An adaptive moving finite element method for steady low Mach number compressible combustion problems

This work surveys an r-adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the computation of such a problem is often extremely time-consuming. Importantly, to capture the significant characteristics of the flame structure when using detailed chemistry, a much more stringent requirement on the spatial resolution of the interior layers of some intermediate species is necessary. Here, we propose a moving mesh method in which the mesh is obtained from the solution of so-called moving mesh partial differential equations. Such equations result from the variational formulation of a minimization problem for a given target functional that characterizes the inherent difficulty in the numerical approximation of the underlying physical equations. Adaptive mesh movement has emerged as an area of intense research in mesh adaptation in the last decade. With this approach, points are only allowed to be shifted in space leaving the topology of the grid unchanged. In contrast to methods with local refinement, data structure hence is unchanged and load balancing is not an issue as grid points remain on the processor where they are. We will demonstrate the high potential of moving mesh methods for effectively optimizing the distribution of grid points to reach the required resolution for chemically reacting flows with extremely thin boundary layers.     

A one-dimensional moving grid solution for the coupled non-linear equations governing multiphase flow in porous media. 1: Model development

A straightforward moving grid finite element method is developed to solve the one-dimensional coupled system of non-linear partial differential equations (PDEs) governing two- and three-phase flow in porous media. The method combines features from a number of self-adaptive grid techniques. These techniques are the equidistribution, the moving grid finite element and the local grid refinement/coarsening methods. Two equidistribution criteria, based on solution gradient and curvature, are employed and nodal distributions are computed iterativcly. Using the developed approach, an intermingle-free nodal distribution is guaranteed. The method involves examination of a single representative gradient to facilitate the application of moving grid algorithms to solve a non-linear coupled set of PDEs and includes a feature to limit mass balance error during nodal redistribution. The finite element part of the developed algorithm is verified against an existing finite difference model. A numerical simulation example involving a single-front two-phase flow problem is presented to illustrate model performance. Additional simulation examples are given in Part 2 of this paper. These examples include single and double moving fronts in two- and three-phase flow systems incorporating source/sink terms. Simulation sensitivity to the moving grid parameters is also explored in Part 2.     

Modeling Three-Dimensional Groundwater Flows by the Body-Fitted Coordinate (BFC) Method: II. Free and Moving Boundary Problems

This paper presents a methodology and solution procedure of the time-dependent body-fitted coordinate (BFC) method for the analysis of transient, three-dimensional groundwater flow problems characterized by free and moving boundaries. The technique consists of numerical grid generation, time-dependent body-fitted coordinate transformation, and application of the finite difference method (FDM) to the transformed partial differential equations. Based on the time-dependent BFC method, a three-dimensional finite-difference computer code, BFC3DGW, was developed and used to solve two unconfined flow problems. The code was verified by comparing numerical results with analytical solutions for a steady-state seepage problem. In order to demonstrate capability of the method in dealing with flow problems with irregular and moving boundary surfaces, an unconfined well-flow problem was solved by the developed code. Difficulties associated with the free and moving irregular boundary have been successfully overcome by employing this method.     

基于边界值方法的微分动力系统数值计算方法

对高维非线性初值问题,微分求积法在每一步的积分过程中需要求解一个更高维的非线性方程组,因而计算量巨大。基于微分求积法与边界值方法两者之间的关系,可以将广义向后差分方法和扩展的隐式梯形积分方法看作是经典微分求积法的稀疏表达形式。将广义向后差分方法以及扩展的隐式梯形积分方法这两类边界值方法应用于微分动力系统的数值计算,提出了一类新的数值计算方法。理论分析及算例结果表明,对高维非线性微分初值问题的数值计算,本文方法相对于经典的微分求积法具有更高的计算效率。     

Motion of a strong shock front in a two-dimensional gas layer with moving internal boundary

We examine the problem of planar one-dimensional motion of a strong shock wave with moving internal boundary in which the initial position of the front, its intensity, the mass of the gas involved in the motion, and the energy contained in this gas are known. The problem is not self-similar and its exact solution, which involves working with partial differential equations, presents serious difficulties. In the following we determine the law of shock-front motion in this problem via the method of [1], which makes it possible to find a system of ordinary differential equations for the problem. The method is based on an initial specification of the power-law coupling between the dimensionless Lagrangian and Eulerian variables and replacement of the energy equation by this coupling and the energy integral. The solution is sought in the first approximation.     

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be stably numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy’s flow problem, the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pressure gradient increasing for the one-dimensional problem.     

Renormalization group symmetry method and gas dynamics

A new scenario in the renormalization group symmetry method is introduced to solve an initial value problem for a system of partial differential equations. As a specific example, we give an exact solution for the adiabatic perfect gas dynamics, which describes a contracting and expanding localized mass of gas.     

基于θ1方法的多体动力学数值算法研究

将结构动力学领域的\theta_1方法拓展到数值求解多体系统运动方程------微分--代数方 程(DAEs), 分别求解指标-3 DAEs形式的运动方程和指标-2超定DAEs (ODAEs)形式的运动方程. 通过数值算例验证了方法的有效性, 并得到\theta _1 方法中参数\theta _1的选取与数值耗散量之间的关系. 数值算例还说明对于同 一个多体系统, 采用指标-3的DAEs 描述时存在速度违约, 用指标-2的ODAEs描述时, 从计算机精度上讲, 位置和速度约束方程 同时满足, 并且\theta_1方法在求解非保守系统DAEs和ODAEs形式的运动方程时 都具有2阶精度. 最后\theta_1 方法与其他直接积分法求解DAEs和ODAEs形式运 动方程的CPU时间进行了比较.     

一种有效的网格自适应方法

针对网格自适应方法中的移动网格变形法进行了研究,该方法通过采用最小二乘有限元法求解div-curl方程组,使雅可比行列式值等于给定的监测函数值,以改变网格节点的位置,从而获得期望的网格边界及大小分布。文末给出的4个二维和三维问题的算例表明,本文方法是非常有效的。     

Precise integration method for LQG optimal measurement feedback control problem

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Flow of a viscous fluid in a cylindrical vessel with a rotating cover

The problem considered arises in solving various technical problems associated with flows of a viscous fluid in a closed space near rotating plane surfaces, turbomachine disks, thrust bearings, rotational viscosimeters, etc. The approximate solution of the problem on the basis of a simplified flow scheme was first obtained by Schultz-Grunow [1], The most complete investigation has been made recently by Grohne [2], who outlined a program for solving the problem by joining several partial solutions on the basis of definite hypotheses concerning the flow core.With the development of electronic digital computers and the necessary numerical methods, the most effective means of solving the considered problem is the use of the grid methods for solving partial differential equations. The present paper is devoted to presenting the results of the solution of the problem using the grid method on a digital computer.     

Solution of the incompressible Navier–Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley & Sons, Ltd.     

多体系统动力学方程违约修正的数值计算方法

多体系统动力学方程为微分代数方程,一般将其转化成常微分方程组进行数值计算,在数值积分的过程中约束方程的违约会逐渐增大.本文对具有完整、定常约束的多体系统,在修改的带乘子Lagrange正则形式的方程的基础上,根据Baumgarte提出的违约修正的方法,给出了一种多体系统微分代数方程违约修正法和系统的动力学方程的矩阵表达式.通过对曲柄-滑块机构的数值仿真,计算结果表明本文给出的方法在计算精度和计算效率上好于Baumgarte提出的两种违约修正的方法.     

Adaptive grids for resolution enhancement

After a brief review of various moving grid methods for resolution enhancement, we present the deformation method originated from differential geometry. The method is used to relocate the nodes of an initial grid according to the solution through a set of deformation differential equations. The node velocity is determined by a monitor function constructed from physical variable(s) through a scalar Poisson equation. It is proved mathematically that the cell volume is proportional to the monitor function at each time step. In particular, the moving grid mapping has positive Jacobian determinant and thus will not fold into itself. The moving grid method is then applied to the Euler equation. The results showed that the grid indeed follows the monitor function closely and that the method significantly improves the resolution compared to uniform grids of the same amount of nodes. Received 27 July 2001 / Accepted 24 April 2002 Published online 8 July 2002     

Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.     

Exact solutions for large elastoplastic deformations of a thick-walled tube under internal pressure

A rate-independent plasticity theory based on the concept of dual variables and dual derivatives is utilized to describe finite elastic-plastic deformations including kinematic and isotropic hardening effects. Application of this theory to the problem of the thick-walled tube under internal pressure leads to a system of partial differential equations of hyperbolic type. The existence and uniqueness of the solution of the boundary value problem is guaranteed, as well as the convergence of its numerical approximation. The exact solution of this problem is calculated by means of an extrapolation technique. This integration method turns out to be applicable for rather general hardening models of rate-independent plasticity. On the basis of the computed solutions the influence of the hardening parameters is investigated. As finite deformations are of special interest, this investigation is carried out not only for the partially yielded tube but also for the completely plastified tube. Furthermore, the onset of secondary plastic flow during unloading as well as residual stress distributions are studied.     

A Singular Initial Value Problem to Construct Density-Equalizing Maps

The diffusion-based algorithm to produce density-equalizing maps interprets diffusion as an advection process. This algorithm uses the dynamics of a flow that is defined by an initial value problem that turns out to be very singular at the initial time. The singularities appear when the initial density has line or angle discontinuities, which is always the case, for example, in area cartogram maps. This singular initial value problem is analyzed mathematically in this article and the conclusion is that despite these singularities, it has a unique solution. This justifies the extensive numerical use of this algorithm in the recent years. The techniques presented in this article use both partial and ordinary differential equations estimates.     

Analysis of a two-grid method for semiconductor device problem

The mathematical model of a semiconductor device is governed by a system of quasi-linear partial differential equations.The electric potential equation is approximated by a mixed finite element method,and the concentration equations are approximated by a standard Galerkin method.We estimate the error of the numerical solutions in the sense of the Lqnorm.To linearize the full discrete scheme of the problem,we present an efficient two-grid method based on the idea of Newton iteration.The main procedures are to solve the small scaled nonlinear equations on the coarse grid and then deal with the linear equations on the fine grid.Error estimation for the two-grid solutions is analyzed in detail.It is shown that this method still achieves asymptotically optimal approximations as long as a mesh size satisfies H=O(h^1/2).Numerical experiments are given to illustrate the efficiency of the two-grid method.     

Integral vorticity transport method on an overset grid

We present an overset grid method for solution of the integro‐differential vorticity–velocity formulation of the Navier–Stokes equations for two‐dimensional, incompressible flow. The method uses a body‐fitted inner grid, on which vorticity is evolved semi‐implicitly, and a Cartesian outer grid with explicit vorticity evolution. The Biot–Savart integral is solved using an adaptive, optimized multipole acceleration method. The Biot–Savart integration is performed over all inner grid cells, over all ‘active cells’ of the outer grid that lie entirely outside of the inner grid, and over sub‐elements of a set of ‘overhanging’ cells of the outer grid that overlap part of the inner grid. A novel method is developed using a level‐set distance function to rapidly and easily partition the overhanging grid cells, which is essential for the Biot–Savart integration in order to avoid double‐counting vorticity in the overhanging region. A similar decomposition into outer, inner and overhanging cells is used in solving for pressure using a boundary‐element formulation, which requires evaluation of an integral over the vorticity field using a method similar to that used for the Biot–Savart integral. The new overset grid method is applied to flow past stationary and moving bodies in two dimensions and found to agree well with prior experimental and numerical results. Copyright © 2011 John Wiley & Sons, Ltd.