An efficient optimality criteria approach to the minimum weight design of elastic structures

This paper is concerned with the optimality criteria approach to the minimum weight design of elastic structures analyzed by finite elements. It is first shown that the classical methods apply the lagrangian multiplier technique to an explicit problem. This one results from high quality, first order approximations of the displacement constraints and cruder, zero order approximations of the stress constraints. A generalized optimality criterion is then proposed as the explicit Kuhn-Tucker conditions of a first order approximate problem. Hence a hybrid optimality criterion is developed by using both zero and first order approximations of the stress constraints, according to their criticality. Efficient solution algorithms of the explicit approximate problem are suggested. Its dual statement generalizes the classical lagrangian approaches. Its primal statement leads to a rigorous definition of the optimality criteria approach, which appears to be closely related to the linearization methods of mathematical programming. Finally some numerical applications clearly illustrate the efficiency of the generalized and hybrid optimality criteria.     

Optimality criteria and mathematical programming in structural weight optimization

It is shown that the optimality criteria approach to the structural weight minimization results from a proper linearization of the displacement constraints but not of the stress constraints in terms of the reciprocal design variables. On the basis of this interpretation, two new ideas are suggested. First, a “mixed method” is proposed, that can be regarded either as a pure mathematical programming or as an optimality criterion approach. It allows for a convergence control of the optimization process. Secondly, a proper linearization of the stress constraints is introduced by considering the stress components as linear combinations of the generalized displacements. The numerical applications presented in the paper show that both modifications of the optimization scheme lead to a significant improvement in the convergence properties.     

Allocating fragments in distributed databases

For a distributed database system to function efficiently, the fragments of the database need to be located, judiciously at various sites across the relevant communications network. The problem of allocating these fragments to the most appropriate sites is a difficult one to solve, however, with most approaches available relying on heuristic techniques. Optimal approaches are usually based on mathematical programming, and formulations available for this problem are based on the linearization of nonlinear binary integer programs and have been observed to be ineffective except on very small problems. This paper presents new integer programming formulations for the nonredundant version of the fragment allocation problem. This formulation is extended to address problems which have both storage and processing capacity constraints; the approach is observed to be particularly effective in the presence of capacity restrictions. Extensive computational tests conducted over a variety of parameter values indicate that the reformulations are very effective even on relatively large problems, thereby reducing the need for heuristic approaches.     

Active control of flexible systems by a mathematical programming approach

A class of control problems for a damped distributed parameter system governed by a system of partial differential equations with side constraints (equality and/or inequality) is considered. The proposed approach approximates each control force of the system by a Fourier-type series. In contrast to standard linear optimal control approaches, the method used here is based on the mathematical programming approach, in which the necessary condition of optimality is derived as a system of linear algebraic equations. The proposed approach is easy to apply to a large class of control problems. A vibrating beam excited by an initial disturbance is studied numerically in which the effectiveness of the control and the amount of force spent in the process are investigated in relation to the reduction to the dynamic response.     

Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods

Structural optimization with frequency constraints is highly nonlinear dynamic optimization problems. Genetic algorithm (GA) has greater advantage in global optimization for nonlinear problem than optimality criteria and mathematical programming methods, but it needs more computational time and numerous eigenvalue reanalysis. To speed up the design process, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented. This reanalysis technique is derived primarily on the Kirsch’s combined approximations method, which is also highly accurate for case of repeated eigenvalues problem. The required number of basis vectors at every generation is adaptively determined and the rules for selecting initial number of basis vectors are given. Numerical examples of truss design are presented to validate the reanalysis-based frequency optimization. The results demonstrate that the adaptive eigenvalue reanalysis affects very slightly the accuracy of the optimal solutions and significantly reduces the computational time involved in the design process of large-scale structures.     

A note on truss design for stress and displacement constraints by optimality criteria methods

One of the difficulties with the optimization of large structural systems by optimality criteria (OC) methods, in which stress constraints are stated in terms of relative displacements, is the need to evaluate a large number of Lagrangians. An improved OC method, which does not require Lagrangians for stress constraints, is outlined in this note in the context of problems with stress constraints and a single displacement constraint. Whilst general formulae giving continuum-type optimality criteria for elastic systems are available (e.g. Rozvany 1989), the criteria for trusses are confirmed here by a simple derivation and verified on a ten-bar truss, for which the proposed technique and a dual programming method are shown to give a twelve-digit agreement.     

Multilevel optimization of laminated composite structures

A two-stage optimization method aiming at the optimal design of shells and plates made of laminated composites has been developed. It is based on a mixture of sensitivity analysis, optimality criteria and mathematical programming techniques. The design variables are the optimality criteria and mathematical programming techniques. The design variables are the macro-element thicknesses and the layers' angles. Weight minimization with material efficiency maximization are the objectives with constraints on stresses and displacements. Maximization of the material efficiency is performed at one level using the conjugated method applied to the angles of the macro-element layers keeping the thicknesses constant. The other level is dedicated to weight reduction using optimality criteria and using as variables the macro-element thicknesses with the angles of the macro-element layers constant.     

Structural optimization using the Newton modified barrier method

The Newton Modified Barrier Method (NMBM) is applied to structural optimization problems with large a number of design variables and constraints. This nonlinear mathematical programming algorithm was based on the Modified Barrier Function (MBF) theory and the Newton method for unconstrained optimization. The distinctive feature of the NMBM method is the rate of convergence that is due to the fact that the design remains in the Newton area after each Lagrange multiplier update. This convergence characteristic is illustrated by application to structural problems with a varying number of design variables and constraints. The results are compared with those obtained by optimality criteria (OC) methods and by the ASTROS program.     

Multidiscipline topology optimization

Topology optimization is used for determining the best layout of structural components to achieve predetermined performance goals. The density method which uses material density of each finite element as the design variable, is employed. Unlike the most common approach which uses the optimality criteria methods, the topology design problem is formulated as a general optimization problem and is solved by the mathematical programming method. One of the major advantages of this approach is its generality; thus it can solve various problems, e.g. multi-objective and multi-constraint problems. In this study, the structural weight is chosen as the objective function and structural responses such as the compliances, displacements and the natural frequencies, are treated as the constraints. The MSC/NASTRAN finite element code is employed for response analyses. One example with four different optimization formulations was used to demonstrate this approach.     

Establishing dominance and potential optimality in multi-criteria analysis with imprecise weight and value

The model presented in this paper does not require exact estimations of decision parameters such as attribute weights and values that may often be considerable cognitive burden of human decision makers. Information on the decision parameters is only assumed to be in the form of arbitrary linear inequalities which form constraints in the model. We consider two criteria, dominance and potential optimality, to check whether or not each alternative is outperform for a fixed feasible region denoted by the constraints. In particular, we develop a method to identify potential optimality of alternatives when all (or subsets) of the attribute values as well as weights are imprecisely know. This formulation becomes a nonlinear programming problem hard to be solved generally so that we provide in this paper how this problem is transformed into a linear programming equivalent.Scope and purposeMost managerial decisions involve choosing an optimal alternative from a number of available alternatives. Researchers have proposed a lot of methods to assist decision makers in choice making with a set of, usually conflicting, criteria or attributes. Many of these approaches require exact (or precise) information about either or both attribute values and/or trade-off weights. In some practice, however, it is not easy for decision makers to provide such exact data because, for example, intangible attributes to reflect social and environmental impacts may be included. To cope with such problem, a mathematical programming model-based approach to multi-criteria decision analysis is presented in this paper when both attribute weights and marginal values are imprecisely identified. A weighted additive rule is used to evaluate the performance of alternatives. We then show how to obtain non-dominated and potentially optimal alternatives in order to support choice making.     

Structural optimization with crashworthiness constraints

An automated structural design methodology has been devised which simultaneously considers design criteria associated with both linear elastic and crashworthiness loading conditions. This method is developed within the context of a nonlinear mathematical programming based structural optimization capability using an efficient two-phased crashworthiness analysis technique. Specially constructed nonlinear approximations for the crashworthiness constraints are employed to further reduce the computational burden during the optimization process. This methodology is demonstrated on an automobile structural design problem. It is shown that more mass efficient designs can be obtained by simultaneously considering elastic and crashworthiness design criteria as compared to a sequential approach in which the structure is first designed for the elastic loads and then modified to satisfy the crashworthiness criteria.     

A hybridization of mathematical programming and dominance-driven enumeration for solving shift-selection and task-sequencing problems

A common problem in production planning is to sequence a series of tasks so as to meet demand while satisfying operational constraints. This problem can be challenging to solve in its own right. It becomes even more challenging when higher-level decisions are also taken into account. For example, determining which shifts to operate clearly impacts how tasks are then scheduled; additionally, reducing the number of shifts that must be operated can have great cost benefits. Integrating the shift-selection and task-sequencing decisions can greatly impact tractability, however, traditional mathematical programming approaches often failing to converge in reasonable run times. Instead, we develop an approach that embeds mathematical programming, as a mechanism for solving simpler feasibility problems, within a larger search-based algorithm that leverages dominance to achieve substantial pruning. In this paper, we introduce the Shift-Selection and Task Sequencing problem (SS-TS), develop the Test-and-Prune algorithm (T&P), and present computational experiments based on a real-world problem in automotive stamping to demonstrate its effectiveness. In particular, we are able to solve to provable optimality, in very short run times, a number of problem instances that could not be solved through traditional integer programming methods.     

A criterion for optimality of design of evop-type experiments—i. optimality criterion for quadratic response surfaces

A long-run objective of maximizing discounted total response over a possibly infinite horizon is formulated. The solution is sought by means of a dynamic programming approach, with parameter uncertainties represented by their posterior (Bayesian) distributions. A linearization of the solution of the infinite-horizon problem is developed, and various approximations for required expectations are introduced and justified. The result is an optimality criterion closely related to that of ‘D-optimal’ experimental designs. The standard computational methods for the latter are adapted to yield a practical method of obtaining the optimal designs in this case.     

Saddle Point Optimality Criteria of Interval Valued Non-Linear Programming Problem

The present paper aims to develop the Kuhn-Tucker and Fritz John criteria for saddle point optimality of interval-valued nonlinear programming problem. To achieve the study objective, we have proposed the definition of minimizer and maximizer of an interval-valued non-linear programming problem. Also, we have introduced the interval-valued Fritz-John and Kuhn Tucker saddle point problems. After that, we have established both the necessary and sufficient optimality conditions of an interval-valued non-linear minimization problem. Next, we have shown that both the saddle point conditions (Fritz-John and Kuhn-Tucker) are sufficient without any convexity requirements. Then with the convexity requirements, we have established that these saddle point optimality criteria are the necessary conditions for optimality of an interval-valued non-linear programming with real-valued constraints. Here, all the results are derived with the help of interval order relations. Finally, we illustrate all the results with the help of a numerical example.     

ε-Optimality and duality for multiobjective fractional programming

Using the scalar ε-parametric approach, we establish the Karush-Kuhn-Tucker (which we call KKT) necessary and sufficient conditions for an ε-Pareto optimum of nondifferentiable multiobjective fractional objective functions subject to nondifferentiable convex inequality constraints, linear equality constraints, and abstract constraints. These optimality criteria are utilized as a basis for constructing one duality model with appropriate duality theorems. Subsequently, we employ scalar exact penalty function to transform the multiobjective fractional programming problem to an unconstrained problem. Under this case, we derive the KKT necessary and sufficient conditions without a constraint qualification for ε-Pareto optimality of multiobjective fractional programming.     

Numerical comparison of nonlinear programming algorithms for structural optimization

For FE-based structural optimization systems, a large variety of different numerical algorithms is available, e.g. sequential linear programming, sequential quadratic programming, convex approximation, generalized reduced gradient, multiplier, penalty or optimality criteria methods, and combinations of these approaches. The purpose of the paper is to present the numerical results of a comparative study of eleven mathematical programming codes which represent typical realizations of the mathematical methods mentioned. They are implemented in the structural optimization system MBB-LAGRANGE, which proceeds from a typical finite element analysis. The comparative results are obtained from a collection of 79 test problems. The majority of them are academic test cases, the others possess some practicalreal life background. Optimization is performed with respect to sizing of trusses and beams, wall thicknesses, etc., subject to stress, displacement, and many other constraints. Numerical comparison is based on reliability and efficiency measured by calculation time and number of analyses needed to reach a certain accuracy level.The research project was sponsored by the Deutsche Forschungsgemeinschaft under research contract DFG-Schi 173/6-1     

A sequential approximate programming strategy for reliability-based structural optimization

Although reliability-based structural optimization (RBSO) is recognized as a rational structural design philosophy that is more advantageous to deterministic optimization, most common RBSO is based on straightforward two-level approach connecting algorithms of reliability calculation and that of design optimization. This is achieved usually with an outer loop for optimization of design variables and an inner loop for reliability analysis. A number of algorithms have been proposed to reduce the computational cost of such optimizations, such as performance measure approach, semi-infinite programming, and mono-level approach. Herein the sequential approximate programming approach, which is well known in structural optimization, is extended as an efficient methodology to solve RBSO problems. In this approach, the optimum design is obtained by solving a sequence of sub-programming problems that usually consist of an approximate objective function subjected to a set of approximate constraint functions. In each sub-programming, rather than direct Taylor expansion of reliability constraints, a new formulation is introduced for approximate reliability constraints at the current design point and its linearization. The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP). It is shown that the approximate MPP, a key component of RBSO problems, is concurrently improved during each sub-programming solution step. Through analytical models and comparative studies over complex examples, it is illustrated that our approach is efficient and that a linearized reliability index is a good approximation of the accurate reliability index. These unique features and the concurrent convergence of design optimization and reliability calculation are demonstrated with several numerical examples.     

Mapping streaming applications on multiprocessors with time-division-multiplexed network-on-chip

This paper addresses mapping of streaming applications (such as MPEG) on multiprocessor platforms with time-division-multiplexed network-on-chip. In particular, we solve processor selection, path selection and router configuration problems. Given the complexity of these problems, state of the art approaches in this area largely rely on greedy heuristics, which do not guarantee optimality. Our approach is based on a constraint programming formulation that merges a number of steps, usually tackled in sequence in classic approaches. Thus, our method has the potential of finding optimal solutions with respect to resource usage under throughput constraints. The experimental evaluation presented in here shows that our approach is capable of exploring a range of solutions while giving the designer the opportunity to emphasize the importance of various design metrics.     

Dynamic programming equations for discounted constrained stochastic control

In this paper, the application of the dynamic programming approach to constrained stochastic control problems with expected value constraints is demonstrated. Specifically, two such problems are analyzed using this approach. The problems analyzed are the problem of minimizing a discounted cost infinite horizon expectation objective subject to an identically structured constraint, and the problem of minimizing a discounted cost infinite horizon minimax objective subject to a discounted expectation constraint. Using the dynamic programming approach, optimality equations, which are the chief contribution of this paper, are obtained for these problems. In particular, the dynamic programming operators for problems with expectation constraints differ significantly from those of standard dynamic programming and problems with worst-case constraints. For the discounted cost infinite horizon cases, existence and uniqueness of solutions to the dynamic programming equations are explicitly shown by using the Banach fixed point theorem to show that the corresponding dynamic programming operators are contractions. The theory developed is illustrated by numerically solving the constrained stochastic control dynamic programming equations derived for simple example problems. The example problems are based on a two-state Markov model that represents an error prone system that is to be maintained.     

Dynamical systems and topology optimization

This paper uses a dynamical systems approach for studying the material distribution (density or SIMP) formulation of topology optimization of structures. Such an approach means that an ordinary differential equation, such that the objective function is decreasing along a solution trajectory of this equation, is constructed. For stiffness optimization two differential equations with this property are considered. By simple explicit Euler approximations of these equations, together with projection techniques to satisfy box constraints, we obtain different iteration formulas. One of these formulas turns out to be the classical optimality criteria algorithm, which, thus, is receiving a new interpretation and framework. Based on this finding we suggest extensions of the optimality criteria algorithm. A second important feature of the dynamical systems approach, besides the purely algorithmic one, is that it points at a connection between optimization problems and natural evolution problems such as bone remodeling and damage evolution. This connection has been hinted at previously but, in the opinion of the authors, not been clearly stated since the dynamical systems concept was missing. To give an explicit example of an evolution problem that is in this way connected to an optimization problem, we study a model of bone remodeling. Numerical examples, related to both the algorithmic issue and the issue of natural evolution represented as bone remodeling, are presented.