In this thesis we study hybrid numerical methods to be usedin computational electromagnetics. We restrict the methods tospectral domain and scattering problems. The hybrids consist ofcombinations of Boundary Element Methods and Geometrical Theoryof Diffraction. In the thesis three hybrid methods will be presented. Onemethod has been developped from a theoretical idea to anindustrial code. The two other methods will be presented mainlyfrom a theoretical perspective. We will also give shortintroductions to the Boundary Element Method and theGeometrical Theory of Diffraction from a theoretical andimplementational point of view. <b>Keywords:</b>Maxwell’s equations, Geometrical Theoryof Diffraction, Boundary Element Method, Hybrid methods,Electromagnetic Scattering / NR 20140805

<div>Quantifying the dynamics and rheology of soft biological suspensions such as red blood cells, vesicles, or capsules is paramount to many biomedical and computational applications. These systems are multiphase flows that can contain a diverse set of deformable cells and rigid bodies with complex wall geometries. For this thesis, we are performing several numerical simulations using boundary element methods (BEM) for biological suspensions in biomedically relevant conditions. Each simulation is devised to answer fundamental questions in modeling these systems.</div><div><br></div><div><br></div><div>Part of this thesis centers around the fluid mechanics of giant unilamellar vesicles (GUVs), fluid droplets surrounded by a phospholipid bilayer. GUVs are important to study because they mimic the dynamics of anuclear cells and are commonly used as a basis for artificial cells. The dynamics of vesicles in simple shear or extensional flows have been extensively studied. However the conditions seen in microfluidic devices or industrial processing are not always described by steady shear or extensional flows alone, and require more investigation. In our first study, we investigate the shape stability of osmotically deflated vesicles in a general linear flow (i.e., linear combinations of extensional and rotational flows). We modeled the vesicles as a droplet with an incompressible interface with a bending resistance. We simulated a range of flow types from purely shear to purely extensional at viscosity ratios ranging from 0.01 to 5.0 and reduced volumes (measured asphericity, higher is more spherical) from 0.60 to 0.70. The vesicle's viscosity ratio appears to play a minimal role in describing its shape and stability for many mixed flows, even in cases when significant flows are present in the vesicle interior. We find in these cases that the bending critical capillary number for shape instabilities collapse onto similar values if the capillary number is scaled by an effective extensional rate. These results contrast with droplet studies where both viscosity ratio and flow type have significant effects on breakup. Our simulations suggest that if the flow type is not close to pure shear flow, one can accurately quantify the shape and stability of vesicles using the results from an equiviscous vesicle in pure extension. Only when the flow type is nearly shear flow, do we start to see deviations in the observations discussed above. In this situation, the vesicle's stationary shape develops a shape deviation, which introduces a stabilizing effect and makes the critical capillary number depend on the viscosity ratio.</div><div><br></div><div><br></div><div>Continuing with our research on single vesicle dynamics, we have performed simulations and experiments on vesicles in large amplitude oscillatory extensional (LAOE) flows. By using LAOE we can probe the non-linear extension and compression of vesicles and how these types of deformation affect dilute suspension microstructure in time-dependent flows through contractions, expansions, or other complex geometries. Our numerical and experimental results for vesicles of reduced volumes from 0.80 to 0.95 have shown there to be three general dynamical regimes differentiated by the amount of deformation that occurs in each half cycle. We have termed the regimes: symmetrical, reorienting, and pulsating in reference to the type of deformation that occurs. We find the deformation of the quasispherical vesicles in the microfluidic experiments and boundary element simulations to be in quantitative agreement. The distinct dynamics observed in each regime result from a competition between the flow frequency, flow time scale, and membrane deformation timescale. Using the numerical results, we calculate the particle coefficient of stresslet and quantify the nonlinear relationship between average vesicle stress and strain rate. We additionally present some results on the dynamics of tubular vesicles in LAOE, showing how the experiments suggest the vesicles undergo a shape transformation over several strain rate cycles. Broadly, our work provides new information regarding the transient dynamics of vesicles in time-dependent flows that directly informs bulk suspension rheology.</div><div><br></div><div><br></div><div>Our most recent project deals with the accuracy of discretized double layer integrals for Stokes flow in the boundary element method.</div><div>In the fluid mechanics literature, the chosen parameterization, meshing procedure, and singularity handling are often selected arbitrarily or based on a convergence study where the number of elements is decreased until the relative error is sufficiently low.</div><div>A practical study on the importance of each of these parameters to the accurate calculation of physically relevant results, such as the particle stresslet, could alleviate some of the guesswork required. The analytical formulas for the eigenfunctions/eigenvalues of the double layer operator of an ellipsoidal particle in a quadratic flow were recently published¹, providing an analytical basis for testing boundary element method discretization accuracy.</div><div>We use these solutions to examine the local and global errors produced by changing the interpolation order of the geometry and the double-layer density. The results show that the local errors can be significant even when the global errors are small, prompting additional study on the distribution of local errors. Interestingly, we find that increasing the interpolation orders for the geometry and the double layer density does not always guarantee smaller errors. Depending on the nature of the meshing near high curvature regions, the number of high aspect ratio elements, and the flatness of the particle geometry, a piecewise-constant density can exhibit lower errors than piecewise-linear density, and there can be little benefit from using curved triangular elements. Overall, this study provides practical insights on how to appropriately discretize and parameterize three-dimensional (3D) boundary-element simulations for elongated particles with prolate-like and oblate-like geometries.</div><div><br></div>

In contrast to the structural modes, which describe the physical motion of vibrating structures, acoustic radiation modes describe the radiated sound power. Radiation modes are beneficial in active noise control because reducing an efficiently radiating radiation mode guarantees the reduction of radiated sound power. Much work has been done to calculate the radiation modes for simple geometries, where analytic solutions are available. In this work, isogeometric analysis (IGA) is used to provide a tool capable of analyzing the radiation modes of arbitrarily complex geometries. IGA offers increased accuracy and efficiency by using basis functions generated from Non-Uniform Rational B-Splines (NURBS) or T-Splines, which can represent geometries exactly. Results showing this increased accuracy and efficiency with IGA using T-Splines are shown for a sphere to validate the method, comparing with the exact analytical solution as well as results from a traditional boundary element method. A free cylindrical shell is also analyzed to show the usefulness of this method. Expected similarities, as well as expected differences, are observed between this free shell and a baffled cylindrical shell.

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Slip on the geometrically complex Rhyolite Ridge Fault System and associated local stresses in the Desert Peak Geothermal Field in Nevada, were modeled with the boundary element method (BEM) implemented in Poly3D. The impact of uncertainty in the fault geometry at depth, the tectonic stresses driving slip, and the potential ranges of frictional strength resisting slip on the likely predictions of fracture slip and formation in the surrounding volume due to these local stresses were systematically explored and quantified. The effect of parameter uncertainty was evaluated by determining the frequency distribution of model predicted values. Alternatively, Bayesian statistics were used to determine the best fitting values for parameters within a probability distribution derived from the difference of the model prediction from the observed data. This approach honors the relative contribution of uncertainties from all existing data that constrains the fault parameters. Lastly, conceptual models for different fault geometries and their evolution were heuristically explored and the predictions of local stress states were compared to available measurements of the local stresses, fault and fracture patterns at the surface and in boreholes, and the spatial extent of the geothermal field. The complex fault geometry leads to a high degree of variability in the locations experiencing stress states that promote fracture, but such locations generally correlate with the main injection and production wells at Desert Peak. In addition, the strongest and most common stress concentrations occur within relays between unconnected fault segments, and at bends and intersections in faults that connect overlapping fault segments associated with relays. The modeling approach in this study tests the conceptual model of the fault geometry at Desert Peak while honoring mechanical constants and available constraints on driving stresses and provides a framework that aids in geothermal exploration by predicting the spatial variations in stresses likely to cause and reactivate fractures necessary to sustain hydrothermal fluid flow. This approach also quantifies the relative sensitivity of such predictions to fault geometry, remote stress, and friction, and determines the best fitting model with its associated probability. / Geology

Melting and vaporization of solids occur very often in nature and in man-made processes. Many analytical and numerical solutions exist for solving the temperature field in the liquid and solid regions, but inaccuracies persist in tracking the phase change interfaces and the numerical solution of the temperature field is usually cumbersome. The Boundary Element Method is proposed as an accurate, efficient way to solve for the temperature field and the interface positions in a phase change problem involving combined melting and vaporization. When comparing to specific one-dimensional test cases, accurate results arc obtained when using a sufficiently small time step. A comparison is made to existing data from a laser drilling experiment. The anticipated physical effects which occur on semi-infinite and finite domains arc confirmed. Consequently, this method can be used to model natural and industrial phenomena involving phase change. / Master of Science