# Browsing by Author "Liu, Qing H"

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Item Open Access 3D Microwave Imaging through Full Wave Methods for Heterogenous Media(2011) Yuan, MengqingIn this thesis, a 3D microwave imaging method is developed for a microwave imaging system with an arbitrary background medium. In the previous study on the breast cancer detection of our research group, a full wave inverse method, the Diagonal Tensor approximation combined with Born Iterative Method (DTA-BIM), was proposed to reconstruct the electrical profile of the inversion domain in a homogenous background medium and a layered background medium. In order to evaluate the performance of the DTA-BIM method in a realistic microwave imaging system, an experimental prototype of an active 3D microwave imaging system with movable antennas is constructed. For the objects immersed in a homogenous background medium or a layered background medium, the inversion results based on the experimental data show that the resolution of the DTA-BIM method can reach finely to a quarter of wavelength of the background medium, and the system's signal-noise-ratio (SNR) requirement is 10 dB. Moreover, the defects of this system make it difficult to be implemented in a realistic application. Thus, another active 3D microwave imaging system is proposed to overcome the problems in the previous system. The new system employs a fix patch antenna array with electric switch to record the data. However, the antenna array makes the inversion system become a non-canonical inhomogeneous background. The analytical Greens' functions used in the original DTA-BIM method become unavailable. Thus, a modified DTA-BIM method, which use the numerical Green's functions combined with measured voltage, is proposed. This modified DTA-BIM method can be used to the inversion in a non-canonical inhomogeneous background with the measured voltages (or $S_{21}$ parameters). In order to verify the performance of this proposed inversion method, we investigate a prototype 3D microwave imaging system with a fix antenna array. The inversion results from the synthetic data show that this method works well with a fix antenna array, and the resolution of reconstructed images can reach to a quarter wavelength even in the presence of a strongly inhomogeneous background medium and antenna couplings. A time-reversal method is introduced as a pre-processing step to reduce the region of interest (ROI) in our inversion. In addition, a Multi-Domain DTA-BIM method is proposed to fit the discontinue inversion regions. With these improvements, the size of the inversion domain and the computational cost can be significantly reduced, and make the DTA-BIM method more feasible for rapid response applications.

Item Open Access A CG-FFT Based Fast Full Wave Imaging Method and its Potential Industrial Applications(2015) Yu, ZhiruThis dissertation focuses on a FFT based forward EM solver and its application in inverse problems. The main contributions of this work are two folded. On the one hand, it presents the first scaled lab experiment system in the oil and gas industry for through casing hydraulic fracture evaluation. This system is established to validate the feasibility of contrasts enhanced fractures evaluation. On the other hand, this work proposes a FFT based VIE solver for hydraulic fracture evaluation. This efficient solver is needed for numerical analysis of such problem. The solver is then generalized to accommodate scattering simulations for anisotropic inhomogeneous magnetodielectric objects. The inverse problem on anisotropic objects are also studied.

Before going into details of specific applications, some background knowledge is presented. This dissertation starts with an introduction to inverse problems. Then algorithms for forward and inverse problems are discussed. The discussion on forward problem focuses on the VIE formulation and a frequency domain solver. Discussion on inverse problems focuses on iterative methods.

The rest of the dissertation is organized by the two categories of inverse problems, namely the inverse source problem and the inverse scattering problem.

The inverse source problem is studied via an application in microelectronics. In this application, a FFT based inverse source solver is applied to process near field data obtained by near field scanners. Examples show that, with the help of this inverse source solver, the resolution of unknown current source images on a device under test is greatly improved. Due to the improvement in resolution, more flexibility is given to the near field scan system.

Both the forward and inverse solver for inverse scattering problems are studied in detail. As a forward solver for inverse scattering problems, a fast FFT based method for solving VIE of magnetodielectric objects with large electromagnetic contrasts are presented due to the increasing interest in contrasts enhanced full wave EM imaging. This newly developed VIE solver assigns different basis functions of different orders to expand flux densities and vector potentials. Thus, it is called the mixed ordered BCGS-FFT method. The mixed order BCGS-FFT method maintains benefits of high order basis functions for VIE while keeping correct boundary conditions for flux densities and vector potentials. Examples show that this method has an excellent performance on both isotropic and anisotropic objects with high contrasts. Examples also verify that this method is valid in both high and low frequencies. Based on the mixed order BCGS-FFT method, an inverse scattering solver for anisotropic objects is studied. The inverse solver is formulated and solved by the variational born iterative method. An example given in this section shows a successful inversion on an anisotropic magnetodielectric object.

Finally, a lab scale hydraulic fractures evaluation system for oil/gas reservoir based on previous discussed inverse solver is presented. This system has been setup to verify the numerical results obtained from previously described inverse solvers. These scaled experiments verify the accuracy of the forward solver as well as the performance of the inverse solver. Examples show that the inverse scattering model is able to evaluate contrasts enhanced hydraulic fractures in a shale formation. Furthermore, this system, for the first time in the oil and gas industry, verifies that hydraulic fractures can be imaged through a metallic casing.

Item Open Access A Hybrid Spectral-Element / Finite-Element Time-Domain Method for Multiscale Electromagnetic Simulations(2010) Chen, JiefuIn this study we propose a fast hybrid spectral-element time-domain (SETD) / finite-element time-domain (FETD) method for transient analysis of multiscale electromagnetic problems, where electrically fine structures with details much smaller than a typical wavelength and electrically coarse structures comparable to or larger than a typical wavelength coexist.

Simulations of multiscale electromagnetic problems, such as electromagnetic interference (EMI), electromagnetic compatibility (EMC), and electronic packaging, can be very challenging for conventional numerical methods. In terms of spatial discretization, conventional methods use a single mesh for the whole structure, thus a high discretization density required to capture the geometric characteristics of electrically fine structures will inevitably lead to a large number of wasted unknowns in the electrically coarse parts. This issue will become especially severe for orthogonal grids used by the popular finite-difference time-domain (FDTD) method. In terms of temporal integration, dense meshes in electrically fine domains will make the time step size extremely small for numerical methods with explicit time-stepping schemes. Implicit schemes can surpass stability criterion limited by the Courant-Friedrichs-Levy (CFL) condition. However, due to the large system matrices generated by conventional methods, it is almost impossible to employ implicit schemes to the whole structure for time-stepping.

To address these challenges, we propose an efficient hybrid SETD/FETD method for transient electromagnetic simulations by taking advantages of the strengths of these two methods while avoiding their weaknesses in multiscale problems. More specifically, a multiscale structure is divided into several subdomains based on the electrical size of each part, and a hybrid spectral-element / finite-element scheme is proposed for spatial discretization. The hexahedron-based spectral elements with higher interpolation degrees are efficient in modeling electrically coarse structures, and the tetrahedron-based finite elements with lower interpolation degrees are flexible in discretizing electrically fine structures with complex shapes. A non-spurious finite element method (FEM) as well as a non-spurious spectral element method (SEM) is proposed to make the hybrid SEM/FEM discretization work. For time integration we employ hybrid implicit / explicit (IMEX) time-stepping schemes, where explicit schemes are used for electrically coarse subdomains discretized by coarse spectral element meshes, and implicit schemes are used to overcome the CFL limit for electrically fine subdomains discretized by dense finite element meshes. Numerical examples show that the proposed hybrid SETD/FETD method is free of spurious modes, is flexible in discretizing sophisticated structure, and is more efficient than conventional methods for multiscale electromagnetic simulations.

Item Open Access Accurate and Efficient Methods for the Scattering Simulation of Dielectric Objects in a Layered Medium(2019) Huang, WeifengElectromagnetic scattering in a layered medium (LM) is important for many engineering applications, including the hydrocarbon exploration. Various computational methods for tackling well logging simulations are summarized. Given their advantages and limitations, main attention is devoted to the surface integral equation (SIE) and its hybridization with the finite element method (FEM).

The thin dielectric sheet (TDS) based SIE, i.e., TDS-SIE, is introduced to the simulation of fractures. Its accuracy and efficiency are extensively demonstrated by simulating both conductive and resistive fractures. Fractures of variant apertures, conductivities, dipping angles, and extensions are also simulated and analyzed. With the aid of layered medium Green's functions (LMGFs), TDS-SIE is extended into the LM, which results in the solver entitled LM-TDS-SIE.

In order to consider the borehole effect, the well-known loop and tree basis functions are utilized to overcome low-frequency breakdown of the Poggio, Miller, Chang, Harrington, Wu, and Tsai (PMCHWT) formulation. This leads to the loop-tree (LT) enhanced PMCHWT, which can be hybridized with TDS-SIE to simulate borehole and fracture together. The resultant solver referred to as LT-TDS is further extended into the LM, which leads to the solver entitled LM-LT-TDS.

For inhomogeneous or complex structures, SIE is not suitable for their scattering simulations. It becomes advantageous to hybridize FEM with SIE in the framework of domain decomposition method (DDM), which allows independent treatment of each subdomain and nonconformal meshes between them. This hybridization can be substantially enhanced by the adoption of LMGFs and loop-tree bases, leading to the solver entitled LM-LT-DDM. In comparison with LM-LT-TDS, this solver is more powerful and able to handle more general low-frequency scattering problems in layered media.

Item Open Access Compatible Subdomain Level Isotropic/Anisotropic Discontinuous Galerkin Time Domain (DGTD) Method for Multiscale Simulation(2015) Ren, QiangDomain decomposition method provides a solution for the very large electromagnetic

system which are impossible for single domain methods. Discontinuous Galerkin

(DG) method can be viewed as an extreme version of the domain decomposition,

i.e., each element is regarded as one subdomain. The whole system is solved element

by element, thus the inversion of the large global system matrix is no longer necessary,

and much larger system can be solved with the DG method compared to the

continuous Galerkin (CG) method.

In this work, the DG method is implemented on a subdomain level, that is, each subdomain contains multiple elements. The numerical flux only applies on the

interfaces between adjacent subdomains. The subodmain level DG method divides

the original large global system into a few smaller ones, which are easier to solve,

and it also provides the possibility of parallelization. Compared to the conventional

element level DG method, the subdomain level DG has the advantage of less total

DoFs and fexibility in interface choice. In addition, the implicit time stepping is

relatively much easier for the subdomain level DG, and the total CPU time can be

much less for the electrically small or multiscale problems.

The hybrid of elements are employed to reduce the total DoF of the system.

Low-order tetrahedrons are used to catch the geometry ne parts and high-order

hexahedrons are used to discretize the homogeneous and/or geometry coarse parts.

In addition, the non-conformal mesh not only allow dierent kinds of elements but

also sharp change of the element size, therefore the DoF can be further decreased.

The DGTD method in this research is based on the EB scheme to replace the

previous EH scheme. Dierent from the requirement of mixed order basis functions

for the led variables E and H in the EH scheme, the EB scheme can suppress the

spurious modes with same order of basis functions for E and B. One order lower in

the basis functions in B brings great benets because the DoFs can be signicantly

reduced, especially for the tetrahedrons parts.

With the basis functions for both E and B, the EB scheme upwind

ux and

EB scheme Maxwellian PML, the eigen-analysis and numerical results shows the

eectiveness of the proposed DGTD method, and multiscale problems are solved

eciently combined with the implicit-explicit hybrid time stepping scheme and multiple

kinds of elements.

The EB scheme DGTD method is further developed to allow arbitrary anisotropic

media via new anisotropic EB scheme upwind

ux and anisotropic EB scheme

Maxwellian PML. The anisotropic M-PML is long time stable and absorb the outgoing

wave eectively. A new TF/SF boundary condition is brought forward to

simulate the half space case. The negative refraction in YVO4 bicrystal is simulated

with the anisotropic DGTD and half space TF/SF condition for the rst time with

numerical methods.

Item Open Access Efficient Electromagnetic Simulation and Experiment Tools for Hydraulic Fracture Evaluation(2019) Fang, YuanHydraulic fracturing is an essential way to improve the production of unconventional oil and gas, especially shale gas. Therefore, it is important to characterize the produced fractures using either acoustic or electromagnetic (EM) methods, and evaluation of hydraulic fractures has been under intensive study since last decade. Electromagnetic techniques, including induction logging and galvanic techniques, have the advantages of nondestructive measurements and high sensitivity to the formation and fracture resistivity. They are widely used for produced fracture evaluation.

However, conventional forward and inverse methods in low frequency range face significant challenges by such multiscale problems where the fracture width (<1cm) is orders of magnitude smaller than its diameters (>100 m). The problem becomes much more complicated when the effects of borehole, casing, and planar stratified medium need to be considered for realistic oil field application.

This dissertation focuses on three aspects. First of all, the application of newly developed efficient forward electromagnetic solvers, hybrid distorted Born approximation and mixed ordered stabilized bi-conjugate gradient FFT (DBA-BCGS-FFT) method, and hybrid numerical mode matching with the stabilized bi-conjugate gradient FFT (NMM-BCGS-FFT) method, are illustrated. For the DBA-BCGS-FFT method, the two components of the solver, distorted Born approximation (DBA) and mixed ordered stabilized bi-conjugate gradient FFT (BCGS-FFT), are separately discussed with their advantages and disadvantages. Then the hybrid DBA-BCGS-FFT will be introduced and explained, including how the combination of the advantages of the two solvers and overcome their disadvantages. For the second forward method, the numerical mode matching (NMM) method is introduced with the procedures of the NMM-BCGS-FFT method for analyzing the effects of the complex cased borehole and planar stratified medium.

Second, the inverse solver, variational Born iterative method (VBIM), is introduced for hydraulic fracture reconstruction. The box constraints in the inversion process is introduced to enhance the fracture reconstruction resolution and avoid unrealistic parameter in the inversion. In the procedures of the inverse solver, the forward solvers are applied to construct the system matrix. In this application, the inverse solvers are applied to process the secondary field data obtained by field scanners and laboratory detectors.

The results will be separate into three sections. First, the validation of the forward and inverse solvers is demonstrated. The commercial software, COMSOL, is used for the validation. Then, induction logging detection and galvanic detection model results show the capability of the forward and inverse solvers. Last, two established experiment systems will be described with details. The laboratory scaled experimental system is established for feasibility study of the electromagnetic induction detection, and the field test control source electromagnetic system is designed and built for hydraulic fracture evaluation. In induction logging detection model, experimental results show that the inverse scattering algorithm is effective for electromagnetic contrast enhanced through-casing hydraulic fracture evaluation. In galvanic detection model, the impact of different hydraulic fracturing material and choices of transmitter/receiver locations on signal response will be discussed to show the application of the forward solvers in field configuration design. Both fracture reconstruction results by the inverse solvers with the experimental data will be discussed in these two chapters.

Item Open Access Numerical Solution of Multiscale Electromagnetic Systems(2013) TOBON, LUIS E.The Discontinuous Galerkin time domain (DGTD) method is promising in modeling of realistic multiscale electromagnetic systems. This method defines the basic concept for implementing the communication between multiple domains with different scales.

Constructing a DGTD system consists of several careful choices: (a) governing equations; (b) element shape and corresponding basis functions for the spatial discretization of each subdomain; (c) numerical fluxes onto interfaces to bond all subdomains together; and (d) time stepping scheme based on properties of a discretized

system. This work present the advances in each one of these steps.

First, a unified framework based on the theory of differential forms and the finite element method is used to analyze the discretization of the Maxwell's equations. Based on this study, field intensities (E and H) are associated to 1-forms and curl-conforming basis functions; flux densities (D and B) are associated to 2-forms and divergence-conforming basis functions; and the constitutive relations are defined by Hodge operators.

A different approach is the study of numerical dispersion. Semidiscrete analysis is the traditional method, but for high order elements modal analysis is prefered. From these analyses, we conclude that a correct discretization of fields belonging to different p-form (e.g., E and B) uses basis functions with same order of interpolation; however, different order of interpolation must be used if two fields belong to the same p-form (e.g., E and H). An alternative method to evaluate numerical dispersion based on evaluation of dispersive Hodge operators is also presented. Both dispersion analyses are equivalent and reveal same fundamental results. Eigenvalues, eigenvector and transient results are studied to verify accuracy and computational costs of different schemes.

Two different approaches are used for implementing the DG Method. The first is based on E and H fields, which use curl-conforming basis functions with different order of interpolation. In this case, the Riemman solver shows the best performance to treat interfaces between subdomains. A new spectral prismatic element, useful for modeling of layer structures, is also implemented for this approach. Furthermore, a new efficient and very accurate time integration method for sequential subdomains is implemented.

The second approach for solving multidomain cases is based on E and B fields, which use curl- and divergence-conforming basis functions, respectively, with same order of interpolation. In this way, higher accuracy and lower memory consumption are obtained with respect to the first approach based on E and H fields. The centered flux is used to treat interfaces with non-conforming meshes, and both explicit Runge-Kutta method and implicit Crank-Nicholson method are implemented for time integration.

Numerical examples and realistic cases are presented to verify that the proposed methods are non-spurious and efficient DGTD schemes.

Item Open Access Spectral Integral Method and Spectral Element Method Domain Decomposition Method for Electromagnetic Field Analysis(2011) Lin, YunIn this work, we proposed a spectral integral method (SIM)-spectral element method (SEM)- finite element method (FEM) domain decomposition method (DDM) for solving inhomogeneous multi-scale problems. The proposed SIM-SEM-FEM domain decomposition algorithm can efficiently handle problems with multi-scale structures,

by using FEM to model electrically small sub-domains and using SEM to model electrically large and smooth sub-domains. The SIM is utilized as an efficient boundary condition. This combination can reduce the total number of elements used in solving multi-scale problems, thus it is more efficient than conventional FEM or conventional FEM domain decomposition method. Another merit of the proposed method is that it is capable of handling arbitrary non-conforming elements. Both geometry modeling and mesh generation are totally independent for different sub-domains, thus the geometry modeling and mesh generation are highly flexible for the proposed SEM-FEM domain decomposition method. As a result, the proposed SIM-SEM-FEM DDM algorithm is very suitable for solving inhomogeneous multi-scale problems.