A shape and topology optimization technique for solving a class of linear complementarity problems in function space

A shape and topology optimization driven solution technique for a class of linear complementarity problems (LCPs) in function space is considered. The main motivating application is given by obstacle problems. Based on the LCP together with its corresponding interface conditions on the boundary between the coincidence or active set and the inactive set, the original problem is reformulated as a shape optimization problem. The topological sensitivity of the new objective functional is used to estimate the “topology” of the active set. Then, for local correction purposes near the interface, a level set based shape sensitivity technique is employed. A numerical algorithm is devised, and a report on numerical test runs ends the paper.     

Range sets for weak efficiency in multiobjective linear programming and a parametric polytopes intersection problem

ABSTRACT

The aim of this paper is to obtain the range set for a given multiobjective linear programming problem and a weakly efficient solution. The range set is the set of all values of a parameter such that a given weakly efficient solution remains efficient when the objective coefficients vary in a given direction. The problem was originally formulated by Benson in 1985 and left to be solved. We formulate an algorithm for determining the range set, based on some hard optimization problems. Due to toughness of these optimization problems, we propose also lower and upper bound approximation techniques. In the second part, we focus on topological properties of the range set. In particular, we prove that a range set is formed by a finite union of intervals and we propose upper bounds on the number of intervals. Our approach to tackle the range set problem is via the intersection problem of parametric polytopes. Thus, our results have much wider area of applicability since the intersection (and separability) problem of convex polyhedra is important in many fields of optimization.     

Modelling of topological derivatives for contact problems

The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization. Partially supported by the grant 4 T11A 01524 of the State Committee for the Scientific Research of the Republic of Poland     

具有动应力和动位移约束的离散变量结构拓扑优化设计方法

讨论了动应力、动位移约束下离散变量结构拓扑优化设计问题.首先给出问题的数学模型,然后用拟静力算法,将结构惯性力极值作为静载荷施加到结构上,求得结构的动位移和动内力,将考虑动应力约束和动位移约束的离散变量结构拓扑设计问题化为静应力和静位移约束的优化问题,然后利用两类变量统一考虑的离散变量结构拓扑优化设计的综合算法进行求解.     

Shape and topology optimization of contact problems by level set methods

This paper deals with the numerical solution of a topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is considered. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. In the paper shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization. Level set based numerical algorithm for the solution of the shape optimization problem is proposed. Level set method is used to describe the position of the boundary of the body and its evolution on a fixed mesh. This evolution is governed by Hamilton – Jacobi equation. The speed vector field driving the propagation of the boundary of the body is given by the shape derivative of a cost functional with respect to the free boundary. Numerical examples are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Algorithms for a multi-level network optimization problem

In this paper we work on a multi-level network optimization problem that integrates into the same model important aspects of: (i) discrete facility location, (ii) topological network design, and (iii) network dimensioning. Potential applications for the model are discussed, stressing its growing importance. The multi-level network optimization problem treated is defined and a mathematical programming formulation is presented. We make use of a branch-and-bound algorithm based on Lagrangean relaxation lower bounds to introduce some new powerful auxiliary algorithms to exactly solve the problem. We conduct a set of computational experiments that indicate the quality of the proposed approach.     

集值优化问题超有效解的Kuhn-Tucker最优性条件

在Hausdorff局部凸拓扑线性空间中考虑约束向量集值优化问题(VP)的超有效性．在近似锥-次类凸假设下,利用择一性定理得到了Kuhn-Tucker型最优性必要条件,利用标量化定理得到了Kuhn-Tucker型最优性充分条件．最后给出了一种与(VP)等价的无约束优化．     

Total unimodularity and the Euler-subgraph problem

We give a short treatment (including proofs) of all known characterizations by way of forbidden structures of totally unimodular matrices, i.e. matrices whose square minors have determinants of 0, +1, or −1 only. A generalization of a known topological characterization gives rise to a new combinatorial optimization problem which is introduced here and called the Euler-subgraph problem.     

应力和位移约束下连续体结构拓扑优化

同时考滤应力和位移约束的连续体结构拓扑优化问题,很难用现有的均匀方法或变密度方法等求解。主要困难在于难以建立应力和位移约束与拓扑设计变量间显式关系式;即使建立了这种关系,也由于优化问题规模过大,利用常规的数学规划方法难以求解。隋允康、杨德庆曾提出了基于独立连续拓扑变量及映射变换（ＩＣＭ）的桁架结构拓扑优化模型。本文在此基础上,建立了以重量为目标,考虑应力和位移约束的连续体结构拓扑优化模型,并推导出     

A level-set topological optimization method to analyze two-dimensional thermal problem using BEM

A level-set based topological optimization approach is proposed using boundary element method (BEM) to solve two-dimensional(2D) thermal problems. The objective function is considered as a function of temperature and thermal flux defined on boundaries with Dirichlet and Neumann boundary conditions. The topological sensitivity is derived combining BEM under the assumption of insulating topological boundaries generated during optimization. Smooth boundaries represented by the level-set function is updated using topological sensitivity with a regularization term. Numerical examples with different objective functions considering the real-world problems are presented to show the effectiveness of the proposed approach. The topological sensitivity, computational time and boundary smoothness are verified by comparing with finite difference method (FDM).     

A new reconstruction method for a parabolic inverse source problem

In this paper, we focus on the detection of the shape and location of a discontinuous source term from the knowledge of boundary measurements. We propose a non-iterative reconstruction algorithm based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from an energy-like cost function. The unknown shape of the term source support is reconstructed using a level-set curve of the topological gradient. The efficiency of our algorithm is illustrated by some numerical simulations.     

Keplerian Action,Convexity Optimization,and the 4-Body Problem

In this paper we introduce a method to construct periodic solutions for the n-body problem with only boundary and topological constraints. Our approach is based on some novel features of the Keplerian action functional, constraint convex optimization techniques, and variational methods. We demonstrate the strength of this method by constructing relative periodic solutions for the planar four-body problem within a special topological class, and our results hold for an open set of masses.     

A property of efficient and ε-efficient solutions in vector optimization

We present a property that is a characterization of the solution to a scalar optimization problem. This property is also considered in vector optimization, and two sufficient conditions are provided: one is in connection with strict efficiency and the other takes into account topological characteristics of the problem.     

近似锥-次类凸集值优化的严有效性

在Hausdorff局部凸拓扑线性空间中考虑约束集值优化问题(VP)的严有效性.在近似锥-次类凸假设下,利用凸集分离定理,分别得到了Kuhn-Tucker型和Lagrange型最优性条件,建立了与(VP)等价的两种形式的无约束优化.     

广义凸拓扑线性空间集值优化的ε-强有效解

在Hausdorff局部凸拓扑线性空间中考虑约束集值优化问题(VP)的ε-强有效性.在内部锥类凸假设下,利用凸集分离定理,分别建立了关于ε-强有效解的标量化定理和ε-Lagrange乘子定理.     

On an equivalence of topological vector space valued cone metric spaces and metric spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces.     

On super efficiency in set-valued optimization

The set-valued optimization problem with constraints is considered in the sense of super efficiency in locally convex linear topological spaces. Under the assumption of iccone-convexlikeness, by applying the seperation theorem, Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions, the necessary conditions are obtained for the set-valued optimization problem to attain its super efficient solutions. Also, the sufficient conditions for Kuhn-Tucker＇s, Lagrange＇s and saddle points optimality conditions are derived.     

On equivalent stability properties in semi-infinite optimization

This paper studies noncompact feasible sets of a semi-infinite optimization problem which are defined by finitely many equality constraints and infinitely many inequality constraints. The main result is the equivalence of the overall validity of the Extended Mangasarian Fromovitz Constraint Qualification with certain (topological) stability conditions. Furthermore, two perturbation theorems being of independent interest are presented.This work was supported by the Deutsche Forschungsgemeinschaft under grant Gu 304/1-2.     

Fast and accurate algorithm for cavities reconstruction in an elasticity problem

The aim of this work is to reconstruct the location and geometry of a cavity embedded in a linear isotropic material Ω via an exterior boundary measurement of the displacement field. The considered problem is governed by the linear elasticity system. This inverse problem of geometry reconstruction (ie, location and shape) is formulated as a topology optimization one and solved by minimizing a Kohn‐Vogelius type functional with the help of the topological sensitivity method. Some numerical results are presented using a noniterative geometric algorithm.