Efficient Poisson Solver for Semiconductor Device Modeling Using the Multi-Grid Preconditioned BiCGSTAB Method

This paper presents the performance results of an efficient algorithm for solving the three dimensional Poisson equation. The multi-grid method exploits the efficient oscillatory error reduction of basic iterative methods by smoothing on a set of progressively coarsened grids. When used as a preconditioner for BiCGSTAB method, a computationally demanding solver can be shown to be effective for large scale simulations. Varying the number of grids used and the level of overrelaxation as well as exploring the benefits of semicoarsening in the multi-grid preconditioner reveals the underlying strengths of this combined scheme. The convergence properties of the developed solver are tested on a 3D split-gate silicon on insulator (SOI) device.     

A parallel computational framework to solve flow and transport in integrated surface–subsurface hydrologic systems

Hydrologic modeling requires the handling of a wide range of highly nonlinear processes from the scale of a hill slope to the continental scale, and thus the computational efficiency of the model becomes a critical issue for water resource management. This work is aimed at implementing and evaluating a flexible parallel computing framework for hydrologic simulations by applying OpenMP in the HydroGeoSphere (HGS) model. HGS is a 3D control-volume finite element model that solves the nonlinear coupled equations describing surface–subsurface water flow, solute migration and energy transport. The computing efficiency of HGS is improved by three parallel computing schemes: 1) parallelization of Jacobian matrix assembly, 2) multi-block node reordering for performing LU solve efficiently, and 3) parameter privatization for reducing memory access latency. Regarding to the accuracy and consistency of the simulation solutions obtained with parallel computing, differences in the solutions are entirely due to use of a finite linear solver iteration tolerance, which produces slightly different solutions which satisfy the convergence tolerance. The maximum difference in the head solution between the serial and parallel simulations is less than 10⁻³ m, using typical convergence tolerances. Using the parallel schemes developed in this work, three key achievements can be summarized: (1) parallelization of a physically-based hydrologic simulator can be performed in a manner that allows the same code to be executed on various shared memory platforms with minimal maintenance; (2) a general, flexible and robust parallel iterative sparse-matrix solver can be implemented in a wide range of numerical models employing either structured or unstructured mesh; and (3) the methodology is flexible, especially for the efficient construction of the coefficient and Jacobian matrices, compared to other parallelized hydrologic models which use parallel library packages.     

Preconditioned Krylov subspace methods for solving radiative transfer problems with scattering and reflection

Two Krylov subspace methods, the GMRES and the BiCGSTAB, are analyzed for solving the linear systems arising from the mixed finite element discretization of the discrete ordinates radiative transfer equation. To increase their convergence rate and stability, the Jacobi and block Jacobi methods are used as preconditioners for both Krylov subspace methods. Numerical experiments, designed to test the effectiveness of the (preconditioned) GMRES and the BiCGSTAB, are performed on various radiative transfer problems: (i) transparent, (ii) absorption dominant, (iii) scattering dominant, and (iv) with specular reflection. It is observed that the BiCGSTAB is superior to the GMRES, with lower iteration counts, solving times, and memory consumption. In particular, the BiCGSTAB preconditioned by the block Jacobi method performed best amongst the set of other solvers. To better understand the discrete systems for radiative problems (i) to (iv), an eigenvalue spectrum analysis has also been performed. It revealed that the linear system conditioning deteriorates for scattering media problems in comparison to absorbing or transparent media problems. This conditioning further deteriorates when reflection is involved.     

Benchmarks of 3D Laplace Equation Solvers in a Cubic Configuration for Streamer Simulation

The aim of this paper is to test a developed SOR R&B method using the Chebyshev accelerator algorithm to solve the Laplace equation in a cubic 3D configuration.Comparisons are made in terms of precision and computing time with other elliptic equation solvers proposed in the open source LIS library.The first results,obtained by using a single core on a HPC,show that the developed SOR R&B method is efficient when the spectral radius needed for the Chebyshev acceleration is carefully pre-estimated.Preliminary results obtained with a parallelized code using the MPI library are also discussed when the calculation is distributed over one hundred cores.     

Flow topology in a steady three-dimensional lid-driven cavity

We present in this paper a thorough investigation of three-dimensional flow in a cubical cavity, subject to a constant velocity lid on its roof. In this steady-state analysis, we adopt the mixed formulation on tri-quadratic elements to preserve mass conservation. To resolve difficulties in the asymmetric and indefinite large-size matrix equations, we apply the BiCGSTAB solution solver. To achieve stability, weighting functions are designed in favor of variables on the upstream side. To achieve accuracy, the weighting functions are properly chosen so that false diffusion errors can be largely suppressed by the equipped streamline operator. Our aim is to gain some physical insight into the vortical flow using a theoretically rigorous topological theory. To broaden our understanding of the vortex dynamics in the cavity, we also study in detail the longitudinal spiralling motion in the flow interior.