Application of Numerical Methods to Study Arrangement and Fracture of Lithium-Ion Microstructure

The focus of this work is to develop and employ numerical methods that provide characterization of granular microstructures, dynamic fragmentation of brittle materials, and dynamic fracture of three-dimensional bodies.

We first propose the fabric tensor formalism to describe the structure and evolution of lithium-ion electrode microstructure during the calendaring process. Fabric tensors are directional measures of particulate assemblies based on inter-particle connectivity, relating to the structural and transport properties of the electrode. Applying this technique to X-ray computed tomography of cathode microstructure, we show that fabric tensors capture the evolution of the inter-particle contact distribution and are therefore good measures for the internal state of and electronic transport within the electrode.

We then shift focus to the development and analysis of fracture models within finite element simulations. A difficult problem to characterize in the realm of fracture modeling is that of fragmentation, wherein brittle materials subjected to a uniform tensile loading break apart into a large number of smaller pieces. We explore the effect of numerical precision in the results of dynamic fragmentation simulations using the cohesive element approach on a one-dimensional domain. By introducing random and non-random field variations, we discern that round-off error plays a significant role in establishing a mesh-convergent solution for uniform fragmentation problems. Further, by using differing magnitudes of randomized material properties and mesh discretizations, we find that employing randomness can improve convergence behavior and provide a computational savings.

The Thick Level-Set model is implemented to describe brittle media undergoing dynamic fragmentation as an alternative to the cohesive element approach. This non-local damage model features a level-set function that defines the extent and severity of degradation and uses a length scale to limit the damage gradient. In terms of energy dissipated by fracture and mean fragment size, we find that the proposed model reproduces the rate-dependent observations of analytical approaches, cohesive element simulations, and experimental studies.

Lastly, the Thick Level-Set model is implemented in three dimensions to describe the dynamic failure of brittle media, such as the active material particles in the battery cathode during manufacturing. The proposed model matches expected behavior from physical experiments, analytical approaches, and numerical models, and mesh convergence is established. We find that the use of an asymmetrical damage model to represent tensile damage is important to producing the expected results for brittle fracture problems.

The impact of this work is that designers of lithium-ion battery components can employ the numerical methods presented herein to analyze the evolving electrode microstructure during manufacturing, operational, and extraordinary loadings. This allows for enhanced designs and manufacturing methods that advance the state of battery technology. Further, these numerical tools have applicability in a broad range of fields, from geotechnical analysis to ice-sheet modeling to armor design to hydraulic fracturing.

Experimental Study of Storage Ring Free-Electron Laser with Novel Capabilities

The Duke Free-electron laser (FEL) system, driven by the Duke electron storage ring, has been at the forefront of developing new light source capabilities over the past two decades. In 1999, the Duke FEL demonstrated the first lasing of a storage ring FEL in the vacuum ultraviolet (VUV) region at $194$ nm using two planar OK-4 undulators. With two helical undulators added to the outboard sides of the planar undulators, in 2005 the highest FEL gain ($47.8\%$) of a storage ring FEL was achieved using the Duke FEL system with a four-undulator configuration. In addition, the Duke FEL has been used as the photon source to drive the High Intensity $\gamma$-ray Source (HIGS) via Compton scattering of the FEL beam and electron beam inside the FEL cavity. Taking advantage of FEL's wavelength tunability as well as the adjustability of the energy of the electron beam in the storage ring, the nearly monochromatic $\gamma$-ray beam has been produced in a wide energy range from $1$ to $100$ MeV at the HIGS. To further push the FEL short wavelength limit and enhance the FEL gain in the VUV regime for high energy $\gamma$-ray production, two additional helical undulators were installed in 2012 using an undulator switchyard system to allow switching between the two planar and two helical undulators in the middle section of the FEL system. Using different undulator configurations made possible by the switchyard, a number of novel capabilities of the storage ring FEL have been developed and exploited for a wide FEL wavelength range from infrared (IR) to VUV. These new capabilities will eventually be made available to the $\gamma$-ray operation, which will greatly enhance the $\gamma$-ray user research program, creating new opportunities for certain types of nuclear physics research.

With the wide wavelength tuning range, the FEL is an intrinsically well-suited device to produce lasing with multiple colors. Taking advantage of the availability of an undulator system with multiple undulators, we have demonstrated the first two-color lasing of a storage ring FEL. Using either a three- or four-undulator configuration with a pair of dual-band high reflectivity mirrors, we have achieved simultaneous lasing in the IR and UV spectral regions. With the low-gain feature of the storage ring FEL, the power generated at the two wavelengths can be equally built up and precisely balanced to reach FEL saturation. A systematic experimental program to characterize this two-color FEL has been carried out, including precise power control, a study of the power stability of two-color lasing, wavelength tuning, and the impact of the FEL mirror degradation. Using this two-color laser, we have started to develop a new two-color $\gamma$-ray beam for scientific research at the HIGS.

Using the undulator switchyard, four helical undulators installed in the beamline can be configured to not only enhance the FEL gain in the VUV regime, but also allow for the full polarization control of the FEL beams. For the accelerator operation, the use of helical undulators is essential to extend the FEL mirror lifetime by reducing radiation damage from harmonic undulator radiation. Using a pair of helical undulators with opposite helicities, we have realized (1) fast helicity switching between left- and right-circular polarizations, and (2) the generation of fully controllable linear polarization. In order to extend these new capabilities of polarization control to the $\gamma$-ray operation in a wide energy range at the HIGS, a set of FEL polarization diagnostic systems need to be developed to cover the entire FEL wavelength range. The preliminary development of the polarization diagnostics for the wavelength range from IR to UV has been carried out.

Efficient Computation of Electromagnetic Waves in Hydrocarbon Exploration Using the Improved Numerical Mode Matching (NMM) Method

In this study, we developed and improved the numerical mode matching (NMM) method which has previously been shown to be a fast and robust semi-analytical solver to investigate the propagation of electromagnetic (EM) waves in an isotropic layered medium. The applicable models, such as cylindrical waveguide, optical fiber, and borehole with earth geological formation, are generally modeled as an axisymmetric structure which is an orthogonal-plano-cylindrically layered (OPCL) medium consisting of materials stratified planarly and layered concentrically in the orthogonal directions.

In this report, several important improvements have been made to extend applications of this efficient solver to the anisotropic OCPL medium. The formulas for anisotropic media with three different diagonal elements in the cylindrical coordinate system are deduced to expand its application to more general materials. The perfectly matched layer (PML) is incorporated along the radial direction as an absorbing boundary condition (ABC) to make the NMM method more accurate and efficient for wave diffusion problems in unbounded media and applicable to scattering problems with lossless media. We manipulate the weak form of Maxwell's equations and impose the correct boundary conditions at the cylindrical axis to solve the singularity problem which is ignored by all previous researchers. The spectral element method (SEM) is introduced to more efficiently compute the eigenmodes of higher accuracy with less unknowns, achieving a faster mode matching procedure between different horizontal layers. We also prove the relationship of the field between opposite mode indices for different types of excitations, which can reduce the computational time by half. The formulas for computing EM fields excited by an electric or magnetic dipole located at any position with an arbitrary orientation are deduced. And the excitation are generalized to line and surface current sources which can extend the application of NMM to the simulations of controlled source electromagnetic techniques. Numerical simulations have demonstrated the efficiency and accuracy of this method.

Finally, the improved numerical mode matching (NMM) method is introduced to efficiently compute the electromagnetic response of the induction tool from orthogonal transverse hydraulic fractures in open or cased boreholes in hydrocarbon exploration. The hydraulic fracture is modeled as a slim circular disk which is symmetric with respect to the borehole axis and filled with electrically conductive or magnetic proppant. The NMM solver is first validated by comparing the normalized secondary field with experimental measurements and a commercial software. Then we analyze quantitatively the induction response sensitivity of the fracture with different parameters, such as length, conductivity and permeability of the filled proppant, to evaluate the effectiveness of the induction logging tool for fracture detection and mapping. Casings with different thicknesses, conductivities and permeabilities are modeled together with the fractures in boreholes to investigate their effects for fracture detection. It reveals that the normalized secondary field will not be weakened at low frequencies, ensuring the induction tool is still applicable for fracture detection, though the attenuation of electromagnetic field through the casing is significant. A hybrid approach combining the NMM method and BCGS-FFT solver based integral equation has been proposed to efficiently simulate the open or cased borehole with tilted fractures which is a non-axisymmetric model.

Dosimetric Evaluation of Metal Artefact Reduction using Metal Artefact Reduction (MAR) Algorithm and Dual-energy Computed Tomography (CT) Method

Purpose: Computed Tomography (CT) is one of the standard diagnostic imaging modalities for the evaluation of a patient’s medical condition. In comparison to other imaging modalities such as Magnetic Resonance Imaging (MRI), CT is a fast acquisition imaging device with higher spatial resolution and higher contrast-to-noise ratio (CNR) for bony structures. CT images are presented through a gray scale of independent values in Hounsfield units (HU). High HU-valued materials represent higher density. High density materials, such as metal, tend to erroneously increase the HU values around it due to reconstruction software limitations. This problem of increased HU values due to metal presence is referred to as metal artefacts. Hip prostheses, dental fillings, aneurysm clips, and spinal clips are a few examples of metal objects that are of clinical relevance. These implants create artefacts such as beam hardening and photon starvation that distort CT images and degrade image quality. This is of great significance because the distortions may cause improper evaluation of images and inaccurate dose calculation in the treatment planning system. Different algorithms are being developed to reduce these artefacts for better image quality for both diagnostic and therapeutic purposes. However, very limited information is available about the effect of artefact correction on dose calculation accuracy. This research study evaluates the dosimetric effect of metal artefact reduction algorithms on severe artefacts on CT images. This study uses Gemstone Spectral Imaging (GSI)-based MAR algorithm, projection-based Metal Artefact Reduction (MAR) algorithm, and the Dual-Energy method.

Materials and Methods: The Gemstone Spectral Imaging (GSI)-based and SMART Metal Artefact Reduction (MAR) algorithms are metal artefact reduction protocols embedded in two different CT scanner models by General Electric (GE), and the Dual-Energy Imaging Method was developed at Duke University. All three approaches were applied in this research for dosimetric evaluation on CT images with severe metal artefacts. The first part of the research used a water phantom with four iodine syringes. Two sets of plans, multi-arc plans and single-arc plans, using the Volumetric Modulated Arc therapy (VMAT) technique were designed to avoid or minimize influences from high-density objects. The second part of the research used projection-based MAR Algorithm and the Dual-Energy Method. Calculated Doses (Mean, Minimum, and Maximum Doses) to the planning treatment volume (PTV) were compared and homogeneity index (HI) calculated.

Results: (1) Without the GSI-based MAR application, a percent error between mean dose and the absolute dose ranging from 3.4-5.7% per fraction was observed. In contrast, the error was decreased to a range of 0.09-2.3% per fraction with the GSI-based MAR algorithm. There was a percent difference ranging from 1.7-4.2% per fraction between with and without using the GSI-based MAR algorithm. (2) A range of 0.1-3.2% difference was observed for the maximum dose values, 1.5-10.4% for minimum dose difference, and 1.4-1.7% difference on the mean doses. Homogeneity indexes (HI) ranging from 0.068-0.065 for dual-energy method and 0.063-0.141 with projection-based MAR algorithm were also calculated.

Conclusion: (1) Percent error without using the GSI-based MAR algorithm may deviate as high as 5.7%. This error invalidates the goal of Radiation Therapy to provide a more precise treatment. Thus, GSI-based MAR algorithm was desirable due to its better dose calculation accuracy. (2) Based on direct numerical observation, there was no apparent deviation between the mean doses of different techniques but deviation was evident on the maximum and minimum doses. The HI for the dual-energy method almost achieved the desirable null values. In conclusion, the Dual-Energy method gave better dose calculation accuracy to the planning treatment volume (PTV) for images with metal artefacts than with or without GE MAR Algorithm.

A New Method for Modeling Free Surface Flows and Fluid-structure Interaction with Ocean Applications

The computational modeling of ocean waves and ocean-faring devices poses numerous challenges. Among these are the need to stably and accurately represent both the fluid-fluid interface between water and air as well as the fluid-structure interfaces arising between solid devices and one or more fluids. As techniques are developed to stably and accurately balance the interactions between fluid and structural solvers at these boundaries, a similarly pressing challenge is the development of algorithms that are massively scalable and capable of performing large-scale three-dimensional simulations on reasonable time scales. This dissertation introduces two separate methods for approaching this problem, with the first focusing on the development of sophisticated fluid-fluid interface representations and the second focusing primarily on scalability and extensibility to higher-order methods.

We begin by introducing the narrow-band gradient-augmented level set method (GALSM) for incompressible multiphase Navier-Stokes flow. This is the first use of the high-order GALSM for a fluid flow application, and its reliability and accuracy in modeling ocean environments is tested extensively. The method demonstrates numerous advantages over the traditional level set method, among these a heightened conservation of fluid volume and the representation of subgrid structures.

Next, we present a finite-volume algorithm for solving the incompressible Euler equations in two and three dimensions in the presence of a flow-driven free surface and a dynamic rigid body. In this development, the chief concerns are efficiency, scalability, and extensibility (to higher-order and truly conservative methods). These priorities informed a number of important choices: The air phase is substituted by a pressure boundary condition in order to greatly reduce the size of the computational domain, a cut-cell finite-volume approach is chosen in order to minimize fluid volume loss and open the door to higher-order methods, and adaptive mesh refinement (AMR) is employed to focus computational effort and make large-scale 3D simulations possible. This algorithm is shown to produce robust and accurate results that are well-suited for the study of ocean waves and the development of wave energy conversion (WEC) devices.