Stabilized Biconjugate Gradient Research Articles

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.

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.

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.

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.

Reconstruction of three-dimensional objects in layered composite structures from multimode orbital angular momentum.

Because of helical phase wavefront distribution, vortex electromagnetic waves are considered to carry more information and additional degrees of freedom than traditional spherical waves. Therefore, a vortex wave carrying orbital angular momentum (OAM) can improve inversion and imaging accuracy. In this work, we revisit the reconstruction of three-dimensional objects in layered composite structures extended with OAM. In forward modeling, the concentric uniform circle array is used to generate electromagnetic vortex beams. To analyze the difference of vortex beams, the electric field radiation pattern and phase pattern distribution of OAM waves with different modes are calculated. Then, the scattered field of layered media illuminated by different OAM beams is determined by the dyadic Green's function and the stabilized biconjugate gradient technique with a fast Fourier transform algorithm. In the inversion, the variational Born iterative method is used to reconstruct targets in layered composite structures, and multiple OAM modes are used to improve the reconstruction results. The numerical results prove that the permittivity of the target can be better reconstructed by using the multiple OAM modes rather than the traditional spherical wave. With the increase of OAM mode number, the reconstructed target parameters are closer to the true value. We expect that our results will provide a better understanding of the OAM and pave the way for the improvement of inversion and optical imaging technology using vortex waves.

Numerical Modelling of TEM Fields by the Edge-Based Finite Element Method Using Tetrahedral Element for Central Rectangular Loops

In this study, we have implemented an edge-based finite element method for the numerical modeling of the transient electromagnetic method. We took the Helmholtz equation of the electric field as the governing equation for the edge-based finite element analysis. The modeling domain was discretized using linear tetrahedral mesh supported by Whitney-type vector basis functions. We inferred the equations by applying the Galerkin method. The system of equation was solved using a corrected version of the Bi Conjugate Gradient Stabilized Method (BiCGStab) algorithm to reduce the computational time. We obtained numerical solution for electric field in the Laplace domain; then the field was transformed into the time domain using the Gaver-Stehfest algorithm. Following this, the impulse response of the magnetic field was obtained through the Faraday law of electromagnetic induction as it is considerably more stable and computationally more efficient than inversion using the Fourier Transform. 3D geoelectric models were used to investigate the convergence of the edge-based finite element method with the analytic solution. The results are in good agreement with the analytical solution value for two resistivity contrasts in the 3D geoelectric brick model. We also compared the results of tetrahedral elements with the brick element in the 3D horizontal sheet and 3D conductive brick model. The results indicated that these two elements show very similar errors, but tetrahedral reflects fewer relative errors. For the low resistivity geoelectric model, numerical checks against the analytical solution, integral-equation method, and finite-difference time-domain solutions showed that the solutions would provide accurate results.

A robust meta-heuristic adaptive Bi-CGSTAB algorithm to online estimation of a three DoF state–space model in the presence of disturbance and uncertainty

Most control systems require a fairly accurate model of dynamic system to design or implement a controller. When system dynamics change, the dynamic model must undergo online or offline re-estimation. The online model estimation algorithms in the time domain, especially for large models in presence of sensor noise, model uncertainty and external disturbance are almost inaccurate and unstable. In this paper, based on the dynamic model characteristics a novel online robust meta-heuristic adaptive Bi Conjugate Gradient Stabilized (Bi-CGSTAB) algorithm is proposed to estimate the model parameters and attitude simultaneously. First, the model is estimated iteratively using the output of attitude estimation from the Kalman filter algorithm, and the attitude is estimated by the output of estimated model from the least square method. The estimation method focuses on the solving algorithm of the matrix equations of the model estimation. The online robust meta-heuristic adaptive Bi-CGSTAB method uses the information of previous iteration in the current iteration to set the solving-steps toward the local optimums. This method leads to a broader and more intelligent search in the Krylov subspace of answers. The numerical results show a higher performance, robustness and more accurate model estimation than the other stated methods in the paper.

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.

First- and second-order unconditionally stable direct discretization methods for multi-component Cahn–Hilliard system on surfaces

This paper proposes a first- and second-order unconditionally stable direct discretization method based on a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation for solving the N-component Cahn–Hilliard system. We define the discretizations of the gradient, divergence, and Laplace–Beltrami operators on triangle surfaces. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. We also prove that our method satisfies the mass conservation. The proposed scheme is solved by the biconjugate gradient stabilized (BiCGSTAB) method, which can be straightforwardly applied to GPU-accelerated biconjugate gradient stabilized implementation by using the Matlab Parallel Computing Toolbox. Several numerical experiments are performed and confirm the accuracy, stability, and efficiency of our proposed algorithm.

A novel local meshless scheme based on the radial basis function for pricing multi-asset options

‎A novel local meshless scheme based on the radial basis function (RBF) is introduced in this article for price multi-asset options of even European and American types based on the Black-Scholes model‎. ‎The proposed approach is obtained by using operator splitting and repeating the schemes of Richardson extrapolation in the time direction and coupling the RBF technology with a finite-difference (FD) method that leads to extremely sparse matrices in the spatial direction‎. ‎Therefore‎, ‎it is free of the ill-conditioned difficulties that are typical of the standard RBF approximation‎. ‎We have used a strong iterative idea named the stabilized Bi-conjugate gradient process (BiCGSTAB) to solve highly sparse systems raised by the new approach‎. ‎Moreover‎, ‎based on a review performed in the current study‎, ‎the presented scheme is unconditionally stable in the case of independent assets when spatial discretization nodes are equispaced‎. ‎As seen in numerical experiments‎, ‎it has a low computational cost and generates higher accuracy‎. ‎Finally‎, ‎the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector‎, ‎as well as in other fields of engineering and science‎. ‎Finally‎, ‎we conclude that the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector‎, ‎as well as in other fields of engineering and science‎.

The GMRES solver for the interpolating meshless local Petrov–Galerkin method applied to heat conduction

PurposeThe work presents a novel implementation of the generalized minimum residual (GMRES) solver in conjunction with the interpolating meshless local Petrov–Galerkin (MLPG) method to solve steady-state heat conduction in 2-D as well as in 3-D domains.Design/methodology/approachThe restarted version of the GMRES solver (with and without preconditioner) is applied to solve an asymmetric system of equations, arising due to the interpolating MLPG formulation. Its performance is compared with the biconjugate gradient stabilized (BiCGSTAB) solver on the basis of computation time and convergence behaviour. Jacobi and successive over-relaxation (SOR) methods are used as the preconditioners in both the solvers.FindingsThe results show that the GMRES solver outperforms the BiCGSTAB solver in terms of smoothness of convergence behaviour, while performs slightly better than the BiCGSTAB method in terms of Central processing Unit (CPU) time.Originality/valueMLPG formulation leads to a non-symmetric system of algebraic equations. Iterative methods such as GMRES and BiCGSTAB methods are required for its solution for large-scale problems. This work presents the use of GMRES solver with the MLPG method for the very first time.

A finite-difference iterative solver of the Helmholtz equation for frequency-domain seismic wave modeling and full-waveform inversion

We have developed a finite-difference iterative solver of the Helmholtz equation for seismic modeling and inversion in the frequency domain. The iterative solver involves the shifted Laplacian operator and two-level preconditioners. It is based on the application of the preconditioners to the Krylov subspace stabilized biconjugate gradient method. A critical factor for the iterative solver is the introduction of a new preconditioner into the Krylov subspace iteration method to solve the linear equation system resulting from the discretization of the Helmholtz equation. This new preconditioner is based on a reformulation of an integral equation-based convergent Born series for the Lippmann-Schwinger equation to an equivalent differential equation. We have determined that our iterative solver combined with the novel preconditioner when incorporated with the finite-difference method accelerates the convergence of the Krylov subspace iteration method for frequency-domain seismic wave modeling. A comparison of a direct solver, a one-level Krylov subspace iterative solver, and our two-level iterative solver verified the accuracy and accelerated convergence of the new scheme. Extensive tests in full-waveform inversion demonstrate the solver’s applicability to such problems.

Numerical simulation of a prostate tumor growth model by the RBF-FD scheme and a semi-implicit time discretization

The aim of this work consists of finding a suitable numerical method for the solution of the mathematical model describing the prostate tumor growth, formulated as a system of time-dependent partial differential equations (PDEs), which plays a key role in the field of mathematical oncology. In the literature on the subject, there are a few numerical methods for solving the proposed mathematical model. Localized prostate cancer growth is known as a moving interface problem, which must be solved in a suitable stable way. The mathematical model considered in this paper is a system of time-dependent nonlinear PDEs that describes the interaction between cancer cells, nutrients, and prostate-specific antigen (PSA). Here, we first derive a non-dimensional form of the studied mathematical model using the well-known non-dimensionalization technique, which makes it easier to implement different numerical techniques. Afterward, the analysis of the numerical method describing the two-dimensional prostate tumor growth problem, based on radial basis function-generated finite difference (RBF-FD) scheme, in combination with a first-order time discretization has been done. The numerical technique we use, does not need the use of any adaptivity techniques to capture the features in the interface. The discretization leads to solving a linear system of algebraic equations solved via the biconjugate gradient stabilized (BiCGSTAB) algorithm with zero-fill incomplete lower–upper (ILU) preconditioner. Comparing the results obtained in this investigation with those reported in the recent literature, the proposed approach confirms the ability of the developed numerical scheme. Besides, the effect of choosing constant parameters in the mathematical model is verified by many simulations on rectangular and circular domains.

BICGSTAB-FFT Method of Moments with NURBS for Analysis of Planar Generic Layouts Embedded in Large Multilayer Structures

BICGSTAB-FFT method of moment (MM) scheme is proposed to analyze several levels of planar generic layouts embedded in large multilayer structures when the layout geometries are modeled by NURBS surfaces. In this scheme, efficient computation of normalized error defined in iterative bi-conjugate gradient stabilized (BICGSTAB) method for large multilayer structure analysis problems is implemented. The efficient computation is based on pulse expansion with dense equi-spaced mesh of generalized rooftop basis functions (BFs) defined on NURBS surfaces and equivalent periodic problem (EPP) in order to apply fast Fourier transforms (FFT). Moreover, efficient computation of Green’s functions for multilayer structure is implemented for near and far field regions. Experimental and numerical validations of whole printed reflect array antennas of electrical size between 8 and 16 times the vacuum wavelengths are shown. In these validations, CPU time consumptions of the proposed method are obtained with results between few minutes and half an hour using a conventional laptop.

Simulation of 3-D Electromagnetic Scattering and Inverse Scattering by Arbitrary Anisotropic Dielectric Objects Embedded in Layered Arbitrary Anisotropic Media

This communication studies 3-D electromagnetic (EM) scattering and inverse scattering of arbitrary anisotropic dielectric objects embedded in multilayered media. Different from previous work, the scatterer is not only arbitrary anisotropic, but also the arbitrary anisotropy of the layered background media is formed by rotation of the principal axes of uniaxial media. In addition, the scatterers are allowed to be placed across layer boundaries. The forward model is formulated by the electric field integral equation (EFIE), solved by the biconjugate gradient stabilized (BCGS) method, and accelerated by the fast Fourier transform (FFT). The time complexity is analyzed for the BCGS-FFT applied to layered arbitrary anisotropic media. In the inversion model, the variational Born iterative method (VBIM) is employed to reconstruct the arbitrary anisotropic profiles of both the permittivity and conductivity. Numerical simulations are performed to verify the feasibility and antinoise ability of the forward and inverse solvers in the circumstance of layered arbitrary anisotropic background media.

Radial basis function-generated finite difference scheme for simulating the brain cancer growth model under radiotherapy in various types of computational domains

Background and ObjectivesWe extend the original mathematical model, i.e., Swanson’s reaction-diffusion equation to the surfaces with no boundary, and we find a new numerical method based on a meshless approach for solving numerically Swanson’s reaction-diffusion model in the square and on the sphere. MethodsTo solve numerically the Swanson’s reaction-diffusion model and its extension version, a collocation meshless technique, namely radial basis function-generated finite difference (RBF-FD) scheme is employed for approximating the spatial variables in the square domain and on the sphere, respectively. Also, to approximate the time variable of the studied models, a first-order semi-implicit backward Euler scheme is used. The resulting fully discrete scheme is a linear system of algebraic equations per time step that is solved via the biconjugate gradient stabilized (BiCGSTAB) iterative algorithm with a zero-fill incomplete lower-upper (ILU) preconditioner. ResultsThe numerical simulations show the growth of untreated and treated brain tumors with radiotherapy using estimated and clinical data (given from magnetic resonance imaging (MRI) scans of patients). Moreover, the results reported here can be used for improving the treatment strategies of the invasive brain tumor. ConclusionsUsing the developed numerical scheme in this paper, we can simulate the behavior of the invasive form of brain tumor response to radiotherapy. Also, we can see the effects of radiation response on the brain tumor cell concentration of individual patients. The proposed meshless technique, which is applied for solving numerically the studied model, does not depend on any background mesh or triangulation for approximation in comparison with mesh-dependent methods. Moreover, we apply this technique to the sphere via any set of distributed points easily.

Minimization of entropy generation due to MHD double diffusive mixed convection in a lid driven trapezoidal cavity with various aspect ratios

Entropy generation minimization has significant importance in fluid flow, heat and mass transfer in an enclosure to get the maximum efficiency of a system and to reduce the loss of energy. In the present study, the analysis of mixed convection fluid flow, heat and mass transfer with heat line and mass line concept and entropy generation due to the effects of fluid flow, heat flow, mass flow and magnetic field in a trapezoidal enclosure with linearly heated and diffusive left wall, uniformly heated and diffusive lower wall, cold and nondiffusive right wall, adiabatic and zero diffusion gradient top wall have been reported. Parametric studies for the wide range of Prandtl number (Pr = 0.7 for air cooling system and Pr = 1000 for the engines filled with olive or engine oils), Rayleigh number (Ra = 103–105), aspect ratio (A = 0.5–1.5) and inclination angle of the cavity (ϕ = 45°–90°) have been performed, which help to construct the perfect shape of cavity in many engineering and physical applications so that the entropy is minimum to get the maximum efficiency of any system. The finite-difference approximation has been used to find out the numerical solutions. Biconjugate Gradient Stabilized (BiCGStab) method is used to solve the discretized nonhomogeneous system of linear equations.

MAGNETIC FIELD EFFECT ON THE HEAT TRANSFER IN A NANOFLUID FILLED LID DRIVEN CAVITY WITH JOULE HEATING

In this paper, the effects of magnetic field, Joule heating and volumetric heat generation on the heat transfer and fluid flow in a Cu-Water nanofluid filled lid driven cavity using enhanced streamfunction–velocity method are investigated. The cavity is heated by a uniform volumetric heat density and side walls have constant temperature. The top wall moves with constant velocity in +x direction, while no-slip boundary conditions are imposed on the other walls of the cavity. An inclined fixed magnetic field is applied to the left side wall of the cavity. The dimensionless governing equations are solved numerically for the stream function and temperature using finite difference method for various Richardson(Ri), Reynolds(Re), Hartmann (Ha), Eckert(Ec)numbers, magnetic field angle(α) and solid volume fraction of the nanofluid() in MATLAB software. To discretize the streamfunction-velocity formulation, a five point constant coefficient second-order compact finite difference approximation which avoids difficulties inherent in the conventional streamfunction–vorticity and primitive variable formulations is used. The stream function equation is solved using fast Poisson's equation solver on a rectangular grid (POICALC function in MATLAB) and the temperature equation is solved using Jacobi bi-conjugate gradient stabilized (BiCGSTAB) method. The heat transfer within the cavity is characterized by Nusselt number (Nu1). The results show that Nu1 is significantly increased by increasing Ri and  and increasing the Reynolds number enhances convective cooling. The heat transfer within the cavity is decreased by increasing Hartmann number which improves conduction heat transfer and reduces Nu1. Joule heating has a negative effect on the convection within the cavity and convection is decreased by increasing the value of Ec. It can be investigated that Nu1 is decreased by increasing Ec due to the strong distortion effect of Joule heating on convection current of heat transfer.

Multiparametric Electromagnetic Inversion of 3-D Biaxial Anisotropic Objects Embedded in Layered Uniaxial Media Using VBIM Enhanced by Structural Consistency Constraint

This article presents the 3-D inversion of biaxial anisotropic objects embedded in layered uniaxial media by the variational Born iterative method (VBIM). Although previously only the isotropic complex permittivity profile was reconstructed, here we retrieve all the six diagonal anisotropic permittivity and conductivity parameters simultaneously. In the forward model, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is used to solve the volume electric field integral equation (EFIE). In the inversion model, VBIM is employed to minimize the cost function. In order to improve the inversion results, we propose a novel structural consistency constraint (SCC) applied to VBIM. The SCC is based on the Monte Carlo method and reduces the computational domain in the Born iteration. The major new contribution of this work is to combine the deterministic inversion algorithm VBIM with the stochastic Monte Carlo method to reconstruct all the diagonal anisotropic parameters of the scatterers. Several typical numerical models are simulated, and the results validate the efficiency, antinoise ability as well as the accuracy of VBIM-SCC for reconstructing 3-D biaxial anisotropic objects embedded in multilayered uniaxial media.