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In order to maintain a continuous power suppply, nowadays relays in transmission systems are required to be able to deal with complicated faults involving non-conventional connections, which poses a challenge to the short circuit analysis performed for the data settings of the relay. The traditional sequence domain method has congenital defects to treat such cases, which leads to a trend of using the actual phase domain method in fault calculation. Although the calculation speed of the phase domain method is not so fast and is memory consumable, it perfomrs well when handling complicated faults. Today more and more commercial software involves phase domain calculation in their short circuit analysis to treat complicated cases. With the advanced development of computers, there is a possibility to totally get rid of the sequence method. In this thesis, a short circuit analysis method based on phase domain is developed. After the three sequence admittance matrices of the system are built, all the data are transformed into phase domain to get the phase domain admittance matrix. The following fault calculations are performed purely in phase domain. The test results of different types of faults in 3 bus, 14 bus, and 30 bus transmission systems are presented and compared with the results of a commercial fault analysis software. The validation of this program is also presented.

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.

Fast and accurate solvers for capacitance extraction are needed by the VLSI industry in order to achieve good design quality in feasible time. With the development of technology, this demand is increasing dramatically. Three-dimensional capacitance extraction algorithms are desired due to their high accuracy. However, the present 3D algorithms are slow and thus their application is limited. In this dissertation, we present several novel techniques to significantly speed up capacitance extraction algorithms based on boundary element methods (BEM) and to compute the capacitance extraction in the presence of floating dummy conductors. We propose the PHiCap algorithm, which is based on a hierarchical refinement algorithm and the wavelet transform. Unlike traditional algorithms which result in dense linear systems, PHiCap converts the coefficient matrix in capacitance extraction problems to a sparse linear system. PHiCap solves the sparse linear system iteratively, with much faster convergence, using an efficient preconditioning technique. We also propose a variant of PHiCap in which the capacitances are solved for directly from a very small linear system. This small system is derived from the original large linear system by reordering the wavelet basis functions and computing an approximate LU factorization. We named the algorithm RedCap. To our knowledge, RedCap is the first capacitance extraction algorithm based on BEM that uses a direct method to solve a reduced linear system. In the presence of floating dummy conductors, the equivalent capacitances among regular conductors are required. For floating dummy conductors, the potential is unknown and the total charge is zero. We embed these requirements into the extraction linear system. Thus, the equivalent capacitance matrix is solved directly. The number of system solves needed is equal to the number of regular conductors. Based on a sensitivity analysis, we propose the selective coefficient enhancement method for increasing the accuracy of selected coupling or self-capacitances with only a small increase in the overall computation time. This method is desirable for applications, such as crosstalk and signal integrity analysis, where the coupling capacitances between some conductors needs high accuracy. We also propose the variable order multipole method which enhances the overall accuracy without raising the overall multipole expansion order. Finally, we apply the multigrid method to capacitance extraction to solve the linear system faster. We present experimental results to show that the techniques are significantly more efficient in comparison to existing techniques.

The thesis proposes a hybrid broadband equivalent model extraction method, and our goal is to combine via structure and irregular transmission line in print circuit board for extraction of broadband SPICE-compatible model by using the time domain algorithm and full wave simulation in frequency domain, respectively. We can construct broadband SPICE-compatible macro-model scalable library with two kind of different extraction methods, tow kind of extraction of equivalent model can construct the circuit structure for designer demand. Every modules of the broadband macro model of the two extraction models are represented by the optimum pole-residue forms. Using a systematic lumped-model extraction technique, all the optimum pole-residue rational functions can be transfered into a corresponding lumped circuit model. The accuracy of Extraction of Broadband Equivalent Models is demonstrated in frequency -domain responses compared with the 3D-FDTD or HFSS simulation. In addition, the extraction model can simulate in commercial tools effectively, ex: Hspice¡BADS. Even the model can simulate signal integrality and power integrality in Hspice or ADS.

One of the important goals in petroleum exploration and production is to make quantitative estimates of a reservoir’s properties from all available but indirectly related surface data, which constitutes an inverse problem. Due to the inherent non-uniqueness of most inverse procedures, a deterministic solution may be impossible, and it makes more sense to formulate the inverse problem in a statistical Bayesian framework and to fully solve it by constructing the Posterior Probability Density (PPD) function using Markov Chain Monte Carlo (MCMC) algorithms. The derived PPD is the complete solution of an inverse problem and describes all the consistent models for the given data. Therefore, the estimated PPD not only leads to the most likely model or solution but also provides a theoretically correct way to quantify corresponding uncertainty. However, for many realistic applications, MCMC can be computationally expensive due to the strong nonlinearity and high dimensionality of the problem. In this research, to address the fundamental issues of efficiency and accuracy in parameter estimation and uncertainty quantification, I have incorporated some new developments and designed a new multiscale MCMC algorithm. The new algorithm is justified using an analytical example, and its performance is evaluated using a nonlinear pre-stack seismic waveform inversion application. I also find that the new technique of multi-scaling is particularly attractive in addressing model parameterization issues especially for the seismic waveform inversion. To derive an accurate reservoir model and therefore to obtain a reliable reservoir performance prediction with as little uncertainty as possible, I propose a workflow to integrate 4D seismic and well production data in a Bayesian framework. This challenging 4D seismic history matching problem is solved using the new multi-scale MCMC algorithm for reasonably accurate reservoir characterization and uncertainty analysis within an acceptable time period. To take advantage of the benefits from both the fine scale and the coarse scale, a 3D reservoir model is parameterized into two different scales. It is demonstrated that the coarse-scale model works like a regularization operator to make the derived fine-scale reservoir model smooth and more realistic. The derived best-fitting static petrophysical model is further used to image the evolution of a reservoir’s dynamic features such as pore pressure and fluid saturation, which provide a direct indication of the internal dynamic fluid flow. / text

A three dimensional viscous/inviscid interactive boundary layer method for predicting the effects of fluid viscosity on the performance of fully wetted propellers is presented. This method is developed by coupling a three dimensional low-order potential based panel method and a two dimensional integral boundary layer analysis method. To simplify the solution procedures, this method applies a reasonable assumption that the effects of the boundary layer along the span wise direction (radially outward for propeller blades) could be negligible compared with those along the stream wise direction (constant radius for propeller blades). One significant development of this method, compared with previous work, is to completely consider the effects of the added sources on the whole blades and wakes rather than evaluate the boundary layer effects along each strip, without interaction among strips. This method is applied to Propeller DTMB4119, Propeller NSRDC4381 and DTMB Duct II for validation. The results show good correlation with experimental measurements or RANS (ANSYS/FLUENT) results. The method is further used to develop a viscous image model for the cases of three dimensional wing blades between two parallel slip walls. An improved method for hydrofoils and propeller blades with non-zero thickness or open trailing edges is presented as well. The method in this thesis follows the idea of Pan (2009, 2011), but applies a new extension scheme, which uses second order polynomials to describe the extension edges. A improved simplified search scheme is also used to find the correct shape of the extension automatically to ensure the two conditions are satisfied. / text

The main aim of this thesis is to design efficient and novel numerical algorithms for a class of deterministic and stochastic dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become, therefore, inefficient when the much longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. This dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three different papers. The framework of the heterogeneous multiscale method (HMM) is considered as a general strategy both for the design and the analysis of multiscale methods. In Chapter 2, we consider a new class of multiscale methods that use a technique related to the construction of a Poincaré map. The main idea is to construct effective paths in the state space whose projection onto the slow subspace shows the correct dynamics. More precisely, we trace the evolution of the invariant manifold M(t), identified by the level sets of slow variables, by introducing a slowly evolving effective path which crosses M(t). The path is locally constructed through interpolation of neighboring points generated from our developed map. This map is qualitatively similar to a Poincaré map, but its construction is based on the procedure which solves two split equations successively backward and forward in time only over a short period. This algorithm does not require an explicit form of any slow variables. In Chapter 3, we present efficient techniques for numerical averaging over the invariant torus defined by ergodic dynamical systems which may not be mixing. These techniques are necessary, for example, in the numerical approximation of the effective slow behavior of highly oscillatory ordinary differential equations in weak near-resonance. In this case, the torus is embedded in a higher dimensional space and is given implicitly as the intersection of level sets of some slow variables, e.g. action variables. In particular, a parametrization of the torus may not be available. Our method constructs an appropriate coordinate system on lifted copies of the torus and uses an iterated convolution with respect to one-dimensional averaging kernels. Non-uniform invariant measures are approximated using a discretization of the Frobenius-Perron operator. These two numerical averaging strategies play a central role in designing multiscale algorithms for dynamical systems, whose fast dynamics is restricted not to a circle, but to the tori. The efficiency of these methods is illustrated by numerical examples. In Chapter 4, we generalize the classical two-scale averaging theory to multiple time scale problems. When more than two time scales are considered, the effective behavior may be described by the new type of slow variables which do not have formally bounded derivatives. Therefore, it is necessary to develop a theory to understand them. Such theory should be applied in the design of multiscale algorithms. In this context, we develop an iterated averaging theory for highly oscillatory dynamical systems involving three separated time scales. The relevant multiscale algorithm is constructed as a family of multilevel solvers which resolve the different time scales and efficiently computes the effective behavior of the slowest time scale. / text