Birna P. Kristinsdottir

Optimal design of large composite panels with varying loads

Abstract This paper presents an optimization formulation for the design of large composite panels when loads vary over the panel. A methodology termed “blending” is introduced and used to ensure that a panel is manufacturable. Two ways of specifying the blending rules in optimal design formulation are set forth and compared. A global optimization algorithm, Improving Hit-and-Run (IHR), is used to find optimal designs. A composite panel is designed with and without using blending rules to demonstrate their effectiveness. The resulting designs show that blending rules are a great assistance in designing large composite panels that are tailored for varying loads in a practical manner.

Hesitant adaptive search: the distribution of the number of iterations to convergence

Abstract.Hesitant adaptive search is a stochastic optimisation procedure which accommodates hesitation, or pausing, at objective function values. It lies between pure adaptive search (which strictly improves at each iteration) and simulated annealing with constant temperature (which allows backtracking, or the acceptance of worse function values). In this paper we build on an earlier work and make two contributions; first, we link hesitant adaptive search to standard counting process theory, and second, we use this to derive the exact distribution of the number of iterations of hesitant adaptive search to termination.

INCORPORATING MANUFACTURING TOLERANCES IN NEAR-OPTIMAL DESIGN OF COMPOSITE STRUCTURES

When a composite structure is manufactured the design variables in the produced structure may vary from the intended design. These variations are called manufacturing tolerances. When optimization is used, the optimal design is often on an active constraint. This may cause the produced structure to fail, even though the variations from the intended design are within tolerances. In this paper a methodology is introduced for finding a near-optimal design that remains feasible for specified manufacturing tolerances. A random search global optimization algorithm, Improving Hit-and-Run, is used to check if a design with specified manufacturing tolerances remains feasible and an algorithm is presented to determine the size of the tolerances for a feasible design. Finally, by changing the right hand side of the constraints and reoptimizing, it is possible to explore near-optimal designs with different tolerances. The methodology is applied to a composites structural design problem.

Optimal estimation of univariate black-box Lipschitz functions with upper and lower error bounds

Abstract We are given an unknown univariate Lipschitz continuous function that we wish to estimate by evaluating the function sequentially at distinct points. We provide a procedure for recursively selecting this sequence of points so that, averaging over points in the domain the resulting worst case error between the estimating and actual functions is minimized. Upper and lower bounds on these errors is also provided. Scope and purpose In engineering and science, it is often necessary to estimate functions based on a small number of evaluations. In this paper we determine which points should be evaluated in order to maximize the information gained at each evaluation. In particular, we prove that the sampling strategy that minimizes worst case error, averaged over points in the domain, is to sample the midpoint of a specific interval. We provide an estimation procedure that bounds a function using a Lipschitz bracket. The resulting estimating function is simple and consists of linear functions only.

Constructing large feasible suboptimal intervals for constrained nonlinear optimization

An algorithm for finding a large feasiblen-dimensional interval for constrained global optimization is presented. Then-dimensional interval is iteratively enlarged about a seed point while maintaining feasibility. An interval subdivision method may be used to check feasibility of the growing box. The resultant feasible interval is constrained to lie within a given level set, thus ensuring it is close to the optimum. The ability to determine such a feasible interval is useful for exploring the neighbourhood of the optimum, and can be practically used in manufacturing considerations. The numerical properties of the algorithm are tested and demonstrated by an example problem.

Discrete Backtracking Adaptive Search for Global Optimization

This paper analyses a random search algorithm for global optimization that allows acceptance of non-improving points with a certain probability. The algorithm is called discrete backtracking adaptive search. We derive upper and lower bounds on the expected number of iterations for the random search algorithm to first sample the global optimum. The bounds are derived by modeling the algorithm using a series of absorbing Markov chains. Finally, upper and lower bounds for the expected number of iterations to find the global optimum are derived for specific forms of the algorithm.

Including manufacturing tolerances in composite design

When composite structures are designed it is important to account for possible variations of design variables. Such variation of design variables are referred to as manufacturing tolerances. Accounting for tolerances early on in the design phase, before the structure is actually manufactured, will reduce the overall lifecycle cost of the structure. In this paper we apply methodologies to incorporate manufacturing tolerances in the optimal design of a composite stiffened panel. A methodology that uses global optimization is used to check if a design has specific manufacturing tolerances. Furthermore, an algorithm for finding the maximal allowable tolerances of a design is a p plied to analyze the tolerances of a design. A procedure to find near-optimal designs that remain feasible within specified manufacturing tolerances is used to obtain designs with tolerances. Designs are obtained for three different objective functions, weight, cost and a combination of weight and cost and a tradeoff analysis is performed.

Optimal sequencing of tasks in an aluminium smelter casthouse

This paper examines the problem of determining the sequence in which to cast aluminium ingots such that setup times are minimized. The aluminium ingots are of different size and consist of different alloys; this poses constraints on the allowable ordering of casting jobs. The sequencing problem can be formulated as an asymmetric traveling salesman problem with additional constraints. Two different methods are used to formulate and solve this problem, a genetic algorithm (GA) and an integer programming approach (IP). A new method for detecting sub-tours in the IP formulation is set forth and used. Both the GA and IP approach are implemented in a software tool that is being used in the aluminium production industry. The results show that even though the integer programming formulation can be used to solve the problem to optimality, the size of the problem is a limiting factor for practical implementation. Therefore, the genetic algorithm is used in the industrial implementation with good results.

Complexity Analysis Integrating Pure Adaptive Search (PAS) and Pure Random Search (PRS)

Pure adaptive search (PAS) is a random search algorithm for global optimization that has promising complexity results. The complexity of pure adaptive search has been analyzed for both continuous and discrete, finite global optimization. Unfortunately, it is not possible at this time to implement pure adaptive search and achieve the ideal computational performance. To model practical random search algorithms more closely, we extend the complexity analysis to integrate pure adaptive search with pure random search. Many practical algorithms have some probability of sampling in the improving region, which is analogous to sampling according to PAS, and a probability of sampling outside the improving region, which is analogous to sampling according to PRS. Simulated annealing also has a probability of accepting a non-improving point. A Markov chain analysis is used to determine the expected number of iterations required to find the global optimum and to provide bounds for the expected number of iterations needed for a combination of PAS and PRS with acceptance probability. The analysis shows that one needs only a small probability of sampling in the improving region in order to dramatically improve performance.