Surface Integral Method Research Articles

FDTD far-field scattering amplitudes: Comparison of surface and volume integration methods

The transformation of the near-field to the corresponding far-field is a key step involved in the application of the finite-difference time-domain (FDTD) technique to the scattering of an electromagnetic wave by a dielectric particle. In practice, either a volume-integral method (VIM) or an equivalent surface-integral method (SIM) can be used to calculate the far-field. Previous studies reported in the literature demonstrated that a fine grid resolution is necessary for implementing the FDTD technique with the aforementioned VIM for electromagnetic scattering of plane waves by a dielectric particle with a large refractive index. A fine grid resolution leads to a significant amount of computational effort in terms of computer memory and CPU time, and consequently, substantially limits the applicability of the FDTD technique. In this study, we show that the SIM allows a much coarser grid resolution than that required by the VIM. The physical mechanism associated with the advantages of the SIM is also discussed. Furthermore, it is demonstrated that the same order of numerical errors for the phase function computation can be achieved using the SIM with a grid size of Δ s = λ / 40 for the two published large refractive indices, namely m = 7.1499 + i 2.914 and 8.2252 + i 1.6808 , whereas the grid size required by the VIM, as reported in the literature, needs to be Δ s = λ / 143 or λ / 165 . With the SIM, the scattering properties of particles with complex refractive indices as large as 7.1499 + i 2.914 can be simulated by the FDTD technique for size parameters up to 20 by using an ordinary desktop computer.

SINGULARITY SUBTRACTION INTEGRAL FORMULAE FOR SURFACE INTEGRAL EQUATIONS WITH RWG, ROOFTOP AND HYBRID BASIS FUNCTIONS

Numerical solution of electromagnetic scattering problems by the surface integral methods leads to numerical integration of singular integrals in the Method of Moments. The heavy numerical cost of a straightforward numerical treatment of these integrals can be avoided by a more efficient and accurate approach based on the singularity subtraction method. In the literature the information of the closed form integral formulae required by the singularity subtraction method is quite fragmented. In this paper we give a uniform presentation of the singularity subtraction method for planar surface elements with RWG, ˆ n×RWG, rooftop, and ˆ n×rooftop basis functions, the latter three cases being novel applications. We also discuss the hybrid use of these functions. The singularity subtraction formulas are derived recursively and can be used to subtract more than one term in the Taylor series of the Green's function.

Life Prediction for Complex Structures

The surface integral method, an indirect boundary element method that represents a crack as a distribution of force dipoles, has been developed to model three-dimensional nonplanar crack growth in complex structures. The finite body was effectively modeled by superposition of stress influence functions for a half-space. As a result of this strategy, only the fracture has to be discretized. Crack propagation was modeled using the maximum circumferential stress theory to predict crack direction and the Forman fatigue equation, modified with an equivalent stress intensity solution for mixed-mode, to predict extension. Comparisons with benchmark solutions and field data verified the computational methodology and defined the limits of its applicability.

A Hybrid FEM/BEM Approach for Designing an Aircraft Engine Structural Health Monitoring

A new hybrid surface-integral-finite- element numerical scheme has been developed to model a three-dimensional crack propagating through a thin, multi-layered coating. The finite element method was used to model the physical state of the coating, and the surface integral method was used to model the fatigue crack growth. The two formulations are coupled through the need to satisfy boundary conditions on the crack and external surface. The coupling is sufficiently weak that the surface integral mesh of the crack surface and the finite element mesh of the uncracked volume can be set up independently. Thus, when modeling crack growth, the finite element mesh can remain fixed for the duration of the simulation as the crack mesh is advanced. This method was developed to evaluate the feasibility of fabricating a structural health monitoring system for real-time detection of surface cracks propagating in aircraft engine components. In this work, the authors formulate the hybrid surface-integral-finite-element method and discuss the mechanical issues of implement- ing a structural health monitoring system in an aircraft engine environment.

중력식 콘크리트 댐에 작용하는 양압력이 응력확대계수에 미치는 영향

The modeling of uplift pressure within dam, on the foundation on which it was constructed, and on the interface between the dam and foundation is a critical aspect in the analysis of concrete gravity dams, i.e. crack stability in concrete dam can correctly be predicted when uplift pressures are accurately modelled. Current models consider a uniform uplift distribution, but recent experimental results show that it varies along the crack faces and the procedures for modeling uplift pressures are well established for the traditional hand-calculation methods, but this is not the case for finite element (FE) analysis. In large structures, such as dams, because of smaller size of the fracture process zone with respect to the structure size, limited errors should occur under the assumptions of linear elastic fracture mechanics (LEFM). In this paper, the fracture behaviour of concrete gravity dams mainly subjected to uplift Pressure at the crack face was studied. Triangular type, trapezoidal type and parabolic type distribution of the uplift pressure including uniform type were considered in case of evaluating stress intensity factor by surface integral method. The effects of body forces, overtopping pressures are also considered and a parametric study of gravity dams under the assumption of LEFM is performed.

Asymptotically exact calculation of the exchange energies of one-active-electron diatomic ions with the surface integral method

We present a general procedure, based on the Holstein–Herring method, for calculating exactly the leading term in the exponentially small exchange energy splitting between two asymptotically degenerate states of a diatomic molecule or molecular ion. The general formulae we have derived are shown to reduce correctly to the previously known exact results for the specific cases of the lowest Σ and Π states of H+2. We then apply our general formulae to calculate the exchange energy splittings between the lowest states of the diatomic alkali cations K+2, Rb+2 and Cs+2, which are isovalent to H+2. Our results are found to be in very good agreement with the best available experimental data and ab initio calculations.

Self-consistent calculations of exciton, biexciton and charged exciton energies in InGaN/GaN quantum dots

We present theoretical calculations of the variation of exciton energies in truncated conical InGaN quantum dots (QDs) in a GaN matrix with dot size and indium composition. We compute the built-in strain-induced and spontaneous piezoelectric fields using a surface integral method that we have recently derived, and confirm that the built-in fields can be of the order of a few MV/cm, resulting in a spatial separation of the electrons and holes. The ground state wavefunctions of the exciton ( X 0 ) , biexciton ( 2 X 0 ) and the two charged excitons ( X − and X + ) are then calculated in the Hartree approximation, using a self-consistent finite difference method. We find that the electron–hole recombination energy is always blue-shifted for the charged excitons X − and X + , with a further blue-shift for the biexciton, and this blue-shift increases with increasing indium content. We describe the trends in interband transition energy and the scale of the blue-shift with dot size, shape and composition. We conclude that spectroscopic studies of the exciton, charged excitons and biexciton should provide a useful probe of the structural and piezoelectric properties of GaN-based QDs.

Coupling of Integral Acoustics Methods with LES for Jet Noise Prediction

This study is focused on developing a Computational Aeroacoustics (CAA) methodology that couples the near field unsteady flow field data computed by a 3-D Large Eddy Simulation (LES) code with various integral acoustic formulations for the far field noise prediction of turbulent jets. The LES code employs state-of-the-art numerical schemes and a localized version of the dynamic Smagorinsky subgrid-scale (SGS) model. The code also has the capability to turn off the SGS model and treat the spatial filter that is needed for numerical stability as an implicit SGS model. Noise computations performed for a Mach 0.9, Reynolds number 400,000 jet using various integral acoustic results are presented and the results are compared against each other as well with those from experiments at similar flow conditions. Our results show that the surface integral acoustics methods (Kirchhoff and Ffowcs Williams – Hawkings) give similar results to the volume integral method (Lighthill's acoustic analogy) at a much lower cost. To the best of our knowledge, Lighthill's acoustic analogy is applied to a Reynolds number 400,000 jet at Mach 0.9 for the first time in this study. The distribution of Lighthill sources that radiate noise in the direction of various observer locations is evaluated. A source decomposition shows significant cancellations among the individual components of the Lighthill source.

Investigation of integral surface formulations for acoustic post-processing of unsteady aerodynamic jet simulations

The acoustic post-processing of unsteady aerodynamic jet simulations using Kirchhoff or Ffowcs Williams and Hawkings (FW–H) surface integral methods is investigated from the theoretical and practical points of view. This analysis is carried out for a supersonic hot jet, starting from the flow fields provided by an unsteady aerodynamic simulation whose characteristics are recalled. Both acoustic integral methods are first compared from a theoretical point of view. In particular, the role of the ‘entropic’ sources on the control surface is pointed out. Various acoustic calculations are then carried out for this hot jet to assess the influence of the type of method and of the position, extent and nature (open or closed) of the control surfaces on the noise predictions. This parametric study shows that the Kirchhoff method using density as input data is poorly suited for acoustic predictions of hot jets. The Kirchhoff method using pressure as input data provides results similar to those obtained with the FW–H surface integral but the latter appears to be more reliable. The study also shows that the use of a downstream closed control surface does not present a real interest, whatever the acoustic method is.

Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship.

Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.

Surface Integral Methods in Computational Aeroacoustics—From the (CFD) Near-Field to the (Acoustic) Far-Field

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.

Light scattering by arbitrary-shaped one-dimensionalparticles

A surface integral method developed for one-dimensional multivaluedsurfaces has been extended to include multiple profiles. This paperis concerned with the numerical implementation of the parametrizationof the surface profile, and the development of an algorithm inorder to determine whether a point is located inside or outsidethe material. This formulation is especially suitable forarbitrary-shaped particles and for particles placed on flat surfaces.The result is a useful and versatile tool for studying the scatteringby irregular particles in both the near and far fields.This method can be used to investigate optical effects associated withmorphology- dependent resonances and multiple scattering by particlesof arbitrary shape.

Experimental and computational investigation of three‐dimensional mixed‐mode fatigue

Experimental and computational methods were developed to model three‐dimensional (3‐D) mixed‐mode crack growth under fatigue loading with the objective of evaluating proposed 3‐D fracture criteria. The experiments utilized 7075‐T73 aluminium forgings cut into modified ASTM E740 surface crack specimens with pre‐cracks orientated at angles of 30, 45 and 60° in separate tests. The progress of the evolving fatigue crack was monitored in real time using an automated visualization system. In addition, the amplitude of the loading was increased at prescribed intervals to mark the location of the 3‐D crack front for post‐test inspection. In order to evaluate proposed crack growth equations, computer simulations of the experiments were conducted using a 3‐D fracture model based on the surface integral method. An automatic mesher advanced the crack front by adding a ring of elements consistent with local application of fracture criteria governing rate and direction of growth. Comparisons of the computational and experimental results showed that the best correlation was obtained when KII and KIII were incorporated in the growth rate equations.

Generalized Heitler-London theory for H 3 : a comparison of the surface integral method with perturbation theory

The generalized Heitler-London (GHL) theory provides a straightforward way to express the potential energy surface of H3 in terms of Coulomb and exchange energies which can be calculated either by perturbation theory or using the surface integral method (SIM). By applying the Rayleigh-Schrodinger perturbation theory, GHL theory for the quartet spin state of H3 is shown to yield results equivalent to the symmetrized Rayleigh-Schrodinger version of symmetry adapted perturbation theory (SAPT). This equivalence allows a comparison with the corresponding results obtained by the surface integral method. The surface integral result calculated with a product of atomic wave functions is found to have certain advantages over the perturbation approach.

New Insight into Exchange Energy of Covalent Chemical Bonds

AbstractThe exchange energy for the H2 molecule is calculated for a wide range of inter nuclear distance R extending down to R = 0.5 a0 using the surface integral method. This method accounts for the physical exchange of the two electrons between the two nuclei. The good agreement of the present results with the standard Heitler‐London calculation demonstrates that the exchange energy can in deed be described in terms of electrons hop ping back and forth between the nuclei, an interpretation which has been frequently criticized in the past.

Tectonomagnetic modeling based on the piezomagnetism: a review

Development of tectonomagnetic modeling on the basis of the piezomagnetic effect is reviewed for the period since the early 1990's. First, the basic theory is briefly summarized, in which the representation theorem or the surface integral representation for the piezomagnetic potential and the Green's function method are presented. In the 1990's, several field observations in earthquakes and volcanoes were interpreted with the aid of analytic solutions based on the Green's function method. A general formula was developed for an inclined rectangular fault with strike-slip, dip-slip and tensile faulting. The surface integral method has been applied to 2D and 3D models, as well as to fault models in the inhomogeneously magnetized crust. When the magnetic field is measured within a bore hole, the effect of magnetic poles around the hole should be taken into account. As a result, tectonomagnetic signals are much enhanced in a bore hole compared with on the ground surface. Finally, piezomagnetic field changes associated with the Parkfield fault model are introduced and the new aspect of the model is discussed.

Three-body exchange energies in H3 and He3 calculated by the surface integral method

Repulsive three-body exchange energies are calculated for H3 and He3 using a new method based on an extension of the surface integral method to three-body interactions. The dependence of the three-body exchange energies on the internuclear distances between 5 and 10 a0 is investigated for equilateral configurations. The angular dependence is also studied for geometries in which two distances are fixed at equal distances. Very good agreement with other calculations at the same level of approximation is obtained for H3 as well as for He3. The relative importance of the different electron permutations in He3 is discussed.

Ground state potentials for alkaline-earth–helium diatoms calculated by the surface integral method

Using a recently derived simple formula the van der Waals ground state potential curves of alkaline-earth–helium pairs BeHe, MgHe, CaHe, SrHe, and BaHe have been calculated. The repulsive potential is based on exchange energies calculated from the surface integral method applied to multielectron systems. With the exception of the long-range dispersion coefficients all the input parameters are atomic properties of the free atoms. The agreement with the available potentials from the literature is reasonable.

Nonlinear Oscillations of Molten Silicon Drops in Electromagnetic Levitator.

Nonlinear oscillations of molten silicon dr ops in an electromagnetic levitator under microgravity are numerically investigated. The electromagnetic field and the magnetic pr essure on the drop surface are calculated by the surface integral method assuming that the skin depth is much smaller than the drop radius. The Galerkin finite element method in combination with the Lagrangian technique is used to analyze the moderate-amplitude axisymmetric oscillations of the silicon dr ops on which the magnetic pressure acts. The effect of the electric curr ent ratio of the heating coils to the levitation coils on the frequency of the oscillations is evaluated for molten silicon dr ops released from either an initially prolate spheroid configuration or four-lobed spherical harmonic configuration. The numerical results show that, in the former case, the frequency decreases with the curr ent ratio. Also, for the latter, the dynamics of the second mode becomes dominant as the curr ent ratio increases, because the inward magnetic pr essure from the heating coil incr eases.

Analysis and capability of a noncompact integration method for rotor noise prediction

Rotor noise computations starting from CFD input data using either the Kirchhoff or the generalized Ffowcs Williams–Hawkings surface integral methods in unsteady conditions may be very lengthy. This is the case for directivity studies or noise level contour calculations for which a high number of observer locations is required. To overcome this drawback, a fast integration method has been designed. It starts from the emission time which avoids having to solve the retarded time equation and uses directly as integration grid the CFD rotating grid (a priori not adapted to acoustic calculations). This is made possible thanks to a noncompact treatment of each grid element. The method has already been successfully applied in a transonic rotor case using indifferently a subsonically or supersonically rotating CFD grid for the surface integration (G. Rahier and J. Prieur, 53rd Forum of the American Helicopter Society, 29 April–1 May 1997). The method is made more understandable based on a new and clearer justification and its possibilities of extension are discussed.