Multigrid Finite Element Method Research Articles

Influence of Geometrical Shape on the Lubrication Performance of Partially Micro-Grooved Gas-Lubricated Parallel Slider Bearings

The influence of grooving parameters and micro-groove cross-section shape on the lubrication performance of partially micro-grooved gas-lubricated parallel slider bearings is investigated in the present study. The solution procedure for the multi-grid finite element method is that the algebraic equations formed by the finite element method are in turn smoothed by the interpolation from the coarse grid to the fine grid and the restriction from the fine grid to the coarse grid. By adopting the multi-grid finite element method, the pressure distribution of partially micro-grooved gas-lubricated parallel slider bearings is obtained. The grooving parameters are optimized to obtain the maximum average pressure for different micro-groove cross-section shapes. The results show that the orientation angle of the micro-grooves, depth of the micro-grooves, width of the micro-grooves, area density of the micro-grooves, micro-grooved fraction, and cross-section shape of the micro-grooves have an important influence on the hydrodynamic pressure of partially micro-grooved gas-lubricated parallel slider bearings. The results of this study demonstrate that the lubrication performance of partially micro-grooved gas-lubricated parallel slider bearings can be obviously enhanced by adopting the appropriate grooving parameters and micro-groove cross-shape. The results of this study is beneficial for the design of partially micro-grooved gas-lubricated parallel slider bearings.

Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour in the Presence of a Nanofluid

A backward facing duct are present in various industrial applications especially those focused on heat transfer. The flow through a curved backward facing duct especially in the presence of nanofluid presents complexities compared to a straight backward facing step (BFS) duct. Therefore, the present numerical study deals a nanofluid flow (Fe3O4-H2O) forced convection in a curved backward facing duct. The objective of this investigation is to visualize at different curvature angles g (0°, 30°, 45°, 60°, 90°) imposed on the top wall of the duct, the effect of Hartmann number Ha (0, 50, 100), magnetic field inclination angle γ (0°, 60°, 90°), Reynolds number Re (10, 100, 200) and nanoparticle volume fraction φ (0 %, 0.05 %, 0.1 %). The dimensionless governing equations are solved using the multigrid finite element method. The results showed that the heat transfer was enhanced at the curved angle g = 90° for large Hartman numbers, thus, the average Nusselt number increased with a ratio of 240.74 % in the case of Hartmann number (Ha = 100).

Investigation and analysis of the numerical approach to solve the multi-term time-fractional advection-diffusion model

<abstract><p>In this paper, a methodical approach is presented to approximate the multi-term fractional advection-diffusion model (MT-FAD). The Lagrange squared interpolation is used to discretize temporal fractional derivatives, and Legendre polynomials are shifted as an operator to discretize the spatial fractional derivatives. The advantage of these numerical techniques lies in the orthogonality of Legendre polynomials and its matrix operations. A quadratic implicit design as well as its stability and convergence analysis are evaluated. It should be noted that the theoretical proof obtained from this study represents the first results for these numerical schemes. Finally, we provide three numerical examples to verify the validity of the proposed methods and demonstrate their accuracy and effectiveness in comparison with previous studies shown in [W. P. Bu, X. T. Liu, Y. F. Tang, J. Y. Yang, Finite element multigrid method for multi-term time fractional advection diffusion equations, <italic>Int. J. Model. Simul. Sci. Comput.</italic>, <bold>6</bold> (2015), 1540001].</p></abstract>

Topological photonics by breaking the degeneracy of line node singularities in semimetal-like photonic crystals.

Degeneracy is an omnipresent phenomenon in various physical systems, which has its roots in the preservation of geometrical symmetry. In electronic and photonic crystal systems, very often this degeneracy can be broken by virtue of strong interactions between photonic modes of the same energy, where the level repulsion and the hybridization between modes causes the emergence of photonic bandgaps. However, most often this phenomenon does not lead to a complete and inverted bandgap formation over the entire Brillouin zone. Here, by systematically breaking the symmetry of a two-dimensional square photonic crystal, we investigate the formation of Dirac points, line node singularities, and inverted bandgaps. The formation of this complete bandgap is due to the level repulsion between degenerate modes along the line nodes of a semimetal-like photonic crystal, over the entire Brillouin zone. Our numerical experiments are performed by a home-build numerical framework based on a multigrid finite element method. The developed numerical toolbox and our observations pave the way towards designing complete bandgap photonic crystals and exploring the role of symmetry on the optical behaviour of even more complicated orders in photonic crystal systems.

Numerical Analysis Of a Falling Circular Particle Passing through a Fluid Channel having Diamond Shaped Obstacles

It has been analyzed that the particle motion inside a vertical channel while passing across diamond shaped obstacles produces severe effects on the fluid. Particle interaction with outer boundary, internal obstacles and with the fluid is inspected. An Eulerian based approach using a computational mesh is used in which solid particles are allowed to move freely in fluid domain. Fluid and particle interaction inside the whole domain is carried using Fictitious boundary method (FBM). A multigrid finite element method combined with the fictitious boundary method (FEM-FBM) is used for the simulation of in-compressible fluid flow along with rigid particle falling and colliding inside a fluid domain. A collision model to treat the Particle-obstacle and particle-wall interactions is used to avoid particle overlapping. The particulate flow is evaluated using an open source multigrid finite element solver FEATFLOW. Numerical investigations are executed in view of different particle positions and different alignment of diamond shaped obstacles. Effects on the movement of the particle and on the interaction of the fluid-particle system due to particle-wall, particle-Obstacle, particle-fluid interactivity has been analyzed.

Effect of the inclination angle of finned cylinder over a BFS on the MHD behavior in the presence of a nanofluid

The present numerical study is based on the forced magnetohydrodynamic (MHD) convection of a ferrofluid through a backward facing step (BFS). A cylinder with two fixed fins and fixed dimensions is implanted inside fluid. The dimensionless governing equations have been solved using the multigrid finite element method. Several parameters were considered, such as the Hartmann number 0≤ Ha ≤100, the magnetic field inclination angle 0°≤ ɣ ≤90°, the Reynolds number 10≤ Re ≤200, the nanoparticle volume fraction 0%≤ φ ≤10%, and the fins inclination angle 0°≤ a ≤180°. The results have shown that the presence of the fins improves the heat transfer, especially at the position a = 90° where the Nuave number increases with a ratio of 113% for Re = 200.

Hydrodynamic Lubrication of Partially Textured Gas Parallel Slider Bearings with Orientation Ellipse Dimples

The hydrodynamic lubrication performance of partially textured gas parallel slider bearings with orientation ellipse dimples is investigated in this paper. By using the multigrid finite element method, the pressure distribution between a partially textured slider and a smooth slider is obtained. The geometric parameters of the ellipse dimples are optimized to maximize the average pressure under a given sliding speed. The numerical results show that geometric parameters such as orientation angle, depth, slender ratio, and area density have an important impact on hydrodynamic pressure. Besides, the effect of textured fraction on hydrodynamic pressure is investigated under a given sliding speed. It is observed that the optimum textured fraction for maximizing the average pressure is dependent on the sliding speed.

Uniform convergence of V-cycle multigrid finite element method for one-dimensional time-dependent fractional problem

Analysing the fractional τ-norm, the uniform convergence of the V-cycle multigrid FEM for the time-dependent fractional problem is strictly proved when τ→0. The numerical experiments are performed to verify the convergence with O(NlogN) complexity by fast Fourier transform method.

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.

A multi-grid technique for coupling fluid flow with porous media flow

In this paper, we consider the coupling of fluid flow with porous media flow. A multi-grid finite element method for the coupled Stokes–Darcy problem with the Beavers–Joseph interface condition is proposed and discussed. The optimal error estimates are obtained. Numerical experiment is given to verify the theoretical analysis and indicate the accuracy and efficiency of the multi-grid method.

Hydrodynamic Lubrication of Micro-Grooved Gas Parallel Slider Bearings with Parabolic Grooves

The hydrodynamic lubrication performance of micro-grooved gas parallel slider bearings with parabolic grooves is investigated in this paper. By using the multi-grid finite element method, the pressure distribution between a micro-grooved slider and a smooth slider is obtained. The geometric parameters of the parabolic grooves are optimized to maximize the average pressure under a given sliding speed. The numerical results show that geometric parameters such as groove depth, width, spacing, and orientation angle have an important influence on hydrodynamic pressure. Furthermore, the effect of sliding speed on hydrodynamic pressure is investigated under a given set of groove depths. It is observed that there is an optimum value of sliding speed to maximize the average pressure for any given groove depth and that the optimum value is dependent of the groove depth. The results of this study indicate that the average pressure could be improved by employing the optimized groove depth according to the practical sliding speed. DOI: http://dx.doi.org/10.5755/j01.mech.23.6.19850

English

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.

Multigrid finite element method in stress analysis of three-dimensional elastic bodies of heterogeneous structure

To calculate the three-dimensional elastic body of heterogeneous structure under static loading, a method of multigrid finite element is provided, when implemented on the basis of algorithms of finite element method (FEM), using homogeneous and composite threedimensional multigrid finite elements (MFE). Peculiarities and differences of MFE from the currently available finite elements (FE) are to develop composite MFE (without increasing their dimensions), arbitrarily small basic partition of composite solids consisting of single-grid homogeneous FE of the first order can be used, i.e. in fact, to use micro approach in finite element form. These small partitions allow one to take into account in MFE, i.e. in the basic discrete models of composite solids, complex heterogeneous and microscopically inhomogeneous structure, shape, the complex nature of the loading and fixation and describe arbitrarily closely the stress and stain state by the equations of three-dimensional elastic theory without any additional simplifying hypotheses. When building the m grid FE, m of nested grids is used. The fine grid is generated by a basic partition of MFE, the other m —1 large grids are applied to reduce MFE dimensionality, when m is increased, MFE dimensionality becomes smaller. The procedures of developing MFE of rectangular parallelepiped, irregular shape, plate and beam types are given. MFE generate the small dimensional discrete models and numerical solutions with a high accuracy. An example of calculating the laminated plate, using three-dimensional 3-grid FE and the reference discrete model is given, with that having 2.2 milliards of FEM nodal unknowns.

A multilevel finite element method for Fredholm integral eigenvalue problems

In this work, we proposed a multigrid finite element (MFE) method for solving the Fredholm integral eigenvalue problems. The main motivation for such studies is to compute the Karhunen–Loève expansions of random fields, which play an important role in the applications of uncertainty quantification. In our MFE framework, solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh. Then, any existing efficient integration scheme can be used for the associated integration process. The error estimates are provided, and the computational complexity is analyzed. It is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh. Several numerical experiments are presented to validate the efficiency of the proposed numerical method.

The mixed finite element multigrid method for stokes equations.

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results.

Finite Element Multigrid Method for the Boundary Value Problem of Fractional Advection Dispersion Equation

The stationary fractional advection dispersion equation is discretized by linear finite element scheme, and a full V-cycle multigrid method (FV-MGM) is proposed to solve the resulting system. Some useful properties of the approximation and smoothing operators are proved. Using these properties we derive the convergence results in both <svg style="vertical-align:-0.0pt;width:16.1625px;" id="M1" height="15.8375" version="1.1" viewBox="0 0 16.1625 15.8375" width="16.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> </svg> norm and energy norm for FV-MGM. Numerical examples are given to demonstrate the convergence rate and efficiency of the method.

Creep analysis of steam turbine welded diaphragm

A procedure for the creep analysis of diaphragms is proposed that is based on the multi-grid finite element method, Rabotnov-type creep aging theory and the method of variable parameters of elasticity. Three-dimensional calculation models of the diaphragm are constructed and the peculiarities of its stress-strain state are found taking into account the reduced resistance to creep in welded joints. It is shown that the axial displacements of the joints of the diaphragm are 30% higher for the end guide vanes than those for the middle ones, while the stress intensity is twice as higher. The area of the elastic strain intensity maximum at the leading edge of the guide vanes moves, with consideration of creep, towards the welded joint area where their intensity reaches 2%.

3-D direct current resistivity forward modeling by adaptive multigrid finite element method

Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid finite element method was proposed. In this algorithm, a-posteriori error estimator was employed to generate adaptively refined mesh on a given initial mesh. On these iterative meshes, V-cycle based multigrid method was adopted to fast solve each linear equation with each initial iterative term interpolated from last mesh. With this error estimator, the unknowns were nearly optimally distributed on the final mesh which guaranteed the accuracy. The numerical results show that the multigrid solver is faster and more stable compared with ICCG solver. Meanwhile, the numerical results obtained from the final model discretization approximate the analytical solutions with maximal relative errors less than 1%, which remarkably validates this algorithm.

MULTIGRID FOURIER ANALYSIS ON SEMI‐STRUCTURED ANISOTROPIC MESHES FOR VECTOR PROBLEMS

An efficient multigrid finite element method for vector problems on triangular anisotropic semi‐structured grids is proposed. This algorithm is based on zebra line‐type smoothers to overcome the difficulties arising when multigrid is applied on stretched meshes. In order to choose the type of multigrid cycle and the number of pre‐ and post‐smoothing steps, a three‐grid Fourier analysis is done. To this end, local Fourier analysis (LFA) on triangular grids for scalar problems is extended to the vector case. To illustrate the good performance of the method, a system of reaction‐diffusion is considered as model problem. A very satisfactory global convergence factor is obtained by using a V(0,2)‐cycle for domains triangulated with highly anisotropic meshes.

A multigrid method for the Cahn–Hilliard equation with obstacle potential

We present a multigrid finite element method for the deep quench obstacle Cahn–Hilliard equation. The non-smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging. The method exhibits mesh-independent convergence properties in practice for arbitrary time step sizes. In addition, numerical evidence shows that this behaviour extends to small values of the interfacial parameter γ . Several numerical examples are given, including comparisons with existing alternative solution methods for the Cahn–Hilliard equation.