The Tikhonov regularization method in elastoplasticity

The numerical simulation of the mechanical behavior of industrial materials is widely used for viability verification, improvement and optimization of designs. Elastoplastic models have been used to forecast the mechanical behavior of different materials. The numerical solution of most elastoplastic models comes across problems of ill-condition matrices. A complete representation of the nonlinear behavior of such structures involves the nonlinear equilibrium path of the body and handling of singular (limit) points and/or bifurcation points. Several techniques to solve numerical problems associated to these points have been disposed in the specialized literature. Two examples are the load-controlled Newton–Raphson method and displacement controlled techniques. However, most of these methods fail due to convergence problems (ill-conditioning) in the neighborhood of limit points, specially when the structure presents snap-through or snap-back equilibrium paths. This study presents the main ideas and formalities of the Tikhonov regularization method and shows how this method can be used in the analysis of dynamic elastoplasticity problems. The study presents a rigorous mathematical demonstration of existence and uniqueness of the solution of well-posed dynamic elastoplasticity problems. The numerical solution of dynamic elastoplasticity problems using Tikhonov regularization is presented in this paper. The Galerkin method is used in this formulation. Effectiveness of Tikhonov’s approach in the regularization of the solution of elastoplasticity problems is demonstrated by means of some simple numerical examples.     

AN EMBEDDED BOUNDARY METHOD FOR ELLIPTIC AND PARABOLIC PROBLEMS WITH INTERFACES AND APPLICATION TO MULTI-MATERIAL SYSTEMS WITH PHASE TRANSITIONS

The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella （1998, J. Comput. Phys. 147, 60） has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems.     

Computing exact solution to nonlinear integer programming: Convergent Lagrangian and objective level cut method

In this paper, we propose a convergent Lagrangian and objective level cut method for computing exact solution to two classes of nonlinear integer programming problems: separable nonlinear integer programming and polynomial zero-one programming. The method exposes an optimal solution to the convex hull of a revised perturbation function by successively reshaping or re-confining the perturbation function. The objective level cut is used to eliminate the duality gap and thus to guarantee the convergence of the Lagrangian method on a revised domain. Computational results are reported for a variety of nonlinear integer programming problems and demonstrate that the proposed method is promising in solving medium-size nonlinear integer programming problems.     

运筹学中若干离散规划问题的相对差分搜索解法

运筹学中有很多离散规划问题。其中的线性规划通常用分枝定界法或割平面法，还有图上作业法求解。不论哪种方法工作量都不小，而且效率低；至于非线性规划大都是用动态规划法求解，也很麻烦、耗时。对于大规模问题，不论线性或非线性离散规划，现有解法都受到问题规模的限制；还有资源分配和背包问题至今没有见到解决方法。本文就是为了解决这些问题，提出了相对差分搜索算法。通过5个算例和其它文献中的一些算例计算验证了本法简单、快速、有效和精确，尤其不受问题规模的限制是其最大的优点。     

弱有限元方法在线弹性问题中的应用

本文考虑弱有限元（简称WG）方法在线弹性问题中的应用.WG方法是传统有限元方法的推广,用于偏微分方程的数值求解.和传统有限元一样,它的基本思想源于变分原理.WG方法的特点是使用在剖分单元内部和剖分单元边界上分别有定义的分片多项式函数（即弱函数）作为近似函数来逼近真解,并针对弱函数定义相应的弱微分算子代入数值格式进行计算.除此之外,WG方法允许在数值格式中引进稳定子以实现近似函数的弱连续性.WG方法具有允许使用任意多边形或多面体剖分,数值格式与逼近函数构造简单,易于满足相应的稳定性条件等优点.本文考虑WG方法在求解线弹性问题中的应用.围绕线弹性问题数值求解中常见的三个问题,即：数值格式的强制性,闭锁性,应力张量的对称性介绍WG方法在线弹性问题求解中的应用.     

Three-level schemes of the alternating triangular method

In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes.     

A Convergent Variant of the Nelder–Mead Algorithm

The Nelder–Mead algorithm (1965) for unconstrained optimization has been used extensively to solve parameter estimation and other problems. Despite its age, it is still the method of choice for many practitioners in the fields of statistics, engineering, and the physical and medical sciences because it is easy to code and very easy to use. It belongs to a class of methods which do not require derivatives and which are often claimed to be robust for problems with discontinuities or where the function values are noisy. Recently (1998), it has been shown that the method can fail to converge or converge to nonsolutions on certain classes of problems. Only very limited convergence results exist for a restricted class of problems in one or two dimensions. In this paper, a provably convergent variant of the Nelder–Mead simplex method is presented and analyzed. Numerical results are included to show that the modified algorithm is effective in practice.     

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.     

Convergent Lagrangian and domain cut method for nonlinear knapsack problems

The nonlinear knapsack problem, which has been widely studied in the OR literature, is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to separable nondecreasing constraints. In this paper we develop a convergent Lagrangian and domain cut method for solving this kind of problems. The proposed method exploits the special structure of the problem by Lagrangian decomposition and dual search. The domain cut is used to eliminate the duality gap and thus to guarantee the finding of an optimal exact solution to the primal problem. The algorithm is first motivated and developed for singly constrained nonlinear knapsack problems and is then extended to multiply constrained nonlinear knapsack problems. Computational results are presented for a variety of medium- or large-size nonlinear knapsack problems. Comparison results with other existing methods are also reported.     

Diagonal discrete gradient bundle method for derivative free nonsmooth optimization

Typically, practical nonsmooth optimization problems involve functions with hundreds of variables. Moreover, there are many practical problems where the computation of even one subgradient is either a difficult or an impossible task. In such cases, the usual subgradient-based optimization methods cannot be used. However, the derivative free methods are applicable since they do not use explicit computation of subgradients. In this paper, we propose an efficient diagonal discrete gradient bundle method for derivative-free, possibly nonconvex, nonsmooth minimization. The convergence of the proposed method is proved for semismooth functions, which are not necessarily differentiable or convex. The method is implemented using Fortran 95, and the numerical experiments confirm the usability and efficiency of the method especially in case of large-scale problems.     

A new method of fundamental solutions applied to nonhomogeneous elliptic problems

The classical method of fundamental solutions (MFS) has only been used to approximate the solution of homogeneous PDE problems. Coupled with other numerical schemes such as domain integration, dual reciprocity method (with polynomial or radial basis functions interpolation), the MFS can be extended to solve the nonhomogeneous problems. This paper presents an extension of the MFS for the direct approximation of Poisson and nonhomogeneous Helmholtz problems. This can be done by using the fundamental solutions of the associated eigenvalue equations as a basis to approximate the nonhomogeneous term. The particular solution of the PDE can then be evaluated. An advantage of this mesh-free method is that the resolution of both homogeneous and nonhomogeneous equations can be combined in a unified way and it can be used for multiscale problems. Numerical simulations are presented and show the quality of the approximations for several test examples. AMS subject classification 35J25, 65N38, 65R20, 74J20     

Quasilinearization and the methods of finite difference and initial values

Methods for the solution of nonlinear boundary-value problems for ordinary differential equations are discussed and classified as either finite-difference methods or initial-value methods. Within this framework, two algorithms, which are generated using the quasilinearization method, are presented and shown to be representative of these two methods. Consequently, both of the most widely used techniques for the solution of these problems can be formulated within the framework of the quasilinearization method. The computational properties of these algorithms are also discussed.     

Two-Level preconditioned Krylov subspace methods for the solution of three-dimensional heterogeneous Helmholtz problems in seismics

In this paper we address the solution of three-dimensional heterogeneous Helmholtz problems discretized with compact fourth-order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large linear systems of equations. We propose an iterative two-grid method where the coarse grid problem is solved inexactly. A single cycle of this method is used as a variable preconditioner for a flexible Krylov subspace method. Numerical results demonstrate the usefulness of the algorithm on a realistic three-dimensional application. The proposed numerical method allows us to solve wave propagation problems with single or multiple sources even at high frequencies on a reasonable number of cores of a distributed memory cluster.     

Trigonometric Tension B-Spline Method for the Solution of Problems in Calculus of Variations

In this paper, the tension B-spline collocation method is used for finding the solution of boundary value problems which arise from the problems of calculus of variations. The problems are reduced to an explicit system of algebraic equations by this approximation. We apply some numerical examples to illustrate the accuracy and implementation of the method.     

An alternative formulation for the fuzzy assignment problem

The existing assignment problems for assigning n jobs to n individuals are limited to the considerations of cost or profit measured as crisp. However, in many real applications, costs are not deterministic numbers. This paper develops a procedure based on Data Envelopment Analysis method to solve the assignment problems with fuzzy costs or fuzzy profits for each possible assignment. It aims to obtain the points with maximum membership values for the fuzzy parameters while maximizing the profit or minimizing the assignment cost. In this method, a discrete approach is presented to rank the fuzzy numbers first. Then, corresponding to each fuzzy number, we introduce a crisp number using the efficiency concept. A numerical example is used to illustrate the usefulness of this new method.     

A Hybrid Method for Solving the Multi-objective Assignment Problem

Most of the well known methods for solving multi-objective combinatorial optimization problems deal with only two objectives. In this paper, we develop a metaheuristic method for solving multi-objective assignment problems with three or more objectives. This method is based on the dominance cost variant of the multi-objective simulated annealing (DCMOSA) and hybridizes neighborhood search techniques which consist of either a local search or a multi-objective branch and bound search (here the multi-objective branch and bound search is used as a local move to a fragment of a solution).     

Multi-scale B-spline method for 2-D elastic problems

In this paper, quadratic B-spline functions are used for solution of 2-D elastic problems. Because B-spline functions are directly used as basis function, there is no need to use meshes and nodes in function approximation. In order to improve the computational efficiency, different scales are used for sub-domains of entire problem domain in function approximation. The modified variational form and Lagrange multipliers method are used for coupling of different scale in function approximation. Compared with meshless methods and other wavelet based methods, this multi-scale B-spline-based method is simple and easy to work with for numerical analysis. Furthermore, the computational efficiency of the multi-scale method is much higher than that of single scale approach. The numerical examples of 2-D elastic problems indicate that the present method is effective and stable for solving complicated problems.     

A variable-metric method using a nonquadratic model

In this paper, a new variable-metric method based on a rational, rather than a quadratic, model is proposed. A switching algorithm is also introduced which selects either the standard quadratic model or the new rational model, depending on which has the smallest condition number. Several functions are used to test the new method, and it is concluded that it is as efficient as the standard model in general and is superior for problems of high dimensionality. Considerable improvement is also obtained for high-dimensional problems when the switching algorithm is used.     

多变量、多约束连续或离散的非线性规划的一个通用算法

利用目标函数对约束函数关于设计变量的一阶微分或差分之比,给出了一个求解非线性规划的通用算法．不论变量和约束有多少,也不论变量是连续的还是离散的,这一算法都比较有效,尤其对离散非线性规划更有效．该方法是一种搜索法,勿需解任何数学方程,只需要计算函数值以及函数对变量的偏微分或差分值．许多数值例题和运筹学中一些经典问题,如1) 一、二维的背包问题;2) 一、二维资源分配问题;3) 复合系统工作可靠性问题;4) 机器负荷问题等,经用此法求解验证均较传统方法更有效和可靠．该方法的主要优点是:1) 不受问题的规模限制;2) 只要在可行域（集）内存在目标函数和约束函数及其一阶导数或差分的值,肯定可以搜索到最优的解,没有不收敛和不稳定的问题．     

The radiosity equation for solving global heat transfer in industrial furnaces

In this paper, a numerical method used in computer graphics to solve global illumination problems is applied to heat radiation. The radiosity equation is solved by an adaptive finite-element method and linked to heat conduction. The method is validated on an analytic benchmark problem. Due to the speed up compared with commonly used methods for heat radiation, this approach might be a way to solve complex three-dimensional problems in high temperature environments.