On sequential approximate simultaneous analysis and design in classical topology optimization

We study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem. The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced. Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.     

On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem

We study the ‘classical’ topology optimization problem, in which minimum compliance is sought, subject to linear constraints. Using a dual statement, we propose two separable and strictly convex subproblems for use in sequential approximate optimization (SAO) algorithms. Respectively, the subproblems use reciprocal and exponential intermediate variables in approximating the non‐linear compliance objective function. Any number of linear constraints (or linearly approximated constraints) are provided for. The relationships between the primal variables and the dual variables are found in analytical form. For the special case when only a single linear constraint on volume is present, we note that application of the ever‐popular optimality criterion (OC) method to the topology optimization problem, combined with arbitrary values for the heuristic numerical damping factor η proposed by Bendsøe, results in an updating scheme for the design variables that is identical to the application of a rudimentary dual SAO algorithm, in which the subproblems are based on exponential intermediate variables. What is more, we show that the popular choice for the damping factor η=0.5 is identical to the use of SAO with reciprocal intervening variables. Finally, computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy, when compared to SAO subproblems based on reciprocal intermediate variables (and hence, the heuristic topology OC method hitherto used). This is attributed to the fact that a different exponent is computed for each design variable in the two‐point exponential approximation we have used, using gradient information at the previously visited point. Copyright © 2007 John Wiley & Sons, Ltd.     

A quadratic approximation for structural topology optimization

In topology optimization, it is customary to use reciprocal‐like approximations, which result in monotonically decreasing approximate objective functions. In this paper, we demonstrate that efficient quadratic approximations for topology optimization can also be derived, if the approximate Hessian terms are chosen with care. To demonstrate this, we construct a dual SAO algorithm for topology optimization based on a strictly convex, diagonal quadratic approximation to the objective function. Although the approximation is purely quadratic, it does contain essential elements of reciprocal‐like approximations: for self‐adjoint problems, our approximation is identical to the quadratic or second‐order Taylor series approximation to the exponential approximation. We present both a single‐point and a two‐point variant of the new quadratic approximation. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical treatment of hemivariational inequalities in mechanics: two methods based on the solution of convex subproblems

Hemivariational inequality problems describe equilibrium points (solutions) for structural systems in mechanics where nonmonotone, possibly multivalued laws or boundary conditions are involved. In the case of problems which admit a potential function this is a nonconvex, nondifferentiable one. In order to avoid the difficulties that arise during the calculation of equilibria for such mechanical systems, methods based on sequential convex approximations have recently been proposed and tested by the authors. The first method is based on ideas developed in the fields of quasidifferential and difference convex (d.c.) optimization and transforms the hemivariational inequality problem into a system of convex variational inequalities, which in turn leads to a multilevel (two-field) approximation technique for the numerical solution. The second method transforms the problem into a sequence of variational inequalities which approximates the nonmonotone problem by an iteratively defined sequence of monotone ones. Both methods lead to convex analysis subproblems and allow for treatment of large-scale nonconvex structural analysis applications.The two methods are compared in this paper with respect to both their theoretical assumptions and implications and their numerical implementation. The comparison is extended to a number of numerical examples which have been solved by both methods.     

Structural optimization of uncertain dynamical systems considering mixed-design variables

In this paper attention is directed to the reliability-based optimization of uncertain structural systems under stochastic excitation involving discrete-continuous sizing type of design variables. The reliability-based optimization problem is formulated as the minimization of an objective function subject to multiple reliability constraints. The probability that design conditions are satisfied within a given time interval is used as a measure of system reliability. The problem is solved by a sequential approximate optimization strategy cast into the framework of conservative convex and separable approximations. To this end, the objective function and the reliability constraints are approximated by using a hybrid form of linear, reciprocal and quadratic approximations. The approximations are combined with an effective sensitivity analysis of the reliability constraints in order to generate explicit expressions of the constraints in terms of the design variables. The explicit approximate sub-optimization problems are solved by an appropriate discrete optimization technique. The optimization scheme exhibits monotonic convergence properties. Two numerical examples showing the effectiveness of the approach reported herein are presented.     

Structural optimization: A new dual method using mixed variables

A new and powerful mathematical programming method is described, which is capable of solving a broad class of structural optimization problems. The method employs mixed direct/reciprocal design variables in order to get conservative, first-order approximations to the objective function and to the constraints. By this approach the primary optimization problem is replaced with a sequence of explicit subproblems. Each subproblem being convex and separable, it can be efficiently solved by using a dual formulation. An attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs. Examples of application to real-life aerospace structures are offered to demonstrate the power and generality of the approach presented.     

Dual methods for discrete structural optimization problems

The purpose of this paper is to present a mathematical programming method developed to solve structural optimization problems involving discrete variables. We work in the following context: the structural responses are computed by the finite elements method and convex and separable approximation schemes are used to generate a sequence of explicit approximate subproblems.Each of them is solved in the dual space with a subgradient‐based algorithm (or with a variant of it) specially developed to maximize the not everywhere differentiable dual function. To show that the application field is large, the presented applications are issued from different domains of structural design, such as sizing of thin‐walled structures, geometrical configuration of trusses, topology optimization of membrane or 3‐D structures and welding points numbering in car bodies. The main drawback of using the dual approach is that the obtained solution is generally not the global optimum. This is linked to the presence of a duality gap, due to the non‐convexity of the primal discrete subproblems. Fortunately, this gap can be quantified: a maximum bound on its value can be computed. Moreover, it turns out that the duality gap is decreasing for higher number of variables; the maximum bound on the duality gap is generally negligible in the treated applications. The developed algorithms are very efficient for 2‐D and 3‐D topology optimization, where applications involving thousands of binary design variables are solved in a very short time. Copyright © 2000 John Wiley & Sons, Ltd.     

一类非凸集值优化问题的对偶定理

对偶理论是数学规划研究领域的重点问题之一,通对偶模型可以实现一个最小化问题与一个最大化问题之间的相互转化.本文的目的是建立一类非凸约束集值优化问题的对偶理论,在逼近多值函数定义的不变凸性假设下,研究了原集值优化问题的Mond-Weir型和Wolfe型对偶问题.利用分析的方法,本文得到了两种对偶模型下关于弱极小元的弱对偶定理,强对偶定理和逆对偶定理.这些对偶定理揭示了原问题与所讨论的Mond-Weir型和Wolfe型对偶问题之间存在着明确的对偶关系.本文所得结果丰富和深化了集值优化理论及其应用的研究内容.     

Optimal Operation of Integrated Heat and Electricity Systems: A Tightening McCormick Approach

Combined heat and electricity operation with variable mass flow rates promotes flexibility, economy, and sustainability through synergies between electric power systems (EPSs) and district heating systems (DHSs). Such combined operation presents a highly nonlinear and nonconvex optimization problem, mainly due to the bilinear terms in the heat flow model—that is, the product of the mass flow rate and the nodal temperature. Existing methods, such as nonlinear optimization, generalized Benders decomposition, and convex relaxation, still present challenges in achieving a satisfactory performance in terms of solution quality and computational efficiency. To resolve this problem, we herein first reformulate the district heating network model through an equivalent transformation and variable substitution. The reformulated model has only one set of nonconvex constraints with reduced bilinear terms, and the remaining constraints are linear. Such a reformulation not only ensures optimality, but also accelerates the solving process. To relax the remaining bilinear constraints, we then apply McCormick envelopes and obtain an objective lower bound of the reformulated model. To improve the quality of the McCormick relaxation, we employ a piecewise McCormick technique that partitions the domain of one of the variables of the bilinear terms into several disjoint regions in order to derive strengthened lower and upper bounds of the partitioned variables. We propose a heuristic tightening method to further constrict the strengthened bounds derived from the piecewise McCormick technique and recover a nearby feasible solution. Case studies show that, compared with the interior point method and the method implemented in a global bilinear solver, the proposed tightening McCormick method quickly solves the heat–electricity operation problem with an acceptable feasibility check and optimality.     

An efficient algorithm for the optimum design of trusses with discrete variables

A method to efficiently solve the problem of minimum weight design of plane and space trusses with discrete or mixed variables is developed. The method can also be applied to continuous variables. The original formulation leads to a non-linear constrained minimization problem with inequality constraints, which is solved by means of a sequence of approximate problems using dual techniques. In the dual space, the objective function is to be maximized, depends on continuous variables, is concave and has first and second order discontinuities. In addition, the constraints deal simply with restricting the dual variables to be non-negative. To solve the problem an ad hoc algorithm from mathematical programming has been adapted. Some examples have been developed to show the effectiveness of the method.     

D.C.集(凸集的差)约束的非凸二次规划的最优解集

本文研究D.C.集(凸集的差)上极小化非凸二次规划问题的最优解。我们首先证明了该问题的Lagrange对偶的稳定性,即不存在对偶间隙;接着利用该性质得到问题的全局最优性条件和最优解集,它可以像凸规划那样,借助它的对偶问题的解集精确地描述出来。最后,通过一个例子来说明这些结论。     

A limited-memory, quasi-Newton preconditioner for nonnegatively constrained image reconstruction

Image reconstruction gives rise to some challenging large-scale constrained optimization problems. We consider a convex minimization problem with nonnegativity constraints that arises in astronomical imaging. To solve this problem, we use an efficient hybrid gradient projection-reduced Newton (active-set) method. By "reduced Newton," we mean that we take Newton steps only in the inactive variables. Owing to the large size of our problem, we compute approximate reduced Newton steps by using the conjugate gradient (CG) iteration. We introduce a limited-memory, quasi-Newton preconditioner that speeds up CG convergence. A numerical comparison is presented that demonstrates the effectiveness of this preconditioner.     

An engineering method for complex structural optimization involving both size and topology design variables

This work presents an engineering method for optimizing structures made of bars, beams, plates, or a combination of those components. Corresponding problems involve both continuous (size) and discrete (topology) variables. Using a branched multipoint approximate function, which involves such mixed variables, a series of sequential approximate problems are constructed to make the primal problem explicit. To solve the approximate problems, genetic algorithm (GA) is utilized to optimize discrete variables, and when calculating individual fitness values in GA, a second-level approximate problem only involving retained continuous variables is built to optimize continuous variables. The solution to the second-level approximate problem can be easily obtained with dual methods. Structural analyses are only needed before improving the branched approximate functions in the iteration cycles. The method aims at optimal design of discrete structures consisting of bars, beams, plates, or other components. Numerical examples are given to illustrate its effectiveness, including frame topology optimization, layout optimization of stiffeners modeled with beams or shells, concurrent layout optimization of beam and shell components, and an application in a microsatellite structure. Optimization results show that the number of structural analyses is dramatically decreased when compared with pure GA while even comparable to pure sizing optimization.     

STRUCTURAL OPTIMIZATION USING A NEW LOCAL APPROXIMATION METHOD

A new method for solving structural optimization problems using a local function approximation algorithm is proposed. This new algorithm, called the Generalized Convex Approximation (GCA), uses the design sensitivity information from the current and previous design points to generate a sequence of convex, separable subproblems. The paper contains the derivation of the parameters associated with the approximation and the formulation of the approximated problem. Numerical results from standard test problems solved using this method are presented. It is observed that this algorithm generates local approximations which lead to faster convergence for structural optimization problems.     

Convex multilevel decomposition algorithms for non-monotone problems

A convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non-monotone, possibly multivalued laws. The theory is developed here for a model structure with non-monotone interface or boundary conditions. First the non-monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non-linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non-convex min-min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,¹ is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non-monotone stick-slip frictional effects illustrate the theory.     

A Combined Convex Approximation—Interior Point Approach for Large Scale Nonlinear Programming

The method of moving asymptotes (MMA) and its globally convergent extension SCP (sequential convex programming) are known to work well in the context of structural optimization. The two main reasons are that the approximation scheme used for the objective function and the constraints fits very well to these applications and that at an iteration point a local optimization model is used such that additional expensive function and gradient evaluations of the original problem are avoided. The subproblems that occur in both methods are special nonlinear convex programs and have traditionally been solved using a dual approach. This is now replaced by an interior point approach. The latter one is more suitable for large problems because sparsity properties of the original problem can be preserved and the separability property of the approximation functions is exploited. The effectiveness of the new method is demonstrated by a few examples dealing with problems of structural optimization.     

TOPOLOGY OPTIMIZATION OF SHEETS IN CONTACT BY A SUBGRADIENT METHOD

We consider the solution of finite element discretized optimum sheet problems by an iterative algorithm. The problem is that of maximizing the stiffness of a sheet subject to constraints on the admissible designs and unilateral contact conditions on the displacements. The model allows for zero design volumes, and thus constitutes a true topology optimization problem. We propose and evaluate a subgradient optimization algorithm for a reformulation into a non-differentiable, convex minimization problem in the displacement variables. The convergence of the method and its low computational complexity are established. An optimal design is derived through a simple averaging scheme which combines the solutions to the linear design problems solved within the subgradient method. To illustrate the efficiency of the algorithm and investigate the properties of the optimal designs, thealgorithm is numerically tested on some medium- and large-scale problems. © 1997 by John Wiley & Sons, Ltd.     

Magnetic resonance tissue quantification using optimal bSSFP pulse-sequence design

We propose a merit function for the expected contrast to noise ratio in tissue quantifications, and formulate a nonlinear, nonconvex semidefinite optimization problem to select locally-optimal balanced steady-state free precession (bSSFP) pulse-sequence design variables. The method could be applied to other pulse sequence types, arbitrary numbers of tissues, and numbers of images. To solve the problem we use a mixture of a grid search to get good starting points, and a sequential, semidefinite, trust-region method, where the subproblems contain only linear and semidefinite constraints. We give the results of numerical experiments for the case of three tissues and three, four or six images, in which we observe a better increase in contrast to noise than would be obtained by averaging the results of repeated experiments. As an illustration, we show how the pulse sequences designed numerically could be applied to the problem of quantifying intraluminal lipid deposits in the carotid artery.     

Multiresponse Optimization and Pareto Frontiers

Methods that can capture evenly distributed solutions along the Pareto frontier are useful for multiresponse optimization problems because they provide a large variety of alternative solutions to the decision maker from among a set of nondominated solutions. However, methods often used for optimizing dual and multiple dual response problems have been rarely evaluated in terms of their ability to capture those solutions. This article provides this information by evaluating a global criterion–based method and the popular weighted mean square error method. Convex and nonconvex response surfaces were considered, and results of the methods were compared with those of a lexicographic approach on the basis of two examples from the literature. Regarding the results, it is shown that the user can be successful in capturing Pareto solutions in convex and nonconvex regions using the global criterion–based method. Moreover, it is shown that the starting point affects the distribution of solutions along the Pareto frontier but is not pivotal to obtain a complete representation of the Pareto frontier. For this purpose, it is necessary to decrease the weight increment and to compute for more solutions. Copyright © 2011 John Wiley & Sons, Ltd.