A class of immersed finite element (IFE) spaces is developed for solving elliptic boundary value problems that have interfaces. IFE spaces are finite element approximation spaces which are based upon meshes that can be independent of interfaces in the domain. Three different quadratic IFE spaces and their related biquadratic IFE spaces are introduced here for the purposes of solving both forward and inverse elliptic interface problems in 1D and 2D. These different spaces are constructed by (i) using a hierarchical approach, (ii) imposing extra continuity requirements or (iii) using a local refinement technique. The interpolation properties of each space are tested against appropriate testing functions in 1D and 2D. The IFE spaces are also used to approximate the solution of a forward elliptic interface problem using the Galerkin finite element method and the mixed least squares finite element method. Finally, one appropriate space is selected to solve an inverse interface problem using either an output least squares approach or the least squares with mixed equation error method. / Ph. D.

In this thesis, we develop the a new immersed finite element(IFE) space formed by piecewise linear polynomials defined on sub-elements cut by the actual interface curve for solving elliptic interface problems on interface independent meshes. A group of geometric identities and estimates on interface elements are derived. Based on these geometric identities and estimates, we establish a multi-point Taylor expansion of the true solutions and show the estimates for the second order terms in the expansion. Then, we construct the local IFE spaces by imposing the weak jump conditions and nodal value conditions on the piecewise polynomials. The unisolvence of the IFE shape functions is proven by the invertibility of the well-known Sherman-Morrison system. Furthermore we derive a group of fundamental identities about the IFE shape functions, which show that the two polynomial components in an IFE shape function are highly related. Finally we employ these fundamental identities and the multi-point Taylor expansion to derive the estimates for IFE interpolation errors in L2 and semi-H1 norms. / Master of Science

The present thesis focuses on evaluating a conjugate heat transfer (CHT) simulation in a ribbed cooling passage with a fully developed flow assumption using LES with the immersed boundary method (IBM-LES-CHT). The IBM with the LES model (IBM-LES) and the IBM with CHT boundary condition (IBM-CHT) frameworks are validated prior to the main simulations by simulating purely convective heat transfer (iso-flux) in the ribbed duct, and a developing laminar boundary layer flow over a two-dimensional flat plate with heat conduction, respectively. For the main conjugate simulations, a ribbed duct geometry with a blockage ratio of 0.3 is simulated at a bulk Reynolds number of 10,000 with a conjugate boundary condition applied to the rib surface. The nominal Biot number is kept at 1, which is similar to the comparative experiment. As a means to overcome a large time scale disparity between the fluid and the solid regions, the use of a high artificial solid thermal diffusivity is compared to the physical diffusivity. It is shown that while the diffusivity impacts the instantaneous fluctuations in temperature, heat transfer and Nusselt numbers, it has an insignificantly small effect on the mean Nusselt number. The comparison between the IBM-LES-CHT and iso-flux simulations shows that the iso-flux case predicts higher local Nusselt numbers at the back face of the rib. Furthermore, the local Nusselt number augmentation ratio (EF) predicted by IBM-LES-CHT is compared to the body fitted grid (BFG) simulation, experiment and another LES conjugate simulation. Even though there is a mismatch between IBM-LES-CHT prediction and other studies at the front face of the rib, the area-averaged EF compares reasonably well in other regions between IBM-LES-CHT prediction and the comparative studies. / Master of Science / The present thesis focuses on the computational study of the conjugate heat transfer (CHT) investigation on the turbine internal ribbed cooling channel. Plenty of prior research on turbine internal cooling channel have been conducted by considering only the convective heat transfer at the wall, which assumes an iso-flux (constant heat flux) boundary condition at the surface. However, applying an iso-flux condition on the surface is far from the realistic heat transfer mechanism occurring in internal cooling systems. In this work, a conjugate heat transfer analysis of the cooling channel, which considers both the conduction within the solid wall and the convection at the ribbed inner wall surface, is conducted for more realistic heat transfer coefficient prediction at the inner ribbed wall. For the simulation, the computational mesh is generated by the immersed boundary method (IBM), which can ease the mesh generation by simply immersing the CAD geometry into the background volume grid. The IBM is combined with the conjugate boundary condition to simulate the internal ribbed cooling channel. The conjugate simulation is compared with the experimental data and another computational study for the validation. Even though there are some discrepancy between the IBM simulation and other comparative studies, overall results are in good agreement. From the thermal prediction comparison between the iso-flux case and the conjugate case v using the IBM, it is found that the heat transfer predicted by the conjugate case is different from the iso-flux case by more than 40 percent at the rib back face. The present study shows the potential of the IBM framework with the conjugate boundary condition for more complicated geometry, such as full turbine blade model with external and internal cooling system.

The delivery of drugs into solid tumors is not trivial due to obstructions in the tumor microenvironment. Innovative drug delivery vehicles are currently being designed to overcome this challenge. In this research, computational fluid dynamics (CFD) simulations were used to evaluate the behavior of several drug delivery vectors in tumor capillaries—specifically motile bacteria, non-motile bacteria, and nanoparticles. Red blood cells, bacteria, and nanoparticles were imposed in the flow using the immersed boundary method. A human capillary model was developed using a novel method of handling deformable red blood cells (RBC). The capillary model was validated with experimental data from the literature. A stochastic model of bacteria motility was defined based on experimentally observed run and tumble behavior. The capillary and bacteria models were combined to simulate the intracapillary transport of bacteria. Non-motile bacteria and nanoparticles of 200 nm, 300 nm, and 405 nm were also simulated in capillary flow for comparison to motile bacteria. Motile bacteria tended to swim into the plasma layer near the capillary wall, while non-motile bacteria tended to get caught in the bolus flow between the RBCs. The nanoparticles were more impacted by Brownian motion and small scale fluid fluctuations, so they did not trend toward a single region of the flow. Motile bacteria were found to have the longest residence time in a 1 mm long capillary as well as the highest average radial velocity. This suggests motile bacteria may enter the interstitium at a higher rate than non-motile bacteria or nanoparticles of diameters between 200–405 nm. / Master of Science

We present immersed discontinuous Galerkin finite element methods for one and two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to be cut by the material interface. The proposed methods use the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one fluid while on interface elements containing more than one fluid they use specially-built piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. Additional curl-free and consistency conditions are added to generate bilinear and biquadratic piecewise shape functions for two dimensional problems. We established the existence and uniqueness of one dimensional immersed finite element shape functions and existence of two dimensional bilinear immersed finite element shape functions for the velocity. The proposed methods are tested on one dimensional problems and are extended to two dimensional problems where the problem is defined on a domain split by an interface into two different media. Our methods exhibit optimal $O(h^{p+1})$ convergence rates for one and two dimensional problems. However it is observed that one of the proposed methods is not stable for two dimensional interface problems with high contrast media such as water/air. We performed an analysis to prove that our immersed Petrov-Galerkin method is stable for interface problems with high jumps across the interface. Local time-stepping and parallel algorithms are used to speed up computation. Several realistic interface problems such as ether/glycerol, water/methyl-alcohol and water/air with a circular interface are solved to show the stability and robustness of our methods. / Ph. D.

This thesis develops an immersed finite element (IFE) space for numerical simulations arising from beam design with multiple materials. This IFE space is based upon meshes that can be independent of interface of the materials used to form a beam. Both the forward and inverse problems associated with the beam equation are considered. The order of accuracy of this IFE space is numerically investigated from the point of view of both the interpolation and finite element solution of the interface boundary value problems. Both single and multiple interfaces are considered in our numerical simulation. The results demonstrate that this IFE space has the optimal order of approximation capability. / Master of Science

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This thesis presents a numerical method for the phase-resolving simulation of rigid particles and deformable bubbles in viscous, magnetohydrodynamic flows. The presented approach features solid robustness and high numerical efficiency. The implementation is three-dimensional and fully parallel suiting the needs of modern high-performance computing. In addition to the steps towards magnetohydrodynamics, the thesis covers method development with respect to the immersed boundary method which can be summarized in simple words by From rigid spherical particles to deformable bubbles. The development comprises the extension of an existing immersed boundary method to non-spherical particles and very low particle-to-fluid density ratios. A detailed study is dedicated to the complex interaction of particle shape, wake and particle dynamics. Furthermore, the representation of deformable bubble shapes, i.e. the coupling of the bubble shape to the fluid loads, is accounted for. The topic of bubble interaction is surveyed including bubble collision and coalescence and a new coalescence model is introduced. The thesis contains applications of the method to simulations of the rise of a single bubble and a bubble chain in liquid metal with and without magnetic field highlighting the major effects of the field on the bubble dynamics and the flow field. The effect of bubble coalescence is quantified for two closely adjacent bubble chains. A framework for large-scale simulations with many bubbles is provided to study complex multiphase phenomena like bubble-turbulence interaction in an efficient manner.

Currently, the concept of multipurpose spacecrafts is being transformed into many small spacecrafts each of them performing specific tasks and thus leading to the realization of pico and nano satellites. No matter what is the application or size, demand for more number of IR channels for earth observation is ever increasing which necessitates significant reduction in the mass, power requirement and cost of the IR detectors. In this scenario, several order of magnitude mass and power savings associated with uncooled IR arrays are advantageous compared to cooled photon detectors. However the poor speed of response of uncooled microbolometer array devices obstruct the total replacement of cooled detectors in thermal imaging applications. This is especially true when the mission requires 50 m to 100 m ground resolution, in which even the "fastest" micro bolometer arrays turns "too slow" to follow the ground trace when looked from low earth orbit (LEO). Hence there is a great and unfulfilled requirement of faster uncooled detector arrays for meeting the demand for future micro and mini satellite projects for advanced missions. The present thesis describes the systematic studies carried out in development of high performance IR and THz detectors for space applications. Ge-Si-O thin films are prepared by ion beam sputtering technique with argon (Ar) alone and argon and oxygen as sputtering species, using sputtering targets of different compositions of Ge and SiO2. The deposited thin films are amorphous in nature and have chemical compositions close to that of the target. The study of electrical properties has shown that the activation energy and hence the thermistor constant (β) and electrical resistivity (ρ) are sensitive to oxygen flow rate, and they are the least for thin films prepared with Ar alone as the sputtering species. Different thermal isolation structures (TIS), consisting of silicon nitride (Si3N4) membrane of different thicknesses, Ge-Si-O thin film and, chromium coating on the rear side of the membrane, are prepared by bulk micro-machining technique, whose thermal conductance (Gth) properties are evaluated from the experimentally determined current-voltage (I-V) characteristics. Gth shows non-linear dependence with respect to raise in temperature of thin film thermistor due to Joule heating. The infrared micro-bolometer detectors, fabricated using one of the TIS structures have shown responsivity (<v) close to 115 V W−1 at a bias voltage of 1.5 V and chopping frequency of 10 Hz, thermal time constant (τth) of 2.5 ms and noise voltage of 255 nV Hz−1⁄2 against the corresponding thermal properties of Gth and thermal capacitance Cth equal to 9.0 × 10−5 W K−1 and 1.95 × 10−7 J K−1 respectively. The detectors are found to have uniform spectral response in the infrared region from 2 µm to 20 µm, and NEDT in the range from 108 mK to 574 mK when used with an F/1 optical system. The detector, in an infrared earth sensor system, is tested before an extended black body which simulates the earth disc in the laboratory and the results are discussed. As an extension of the single element detector to array device, design of a microbolometer array for earth sensor dispensing of scanning mechanisms is presented. It makes use of four microbolometer arrays with in-line staggered configuration that stare at the earth horizons, perceiving IR radiation in the spectral band of 14 µm to 16 µm. Design of the microbolometer has been carried out keeping in mind low power, lightweight, without compromising on the performance. An array configuration of 16 × 2 pixels is designed and developed for this purpose. Finite elemental analysis is carried out for design optimization to yield best thermal properties and thus high performance of the detectors. Suitable optical design configuration was arrived to image the earth horizon on to array. Using this optimum design, prototype arrays have been fabricated, packaged and tested in front of the black body radiation source and found to have Responsivity, NEP, and D∗ of 120 V W−1, 5.0 W Hz−1⁄2, 1.10 × 107 cm Hz1⁄2 W−1 respectively. The pixels show a uniform response within a spread of ±6 % and the pixel resistances are within a range of ±5 %. Optically Immersed Bolometer IR detectors are fabricated using electron beam evaporated Vanadium Oxide as the sensing material. Spin coated polyimide is used as medium to optically immerse the sensing element to the flat surface of a hemispherical germanium lens. This optical immersion layer also serves as the thermal impedance control layer and decides the performance of the devices in terms of responsivity and noise parameters. The devices have been packaged in suitable electro-optical packages and the detector parameters are studied in detail. Thermal time constant varies from 0.57 ms to 6.1 ms and responsivity from 75VW−1 to 757VW−1 corresponding to polyimide thickness in the range 2.0 μm to 70 μm for a detector bias of 9V. Highest D obtained was 1.28 × 108 cm Hz1⁄2W−1. Noise Equivalent Temperature Difference (NETD) of 20mK is achieved for devices with polyimide thickness of 32 μm, whereas the NETD × th product is the lowest for devices with moderate thickness of thermal impedance layer. Bolometric THz detectors were fabricated using V2O5 as sensing element immersed onto germanium hemispherical lens using polyimide as immersion media. These detectors were characterized for their efficiency in detection of THz radiation in the range 10 THz to 35 THz emitted by a black body radiator. The responsivity of the devices determined in four different frequency bands covering the spectrum of interest and a maximum responsivity of 398VW−1 was observed. A variation in the responsivity is observed which is due to the characteristics absorption of polyimide in the THz region of interest and can be avoided by replacing with HDPE which has less attenuation. NEP of 6.8 × 10−10WHz−1⁄2 was observed which is very close to the state of art in the case of uncooled detectors which entitles the detectors for spectroscopic applications. Specific Detectivity D* was observed to be much higher than the conventional detectors thanks to the benefits of immersion. NETD of 26mK was observed which is advantageous of application of these detectors in imaging applications These studies have lead to development of a new technology for fabrication of high performance IR and THz detectors which can be used for spectroscopic and imaging applications. Further, this technology can be scaled for development of linear and area arrays finding applications where the speed of respnose as well as sensitivity are of equal importance. from 0.57 ms to 6.1 ms and responsivity from 75 V W−1 to 757 V W−1 corresponding to polyimide thickness in the range 2.0 µm to 70 µm for a detector bias of 9 V. Highest D∗ obtained was 1.28 × 108 cm Hz1⁄2 W−1. Noise Equivalent Temperature Difference (NETD) of 20 mK is achieved for devices with polyimide thickness of 32 µm, whereas the NETD × τth product is the lowest for devices with moderate thickness of thermal impedance layer. Bolometric THz detectors were fabricated using V2O5 as sensing element immersed onto germanium hemispherical lens using polyimide as immersion media. These detectors were characterized for their efficiency in detection of THz radiation in the range 10 THz to 35 THz emitted by a black body radiator. The responsivity of the devices determined in four different frequency bands covering the spectrum of interest and a maximum responsivity of 398 V W−1 was observed. A variation in the responsivity is observed which is due to the characteristics absorption of polyimide in the THz region of interest and can be avoided by replacing with HDPE which has less attenuation. NEP of 6.8 × 10−10 W Hz−1⁄2 was observed which is very close to the state of art in the case of uncooled detectors which entitles the detectors for spectroscopic applications. Specific Detectivity D* was observed to be much higher than the conventional detectors thanks to the benefits of immersion. NETD of 26 mK was observed which is advantageous of application of these detectors in imaging applications These studies have lead to development of a new technology for fabrication of high performance IR and THz detectors which can be used for spectroscopic and imaging applications. Further, this technology can be scaled for development of linear and area arrays finding applications where the speed of respnose as well as sensitivity are of equal importance.

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