Vivek Sarin

Fluidization of 1204 spheres: simulation and experiment

In this paper we study the fluidization of 1204 spheres at Reynolds numbers in the thousands using the method of distributed Lagrange multipliers. The results of the simulation are compared with a real experiment. This is the first direct numerical simulation of a real fluidized bed at the finite Reynolds number encountered in the applications. The simulations are processed like real experiments for straight lines in lot-log plots leading to power laws as in celebrated correlations of Richardson and Zaki [1954]. The numerical method allows for the first ever direct calculation of the slip velocity and other averaged values used in two-fluid continuum models. The computation and the experiment show that a single particle may be in balance under weight and drag for an interval of fluidizing velocities; the expectation that the fluidizing velocity is unique is not realized. The numerical method reveals that the dynamic pressure actually decreases slowly with the fluidizing velocity. Tentative interpretations of these new results are discussed.

An Efficient Iterative Method for the Generalized Stokes Problem

The generalized Stokes problem, which arises frequently in the simulation of time-dependent Navier--Stokes equations for incompressible fluid flow, gives rise to symmetric linear systems of equations. These systems are indefinite due to a set of linear constraints on the velocity, causing difficulty for most preconditioners and iterative methods. This paper presents a novel method to obtain a preconditioned linear system from the original one which is then solved by an iterative method. This new method generates a basis for the velocity space and solves a reduced system which is symmetric and positive definite. Numerical experiments indicating superior convergence compared to existing methods are presented. A natural extension of this method to elliptic problems is also proposed, along with theoretical bounds on the rate of convergence, and results of experiments demonstrating robust and effective preconditioning.

Numerical Simulation of Mixed Convective Flow Over a Three-Dimensional Horizontal Backward Facing Step

Laminar mixed convective flow over a three-dimensional horizontal backward-facing step heated from below at a constant temperature was numerically simulated using a finite volume technique and the most relevant hydrodynamic and thermal features for air flowing through the channel are presented in this work. The channel considered in this work has an aspect ratio AR = 4, and an expansion ratio ER = 2, while the total length in the streamwise direction is 52 times the step height (L = 52s) and the step length is equal to 2 times the step height (l = 2s). The flow at the duct entrance was considered to be hydro-dynamically fully developed and isothermal. The bottom wall of the channel was subjected to a constant high temperature while the other walls were treated to be adiabatic. The step was considered to be a thermal conductive block.Copyright

Sparse transformations and preconditioners for 3-D capacitance extraction

Three-dimensional (3-D) capacitance-extraction algorithms are important due to their high accuracy. However, the current 3-D algorithms are slow and thus their application is limited. In this paper, we present a novel method to significantly speed up capacitance-extraction algorithms based on boundary element methods (BEMs), under uniform and multiple dielectrics. The n/spl times/n coefficient matrix in the BEM is dense, even when approximated with the fast multipole method or hierarchical-refinement method, where n is the number of panels needed to discretize the conductor surfaces and dielectric interfaces. As a result, effective preconditioners are hard to obtain and iterative solvers converge slowly. In this paper, we introduce a linear transformation to convert the n/spl times/n dense coefficient matrix into a sparse matrix with O(n) nonzero entries, and then use incomplete factorization to produce a very effective preconditioner. For the k/spl times/k bus-crossing benchmark, our method requires at most four iterations, whereas previous best methods such as FastCap and HiCap require 10-20 iterations. As a result, our algorithm is up to 70 times faster than FastCap and up to 2 times faster than HiCap on these benchmarks. Additional experiments illustrate that our method consistently outperforms previous best methods by a large magnitude on complex industrial problems with multiple dielectrics.

Hybrid Parallel Linear System Solvers

This paper presents a new approach to the solution of nonsymmetric linear systems that uses hybrid techniques based on both direct and iterative methods. An implicitly preconditioned modified system is obtained by applying projections onto block rows of the original system. Our technique provides the flexibility of using either direct or iterative methods for the solution of the preconditioned system. The resulting algorithms are robust, and can be implemented with high efficiency on a variety of parallel architectures. The algorithms are used to solve linear systems arising from the discretization of convection-diffusion equations as well as those systems that arise from the simulation of particulate flows. Experiments are presented to illustrate the robustness and parallel efficiency of these methods.

Forced Convection Over a Three-Dimensional Horizontal Backward Facing Step

Forced convective flow over a 3-D backward-facing step is studied numerically. The momentum and energy equations were discretized by means of a finite volume technique. The SIMPLE algorithm scheme was used to link the pressure and velocity fields in the entire domain and a line-by-line scheme was used in each plane to compute the velocity, pressure, and temperature field distributions. The code was validated by comparing numerical predictions with experimental data for flow over a 3-D backward facing step that is available in the literature. Flow of air (Pr = 0.70) over a three-dimensional horizontal backward-facing step geometry with an aspect ratio AR = 8 and an expansion ratio ER = 2 was considered. The stepped wall downstream of the expansion was heated by subjecting it to a constant heat flux (qw = 50 Wm−2) and the other walls were considered as insulated. The inlet flow was taken to be hydro-dynamically fully developed with a uniform temperature profile. Locations where the streamwise velocity and the spanwise velocity components are zero for the nearest plane adjacent to the stepped wall were plotted for different Reynolds numbers. Distributions for local and average Nusselt number for the stepped wall, and graphical representations for u, v, and w velocities components obtained in these simulations are presented in the paper.

Sparse transformations and preconditioners for hierarchical 3-D capacitance extraction with multiple dielectrics

Capacitance extraction is an important problem that has been extensively studied. This paper presents a significant improvement for the fast multipole accelerated boundary element method. We first introduce an algebraic transformation to convert the n x n dense capacitance coefficient matrix into a sparse matrix with O(n) nonzero entries. We then use incomplete Cholesky factorization or incomplete LU factorization to produce an effective preconditioner for the sparse linear system. Simulation results show that our algorithm drastically reduces the number of iterations needed to solve the linear system associated with the boundary element method. For the k x k bus crossing benchmark, our algorithm uses 3-4 iterations, compared to 10-20 iterations used by the previous algorithms such as FastCap [1] and HiCap [2]. As a result, our algorithm is 2-20 times faster than those algorithms. Our algorithm is also superior to the multi-scale method [3] because our preconditioner reduces the number of iterations further and applies to multiple dielectrics.

Parallel Simulation of Particulate Flows

We present the design of software packages called Particle Movers that have been developed to simulate the motion of particles in two dimensional domains. These simulations require the solution of nonlinear Navier-Stokes equations for fluids coupled with Newtons equations for particle dynamics. Furthermore, realistic simulations are extremely computationally intensive, and are feasible only with algorithms that can exploit parallelism effectively. We describe the computational structure of the simulation, including a distributed multilevel preconditioner for the inner systems arising from the nonlinear iteration. A software framework, GVec, has also been developed to support this new algorithm set. It is highly extensible and portable, while maintaining excellent performance and scalability. Large scale simulations, with thousands of particles, demonstrate very good speedup on a large number of processors.

A Preconditioned Hierarchical Algorithm for Impedance Extraction of Three-Dimensional Structures With Multiple Dielectrics

This paper presents the first boundary element method (BEM) impedance extraction algorithm for interconnects with multiple dielectrics. Multiple dielectrics are common in integrated circuits and packages. However, previous BEM algorithms, including FastImp and FastPep, assume uniform dielectric due to their limitation, thus causing considerable errors. Our algorithm introduces a circuit formulation which makes it possible to utilize either multilayer Greens function or equivalent charge method to extract impedance in multiple dielectrics. The novelty of the formulation is the reduction of the unknowns and the application of hierarchical data structure. The hierarchical data structure permits efficient sparsification transformation and preconditioners to accelerate the linear equation solver. Experimental results demonstrate that the new algorithm is accurate and efficient. For uniform dielectric problems, our algorithm is more accurate than FastImp while its number of unknowns is ten times less than that of FastImp. For multiple dielectric problems, its relative error with respect to HFSS is below 3%.

Fast 3-D Capacitance Extraction by Inexact Factorization and Reduction

Capacitance-extraction algorithms based on the boundary element method (BEM) have to solve large linear systems. The number of unknowns equals the number of discretization panels n, which is much greater than the number of conductors m. The authors present a capacitance-extraction algorithm RedCap that first reduces the BEM system of size n into a small system of size O(m) and then solves the small system to compute the capacitances. RedCap uses a number of techniques, including the hierarchical-refinement technique of HiCap [Shi, 2002], the dense-to-sparse transformation of PHiCap [Yan, 2005], a reordering of the sparse linear system, and an incomplete LU factorization, to obtain the reduced system. RedCap achieves a significant speed improvement over previous methods. On benchmark problems with conductors in uniform and multilayer dielectrics, RedCap is up to 100 times faster than FastCap [Nabors and White, 1991] and up to four times faster than PHiCap [Yan, 2005], while restricting error to within 2% of FastCap