3. Integration of subsystems design in a collaborative MDO process
3.4 Step 3: Formulate design problem and solution strategy
Once all the design competences required to design the new product have been collected, an MDO architecture is formulated by the architect to solve the MDO problem. The formulation of the design problem is the main aim of the third step of the development process. This step assumes great importance due to the fact that MDO problems are growing in size and complexity. This happens because of the high number of the coupling variables resulting from the disciplines involved within the design process. Therefore, the MDO problem should be organized to obtain the desired solution minimizing costs, time and effort. In other words, an MDO architecture should be formulated defining how the disciplinary analysis models are coupled together and how the optimization problem shall be addressed [4]. Formulating an MDO architecture means to organize the disciplinary competences – e.g. ordering their sequence during the analyses, defining the data flow – and determining the design variables, the objective functions and the constraints. Due to the high complexity of an MDO architecture, a visual description might be helpful to comprehend the MDO problem. An example of visual description is given by the eXtended Design Structure Matrix (XDSM), a graph of which an example is depicted in Figure 24 [4]. The XDSM is proposed by Martins et al. [4] to enhance the visualization of the interconnection among the modules integrated within a MDO process.
Examples of this type of diagram will be described in Section 4.3. In particular, more than one XDSM will be presented, according to the kind of design problem.
The XDSM depicts the connections among the design disciplines (or disciplinary tools), showing both the data and process flows. The data flow is represented by the grey lines. Each vertical line connected to a discipline model represents inputs, while horizontal lines denote outputs. The design variables 𝑥𝑖 represent the parameters under the control of an optimizer. In other word, an optimization code may vary the values of the design variables within a predetermined range to comply with the objective functions, for instance minimizing costs or masses. The optimized results are denoted as 𝑦𝑖∗, corresponding to the input variables 𝑥𝑖∗ of the design space. The outputs of each design discipline are represented by the data flowing on the horizontal lines. These parameters are denoted as state variables. If these values are exchanged with other disciplines, they are referred as coupling variables 𝑦𝑖.
Step 3: Formulate design problem and solution strategy 79
Figure 24: Example of XDSM [4].
Several design problems might be assessed. Un-converged Multidisciplinary Design Analysis (MDA) processes entail the assessment of the response of a set of disciplinary modules given a certain ensemble of design variables. Iterations and backward loops would be necessary to converge to the design solution (converged MDA). Differently from the MDO problem, MDA processes don’t entail the determination of an optimized solution. Single and multi-objective MDO problems indeed might be setup to obtain an optimized solution. In this regard, a multi- objective MDO problem based on the Fuzzy Logic will be presented in subsection 3.4.1. The last design problem considered in the present dissertation is the Design of Experiments (DOE). The DOE entails the investigation of the design space by means of tests – named experiments – executed to assess the response of a design process according to certain design variables believed to influence it.
3.4.1 Fuzzy Logic multi-objective optimization
The main idea at the base of this method is the optimization of the designed airplane through the negotiation and relax of some high level requirements, as range and payload. The negotiation of the product requirements shall be done in accordance with the system’s users. Some TLARs might be perceived excessive by the customer. The relaxing of these requirements can bring to an improved (optimal) product, preferable from a user perspective.
80 Integration of subsystems design in a collaborative MDO process This optimization problem can be multi-objective, as the goals of the development process might be, for example in the case of a hybrid powered aircraft, the minimization of the fuel consumption together with the maximization of the safety level (i.e. minimization of the minimum safety altitude). In the context of the doctoral research activities, an optimization method based on the Fuzzy Logic approach is developed [148]. The Fuzzy Logic represents a logic employed to deal with statements that are partially true, i.e. neither completely true or completely false [149]. The Fuzzy Logic has been formalized for the first time by Zadeh in 1965 [150]. Since then, many applications of this new concept of logic have been developed in several fields, mostly regarding control systems. Fuzzy Logic controllers are mainly studied and employed in electronics ( [151], [152], [153], [154], [155]). However, control systems based on Fuzzy Logic are spread also in other fields, for example in robotics ( [156], [157]), telecommunications [158], air- conditioning applications [159] and medicine [160]. The Fuzzy Logic is rooted in the Aeronautical Engineering, too. Several application studies deal with flight control systems ( [161], [162], [163], [164], [165], [166]) but also fire detection systems [167], propulsion systems ( [168], [169]) and braking systems [170]. However, other than control system applications, the Fuzzy Logic can be adopted in other contexts. In this regard, a Fuzzy Logic approach has been proposed for multi-objective optimizations in the magnetics field [171]. However, the application of Fuzzy Logic for optimization in aeronautics is still lacking. Moreover, the present subsection proposes an integration between the aircraft multi-objective optimization and the negotiation of the high level requirements.
From a literature review, several multi-objective optimization methods can be disclosed. However, the Fuzzy Logic is considered the most suitable methodology for this kind of problem. The Pareto methods are one of the most well-known multi- objective optimization methodologies. Nevertheless, the Pareto methods allow the determination of a set of optimal solutions all equivalent from a mathematical perspective, without identifying the best one [172]. Furthermore, other two motivations are behind the selection of the Fuzzy Logic approach. Firstly, objectives and requirements defined by quantities measured on different scales can be assessed by means of the Fuzzy Logic. The Fuzzy Logic brings to a transformation of these quantities in scores thanks to which they could be handled together. Secondly, all the design objectives might be ordered following a hierarchy of importance defined by the stakeholders. In other words, the user could retain an objective function more important than another one. Hence, the design process should be led by this more important objective function.
Step 3: Formulate design problem and solution strategy 81 The concept of fuzzy set is at the base of the Fuzzy Logic theory. According to its definition given by Zadeh [150], let y being a vector of generic elements of a space of points Y. The vector y can include requirements of the design case and design results. A fuzzy set A in Y is characterized by a Membership Function (MF)
μ(y), representing the “grade of membership” of y in A. The MF μ(y) associates each
point in Y with a real number in the interval [0, 1]. In other words, μ(y) expresses the “degree of satisfaction” of a considered objective as perceived by the customer, on the basis of the value y resulting from the design. In the current dissertation, only piecewise linear functions are considered.
By way of example and with reference to Figure 25, let y being one of the objective functions of a certain MDO problem. The customer is supposed to consider unacceptable the values of y resulting from the design if equal or greater to ymax. When this parameter results at least ymax, a MF μ(ymax)=0 is set, meaning that the resulting value should be lowered. Vice-versa, results of y equal or below to ymin are fully accepted. In this case a MF μ(ymin)=1 is defined, representing the maximum degree of satisfaction of the customer. Hence, μ(ymin)=1 and μ(ymax)=0
state the boundary values of the MF. In the current example, if a solution of the design problem brings to a value of y equal to y*, a degree of satisfaction μ(y*)=0.6 is returned, as represented in Figure 25. The MF of Figure 25 depicts a decreasing linear function: the degree of satisfaction μ reduces for higher values of y. On the contrary, μ might assume higher results in case of higher values of the objective function, as represented in Figure 26 (a). Additionally, the degree of satisfaction can be characterized by a maximum when y assumes a certain value, as shown in Figure 26 (b).
82 Integration of subsystems design in a collaborative MDO process
Figure 26: Additional examples of Membership Functions of y.
Once the Fuzzy Logic theory has been applied to all the requirements and objectives of the MDO problem, a global degree of satisfaction is estimated. The global degree of satisfaction combines all the MFs through the intersection of the fuzzy sets ( [171], [173]). Given the vector of design parameters x, a number n of objective functions y(x) and relative MFs μi(yi(x)), the global degree of satisfaction G(x) results from:
𝐺(𝑥) = min
𝑖=1,…,𝑛{𝜇𝑖(𝑦𝑖(𝑥))} eq. 21
If multiple design problems are performed, each one with different vectors x of design parameters, several values of G(X) are derived. The optimal solution is the one characterized by the highest degree of satisfaction. This optimal solution can be discovered by an optimizer investigating the design space, looking for the set of design variables constituting the vector x, by which the maximum value of G(x) is derived.