Optimally efficient multigrid algorithms for incompressible Euler equations

A full multigrid/full approximation storage (FMG/FAS) method is developed along with a unified pressure-based algorithm to handle the complex fluid flow problems encompassing both compressible and incompressible regimes. Besides highlighting the key elements of the present algorithm on a curvilinear staggered grid system, several flow problems, ranging from incompressible to hypersonic speeds, are computed, in conjunction with an adaptive grid method, to demonstrate its computational capability. With wide variations in geometry, fluid physics, and grid distribution, speedup between the single- and multigrid procedures can be obtained. Because different pressure boundary conditions are needed for incompressible, subsonic, and supersonic inlet and outlet, the number of grid levels that can be effectively used by the present multigrid method appears flow dependent; the implications of this observation are discussed.

Objectives: The present investigation is devoted to analyzing the behavior of pressure and load carrying capacity of the hydrodynamic lubrication problem. The corresponding Reynolds equation of the considered hydrodynamic lubrication problem is acquired by the application of the Stokes equations of motion to the microcontinuum theory. This modification takes into consideration the couple of stress effects that arise from the combination of various additives with the lubricant. Method: The Daubechies wavelet multigrid method is used to find the solution of the non-dimensional Reynolds equation. Findings: Initially to validate the efficiency and accuracy of the proposed strategy, the test problem is considered. The findings of the test problems are presented in tables and captured in terms of the figures to ensure that the proposed method is more accurate and consumes less computational time as compared with the Finite difference method, Multigrid Method. Later, the effect of non-dimensional pressure and load carrying capacity for the considered geometry is calculated for varied parameters. Novelty: Most of the lubrication problems that are available in the literature are solved by the Finite difference method and this Finite difference method has errors (truncation and roundup) and also consumes a lot of time to find the solution. In order to overcome this, the multigrid method was introduced. Again to obtain more accuracy in the solution we used the Daubechies wavelet multigrid method. When a couple stress parameters changes from 0.0 to 0.2 pressure distribution increases to 111.03%. Keywords: Lubrication, Daubechies Wavelet Multigrid Method, Finite Journal Bearing, Couple Stresses, Reynolds Equation

In this thesis, we discuss a robust preconditioner (the AGKS preconditioner) for solving linear systems arising from approximations of partial differential equations (PDEs) with high-contrast coefficients. The problems considered here include the standard second and higher order elliptic PDEs such as high-contrast diffusion equation, Stokes' equation and biharmonic-plate equation. The goal of this study is the development of robust and parallelizable preconditioners that can easily be integrated to treat large configurations. The construction of the preconditioner consists of two phases. The first one is an algebraic phase which partitions the degrees of freedom into high and low permeability regions which may be of arbitrary geometry. This yields a corresponding block partitioning of the stiffness matrix allowing us to use a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. Singular perturbation analysis plays a big role to analyze the structure of the required subblock inverses in the high contrast case which shows that for high enough contrast each of the subblock inverses can be approximated well by solving only systems with constant coefficients. The second phase involves an efficient multigrid approximation of this exact inverse. After applying singular perturbation theory to each of the sub-blocks, we obtain that inverses of each of the subblocks with high contrast entries can be approximated efficiently using geometric multigrid methods, and that this approximation is robust with respect to both the contrast and the mesh size. The result is a multigrid method for high contrast problems which is provably optimal to both contrast and mesh size. We demonstrate the advantageous properties of the AGKS preconditioner using experiments on model high-contrast problems. We examine its performance against multigrid method under varying discretizations of diffusion equation, Stokes equation and biharmonic-plate equation. Thus, we show that we accomplished a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations.

Most Computational Fluid Dynamics (CFD) applications solve the Naiver-Stokes equations using various discretization methods like Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), etc. The most compute-intensive process in CFD algorithms is the pressure Poisson solver. Solving the pressure Poisson equation requires the solution of a set of linear simultaneous equations. As the problem size/complexity increases, so does the effort and time invested in solving the equations. This study presents fast and robust iterative solvers and the acceleration of the same using the Graphics Processing Unit (GPU) architecture. Serial versions of the codes are implemented in the C language and the parallelization on GPU is achieved using Compute Unified Device Architecture (CUDA). For this study, a two-dimensional heat conduction problem is considered. FDM is used for discretization. Iterative solvers Conjugate Gradient and Multi Grid methods have been implemented in serial and parallel. The performance enhancement with single GPU and CPU system over a single CPU in terms of computing time is reported. It is seen that multigrid algorithms are superior in convergence and nearly 13 times speed-up has been obtained through GPU acceleration using CUDA, for a grid size of 2048 X 2048. The tests have been done on a cluster with Intel(R) Xeon(R) CPU E5-2670 CPU and NVIDIA K10 GPU.

The direct numerical simulation of transitional and turbulent incompressible flows is an area that is increasing with the advance in computational resources. Code parallelization has became a useful tool in these simulations. In the present paper the unsteady two dimensional Navier–Stokes equations are used as physical model. Tollmien–Schlichting waves propagating in a Poiseuille flow is adopted as test case. Three frequencies showing different behavior according to the linear stability theory (LST) are used: unstable, nearly neutral and stable waves. The parallelization is done via domain decomposition in the streamwise direction. The time derivative is integrated by a classical fourth-order Runge–Kutta method. High-order compact finite difference schemes are used for the spatial derivatives discretization. The Poisson equation is solved by multigrid methods. The present paper explores different parallelization techniques for solving the tridiagonal system, and the multigrid method. The results are compared with LST, and also between the different parallel strategies to show the advantages and disadvantages of each one.

As analytical solutions to heat transfer problems are difficult to obtain. Computational methods are presented as important analysis tools but conventional computational methods like Gauss-Seidel iteration are slow to converge. Therefore, a multigrid solver is introduced to address this issue. This report covers the two-dimensional rectangular heat conduction being solved using the finite-difference method and accelerated by the multigrid method. A brief explanation of multigrid will be presented. The result obtained from the analytical solutions were used as the baseline for comparison with the multigrid method. Once the results from the multigrid are validated, the single-grid method (Gauss-Seidel) is compared with the multigrid method in term of convergence rate and accuracy of results.

Solving singularly perturbed convection-diffusion equations, especially for 3D problems, is a challenging problem. In this paper, we extend our work on the extrapolation cascadic multigrid (EXCMG) method for solving the 3D Poisson equation — cf. [Pan et al., J. Sci. Comput. 2017], to 3D convection-diffusion equations with singularly perturbed parameters. First, we introduce an exponential higher order compact finite difference scheme to discretize the 3D convection-diffusion equation with variable convection coefficients, resulting in a larger-scale nonsymmetric linear system. Then, we propose an EXCMG method combined with the biconjugate gradient stabilized smoother to solve the larger-scale nonsymmetric system efficiently. Numerical experiments demonstrate that the EXCMG method is a highly effective solver for convection-dominated problems, which outperforms the existing multigrid methods such as aggregation-based algebraic multigrid method.

In order to limit the computational cost of the variational data assimilation process, we investigate the use of multigrid methods to solve the associated optimal control system. On a linear advection equation, we study the impact of the regularization term and the discretization errors on the efficiency of the coarse grid correction step introduced by the multigrid method. We show that even if for a perfect numerical model the optimal control problem leads to the solution of an elliptic system, discretization errors introduce implicit diffusion that can alter the success of the multigrid methods. Then we test the multigrids configuration and the influence of the algorithmic parameters on a non-linear Burgers equation to show that the algorithm is robust and converges much faster than the monogrid one. Afin de limiter le coût de calcul lié aux méthodes variationnelles d’assimilation de données, nous nous intéressons ici à l’utilisation de méthodes multigrilles pour la résolution de systèmes de contrôle optimal. Sur un modèle simple d’advection linéaire, nous étudions l’impact du terme de régularisation du contrôle optimal ainsi que l’impact des erreurs de discrétisation sur l’efficacité de la correction grille grossière introduite par cette méthode. En particulier, nous montrons que pour un modèle numérique parfait, le problème de contrôle optimal est elliptique mais que les erreurs de discrétisation introduisant une diffusion implicite peuvent altérer les performances de la méthode multigrille. Enfin, sur une équation de Burgers, non linéaire, nous étudions l’influence des différents paramètres inhérents aux méthodes multigrilles et montrons que ces méthodes sont robustes et convergent beaucoup plus rapidement que les méthodes monogrilles.

An efficient finite-difference method for solving Helmholtz equation depends on two points: one is discrete scheme, the other is efficient algorithm. In this paper, we adopt the average-derivative scheme, which owns three advantages : Firstly, it can be applied to unequal directional sampling intervals for Helmholtz equation. Secondly, the scheme is pointwise consistent with Helmholtz equation in a perfect matched layer. And thirdly, it requires less than 4 grid points sampling per wavelength. To solve the discrete Helmholtz equation, which is extremely large and indefinite, direct methods cannot resolve well, and the Krylov subspace iterative methods, such as Bi-CGSTAB and GMRES combining a multigrid-based preconditioner, are good choices. However, the standard multigrid algorithm fails to converge when it encounters unequal directional sampling intervals, which is called anisotropy in multigrid. We analyze the most important three parts of standard multigrid : full weighting restriction operator, point relaxation methods and bilinear interpolation operator, and then we replace them with semi-coarsening, line relaxation and operator-dependent interpolation to make it convergent in anisotropic problems. Consequently, we obtain a satisfactory convergence speed for low and moderate frequency iterative problems in heterogeneous media.

BackgroundThe aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.MethodsWhile other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field.ResultsIt starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.ConclusionSolving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.

During the recent decades, multigrid methods have been extensively used in order to solve large scale linear systems derived from the discretization of partial differential equations using the finite difference method. The purpose of this work is the derivation of hybrid heterogeneous schemes based on time implicit approximating schemes, namely backward differences and Crank–Nicolson method, with the multigrid method in conjunction with explicit approximate inverses for the efficient iterative solution of unsymmetric linear systems. Thus, a new class of smoothers, based on approximate inverses, has been derived. Moreover, a dynamic relaxation scheme is proposed based on the dynamic over - under relaxation (DOUR) algorithm. In addition, a new class of Krylov subspace iterative methods in conjunction with multigrid preconditioning based on explicit approximate inverses is used for solving initial value problems. Finally, the applicability of the new proposed hybrid heterogeneous implicit time - explicit approximate inverses combined with multigrid methods for parabolic PDEs is discussed by solving characteristic time-dependent problems in two dimensions and the numerical results are given.

The purpose of this work is to develop a numerical code for modeling the performance of thin film plasmonic solar cells performing a detailed analysis of both the light interaction with the solar cell and the resulting charging transport due to the photovoltaic effect. The continuity and Poisson equations were modeled in each layer of thin film silicon solar cells using finite difference method. In each layer the system of equations were solved numerically. The mesh size was chosen to be smaller than the Debye length. Because of the large bandgap of amorphous silicon the intrinsic carrier concentration is lower in thin film silicon solar cells. Different parameters like energy bands, electron and hole concentrations, electric field, voltage, current densities etc. were calculated at each mesh point. The generation rate for carrier transport modeling was calculated from full wave electromagnetic simulations.

SUMMARYBootstrap algebraic multigrid (BAMG) is a multigrid‐based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0. This paper studies an improved form of BAMG, called relaxation‐corrected bootstrap algebraic multigrid (rBAMG), that involves adding scaled residuals of the test vectors to the least‐squares equations.The basic rBAMG scheme was introduced in an earlier paper [1] and analyzed on a simple model problem. The purpose of the current paper is to further develop this algorithm by incorporating several new critical components and to systematically study its performance on an interesting model problem from quantum chromodynamics. Whereas the earlier paper introduced a new least‐squares principle involving the residuals of the test vectors, a simple extrapolation scheme is developed here to accurately estimate the convergence factors of the evolving algebraic multigrid solver. Such a capability is essential to the effective development of a fast solver, and the approach introduced here is shown numerically to be much more effective than the conventional approach of just observing successive error reduction factors. Another component of the setup process developed here is an adaptive cycling process. This component assesses the effectiveness of the V‐cycle constructed in the initial rBAMG phase by applying it to the homogeneous equation. When poor convergence is observed, the set of test vectors is enhanced with the resulting error, enabling the subsequent least‐squares fit of interpolation to produce an improved V‐cycle. A related component is the scaling and recombination Ritz process that targets the so‐called weak approximation property in an attempt to reveal the important elements of these evolving error and test vector spaces.The aim of the numerical study documented here is to provide insight into the various design choices that arise in the development of an rBAMG algorithm. With this in mind, the results for quantum chromodynamics focus on the behavior of rBAMG in terms of the number of initial test vectors used, the number of relaxation sweeps applied to them, and the size of the target matrices. Copyright © 2012 John Wiley & Sons, Ltd.

Elastohydrodynamic lubrication (EHL) in point contacts can be numerically solved with various iteration methods, but so far the flow continuity of such solutions has not been explicitly verified. A series of closed regions with the same inlet side boundary is defined, and two treatments to total all flows related to the other boundaries of the closed regions are defined to enable flow-continuity verifications. The multigrid method and the traditional single mesh method with different relaxation configurations are utilized to solve different cases to evaluate computation efficiencies. For the multigrid method, the combination of a pointwise solver together with hybrid-relaxation factors is identified to perform better than other combinations. The single mesh method has inferior degrees of flow continuity than the multigrid method and needs much smaller error control values of pressure to achieve a decent level of flow continuity. Because flow continuity has a physical meaning, its verifications should be routinely included in any self-validation process for any EHL results. Effects of control errors of pressure, mesh sizes, differential schemes, and operating conditions on flow continuities are studied. Then, trends of film thickness with respect to speed are briefly discussed with meshes up to 4097 by 4097.

Four problems of fluid flow and heat transfer were designed in which non‐orthogonal, boundary‐fitted grids were to be used. These are proposed to serve as test cases for testing new solution methods. This paper presents solutions of the test problems obtained by using a multigrid finite volume method with grids of up to 320 × 320 control volumes. Starting from zero fields, iterations were performed until the sum of the absolute residuals had fallen seven orders of magnitude, thus ensuring that the variable values did not change to six most significant digits. By comparing the solutions for successive grids at moderate Reynolds and Rayleigh numbers, the discretization errors were estimated to be lower than 0·1%. The results presented in this paper may thus serve for comparison purposes as bench‐mark solutions.