Advanced three-dimensional electromagnetic modelling using a nested integral equation approach

New parallel stochastic global integral and local differential equation modeling and inversion

Accurate electromagnetic (EM) modeling for packaging structures with lossy interconnects requires to consider the finite conductivity of the conducting interconnects. In the integral equation approach, the conducting loss is usually accounted for through an approximate surface impedance when the skin depth of current is small, but this approximation may not be valid for large skin depth caused by low frequencies or small conductivities. In this paper, we treat the lossy interconnects as dielectric-like media and use the two-region electric field integral equations (EFIEs) to characterize the EM feature of the interconnects. The EFIEs are solved with the method of moments in which the Rao-Wilton-Glisson (RWG) basis function and dual basis function (DBF) are used to represent the electric and magnetic current densities, respectively. The joint use of RWG and DBF can remove the need of wisely formulating integral equations and selecting testing schemes and also alleviate the low-frequency effect in a wideband modeling for multiscale interconnect structures. Numerical examples are presented to demonstrate the approach and its robustness has been verified.

The integral equation model (IEM) developed by A.K. Fung and co-workers has arguably been the most cited and implemented rough surface scattering model in the field of radar remote sensing. It was obtained as a second-order iteration of the incident electromagnetic field to the integral equations of the surface fields. Therefore, it is an extension of the first-order Born approximation of the corresponding Neumann series that produce the classical Kirchhoff approximation. The IEM was tuned many times over the last twenty years due to the imperfect introduction and handling of the Weyl representation of the spherical wave in its first version. The work presented here is a development of the contribution made by the same author in 2001 (IEM2M) and reported extensively in [1] (IEM2Mc). It has been the first version of IEM able to blend analytically both the Kirchhoff and the small-perturbation approximations for the bistatic case and dielectric surfaces. IEM2M was celebrated by those who found the original IEM not enough rigorous but also criticized by Fung and his co-workers. In this paper an extension of IEM2M is presented that extends it in two aspects: the inclusion of evanescent waves in the formulation of the model and the extension of the range of applicability of the second-order scattering terms to interfaces with complex permittivity scattering media. These two issues had only been addressed in the IEM version developed by Du. However, Du needed to make a mathematical assumption on the correlation function which was not explained from a physical point of view. In the work proposed here this assumption is not necessary. Although all these issues are dealt with extensively in [1], this paper focuses on the comparison of IEM2M, IEM2Mc, the Kirchhoff approximation (KA) and the Small Perturbation Method (SPM). account of them. Regarding IEM, Fung's last update on his version of IEM is available in [2]

This thesis consists of two parts. The first one is "Scattering of Waves in a Random Medium," and the second one is "Radiative Transfer in a Sphere Illuminated by a Parallel Beam: An Integral Equation Approach." In the first part, a new formalism tor partially coherent wave scattering in a random medium is developed. In this formalism the coherent wave is the solution of a phenomenological wave equation, and the mutual coherence function of the wave field satisfies a simple integral equation. Using this formalism, the Peierls equation can be readily derived. Also, an improved version of the Peierls equation is derived in which the intensity of the wave field and the first order derivative of the mutual coherence function are calculated at the same time. A simple problem is solved to find the mutual coherence function produced by a laser beam in the atmosphere. The similarity between the mutual coherence function and the density matrix or quantum mechanics is explored and a measure of the randomness is defined for the partially coherent wave field. In the second part of this work, the problem of multiple scattering of non-polarized light in a planetary body of arbitrary shape illuminated by a parallel beam is formulated using the integral equation approach. There exists a simple functional whose stationarity condition is equivalent to solving the equation of radiative transfer and whose value at the stationary point is proportional to the differential cross section. Our analysis reveals a direct relation between the microscopic symmetry of the. Phase function for each scattering event and the macroscopic symmetry of the differential cross section for the entire planetary body, and the intimate connection between these symmetry relations and the variational principle. The case of a homogeneous sphere containing isotropic scatterers is investigated in detail. It is shown that the solution can be expanded in a multipole series such that the general spherical problem is reduced to solving a set of decoupled integral equations in one dimension. Computations have been performed for a range of parameters of interest, and illustrative examples of applications to planetary problems are provided.

Iterative Solution Method for the Linearized Poisson–Boltzmann Equation: Indirect Boundary Integral Equation Approach

A small box Fast Fourier Transformation method for fast Poisson solutions in large systems

We develop a new formulation of the integral equation (IE) method for three-dimensional (3D) electromagnetic (EM) field computation in large-scale models with multiple inhomogeneous domains. This problem arises in many practical appli- cations including modeling the EM fields within the complex geoelectrical structures in geophysical exploration. In geophysical applications, it is difficult to describe an earth structure using the horizontally layered background conductivity model, which is required for the efficient implementation of the conventional IE approach. As a re- sult, a large domain of interest with anomalous conductivity distribution needs to be discretized, which complicates the computations. The new method allows us to con- sider multiple inhomogeneous domains, where the conductivity distribution is differ- ent from that of the background, and to use independent discretizations for different domains. This reduces dramatically the computational resources required for large- scale modeling. In addition, using this method, we can analyze the response of each domain separately without an inappropriate use of the superposition principle for the EM field calculations. The method was carefully tested for the modeling the marine controlled-source electromagnetic (MCSEM) fields for complex geoelectric structures with multiple inhomogeneous domains, such as a seafloor with the rough bathymetry, salt domes, and reservoirs. We have also used this technique to investigate the return induction effects from regional geoelectrical structures, e.g., seafloor bathymetry and salt domes, which can distort the EM response from the geophysical exploration target.

Integral equation approaches are an important tool for the solution of crack problems in plane elasticity [1–5]. The description generally used in the mentioned papers is the singular integral equation approach. Recently, alternatives were suggested. In [6,7], taking the tractions and the dislocation function respectively along the crack to be polynomials, the multiple crack problem and the curved crack problem in an infinite plate were solved. The regularization problem of the singular integral equation for the crack problem was considered in [8,9].

The rapid development of communication systems and the impending traffic congestion of RF spectrum have led to an expansion in frequency usage towards the millimeter wave and sub-millimeter wave bands. In dealing with high frequency spectra in these bands, the conventional fabrication technology for microwave integrated circuit (MWIC) such as microstrip technology no longer applies, primarily because of high transmission loss. Although low-loss transmission media such as dielectric waveguide structures can be used as alternatives, they are difficult to fabricate and integrate with passive or active devices. This results in a growing interest in combining the advantages of microstrip and dielectric waveguide technologies. One of the most common implementations of interconnecting these two dissimilar structures is offered by aperture coupling. The theme of this dissertation is to introduce a rigorous formulation for modeling the coupling between microstrip and open-dielectric related structures (rectangular dielectric resonator antenna and image guide) via an aperture. The model is the first major rigorous method published for analyzing such types of coupling structures. The method is based on a surface integral equation approach by which the dielectric waveguide is modeled as equivalent surface electric and magnetic currents and subsequently treated via a set of combined field integral equations (CFIE). The microstrip structure and the associated coupling mechanism are formulated using the mixed potential integral equation (MPIE) approach. The coupled integral equations are then solved by the method of moments (MoM) in the spatial domain using Galerkin's procedure. Closed-form Green's functions are employed in the calculation to enhance numerical efficiency. The model is demonstrated to be a very versatile and is a practical design tool with the ability to perform accurate full-wave analysis very efficiently. The key to formulating 3-D open-dielectric waveguide structures is the introduction of the hybrid basis functions technique for modeling propagating and scattered fields associated with a guiding structure. It is based on the use of a combination of sub-domain (rooftop) and travelling wave basis functions to represent the irregular and modal surface currents on the dielectric. The technique was first applied to a 2-D planar guided structure (microstrip leaky wave antenna) before it was expanded to the 3-D geometry. The consequence of this technique is far-reaching as it has made possible a MoM analysis of 3-D finite open-dielectric waveguides that has not been reported.

Electromagnetic propagation in media with time-varying parameters is of interest for many applications: shock waves, strong laser fields, nuclear burst, atmospheric or environmental change and others. As a rule, an exact consideration of electromagnetic processes in media with the time-varying parameters is possible in special cases only. In the general case, an initial electromagnetic problem can be investigated by numerical methods. FDTD methods are popular due to their clarity and elegant realization in numerical algorithms. The integral equations approach seems to have a more promising future in terms of both reflection-free realization and stability. An advantage of the integral equation method is that it can be generalized to nonlinear dispersive materials. In this paper both FDTD and the integral equation approaches are investigated, accuracy and stability comparison of these methods are examined. Special computer modeling software is developed for dispatching the numerical experiments.

The Design of Linear Regulators by Kalaba's New Integral Equation Approach and the Relations to Chandrasekhar-type Filters

The accurate interpretation of three-dimensional borehole to surface electromagnetic (BSEM) data has two challenging problems: the accuracy of forward modeling and the efficiency of the inversion process. In this paper, we used the least squares method to develop a preconditioned algorithm combined parametric inversion and resistivity image to overcome these difficulties for reservoir exploration. To improve accuracy, we suggest using the full integral equation (IE) solution for forward modeling. Taking into account that the IE method requires the mesh discretization of an anomalous body only, the parametric inversion based on the IE method would provide an preconditioned inverse domain consisting of the complex structure via a few parameters. Based on the predicted preconditioned inverse domain, the conventional Green’s functions can be recomputed only once and kept unchanged in iterative inversion process of resistivity image, which increases significantly the size of the inverse domain and the computation for IE modeling. The resistivity of background and complex structures embedded in layered stratum are considered to simulate the practical application in preconditioned inversion. Synthetic examples with noise studies are conducted to demonstrate that the preconditioned algorithm can be applied quickly and effectively in a complex structure inversion for BSEM in reservoir exploration.

An efficient numerical scheme for solving Hammerstein integral equation arisen in chemical phenomenon

Reconstruction of objects by microwave illumination can be described by integral equation approach and volume integral equations (VIEs) are indispensable for reconstructing penetrable objects due to the inhomogeneity of materials in the imaging domain. Traditionally, the objects are assumed to be nonmagnetic, so only the permittivity distribution is solved from the iterative process of reconstruction. This work considers the reconstruction of penetrable objects with both dielectric and magnetic materials and the full VIEs are needed to describe the problem. The forward scattering VIEs (FSVIEs) are solved by Nyström method while the inverse scattering VIEs (ISVIEs) are solved with a Gauss-Newton method. A numerical example is used to verify the approach.

The problem of the excitation, detection, and scattering of electroelastic surface (Bleustein) waves is solved exactly by determining the charge distribution on the fingers of an interdigital transducer. The approach is to solve an integral equation, in the Fourier transform domain, that relates the charge density on the fingers to the electric potential of the fingers. The solution of the integral equation is accomplished by expanding the charge distribution in a series of pulses and then transforming the problem to a vector matrix,one which is readily handled by a computer. In this manner the charge distribution is determined for a variety of conditions. Subject Classification: [43]35,54.