Adaptive and nonadaptive Hermitian operator methods for combustion phenomena

Adaptive and nonadaptive, three-point, fourth-order accurate, compact or Hermitian operator methods are developed and used to study one-dimensional combustion phenomena. The nonadaptive Hermitian operator methods are based on the time linearization of the nonlinear partial differential equations, and employ an approximate factorization technique to reduce a three-dimensional reaction-diffusion operator to a sequence of three one-dimensional, linear, second-order differential operators in space. The three adaptive Hermitian operator techniques presented in this paper are based on the equidistribution of the arc length of the vector of dependent variables and use a subequidistribution principle to obtain smooth grids. The first adaptive technique uses quasilinearization and yields a block tridiagonal matrix for the values of the dependent variables at each iteration. The second technique employs partial quasilinearization and yields a system of uncoupled, linear algebraic equations for each dependent variable at each iteration. The third technique employs a predictor-corrector method to predict the grid point locations and a time linearization procedure to obtain the values of the dependent variables. It is shown that the efficiency and accuracy of adaptive Hermitian operator methods depend on the time step and number of grid points used in the calculations. It is also shown that adaptive methods which use a Crank-Nicolson scheme in time may yield oscillatory solutions, and that nonadaptive Hermitian operator methods require a much larger number of grid points than nonadaptive techniques if the solution of the governing equations is characterized by fast ignition phenomena and/or steep, fast moving flame fronts.     

A new adaptive grid generation by elliptic equations with orthogonality at all of the boundaries

A new adaptive grid generation procedure is proposed, which combines a previously developed method that used Anderson's adaptive grid scheme along with a method for controlling grid spacing and orthogonality on all of the boundaries. The proposed method assigns the desired grid stretching over the smooth region during initial grid system generation and before grid adaptation is performed. After properly interpreting the smoothing term of the weighting function, the desired grid stretching is added to the adaptive grid scheme. Several test cases illustrate the method's feasibility.     

Generalization of the CABARET scheme to two-dimensional orthogonal computational grids

It is proposed to generalize the CABARET method to a two-dimensional system of Euler equations. The transition from one-dimensional problems of gas dynamics to multidimensional ones for the CABARET scheme involves a number of innovative aspects. The first is a procedure for spatial splitting of an algorithm in order to calculate new flow variables. The second one is the specific application of the maximum principle aimed at regularizing the solutions for inhomogeneous equations of transfer of local invariants in different directions. Examples are given of test and model calculations.     

2维载频条纹2维窗口傅里叶变换轮廓术

为了解决3维形貌测量中单一频率光栅条纹的测量信息少、1维双频光栅两个同方向载频容易产生频率混叠及两个载频条纹测量精度不同、连续相位解包法在复杂物体相位解包所存在的问题,提出采用2维网格光栅作为空间载频光栅条纹联合2维窗口傅里叶变换的傅里叶变换轮廓术。两个载频相互垂直,减少可能的频谱混叠;应用2维窗口傅里叶变换进行频谱分析和设计带宽自动调整的滤波器分离出两个1维变形光栅条纹,在一幅变形网格光栅图像中得到两个方向光栅各自所对应的包裹相位分布;并且应用查表法解包得到真实调制相位分布。与1维单频和1维双频比较,这种技术可以获得两倍相同精度的测量信息,并能对复杂物体的相位分布进行解包。给出了详细的理论推导、计算机模拟和实验结果,证实了该方法的可行性。     

An adaptive method of lines solution of the Korteweg-de Vries equation

Following a method of lines formulation, the Korteweg-de Vries equation is solved using a static spatial remeshing algorithm based on the equidistribution principle, which allows the number of nodes to be significantly reduced as compared to a fixed-grid solution. Several finite difference schemes, including direct and stagewise procedures, are compared and the results of a large number of computational experiments are presented, which demonstrate that the selection of a spatial approximation scheme for the third-order derivative term is the primary determinant of solution accuracy.     

Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids

In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform grids. The nonuniform grids are obtained by equidistribution of a positive monitor function, which is a linear combination of a constant and the second-order spatial derivative of the singular component of the solution on every temporal level. Truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.     

The Finite Volume-Complete Flux Scheme for Advection-Diffusion-Reaction Equations

We present a new finite volume scheme for the advection-diffusion-reaction equation. The scheme is second order accurate in the grid size, both for dominant diffusion and dominant advection, and has only a three-point coupling in each spatial direction. Our scheme is based on a new integral representation for the flux of the one-dimensional advection-diffusion-reaction equation, which is derived from the solution of a local boundary value problem for the entire equation, including the source term. The flux therefore consists of two parts, corresponding to the homogeneous and particular solution of the boundary value problem. Applying suitable quadrature rules to the integral representation gives the complete flux scheme. Extensions of the complete flux scheme to two-dimensional and time-dependent problems are derived, containing the cross flux term or the time derivative in the inhomogeneous flux, respectively. The resulting finite volume-complete flux scheme is validated for several test problems.     

Uniform convergence analysis of finite difference approximations for advection–reaction–diffusion problem on adaptive grids

A kind of singularly perturbed advection–reaction–diffusion problems with the exponential boundary layer are considered on an adaptive mesh. The existence, uniqueness and stability for the solution of the discrete problem are analysed with the maximum principle. The stability of the continuous problem is also considered. For the equidistribution problem composed by the difference scheme and equidistribution mesh equations, we establish a first-order ?-independent convergence rate for the numerical scheme defined on the equidistribution mesh and also an estimation for the accuracy of the solution computed on the final mesh generated by the adaptive algorithm. Numerical results are given to examine the validity of our theoretical analysis and the efficiency of the adaptive algorithm.     

A multi-dimensional solver for the steady Euler equations

In this paper a numerical algorithm for the solution of the multi-dimensional steady Euler equations in conservative and non-conservative form is presented. Most existing standard and multi-dimensional schemes use flux balances with assumed constant distribution of variables along each cell edge, which interfaces two grid cells. This assumption is believed to be one of the main reasons for the limited advantage gained from multi-dimensional high order discretisations compared to standard one-dimensional ones. The present algorithm is based on the optimisation of polynomials describing the distribution of flow variables in grid cells, where only polynomials that satisfy the Euler equations in the entire grid cell can be selected. The global solution is achieved if all polynomials and by that the flow variables are continuous along edges interfacing neighbouring grid cells. A discrete approximation of a given spatial order is converged if the deviation between polynomial distributions of adjacent grid cells along the interfacing edge of the cells is minimal. Results from the present scheme between first and fifth order spatial accuracy are compared to standard first and second order Roe computations for simple test cases demonstrating the gain in accuracy for a number of sub- and supersonic flow problems.     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

A Non-oscillatory Multi-Moment Finite Volume Scheme with Boundary Gradient Switching

In this work we propose a new formulation for high-order multi-moment constrained finite volume (MCV) method. In the one-dimensional building-block scheme, three local degrees of freedom (DOFs) are equidistantly defined within a grid cell. Two candidate polynomials for spatial reconstruction of third-order are built by adopting one additional constraint condition from the adjacent cells, i.e. the DOF at middle point of left or right neighbour. A boundary gradient switching (BGS) algorithm based on the variation-minimization principle is devised to determine the spatial reconstruction from the two candidates, so as to remove the spurious oscillations around the discontinuities. The resulted non-oscillatory MCV3-BGS scheme is of fourth-order accuracy and completely free of case-dependent ad hoc parameters. The widely used benchmark tests of one- and two-dimensional scalar and Euler hyperbolic conservation laws are solved to verify the performance of the proposed scheme in this paper. The MCV3-BGS scheme is very promising for the practical applications due to its accuracy, non-oscillatory feature and algorithmic simplicity.     

基于特征识别的多尺度构件网格自动生成算法

随着结构力学领域待解决问题复杂程度不断提高,多尺度构件的高质量网格生成对于其数值模拟的计算精度起着至关重要的作用。本文提出一种基于特征识别的网格自动生成技术方法,该方法将多尺度构件的不同量级尺度几何特征识别出来,根据其不同尺度尺寸设置相关区域的网格尺寸值,利用Delaunay三角化算法和前沿推进法生成能够反映不同尺度几何特征的网格单元,再对小尺度区域周围进行加密处理,最后通过几何指数控制函数将不同尺度网格过渡连接起来,形成多尺度构件的整体网格划分模型。通过2个几何模型的测试表明该方法生成的整体网格质量好,不同尺度区域网格过渡合理,自动化程度较高。     

Bicompact monotonic schemes for a multidimensional linear transport equation

Bicompact difference schemes, previously proposed by the authors for linear one-dimensional transport equations are generalized to the multidimensional case by using a coordinate-wise splitting of the multidimensional problem. The scheme stencil for each of the spatial directions is minimal and consists of two points. The schemes are efficient and can be solved by the running calculation method. The proposed difference schemes have the fourth-order approximation in space variables and first- or third-order time approximation for smooth solutions. The schemes for solving multidimensional problems have inherited the monotonicity property of one-dimensional bicompact schemes. Numerical examples are given illustrating the actual accuracy order of bicompact schemes for smooth solutions and the scheme monotonicity for discontinuous solutions.     

A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time Taylor expansion at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the Cauchy-Kovalevskaya procedure. The surface and volume integrals in the variational formulation are approximated by Gaussian quadrature with the values of the space-time approximate solution. The numerical fluxes at grid cell interfaces are based on the approximate solution of generalized Riemann problems for both, the inviscid and viscous part. The presented scheme has to satisfy a stability restriction similar to all other explicit DG schemes which becomes more restrictive for higher orders. The loss of efficiency, especially in the case of strongly varying sizes of grid cells is circumvented by use of different time steps in different grid cells. The presented time accurate numerical simulations run with local time steps adopted to the local stability restriction in each grid cell. In numerical simulations for the two-dimensional compressible Navier-Stokes equations we show the efficiency and the optimal order of convergence being p+1, when a polynomial approximation of degree p is used.     

分布式电源并网对配电网电流保护影响的研究

分析了分布式电源并网对传统的配电网电流保护的影响,采用Matlab/Simulink仿真软件验证了分布式电源容量不同、并网位置不同对配电网各段电流保护的影响,提出了一种能够满足分布式电源并网后配电网保护要求的改进方案。该改进方案实现原理:对于分布式电源下游的保护,将供电系统电源和分布式电源作为一个整体,重新整定各段电流保护的整定值;对于分布式电源上游的保护,需在各保护装置上安装一个基于故障电流分量的方向元件,只有当保护中的过流元件和方向元件同时启动时,保护才能够可靠动作。     

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.     

An immersed boundary technique for simulating complex flows with rigid boundary

A new immersed boundary (IB) technique for the simulation of flow interacting with solid boundary is presented. The present formulation employs a mixture of Eulerian and Lagrangian variables, where the solid boundary is represented by discrete Lagrangian markers embedding in and exerting forces to the Eulerian fluid domain. The interactions between the Lagrangian markers and the fluid variables are linked by a simple discretized delta function. The numerical integration is based on a second-order fractional step method under the staggered grid spatial framework. Based on the direct momentum forcing on the Eulerian grids, a new force formulation on the Lagrangian marker is proposed, which ensures the satisfaction of the no-slip boundary condition on the immersed boundary in the intermediate time step. This forcing procedure involves solving a banded linear system of equations whose unknowns consist of the boundary forces on the Lagrangian markers; thus, the order of the unknowns is one-dimensional lower than the fluid variables. Numerical experiments show that the stability limit is not altered by the proposed force formulation, though the second-order accuracy of the adopted numerical scheme is degraded to 1.5 order. Four different test problems are simulated using the present technique (rotating ring flow, lid-driven cavity and flows over a stationary cylinder and an in-line oscillating cylinder), and the results are compared with previous experimental and numerical results. The numerical evidences show the accuracy and the capability of the proposed method for solving complex geometry flow problems both with stationary and moving boundaries.     

A finite difference non-matching domain decomposition algorithm for the parabolic equation

A finite difference domain decomposition algorithm on a non-overlapping non-matching grid for the parabolic equation is discussed. The basic procedure is to define the explicit scheme at the interface points with a larger mesh spacing H, then the implicit schemes with different mesh spacings are applied on the non-matching subdomains, respectively. The stability bound is released both for the one-dimensional and two-dimensional parabolic problem. Finally, numerical experiments are also presented.     

Automatic mesh generation on a regular background grid

This paper presents and automatic mesh generation procedure on a 2D domain based on a regular background grid.The idea is to devise a robust mesh generation scheme with equal emphasis on quality and efficiency,Instead of using a traditional regular rectangular grid,a mesh of equilateral triangles is employed to ensure triangular element of the best quality will be preserved in the interior of the domain.As for the boundary,it is to be generated by a node/segment insertion process.Nodes are inserted into the background mesh one by one following the sequence of the domain boundary.The local strcture of the mesh is modified based on the Delaunay criterion with the introduction of each node.Those boundary segments.which are not produced in the phase of node insertion,will be recovered through a systematic element swap produced in the phase of node insertion will be recovered through a systematic element swap process.Two theorems will be presented and proved to set up the theoretical basic of the boundary recovery part.Examples will be presented to demonstrate the robustness and the quality of the mesh generated by the proposed technique.