Red Black Successive Overrelaxation Research Articles

Optimal Complex Relaxation Parameters in Multigrid for Complex-Shifted Linear Systems

We derive optimal complex relaxation parameters minimizing smoothing factors associated with multigrid using red-black successive overrelaxation or damped Jacobi smoothing applied to a class of lin...

A heterogeneous parallel Red–Black SOR technique and the numerical study on SIMPLE

A basic heterogeneous parallel Red–Black successive over-relaxation (SOR) implement, the mono-color floating-point scheme, was developed on graphics processing units (GPU) with OpenCL platform. Designed in fine granularity, compact data structure, and stencil function, a concise mapping relationship was created to implicitly describe the complex rules for searching neighbor elements, which could avoid low utilization of GPU in the traditional scheme of Red–Black SOR. The new mono-color floating-point scheme was applied to build fast Semi-Implicit Method for Pressure Linked Equations (SIMPLE) solver with OpenCL and OpenMP on the heterogeneous parallel computing device. Compared with SIMPLE solver in the traditional Red–Black SOR scheme, the new scheme can achieve 1.7 to 1.8 faster accelerative performance on the same GPU. And this scheme can eliminate the complex searching module in mono-color logical scheme and behave better than the mono-color logical scheme by 20–30% acceleration. Numerical cases in double precision showed that SIMPLE solver on GPU with new scheme of Red–Black SOR could save up to 92% computing time compared with the serial solver on CPU.

A High-Performance Dense Optical Flow Architecture Based on Red-Black SOR Solver

Optical flow (OF) is an integral part of many vision systems, especially in the embedded and mobile application with ever-increasing challenges in achieving higher speed, minimal resource and lower power consumption. The work introduces a Dense High Throughput Optical Flow (DHTOF) architecture based on a novel fast converging Red-Black Successive Over Relaxation (RBSOR) solver architecture for computing dense and accurate OF using Horn and Schunck Optical Flow (HSOF) algorithm from Full High Definition (FHD) frames in real-time. The DHTOF architecture can capture dense OF from Ultra High Definition (UHD) frames at 48 Frames Per Second (FPS) with a throughput of 406 Megapixels/sec achieving a Throughput Per Watt (TPW) of 43 Giga Operation Per Second Per Watt (GOPS/Watt). The superscalar and deeply pipelined DHTOF architecture achieve same or lower Average Angular Error (AAE) with ≈ 4 × lesser number of RBSOR solver iterations as compared to the prior HSOF implementations based on Jacobi solver. It consumes 12.5 × lesser resources and 29.3% lower power for FHD resolution when compared to prior architectures. The proposed DHTOF architecture achieves highest area delay normalized speedup (at least by 28.2 ×) among the state of the art HSOF architectures. The successful evaluation of the proposed architecture for real-time OF sensor is demonstrated in Xilinx Virtex-VC707 Field Programmable Gate Array (FPGA) evaluation board.

Implementation of scalable bidomain-based 3D cardiac simulations on a graphics processing unit cluster

Computational models of the human cardiac cells provide detailed properties of human ventricular cells. The execution time for a realistic 3D heart simulation based on these models is a major barrier for physicians to study and understand the heart diseases, and evaluate hypotheses rapidly toward developing treatments. Graphics processing unit (GPU)-based parallelization efforts to this date have been shown to be more effective than parallelization efforts on the CPU-based clusters in terms of addressing the 3D cardiac simulation time challenge. In this paper, we review all GPU-based studies and investigate both the cardiac cell models and cardiac tissue models in 3D space. We propose algorithmic optimizations based on red black successive over-relaxation method for reducing the number of simulation iterations and convergence method for dependence elimination between neighboring cells of the heart tissue. We investigate data transfer reduction and 2D mesh partitioning strategies, evaluate their impact on thread utilization, and propose a strongly scalable cardiac simulation. Our implementation results with reducing the execution time by a factor of five compared to the state-of-the-art baseline implementation. More importantly, our implementation is an important step toward achieving real-time cardiac simulations as it achieves the strongest scalability among all other cluster-based implementations.

High-Efficiency Computing Technology for Thermohydrodynamic Lubrication Analysis

ABSTRACTNumerical analysis of thermal hydrodynamic (THD) lubrication is complex and time consuming because of the three-dimensional numerical domain of the solution of the energy equation. To reduce the execution time, many methods for determining solutions for bearing design problems are available; parallel computing technology is crucial for achieving high computational capability and involves using workstations equipped with multicore central processing units and many-core graphics processing units (GPUs). High-performance GPUs have emerged as powerful computing tools for modeling purposes in engineering and science. High-level GPUs depend on thousands of shaders, namely, processor cores. In this study, the traditional successive overrelaxation (SOR) method for solving the Reynolds equation and energy equation was replaced with the two- and three-dimensional red–black SOR methods for parallelizing the governing equations of a GPU system. The results show that a high-level GPU can be used to increase the parallel computing speed in the process of solving THD problems.

Effects of Shaft Axial Motion and Misalignment on the Lubrication Performance of Journal Bearings Via a Fast Mixed EHL Computing Technology

A mixed elastohydrodrynamic (EHL) model for journal bearings considering an axial flow due to shaft axial motion and misalignment is developed for lubrication performance evaluation. A new, faster mixed EHL computing technology utilizing the odd–even successive overrelaxation (OESOR) parallel numerical iterative method is proposed based on the red–black successive overrelaxation (RBSOR) method to minimize the execution time prolonged by the complexity caused by the axial flow and misalignments. The multithreaded computing scheme conducted by the OpenMP directive using different meshes and threads suggests that the OESOR method exhibits better efficiency. A series of transient analyses was conducted to solve the mixed EHL model with the parallel OESOR method. The results show that the axial flow and misalignments significantly affect the average pressure, hydrodynamic and asperity contact pressure, elastic deformation, and other characteristics of the journal bearings.

Exploration on a Fast EHL Computing Technology for Analyzing Journal Bearings with Engineered Surface Textures

Solving elastohydrodynamic lubrication (EHL) problems is a complex and time-consuming process due to the interactive solutions of the Reynolds equation and contact elasticity. Analyzing journal bearing EHL problems is even more difficult due to the scale difference in the structural and surface features, which may span four orders of magnitude. This article presents a fast EHL computing technology utilizing a parallel numerical iterative method (the red–black successive overrelaxation method) and multithreaded computing scheme conducted by OpenMP directives. The fast computational approaches allow the construction of high-density EHL meshes for effective descriptions of important texture features of journal bearing surfaces.

Parallel Iterative Solution Schemes for the Analysis of Air Foil Bearings

AbstractIn an air foil bearing analysis the model is usually solved iteratively due in part to the nonlinearity of the modeling Reynolds equation and the compliance of the bearing surface. The solution procedure requires a multiple-level-deep nested iteration, which involves extended solution time and convergence difficulty. In this study, a simple air foil bearing model is used and the compressible-fluid Reynolds equation for modeling gas lubrication is linearized by Newton's method. The discretized equation is solved by one of the two parallel iterative methods, red-black or strip partition successive-over-relaxation (SOR) method. The parallel programming is conducted using OpenMP programming in an eight-core work-station. Then, a numerical damping scheme for the film-profile convergence is presented. Finally, a root-finding process is conducted to iteratively attain the eccentricity of the bearing for a given load. It is found that the numerical damping step is crucial, which allows the use of a larger relaxation factor to have a fast rate of convergence. Both the parallel SOR methods are easy to implement and the red-black SOR method exhibits better efficiency in the studied cases. This study presents a parallel computing scheme for analyzing air foil bearing of bump-type by today's shared-memory multicore platforms.

A fully vectorizable SOR variant

The implementation of a variant of the Successive Overrelaxation (SOR) method for the numerical solution of elliptic partial differential equations is discussed. The convergence of this variant (SORV) is faster than that of SOR for a fixed prescribed error bound, and it is asymptotically as fast as SOR. In addition, SORV is much better suited for vector computers than the classical SOR or SOR with so-called red-black ordering. In the following, we compare SORV with SOR, red-black SOR (SORB1), SORB1 with stride 1 (SORB2) and a one-dimensional long-vector version of SORB2 (SORB3). Implementation is discussed and results are presented for the IBM 3090-200VF and the CRAY-2.