Adaptive Grid Method Research Articles

A Hierarchical Multiresolution Adaptive Mesh Refinement for the Solution of Evolution PDEs

In this paper, we propose a novel multiresolution adaptive mesh refinement algorithm for solving initial-boundary value problems (IBVP) for evolution PDEs. The proposed algorithm dynamically adapts the grid to any existing or emerging irregularities in the solution, by refining the grid only at those places where the solution exhibits sharp features. The main advantage of the proposed grid adaptation method is that it results in a grid with a fewer number of nodes when compared to adaptive grids generated by existing multiresolution-based mesh refinement techniques. Several examples show the robustness and stability of the proposed algorithm.

A grid redistribution method for singular problems

Many physical phenomena develop singular, or nearly singular behavior in localized regions, e.g. boundary layers or blowup solutions. Using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity. Ren and Wang developed a semi-static adaptive grid method [W. Ren, X.P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions, J. Comput. Phys. 159 (2000) 246–273] for the solution of these problems, known as the iterative grid redistribution (IGR) method. In this study we develop a theoretical basis for semi-static adaptive grid method for singular problems. Based on this theory, we obtain the key result of this study – a methodology for designing robust weight functionals which ensures grid resolution in the singular region, as well as control of the maximal grid spacing in the outer region. Using this methodology, we introduce a semi-static adaptive grid method, which does not involve an iterative procedure for grid redistribution, as in the IGR method. We demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude.

Implicit adaptive mesh refinement for 2D reduced resistive magnetohydrodynamics

An implicit structured adaptive mesh refinement (SAMR) solver for 2D reduced magnetohydrodynamics (MHD) is described. The time-implicit discretization is able to step over fast normal modes, while the spatial adaptivity resolves thin, dynamically evolving features. A Jacobian-free Newton–Krylov method is used for the nonlinear solver engine. For preconditioning, we have extended the optimal “physics-based” approach developed in [L. Chacón, D.A. Knoll, J.M. Finn, An implicit, nonlinear reduced resistive MHD solver, J. Comput. Phys. 178 (2002) 15–36] (which employed multigrid solver technology in the preconditioner for scalability) to SAMR grids using the well-known Fast Adaptive Composite grid (FAC) method [S. McCormick, Multilevel Adaptive Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1989]. A grid convergence study demonstrates that the solver performance is independent of the number of grid levels and only depends on the finest resolution considered, and that it scales well with grid refinement. The study of error generation and propagation in our SAMR implementation demonstrates that high-order (cubic) interpolation during regridding, combined with a robustly damping second-order temporal scheme such as BDF2, is required to minimize impact of grid errors at coarse-fine interfaces on the overall error of the computation for this MHD application. We also demonstrate that our implementation features the desired property that the overall numerical error is dependent only on the finest resolution level considered, and not on the base-grid resolution or on the number of refinement levels present during the simulation. We demonstrate the effectiveness of the tool on several challenging problems.

Modeling the Dust Properties ofz∼ 6 Quasars with ART2—All‐Wavelength Radiative Transfer with Adaptive Refinement Tree

The detection of large quantities of dust in z ∼ 6 quasars by infrared and radio surveys presents puzzles for the formation and evolution of dust in these early systems. Previously, Li et al. showed that luminous quasars at z≳ 6 can form through hierarchical mergers of gas-rich galaxies, and that these systems are expected to evolve from starburst through quasar phases. Here, we calculate the dust properties of simulated quasars and their progenitors using a three-dimensional Monte Carlo radiative transfer code, ART2 (All-wavelength Radiative Transfer with Adaptive Refinement Tree). ART2 incorporates a radiative equilibrium algorithm which treats dust emission self-consistently, an adaptive grid method which can efficiently cover a large dynamic range in both spatial and density scales, a multiphase model of the interstellar medium which accounts for the observed scaling relations of molecular clouds, and a supernova-origin model for dust which can explain the existence of dust in cosmologically young objects. By applying ART2 to the hydrodynamic simulations of Li et al., we reproduce the observed spectral energy distribution (SED) and inferred dust properties of SDSS J1148+5251, the most distant Sloan quasar. We find that the dust and infrared emission are closely associated with the formation and evolution of the quasar host. The system evolves from a cold to a warm ultraluminous infrared galaxy (ULIRG) owing to heating and feedback from stars and the active galactic nucleus (AGN). Furthermore, the AGN activity has significant implications for the interpretation of observation of the hosts. Our results suggest that vigorous star formation in merging progenitors is necessary to reproduce the observed dust properties of z ∼ 6 quasars, supporting a merger-driven origin for luminous quasars at high redshifts and the starburst-to-quasar evolutionary hypothesis.

Overset Adaptive Cartesian/Prism Grid Method for Stationary and Moving-Boundary Flow Problems

T HE use of unstructured grids in computational fluid dynamics (CFD) has become widespread during the last two decades, due to their ability to discretize arbitrarily complex geometries and the flexibility in supporting solution-based grid adaptations to enhance the solution accuracy and efficiency [1–7]. In the early days of unstructured grid development, triangular/tetrahedral grids were employed the most in dealing with complex geometries. Recently, mixed or hybrid grids including many different cell types have gained popularity because of the improved efficiency and accuracy over pure tetrahedral grids. For example, hybrid prism/tetrahedral grids [8], mixed grids including tetrahedral/prism/pyramid/ hexahedral cells [9], and adaptive Cartesian grid methods [10–15] have been used in many applications with complex configurations. The success demonstrated by unstructured grids for steady flow problems has prompted their applications to unsteady, movingboundary, flow problems. A powerful approach for moving-boundary flow problems is the overset Chimera grid method [16]. Originally, the Chimera grid method was used to simplify domain decomposition for complex geometries using structured grids. The method is particularly useful for moving-boundary flow simulations because grid remeshing can be avoided [17]. However, frequent hole-cutting and donor-cell searching may be necessary to facilitate communications between the moving Chimera grids. With continuous improvement over the last decade and a half, the Chimera grid method has achieved tremendous success in handling very complex moving-boundary flow problems.More recently, to further simplify the grid-generation process, unstructured grids are also used in aChimera grid system for moving-boundary flow computations, making the approach even more flexible in handling complex geometries [18]. In this Technical Note, we advocate the use of an overset adaptive Cartesian/prism grid method for moving-boundary flow computations. Themethod combines the advantage of the adaptive Cartesian/ prism grid in geometry flexibility with that of the Chimera approach in tacklingmoving-boundary flowwithout grid remeshing. There are several reasons why an adaptive Cartesian grid is used for movingboundary problems: 1) Cartesian cells are more efficient at filling space given a certain length scale than triangular/tetrahedral cells. 2) Searching operations can be performed very efficiently with the octree data structure. 3) Solution-based and geometry-based grid adaptations are straightforward to carry out. The grid-generation process is as follows. Body-fitted prism grids are generated first near solid bodies to resolve viscous boundary layers using the advancing layer approach by marching the prism-layer grid in the approximate surface normal directions. The prism-grid-generation approaches (and their limitations) have been well researched in the literature in the past two decades, for example, in [6,8,18,19], and the current implementation follows similar ideas. Therefore, interested readers should consult these references for prism-grid-generation methods and their limitations. An adaptive Cartesian grid is then generated to cover the outer domain and serve as the background grid for bridging the “gaps” between the prism grids. The outer boundaries of the prism grids are used to generate holes in the adaptive Cartesian grid to facilitate data communication. If the bodies move, the prism grids move with the bodies, whereas the Cartesian grid remains stationary. After a few (tens of) time steps, new holes are cut out of the Cartesian grids, and new donor cells are also identified. Solution fields are interpolated from the old Cartesian grid to the new grid using a cellwise linear reconstruction technique. This Technical Note is organized as follows. In the next section, the overset adaptive Cartesian/prism grid-generation and holecutting approaches will be presented, together with illustrative examples. In Sec. III, several steady and unsteady moving-boundary flow problems are computed. Grid-refinement studies are performed to ensure the computational solutions are grid-independent. Computational results are compared with experimental data and other simulations whenever possible. Finally, conclusions from this study are summarized in Sec. IV.

Experiments with a local adaptive grid h-refinement for the finite-difference solution of BVPs in singularly perturbed second-order ODEs

Despite many years of the development of adaptive grid methods, there is still a need for re-evaluation, tuning, and comparisons of the various approaches to grid adaptation. In this study we seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement/de-refinement algorithm for the finite-difference solution of singularly perturbed BVPs in second-order ODEs, that is also easily extendable to evolutionary PDEs. Conventional three-point discretisations of the first and second derivatives are used. Errors of the discrete solution and its first derivative are simultaneously controlled. The grid refinement is accomplished by adding nodes at inter-nodal positions, and de-refinement by removing previously added nodes in reverse order. Computational experiments are performed for 15 examples of ODEs, assuming several alternative choices of the a posteriori error estimators, grid refinement indicators, and regridding strategies. The most satisfactory results are obtained by combining: deferred approach to error estimation, based on higher-order reference derivative approximations for truncation error evaluation; interpolation-based indicator for grid refinement; and regridding strategy that uses the mean indicator value as the indicator threshold for refinement.

An adaptive line-by-line—statistical model for fast and accurate spectral simulations in low-pressure plasmas

An adaptive grid method is proposed for the simulation of low-pressure plasma radiation. The method relies on two complementary approaches which significantly reduce calculation times and the size of the obtained grids: Weak lines are calculated as a so-called “pseudo-continuum”, hence reducing the number of calculated lines, and a numerical algorithm has been developed for accurately calculating Voigt lineshapes using a minimum number of points. The method is fully user-parametric, allowing the choice of privileging calculation efficiency, or alternatively privileging the accuracy of the computed spectra. Sample radiative transfer calculations are presented, which show the efficiency of the method, also providing some guidelines on how to define lineshape calculation parameters, depending on the problem to be solved.

Three-dimensional adaptive Cartesian grid method with conservative interface restructuring and reconstruction

Multiphase flows associated with interfacial dynamics, steep jumps in fluid properties and moving boundaries between different phases pose substantial computational challenges in terms of both modeling as well as computational efficiency. The present work extends a marker-based immersed boundary, or front tracking, technique to model the three-dimensional interfacial dynamics. It tracks the moving boundary using triangulated surface grids and solves the flow governing equations on a stationary Cartesian grid. A locally adaptive grid is employed to help meet the resolution requirements based on the interface location and solution features. The interface resolution is controlled via a conservative restructuring technique satisfying mass continuity. An improved level contour reconstruction algorithm for topology change, preserving the interface connectivity information, is presented highlighting various algorithmic difficulties and implemented remedies. The outlines of a finite-volume, staggered grid Navier–Stokes solution using the projection method are discussed. The impact of conservative interface restructuring and reconstruction has been assessed against mass-conservation and spurious velocity errors. The overall capabilities of the developed algorithms have been demonstrated for large density ratios, O(1000), interfacial flows using various rising bubbles and drop collision/coalescence computations involving coalescence and break-up dynamics.

Numerical Simulations for Transport Aircraft High-Lift Configurations Using Adaptive Cartesian Grid Methods

Computations of complex flow fields around three-dimensional high-lift configurations using an adaptive Cartesian grid method and a cell-center finite-volume method are carried out and compared to wind-tunnel experiments. Because of complexities in both configuration and flowfield, there are great difficulties in numerical simulation of flowfields around transport aircraft high-lift configurations with leading slats and trailing flaps. In this paper, a modified Cartesian grid method is introduced to simulate the complex flow using the adaptive grid technique. The main emphasis is given to a multizone technique and a face-to-face algorithm and its applications. The multizone technique is employed to simplify the great difficulties in grid generation. The face-to-face algorithm is constructed to precisely establish the information exchange of flowfields on interfaces between zones. We solve Euler equations with a conventional algorithm, which includes a cell-center finite-volume method and an explicit four-stage Runge-Kutta time-stepping scheme in combination with a dual-time stepping approach. For three three-dimensional high-lift configurations, the numerical results are presented and analyzed. The computational results are in good agreement with the available experimental data show that the present method is feasible and effective for solving the flow around complex three-dimensional high-lift systems.

An adaptive grid refinement strategy for the simulation of negative streamers

The evolution of negative streamers during electric breakdown of a non-attaching gas can be described by a two-fluid model for electrons and positive ions. It consists of continuity equations for the charged particles including drift, diffusion and reaction in the local electric field, coupled to the Poisson equation for the electric potential. The model generates field enhancement and steep propagating ionization fronts at the tip of growing ionized filaments. An adaptive grid refinement method for the simulation of these structures is presented. It uses finite volume spatial discretizations and explicit time stepping, which allows the decoupling of the grids for the continuity equations from those for the Poisson equation. Standard refinement methods in which the refinement criterion is based on local error monitors fail due to the pulled character of the streamer front that propagates into a linearly unstable state. We present a refinement method which deals with all these features. Tests on one-dimensional streamer fronts as well as on three-dimensional streamers with cylindrical symmetry (hence effectively 2D for numerical purposes) are carried out successfully. Results on fine grids are presented, they show that such an adaptive grid method is needed to capture the streamer characteristics well. This refinement strategy enables us to adequately compute negative streamers in pure gases in the parameter regime where a physical instability appears: branching streamers.

A grid and flow adaptive wall-function method for RANS turbulence modelling

This paper presents a grid and flow adaptive wall-function method for RANS turbulence modelling with emphasis on aerodynamic flows. A near-wall grid adaptation technique ensures a locally appropriate resolution depending on both the near-wall flow physics to be captured and the range of validity of the wall-function model. The near-wall RANS solutions of the Spalart–Allmaras and SST k– ω turbulence model are investigated near stagnation points and subsequent not yet fully developed turbulent flow, and in regions of adverse pressure gradient before separation. These are compared with the corresponding turbulence model specific universal wall-functions and suggestions for the design of wall-function methods for non-equilibrium flows are given. Regions of non-equilibrium flow are detected by a flow based sensor and near-wall grid adaptation is then made possible due to the hybrid character of the wall-functions.

A numerical study on propagation of premixed flames in small tubes

A premixed flame in a tube suffers strong variation in its shape and structure depending on boundary conditions. The effects of thermal boundary conditions and flow fields on flame propagation are numerically investigated. This study employs eight combinations of thermal and velocity boundary conditions. Navier–Stokes equations and species equations are solved with a one-step irreversible global reaction model of methane–air mixtures. Finite volume method using an adaptive grid method is applied to investigate the flame structure. In the case of an adiabatic wall, friction force on the wall significantly affected the flame structure while in the case of an isothermal wall, local quenching near the wall dominated flame shape and propagation. In both cases, variations of flow fields occurred not only in the near field of the flame but also within the flame itself, which affected propagation velocities. Near the quenching conditions, strong similarity in the flame structure was found regardless of the boundary velocity profiles due to self-induced velocity deformation. This study provides an overview of the characteristics of flames in small tubes at a steady state.

An adaptive grid method for two-dimensional viscous flows

This paper extends the gas-kinetic BGK-NS scheme to an adaptive grid for the viscous flow simulations. The grid movement and adaptation is controlled by a monitor function which may depend on velocity gradient or other flow variables, such as density or pressure. For the viscous flow computation, the use of adaptive mesh much improves the efficiency and accuracy of the method in comparison with the methods with static mesh points. The current method is an accurate and efficient method for the viscous flow computation, where the grid points can be easily moved and concentrated on the regions with large velocity and density gradients, such as the boundary layer and multi-material interface. Many numerical examples validate the current approach for the viscous flow simulations.

Resolving small-scale structures in Boussinesq convection by adaptive grid methods

Inviscid Boussinesq convection is a challenging problem both analytically and numerically. Due to the complex dynamic development of small scales and the rapid loss of solution regularity, the Boussinesq convection pushes any numerical strategy to the limit. In E and Shu (Phys. Fluids 6 (1994) 49), a detailed numerical study of the Boussinesq convection in the absence of viscous effects is carried out using filtered pseudospectral method and a high-order accurate ENO schemes. In their computations, very fine grids have to be used in order to resolve the small-structures of the Boussinesq fluid. In this work, we will develop an efficient adaptive grid method for solving the inviscid incompressible flows, which can be useful in resolving extremely small-structures with reasonably small number of grid points. To demonstrate the effectiveness of the proposed method, the Boussinesq convection problem will be computed using the adaptive grid method.

Flow feature aligned grid adaptation

AbstractA flow feature aligned grid adaptation method is proposed for the solution of Euler and Navier–Stokes equations for compressible flows, motivated by the desire for an efficient grid system for an accurate and robust solution method to best resolve flow features of interest. The method includes extraction of the flow features; generation of the embedded flow feature aligned structured blocks combined with unstructured grid generation for the rest of the flowfield; and adaptation of the hybrid grid for high flow feature resolution. The feature alignment makes it possible to maintain the high resolution property for both shock waves and shear layers of the approximate Riemann solvers and the higher order reconstruction schemes based on one‐dimensional derivation and dimensional splitting. High grid efficiency is obtained with highly anisotropic directional grid corresponding to the feature directions. The computational procedure is described in details in the paper and its application to flow solutions involving shock waves, boundary layers, wakes and shock boundary layer interaction are demonstrated. Its accuracy, efficiency and robustness are discussed in comparison with an anisotropic unstructured grid adaptations for the shock boundary layer interaction case. Copyright © 2006 John Wiley & Sons, Ltd.

Adaptive refinement techniques based on tetrahedral and hexahedral grids for finite element analysis of mold filling in casting processes

A 3-D finite element model combined with a volume tracking method is presented in this work to simulate the mold filling for casting processes. Especially, the analysis involves an adaptive grid method that is created under a criterion of element categorization of filling states and locations in the total region at each time step. By using an adaptive grid wherein the elements, finer than those in internal and external regions, are distributed at the surface region through refinement and coarsening procedures, a more efficient analysis of transient fluid flow with free surface is achieved. Adaptive grid based on VOF method is developed in tetrahedral and hexahedral elements. Through a 3-D analysis of the benchmark test of the casting process using two types of elements, the efficiency of the proposed adaptive grid method is verified.

Uniform convergence analysis of finite difference approximations for singular perturbation problems on an adapted grid

A singularly perturbed two-point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the approximated solutions. A first-order rate of convergence, independent of the perturbation parameter, is established by using the theory of the discrete Green's function. Unlike some previous analysis for the fully discretized approach, the present problem does not require the conservative form of the underlying boundary value problem.

Solution of Equilibrium Radiation Diffusion Problems Using Implicit Adaptive Mesh Refinement

Diffusion approximations to radiation transport feature a nonlinear conduction coefficient that leads to formation of a sharp front, or Marshak wave, under suitable initial and boundary conditions. The front can vary several orders of magnitude over a very short distance. Resolving the shape of the Marshak wave is essential, but using a global fine mesh can be prohibitively expensive. In such circumstances it is natural to consider using adaptive mesh refinement (AMR) to place a fine mesh only in the vicinity of the propagating front. In addition, to avoid any loss of accuracy due to linearization, implicit time integration should be used to solve the equilibrium radiation diffusion equation. Implicit time integration on AMR grids introduces a new challenge, as algorithmic complexity must be controlled to fully realize the performance benefits of AMR\@. A Newton--Krylov method together with a multigrid preconditioner addresses this latter issue on a uniform grid. A straightforward generalization is to use a multilevel preconditioner that is tuned to the structure of the AMR grid, such as the fast adaptive composite grid (FAC) method. We describe the resulting Newton--Krylov-FAC method and demonstrate its performance on simple equilibrium radiation diffusion problems.

A time-dependent Green's function approach to study the transient phenomena in metamaterial lens focusing

We applied a time-dependent Green's function approach to study the transient behaviors in metamaterial lens focusing. An adaptive-grid method is developed to deal with the singularities encountered in the numerical integrations and a benchmark result is presented to demonstrate the validity of our theory. We then implement the method to study several typical focusing processes with metamaterial-based lenses.

중첩격자계와 접합격자계를 이용한 적응격자 기법

중첩격자계와 접합격자계를 이용한 적응격자 기법이 개발되었다. 유동장은 물체와 근접한 영역과 떨어진 영역으로 구분된다. 근접한 영역은 곡선 격자계로 채워지며 중첩격자기법으로 영역이 연결되고 떨어진 영역은 다양한 적응 단계를 가진 직교 격자계로 채워지며 접합격자기법으로 연결된다. 본 적응격자기법은 격자생성에 있어서의 유연성과 효과적인 격자적응 기능을 보여준다. 2차원 스토어 분리 해석을 포함하는 몇 가지 수치해석을 통해 본 적응격자기법의 성능을 검증하였다. A grid adaptation method within systems of chimera and patched grids is presented. Problem domains are divided into near-body and off-body fields. Near-body field is filled with curvilinear body-fitted grids that extend only a short distance from body surfaces and connected to other grid systems via chimera domain connectivity method. Off-body field is filled with patched uniform cartesian grids of varying levels of refinement. This method gives flexibility in grid generation and efficient adaptation capability. Several numerical experiments including 2D store separation were performed to show the performance of the proposed adaptation method.