New exact penalty function for solving constrained finite min-max problems

This paper introduces a new exact and smooth penalty function to tackleconstrained min-max problems.By using this new penalty function and adding justone extra variable,a constrained min-max problem is transformed into an unconstrainedoptimization one.It is proved that,under certain reasonable assumptions and when thepenalty parameter is sufficiently large,the minimizer of this unconstrained optimizationproblem is equivalent to the minimizer of the original constrained one.Numerical resultsdemonstrate that this penalty function method is an effective and promising approach forsolving constrained finite min-max problems.     

New simple exact penalty function for constrained minimization

By adding one variable to the equality-or inequality-constrained minimization problems, a new simple penalty function is proposed. It is proved to be exact in the sense that under mild assumptions, the local minimizers of this penalty function are precisely the local minimizers of the original problem, when the penalty parameter is sufficiently large.     

APPLICATION OF PENALTY FUNCTION METHOD IN ISOPARANIETRIC HYBRID FINITE ELEMENT ANALYSIS

IntroductionSince T.H.H.Pain firstly puts forward hybrid element method[1]in1964, the researchand application of hybrid element have got great development. T.H.H.Painet al.havemade important pioneer works in the domain of the research on incompatible displacementstructure hybrid elementstress pattern[2,3]in recentdecades. Reference [4] had provided theoptimizing design concept of hybrid element further and established standardization methodof incompatible displacement structure hybrid eleme…     

EXACT AUGMENTED LAGRANGIAN FUNCTION FOR NONLINEAR PROGRAMMING PROBLEMS WITH INEQUALITY CONSTRAINTS

An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, from the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.     

A boundary element method for steady viscous fluid flow using penalty function formulation

A new boundary element method is presented for steady incompressible flow at moderate and high Reynolds numbers. The whole domain is discretized into a number of eight-noded cells, for each of which the governing boundary integral equation is formulated exclusively in terms of velocities and tractions. The kernels used in this paper are the fundamental solutions of the linearized Navier–Stokes equations with artificial compressibility. Significant attention is given to the numerical evaluation of the integrals over quadratic boundary elements as well as over quadratic quadrilateral volume cells in order to ensure a high accuracy level at high Reynolds numbers. As an illustration, square driven cavity flows are considered for Reynolds numbers up to 1000. Numerical results demonstrate both the high convergence rate, even when using simple (direct) iterations, and the appropriate level of accuracy of the proposed method. Although the method yields a high level of accuracy in the primary vortex region, the secondary vortices are not properly resolved. © 1997 John Wiley & Sons, Ltd.     

AN EXACT ELEMENT METHOD FOR THE BENDING OF NONHOMOGENEOUS REISSNER’S PLATE

In this paper,based on the step reduction method and exact analytic method,a new method,the exact element method for constructing finite element,is presented.Since the new method doesn’t need variational principle,it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficients.By this method,a triangle noncompatible element with15 degrees of freedom is derived to solve the bending of nonhomogenous Reissner’s plate.Because the displacement parameters at the nodal point only contain deflection and rotation angle.it is convenient to deal with arbitrary boundary conditions.In this paper,the convergence of displacement and stress resultants is proved.The element obtained by the present method can be used for thin and thick plates as well,Four numerical examples are given at the end of this paper,which indicates that we can obtain satisfactory results and have higher numerical precision.     

纯幂律全塑性罚函数有限元

从非线性连续介质力学出发导出一种新的形式较简单的纯幂律体积不可压缩一构方程,形成纯幂律罚函数有限元。提出迭代解分析方法论羰应变条件下罚函数方法有效性。再结合计算结果讨论不同条件下罚因子对应在力应变场的影响并对迭代特性进行分析。结果发现纯幂律全塑性罚函数有限元存在;１）迭代收敛速度随害虫律指数增加而减慢,并且εｅ〉ε０时减幅很大;εｅ〈ε０时不明显。２）应力应变的计算精度随幂律指数变化而变化,当ε〉     

A high convergent precision exact analytic method for differential equation with variable coefficients

The exact analytic method was given by[1].It can be used for arbitrary variable coefficient differential equations and the solution obtained can have the second order convergent precision.In this paper,a new high precision algorithm is given based on[1],through a bending problem of variable cross-section beams.It can have the fourth convergent precision without increasing computation work.The present computation method is not only simple but also fast.The numerical examples are given at the end of this paper which indicate that the high convergent precision can be obtained using only a few elements.The correctness of the theory in this paper is confirmed.     

An exact element method for bending of nonhomogeneous thin plates

In this paper, based on the step reduction method, a new method, the exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a triangle noncompatible element with 6 degrees of freedom is derived to solve the bending of nonhomogeneous plate. The convergence of displacements and stress resultants which have satisfactory numerical precision is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.     

区域无线电定位系统基站的最优几何布局

地面基站的几何布局方式是影响区域无线电定位系统定位精度的重要因素,水平几何精度递变因子（HDOP）是衡量地面基站几何布局对系统定位精度影响的基本指标。为求解区域无线电定位系统基站的最优几何布局,给出了区域无线电定位系统HDOP的计算方法,建立了求解区域无线电定位系统地面基站最优几何布局的数学模型;提出了应用惩罚函数对数学模型进行求解的方法,将约束优化问题转化为无约束优化问题;并针对求解过程中目标函数难于求导的问题,提出了采用模拟退火算法进行全局寻优,使求得的结果逼近全局最优解。仿真结果表明,这种结合惩罚函数法和模拟退火算法进行全局寻优的方法,对于区域无线电定位系统求解地面基站的最优布局方案具有工程实用价值。     

A novel method of solving the exact equations for capillary-gravity waves

A new method is presented for the computation of two-dimensional periodicprogressive surface waves propagating under the combined influence of gravity and surfacetension.The nonlinear surface is expressed by Fourier series with finite number of terms,after the computational domain is transformed into a unit circle.The dynamic boundaryequation is used in its exact nonlinear form and the coefficients of Fourier series are foundby the Nweton-Raphson method successively.This is a neat method,Yielding highprescision with little computational effort.     

A method for finding the invariants and exact solutions of coupled non-linear differential equations with applications to dynamical systems

The polynomial invariants of (a set) non-linear differential equations are found by using a direct approach. The integrability of these invariants deserves the integrability of the given set of coupled differential equations. As applications, the Lorenz and Rikitake sets, among others, are studied. New invariants are obtained.     

轴对称热冲击SIF过渡过程分析的热权函数法

热权函数法利用温度与热权函数的乘积的积分来直接计算热冲击过程中裂纹尖端的应力强度因子过渡过程。热权函数与时间τ无关,由于免除了对每一时刻τ所需作的有限元或边界元分析,计算过程大大简化,计算效率得到极大提高,本文将热权函数与有限元法直接耦合,给出了基于刚度阵导数法的轴对称问题的热权函数计算格式,实例计算表明,本文给出的热权函数计算格式具有满意的计算精度。     

SUPG finite element method based on penalty function for lid-driven cavity flow up to $$Re = 27500$$

A streamline upwind/Petrov–Galerkin(SUPG)finite element method based on a penalty function is proposed for steady incompressible Navier–Stokes equations.The SUPG stabilization technique is employed for the formulation of momentum equations. Using the penalty function method, the continuity equation is simplified and the pressure of the momentum equations is eliminated. The lid-driven cavity flow problem is solved using the present model. It is shown that steady flow simulations are computable up to Re = 27500, and the present results agree well with previous solutions. Tabulated results for the properties of the primary vortex are also provided for benchmarking purposes.     

连续体结构拓扑优化的一种改进变密度法及其应用

针对连续体结构拓扑优化设计变密度方法SIMP和RAMP,因惩罚函数选取的不合理而导致拓扑结构形式不甚合理的问题,本文提出了一种新的惩罚函数,并基于此函数导出了相应的迭代设计公式,几个典型考题的数值结果,说明了方法的可行性和有效性.     

可靠性优化问题中遗传算法适应值函数的建立

应用遗传算法进行优化,约束的处理成为建立适应值函数和算法进行的关键.可靠性优化是以系统可拿性指标作为优化问题的约束条件.首先结合外罚函数法建立数学模型,处理约束的惩罚因子时根据种群情况白适应取值,构造适应值函数的映射公式.随后采用拉格朗日秉子法建立了新的约束与目标函数向适应值函数的映射公式,该公式可以避免因罚函数病态所导致的搜索终止,收敛更加快速,使遗传算法得以成功应用于可靠性优化问题中.分析计算结果表明乘子法具有更好地收敛效果,两个公式构造合理.     

Application of subregion function method for solving beam-board structure

In this paper,the solution to the structure consisting of a bead and a board is given as aresult of the application of the subregion function method which was suggested in ref.[1].The same problem is also computed with finite element method.The comparison betweenthe two results shows that the application of the subregion function in the method of weightedresiduals is practical and effective,especially for solving compound structures.     

三维数值流形方法的点-面接触模型

建立了三维流形元的点-面接触模型,给出了嵌入准则并应用罚方法强迫接触界面满足不嵌入和无拉伸条件约束求解,详细推导了法向弹簧、切向弹簧和摩擦力子矩阵并给出了接触状态(张开、滑动和固定)的模式变化准则,算例表明该模型的有效性。     

阶跃函数高精度逼近的结构拓扑优化方法

为了提高ICM(Independent Continuous and Mapping,即独立、连续及映射)方法求解结构拓扑优化问题的效率,本文改进了阶跃函数及其反函数的近似逼近函数——磨光函数和过滤函数。首先,分别对ICM方法的磨光函数和过滤函数按其近似性质进行了分类,分别提出了左磨函数及上磨函数和快滤函数、慢滤函数诸概念。然后得到了区分左磨函数和上磨函数、快滤函数和慢滤函数的两个判别定理;并得到了上磨函数、快滤函数、左磨函数及慢滤函数的对应定理。进而给出了磨光函数和过滤函数的使用准则及构造方法。采用高精度逼近阶跃函数的指数类函数做左磨函数,建立近似程度更高的结构拓扑优化模型。上述策略带来了模型非线性程度的提高,增加了求解难度。为此,针对该模型给出了精确对偶映射下的序列二次近似解法。最后,以位移约束下结构重量最轻化问题为例,叙述了相应的算法。与以往采用幂函数做磨光函数时算例结果的比较表明,该模型的提法合理,算法更加有效。由于提高了对阶跃函数及其反函数的逼近程度,从而显著减少了优化迭代的次数。     

Weak imposition of boundary conditions for the Navier–Stokes equations by a penalty method

We prove convergence of the finite element method for the Navier–Stokes equations in which the no‐slip condition and no‐penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier–Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd.