Local Refinement Method Research Articles

Computational and experimental study of oxygen-enhanced axisymmetric laminar methane flames

Three axisymmetric laminar coflow diffusion flames, one of which is a nitrogen-diluted methane/air flame (the ‘base case’) and the other two of which consist of nitrogen-diluted methane vs. pure oxygen, are examined both computationally and experimentally. Computationally, the local rectangular refinement method is used to solve the fully coupled nonlinear conservation equations on solution-adaptive grids. The model includes C2 chemistry (GRI 2.11 and GRI 3.0 chemical mechanisms), detailed transport, and optically thin radiation. Because two of the flames are attached to the burner, thermal boundary conditions at the burner surface are constructed from smoothed functional fits to temperature measurements. Experimentally, Raman scattering is used to measure temperature and major species concentrations as functions of the radial coordinate at various axial positions. As compared to the base case flame, which is lifted, the two oxygen-enhanced flames are shorter, hotter, and attached to the burner. Computational and experimental flame lengths show excellent agreement, as do the maximum centreline temperatures. For each flame, radial profiles of temperature and major species also show excellent agreement between computations and experiments, when plotted at fixed values of a dimensionless axial coordinate. Computational results indicate peak NO levels in the oxygen-enhanced flames to be very high. The majority of the NO in these flames is shown to be produced via the thermal route, whereas prompt NO dominates for the base case flame.

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

A new adaptive local mesh refinement method is presented for thin film flow problems containing moving contact lines. Based on adaptation on an optimal interpolation error estimate in the Lp norm (1<p⩽∞) [L. Chen, P. Sun, J. Xu, Multilevel homotopic adaptive finite element methods for convection dominated problems, in: Domain Decomposition Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 40 (2004) 459–468], we obtain the optimal anisotropic adaptive meshes in terms of the Hessian matrix of the numerical solution. Such an anisotropic mesh is optimal for anisotropic solutions like the solution of thin film equations on moving contact lines. Thin film flow is described by an important type of nonlinear degenerate fourth order parabolic PDE. In this paper, we address the algorithms and implementation of the new adaptive finite element method for solving such fourth order thin film equations. By means of the resulting algorithm, we are able to capture and resolve the moving contact lines very precisely and efficiently without using any regularization method, even for the extreme degenerate cases, but with fewer grid points and degrees of freedom in contrast to methods on a fixed mesh. As well, we compare the method theoretically and computationally to the positivity-preserving finite difference scheme on a fixed uniform mesh which has proven useful for solving the thin film problem.

Structured mesh generation by kriging with local refinement with a new elliptic scheme

Accurate results in finite element analysis are strongly related to mesh quality. In this paper, an automatic quadrilateral mesh generation methodology using kriging interpolation is described, a quality mesh study is conducted, and the development of a new local refinement scheme, called the elliptic scheme, is presented. The new elliptic refinement scheme is evaluated using four standard structural cases, and it is shown that it compares very well with octree-based refinement schemes and other local refinement methods.

Improving the capabilities of a continuum laser plasma interaction code

The numerical simulation of plasmas is a critical tool for inertial confinement fusion (ICF). We have been working to improve the predictive capability of a continuum laser plasma interaction code pF3d, which couples a continuum hydrodynamic model of an unmagnetized plasma to paraxial wave equations modeling the laser light. Advanced numerical techniques such as local mesh refinement, multigrid, and multifluid Godunov methods have been adapted and applied to nonlinear heat conduction and to multifluid plasma models. We describe these algorithms and briefly demonstrate their capabilities.

Second-order accurate FDTD space and time grid refinement method in three space dimensions

We present an algorithm based on the finite-difference time-domain method for local refinement of a three-dimensional computational grid in space and time. The method has second-order accuracy in space and time as verified in the numerical examples. A number of test cases with material traverse normal to the grid interfaces were used to assess the long integration time stability of the algorithm. Resulting improvements in the computation time are discussed for a photonic crystal microcavity design that exhibits a sensitive dependence of the quality factor on subwavelength geometrical features.

High-Performance Medical Image Registration Using New Optimization Techniques

Optimization of a similarity metric is an essential component in intensity-based medical image registration. The increasing availability of parallel computers makes parallelizing some registration tasks an attractive option to increase speed. In this paper, two new deterministic, derivative-free, and intrinsically parallel optimization methods are adapted for image registration. DIviding RECTangles (DIRECT) is a global technique for linearly bounded problems, and multidirectional search (MDS) is a recent local method. The performance of DIRECT, MDS, and hybrid methods using a parallel implementation of Powell's method for local refinement, are compared. Experimental results demonstrate that DIRECT and MDS are robust, accurate, and substantially reduce computation time in parallel implementations.

An anelastic allspeed projection method for gravitationally stratified flows

This paper looks at gravitationally stratified atmospheric flows at low Mach and Froude numbers and proposes a new algorithm to solve the compressible Euler equations, in which the asymptotic limits are recovered numerically and the boundary conditions for block-structured local refinement methods are well-posed. The model is non-hydrostatic and the numerical algorithm uses a splitting to separate the fast acoustic dynamics from the slower anelastic dynamics. The acoustic waves are treated implicitly while the anelastic dynamics is treated semi-implicitly and an embedded-boundary method is used to represent orography. We present an example that verifies our asymptotic analysis and a set of results that compares very well with the classical gravity wave results presented by Durran.

Development and validation of a consistency based multiple structure alignment algorithm

We introduce an algorithm that uses the information gained from simultaneous consideration of an entire group of related proteins to create multiple structure alignments (MSTAs). Consistency-based alignment (CBA) first harnesses the information contained within regions that are consistently aligned among a set of pairwise superpositions in order to realign pairs of proteins through both global and local refinement methods. It then constructs a multiple alignment that is maximally consistent with the improved pairwise alignments. We validate CBA's alignments by assessing their accuracy in regions where at least two of the aligned structures contain the same conserved sequence motif. CBA correctly aligns well over 90% of motif residues in superpositions of proteins belonging to the same family or superfamily, and it outperforms a number of previously reported MSTA algorithms.

A higher-order implicit IDO scheme and its CFD application to local mesh refinement method

The Interpolated Differential Operator (IDO) scheme has been developed for the numerical solution of the fluid motion equations, and allows to produce highly accurate results by introducing the spatial derivative of the physical value as an additional dependent variable. For incompressible flows, semi-implicit time integration is strongly affected by the Courant and diffusion number limitation. A high-order fully-implicit IDO scheme is presented, and the two-stage implicit Runge-Kutta time integration keeps over third-order accuracy. The application of the method to the direct numerical simulation of turbulence demonstrates that the proposed scheme retains a resolution comparable to that of spectral methods even for relatively large Courant numbers. The scheme is further applied to the Local Mesh Refinement (LMR) method, where the size of the time step is often restricted by the dimension of the smallest meshes. In the computation of the Karman vortex street problem, the implicit IDO scheme with LMR is shown to allow a conspicuous saving of computational resources.

A fourth-order accurate local refinement method for Poisson’s equation

We present a block-structured local refinement method for computing solutions to Poisson’s equation in two and three dimensions. It is based on a conservative, finite-volume formulation of the classical Mehrstellen methods. This is combined with finite volume local refinement discretizations to obtain a method that is fourth-order accurate in solution error, and with easily verifiable solvability conditions for Neumann and periodic boundary conditions.

Simulation of sharp gas–liquid interface using VOF method and adaptive grid local refinement around the interface

AbstractThe volume of fluid (VOF) method is used to perform two‐phase simulations (gas–liquid). The governing Navier–Stokes conservation equations of the flow field are numerically solved on two‐dimensional axisymmetric or three‐dimensional unstructured grids, using Cartesian velocity components, following the finite volume approximation and a pressure correction method. A new method of adaptive grid local refinement is developed in order to enhance the accuracy of the predictions, to capture the sharp gas–liquid interface and to speed up the calculations. Results are compared with experimental measurements in order to assess the efficiency of the method. Copyright © 2004 John Wiley &amp; Sons, Ltd.

A Numerical Method on Cartesian Grid for Image-based Blood Flow Simulation

We have developed a numerical method on Cartesian grid for image-based blood flow simulation. Cartesian grid has advantages for grid generation for the complex and moving boundary problems. Interpolated Differential Operator (IDO) scheme with Local Mesh Refinement method are effectively introduced for blood flow simulations. In order to simulate flow interacting with elastic vessel on Cartesian grid, a new technique INTERGRID has been developed and it is shown that the simulation of fluid-structure interaction problems that structure does not displace over the resolution of grid is carried out.

FDTD based second-order accurate local mesh refinement method for Maxwell's equations in two space dimensions

An algorithm is presented for local space-time mesh refinement appropriate for electromagnetic simulations based on the space-time staggered FDTD method. The method is based on the adaptive mesh re.nement algorithm originally developed for hyperbolic conservation laws. Analysis of the dispersion relation and of the numerical reflection and transmission coefficients in one and two space dimensions shows that a scheme based on linear interpolation at the grid interfaces is unstable due to reflection coefficient > 1 at frequencies above the cutoff frequency of the coarse grid. A second-order accurate algorithm based on higher-order interpolations that enforces conservation of the magnetic field circulation at the fine-coarse grid boundaries is constructed. The new algorithm is shown to be stable and accurate for long time integration. A numerical simulation of an optical ring microcavity resonator using multilevel grid refinement in two space dimensions is presented.

Development and evaluation of a local grid refinement method for block-centered finite-difference groundwater models using shared nodes

A new method of local grid refinement for two-dimensional block-centered finite-difference meshes is presented in the context of steady-state groundwater-flow modeling. The method uses an iteration-based feedback with shared nodes to couple two separate grids. The new method is evaluated by comparison with results using a uniform fine mesh, a variably spaced mesh, and a traditional method of local grid refinement without a feedback. Results indicate: (1) The new method exhibits quadratic convergence for homogenous systems and convergence equivalent to uniform-grid refinement for heterogeneous systems. (2) Coupling the coarse grid with the refined grid in a numerically rigorous way allowed for improvement in the coarse-grid results. (3) For heterogeneous systems, commonly used linear interpolation of heads from the large model onto the boundary of the refined model produced heads that are inconsistent with the physics of the flow field. (4) The traditional method works well in situations where the better resolution of the locally refined grid has little influence on the overall flow-system dynamics, but if this is not true, lack of a feedback mechanism produced errors in head up to 3.6% and errors in cell-to-cell flows up to 25%.

The effect of overall discretization scheme on Jacobian structure, convergence rate, and solution accuracy within the local rectangular refinement method

AbstractThe local rectangular refinement (LRR) solution‐adaptive gridding method automatically produces orthogonal unstructured adaptive grids and incorporates multiple‐scale finite differences to discretize systems of elliptic governing partial differential equations (PDEs). The coupled non‐linear discretized equations are solved simultaneously via Newton's method with a Bi‐CGSTAB linear system solver. The grids' unstructured nature produces a nonstandard sparsity pattern within the Jacobian.The effects of two discretization schemes (LRR multiple‐scale stencils and traditional single‐scale stencils) on Jacobian bandwidth, convergence speed, and solution accuracy are studied. With various point orderings, for two simple problems with analytical solutions, the LRR multiple‐scale stencils are seen to: (1) produce Jacobians of smaller bandwidths than those resulting from the traditional single‐scale stencils; (2) lead to significantly faster Newton's method convergence than the single‐scale stencils; and (3) produce more accurate solutions than the single‐scale stencils.The LRR method, including the LRR multiple‐scale stencils, is finally applied to an engineering problem governed by strongly coupled, highly non‐linear PDEs: a steady‐state lean Bunsen flame with complex chemistry, multicomponent transport, and radiation modeling. Very good agreement is observed between the computed flame height and previously published experimental data. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Some a posteriori error estimates of the finite-difference streamline-diffusion method for convection-dominated diffusion equations ∗

In this paper, some a posteriori error estimates of the finite-difference streamline-diffusion method for one- and two-dimensional time-dependent convection-dominated diffusion equations are presented, which can be used to reasonably adjust space mesh. Numerical results show that this method of local refinement is feasible and effective.

A Local Space-Time Adaptive Scheme in Solving Two-Dimensional Parabolic Problems Based on Domain Decomposition Methods

We describe a local space-time adaptive refinement and derefinement method for solving systems of parabolic partial differential equations. Solutions are calculated on local space-time meshes using a domain decomposition finite element algorithm in space and an extrapolation algorithm in time. An a posteriori spatial error estimate controls a local spatial refinement strategy and an a posteriori temporal error estimate monitors temporal refinement and order of the integration. A multiplicative Schwarz algorithm is used to coordinate solutions between overlapping grids. We use nearly minimal fine meshes which follow the dynamics of the problems in an efficient manner.

Local Rectangular Refinement with Application to Nonreacting and Reacting Fluid Flow Problems

A new solution-adaptive gridding method has been developed for the solution of discretized systems of coupled nonlinear elliptic partial differential equations on rectangular domains. Such a method is required for the numerical solution of realistic combustion problems, in which physical quantities may vary by orders of magnitude over one-tenth of a millimeter at atmospheric pressure, or over micrometers at higher pressures. The local rectangular refinement (LRR) method maintains orthogonality at grid-line intersections but lifts the tensor product restriction common to traditional grids, producing unstructured grids. Governing equations are discretized throughout the domain using newly derived forms, and Newton's method is used to solve the resulting system. On a simple test case with a known solution, the LRR method and its new discretizations are found to be more accurate than gridding methods representative of those appearing previously in the literature. For the more realistic problem of nonreacting driven square cavity flow, the LRR solution agrees very well with previously published data. When the LRR method is applied to a practical reacting flow (a rich axisymmetric laminar Bunsen flame with complex chemistry, multicomponent transport, and an optically thin radiation submodel), grid spacing highly influences the inner flame's position, which stabilizes only with adequate refinement. The vorticity–velocity formulation of the governing equations is shown to produce valid results when used in conjunction with the LRR gridding technique. Furthermore, each LRR grid is used to form a nonuniform equivalent tensor product (ETP) grid and also, in most cases, an equispaced fully refined (FR) grid; these additional grids are supersets of the LRR grids and thus contain refinement in exactly the same regions. Performance comparisons between the LRR, ETP, and FR grids indicate that the LRR method provides substantial savings in execution time and computer memory requirements, without compromising solution accuracy.

A characteristic domain decomposition and space-time local refinement method for first-order linear hyperbolic equations with interfaces

We develop a characteristic-based domain decomposition and space{time local refinement method for firstorder linear hyperbolic equations. The method naturally incorporates various physical and numerical interfaces into its formulation and generates accurate numerical solutions even if large time-steps are used. The method fully utilizes the transient and strongly local behavior of the solutions of hyperbolic equations and provides solutions with significantly improved accuracy and efficiency. Several numerical experiments are presented to illustrate the performance of the method and for comparison with other domain decomposition and local refinement schemes. c 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 1{28, 1999

Application of a Hybrid Method to Jet Impinging Phenomena with a Cross Flow. 1st Report. Effects of a Local Refinement Method with a Truncation Error Estimation.

The local refinement technique is very useful method to get high-precision solution with lower efforts in the structural grid system and has been applied to many cases. In using this method, some methods are needed to specify the regions on which the mesh should be fined on the calculation domain and a method based on the truncation error estimation theory was proposed. This method makes it possible to determine the regions with only one, non-iterative calculation and is useful for implicit method. A local mesh refinement method with SIMPLE algorithm was applied to the abovementioned region. This method basically satisfies the conservation law on the fine/coarse cell boundaries. As a result, high-precision solution was got with shorter time and smaller memory in comparison with the usual method.