Approximated Computation of Belief Functions for Robust Design Optimization

This paper presents some ideas to reduce the computational cost of evidence-based robust design optimization. Evidence Theory crystallizes both the aleatory and epistemic uncertainties in the design parameters, providing two quantitative measures, Belief and Plausibility, of the credibility of the computed value of the design budgets. The paper proposes some techniques to compute an approximation of Belief and Plausibility at a cost that is a fraction of the one required for an accurate calculation of the two values. Some simple test cases will show how the proposed techniques scale with the dimension of the problem. Finally a simple example of spacecraft system design is presented.Comment: AIAA-2012-1932 14th AIAA Non-Deterministic Approaches Conference. 23-26 April 2012 Sheraton Waikiki, Honolulu, Hawai

Structural Reliability Assessment Based on the Improved Constrained Differential Evolution Algorithm

In this work, the reliability analysis is employed to take into account the uncertainties in a structure. Reliability analysis is a tool to compute the probability of failure corresponding to a given failure mode. In this study, one of the most commonly used reliability analysis method namely first order reliability method is used to calculate the probability of failure. Since finding the most probable point (MPP) or design point is a constrained optimization problem, in contrast to all the previous studies based on the penalty function method or the preference of the feasible solutions technique, in this study one of the latest versions of the differential evolution metaheuristic algorithm named improved (μ+λ)-constrained differential evolution (ICDE) based on the multi-objective constraint-handling technique is utilized. The ICDE is very easy to implement because there is no need to the time-consuming task of fine tuning of the penalty parameters. Several test problems are used to verify the accuracy and efficiency of the ICDE. The statistical comparisons revealed that the performance of ICDE is better than or comparable with the other considered methods. Also, it shows acceptable convergence rate in the process of finding the design point. According to the results and easier implementation of ICDE, it can be expected that the proposed method would become a robust alternative to the penalty function based methods for the reliability assessment problems in the future works

Efficient Design Optimization Methodology for Manufacturable Variable Stiffness Laminated Composite Structures

Because of their superior mechanical and environmental properties compared to traditional metals, fiber-reinforced composite materials have earned a widespread acceptance for different structural applications. The tailoring potential of composites to achieve high specific stiffness and strength has promoted them as promising candidates for constructing lightweight structures. From that aspect, designers have tackled the problem of designing composite laminates, which is inherently challenging due to the presence of non-linear, non-convex, and multi-dimensional optimization problems with discrete and continuous design variables. However, despite their increased usage, the possible improvements that can be achieved by composite laminates have not been fully exploited. With the introduction of new manufacturing technologies such as advanced fiber placement, engineers now have the capability to harness the full potential of nonconventional variable stiffness composite laminates using in-plane fiber steering. This can be a blessing as well as a curse for the designer, where the additional improvements can be attained at the expense of an increased complexity of the design problem. To circumvent this difficulty, this research aims to develop appropriate design tools to help unlock the advancements achieved by nonconventional variable stiffness laminates. The purpose is to adopt an efficient design optimization methodology to abandon the traditional usage of straight fiber composite laminates in the favor of exploring the structural improvements that can be achieved by steered laminated composite structures, subject to manufacturing constraints and industry design guidelines. This represents a remarkable step in the development of energy-efficient light-weight structures and in their certification. The complexity of the optimization problem imposes the need for an efficient multi-level optimization approach to achieve a global optimum design. In this work, the importance of including a design-manufacturing mesh is demonstrated in each optimization step of the multi-level optimization framework. In the first step (Stiffness Optimization), a theoretical optimum stiffness distribution parameterized in terms of lamination parameters is achieved that accounts for optimum structural performance while maintaining smoothness and robustness. The design-manufacturing mesh allows the spatial stiffness distribution to be expressed as a B-spline or NURBS surface defined by the control points of the design-manufacturing mesh. The fiber angle distribution is then obtained in the second optimization step (Stacking Sequence Retrieval) to match the optimum stiffness properties from the first optimization step while accounting for the maximum steering constraint and laminate design guidelines to attain manufacturability and feasibility. A bilinear sine angle variation is presented to obtain smooth fiber angle distributions, and the maximum steering constraint is derived to guarantee a certain degree of manufacturability at the second optimization step. Using the design-manufacturing mesh, a constant curvature arc solution is developed in the third optimization step (Fiber Path Construction) to generate manufacturable fiber paths with piecewise constant curvature arcs that match the optimal fiber orientation angles from the second optimization step while locally satisfying the maximum curvature constraint. To minimize gaps and overlaps obtained due to fiber steering, a design-for-manufacturing tool is developed to generate tow-by-tow descriptions of the steered plies in the form of manufacturing boundaries for the AFP machine with optimized cut and restart positions. The design of cylindrical shells under bending with a specified cutout is chosen as an aerospace application to demonstrate the effectiveness of using nonconventional variable stiffness laminates compared to traditional conventional laminates. The presence of the cutout in the cylindrical shell imposes severe stress concentrations yielding a need to use variable stiffness laminates that have continuously varying fiber orientation angles to redistribute the stresses and obtain a structurally optimal design. A design-manufacturing mesh was introduced to perform the buckling load optimization, where both circumferential and longitudinal stiffness variations were considered to physically understand the importance of the stiffness tailoring mechanism in efficient load redistribution and local reinforcements around the regions of the cutouts. The multi-level optimization framework is utilized to obtain a manufacturable fiber-steered laminate that improves the buckling load significantly. The design-for-manufacturing tool developed then generates the tow-level information in the form of exported AFP boundaries. The designed cylindrical shell is imported into CATIA V5® for composite design programming to demonstrate the applicability of the design-for-manufacturing tool developed

The optimisation of brass instruments to include wall vibration effects

This thesis focuses on the design optimisation of a brass instrument. The bore profile of such an instrument is known to be the primary influence on the sound of the instrument as it directly controls the shape of the air-column contained within the instruments' walls. It has long been claimed, however, that other factors, such as the wall material and wall vibrations, are also significant, although to a lesser degree. In recent years, it has been proven that wall vibrations do indeed have an audible effect on the sound (Moore et al 2005, Kausel et al 2007, Nachtmann et al 2007, Kausel, Zietlow and Moore 2010). This effect corresponds to a relative increase in the power of upper harmonics of the sound spectrum when vibrations are greatest, and relative increase in the power of the lower harmonics, in particular the fundamental, when vibrations are at their least. The result is a timbral difference where a greater relative power in the upper harmonics results in a 'brighter' sound, and where the opposite results in a 'darker' sound. Studies have also found that the degree of the wall vibration is increased when the resonant frequencies of the air-column and those of the instruments' structure align. It is this principle that this work is based on. The primary objective of this work was to devise a suitable approach for incorporating the wall vibration effect into an optimisation method to investigate the optimum designs for two scenarios: maximum wall vibration and minimum wall vibration. It was also of interest to investigate if there were any design characteristics for each scenario. Two analysis methods were investigated for their suitability, namely free and forced vibration using finite element analysis (FEA). Different approaches to defining the design variables were explored and the suitability of different optimisation algorithms was investigated. The free vibration approach was found to be inadequate for this application due to the inherent omission of valuable magnitude information. The forced vibration approach was found to be more successful, although it was not possible to align a resonance with each frequency of interest

Optimization of a Bi-dimensional Intake for a Supersonic Aircraft Using a Genetic Algorithm

Since the beginning of jet flight, the intake has been subject to many studies due to its importance. From its position on the wings, in the wing root, nose and being mounted on the fuselage, many decades were spent on the research and solution of constraints. These are still present in the form of space and weight saving, lower drag, centre of gravity, lower radar visibility, etc. Although theoretically intakes with an infinite number of weak shocks are better, these fail to produce better results. An intake with small number of shocks is preferred as it is smaller in size, thus saving weight, and has better off-design performance. In the present work, a bi-dimensional mixed compression intake is designed having the Panavia’s Tornado engine requirements in mind and the aircrafts performance values. The designed intake results were in line with other studies. A comparison is made between the parametrical result and the results of a genetic optimization algorithm to minimize length and maximize total pressure recovery (thus saving weight while having high performance) by using a variation on Mach numbers around the designated speed for the Tornado, while using a mixed compression intake. Accordingly, a longer intake had better performance, but a shorter intake comes with lower weight. With lower speeds, a higher total pressure recovery and lower length is reached. This, along with the loss of importance of high speeds for most mission profiles, suggests that an aircraft with a lower top speed is advantageous as its weight lowers and performance increases.Desde as primeiras aeronaves supersónicas, as admissões foram objeto de muitos estudos devido à sua importância. Desde a sua posição nas asas, à sua passagem para as raízes destas, nariz e finalmente a montagem na sua fuselagem, foram necessárias décadas no estudo das admissões e resolução das suas restrições. Estas restrições continuam a estar presentes no projeto, quer sejam em relação ao espaço, peso, arrasto, centro de gravidade, visibilidade no radar, entre outras. Embora admissões com um número infinito de choques fracos sejam teoricamente preferíveis, na prática, uma admissão com um número pequeno de choques é superior, pois é mais leve, devido a menores dimensões, e traz consigo um melhor desempenho em condições off-design. No presente trabalho é projetada uma admissão bi-dimensional com compressão mista, tendo os requisitos de motor e outros constrangimentos do avião Panavia Tornado em mente. Os resultados obtidos são corroborados com resultados obtidos em artigos relacionados e posteriormente comparados com os resultados de uma otimização multiobjectivos para minimizar o comprimento e maximizar a recuperação de pressão total. Deste modo, é possível alcançar um desempenho alto enquanto se mantém o peso baixo. Estes objetivos são alcançados ao usar um intervalo de velocidade próximo da operação do Tornado. De acordo com o esperado, uma admissão com menores dimensões tem uma recuperação total de pressão inferior em condições on-design. No entanto, também é sinónimo de baixo peso estrutural. Outra ligação encontrada é o aumento de recuperação de pressão total em admissões projetadas para menores velocidades, além de um comprimento inferior. Este resultado, além da redução na importância da velocidade máxima da aeronave na maioria dos perfis de missão, sugere que uma aeronave com uma velocidade máxima inferior é vantajosa, pois a sua agilidade e desempenho será superior

Optimisation over the non-dominated set of a multi-objective optimisation problem

In this thesis we are concerned with optimisation over the non-dominated set of a multiobjective optimisation problem. A multi-objective optimisation problem (MOP) involves multiple conflicting objective functions. The non-dominated set of this problem is of interest because it is composed of the “best” trade-off for a decision maker to choose according to his preference. We assume that this selection process can be modelled by maximising a function over the non-dominated set. We present two new algorithms for the optimisation of a linear function over the non-dominated set of a multi-objective linear programme (MOLP). A primal method is developed based on a revised version of Benson’s outer approximation algorithm. A dual method derived from the dual variant of the outer approximation algorithm is proposed. Taking advantage of some special properties of the problem, the new methods are designed to achieve better computational efficiency. We compare the two new algorithms with several algorithms from the literature on a set of randomly generated instances. The results show that the new algorithms are considerably faster than the competitors. We adapt the two new methods for the determination of the nadir point of (MOLP). The nadir point is characterized by the componentwise worst values of the non-dominated points of (MOP). This point is a prerequisite for many multi-criteria decision making (MCDM) procedures. Computational experiments against another exact method for this purpose from the literature reveal that the new methods are faster than the competitor. The last section of the thesis is devoted to optimising a linear function over the non-dominated set of a convex multi-objective problem. A convex multi-objective problem (CMOP) often involves nonlinear objective functions or constraints. We extend the primal and the dual methods to solve this problem. We compare the two algorithms with several existing algorithms from the literature on a set of randomly generated instances. The results reveal that the new methods are much faster than the others