Optimal Flying Wings: A Numerical Optimization Study

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/97067/1/AIAA2012-1758.pd

I An Intuitive Approach to Visualize Multidimensional Data and Relationships

Multidimensional relationships exist in almost any discipline. There are numerous visualization methods that enable us to ‘see ’ multidimensional relationships in an easy and intuitive manner for two or three variables, using 2-d and 3-d representations. It is substantially more difficult, and sometimes even impossible, to visualize more than three dimensions – at least easily or in a way that is intuitive to the user. In this paper, a visualization methodology is presented in which multidimensional relationships can be viewed in an intuitive and straightforward manner. Based on this visualization, it is possible to quickly identify regions of interest for functional relationships, optimization applications, or for high dimensional datasets, regardless of the complexity of space or data. The technology provides a new approach to visualize more than three dimensions in a way that looks remarkably similar to traditional 2-d and 3-d representations. The method uses a new technique of ‘lossless ’ dimension blending, termed the Hyper-Space Diagonal Counting (HSDC). The methodology developed here provides a unique way to visually represent the relationships for n-dimensional problems. What is described here represents a totally new methodology that has the potential to greatly impact numerous industries, as well as the educational enterprise. I

Intuitive Design Selection using Visualized n-Dimensional Pareto Frontier

Abstract A visualization methodology is presented in which a Pareto Frontier can be visualized in an intuitive and straightforward manner for an n-dimensional performance space. An approach for preference incorporation is presented that enables a designer to quickly identify 'good' points and regions of the performance spaces for a multi-objective optimization application, regardless of space complexity, numbers of objectives, or numbers of Pareto points. Visualizing Pareto solutions for more than three objectives has long been a significant challenge to the multi-objective optimization community. The Hyper-space Diagonal Counting (HSDC) method described here enables the lossless visualization to be implemented to achieve a hyperspace Pareto frontier. In this paper, we demonstrate the incredible power of using the hyperspace Pareto frontier as a visualization tool for design concept selection in a multiobjective optimization environment

Ordering Design Tasks Based on Coupling Strengths

The design process associated with large engineering systems requires an initial decomposition of the complex system into modules of design tasks which are coupled through the transference of output data. In analyzing or optimizing such a coupled system, it is essential to be able to determine which interactions figure prominently enough to significantly affect the accuracy of the system solution. Many decomposition approaches assume the capability is available to determine what design tasks and interactions exist and what order of execution will be imposed during the analysis process. Unfortunately, this is often a complex problem and beyond the capabilities of a human design manager. A new feature for DeMAID (Design Manager's Aid for Intelligent Decomposition) will allow the design manager to use coupling strength information to find a proper sequence for ordering the design tasks. In addition, these coupling strengths aid in deciding if certain tasks or couplings could be removed (o..

A Estimation of Multi-Objective Pareto Frontier using Hyperspace Diagonal Counting

The Hyper-Space Diagonal Counting (HSDC) method was previously proposed for generating representations of n-dimensional data in 2- or 3-dimensions. Since its inception, the HSDC has been used to visualize the n-dimensional performance space in an intuitive fashion for Multiobjective Optimization Applications, through the incorporation of the HSDC into the Hyperspace Pareto Frontier (HPF) visualization approach. This paper presents a newer application of the HSDC method that enables estimation of the Pareto frontier without performing a formal optimization. Further, it is demonstrated that this estimated Pareto frontier can be represented in the design space, with a different representation associated with each objective. This very different type of visualization provides designers with a means of investigating the trade-offs for both objectives as well as design points using only the design space. I

A Intuitive Visualization of Pareto Frontier for Multi- Objective Optimization in n-Dimensional Performance Space

A visualization methodology is presented in which a Pareto Frontier can be visualized in an intuitive and straightforward manner for an n-dimensional performance space. Based on this visualization, it is possible to quickly identify ‘good ’ regions of the performance and optimal design spaces for a multi-objective optimization application, regardless of space complexity. Visualizing Pareto solutions for more than three objectives has long been a significant challenge to the multi-objective optimization community. The Hyper-space Diagonal Counting (HSDC) method described here enables the lossless visualization to be implemented. The proposed method requires no dimension fixing. In this paper, we demonstrate the usefulness of visualizing n-f space (i.e. for more than three objective functions in a multiobjective optimization problem). The visualization is shown to aid in the final decision of what potential optimal design point should be chosen amongst all possible Pareto solutions. I