Homotopy continuation enhanced branch and bound algorithms for strongly nonconvex mixed-integer nonlinear optimization

Large-scale strongly nonlinear and nonconvex mixed-integer nonlinear programming (MINLP) models frequently appear in optimization-based process synthesis, integration, intensification, and process control. However, they are usually difficult to solve by existing algorithms within acceptable time. In this study, we propose two robust homotopy continuation enhanced branch and bound (HCBB) algorithms (denoted as HCBB-FP and HCBB-RB) where the homotopy continuation method is employed to gradually approach the optimum of the NLP subproblem at a node from the solution at its parent node. A variable step length is adapted to effectively balance feasibility and computational efficiency. The computational results from solving four existing process synthesis problems demonstrate that the proposed HCBB algorithms can find the same optimal solution from different initial points, while the existing MINLP algorithms fail or find much worse solutions. In addition, HCBB-RB is superior to HCBB-FP due to much lower computational effort required for the same locally optimal solution.     

Second order velocity slip and thermal jump of Cu–water nanofluid over a cone in the presence of nonlinear radiation and nonuniform heat source/sink using homotopy analysis method

The purpose of the present paper is to explore the second order slip effects on nanofluid flow over a vertical cone. The effects of nonlinear thermal radiation and nonuniform heat source/sink are also taken into account. Water with copper nanoparticles is used as nanofluid in this investigation. The governing partial differential equations for the flow are converted into ordinary differential equations by using transformations and then are solved using homotopy analysis method. The influence of various important parameters on velocity, temperature, skin‐friction, and Nusselt number are presented through graphs. Results indicate that the velocity and magnitude of skin friction decrease with a rise in first and second order velocity slips. A raise in either first or second order temperature jump causes a fall in temperature. Nonlinear radiation increases the more rapidly when compared to the linear radiation case.     

用同伦方法反演流体饱和孔隙介质的参数

从材料响应的理论合成应与实际测量数据相拟合这一出发点，将参数反演问题转化为非线性方程组的零点求解问题，从而应用一种大范围收敛的同伦方法来求解非线性方程组，并把这种方法用于Simon1984年给出的具有解析解的一维双相介质模型的数值模拟。数值模拟实例的结果表明了同伦方法的可行性和稳健性。     

黏弹性介质参数反演的同伦方法

以介质的动态时域位移响应作为参数反演的依据，对黏弹性介质的材料参数进行反分析研究。动态时域位移响应比拟静态和频域位移响应含有更丰富的信息，且在实际工程中所测得的信息也多为动态时域信号，因此将时间全过程的位移信息用于黏弹性介质参数反分析，将使黏弹性介质参数反演的结果更为合理、更接近实际。根据位移响应的理论合成值应与实际测量数据相拟合的原则，将黏弹性介质参数反演问题转化为非线性算予方程的零点求解问题；然后，应用大范围收敛的同伦方法求出非线性算予等于0的根作为反问题的解。以黏弹性半空间介质为例进行数值模拟。模拟结果表明，反演结果不依赖于初始值的选取，且反算过程稳定收敛，说明同伦反演方法对于黏弹性参数的动态反演是适用、有效的。     

A new analytical method for solving systems of Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.     

A homotopy analysis method for the nonlinear partial differential equations arising in engineering

In this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.     

Identification of diffusion parameters in a non-linear convection–diffusion equation using adaptive homotopy perturbation method

This paper is concerned with the numerical identification of diffusion parameters in a non-linear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. In order to overcome the defect of the local convergence of traditional methods, an adaptive homotopy perturbation method is applied to solve this parameter identification inverse problem. The adaptive homotopy perturbation method provides a simple way to adapt computational refinement to the choice of the homotopy parameter. Numerical simulations illustrate that the proposed algorithm is globally convergent and computationallyefficient.     

电力系统潮流多解的同伦算法

电力系统潮流方程大多是亏欠的多元多项式方程组,可利用同伦方法求其全部解.将变元进行分组,找出潮流方程的最小m-Be(z)out数并恰当构造同伦映射,达到全部或部分排除发散路径,同时避免漏解现象.实例验证了该算法的有效性.     

Stagnation Point Flow of a Nanofluid Past a Permeable Cylinder with Chemical Reaction

The primary objective of the present paper is to investigate the novel aspect of nanofluid flow near the stagnation‐point past a permeable cylinder with chemical reaction. The prescribed surface heat and nanoparticle fluxes are also taken into account. The improved homotopy analysis method is introduced to obtain the recursively analytic solutions with high precision. The convergence of the obtained series solution is discussed explicitly. Besides, the effects of physically significant parameters on skin friction coefficient, local Nusselt number, local Sherwood number, as well as profiles of velocity, temperature, and nanoparticle volume fraction are examined and discussed in detail. It is found that the local Sherwood number increases when a chemical reaction occurs in the nanofluid. It is also indicated that the increase of the reaction rate parameter leads to a higher temperature and a lower nanoparticle volume fraction.     

用近似同伦对称法求解阻尼KdV方程

应用近似同伦对称方法对阻尼KdV方程进行约化并求得近似解.为阻尼KdV方程选取一个同伦模型,其近似解可以用级数形式表示;再应用近似对称法,可以将其方程约化为若干个3阶ODE方程.近似同伦对称方法适用于一类强扰动非线性微分方程.