enthalpy of vaporization (ΔHv) of typical biogenic and anthropogenic SOA components by the integrated volume method. Determined ΔHv and P0sat values for succinic acid, adipic acid, pimelic acid, phthalic acid, pinic acid, and ketopinic acid were 86.6 ± 2.5 kJ mol−1, 146.1 kJ mol−1, 139.2 kJ mol−1, 135.9 kJ mol−1, 93.8 ± 1.8 kJ mol−1, and 136.6 kJ mol−1 and 4.70 ± 0.7 x 10−5 Pa, 1.98 x 10−5 Pa, 8.91 x 10−5 Pa, 8.50 x 10−5 Pa, 1.61 ± 0.5 x 10−4 Pa, and 5.81 x 10−5 Pa, respectively. These values were comparable with the literature data.

An Integral Face Formulation for Thin Non-Conductive Magnetic Regions

This paper deals with the use of the volume integral method to compute the magnetic anomaly created by a non-conductive ferromagnetic thin shell placed in a static inductor field. An original facet integral formulation considering the magnetic induction as unknowns is presented. The use of thin shell element assumption leads to a surface mesh decreasing highly the number of elements so the computation cost. The method is very performing in terms of speed and accuracy.