Browsing by Subject "Boundary element method"
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Item Efficient numerical methods for capacitance extraction based on boundary element method(Texas A&M University, 2006-04-12) Yan, ShuFast 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.Item Leading edge vortex modeling and its effect on propulsor performance(2014-12) Tian, Ye, active 21st century; Kinnas, Spyros A.A novel numerical method solves the VIScous Vorticity Equation (VISVE) in 3D in order to model the Leading Edge Vortex (LEV) of propellers is proposed and implemented in this dissertation. The spatial concentration of the vorticity is exploited in the method, which is designed to be spatially compact and numerically efficient, in the meantime, capable of modeling complicated vorticity/solid boundary interaction in 2D and 3D. The numerical model can work as a viscous correction on top of the traditional Boundary Element Method (BEM) results. The proposed method is first applied in the case of a 2D hydrofoil at high angle of attack. The results are correlated with those from Navier-Stokes (N-S) simulation. The method is then used to model the LEV and tip vortex of a 3D swept wing. The results of the 3D simulation show great similarity to those from N-S. In the end, the method is applied in the case of propellers at low advance ratios. All the essential flow characteristics (LEV and tip vortex) are predicted. The objective of this dissertation is not developing a mathematically equivalent numerical method to the full-blown Reynolds-Averaged Navier-Stokes (RANS) solver, but inventing an accurate and computationally efficient tool to model the effects of the LEV on the propeller performance for engineering's purpose.Item Numerical simulation of unsteady rotor/stator interaction and application to propeller/rudder combination(2010-08) He, Lei, doctor of civil engineering; Kinnas, Spyros A.; Liljestrand, Howard M. (Howard Michael); Hodges, Ben; Kallivokas, Loukas; Moser, Robert; Lee, HanseongIn this thesis, a numerical approach based on a potential flow method has been developed in order to simulate unsteady rotor/stator interaction, and to predict the unsteady performance of a propeller and its rudder. The method is first developed and tested in two-dimensions by using a boundary element method in which a front hydrofoil is moving downward, while a back hydrofoil is stationary. The wakes of the two hydrofoils are modeled by continuous dipole sheets and determined in time by applying a force free-condition on each wake surface. The wake/hydrofoil interaction is de-singularized by applying a numerical fence on the surface of the back hydrofoil. The viscous wake/hydrofoil interaction is considered by employing a viscous wake vorticity model on the rotor's wake surface. The present method is validated by comparison with analytical solutions, experimental data and by using the results from a commercial Reynolds Averaged Navier-Stokes (RANS) solver for the same set-up and conditions. The numerical approach is further extended to three-dimensions to predict the mutual interaction between a propeller and rudder. A fully unsteady wake alignment algorithm is implemented into a Vortex Lattice Method to simulate the unsteady propeller flow. The interaction between propeller and rudder is investigated in a fully unsteady manner, where a panel method is used to solve the flow around the rudder, and a vortex lattice method is used to solve the flow around the propeller. The interaction between a propeller and its rudder is considered in an iterative manner by solving the propeller and the rudder problems separately and by including the unsteady effects of one component on the other. The effect of the unsteady propeller-rudder interaction on the mean and on the unsteady propeller/rudder performance, including sheet cavitation on the rudder, is studied.Item Simulation of the growth of multiple interacting 2D hydraulic fractures driven by an inviscid fluid(2012-12) Erickson, Andrew Jay; Mear, Mark E.; Landis, Chad MIn this paper we develop a computational procedure to investigate linear fracture of two-dimensional problems in isotropic linearly elastic media. A symmetric Galerkin boundary element method (SGBEM), based on a weakly singular, weak-form traction integral equation, is adopted to model these fractures. In particular we consider multiple interacting cracks in an unbounded domain subject to internal pressure and remote stress. The growth of the cracks is driven by either linearly dependent injection pressures or volumes in each crack. A variety of crack geometries are investigated.Item Solving three-dimensional problems in natural and hydraulic fracture development : insight from displacement discontinuity modeling(2013-08) Sheibani, Farrokh; Olson, Jon E.Although many fracture models are based on two-dimensional plane strain approximations, accurately predicting fracture propagation geometry requires accounting for the three-dimensional aspects of fractures. In this study, we implemented 3-D displacement discontinuity (DD) boundary element modeling to investigate the following intrinsically 3-D natural or hydraulic fracture propagation problems: the effect of fracture height on lateral propagation of vertical natural fractures, joint development in the vicinity of normal faults, and hydraulic fracture height growth and non-planar propagation paths. Fracture propagation is controlled by stress intensity factor (SIF) and its determination plays a central role in LEFM. The DD modeling is used to evaluate SIF in Mode I, II and III at the tip of an arbitrarily-shaped embedded crack by using crack-tip element displacement discontinuity. We examine the accuracy of SIF calculation is for rectangular, penny-shaped, and elliptical planar cracks. Using the aforementioned model for lateral propagation of overlapping fractures shows that the curving path of overlapping fractures is strongly influenced by the spacing-to-height ratio of fractures, as well as the differential stress magnitude. We show that the angle of intersection between two non-coincident but parallel en-echelon fractures depends strongly on the fracture height-to-spacing ratio, with intersection angles being asymptotic for "tall" fractures (large height-to-spacing ratios) and nearly orthogonal for "short" fractures. Stress perturbation around normal faults is three-dimensionally heterogeneous. That perturbation can result in joint development at the vicinity of normal faults. We examine the geometrical relationship between genetically related normal faults and joints in various geologic environments by considering a published case study of fault-related joints in the Arches National Park region, Utah. The results show that joint orientation is dependent on vertical position with respect to the normal fault, the spacing-to-height ratio of sub-parallel normal faults, and Poisson's ratio of the media. Our calculations represent a more physically reasonable match to measured field data than previously published, and we also identify a new mechanism to explain the driving stress for opening mode fracture propagation upon burial of quasi-elastic rocks. Hydraulic fractures may not necessarily start perpendicular to the minimum horizontal remote stress. We use the developed fracture propagation model to explain abnormality in the geometry of fracturing from misaligned horizontal wellbores. Results show that the misalignment causes non-planar lateral propagation and restriction in fracture height and fracture width in wellbore part.Item Speed and accuracy tradeoffs in molecular electrostatic computation(2009-12) Chen, Shun-Chuan, 1979-; Bajaj, Chandrajit; Arikan, Okan; Dhillon, Inderjit; Rossky, Peter; Ying, LexingIn this study, we consider electrostatics contributed from the molecules in the ionic solution. It plays a significant role in determining the binding affinity of molecules and drugs. We develop the overall framework of computing electrostatic properties for three-dimensional molecular structures, including potential, energy, and forces. These properties are derived from Poisson-Boltzmann equation, a partial differential equation that describes the electrostatic behavior of molecules in ionic solutions. In order to compute these properties, we derived new boundary integral equations and designed a boundary element algorithm based on the linear time fast multipole method for solving the linearized Poisson-Boltzmann equation. Meanwhile, a higher-order parametric formulation called algebraic spline model is used for accurate approximation of the unknown solution of the linearized Poisson-Boltzmann equation. Based on algebraic spline model, we represent the normal derivative of electrostatic potential by surrounding electrostatic potential. This representation guarantees the consistent relation between electrostatic potential and its normal derivative. In addition, accurate numerical solution and fast computation for electrostatic energy and forces are also discussed. In addition, we described our hierarchical modeling and parameter optimization of molecular structures. Based on this technique, we can control the scalability of molecular models for electrostatic computation. The numerical test and experimental results show that the proposed techniques offer an efficient and accurate solution for solving the electrostatic problem of molecules.Item Thermo-Poroelastic Fracture Propagation Modeling with Displacement Discontinuity Boundary Element Method(2013-08-01) Chun, Kwang HeeThe effect of coupled thermo-poroelastic behavior on hydraulic fracture propagation is of much interest in geothermal- and petroleum-related geomechanics problems such as wellbore stability and hydraulic fracturing as pore pressure and temperature variations can significantly induce rock deformation, fracture initiation, and propagation. In this dissertation, a two-dimensional (2D) boundary element method (BEM) was developed to simulate the fully coupled thermo-poroelastic fracture propagation process. The influence of pore pressure and temperature changes on the fracture propagation length and path, as well as on stress and pore pressure distribution near wellbores and fractures, was considered in isotropic and homogeneous rock formations. The BEM used in this work consists of the displacement discontinuity (DD) method and the fictitious stress (FS) method. Also, a combined FS-DD numerical model was implemented for the hydraulically or thermally-induced fractures in the vicinity of a wellbore. The linear elastic fracture mechanics (LEFM) theory was adopted to numerically model within the framework of poroelasticity and thermo-poroelasticity theory. For high accuracy of crack tip modeling, a special displacement discontinuity tip element was developed and extended to capture the pore pressure and temperature influence at the tip. For poroelastic fracture propagation, a steadily propagating crack driven by fluid pressure was modeled to find the effect of pore pressure on crack path under the two limiting poroelastic conditions (undrained and drained). The results indicate that the pore pressure diffusion has no influence on the crack growth under the undrained condition because the crack propagation velocity is too fast for the diffusion effect to take place. On the other hand, its influence on the crack path under the drained condition with its low propagation velocity has significance because it induces a change in principal stress direction, resulting in an alteration of fracture orientation. For the thermal fracturing, when the rock around a wellbore and a main fracture is cooled by injecting cold water in a hot reservoir, the rapid decrease in temperature gives rise to thermal stress, which causes a crack to initiate and propagate into the rock matrix. The single and multiple fracture propagation caused by transient cooling in both thermoelastic and poro-thermoelastic rock were numerically modeled. The results of this study indicate that the thermal stresses induced by cooling may exceed the in-situ stress in the reservoir, creating secondary fractures perpendicular to main fracture. Furthermore, the faster cooling rate produces longer crack extension of the secondary thermal fractures. This implies that the faster cooling induces a higher tensile stress zone around the fracture, which tends to produce larger driving forces to make the secondary fractures penetrate deeper into the geothermal reservoir.