Line Integral Equations Research Articles

A slender body theory for the motion of special Cosserat filaments in Stokes flow

The motion of filament-like structures in fluid media has been a topic of interest since long. In this regard, a well known slender body theory exists, wherein the fluid flow is assumed to be Stokesian while the filament is modeled as a Kirchhoff rod which can bend and twist but remains inextensible and unshearable. In this work, we relax the inextensibility and unshearability constraints on filaments, i.e., the filament is modeled as a special Cosserat rod. Starting with the boundary integral formulation of Stokes flow involving the filament’s surface velocity and fluid traction that acts on the filament surface, the method of matched asymptotic expansion is used to first obtain a leading-order representation of the boundary integral kernels in the filament’s aspect ratio. We then substitute Fourier series expansion (in filament’s circumferential coordinate) of both the filament’s surface velocity and fluid traction in the aforementioned leading-order representation and further linearize it in the rod’s shear strains to reduce the two-dimensional boundary integral over the filament surface into a line integral over the filament’s centerline. Upon further collecting the coefficients of sine and cosine terms, the zeroth-order Fourier mode yields a line integral equation relating the rod’s centerline velocity with the distributed fluid force that acts on the filament. The presence of line integral makes the relation non-local in nature. On the contrary, the first-order Fourier mode yields a simpler local relation between the rod’s angular velocity and the distributed fluid couple. The line integral equation is shown to reduce to the classical slender body theory when shear strains and axial strain are set to zero. The non-dimensional governing equations of the special Cosserat rod are also derived accounting for the distributed fluid force and distributed fluid couple in them which are solved to obtain the filament motion. The presented theory is demonstrated with an example problem of the tumbling of filaments in background shear flow. We show that for relatively shorter filaments where the effect of shear and axial stretch is more dominant, the obtained results deviate from the ones based on the classical slender body theory.

Force–moment line element method for Stokes flow around a slender body

We present a higher-order line element method for Stokes flow around a slender body by taking into account effects of both net force and couple moment densities along its center line. The numerical technique is based on a line integral equation, which in turn is derived by reducing from a boundary integral equation of Stokes flow. The line integral equation of velocity gradient is employed along with that of velocity to close the formulation. Numerical examples of rigid slender bodies are presented to demonstrate the capability and validity of the present method, including passive motion of a slender body suspended in a simple shear flow and active motion of a slender body driven by a body moment. In the first case, it is shown that the moment density due to shear rigidity mismatch between the fluid and solid can be finite. Meanwhile, the transverse net force density is rather induced by a small velocity lag due to the impeding moment density component. Effects of slip and no-slip interfacial conditions are examined, showing their important role in determining the slender-body hydrodynamics. The present force–moment line element method provides a capable tool for solving the problems of strong and weak interactions between a slender body and a viscous fluid at low Reynolds number.

Three-Dimensional Magnetic Field and NMR Sensitivity Computations Incorporating Conductivity Anomalies and Variable-Surface Topography

We have developed a numerical algorithm for computing the magnetic field distribution and the nuclear magnetic resonance (NMR) sensitivity function for arbitrary topography overlying a known 3-D conductivity structure. The magnetic vector potential is split into primary and secondary terms. The primary term is obtained using a thin-wire line integral equation that accounts for arbitrary loop shape and position. It allows the singularity of the source field to be effectively removed. The secondary potential is obtained by solving the second-order partial differential equations on an unstructured tetrahedral mesh using the finite element technique. We validate the results of applying our algorithm against an explicit infinite integral solution for circular loops on a layered earth and against the results of applying a commercial simulation tool. The spatially oscillating NMR sensitivity functions to hydrogen protons (i.e., unbounded water molecules) in the sub-surface are computed on a refined unstructured grid. We apply the numerical algorithm to a number of synthetic examples in surface NMR tomography of hydrological relevance.

Eddy current analysis of thin metal container in induction heating by line integral equations

AbstractIn recent years, induction‐heating cookers have spread rapidly. It is desirable to commercialize flexible and disposable food containers that are suitable for induction heating. In order to develop a good‐quality food container for moderate heating, the eddy currents induced in a thin metal plate must be accurately analyzed. The integral equation method is widely used for solving induction‐heating problems. If the plate thickness approaches zero, the surface integral equations on the upper and lower plate surfaces tend to become the same and the equations become ill‐conditioned. In this paper, we first derive line integral equations from the boundary integral equations under the assumption that the electromagnetic fields in the metal are attenuated more rapidly than those along the metal surface. Next, to test the validity of the line integral equations, we solve for the eddy current induced in a thin metal container in induction heating and obtain the power density imparted to the container and the impedance characteristics of the heating coil. We then compare the computed results with those obtained by the FEM. © 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 168(2): 20–27, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20813

Nonlinear eddy current analysis by line integral equations utilizing integral formulas of electromagnetic fields

PurposeThe purpose of this paper is to supply a numerical analysis tool to solve eddy currents induced in nonlinear materials such as steel by boundary element method (BEM), and then apply it to design and analysis of power devices.Design/methodology/approachUtilizing integral formulas derived on the basis of rapid attenuation of the electromagnetic fields, the paper formulates eddy currents in steel. In the formulation, nonlinear terms are regarded as virtual sources, which are improved iteratively with the electromagnetic fields on the surface. The periodic electromagnetic fields are expanded in Fourier series and each harmonic is analyzed by BEM. The surface and internal electromagnetic fields are obtained numerically one after the other until convergence by the Newton‐Raphson method.FindingsIt is confirmed that this approach gives accurate solutions with meshes much larger than the skin depth and therefore is adequate to apply to a large‐scale application.Research limitations/implicationsThe eddy current is formulated by utilizing the impedance boundary condition in order to meet a large‐scale application, and so solutions near the edge are poor. In the case of better solutions being required, some modifications are necessary.Originality/valueTo lessen computer memory consumption, the parallel component of the currents to the steel surface is analyzed as a 2D problem and the normal component is obtained from the parallel component. One 2D equation for one analyzing region is discretized by dividing the region into layers adaptively and then solved. Next, another is solved sequentially. This method gives a compatible numerical analysis tool with finite element method.

Eddy Current Analysis of Thin Metal Container in Induction Heating by Line Integral Equations

In recent years, induction-heating cookers have been disseminated explosively. It is wished to commercialize flexible and disposable food containers that are available for induction heating. In order to develop a good quality food container that is heated moderately, it is necessary to analyze accurately eddy currents induced in a thin metal plate. The integral equation method is widely used for solving induction-heating problems. If the plate thickness approaches zero, the surface integral equations on the upper and lower plate surfaces tend to become the same and the equations become ill conditioned. In this paper, firstly, we derive line integral equations from the boundary integral equations on the assumption that the electromagnetic fields in metal are attenuated rapidly compared with those along the metal surface. Next, so as to test validity of the line integral equations, we solve the eddy current induced in a thin metal container in induction heating and obtain power density given to the container and impedance characteristics of the heating coil. We compare computed results with those by FEM.

Eddy-current analysis of a conductor with a crack by line integral equations with loop electric and magnetic currents as unknowns

It is difficult to calculate eddy currents induced in a conductor with a crack by the surface integral equations because its adjacent edges and corners, where the equations become singular, are close, and moreover, the equations become ill-conditioned in cases where the crack gap is narrow. In this paper, we propose a method employing line integral equations with loop electric and magnetic currents as unknowns fully utilizing the boundary conditions of the surface electromagnetic fields. Solving eddy-current problems and comparing the results with those by FEM, we study how to calculate accurately the eddy currents induced in a conductor with a crack

Analysis of the base-excited response of intake-outlet towers by a Green's function approach

The dynamic response of a circular, intake-outlet tower structure subjected to high-frequency horizontal ground excitation is investigated theoretically. The tower is idealized as a one-dimensional beam of uniform flexural rigidity and mass per unit length, the fluid is assumed to be linearly compressible and to undergo small-amplitude irrotational motion. The presence of the fluid contained within, as well as the surrounding the structure, is considered in a consistent manner. Both the structural and fluid motions are expressed in terms of appropriate Green's functions leading to a pair of coupled line integral equations which are solved numerically for the fluid velocity potentials on the structure. It is found that the presence of both the interior and exterior fluids can significantly influence the dynamic response, especially for those structures possessing larger diameter to water depth ratios (squatty towers).

A thin Wire Approximation

A line integral equation is generated for the scattering of a harmonic electromagnetic wave by a thin wire whose cross section is not constant along its length. Comparison with experiment shows that the solution is practically useful and that the scattering depends on the cross section more than one would expect at first sight.

Earthquake-induced hydrodynamic pressures on submerged cylindrical storage tanks

The boundary integral method is utilized to calculate the hydrodynamic pressure distribution on a rigid submerged cylindrical storage tank subjected to horizontal or vertical harmonic ground excitation. The fluid is assumed to be linearly compressible and to undergo small-amplitude, irrotational motion. The vertical axisymmetry of the structure is exploited utilizing a fluid Green's function which is separable in the cylindrical polar coordinate system and so leads in each case to a line integral equation for the fluid velocity potential on the surface of the structure. Numerical results are presented which illustrate the influence of the direction and frequency of ground motion and the body dimensions on the hydrodynamic pressure distribution. For both horizontal and vertical ground excitation, the numerical values from the boundary integral method have been verified by a comparison with those obtained by an eigenfunction expansion approach.

Green's function solution for axisymmetric vibration of flexible liquid-filled storage tanks

The dynamic response of a flexible, liquid-filled cylindrical storage tank subjected to high frequency vertical ground motions is investigated theoretically. The tank motion is idealized as that of a uniform, elastic, thin-walled cylindrical shell undergoing radial displacements; all axial deformations are neglected. The fluid is assumed to be inviscid and linearly compressible and to undergo small-amplitude, irrotational motion. Both the structural and fluid motions are expressed in terms of appropriate Green's functions. The axisymmetry of the response then leads to a pair of coupled line integral equations for the fluid velocity potential and its normal derivative on the walls of the tank. These equations are then solved numerically. Results are presented which illustrate the influence of the frequency of ground excitation and the various geometric and material parameters on the hydrodynamic pressure distribution and associated dynamic response of several example structures.

Earthquake response of submerged circular cylinder

The boundary integral method is utilized to calculate the dynamic response of a surface-piercing, vertical, circular cylinder subjected to high-frequency horizontal ground excitation. The cylinder is idealized as a one-dimensional beam of uniform flexural rigidity and uniform mass per unit length, the fluid is assumed linearly compressible and to undergo small-amplitude irrotational motion. The axisymmetry of the structure is exploited by utilizing a new Green's function for linearly compressible fluid flow which is separable in the cylindrical polar coordinate system, thus reducing the surface integral equation to a series of line integral equations. The motion of the cylinder may also be described in terms of a line integral equation through an appropriate Green's function. These two coupled line integral equations are then solved approximately by a numerical approach which involves discretising the region of integration and replacing the continuous problems by systems of algebraic equations which are then solved by standard matrix inversion techniques. Numerical examples are presented which show the influence of the frequency of ground excitation and the various geometric and material parameters on the hydrodynamic force and associated dynamic response of the cylinder.

Wave forces on vertical bodies of revolution

The axisymmetry of a body which is diffracting water waves may be exploited to give a line integral equation to be solved for the scattered wave field and forces on the body. Each term in a previously established surface integral equation is shown to be expressible as a Fourier series, which is then integrated once analytically. The resulting one-dimensional equation is shown to possess singularities, previously ignored by Black (1975). This equation, with series transformations and subtraction of singularities such that all series are quickly convergent and that it has to be solved only along a curve, reduces computational effort by some three orders of magnitude. Results obtained by this method give good agreement with previous analytical and experimental results, even if a rather coarse numerical approximation is used.

A solution for the narrow rectangular punch

This paper considers the problem of a rectangular flat ended punch acting on an elastic half-space. An approximate solution is generated through application of the elastic line integral equations. The results produced by this method are then compared with another approximate solution already available.

The non-elastic cross section of 9Be for neutrons between 2·4 and 3·9 MeV

Shell transmission measurements were made in beryllium with neutrons between 2·4 and 3·9 MeV and analysed by a new method based on the line integral transport equation. The non-elastic cross section for 9Be was derived.