BiCGSTAB Research Articles

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.

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.

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.

A state-based peridynamic analysis in a finite element framework

This study presents a PeriDynamic (PD) element to perform deformation and fracture analysis within a finite element framework. The PD interactions among the finite element nodes are achieved through a truss element. This element permits non-uniform discretization with an irregular shape domain of interaction and a variable horizon. The size and shape of the interaction domain dictates the element connectivity. Such connectivity results in a sparsely populated global system stiffness matrix. The solution of such a system of equations is achieved by employing the BiConjugate Gradient Stabilized (BICGSTAB) method within the in-house program. The explicit analysis is performed by constructing a global lumped mass matrix along with a hybrid implicit/explicit time integration scheme. The solution of resulting system of equations is achieved through an implicit solver until crack initiation, and it continues with an explicit time integration algorithm during crack growth. Crack nucleation and its growth occur when the maximum principal stress in an element exceeds the uniaxial tensile strength of the material or the visibility criteria is not satisfied. The capability of this truss element and failure criteria is established by considering four distinct geometric configurations with and without a crack, and loading conditions. In the absence of crack propagation, the peridynamic truss element predictions are compared with those of analytical and finite element results. In presence of crack propagation, the PD damage predictions are compared with the available experimental observations.

Weak form of peridynamics for nonlocal essential and natural boundary conditions

This study presents the weak form of peridynamic (PD) governing equations which permit the direct imposition of nonlocal essential and natural boundary conditions. It also presents a variational approach to derive the PD form of first- and second-order derivatives of a field variable at a point which is not symmetrically located in its domain of interaction. This capability enables the nonlocal PD representation of the internal force vector and the stress components without any calibration procedure. Furthermore, it removes the concern of truncated domain of interaction for a point near the surface. Thus, the solution is free of nonlocal boundary forces and surface effects. The numerical solution of the resulting equations can be achieved by considering an unstructured nonuniform discretization. The implicit solution to the discrete form of the equations is achieved by employing BiConjugate Gradient Stabilized (BICGSTAB) method which is an iterative technique for solving sparse non-symmetric linear systems. The explicit analysis is performed by constructing a global diagonal mass matrix, and using a hybrid implicit/explicit time integration scheme. The accuracy of this approach is demonstrated by considering an elastic isotropic plate with or without a cutout subjected to a combination of different types of boundary conditions under plane stress conditions.

2D Zero-Inertia Model for Solution of Overland Flow Problems in Flexible Meshes

AbstractA study of the efficiency of a zero-inertia model (ZI) for two-dimensional (2D) overland flow simulation is presented in this work. An upwind numerical scheme is used for the spatial discretization in the frame of finite-volume methods and an implicit formulation is chosen to avoid numerical instability. The scheme is applied in both structured and unstructured meshes, focusing in the latter ones due to their good adaptability. The ZI equation has a nonlinear character; hence, a linearization is required in the implicit procedure. This is carried out by means of Picard iterations method as a previous step to the system matrix resolution, characteristic of implicit techniques. The BiConjugate Gradient Stabilized (BiCGStab) method combined with sparse storage strategies is selected for the system resolution. A dual-threshold incomplete lower upper factorization (ILUT) is chosen as matrix preconditioner. Computational efficiency of the implicit temporal discretization for ZI model is explored under b...

Анализ эффективности итерационных методов решения систем линейных алгебраических уравнений, реализованных в пакете OpenFOAM

Significant part of the computational work in numerical simulations of technical systems, physical phenomena and technological processes is solving of linear algebraic equations systems arising from the discretization of the corresponding differential or integrodifferential equations. There are several classes of iterative methods for linear algebraic equations systems solving, which differ by the approach to the construction of the next iterative approximation. These classes are methods based on splitting, variational-type methods and projection-type methods. The aim of this study is the approach development for computational efficiency analysis of iterative methods for linear algebraic equations systems solving, which are difference analogues of continuum mechanics equations, and the approaches for method a’priori choice for linear algebraic equation solving with high computational efficiency. To choose the optimal numerical method for linear systems solving, in addition to the rate of convergence such characteristics of a linear system and numerical method, as condition number, smoothing factor and cost-coefficient should be considered. The smoothing factor and cost-coefficient can be computed through the amplification factors of the modes. The performance of a smoothing method is measured by its smoothing factor, but the cost of a numerical method is measured through its costcoefficient which shows the difference between amplitudes vanishing speeds of smooth modes and rough modes. The method for modes amplification factors computing using the discrete Fourier transform is proposed. The cost-coefficient usage allows to choose the optimal parameters of the multigrid preconditioner. Some test problems are considered and the efficiency of BiCGStab (BiConjugate Gradient Stabilized) method with the Incomplete LU and multigrid preconditioners is investigated for linear systems solving which follow from discrete forms of Helmholtz and Poisson equations. These linear algebraic equations systems arise in numerical simulation of incompressible viscous flow in a square cavity by using the LS-STAG cut-cell immersed boundary method with level-set function.

ML(n)BiCGStab: Reformulation, Analysis and Implementation

With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in [39] more systematically. There are n ways to define the ML(n)BiCGStab residual vector. Each definition leads to a different ML(n)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG is a bridge connecting BiCG and FOM. We also analyze the breakdown situation from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab. Implementation issues are also addressed.

A semi-implicit 3-D numerical model using sigam-coordinate for non-hydrostatic pressure free-surface flows

A 3-D numerical formulation is proposed on the horizontal Cartesian, vertical sigma-coordinate grid for modeling non-hydrostatic pressure free-surface flows. The pressure decomposition technique and semi-implicit method are used, with the solution procedure being split into two steps. First, with the implicit parts of non-hydrostatic pressures excluded, the provisional velocity field and free surface are obtained by solving a 2-D Poisson equation. Second, the theory of the differential operator is employed to derive the 3-D Poisson equation for non-hydrostatic pressures, which is solved to obtain the non-hydrostatic pressures and to update the provisional velocity field. When the non-orthogonal sigma-coordinate transformation is introduced, additional terms come into being, resulting in a 15-diagonal, diagonally dominant but unsymmetric linear system in the 3-D Poisson equation for non-hydrostatic pressures. The Biconjugate Gradient Stabilized (BiCGstab) method is used to solve the resulting 3-D unsymmetric linear system instead of the conjugate gradient method, which can only be used for symmetric, positive-definite linear systems. Three test cases are used for validations. The successful simulations of the small-amplitude wave, a supercritical flow over a ramp and a turbulent flow in the open channel indicate that the new model can simulate well non-hydrostatic flows, supercritical flows and turbulent flows.

Optimization of the iteration parameters of the simulation of incompressible flow

Abstract In optimizing the iteration parameters of the SIMPLE numerical procedure, a genetic algorithm (GA) that searches for a minimum calculation time in a space of iteration numbers was developed. A methodology has been presented for the numerical solution of natural convection in a squeezed cavity at Rayleigh numbers of 10 6. A double-staggered-grid numerical method which leads to generalization of the SIMPLE-like method for external and internal problems was used to optimize the iteration parameters for a single grid. The momentum equation was treated by the successive overrelaxation and the Gauss-Seidel methods. The pressure correction equation was solved by the conjugate gradient method with an incomplete factorization preconditioner. The temperature equation was solved by the bi-conjugate gradient-stabilized method with incomplete factorization preconditioner.

내부조파기법을 활용한 Navier-Stokes 방정식 모형의 고립파 처오름 수치모의

급경사에서의 고립파의 처오름을 예측하기 위해 3차원 수치모형에 내부조파기법을 도입하여 수치모형실험을 수행하였다. 수치모형은 Navier-Stokes 방정식을 유한차분법을 이용하여 계산하는 동수압 모형으로서, 난류의 해석을 위해서 상대적으로 큰 에디(eddy)만을 고려하는 SANS(spatially averaged Navier-Stokes) 방정식을 푸는 LES(large-eddy-simulation) 기반의 수치모형을 사용한다. 엇갈림 격자체계에서 유한차분법을 사용하여 지배방정식을 해석하는 모형으로서 수치기법으로 Two-step projection 기법을 사용하여 SANS 방정식을 풀었으며, Poisson 방정식을Bi-CGSTAB 기법을 이용하여 풀고 압력장을 계산하였다. 또한, 자유수면의 추적을 위하여 2차 정확도의 VOF(volume-of-fluid) 기법을 사용하였다. 먼저 고립파를 3차원 공간의 일정 수심상에서 내부조파하여 해석해와 비교한 후 분산오차에 대해 분석하였다. 그리고 고립파를 내부조파하여 급경사에서의 고립파의 처오름 및 처내림 현상을 예측하고 수리모형 실험결과와 비교 및 분석하였다. A three-dimensional numerical model called NEWTANK is employed to investigate solitary wave run-up with an internal wave-maker on a steep slope. The numerical model solves the spatially averaged Navier-Stokes equations for two-phase flows. The LES (large-eddy-simulation) approach is adopted to model the turbulence effect by using the Smagorinsky SGS (sub-grid scale) closure model. A two-step projection method is adopted in numerical solutions, aided by the Bi-CGSTAB (Bi-Conjugate Gradient Stabilized) method to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate VOF (volume-of-fluid) method is used to track the distorted and broken free surface. A solitary wave is first internally generated and propagated over a constant water depth in the three-dimensional domain. Numerically predicted results are compared with analytical solutions and numerical errors are analyzed in detail. The model is then applied to study solitary wave run-up on a steep slope and the obtained results are compared with available laboratory measurements.

Solution Methods for Nonsymmetric Linear Systems with Large off-Diagonal Elements and Discontinuous Coefficients

Linear systems with very large off-diagonal elements and discontinuous coefficients (LODC systems) arise in some modeling cases, such as those involving heterogeneous media. Such problems are usually solved by domain decomposition methods, but these can be difficult to implement on unstructured grids or when the boundaries between subdomains have a complicated geometry. Gordon and Gordon have shown that Bjorck and Elfving’s (sequential) CGMN algorithm and their own block-parallel CARP-CG are very robust and efficient on strongly convection dominated cases (but without discontinuous coefficients). They have also shown that scaling the equations by dividing each equation by the L2-norm of its coefficients, called “geometric row scaling” (GRS), improves the convergence properties of Bi-CGSTAB and GMRES on nonsymmetric systems with discontinuous coefficients, provided the convection terms are only small to moderate. Given a system Ax = b, it is shown that if C is obtained from A by applying GRS, then the diagonal elements of CCT are larger than the off-diagonal ones, so the normal equations system is manageable. These two operations are inherent in the Kaczmarz algorithm, and hence also in CGMN and CARP-CG (which are CG-accelerations of Kaczmarz). It is shown that these two methods are also very effective on systems with discontinuous coefficients derived from strongly convection dominated elliptic PDEs. CGNR and CGNE also benefit greatly from this approach, but they are much less efficient.

Application of Iterative Techniques for Electromagnetic Scattering From Dielectric Random and Reentrant Rough Surfaces

Stationary [e.g., forward-backward method (FBM)] and nonstationary [e.g., conjugate gradient squared, quasi-minimal residual, and biconjugate gradient stabilized (Bi-CGSTAB)] iterative techniques are applied to the solution of electromagnetic wave scattering from dielectric random rough surfaces with arbitrary complex dielectric constants. The convergence issues as well as the efficiency and accuracy of all the approaches considered in this paper are investigated by comparing obtained scattering (in the form of normalized radar cross section) and surface field values with the numerically exact solution, computed by employing the conventional method of moments. It has been observed that similar to perfectly and imperfectly conducting rough surface cases, the stationary iterative FBM converges faster when applied to geometries yielding best conditioned systems but exhibits convergence difficulties for general geometries due to its inherit limitations. However, nonstationary techniques are, in general, more robust when applied to arbitrarily general dielectric random rough surfaces, which yield more ill-conditioned systems. Therefore, they might prove to be more suitable for general scattering problems. Besides, as opposed to the perfectly and imperfectly conducting rough surface cases, the Bi-CGSTAB method and FBM show two interesting behaviors for dielectric rough surface profiles: 1) FBM generally converges for reentrant surfaces when the vertical polarization is considered and 2) the Bi-CGSTAB method has a peculiar convergence problem for horizontal polarization. Unlike the other nonstationary iterative techniques used in this paper, where a Jacobi preconditioner is used, convergent results are obtained by using a block-diagonal preconditioner