Local Fine Grid Research Articles

Local and Parallel Stabilized Finite Element Methods Based on the Lowest Equal-Order Elements for the Stokes–Darcy Model

In this article, two kinds of local and parallel stabilized finite element methods based upon two grid discretizations are proposed and investigated for the Stokes–Darcy model. The lowest equal-order finite element pairs (P1-P1-P1) are taken into account to approximate the velocity, pressure, and piezometric head, respectively. To circumvent the inf-sup condition, the stabilized term is chosen as the difference between a consistent and an under-integrated mass matrix. The proposed algorithms consist of approximating the low-frequency component on the global coarse grid and the high-frequency component on the local fine grid and assembling them to obtain the final approximation. To obtain a global continuous solution, the technique tool of the partition of unity is used. A rigorous theoretical analysis for the algorithms was conducted and numerical experiments were carried out to indicate the validity and efficiency of the algorithms.

Three-dimensional numerical analysis and experimental confirmation for investigating the ground-based lateral droplet ejection toward microgravity simulation

Rapid in situ manufacturing is significant in space exploration. Droplet-based printing technology with micrometer accuracy has great potential in space due to the advantages of convenient transportation, customized metal material, and good environmental adaptability. It could achieve the ground microgravity simulation by a small Bond number (Bo < 1). The present work proposes a new method to evaluate the reliability of the ground microgravity simulation in the lateral metallic droplet-based ejection. The three-dimensional model is developed to numerically analyze the droplet ejection dynamic process coupled with the volume of fluid method and the k–ω shear stress transport model. The model accuracy and efficiency are improved by the local fine grid. In addition, the computation is validated by the cryogenic alloy droplet ejection experiments and theoretical analysis. The proposed theoretical analysis equation has good agreement with the SnPb alloy droplet ejection trajectory. Reynolds number (Re), Weber number (We), Froude number (Fr), Ohnesorge number (Oh), and breakup length (Lb) are used to analyze the gravity influences on the droplet ejection process of different materials, nozzle length–diameter ratios, and crucible fluid unfilled heights. The ejection direction has little effect on the aluminum droplet formation time and breakup length and the gravity effect increases with the length–diameter ratio and unfilled heights. In simulated results, the minimum We number of the aluminum droplet formation is 0.22 and the cryogenic alloy droplet formation is 0.19. The reliability of ground physical microgravity simulation is dependent on material selection, and aluminum is more suitable than the cryogenic and SnPb alloys.

Convergence Properties of Local Defect Correction Algorithm for the Boundary Element Method

Sometimes boundary value problems have isolated regions where the solution changes rapidly. Therefore, when solving numerically, one needs a fine grid to capture the high activity. The fine grid can be implemented as a composite coarse-fine grid or as a global fine grid. One cheaper way of obtaining the composite grid solution is the use of the local defect correction technique. The technique is an algorithm that combines a global coarse grid solution and a local fine grid solution in an iterative way to estimate the solution on the corresponding composite grid. The algorithm is relatively new and its convergence properties have not been studied for the boundary element method. In this paper the objective is to determine convergence properties of the algorithm for the boundary element method. First, we formulate the algorithm as a fixed point iterative scheme, which has also not been done before for the boundary element method, and then study the properties of the iteration matrix. Results show that we can always expect convergence. Therefore, the algorithm opens up a real alternative for application in the boundary element method for problems with localised regions of high activity.

Hybrid sub-gridding ADE–FDTD method of modeling periodic metallic nanoparticle arrays**Project supported by the National Natural Science Foundation of China (Grant Nos. 61471105 and 61331007).

In this paper, a modified sub-gridding scheme that hybridizes the conventional finite-difference time-domain (FDTD) method and the unconditionally stable locally one-dimensional (LOD) FDTD is developed for analyzing the periodic metallic nanoparticle arrays. The dispersion of the metal, caused by the evanescent wave propagating along the metal-dielectric interface, is expressed by the Drude model and solved with a generalized auxiliary differential equation (ADE) technique. In the sub-gridding scheme, the ADE–FDTD is applied to the global coarse grids while the ADE–LOD–FDTD is applied to the local fine grids. The time step sizes in the fine-grid region and coarse-grid region can be synchronized, and thus obviating the temporal interpolation of the fields in the time-marching process. Numerical examples about extraordinary optical transmission through the periodic metallic nanoparticle array are provided to show the accuracy and efficiency of the proposed method.

A multiscale mixed finite element method with oversampling for modeling flow in fractured reservoirs using discrete fracture model

Fractures play a significant effect on the macro-scale flow, thus should be described exactly. Accurate modeling of flow in fractured media is usually done by discrete fracture model (DFM), as it provides a detailed representation of flow characteristic. However, considering the computational efficiency and accuracy, traditional numerical methods are not suitable for DFM. In this study, a multiscale mixed finite element method (MsMFEM) is proposed for detailed modeling of two-phase oil–water flow in fractured reservoirs. In MsMFEM, the velocity and pressure are first obtained on coarse grid. The interaction between the fractures and the matrix is captured through multiscale basis functions calculated numerically by solving DFM on the local fine grid. Through multiscale basis functions, this method can not only reach a high efficiency as upscaling technology, but also finally generate a more accurate and conservative velocity field on the full fine-scale grid. In our approach, oversampling technique is applied to get more accurate small-scale details. Triangular fine-scale grid is applied, making it possible to consider fractures in arbitrary directions. The validity of MsMFEM is proved through comparing experimental and numerical results. Comparisons of the multiscale solutions with the full fine-scale solutions indicate that the later one can be totally replaced by the former one. The results demonstrate that the multiscale technology is a promising method for multiscale flows in high-resolution fractured media.

A Simplified Parallel Two-Level Iterative Method for Simulation of Incompressible Navier-Stokes Equations

AbstractBased on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.

A–[formula omitted] finite element method with composite grids for time-dependent eddy current problem

We investigate the A–ϕ finite element method with the global coarse grids and the local fine grids (composite grids) for solving a time-dependent eddy current problem, which can improve the accuracy of the coarse grid solutions in some subdomains of interest in the case when properly increasing computational costs. Meanwhile, to decrease calculation complexity and avoid dealing with a saddle-point problem from the traditional A–ϕ scheme, we design an iteration which combines the composite grid method with classic steepest descent such that the solutions of the unknowns A and ϕ for the global coarse grids domain are decoupled in two separate equations. We prove it converges with a bounded rate independent of the mesh sizes.

Local defect correction for boundary integral equation methods

The aim in this paper is to develop a new local defect correction approach to gridding for problems with localised regions of high activity in the boundary element method. The technique of local defect correction has been studied for other methods as finite difference methods and finite volume methods. The initial attempts to developing such a technique by the authors for the boundary element method was based on block decomposition and manipulation of the coefficient matrix and right hand side of the system of equations in three dimension. It ignored the inherent global nature of the boundary integral equation, that is, each node of the grid contributes to all the others in the grid through integration. In this paper we present a better approach that takes this into account. We use a new integral approach to defect correction and develop the technique for the boundary element method. The technique offers an iterative way for obtaining the solution on an equivalent composite grid. It uses two grids: a global uniform coarse grid covering the whole boundary and a local fine grid covering the local active boundary. The solution of the local problem on the local fine grid is used to estimate the defect on the fine grid. The effect of the defect onto the right hand side is then considered. We demonstrate the technique’s strength using an example and show that it offers a cheaper alternative to either solving on a global uniform grid or directly on a composite grid. Keywords: integral equations, boundary elements, local error, local defect correction, local activity, gridding.

Adaptive variational multiscale method for the Stokes equations

SUMMARYAn adaptive variational multiscale method for the Stokes equations is presented in this paper. We solve the coarse scale problem on the coarse mesh and approximate the fine scale solution by solving a series of local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, numerical examples are presented to verify the algorithm.Copyright © 2012 John Wiley & Sons, Ltd.

Adaptive Local Postprocessing Finite Element Method for the Navier-Stokes Equations

An adaptive local postprocessing finite element method for the Navier-Stokes equations is presented in this paper. We firstly solve the problem on a relative coarse grid to get a rough approximation. Then, we correct the rough approximation by solving a series of approximate local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, some numerical examples are presented to verify the algorithm.

A multiscale adjoint method to compute sensitivity coefficients for flow in heterogeneous porous media

A multiscale adjoint (MSADJ) method is developed to compute high-resolution sensitivity coefficients for subsurface flow in large-scale heterogeneous geologic formations. In this method, the original fine-scale problem is partitioned into a set of coupled subgrid problems, such that the global adjoint problem can be efficiently solved on a coarse grid. Then, the coarse-scale sensitivities are interpolated to the local fine grid by reconstructing the local variability of the model parameters with the aid of solving embedded adjoint subproblems. The approach employs the multiscale finite-volume (MSFV) formulation to accurately and efficiently solve the highly detailed flow problem. The MSFV method couples a global coarse-scale solution with local fine-scale reconstruction operators, hence yielding model responses that are quite accurate at both scales. The MSADJ method is equally efficient in computing the gradient of the objective function with respect to model parameters. Several examples demonstrate that the approach is accurate and computationally efficient. The accuracy of our multiscale method for inverse problems is twofold: the sensitivity coefficients computed by this approach are more accurate than the traditional finite-difference-based numerical method for computing derivatives, and the calibrated models after history matching honor the available dynamic data on the fine scale. In other words, the multiscale based adjoint scheme can be used to history match fine-scale models quite effectively.

Local and parallel finite element algorithms for time-dependent convection-diffusion equations

Local and parallel finite element algorithms based on two-grid discretization for the time-dependent convection-diffusion equations are presented. These algorithms are motivated by the observation that, for a solution to the convection-diffusion problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedures. Hence, these local and parallel algorithms only involve one small original problem on the coarse mesh and some correction problems on the local fine grid. One technical tool for the analysis is the local a priori estimates that are also obtained. Some numerical examples are given to support our theoretical analysis.

A finite volume local defect correction method for solving the transport equation

The local defect correction (LDC) method is applied in combination with standard finite volume discretizations to solve the advection–diffusion equation for a passive tracer. The solution is computed on a composite grid, i.e. a union of a global coarse grid and local fine grids. For the test a dipole colliding with a no-slip wall is used to provide an actively changing velocity field. The LDC method is tested for the problem of localized patch of tracer material that is transported by the provided velocity field. The LDC algorithm can be formulated to conserve the total amount of tracer material. However, if the local fine grids are moved to adaptively follow the behaviour of the solution, a loss or gain in the total amount of tracer material is produced. This deficit in tracer material is created when the solution is interpolated to obtain data for the moved fine grid. The data obtained by the interpolation scheme in the new refined region can be adapted in such a way that the surplus or deficit is spread over the new grid points and conservation of tracer material is satisfied. Finally, the results of the conservative finite volume LDC method are compared and validated with results from a spectral method.

Accurate local upscaling with variable compact multipoint transmissibility calculations

We propose a new single-phase local upscaling method that uses spatially varying multipoint transmissibility calculations. The method is demonstrated on two-dimensional Cartesian and adaptive Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve two generic local flow problems. The multipoint stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to ensure that the flux approximation is as close as possible to being two-point. The resulting multipoint flux approximations are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to two-point expressions in cases without full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local–global upscaling method.

Local defect correction in numerical simulation of surface remelting

We study the efficient numerical simulation of laser surface remelting, a process to improve the surface quality of steel components. To this end we use adaptive grids, which are well-suited for problems with moving heat sources. To account for the local high activity due to the heat source, we introduce local uniform grids and couple the solutions on the global coarse and local fine grids using local defect correction (LDC). LDC is based on simple data structures and simple discretization stencils on uniform or tensor-product grids.

A finite volume scheme for solving elliptic boundary value problems on composite grids

We present a finite volume scheme for solving elliptic boundary value problems with solutions that have one or a few small regions with high activity. The scheme results from combining the local defect correction method (LDC), introduced in [11], with standard finite volume discretizations on a global coarse and on local fine uniform grids. The iterative discretization method that is obtained in this way yields a discrete approximation of the continuous solution on a composite grid. For the LDC method in its standard form, the discrete conservation property, which is one of the main attractive features of a finite volume method, is lost for the composite grid approximation. For the modified LDC method we present here, discrete conservation holds for the composite grid solution, too.

A finite difference discretization method for elliptic problems on composite grids

In this paper we discusss a simple finite difference method for the discretization of elliptic boundary value problems on composite grids. For the model problem of the Poisson equation we prove stability of the discrete operator and bounds for the global discretization error. These bounds clearly show how the discretization error depends on the grid size of the coarse grid, on the grid size of the local fine grid and on the order of the interpolation used on the interface. Furthermore, the constants in these bounds do not depend on the quotient of coarse grid size and fine grid size. We also discuss an efficient solution method for the resulting composite grid algebraic problem.

Further analysis of the local defect correction method

We analyze a special case of the Local Defect Correction (LDC) method introduced in [4]. We restrict ourselves to finite difference discretizations of elliptic boundary value problems. The LDC method uses the discretization on a uniform global coarse grid and on one or more uniform local fine grids for approximating the continuous solution. We prove that this LDC method can be seen as an iterative method for solving an underlying composite grid discretization. This result makes it possible to explain important properties of the LDC method, e.g. concerning the size of the discretization error. Furthermore, the formulation of LDC as an iterative solver for a given composite grid problem makes it possible to prove a close correspondence between LDC and the Fast Adaptive Composite grid (FAC) method from [8–10].

Multilevel adaptive methods for laminar diffusion flames

A multilevel adaptive method is applied to the numerical simulation of laminar diffusion flames. A local fine grid is embedded near the jet inlet of the simulation to provide increased resolution and accuracy. Computational results confirm that the multilevel local refinement process substantially increases local and global accuracy with little added cost.

Domain Decomposition with Local Mesh Refinement

A preconditioned Krylov iterative algorithm based on domain decomposition for linear systems arising from implicit finite-difference or finite-element discretizations of partial differential equation problems requiring local mesh refinement is described. To keep data structures as simple as possible for parallel computing applications, the fundamental computational unit in the algorithm is defined as a subregion of the domain spanned by a locally uniform tensor-product grid, called a tile. In the tile-based domain decomposition approach, two levels of discretization are considered at each point of the domain: a global coarse grid defined by tile vertices only, and a local fine grid where the degree of resolution can vary from tile to tile. One global level and one local level provide the flexibility required to adaptively discretize a diverse collection of problems on irregular regions and solve them at convergence rates that deteriorate only logarithmically in the finest mesh parameter, with the coarse tessellation held fixed. A logarithmic departure from optimality seems to be a reasonable compromise for the simplicity of the composite grid data structure and concomitant regular data exchange patterns in a multiprocessor environment. Some experiments with up to 1024 tiles are reported, and the evolution of the algorithm is commented on and contrasted with optimal nonrefining two-level algorithms and optimal refining multilevel algorithms. Computational comparisons with some other popular methods are presented.