# Browsing by Subject "preconditioning"

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Item Efficient numerical methods for capacitance extraction based on boundary element method(Texas A&M University, 2006-04-12) Yan, ShuShow more 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.Show more Item Solution Techniques for Single-Phase Subchannel Equations(2013-04-11) Hansel, Joshua EdmundShow more A steady-state, single phase subchannel solver was created for the purpose of integration into a multi-physics nuclear fuel performance code. Since applications of such a code include full nuclear reactor core flow simulation, a thorough investigation of efficient solution techniques is a requirement. Execution time profiling found that formation of the Jacobian matrix required by the nonlinear Newton solve was found to be the most time-consuming step in solution of the subchannel equations, so several techniques were tested to minimize the time spent on this task, such as finite difference and the formation of an approximate Jacobian. Simple Jacobian lagging was shown to be very effective at reducing the total time computing the Jacobian throughout the Newton iteration process. Various linear solution techniques were investigated with the subchannel equations, such as the generalized minimal residual method (GMRES) and the aggregation- based algebraic multigrid method (AGMG). A number of physics-based preconditioners were created, based on a simplified formulation with no crossflow between subchannels, and it was found that of the preconditioners developed for this research, the most promising was a preconditioner that fully decoupled the subchannels by ignoring crossflow, conduction, and turbulent momentum exchange between subchannels. This independence between subchannels makes the task of parallelization in the preconditioner to be very feasible. However, AGMG clearly proved to be the most efficient linear solution technique for the subchannel equations, solving the linear systems in less than 5 percent of the time required for preconditioned GMRES.Show more Item Support graph preconditioners for sparse linear systems(Texas A&M University, 2005-02-17) Gupta, RadhikaShow more Elliptic partial differential equations that are used to model physical phenomena give rise to large sparse linear systems. Such systems can be symmetric positive de?nite and can be solved by the preconditioned conjugate gradients method. In this thesis, we develop support graph preconditioners for symmetric positive de?nite matrices that arise from the ?nite element discretization of elliptic partial di?erential 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 ?nite element matrices.Show more