Maximum principle preserving high order schemes for convection-dominated diffusion equations

The maximum principle is an important property of solutions to PDE. Correspondingly, it's of great interest for people to design a high order numerical scheme solving PDE with this property maintained. In this thesis, our particular interest is solving convection-dominated diffusion equation. We first review a nonconventional maximum principle preserving(MPP) high order finite volume(FV) WENO scheme, and then propose a new parametrized MPP high order finite difference(FD) WENO framework, which is generalized from the one solving hyperbolic conservation laws. A formal analysis is presented to show that a third order finite difference scheme with this parametrized MPP flux limiters maintains the third order accuracy without extra CFL constraint when the low order monotone flux is chosen appropriately. Numerical tests in both one and two dimensional cases are performed on the simulation of the incompressible Navier-Stokes equations in vorticity stream-function formulation and several other problems to show the effectiveness of the proposed method.

Effect of mesh distortion on the accuracy of high order vorticity-velocity CFD approaches

The numerical solution of the incompressible Navier-Stokes equations offers an alternative to experimental analysis of fluid-structure interaction (FSI). We would save a lot of time and effort and help cut back on costs, if we are able to accurately model systems by these numerical solutions. These advantages are even more obvious when considering huge structures like bridges, high rise buildings or even wind turbine blades with diameters as large as 200 meters. The modeling of such processes, however, involves complex multiphysics problems along with complex geometries. This thesis focuses on a novel vorticity-velocity formulation called the Kinematic Laplacian Equation (KLE) to solve the incompressible Navier-stokes equations for such FSI problems. This scheme allows for the implementation of robust adaptive ordinary differential equations (ODE) time integration schemes, allowing us to tackle each problem as a separate module. The current algortihm for the KLE uses an unstructured quadrilateral mesh, formed by dividing each triangle of an unstructured triangular mesh into three quadrilaterals for spatial discretization. This research deals with determining a suitable measure of mesh quality based on the physics of the problems being tackled. This is followed by exploring methods to improve the quality of quadrilateral elements obtained from the triangles and thereby improving the overall mesh quality. A series of numerical experiments were designed and conducted for this purpose and the results obtained were tested on different geometries with varying degrees of mesh density.

Effect of high order interpolation in the stability and efficiency of the time-integration process in vorticity-velocity CFD algorithms

The numerical solution of the incompressible Navier-Stokes Equations offers an effective alternative to the experimental analysis of Fluid-Structure interaction i.e. dynamical coupling between a fluid and a solid which otherwise is very complex, time consuming and very expensive. To have a method which can accurately model these types of mechanical systems by numerical solutions becomes a great option, since these advantages are even more obvious when considering huge structures like bridges, high rise buildings, or even wind turbine blades with diameters as large as 200 meters. The modeling of such processes, however, involves complex multiphysics problems along with complex geometries. This thesis focuses on a novel vorticity-velocity formulation called the KLE to solve the incompressible Navier-stokes equations for such FSI problems. This scheme allows for the implementation of robust adaptive ODE time integration schemes and thus allows us to tackle the various multiphysics problems as separate modules. The current algorithm for KLE employs a structured or unstructured mesh for spatial discretization and it allows the use of a self-adaptive or fixed time step ODE solver while dealing with unsteady problems. This research deals with the analysis of the effects of the Courant-Friedrichs-Lewy (CFL) condition for KLE when applied to unsteady Stoke’s problem. The objective is to conduct a numerical analysis for stability and, hence, for convergence. Our results confirmthat the time step ∆t is constrained by the CFL-like condition ∆t ≤ const. hα, where h denotes the variable that represents spatial discretization.

Implementation of monotonic higher order upwind scheme in KIVA 4

KIVA is a FORTRAN code developed by Los Alamos national lab to simulate complete engine cycle. KIVA is a flow solver code which is used to perform calculation of properties in a fluid flow field. It involves using various numerical schemes and methods to solve the Navier-Stokes equation. This project involves improving the accuracy of one such scheme by upgrading it to a higher order scheme. The numerical scheme to be modified is used in the critical final stage calculation called as rezoning phase. The primitive objective of this project is to implement a higher order numerical scheme, to validate and verify that the new scheme is better than the existing scheme. The latest version of the KIVA family (KIVA 4) is used for implementing the higher order scheme to support handling the unstructured mesh. The code is validated using the traditional shock tube problem and the results are verified to be more accurate than the existing schemes in reference with the analytical result. The convection test is performed to compare the computational accuracy on convective transfer; it is found that the new scheme has less numerical diffusion compared to the existing schemes. A four valve pentroof engine, an example case of KIVA package is used as application to ensure the stability of the scheme in practical application. The results are compared for the temperature profile. In spite of all the positive results, the numerical scheme implemented has a downside of consuming more CPU time for the computational analysis. The detailed comparison is provided. However, in an overview, the implementation of the higher order scheme in the latest code KIVA 4 is verified to be successful and it gives better results than the existing scheme which satisfies the objective of this project.

MAGNETO-PHOTONIC CRYSTALS FOR OPTICAL SENSING APPLICATIONS

Among the optical structures investigated for optical sensing purpose, a significant amount of research has been conducted on photonic crystal based sensors. A particular advantage of photonic crystal based sensors is that they show superior sensitivity for ultra-small volume sensing. In this study we investigate polarization changes in response to the changes in the cover index of magneto-optic active photonic band gap structures. One-dimensional photonic-band gap structures fabricated on iron garnet materials yield large polarization rotations at the band gap edges. The enhanced polarization effects serve as an excellent tool for chemical sensing showing high degree of sensitivity for photonic crystal cover refractive index changes. The one dimensional waveguide photonic crystals are fabricated on single-layer bismuth-substituted rare earth iron garnet films ((Bi, Y, Lu)3(Fe, Ga)5O12 ) grown by liquid phase epitaxy on gadolinium gallium garnet substrates. Band gaps have been observed where Bragg scattering conditions links forward-going fundamental waveguide modes to backscattered high-order waveguide modes. Large near-band-edge polarization rotations which increase progressively with backscattered-mode order have been experimentally demonstrated for multiple samples with different composition, film thickness and fabrication parameters. Experimental findings are supported by theoretical analysis of Bloch modes polarization states showing that large near stop-band edge rotations are induced by the magneto-photonic crystal. Theoretical and experimental analysis conducted on polarization rotation sensitivity to waveguide photonic crystal cover refractive index changes shows a monotonic enhancement of the rotation with cover index. The sensor is further developed for selective chemical sensing by employing Polypyrrole as the photonic crystal cover layer. Polypyrrole is one of the extensively studied conducting polymers for selective analyte detection. Successful detection of aqueous ammonia and methanol has been achieved with Polypyrrole deposited magneto-photonic crystals.

Transformer modeling ATP : internal faults & high-frequency discretization

Power transformers are key components of the power grid and are also one of the most subjected to a variety of power system transients. The failure of a large transformer can cause severe monetary losses to a utility, thus adequate protection schemes are of great importance to avoid transformer damage and maximize the continuity of service. Computer modeling can be used as an efficient tool to improve the reliability of a transformer protective relay application. Unfortunately, transformer models presently available in commercial software lack completeness in the representation of several aspects such as internal winding faults, which is a common cause of transformer failure. It is also important to adequately represent the transformer at frequencies higher than the power frequency for a more accurate simulation of switching transients since these are a well known cause for the unwanted tripping of protective relays. This work develops new capabilities for the Hybrid Transformer Model (XFMR) implemented in ATPDraw to allow the representation of internal winding faults and slow-front transients up to 10 kHz. The new model can be developed using any of two sources of information: 1) test report data and 2) design data. When only test-report data is available, a higher-order leakage inductance matrix is created from standard measurements. If design information is available, a Finite Element Model is created to calculate the leakage parameters for the higher-order model. An analytical model is also implemented as an alternative to FEM modeling. Measurements on 15-kVA 240?/208Y V and 500-kVA 11430Y/235Y V distribution transformers were performed to validate the model. A transformer model that is valid for simulations for frequencies above the power frequency was developed after continuing the division of windings into multiple sections and including a higher-order capacitance matrix. Frequency-scan laboratory measurements were used to benchmark the simulations. Finally, a stability analysis of the higher-order model was made by analyzing the trapezoidal rule for numerical integration as used in ATP. Numerical damping was also added to suppress oscillations locally when discontinuities occurred in the solution. A maximum error magnitude of 7.84% was encountered in the simulated currents for different turn-to-ground and turn-to-turn faults. The FEM approach provided the most accurate means to determine the leakage parameters for the ATP model. The higher-order model was found to reproduce the short-circuit impedance acceptably up to about 10 kHz and the behavior at the first anti-resonant frequency was better matched with the measurements.