Biconjugate Gradient Research Articles

Research on icing model and calculation methods

Aircraft icing seriously threatens flight safety. This paper describes a modified shallow-water icing thermodynamic model that is applicable to unstructured grids and considers the effects of changes in water physical parameters and sublimation on icing. An icing calculation method enables the automatic determination of whether freezing occurs and the type of icing. An autonomous icing calculation program is developed to extract the geometric coordinates and mesh information of the icing surface and combine this information with the multiphase flow field of air-supercooled water droplets. The icing equations in the Godnov format are discretized to produce a large-scale sparse matrix that is solved iteratively using the biconjugate gradient stabilized method. The output includes parameter distributions for the ice thickness, water film thickness, and equilibrium temperature. Taking the National Advisory Committee for Aeronautic 0012 airfoil as the study object, the results of icing simulations are compared with experimental data from an ice wind tunnel and the ice shapes calculated by the FENSAP-ICE software. The results are found to be in good agreement with the experimental data, and the ice height errors at stagnation points are less than 15%. In the case of rime ice, single-horned ice shapes are simulated for both conventional and large supercooled droplets. For mixed ice and glaze ice, double-horned ice shapes are simulated for both droplet conditions. FENSAP-ICE fails to simulate the ice horns and produces large errors in ice thickness and ice range.

Numerical Simulation of Non-Darcy Flow in Naturally Fractured Tight Gas Reservoirs for Enhanced Gas Recovery

In this work, we analyze non-Darcy two-component single-phase isothermal flow in naturally fractured tight gas reservoirs. The model is applied in a scenario of enhanced gas recovery (EGR) with the possibility of carbon dioxide storage. The properties of the gases are obtained via the Peng–Robinson equation of state. The finite volume method is used to solve the governing partial differential equations. This process leads to two subsystems of algebraic equations, which, after linearization and use of an operator splitting method, are solved by the conjugate gradient (CG) and biconjugate gradient stabilized (BiCGSTAB) methods for determining the pressure and fraction molar, respectively. We include inertial effects using the Barree and Conway model and gas slippage via a more recent model than Klinkenberg’s, and we use a simplified model for the effects of effective stress. We also utilize a mesh refinement technique to represent the discrete fractures. Finally, several simulations show the influence of inertial, slippage and stress effects on production in fractured tight gas reservoirs.

The localized RBF interpolation with its modifications for solving the incompressible two-phase fluid flows: A conservative Allen–Cahn–Navier–Stokes system

In this research work, we apply a numerical scheme based on the first-order time integration approach combined with the modifications of the meshless approximation for solving the conservative Allen–Cahn–Navier–Stokes equations. More precisely, we first utilize a first-order time discretization for the Navier–Stokes equations and the time-splitting technique of order one for the dynamics of the phase-field variable. Besides, we use the local interpolation based on the Matérn radial function for spatial discretization. We should solve a Poisson equation with the proper boundary conditions to have the divergence-free property during the numerical algorithm. Accordingly, the applied numerical procedure could not give a stable and accurate solution. Instead, we solve a regularization system in a discrete form. To prevent the instability of the numerical solution concerning the convection term, a biharmonic term with a small coefficient based on the high-order hyperviscosity formulation has been added, which has been approximated by a scalable interpolation based on the combination of polyharmonic spline with polynomials (known as the PHS+poly). The obtained full-discrete problem is solved using the biconjugate gradient stabilized method considering a proper preconditioner. We investigate the potency of the numerical scheme by presenting some simulations via uniform, hexagonal, and quasi-uniform nodes on rectangular and irregular domains. Besides, we have compared the proposed meshless method with the standard finite element method due to the used CPU time.

Flow dynamics and heat transfer in the wake of an rotary oscillating circular cylinder with an isothermal control plate

This research explores the flow dynamics and heat transfer within the wake of a rotary oscillating circular cylinder in the presence of an isothermal control plate. Two-dimensional, unsteady, viscous and laminar flow of a Newtonian fluid is considered using the Higher Order Compact Scheme (HOC) to discretize the governing equations and the Bi-Conjugate Gradient Stabilized method to solve the resulting system. Simulations are performed for various gap ratios, maximum angular velocities, and frequency ratios of oscillation at Prandtl number 0.7 and Reynolds number 150 using an in-house code. Results show a significant increase in heat transmission for d / R 0 = 0.5 and f / f 0 = 0.5 for all αm . Drag and lift coefficients are also analyzed, with the maximum peak of the drag coefficient decreasing by 9.88% for d / R 0 = 3 . This research offers valuable insights for advancing and optimizing aerodynamic forces and heat transfer processes, particularly in the field of fluid dynamics. The study focuses on enhancing the efficiency of heat transfer mechanisms around rotary oscillating circular cylinders, contributing to the development of cutting-edge technologies in aerodynamics and thermal management.

A GPU based accelerated solver for simulation of heat transfer during metal casting process

The metal casting process, which is one of the key drivers of the manufacturing industry, involves several physical phenomena occurring simultaneously like fluid flow, phase change, and heat transfer which affect the casting yield and quality. Casting process modeling involves numerical modeling of these phenomena on a computer. In recent decades, this has become an inevitable tool for foundry engineers to make defect-free castings. To expedite computational time graphics processing units (GPUs) are being increasingly used in the numerical modeling of heat transfer and fluid flow. Initially, in this work a CPU based implicit solver code is developed for solving the 3D unsteady energy equation including phase change numerically using finite volume method which predicts the thermal profile during solidification in the metal casting process in a completely filled mold. To address the computational bottleneck, which is identified as the linear algebraic solver based on the bi-conjugate gradient stabilized method, a GPU-based code is developed using Compute Unified Device Architecture toolkit and was implemented on the GPU. The CPU and GPU based codes are then validated against a commercial casting simulation code FLOW-3D CAST® for a simple casting part and against in-house experimental results for gravity die casting of a simple geometry. Parallel performance is analyzed for grid sizes ranging from 10 × 10 × 10 to 210 × 210 × 210 and for three time-step sizes. The performance of the GPU code based on occupancy and throughput is also investigated. The GPU code exhibits a maximum speedup of 308× compared to the CPU code for a grid size of 210 × 210 × 210 and a time-step size of 2 s.

An efficient multi-field dynamic model for 3D wave propagation in saturated anisotropic porous media

An efficient multi–field coupled dynamic model for saturated anisotropic porous materials is proposed here. The mathematical formulation of the dynamic deformation–diffusion problem is developed starting from the mixture theory and the definition of the effective stress for anisotropic poro–elasticity, taking into account also the fluid phase compressibility and anisotropic permeability. The effective stress principle is properly extended through the Biot tensor to predict the coupling between the shear stress of the solid skeleton and the pore fluid pressure. Numerical solution of the coupled problem is obtained by inf-sup stable Finite Element spaces. A fundamental issue for the computational efficiency of the coupled model is the numerical solution of the resulting large–size and non–symmetric discrete problem. In this work, we develop a fully implicit monolithic solver based on the Bi–Conjugate Gradient Stabilized (Bi-CGStab) algorithm accelerated via an ad–hoc Multi–Physics Reduction (MPR) preconditioning technique. The proposed advances are implemented in the 3D GeoMatFem research code and several numerical analyses are performed to test the potential and computational efficiency of the proposed tool. In particular, we focus on 3D wave propagation applications in fully saturated single and multi-layered anisotropic media. Numerical results show that our implementation is able to identify the shear wave splitting phenomenon due to the different degree of material symmetry between the solid phase and the whole porous material.

A weighted combination of reproducing kernel particle shape functions with cardinal functions of scalable polyharmonic spline radial kernel utilized in Galerkin weak form of a mathematical model related to anti-angiogenic therapy

In this manuscript, a new localized meshfree (meshless) approximation, i.e., a combination of the reproducing kernel particle (RKP) shape functions with the cardinal functions of the scalable polyharmonic spline radial kernel with polynomial augmentation (PHS+poly) is introduced. It is called the RKP+PHS+poly approximation, and its convergence rate is of order O(hm+1), where m is the total degree of polynomials. We apply this method to construct the spaces of trial and test functions in a Galerkin scheme of an extended version of the biological mathematical model in two dimensions describing the interactions between endothelial cells, fibronectin, angiogenic growth factors, and fasentin concentrations. By considering the row-sum method, the eigenvalue stability is also numerically carried out for the discrete equations corresponding to the obtained weak formulation. Accordingly, a semi-implicit form of the backward difference method of order 1 (SBDF1) has been utilized to approximate the weak form in time. We complete our numerical algorithm by solving the obtained full-discretized problem overtime via the biconjugate gradient stabilized (BiCGSTAB) solver with a proper preconditioner. Some simulation results are investigated by estimating the maximum velocity parameter of an enzymatic reaction from the experimental dose–response curve of fasentin to demonstrate the effect of using fasentin drug.

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.

Composite Scattering Study of Layered Rough Surface withTarget based on CCIA

In this paper, a fast Cross Coupling iterative Approach (CCIA) is proposed for studying the composite scattering of the layered rough surfaces with buried target, which uses forward backward method (FBM) to solve the electric field integral equations (EFIE) of the layered rough surface and bi-conjugate gradient method (BI-CG) to solve the EFIE of the target, and the interaction between the rough surface and the target is achieved by updating the excitation term. The algorithm is applied to calculate the composite scattering coefficients of the rough surface with a buried target, the results match with those of the traditional numerical algorithm MOM while the error can be reduced to 10−3 by 6 iterations, and the convergence speed and calculation accuracy meet the requirements. The composite scattering coefficients and Angular Correlation Function (ACF) amplitudes of layered rough surface and dielectric targets with different conditions are calculated, and the effects of various factors such as target size and burial depth on the composite scattering characteristics are discussed. It is found that the buried targets will have a great influence on the scattering characteristics, weakening or neglecting the coupling between them will lead to larger errors. Moreover, the results show that ACF can suppress scattering from rough surfaces well, making the scattering characteristics of the target more obvious, which is important for detecting underground targets.

Precision and efficiency of an interpolation approach to weakly singular integral equations

PurposeThis study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.Design/methodology/approachThe authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.FindingsApproaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.Research limitations/implicationsThe convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.Practical implicationsThere is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.Social implicationsThis work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.Originality/valueTo the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

A Hybrid Chebyshev Pseudo-Spectral Finite-Difference Time-Domain Method for Numerical Simulation of 2D Acoustic Wave Propagation

In this study, a hybrid Chebyshev pseudo-spectral finite-difference time-domain (CPS-FDTD) algorithm is proposed for simulating 2D acoustic wave propagation in heterogeneous media, which is different from the other traditional numerical schemes such as finite element and finite difference. This proposed hybrid method integrates the efficiency of the FDTD approach in the time domain and the high accuracy of the CPS technique in the spatial domain. We present the calculation formulas of this novel approach and conduct simulation experiments to test it. The biconjugate gradient is solved by combining the large symmetric sparse systems stabilized algorithm with an incomplete LU factorization. Three numerical experiments are further presented to illustrate the accuracy, efficiency, and flexibility of the hybrid CPS-FDTD algorithm.

Preconditioned BiCGSTAB and BiCRSTAB methods for solving the Sylvester tensor equation

This paper presents a biconjugate gradient stabilized (BiCGSTAB) method and a biconjugate residual stabilized (BiCRSTAB) method for solving the Sylvester tensor equation, respectively. Preconditioned BiCGSTAB and BiCRSTAB algorithms are also developed to solve the Sylvester tensor equation. The convergence of each proposed iterative algorithm is proved. Several numerical examples are shown to illustrate the effectiveness of the proposed methods.

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.

Fast large-scale gravity modeling using a high-order compact cascadic multigrid strategy

A new high-order compact (HOC) cascadic multigrid (MG) strategy is developed for modeling large-scale complex gravity problems on nonuniform rectilinear grids. First, the governing 3D Poisson’s gravitational potential equation is discretized by a novel HOC difference scheme, which allows for Robin boundary conditions to further reduce the computational domain. Then, the resulting asymmetric linear system of equations is solved by a generalized extrapolation cascadic MG (gEXCMG) method augmented with a novel MG prolongation operator. The quartic interpolation and completed Richardson extrapolation assist this MG prolongation operator to produce a good initial guess for the biconjugate gradient stabilized smoother. Two synthetic and SEG/EAGE salt models are examined to verify the presented approach. Results indicate that our algorithm can efficiently and accurately produce the gravity signals for complex geologic models. Moreover, the comparisons with the state-of-the-art algebraic MG solvers demonstrate that our gEXCMG solver offers substantially better efficiency with significantly less memory consumption. Therefore, our newly developed method has the potential for serving as a forward engine in large-scale gravity inversions.

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).

Microseismic wavefield modelling in anisotropic elastic media using integral equation method

AbstractIn this paper, we present a frequency‐domain volume integral method to model the microseismic wavefield in heterogeneous anisotropic‐elastic media. The elastic wave equation is written as an integral equation of the Lippmann–Schwinger type, and the seismic source is represented as a general moment tensor. The actual medium is split into a background medium and a scattered medium. The background part of the displacement field is computed analytically, but the scattered part requires a numerical solution. The existing matrix‐based implementation of the integral equation is computationally inefficient to model the wavefield in three‐dimensional earth. An integral equation for the particle displacement is, hence, formulated in a matrix‐free manner through the application of the Fourier transform. The biconjugate gradient stabilized method is used to iteratively obtain the solution of this equation. The integral equation method is naturally target oriented, and it is not necessary to fully discretize the model. This is very helpful in the microseismic wavefield computation at receivers in the borehole in many cases; say, for example, we want to focus only on the fluid injection zone in the reservoir–overburden system and not on the whole subsurface region. Additionally, the integral equation system matrix has a low condition number. This provides us flexibility in the selection of the grid size, especially at low frequencies for given wave velocities. Considering all these factors, we apply the numerical scheme to three different models in order of increasing geological complexity. We obtain the elastic displacement fields corresponding to the different types of moment tensor sources, which prove the utility of this method in microseismic. The generated synthetic data are intended to be used in inversion for the microseismic source and model parameters.

Efficient 2D Tikhonov smoothness regularization with recursive filtering

In this work, we describe an implementation of the 2D Tikhonov regularization filter which scales linearly with the input signal’s size. In the homogeneous case, we propose a novel algorithm to decompose the filter’s 2D kernel as a sum of axis-aligned Gaussians. Our algorithm uses symmetries of the kernel to provide a fast computation of the Gaussian decomposition in the frequency domain, where the 2D Tikhonov kernel has a closed-form expression. The convolution with each Gaussian is then computed using linear-time separable recursive filtering. In the non-homogeneous case, we also decompose the 2D problem as a series of iterated linear-time separable recursive filters, which can be combined with the Biconjugate Gradient Stabilized method for fast convergence. In this way, a fast solution to the 2D Tikhonov regularization problem is obtained.

From BEYONDPLANCK to COSMOGLOBE: Preliminary WMAP Q-band analysis

We present the first application of the COSMOGLOBE analysis framework by analyzing nine-year WMAP time-ordered observations that uses similar machinery to that of BEYONDPLANCK for the Planck Low Frequency Instrument (LFI). We analyzed only the Q-band (41 GHz) data and report on the low-level analysis process based on uncalibrated time-ordered data to calibrated maps. Most of the existing BEYONDPLANCK pipeline may be reused for WMAP analysis with minimal changes to the existing codebase. The main modification is the implementation of the same preconditioned biconjugate gradient mapmaker used by the WMAP team. Producing a single WMAP Q1-band sample requires 22 CPU-hrs, which is slightly more than the cost of a Planck 44 GHz sample of 17 CPU-hrs; this demonstrates that a full end-to-end Bayesian processing of the WMAP data is computationally feasible. In general, our recovered maps are very similar to the maps released by the WMAP team, although with two notable differences. In terms of temperature, we find a ∼2 μK quadrupole difference that most likely is caused by different gain modeling, while in polarization we find a distinct 2.5 μK signal that has been previously referred to as poorly measured modes by the WMAP team. In the COSMOGLOBE processing, this pattern arises from temperature-to-polarization leakage from the coupling between the CMB Solar dipole, transmission imbalance, and sidelobes. No traces of this pattern are found in either the frequency map or TOD residual map, suggesting that the current processing has succeeded in modeling these poorly measured modes within the assumed parametric model by using Planck information to break the sky-synchronous degeneracies inherent in the WMAP scanning strategy.

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.

Preconditioned tensor format conjugate gradient squared and biconjugate gradient stabilized methods for solving stein tensor equations

AbstractThis article is concerned with solving the high order Stein tensor equation arising in control theory. The conjugate gradient squared (CGS) method and the biconjugate gradient stabilized (BiCGSTAB) method are attractive methods for solving linear systems. Compared with the large‐scale matrix equation, the equivalent tensor equation needs less storage space and computational costs. Therefore, we present the tensor formats of CGS and BiCGSTAB methods for solving high order Stein tensor equations. Moreover, a nearest Kronecker product preconditioner is given and the preconditioned tensor format methods are studied. Finally, the feasibility and effectiveness of the new methods are verified by some numerical examples.