The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework

This paper presents a comprehensive overview of the element-wise locally conservative Galerkin (LCG) method. The LCG method was developed to find a method that had the advantages of the discontinuous Galerkin methods, without the large computational and memory requirements. The initial application of the method is discussed, to the simple scalar transient convection-diffusion equation, along with its extension to the Navier-Stokes equations utilising the Characteristic Based Split (CBS) scheme. The element-by-element solution approach removes the standard finite element assembly necessity, with an face flux providing continuity between these elemental subdomains. This face flux provides explicit local conservation and can be determined via a simple small post-processing calculation. The LCG method obtains a unique solution from the elemental contributions through the use of simple averaging. It is shown within this paper that the LCG method provides equivalent solutions to the continuous (global) Galerkin method for both steady state and transient solutions. Several numerical examples are provided to demonstrate the abilities of the LCG method.     

Thermal analysis for transient radiative cooling of a conducting semitransparent layer of ceramic in high-temperature applications

A method is developed for obtaining transient temperature distribution in a cooling semitransparent layer of ceramic. The layer is emitting, absorbing, isotropically scattering and heat conducting with a refractive index ranging from 1 to 2. The solution involves solving simultaneously the energy equation and the integral equation for the radiative flux gradient. The energy equation is solved using an implicit finite volume scheme and the integral equation of radiative heat transfer is solved using the singularity technique and Gaussian integration. The effects of scattering are investigated. It is shown that scattering has a significant effect on the transient temperature distribution and the transient mean temperature of the layer.     

含对流与蒸发表面的有限厚度材料的双曲型热传导分析

本文对于含有对流与蒸发表面的有限厚度材料,在遭受一脉冲表面热流作用时的双曲型热传导进行了分析,采用热势函数描述的双曲型热传导方程来描述该问题,并用有限差分法进行数值求解．同时,本文还就表面对流与蒸发对材料表面与内部瞬态温度变化的影响以及材料双曲型热传导对表面蒸发的影响进行了分析．     

Natural convection with damped thermal flux in a vertical circular cylinder

Natural convection flows of an incompressible Newtonian fluid inside a circular cylinder are studied. The heat transfer process is described by a generalized fractional constitutive equation for the thermal flux-temperature gradient. Caputo time-fractional derivative operator, which provides the damping of thermal flux, is considered into the studied model.Analytical solutions to the fluid temperature, thermal flux, fluid velocity and volume flow rate are obtained with the integral transforms method (Laplace transform and finite Hankel transform).Temperature behaviors for small and large values of the time t, as well as the post-transient and transient velocity components are determined. The influence of the memory parameter (the order of the time-fractional derivative) on the temperature, thermal flux, velocity and the volume flow rate is numerically and graphically studied.     

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.     

Radiation transfer in vegetative canopies and other layered media: Rapidly solvable exact integral equation not requiring fourier resolution

We develop an exact integral equation for the angle-integrated flux interception by a leaf or a volume element in a layered medium. Iterative numerical solutions converge well. Any converged solution can be simply integrated to give the exact, angle-dependent flux at any depth. The method is presented for an axially symmetric input and an isotropic canopy, with results indicating quantitative significance in photosynthetic modeling and remote sensing. The generalization to arbitrary canopy anisotropy is developed without numerical results.     

用改进的准静态方法求解反应堆三维瞬态方程

本文介绍了三维中子瞬态方程的改进准静态求解方法。给出了两个瞬态问题计算结果的比较;TWIGL结果与基准直接数值结果比较,其功率最大偏差为1.57%;CANDU堆模型,在功率上升达到最大值时的热中子通量与参考解的偏差为0.88%。因此,用改进的准静态方法求解三维中子瞬变问题,其结果是满意的,大大节省了计算时间。     

大平板瞬态热传导问题的一种新的近似解法

１引言大平板瞬态热传导问题有着广泛的工程应用背景。对于复杂的初边值条件或含内热源问题,以及工程上常见的多层复合平壁对象,分析求解难度很大甚至无法求解。在此类情况下往往采用数值方法。但是单纯的数值解不便于理解影响该问题的各种参数的物理意义。因此,各种近似分析方法得到了发展［１,２］。但在近似精度上,往往难以对整个时间坐标范围都达到较高的精度,这就使得近似解更多地局限于定性分析。此外,对于不同的初边值条件或含内热源问题,近似解的形式相异,降低了解的通用性。增加了求解的工作量。本文提出一种基于矩阵理论…     

Two-dimensional radiative equilibrium: A semi-infinite medium subjected to a finite strip of radiation

Exact numerical results are presented for the emissive power and radiative flux at the boundary of a two-dimensional, absorbing-emitting, semi-infinite medium bounded by (1) a strip of collimated radiation and (2) a constant temperature black strip. The method of super-position is used to obtain the finite strip solutions in terms of cosine varying solutions. The infinite integrals arising in the solutions are converted to an alternating series of finite integrals. The Euler transformation is then applied to speed up convergence. Error bounds are determined whereby the two-dimensional finite strip analysis can be approximated by the simpler one-dimensional solution.     

Zu Abrikosovs Theorie der periodischen Lösungen der GL-Gleichungen für Supraleiter 2. Art

Abrikosov's method to get periodic solutions of the GL-equations for superconductors of type II is reformulated in order to make higher approximations possible. It is shown that instead of the external magnetic field, the average field in the superconductor must be used as fixed quantity in the approximations; then the method can be extended to a general iteration procedure with a fixed period-parallelogram for the solution. In the first approximation every periodic solution is given by a simple product of ?-functions; among those solutions which possess one flux quantum per period-parallelogram only, the hexagonal modification has the lowest free energy as already proposed byKleiner et al.     

Modeling and Measurement of the Initial Anode Heat Fluxes in Pulsed High-Current Arcs

An anode heat flux model has been developed for pulsed high-intensity dc arcs. The model is based on temperature-time-history measurements of the rear face of a very thin plane anode and high-speed streak photographs of the arc. The arc heat flux model is derived from a comparison of experimental data with an analytical solution of the one-dimensional heat conduction equation and the arc intensity and timing information obtained from high-speed photographs. A simplified input heat flux model consisting of connected segments of linearly varying heat fluxes with respect to time is used. Duration of the individual segments is determined from the streak photographs and the graphical match of measured rear-face temperature history and the numerical solution. Results using argon gas at atmospheric pressure indicate an initial transient heat flux regime of 100-?s duration with a peak heat flux of 2 × 109 W/m2 followed by a quasi-steady heat flux regime with a heat flux of 1 × 108 W/m2.     

线性瞬态涡流电磁场定解问题解的唯一性和稳定性

线性瞬态涡流电磁场定解问题的主要特点是边界条件使用磁感应强度的法向分量边界条件代替了电场强度的切向分量边界条件，约束方程中忽略了位移电流.这种具有特殊性的定解问题的解是否唯一和稳定对于求解瞬态涡流电磁场而言是一个基本问题.本文在非涡流区引入标量位函数，证明了在推导过程中起重要作用的辅助函数的存在性.通过推导线性瞬态涡流电磁场定解问题的能量估计式，证明了该定解问题的解是唯一的，并且关于初始条件和外源项是稳定的.本结果对于线性瞬态涡流电磁场的求解有一定的指导意义.作为应用，给出了通有单脉冲电流的单匝圆环线圈与球形导体共轴的涡流问题的解析解. 关键词： 瞬态涡流电磁场 能量估计式 唯一性 稳定性     

采用分子动力学方法研究氯化钠水溶液的微观结构

溶液的微观结构对溶液的宏观性质具有决定性影响,团簇的存在和溶液不均性的认知是该领域研究的重要进展之一,也是关注的热点.为了考察溶液的微观结构,本文采用分子动力学方法对浓度适中的氯化钠水溶液进行了模拟,获得了各原子(离子)间的径向分布函数和溶液的瞬态图像.通过对比分析Na~+-Cl~-和Cl~--Na~+的径向分布函数,并结合已有研究结果,表明该径向分布函数上第三峰也来源于离子对,发现氯化钠水溶液存在着第三种离子对;上述径向分布函数上的三个峰分别与直接接触离子对、部分间隔离子对和完全间隔离子对相对应,并给出了这三种离子对的瞬态图像.三个时刻下系统瞬态图像的分析结果表明,溶液中存在水分子连续分布的区域和由离子与水分子共同构成的离子团簇.各瞬态图像中水分子连续分布的最大区域的平均尺寸至少为1.43 nm,溶液中部分间隔离子对的比例最大,约为0.68.在同时考虑直接接触离子对和部分间隔离子对时,三个时刻中最大团簇尺度的平均值为1.44 nm,该值约为部分间隔离子对最概然尺寸的3倍;团簇尺寸随团簇中离子数的增加而增大,在团簇中离子数较大时,满足无规行走(自由连接链)模型的结果.这些说明氯化钠水溶...     

An iterative multiscale finite volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).     

Exact inhomogeneous Einstein-Liouville solutions in Robertson-Walker space-times

We give the general solution of Liouville's equation in Robertson-Walker space-times, and use this to find exact inhomogeneous Einstein-Liouville solutions, using the covariant harmonic method of Ellis, Matravers and Treciokas [1]. The average four-velocity of the gas is tilted, and the gas has nonzero acceleration, shear, energy flux, and anisotropic stress.     

辐射传输方程的三阶球谐展开(P3)近似的有限元法求解

从辐射传输方程出发,应用球谐函数方法对辐射率进行多项式展开,并利用球谐函数的正交性和递推性,在二维笛卡儿坐标下,导出了三阶展开的球谐函数微分方程组（P3近似）,改进了以往成像文献中忽略各项异性因子的P3近似,并用有限元方法对二维圆域均匀和非均匀两种情况做了数值模拟.与漫射近似模型相比较,P3近似能更准确地描述光源附近及吸收较强情况下边界的光辐射分布情况.     

A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities

We introduce a multidimensional peridynamic formulation for transient heat-transfer. The model does not contain spatial derivatives and uses instead an integral over a region around a material point. By construction, the formulation converges to the classical heat transfer equations in the limit of the horizon (the nonlocal region around a point) going to zero. The new model, however, is suitable for modeling, for example, heat flow in bodies with evolving discontinuities such as growing insulated cracks. We introduce the peridynamic heat flux which exists even at sharp corners or when the isotherms are not smooth surfaces. The peridynamic heat flux coincides with the classical one in simple cases and, in general, it converges to it in the limit of the peridynamic horizon going to zero. We solve test problems and compare results with analytical solutions of the classical model or with other numerical solutions. Convergence to the classical solutions is seen in the limit of the horizon going to zero. We then solve the problem of transient heat flow in a plate in which insulated cracks grow and intersect thus changing the heat flow patterns. We also model heat transfer in a fiber-reinforced composite and observe transient but steep thermal gradients at the interfaces between the highly conductive fibers and the low conductivity matrix. Such thermal gradients can lead to delamination cracks in composites from thermal fatigue. The formulation may be used to, for example, evaluate effective thermal conductivities in bodies with an evolving distribution of insulating or permeable, possibly intersecting, cracks of arbitrary shapes.     

Transient torsional vibration responses of finite, semi-infinite and infinite hollow cylinders

Torsional guided waves are often used to detect the defects in a hollow cylinder. To realize the excitation of the torsional guided waves with high efficiency, the transient vibration responses of finite, semi-infinite and infinite hollow cylinders to external torsional forces must be clarified theoretically. In this study, the method of eigenfunction expansion is employed to solve the above problems. The exact analytical solutions derived by this method are not only explicit but also concise. Furthermore, the analytical solution of the transient torsional vibration of the finite hollow cylinder is numerically evaluated. The results obtained agree well with those simulated by the finite element method.     

一维含化学反应流体力学方程组的ALE间断有限元方法

研究一维含化学反应流体力学方程组的数值模拟方法.结合理想气体状态方程并利用HLLC解法器在各个单元边界处的数值通量,给出ALE间断有限元方法.高阶计算时,使用TVD斜率限制器对数值解可能产生的非物理振荡进行抑制.结果表明：该算法能够保持物理量的守恒性和高精度,并能够清晰地捕捉爆轰波的结构特征.     

Taylor与Agrawal模型的解析求解与模型比较

 采用解析方法对双导传输线Taylor模型和Agrawal模型进行研究,获得了这两个模型基于平面电磁波激励下的终端负载响应的频域解析解,并证明它们的解析解是相同的,从而得到双导传输线基于平面电磁波激励的终端负载响应的频域解析解公式。Taylor模型和Agrawal模型其实是对同一个解的不同描述,由于Agrawal模型比Taylor模型相对简单,因此在实际应用时可以采用Agrawal模型或频域解析解公式进行求解。应用频域解析解公式和Fourier变换与逆变换技术,可以求得终端电压或电流的瞬时响应解。