不同二维声学边界单元在敏感度分析上的数值表现

非连续边界单元可以有效提高边界元法在具有复杂边界形状的实际问题上的应用能力。本文发展基于Burton-Miller法的无奇异二维声学非连续边界元法,通过带有解析解数值算例,对比不同类型单元的计算精度以得到最有效的单元类型,并考察非连续线性和二次单元的优化节点位置。同时本文采用伴随变量法进行声学敏感度计算,并结合MMA算法(The method of moving asymptotes)对Y型声屏障进行结构优化分析。     

变系数各向异性热传导问题边界元解法

提出了一种用边界元法求解一般变系数各向异性热传导问题时建立基本解的方法,并导出了求解一般二维和三维各向异性稳态热传导问题的纯边界积分方程。所建立的基本解考虑了热导率是空间坐标的函数,因此所导出的积分方程可用于求解非均质材料传热问题。由热源项引起的域积分,运用径向积分法将其转换成边界积分,形成不需要内部点的纯边界元算法。给出了二维和三维问题3个分析算例,并通过将边界元法结果与有限元法结果进行对比,证明了方法的正确性和有效性。     

变系数各向异性热传导问题边界元解法

提出了一种用边界元法求解一般变系数各向异性热传导问题时建立基本解的方法,并导出了求解一般二维和三维各向异性稳态热传导问题的纯边界积分方程。所建立的基本解考虑了热导率是空间坐标的函数,因此所导出的积分方程可用于求解非均质材料传热问题。由热源项引起的域积分,运用径向积分法将其转换成边界积分,形成不需要内部点的纯边界元算法。给出了二维和三维问题3个分析算例,并通过将边界元法结果与有限元法结果进行对比,证明了方法的正确性和有效性。     

二维弹性随机边界元中的奇异性问题

研究了二维弹性随机边界元问题,讨论了弹性模量、泊松比及载荷为随机变量时的随机边界积分方程的列式与推导,当采用线性单元离散化边界时,计算影响系数矩阵偏导数的对角元时会产生奇异性。推荐了一种方法能解决这个问题,且具有较好的效率和精度。     

浅谈有限差分,有限单元及边界元在位场向上延拓中的应用比较

地球物理数值模拟法即根据地球物理中的偏微分方程和边界条件,用数值方法求场值的近似解。地球物理正演的数值计算方法种类有很多,较常用的方法有三种：有限差分法、有限单元法和边界单元法。本文结合实例对这三种方法的原理及一般步骤给予介绍,然后以位场向上延拓为算例,分别用有限差分,有限元,边界元法对其求解并编制相关程序进行了计算。最后分析总结了这三种方法在数值模拟中的优缺点。     

二维位势边界条件反识别TSVD正则化法

二维各向同性材料Cauchy位势边界条件反识别问题是不适定的,通过边界元方法得到线性方程组的系数矩阵呈现病态,测量数据的随机偏差会影响分析结果的稳定性和精确性。文章运用截断奇异值分解正则化方法来处理该反问题,借助L曲线法选择奇异值截断位置,进一步可求解得到未知边界条件。圆环和方形板区域热传导2个算例结果分析表明:获取数据的偏差越小,边界划分单元越细密,数值解越接近解析值,正则误差也越小。     

注塑模典型截面温度场的数值分析

通过二维典型截面简化模具的三维结构,建立了注塑模典型截面温度场的边界积分方程,并进行数值求解。将模具温度分为稳态和波动两部分,稳态部分是采用循环平均假设,推导出求解模具典型截面二维稳态温度场的边界积分方程。然后利用边界元法,分别对动模和定模进行传热分析,根据分型面边界相容性条件进行耦合;波动部分是在给出温度波动的微分方程后,利用有限差分法结合传热学对型腔表面温度波动进行数值求解。最后通过实例验证了文中算法的正确性与有效性。     

圆柱壳开孔应力集中问题的边界元解法

本文将边界元法应用于圆柱壳。在建立积分方程时,作者建议用板的基本解叠加三角级数解作为圆柱壳的基本解,并导出了圆柱壳的边界元解法的基本公式。计算表明,使用该基本解,提高了计算精度。文中还提出了区域积分的处理方法。数值计算显示,用边界元法分析圆柱壳的开孔问题是十分有效的。     

注塑模典型截面温度场的数值分析

通过二维典型截面简化模具的三维结构，建立了注塑模典型截面温度场的边界积分方程。利用边界元法分别对动模和定模进行传热分析，根据分型面边界相容性条件进行耦合，最终求出型腔表面温度场的数值解。通过实例验证了文中算法的正确性与有效性。     

混凝土中氯离子二维扩散分析的边界元法

混凝土二维扩散分析的有限元法等常用数值法通常需要在空间域和时间域中同时采取细密的离散网格,计算量大.针对混凝土中氯离子二维扩散问题,提出了计算长度的概念及其计算表达式,首次建立了相应的边界元计算方法,确定了时间域离散的步长.通过该计算模型研究了混凝土结构的拐角等几何形状复杂位置的氯离子分布规律.由于边界元法可以将二维问题简化为一维离散问题,而且该计算模型在时间域内的离散网格非常稀疏,因此,相对于其他数值方法,该方法计算量很小,算例分析验证了该方法具有很高的计算精度和计算效率.同时,对角点等位置的氯离子浓度的计算结果表明,二维方法能够更准确地反映氯离子在混凝土结构中的扩散和积聚规律.     

Development of a computer algorithm based on a conjugate gradient approach for optimization of fed-batch fermentations

The problem of optimization of fed-batch fermentations using the substrate feed rate as the control variable is singular in nature. Previous approaches, including the boundary condition iteration method and transformation to a nonsingular problem using a different control variable, do not work well for solving optimization of systems governed by more than four differential equations. The applicability of a first-order conjugate gradient algorithm for optimizing fed-batch fermentations was tested for systems of varing complexity. This approach does not need any variable transformation ora priori knowledge of the control arc sequence. Constraints on the feed rate are handled in a simple and direct manner. The algorithm worked very well for three, four, and five-dimensional singular systems. The correctness of the optimal profile was judged by observing the variation in the sign of the gradient of the Hamiltonian. The gradient was found to be zero during the singular period and had the appropriate sign on the boundary arcs. The optimization method based on conjugated gradient approach can be complementary to the boundary condition iteration method for determination of the exact optimum profile.     

An agent-based solution of Lagrange equations for adsorption processes

A simple numerical method for solving the rate equation of adsorption processes is presented. The method starts with the mass balance for the fluid phase in its lagrangian form and the corresponding equation for the solid phase; these equations are then used to specify the governing interaction rules of discrete elements, dubbed agents. In the calculation code, ALEAP, the calculation is carried out as a series of cycles in which the agents, representing the adsorption process, interact according to these rules. In this paper we present the results obtained for linear isotherms from no transfer to high transfer rate. The method is surprisingly efficient for finding the right solution for the problem of dispersion with no adsorption and superior, in terms of computer processing time, to other methods for the simulation of the adsorption process with linear or non-linear isotherms.     

The instrument spreading correction in GPC. III. The general shape function using singular value decomposition with a nonlinear calibration curve

The Gram-Charlier series was suggested as an empirical instrument spreading function in the first paper (part I) of this series. In the second paper (part II) of this series, the Fourier transform method was used together with the suggested series to solve Tung's integral equation. In this paper, an alternate method for solving Tung's equation is proposed which eliminates some of the limitations of the Fourier transform method. In the approach used in this study, Tung's integral equation is approximated by a set of linear equations. Since no unique least-squares solution can be computed, a closely related problem whose solution closely approximates the original problem is formulated and solved using singular value decomposition. By avoiding the use of the smallest singular values and forcing the equality of the areas of the corrected and the uncorrected chromatograms, an approximate solution to the original problem is obtained in which the oscillations inherently present due to the ill-posed nature of the problem are filtered out. The performance of the method with the experimental data given in Part II is indicated.     

受扰双线性系统的近似最优扰动抑制方法

研究具有外界持续扰动作用下双线性系统的最优控制问题.关于二次型性能指标给出了一种设计最优扰动抑制控制律的逐次逼近方法.利用该算法可将在扰动作用下双线性系统的最优控制问题转化为求解一组线性非齐次两点边值序列问题.通过迭代序列得到的最优扰动抑制控制律由解析的线性前馈-反馈项和序列极限形式的非线性补偿项组成.通过截取非线性补偿序列的有限项,可以得到近似最优扰动抑制控制律.仿真结果表明,该方法抑制外部持续扰动的鲁棒性优于经典反馈最优控制.     

An interior point method implementation for solving large planning problems in the oil refinery industry

Multi-period planning problems in the oil and refinery industry are typically large, sparse, staircase/band diagonal structured and nonlinear optimization problems. Successive linear programming (SLP) type methods have been widely used for solving these planning problems. But, it has long been recognized that the simplex method used in solving linear programs requires a large number of iterations for staircase/band diagonal structured problems. In this paper, we report results of an application of a recently developed interior point method that promises to be many times faster than the simplex method for multi-period planning problems. However, to facilitate the use of interior point method in the current SLP algorithms a hybrid method combining the interior point method and the simplex method is developed. Therefore, the results determined with this hybrid method are qualitatively equivalent to that obtained with the simplex method alone. The CPU times corresponding to the hybrid method are compared with the CPU times of simplex and dual affine methods. The new hybrid method generates a basic feasible solution of the linear programming problem and is approximately 7 times faster than the simplex method on the tested planning problems. Moreover, the interior point and hybrid methods become faster as the problem size increases.     

Mass transfer with axial diffusion and chemical reaction

Mass transfer with axial diffusion and chemical reaction is studied for three different cases. The first one is plug flow mass transfer in semi-infinite region. The second one is laminar flow mass transfer in a semi-infinite region. The third one is laminar flow convective diffusion in the region of x ≤ 0 while mass transfer with chemical reaction in the region of x ≥ 0. The first problem is solved in terms of Bessel functions while the second in terms of confluent hypergeometric functions. In order to solve the third problem, orthonormal functions are constructed by linear combination of the eigenfunctions of the two semi-infinite regions respectively. The series expansion coefficients of the solutions are determined by solving a set of forty simultaneous equations obtained through matching boundary conditions at x = 0.Solutions are generally obtained for low Peclet number (Pe = 1, 5, 10) and high reaction rate parameters. Effects of axial diffusion and chemical reaction on the concentration field are discussed.     

COMPARISON OF APPROXIMATE METHODS FOR SOLVING NON-LINEAR CONVECTIVE DIFFUSION PROBLEMS

A comparison is made of the approximate methods of solving a high Schmidt number non-linear convective diffusion equation and boundary conditions similar to those which arise in various separation problems including reverse osmosis and directional solidification. The series expansion method gives the best agreement with exact numerical solutions. The film theory, which provides very simple results, yields surprisingly good estimates which are second in accuracy only to the series expansion. Several more sophisticated techniques including rapidly varying boundary conditions, the integral method and local non-similarity yield results of somewhat disappoinling accuracy. The linear problem, B₂ = 0, for which exact analytical and numerical solutions exist, provides a discriminating test of the accuracy of the methods.     

A simultaneous approach for singular optimal control based on partial moving grid

Singular optimal control plays an important role in process engineering, including optimal operation of batch and semi-batch reactions. However, for many practical applications, accurate solution of singular optimal control profiles is still an open issue. In particular, numerical optimization must deal with an ill-conditioned problem that often leads to very slow convergence or failure. Starting from the nested approach in our previous work in 2016, this study develops a more efficient strategy for singular control through a heuristic approach for the outer problem. The approach includes three stages. Starting from a coarse distribution of finite elements, sufficiently many finite elements are inserted where control profiles are steep and fixed gridpoints are inserted on the basis of error estimation of state profiles. Then, moving gridpoints are inserted where the modified switching function is violated. Initial junctions are obtained by moving the latest inserted gridpoints. Moreover, further mesh refinements are considered based on switching point detection and a moving grid point update strategy, until modified switching conditions are satisfied over the whole-time span. A key feature of this approach is that only a subset of finite elements needs to move during optimization. Complexity of the optimization formulation is considerably decreased compared to our previous work. This approach is demonstrated on eight classical singular control problems with known solutions, as well as six complex singular control problems drawn from the chemical engineering literature.     

Solution techniques for transport problems involving steep concentration gradients: application to noncatalytic fluid solid reactions

Some efficient solution techniques for solving models of noncatalytic gas–solid and fluid–solid reactions are presented. These models include those with non-constant diffusivities for which the formulation reduces to that of a convection–diffusion problem. A singular perturbation problem results for such models in the presence of a large Thiele modulus, for which the classical numerical methods can present difficulties. For the convection–diffusion like case, the time-dependent partial differential equations are transformed by a semi-discrete Petrov–Galerkin finite element method into a system of ordinary differential equations of the initial-value type that can be readily solved. In the presence of a constant diffusivity, in slab geometry the convection-like terms are absent, and the combination of a fitted mesh finite difference method with a predictor–corrector method is used to solve the problem. Both the methods are found to converge, and general reaction rate forms can be treated. These methods are simple and highly efficient for arbitrary particle geometry and parameters, including a large Thiele modulus.