BiCGStab Method Research Articles

Simulation study on magneto-acoustic concentration tomography of magnetic nanoparticles based on vectorial acoustic source and BICGSTAB method

Abstract To overcome the vulnerability to noise of the reconstructed image and simplify the cumbersome iteration process of algorithm in the magneto-acoustic concentration tomography with magnetic induction (MACT-MI) for magnetic nanoparticles (MNPs), we established the matrix relationship between the concentration of MNPs and the first-order derivative of sound pressure based on the reconstruction method of vectorial acoustic source, and proposed the application of BICGSTAB method in solving the concentration distribution. Firstly, a simulation model was established in COMSOL Multiphysics. Secondly, the obtained data were substituted into the derived formula for imaging reconstruction. Finally, the quality of the reconstructed image was analyzed. The effects of MNP radius, shape, asymptotic concentration, and SNR on the reconstruction results were studied. Simulation results show that under the same noise condition, compared with the reconstruction method based on the LSQR-trapezoidal method, the average correlation coefficient increased by 32.9%, the average relative error decreased by 48.5%, the average structural similarity increased by 48.2%, and the average iterations decreased by 58.5%. The proposed method shows superior imaging quality and noise immunity. The research provides a theoretical basis for subsequent experimental research.

Numerical simulation and entropy optimization of hybrid nanofluid flow in an inclined wavy enclosure subjected to thermal radiation

The current study investigates two-dimensional natural convective heat transference and entropy production in a tilted wavy-walled enclosure under the magnetic field and thermal radiation effect. The enclosure is filled with Cu–Al2O3/H2O hybrid nanofluid and subjected to a non-uniformly heated curved left wall, constant cold right curved side, uniformly heated bottom side, and insulated upper side. The regulating formulas in the non-dimensional form are reduced to streaming function-velocity expression and are numerically solved based on the Bi-CGStab method. The simulations are behaved with diverse Rayleigh amounts, Hartmann numbers, an incline angle of the enclosure, radiation parameters, diverse amplitude of the curved wall, and volume fraction of hybrid nanoparticles. Numerical code validation with other published results agrees well with the present outcomes. Heat and entropy generation were improved with the change in dimensionless parameters, and the findings have been clarified through discussion. The observations indicate that the Rayleigh numbers, nanoparticle fractional size, amplitude, and radiation parameter influence the convective effect and entropy production inside the enclosure. In contrast, the Hartmann number detracts from the convective effect. The findings suggest that a rise in the cavity angle may result in a corresponding boost or decline in heat transference. The minimum Nusselt numbers is obtained at [Formula: see text], as the angle of incline of the enclosure restraints the fluid rapidity and diminishes the heat transference rates. To design heat exchangers, this particular study may serve as a guide.

MHD double diffusive mixed convection and heat generation / absorption in a lid-driven inclined wavy enclosure filled with a ferrofluid

The magneto-hydrodynamics (MHD) double-diffusive mixed convection and heat generation/absorption in a lid-driven inclined wavy enclosure filled with (Fe3O4/ water) ferrofluid is quantitatively investigated in this paper. The present study focuses on improving the efficiency of mass and thermal performance of a system. Both the left and right sidewalls of the cavity are allowed to move with a constant velocity in the upward and downward directions, respectively. The finite difference approach was applied to discretize the subsequent governing equations followed by the Bi-Conjugate Gradient Stabilized (Bi-CGStab) method to solve them. The numerical simulation was performed for a variety of parameters, including the Hartmann number varied as 0 ≤ Ha ≤ 45, the inclination angle of the enclosure varied as 0° ≤ δ ≤ 180°, the buoyancy ratio varied as -2 ≤ N ≤ 2, heat generation or absorption parameter varied as −10 ≤ Qo ≤ 10, Richardson number varied as 0.01 ≤ Ri ≤ 10, and solid volume fraction varied as 0 ≤ ϕ ≤ 0.06. The numerical simulation results were presented in terms of streamlines, isotherms, isoconcentrations, average Nusselt number, and average Sherwood number. The heat and mass transfer rates were found to decrease with the increase in Ha but increase with N, Ri, and Φ. Also, both of them reach their peak values at Ri = 10. In addition, the heat generation parameter enhances both thermal and mass performance as they reach their maximum values at Qo = 10. Increasing the heat generation factor from Qo = 5 to Qo = 10 increases the Nusselt number by 3.5 times. The outcomes of the study have significant importance for modern industrial applications specifically in the discipline of electronic device cooling.

Shifted LOPBiCG: A locally orthogonal product‐type method for solving nonsymmetric shifted linear systems based on Bi‐CGSTAB

AbstractWhen solving shifted linear systems using shifted Krylov subspace methods, selecting a seed system is necessary, and an unsuitable seed may result in many shifted systems being unsolved. To avoid this problem, a seed‐switching technique has been proposed to help switch the seed system to another linear system as a new seed system without losing the dimension of the constructed Krylov subspace. Nevertheless, this technique requires collinear residual vectors when applying Krylov subspace methods to the seed and shifted systems. Since the product‐type shifted Krylov subspace methods cannot provide such collinearity, these methods cannot use this technique. In this article, we propose a variant of the shifted BiCGstab method, which possesses the collinearity of residuals, and apply the seed‐switching technique to it. Some numerical experiments show that the problem of choosing the initial seed system is circumvented.

General framework for re-assuring numerical reliability in parallel Krylov solvers: A case of bi-conjugate gradient stabilized methods

Parallel implementations of Krylov subspace methods often help to accelerate the procedure of finding an approximate solution of a linear system. However, such parallelization coupled with asynchronous and out-of-order execution often makes more visible the non-associativity impact in floating-point operations. These problems are even amplified when communication-hiding pipelined algorithms are used to improve the parallelization of Krylov subspace methods. Introducing reproducibility in the implementations avoids these problems by getting more robust and correct solutions. This paper proposes a general framework for deriving reproducible and accurate variants of Krylov subspace methods. The proposed algorithmic strategies are reinforced by programmability suggestions to assure deterministic and accurate executions. The framework is illustrated on the preconditioned BiCGStab method and its pipelined modification, which in fact is a distinctive method from the Krylov subspace family, for the solution of non-symmetric linear systems with message-passing. Finally, we verify the numerical behavior of the two reproducible variants of BiCGStab on a set of matrices from the SuiteSparse Matrix Collection and a 3D Poisson’s equation.

An Enhancement of the Accuracy of the BiCGStab Method for Solving Linear Systems with Single or Multiple Right-Hand Sides

In this paper, we present a technique to improve the convergence of the biconjugate gradient stabilized (BiCGStab) method. This method was developed by Van der Vorst for solving nonsymmetric linear systems with a single right-hand side. The global and block versions of the BiCGStab method have been proposed for solving nonsymmetric linear systems with multiple right-hand sides. Using orthogonal projectors to minimize the residual norm in each step, we get an enhancement of the convergence of each version of the BiCGStab method. The considered methods are BiCGStab, global BiCGStab, and block BiCGStab methods, noted, respectively, as Gl-BiCGStab and Bl-BiCGStab. To show the performance of our enhanced algorithms, we compare them with the standard, global, and block versions of the well-known generalized minimal residual method (GMRES).

Communication-hiding pipelined BiCGSafe methods for solving large linear systems

Recently, a new variant of the BiCGStab method, known as the pipelined BiCGStab, has been proposed. This method can achieve a higher degree of scalability and speed-up rates through a mechanism in which the communication phase for the computation of the inner product can be overlapped with the computation of the matrix-vector product. Meanwhile, several generalized iteration methods with better convergence behavior than BiCGStab exist, such as ssBiCGSafe, BiCGSafe, and GPBi-CG. Among these methods, ssBiCGSafe, which requires a single phase of computing inner products per iteration, is best suited for high-performance computing systems. As described herein, inspired by the success of the pipelined BiCGStab method, we propose pipelined variations of the ssBiCGSafe method in which only one phase of inner product computation per iteration is required and this phase of inner product computation can be overlapped with the matrix-vector computation. Through numerical experimentation, we demonstrate that the proposed methods engender improvements in convergence behavior and execution time compared to the pipelined BiCGStab and ssBiCGSafe methods.

Tensor product-type methods for solving Sylvester tensor equations

The tensor biconjugate gradient (TBiCG) method has recently been proposed for solving Sylvester tensor equations. The TBiCG method is based on the BiCG method that may exhibit irregular convergence behavior. To overcome the limitations, product-type methods, such as BiCGSTAB and GPBiCG, have been proposed. In this study, we apply the idea of product-type methods to solve Sylvester tensor equations and propose tensor GPBiCG and BiCGSTAB methods. Furthermore, we consider preconditioned algorithms of the tensor GPBiCG and BiCGSTAB methods using the nearest Kronecker product preconditioner. Numerical experiments illustrate that the proposed methods are competitive with some existing methods.

The immersed boundary method: A SIMPLE approach

A novel formulation of the direct forcing immersed boundary (IB) method, that treats it as an integral part of a SIMPLE method is presented for the simulation of incompressible flows. The incompressibility and no-slip kinematic constraints are treated implicitly as distributed Lagrange multipliers and are fully coupled with each other by combining them into a single Poisson-body forces system of equations that constitutes a regularized saddle point problem. A physically justified approximation of the system, resembling a technique typical of the penalty method, is carried out to facilitate the solution. By utilizing the Schur complement approach, the approximated system is decomposed into two separate systems of equations, allowing us to compute the values for the volumetric force and pressure corrections. The first system, which is characterized by a small (only O(10)) value of the condition number, is conveniently solved by the BiCgStab method [1], converging within 2-3 iterations, while the second system is addressed by the direct TPF solver [2], characterized by O(N4/3) complexity. The entire methodology is designed to be highly portable, which facilitates the use of any available solver that was designed to simulate incompressible flows governed by the Helmholtz and Laplace operators but could not benefit from the immersed boundary formalism. The capabilities of the developed methodology applied to the simulation of representative shear- and buoyancy-driven confined flows developing in the presence of stationary immersed bodies are demonstrated. A further application of the developed approach to moving boundary and two-way coupled fluid-structure interaction problems is discussed.

Subspace Correction Preconditioning for Solving a Sequence of Asymmetric Linear Systems Using the Bi-CGSTAB Method

Subspace Correction Preconditioning for Solving a Sequence of Asymmetric Linear Systems Using the Bi-CGSTAB Method

Multiphysics Computing of Challenging Antenna Arrays Under a Supercomputer Framework

A parallel multiphysics simulation solver is developed to solve electromagnetic-thermal-mechanical coupling for some challenging large-scale antenna arrays. To achieve high scalability of supercomputer architectures, we reconstruct the preconditioned BiCGSTAB method and the non-overlapping domain decomposition method, so that the most resource-intensive matrix factorization steps can be performed in parallel independently within subdomains. The electromagnetic and thermal fields are solved separately, while coupled through the dissipated power and the temperature-dependent material parameters; after thermal steady state is reached, the mechanical simulation is stimulated subject to the temperature rise. The accuracy of electromagnetic-thermal coupling and thermal stress solution are first validated, and then the strong/weak parallel scalability experiments of the developed multiphysics solver are performed on supercomputer. Finally, an extremely challenging antenna array is simulated using the proposed solver, where to our best knowledge we bring the scale of multiphysics simulations excited by frequency-domain electromagnetic fields to the order of billion unknowns for the first time.

On the Numerical Solution of Sparse Linear Systems Emerging in Finite Volume Discretizations of 2D Boussinesq-Type Models on Unstructured Grids

This work aims to supplement the realization and validation of a higher-order well-balanced unstructured finite volume (FV) scheme, that has been relatively recently presented, for numerically simulating weakly non-linear weakly dispersive water waves over varying bathymetries. We investigate and develop solution strategies for the sparse linear system that appears during this FV discretisation of a set of extended Boussinesq-type equations on unstructured meshes. The resultant linear system of equations must be solved at each discrete time step as to recover the actual velocity field of the flow and advance in time. The system’s coefficient matrix is sparse, un-symmetric and often ill-conditioned. Its characteristics are affected by physical quantities of the problem to be solved, such as the undisturbed water depth and the mesh topology. To this end, we investigate the application of different well-known iterative techniques, with and without the usage of preconditioners and reordering, for the solution of this sparse linear system. The iiterative methods considered are the GMRES and the BiCGSTAB, three preconditioning techniques, including different ILU factorizations and two different reordering techniques are implemented and discussed. An optimal strategy, in terms of computational efficiency and robustness, is finally proposed which combines the use of the BiCGSTAB method with the ILUT preconditioner and the Reverse Cuthill–McKee reordering.

Numerical Calculation of Fluid Film Force on Journal Bearings Based on a Biconjugate Gradient-Stabilized Algorithm

Abstract The aim of this work is to present a technical numerical method to improve the efficiency of solving the Reynolds equation for hydrodynamic bearings. Hydrodynamic bearings are largely employed in the shafting of high-speed rotating machinery to provide adequate support. The distribution of hydrodynamic pressure inside the bearing can be obtained by solving the Reynolds equation. Improving the efficiency of solving the Reynolds equation plays an essential role in the design and analysis of journal bearings. A numerical model has been developed to support the analysis and future design of hydrodynamic bearings. The primary objective of the model is to improve the efficiency of solving the Reynold equation during the steady-state and dynamic analysis. The developed method effectively combines the bi-conjugate gradient stabilized (Bi-CGSTAB) algorithm with the Reynolds boundary conditions, resulting in an effective methodology to characterize the pressure distribution within the lubricant for hydrodynamic bearings. A c++ program was implemented, and the difference between the BI-CGSTAB method and the successive over relaxation (SOR) method was evaluated against accuracy, convergence, and computational consumption. The Bi-CGSTAB algorithm has shown promising application in steady-state and dynamic analyses of hydrodynamic bearings. Validation of the results has been made with reference and analytical solutions.

Heat and mass source effect on MHD double-diffusive mixed convection and entropy generation in a curved enclosure filled with nanofluid

This paper examines the two-dimensional laminar steady magnetohydrodynamic doublediffusive mixed convection in a curved enclosure filled with different types of nanofluids. The enclosure is differentially heated and concentrated, and the heat and mass source are embedded in a part of the left wall having temperature Th (>Tc) and concentration ch (>cc). The right vertical wall is allowed to move with constant velocity in a vertically upward direction to cause a shear-driven flow. The governing equations along with the boundary conditions are transformed into a nondimensional form and are written in stream function-velocity formulation, which is then solved numerically using the Bi-CGStab method. Based on the numerical results, the effects of the dominant parameters such as Richardson number (1 ≤ Ri ≤ 50), Hartmann number (0 ≤ Ha ≤ 60), solid volume fraction of nanoparticles (0.0 ≤ ϕ ≤ 0.02), location and length of the heat and mass source are examined. Results indicate that the augmentation of Richardson number, heat and mass source length and location cause heat and mass transfer to increase, while it decreases when Hartmann number and volume fraction of the nanoparticles increase. The total entropy generation rises by 1.32 times with the growing Richardson number, decreases by 1.21 times and 1.02 times with the rise in Hartmann number and nanoparticles volume fraction, respectively.

Double Diffusive Magnetohydrodynamic (MHD) Natural Convection and Entropy Generation in a Discretely Heated Inclined Dome-Shaped Enclosure Filled with Cu-Water Nanofluid

This research presents an investigation of laminar two-dimensional double-diffusive free convection and entropy formation in an inclined enclosure under the influence of an inclined magnetic field. The performance of natural convective heat transfer can be improved by doing modifications in enclosure geometry that impact the flow structure. We have considered a dome-shaped enclosure to examine the heat and mass transfer performance. The enclosure is saturated with Cu-water nanofluid and the two sidewalls of the enclosure are maintained at constant temperature Tc(<Th) and concentration cc(<ch). The top-curved wall is adiabatic, and the lower wall is discretely heated and concentrated. The governing equations are first non-dimensionalized and then written in stream function-velocity formulation that is solved numerically using the Bi-CGStab method. A comparison with previously published work in literature is presented and found to be in excellent agreement. Numerical simulations are performed for various values of considered parameters such as Rayleigh number (Ra), Hartmann number (Ha), the orientation of magnetic field (γ), volume fraction of nanoparticles (Φ), and inclination angle of the enclosure (δ). The mentioned parameters have a substantial impact on the cavity flow characteristics. The obtained results demonstrate that the average Sherwood number and Nusselt number are decreasing functions of both the Hartmann number and inclination angle of the enclosure. The minimum heat and mass transfer took place at δ = 135° as the angle of inclination of the enclosure restrains the fluid velocity and reduces the heat transfer rate. Also, entropy generation analysis is conducted for all the considered parameters. The results show that the dome-shaped enclosure has a substantial impact on the fluid flow that enables a smoother and more effective flow inside the cavity, which improves the natural convective heat and mass transmission.

Numerical solutions to linear transfer problems of polarized radiation

Context. Numerical solutions to transfer problems of polarized radiation in solar and stellar atmospheres commonly rely on stationary iterative methods, which often perform poorly when applied to large problems. In recent times, stationary iterative methods have been replaced by state-of-the-art preconditioned Krylov iterative methods for many applications. However, a general description and a convergence analysis of Krylov methods in the polarized radiative transfer context are still lacking. Aims. We describe the practical application of preconditioned Krylov methods to linear transfer problems of polarized radiation, possibly in a matrix-free context. The main aim is to clarify the advantages and drawbacks of various Krylov accelerators with respect to stationary iterative methods and direct solution strategies. Methods. After a brief introduction to the concept of Krylov methods, we report the convergence rate and the run time of various Krylov-accelerated techniques combined with different formal solvers when applied to a 1D benchmark transfer problem of polarized radiation. In particular, we analyze the GMRES, BICGSTAB, and CGS Krylov methods, preconditioned with Jacobi, (S)SOR, or an incomplete LU factorization. Furthermore, specific numerical tests were performed to study the robustness of the various methods as the problem size grew. Results. Krylov methods accelerate the convergence, reduce the run time, and improve the robustness (with respect to the problem size) of standard stationary iterative methods. Jacobi-preconditioned Krylov methods outperform SOR-preconditioned stationary iterations in all respects. In particular, the Jacobi-GMRES method offers the best overall performance for the problem setting in use. Conclusions. Krylov methods can be more challenging to implement than stationary iterative methods. However, an algebraic formulation of the radiative transfer problem allows one to apply and study Krylov acceleration strategies with little effort. Furthermore, many available numerical libraries implement matrix-free Krylov routines, enabling an almost effortless transition to Krylov methods.

Study of Entropy Generation and Magnetohydrodynamic (MHD) Natural Convection in a Curved Enclosure Having Various Amplitude and Filled with Cu–TiO2/Water Hybrid Nanofluid

We have studied the buoyancy-driven convection enhancement and entropy production in a Cu–TiO2/water (water with copper and titanium dioxide nanoparticles) hybrid nanofluid filled curved enclosure subjected to a uniform magnetic field. The enclosure has a sinusoidally heated right wall, cold left wall, uniformly heated bottom wall, and thermally insulated upper curved wall. The effect of different amplitudes (concave, square, and convex) of the upper curved wall is considered. The non-linear governing equations are non-dimensionalized and written in stream function-velocity formulation. Bi Conjugate Gradient Stabilized (BiCGStab) method is employed followed by a Tri-diagonal matrix algorithm (TDMA) for the numerical simulation. The considered parameters are as follows: Rayleigh number (Ra), Hartmann number (Ha), phase angle (ε), the amplitude of the curved wall (a), and nanoparticle volume fraction (Φ). The influence of the model parameters on entropy production, fluid flow, and heat transfer phenomenon has been investigated, and the simulated results are displayed in terms of flow fields and temperature fields. According to the studies, increasing the Rayleigh number and volume percentage of nanoparticles has a significant impact on heat transmission and hence dominates the convection effect, whereas increasing the Hartmann number has the opposite effect.

Preconditioned global GPBiCG method for solving saddle point problems with multiple right-hand sides and its convergence analysis

In this paper, we propose the preconditioned global generalized product-type method based on the preconditioned global BiCG method to solve nonsymmetric saddle point problems with multiple right-hand sides. We apply an indefinite preconditioner to enhance the convergence rate of the method. We also present some theoretical analyses and discuss the convergence of the PGl-GPBiCG method. Some useful properties of the preconditioned matrix are established. Moreover, we present the bounds for the residual norm of the PGl-GPBiCG method according to the residual norm of the global GMRES method that guarantees convergence. Finally, some numerical examples are presented to show the efficiency of the new method in comparison with the preconditioned global BiCGSTAB method, and a comparison with another preconditioner is also provided.

A high-throughput hybrid task and data parallel Poisson solver for large-scale simulations of incompressible turbulent flows on distributed GPUs

The solution of the pressure Poisson equation arising in the numerical solution of incompressible Navier–Stokes equations (INSE) is by far the most expensive part of the computational procedure, and often the major restricting factor for parallel implementations. Improvements in iterative linear solvers, e.g. deploying Krylov-based techniques and multigrid preconditioners, have been successfully applied for solving the INSE on CPU-based parallel computers. These numerical schemes, however, do not necessarily perform well on GPUs, mainly due to differences in the hardware architecture. Our previous work using many P100 GPUs of a flagship supercomputer showed that porting a highly optimized MPI-parallel CPU-based INSE solver to GPUs, accelerates significantly the underlying numerical algorithms, while the overall acceleration remains limited (Zolfaghari et al. [3]). The performance loss was mainly due to the Poisson solver, particularly the V-cycle geometric multigrid preconditioner. We also observed that the pure compute time for the GPU kernels remained nearly constant as grid size was increased. Motivated by these observations, we present herein an algebraically simpler, yet more advanced parallel implementation for the solution of the Poisson problem on large numbers of distributed GPUs. Data parallelism is achieved by using the classical Jacobi method with successive over-relaxation and an optimized iterative driver routine. Task parallelism is enhanced via minimizing GPU-GPU data exchanges as iterations proceed to reduce the communication overhead. The hybrid parallelism results in nearly 300 times less time-to-solution and thus computational cost (measured in node-hours) for the Poisson problem, compared to our best-case scenario CPU-based parallel implementation which uses a preconditioned BiCGstab method. The Poisson solver is then embedded in a flow solver with explicit third-order Runge-Kutta scheme for time-integration, which has been previously ported to GPUs. The flow solver is validated and computationally benchmarked for the transition and decay of the Taylor-Green Vortex at Re=1600 and the flow around a solid sphere at ReD=3700. Good strong scaling is demonstrated for both benchmarks. Further, nearly 70% lower electrical energy consumption than the CPU implementation is reported for Taylor-Green vortex case. We finally deploy the solver for DNS of systolic flow in a bileaflet mechanical heart valve, and present new insight into the complex laminar-turbulent transition process in this prosthesis.

A Parallel Preconditioned Solver for the 3D Helmholtz Equation

Discretization of the 3D Helmholtz equation leads to a linear system of a large size. The resulting linear system is difficult to be solved by a sequential method. In this paper, we propose a parallel preconditioned Bi-CGSTAB method for solving the system based on MPI(Message Passing Interface). We precondition the 3D Helmholtz equation by the complex shifted-Laplacian preconditioner, and employ the Krylov subspace method Bi-CGSTAB combined with the multi-grid to solve the resulting system. Numerical experiments are presented to illustrate the efficiency of the parallel Preconditioned solver.