Elliptic partial differential equations that are used to model physical phenomena give rise to large sparse linear systems. Such systems can be symmetric positive definite and can be solved by the preconditioned conjugate gradients method. In this thesis, we develop support graph preconditioners for symmetric positive definite matrices that arise from the finite element discretization of elliptic partial differential equations. An object oriented code is developed for the construction, integration and application of these preconditioners. Experimental results show that the advantages of support graph preconditioners are retained in the proposed extension to the finite element matrices.

A new format for microfluidic based DNA biosensors is presented in which the biorecognition element (single stranded DNA probes) is immobilized as a continuous density gradient of probes along the length of a microfluidic channel instead of a standard array format commonly used in microarray technologies or DNA based biosensors. The development of continuous density gradients of immobilized probe was achieved by electrokinetically subjecting probes that were terminated with an appropriate functional group for a surface coupling reaction to increasing convective velocity along the length of the microfluidic channel. This gradient format was able to discriminate between a fully complementary target and one containing 3 BPM based on the spatial pattern of hybridization for picomole quantities of DNA targets. Temperature mediated destabilization of DNA hybrids demonstrated that the density of immobilized probes plays an important role in the thermodynamic stability of DNA hybrids. In addition, it was found that efficiency, selectivity and melt temperature of DNA hybrids for surface based hybridization is dependent on the density of the probe molecules.

A new format for microfluidic based DNA biosensors is presented in which the biorecognition element (single stranded DNA probes) is immobilized as a continuous density gradient of probes along the length of a microfluidic channel instead of a standard array format commonly used in microarray technologies or DNA based biosensors. The development of continuous density gradients of immobilized probe was achieved by electrokinetically subjecting probes that were terminated with an appropriate functional group for a surface coupling reaction to increasing convective velocity along the length of the microfluidic channel. This gradient format was able to discriminate between a fully complementary target and one containing 3 BPM based on the spatial pattern of hybridization for picomole quantities of DNA targets. Temperature mediated destabilization of DNA hybrids demonstrated that the density of immobilized probes plays an important role in the thermodynamic stability of DNA hybrids. In addition, it was found that efficiency, selectivity and melt temperature of DNA hybrids for surface based hybridization is dependent on the density of the probe molecules.

An experimental program is designed to study the salient features of separated and reattached flows in pressure gradients generated in asymmetric diverging and converging channels. The channels comprised a straight flat floor and a curved roof that was preceded and followed by straight parallel walls. Reference measurements were also made in a parallel-wall channel to facilitate the interpretation of the pressure gradient flows. A transverse square rib located at the start of convergence/divergence was used to create separation inside the channels. In order to simplify the interpretation of the relatively complex separated and reattached flows in the asymmetric converging and diverging channels, measurements were made in the plain converging and diverging channel without the rib on the channel wall. All the measurements were obtained using a high resolution particle image velocimetry technique. The experiments without the ribs were conducted in the diverging channel at Reynolds number based on half channel depth (Reh) of 27050 and 12450 and in the converging channel at Reh = 19280. For each of these three test conditions, a high resolution particle image velocimetry technique (PIV) was used to conduct detailed velocity measurements in the upstream parallel section, within the converging and diverging section, and downstream of the converging and diverging sections. From these measurements, the boundary layer parameters and profiles of the mean velocities, turbulent quantities as well as terms in the transport equations for turbulent kinetic energy and Reynolds stresses were obtained to document the effects of pressure gradient on the flow. In the adverse pressure gradient case, the turbulent quantities were enhanced more significantly in the lower boundary layer than the upper boundary layer. On the other hand, favorable pressure gradient attenuated the turbulence levels and the effect was found to be similar on both the upper and the lower boundary layers. For the separated and reattached flows in the converging, diverging and parallel-wall channels at Reh = 19440, 12420 and 15350, respectively. The Reynolds number based on the approach velocity and rib height was Rek  2700. From these measurements, profiles of the mean velocities, turbulent quantities and the various terms in the transport equations for turbulent kinetic energy and Reynolds stresses were also obtained. The flow dynamics in the upper boundary layer in the separated region and the early stages of flow redevelopment were observed to be insensitive to the pressure gradients. In the lower boundary layer, however, the flow dynamics were entirely dominated by the separated shear layer in the separated region as well as the early region of flow redevelopment. The effects of the separated shear layer diminished in the redevelopment region so that the dynamics of the flow were dictated by the pressure gradients. The proper orthogonal decomposition (POD) was applied to educe the dominant large scale structures in the separated and reattached flows. These dominant scales were used to document structural differences between the canonical upstream flow and the flow field within the separated and redeveloping region. The contributions of these dominant structures to the dynamics of the Reynolds normal and shear stresses are also presented and discussed. It was observed that the POD recovers Reynolds shear stress more efficiently than the turbulent kinetic energy. The reconstruction reveals that large scales contribute more to the Reynolds shear stress than the turbulent kinetic energy. / February 2009

µL

Elliptic partial differential equations that are used to model physical phenomena give rise to large sparse linear systems. Such systems can be symmetric positive definite and can be solved by the preconditioned conjugate gradients method. In this thesis, we develop support graph preconditioners for symmetric positive definite matrices that arise from the finite element discretization of elliptic partial differential equations. An object oriented code is developed for the construction, integration and application of these preconditioners. Experimental results show that the advantages of support graph preconditioners are retained in the proposed extension to the finite element matrices.

We present a novel method for building 3D models from a user sketch. Given a 2D sketch as input, the approach aligns and deforms a chosen 3D template model to match the sketch. This is guided by a set of user-specified correspondences and an algorithm that deforms the 3D model to match the sketched profile. Our primary contribution is related to fitting the 3D deformable geometry to the 2D user sketch. We demonstrate our technique on several examples.

No description available.

An experimental program is designed to study the salient features of separated and reattached flows in pressure gradients generated in asymmetric diverging and converging channels. The channels comprised a straight flat floor and a curved roof that was preceded and followed by straight parallel walls. Reference measurements were also made in a parallel-wall channel to facilitate the interpretation of the pressure gradient flows. A transverse square rib located at the start of convergence/divergence was used to create separation inside the channels. In order to simplify the interpretation of the relatively complex separated and reattached flows in the asymmetric converging and diverging channels, measurements were made in the plain converging and diverging channel without the rib on the channel wall. All the measurements were obtained using a high resolution particle image velocimetry technique. The experiments without the ribs were conducted in the diverging channel at Reynolds number based on half channel depth (Reh) of 27050 and 12450 and in the converging channel at Reh = 19280. For each of these three test conditions, a high resolution particle image velocimetry technique (PIV) was used to conduct detailed velocity measurements in the upstream parallel section, within the converging and diverging section, and downstream of the converging and diverging sections. From these measurements, the boundary layer parameters and profiles of the mean velocities, turbulent quantities as well as terms in the transport equations for turbulent kinetic energy and Reynolds stresses were obtained to document the effects of pressure gradient on the flow. In the adverse pressure gradient case, the turbulent quantities were enhanced more significantly in the lower boundary layer than the upper boundary layer. On the other hand, favorable pressure gradient attenuated the turbulence levels and the effect was found to be similar on both the upper and the lower boundary layers. For the separated and reattached flows in the converging, diverging and parallel-wall channels at Reh = 19440, 12420 and 15350, respectively. The Reynolds number based on the approach velocity and rib height was Rek  2700. From these measurements, profiles of the mean velocities, turbulent quantities and the various terms in the transport equations for turbulent kinetic energy and Reynolds stresses were also obtained. The flow dynamics in the upper boundary layer in the separated region and the early stages of flow redevelopment were observed to be insensitive to the pressure gradients. In the lower boundary layer, however, the flow dynamics were entirely dominated by the separated shear layer in the separated region as well as the early region of flow redevelopment. The effects of the separated shear layer diminished in the redevelopment region so that the dynamics of the flow were dictated by the pressure gradients. The proper orthogonal decomposition (POD) was applied to educe the dominant large scale structures in the separated and reattached flows. These dominant scales were used to document structural differences between the canonical upstream flow and the flow field within the separated and redeveloping region. The contributions of these dominant structures to the dynamics of the Reynolds normal and shear stresses are also presented and discussed. It was observed that the POD recovers Reynolds shear stress more efficiently than the turbulent kinetic energy. The reconstruction reveals that large scales contribute more to the Reynolds shear stress than the turbulent kinetic energy.

In each iteration of the interior point method (IPM) at least one linear system has to be solved. The main computational effort of IPMs consists in the computation of these linear systems. Solving the corresponding linear systems with a direct method becomes very expensive for large scale problems. In this thesis, we have been concerned with using an iterative method for solving the reduced KKT systems arising in IPMs for linear programming. The augmented system form of this linear system has a number of advantages, notably a higher degree of sparsity than the normal equations form. We design a block triangular preconditioner for this system which is constructed by using a nonsingular basis matrix identified from an estimate of the optimal partition in the linear program. We use the preconditioned conjugate gradients (PCG) method to solve the augmented system. Although the augmented system is indefinite, short recurrence iterative methods such as PCG can be applied to indefinite system in certain situations. This approach has been implemented within the HOPDM interior point solver. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of IPM for this inexact case. We present the convergence analysis of the inexact infeasible path-following algorithm, prove the global convergence of this method and provide complexity analysis.