Adaptive Grid Refinement Technique Research Articles

Adaptive Finite Element Simulation of Double-Diffusive Convection

Double-diffusive convection plays an important role in many physical phenomena of practical importance. However, the numerical simulation of these phenomena is challenging since fine meshes are often required to capture the flow physics. Hence, several different numerical methods have been employed in the past. This work reports the development and application of an adaptive finite element method for the simulation of these phenomena, thereby avoiding the need for the use of very fine meshes over the whole domain. The weak formulation of the conservation equations for mass, momentum, energy and species concentration is used. The Boussinesq approximation relates the density of the fluid to the temperature and/or the species concentration. A second-order backward difference method is used for time discretization and the Galerkin method is employed for spatial discretization. Both adaptive time step and grid refinement techniques are employed, and the code is parallelized using MPI. Three different stabilization methods of the convective-diffusion equations are compared; namely, the streamline upwind Petrov–Galerkin (SUPG) method, and two modified methods aimed at diminishing spurious oscillations that include an artificial diffusion term. This diffusion term may be either isotropic or orthogonal to the streamlines. The addition of artificial isotropic diffusion to the SUPG method provides enhanced stability. The method is applied to double-diffusive finger convection in a sucrose-salt aqueous mixture and a stratified salt solution heated from below. The method accurately reproduces the experimentally observed temporal evolution of the salt fingers in the former case and the location of the interfaces between convective and non-convective zones in the latter.

A 3D fully‐nonlinear potential‐flow solver for efficient simulations of large‐scale free‐surface waves

AbstractIn the quest for a numerical method for surface waves and wave‐induced effects applicable when linear or weakly nonlinear methods are insufficient, a three‐dimensional numerical wave tank assuming fully‐nonlinear potential‐flow theory is proposed. When viscous‐flow effects, breaking waves or other violent flow‐phenomena are not of primary importance, potential‐flow methods may have similar capability in capturing the involved physics as Navier–Stokes solvers while being potentially more accurate in handling wave‐propagation mechanism and more computationally efficient. If made sufficiently accurate, efficient and numerically robust, fully‐nonlinear potential flow models can therefore represent a powerful tool in the study of ocean waves and their interaction with marine structures, which is the main motivation behind the present work. The governing Laplace equation for the velocity potential is solved using the harmonic polynomial cell method, which is a field method giving high‐order accuracy provided that the cells used to describe the water domain have no stretching or distortion. This can only be achieved in a grid with cubic cells, which leads to poor numerical efficiency unless measures are introduced to refine the grid locally. Here, to improve the efficiency using strictly cubic cells, an adaptive grid refinement technique is introduced. It is shown that this has the ability to improve the computational speed with a factor of up to 20 without sacrificing accuracy. Numerical results are shown to be in good agreement with highly accurate nonlinear reference solutions for regular and irregular waves of various steepness up to the limit of theoretical wave breaking. For long‐crested irregular waves, significant discrepancies with a second‐order theory for the crest‐height distribution are identified, while the second‐order theory appears to provide a better description of the crest height for the single short‐crested irregular sea state simulated. Having demonstrated that the proposed numerical method accurately models nonlinear wave phenomena up to the limit of wave breaking, future work should seek to implement wave‐body interaction capabilities. The adaptive grid refinement technique, which refines the grid dynamically depending on the position of boundaries of interest, is developed with this application in mind. Except from providing a robust way of dealing with wave‐body intersection points, extending the method to account for wave‐body interactions should therefore involve limited difficulty.

High‐resolution numerical simulations of electrophoresis using the Fourier pseudo‐spectral method

We present the formulation, implementation, and performance evaluation of the Fourier pseudo-spectral method for performing fast and accurate simulations of electrophoresis. We demonstrate the applicability of this method for simulating a wide variety of electrophoretic processes such as capillary zone electrophoresis, transient-isotachophoresis, field amplified sample stacking, and oscillating electrolytes. Through these simulations, we show that the Fourier pseudo-spectral method yields accurate and stable solutions on coarser computational grids compared with other nondissipative spatial discretization schemes. Moreover, due to the use of coarser grids, the Fourier pseudo-spectral method requires lower computational time to achieve the same degree of accuracy. We have demonstrated the application of the Fourier pseudo-spectral method for simulating realistic electrophoresis problems with current densities as high as 5000 A/m2 with over tenfold speed-up compared to the commonly used second-order central difference scheme, to achieve a given degree of accuracy. The Fourier pseudo-spectral method is also suitable for simulating electrophoretic processes involving a large number of concentration gradients, which render the adaptive grid-refinement techniques ineffective. We have integrated the numerical scheme in a new electrophoresis simulator named SPYCE, which we offer to the community as open-source code.

Prediction of burning velocity and quenching distance of hydrogen flames

Atmospheric, laminar, stoichiometric, premixed hydrogen-air flames in a diverging channel are investigated by means of Computational Fluid Dynamics. This configuration has been recently used in a series of experimental investigations to determine the burning velocities and quenching distances for premixed flames of different fuels. The purpose of the present investigation is the validation of the prediction procedures for the burning speeds and quenching distances for hydrogen flames by comparing them with these measurements. Global and detailed reaction mechanisms are applied to describe the combustion process. For assuring an adequately fine resolution of the flame fronts, adaptive grid refinement techniques are applied. A reasonable agreement is observed with the experiments, where the detailed and global mechanisms are slightly overpredicting and underpredicting the quenching distance, respectively.

Sharp interface approaches and deep learning techniques for multiphase flows

We present a review on numerical methods for simulating multiphase and free surface flows. We focus in particular on numerical methods that seek to preserve the discontinuous nature of the solutions across the interface between phases. We provide a discussion on the Ghost-Fluid and Voronoi Interface methods, on the treatment of surface tension forces that avoid stringent time step restrictions, on adaptive grid refinement techniques for improved efficiency and on parallel computing approaches. We present the results of some simulations obtained with these treatments in two and three spatial dimensions. We also provide a discussion of Machine Learning and Deep Learning techniques in the context of multiphase flows and propose several future potential research thrusts for using deep learning to enhance the study and simulation of multiphase flows.

Analysis of adaptive grid refinement technique for simulations of ES-SAGD in heavy oil reservoirs

The intrinsic advantages of hybrid steam-solvent or solvent-based thermal recovery methods for heavy oil include faster oil production, higher ultimate recovery factor, reduction of energy, and water treatment expenses, better control of GHG emission. Light hydrocarbon injection implies improved heavy oil mobilization, steam and solvent condensation, solvent to oil mixing, and oil viscosity effect at moderate temperature. Typically, field-scale reservoir simulation models are built on numerical grids with cell sizes making practically difficult accurate description of heat and mass transport at the edge of the gas chamber. Outside this zone, this typical grid can be acceptable. The adaptive grid refinement may offer a compromise solution if the numerical model accepts the necessary degree of refinement. In our current work, we present a methodology of large-scale numerical simulations for so-called expanding solvent SAGD (ES-SAGD) processes using adaptive dynamic gridding at different degrees of the refinement. The adaptive grid refinement is a Cartesian grid amalgamation process based on threshold values of predefined trigger variables. Comparison is performed between oil production mechanism interactions for typical Athabasca bitumen and illustrated with temperature and solvent concentration fields for different discretization sizes. Good adaptive grid refinement results have been obtained for the models based on adaptive implicit approach with high refinement degree, in heterogeneous reservoir (at relatively small correlation length) in vertical 2D cross-section. The numerical performance aspects including the CPU time analysis with and without the use of dynamic gridding are presented in some detail.

Adaptive Finite Difference Methods for Nonlinear Elliptic and Parabolic Partial Differential Equations with Free Boundaries

Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic partial differential equations (PDEs). These methods are best suited to regular rectangular grids, which leads to low accuracy near curved boundaries or singularities of solutions. In this article we combine monotone finite difference methods with an adaptive grid refinement technique to produce a PDE discretization and solver which is applied to a broad class of equations, in curved or unbounded domains which include free boundaries. The grid refinement is flexible and adaptive. The discretization is combined with a fast solution method, which incorporates asynchronous time stepping adapted to the spatial scale. The framework is validated on linear problems in curved and unbounded domains. Key applications include the obstacle problem and the one-phase Stefan free boundary problem.

Coupling a local adaptive grid refinement technique with an interface sharpening scheme for the simulation of two-phase flow and free-surface flows using VOF methodology

This study presents the implementation of an interface sharpening scheme on the basis of the Volume of Fluid (VOF) method, as well as its application in a number of theoretical and real cases usually modelled in literature. More specifically, the solution of an additional sharpening equation along with the standard VOF model equations is proposed, offering the advantage of “restraining” interface numerical diffusion, while also keeping a quite smooth induced velocity field around the interface. This sharpening equation is solved right after volume fraction advection; however a novel method for its coupling with the momentum equation has been applied in order to save computational time. The advantages of the proposed sharpening scheme lie on the facts that a) it is mass conservative thus its application does not have a negative impact on one of the most important benefits of VOF method and b) it can be used in coarser grids as now the suppression of the numerical diffusion is grid independent. The coupling of the solved equation with an adaptive local grid refinement technique is used for further decrease of computational time, while keeping high levels of accuracy at the area of maximum interest (interface). The numerical algorithm is initially tested against two theoretical benchmark cases for interface tracking methodologies followed by its validation for the case of a free-falling water droplet accelerated by gravity, as well as the normal liquid droplet impingement onto a flat substrate. Results indicate that the coupling of the interface sharpening equation with the HRIC discretization scheme used for volume fraction flux term, not only decreases the interface numerical diffusion, but also allows the induced velocity field to be less perturbed owed to spurious velocities across the liquid–gas interface. With the use of the proposed algorithmic flow path, coarser grids can replace finer ones at the slight expense of accuracy.

Cooling effectiveness of droplets at low Weber numbers: Effect of temperature

The impact of water droplets onto a heated non-isothermal surface is studied numerically using the VOF methodology and assuming a 2D-axisymmetric computational domain. An adaptive grid refinement technique in both the fluid and the solid phases is used to solve the conjugate problem of fluid flow and heat transfer. Parametric studies examine the effect of initial solid surface temperature and initial droplet temperature, while all the other parameters which affect the temporal evolution of the phenomenon are kept constant; the wall temperatures examined are low enough to prevent the onset of nucleate boiling. It is proved that the variables describing the temporal evolution of the phenomenon evolve with the same way when they are non-dimensionalised with expressions arising from the transient heat conduction theory. Additionally, a formula for the maximum droplet spreading which accounts for the droplet heating during spreading is proposed, as also semi-analytical expressions which describe the thermal behaviour of the droplet

Parallel finite element density functional computations exploiting grid refinement and subspace recycling

In this communication computational methods that facilitate finite element analysis of density functional computations are developed. They are: (i) h–adaptive grid refinement techniques that reduce the total number of degrees of freedom in the real space grid while improving on the approximate resolution of the wanted solution; and (ii) subspace recycling of the approximate solution in self-consistent cycles with the aim of improving the performance of the generalized eigenproblem solver. These techniques are shown to give a convincing speed-up in the computation process by alleviating the overhead normally associated with computing systems with many degrees-of-freedom.

Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology

In this work adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner. The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction–diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice.

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Finite element analysis of filling stage in die‐casting process using marker surface method and adaptive grid refinement technique

AbstractThe marker surface method and the adaptive grid refinement technique have been applied to the three‐dimensional (3‐D) finite element analysis of the filling stage in the die‐casting process. Especially, the marker surface plugging technique and the marker surface regeneration technique incorporated in the marker surface method have been proposed for the efficient analysis of 3‐D practical problems. Through the marker surface plugging technique, new parts of marker surface are effective lycreated in order to eliminate the gaps between the parts of marker surface or between the edge of marker surface and cavity wall. By using the marker surface regeneration technique, the marker surface including a great number of marker elements is recreated on the basis of its original shape in order to decrease the number of marker elements and computational time. A3‐D example used as the benchmark test and a typical industrial problem of the die‐casting process have been analysed. The numerical results have been in good agreement with the experimental results and the efficiency of the adaptive grid refinement technique has been verified. It has been shown that the proposed techniques incorporated in the marker surface method and the adaptive grid refinement technique can be effectively applied to general industrial problems. Copyright © 2004 John Wiley & Sons, Ltd.

Application of an adaptive grid refinement technique to three-dimensional finite element analysis of the filling stage in the die-casting process

In order to obtain a higher resolution of the free surface configuration, an adaptive grid refinement technique is incorporated into the volume of fluid (VOF) and the marker surface methods for three-dimensional finite element analysis of the filling stage in the die-casting process. The adaptive grid is created at each time step through the grid refinement and coarsening procedures, as opposed to employing the fixed grid in the classical VOF and marker particle methods. In the adaptive grid, the elements finer than those of the internal and external regions are distributed over the surface region. The adaptive grid is used for generating the fluid region grid that better describes the shape of the predicted free surface at each time step, and for advancing the predicted free surface more precisely. To verify the efficiency of the adaptive grid refinement technique rigorously, a benchmark problem of the conference on the “Modeling of Casting, Welding and Advanced Solidification Processes VII” (MCWASP VII) has been analyzed and the numerical results are found to be in good agreement with the experimental results. A typical industrial problem of the die-casting process such as a cover of transmission case has been analyzed and it has been shown that the adaptive grid incorporated into the VOF and marker surface methods can be effectively applied to general die-casting process analysis.

Incomplete Multilevel Cholesky Factorizations

Adaptive in-time local grid refinement techniques use multilevel local discretizations designed to achieve local accuracy. The changing nature of the matrix structure of the linear systems arising from the multilevel local discretizations requires flexible approximate factorizations that focus on local components and coordinate their interaction. The solution of these composite grid systems with Krylov solvers is considered. The selection of an adequate preconditioner is crucial. The incomplete Cholesky (IC) factorization of the composite matrix and the inexact BEPS preconditioner are two such potential preconditioners. The inexact BEPS preconditioner can be constructed and applied with significantly more flexibility than the IC factorizations of the composite multilevel grid matrix. An extension of the IC factorizations for matrices arising from discretizations of self-adjoint PDEs on multilevel composite grids is proposed. The resulting factorization is referred to as the incomplete multilevel Cholesky (IMC) factorization. The IMC factorization is spectrally equivalent to the IC factorization of the matrix of the composite grid system. IMC factorization can be constructed with the same flexibility as the inexact BEPS preconditioner. The application of IMC factorization is achieved via a multilevel LLT-cycle consisting of a forward elimination pass proceeding downward on the grids from fine to coarse followed by an reverse-order upward back substitution pass. The application of multilevel factorization as a preconditioner requires roughly one-half as many operations as the inexact BEPS preconditioner. The numerical results illustrate the potential of the method.

A Solution-Adaptive Unstructured Grid Method by Grid Subdivision and Local Remeshing

An adaptive unstructured grid refinement technique has been developed and successfully applied to several three-dimensional inviscid flow test cases. The method is based on a combination of surface mesh subdivision and local remeshing of the volume grid. Simple functions of flow quantities are employed to detect dominant features of the flowfield. The method is designed for modular coupling with various error/feature analyzers and flow solvers. Several steady-state, inviscid flow test cases are presented to demonstrate the applicability of the method for solving practical three-dimensional problems. In all cases accurate solutions featuring complex, nonlinear flow phenomena such as shock waves and vortices have been generated automatically and efficiently

Preconditioned conjugate gradient methods for large-scale fluid flow applications

The simulation of large-scale fluid flow applications often requires the efficient solution of extremely large nonsymmetric linear and nonlinear sparse systems of equations arising from the discretization of systems of partial differential equations. While preconditioned conjugate gradient methods work well for symmetric, positive-definite matrices, other methods are necessary to treat large, nonsymmetric matrices. The applications may also involve highly localized phenomena which can be addressed via local and adaptive grid refinement techniques. These local refinement methods usually cause non-standard grid connections which destroy the bandedness of the matrices and the associated ease of solution and vectorization of the algorithms. The use of preconditioned conjugate gradient or conjugate-gradient-like iterative methods in large-scale reservoir simulation applications is briefly surveyed. Then, some block preconditioning methods for adaptive grid refinement via domain decomposition techniques are presented and compared. These techniques are being used efficiently in existing large-scale simulation codes.