Design of Experiments DoE

13 4. EXECUTION OF THE DESIGN EXPERIMENT 5. DATA CONSISTENCY CHECK WITH ASSUMPTIONS are the results reproducible? 6. ANALYSIS MODELING OF THE RESULTS examine the results for outliers, typographical errors and obvious problems, create the model from the data, check the model residuals and use the results to answer the questions set in the objectives Using simple mathematical functions a.k.a. regression functions to model the effects of the input variables on the output of the system, the DoE response surface models differ significantly from other approaches, such as ANNs or phenomeno- logical models. Both ANN and DoE approaches use “black-box” concepts to model the system behavior, and while ANN models are capable of approximating any con- tinuous function describing the inputoutput correlations, DoE models are not. Applications Examples Despite the limitations in generality, given the structured procedure and the possibil- ity to reduce the number of experiments necessary, DoE approaches are commonly used in industrial applications, such as combustion engine RD [18]. Examples range from comparative studies of engine components and injection strategies for an automotive DI diesel engine [13], to screening and modeling studies of the in-cylin- der flow field and combustion chamber geometry for a medium-duty DI diesel engine [77].

2.4 Optimization

Optimization can be defined as the search for the best possible solutions to a given problem. In general, the n-dimensional optimization problem is expressed by , 2.3 where is the parameter vector minimizing 1 the single-objective function , subject to the equality and inequality constraints given. Additionally, the limits highlow for the parameter vector values are defined as , where stands for the lower and for the higher limit respectively. 1. In practice, the optimum is generally defined as the minimum of an objective function. Maximum optimization problems are therefore transformed into minimization problem using max f x = - min - f x . min f x g i x 0 i 1 … p , , = , = h j x 0 j 1 … q , , = , ≤ ⎩ ⎪ ⎨ ⎪ ⎧ x ℜ n ∈ f x ℜ ∈ g i x h j x x k l x x k h k 1 … n , , = , ≤ ≤ l h 14 Given that in engineering applications, e.g. gas turbine design [20], multiple con- flicting objectives have to be optimized, the result of an optimization is a set of “trade-off” solutions i.e. pareto optimal set 1 , rather than one single best solution. A mutual comparison of at least two solutions from the pareto set thereby shows, that both of them are better and worse in at least one objective at the same time [97]. Various solutions exist to tackle the optimization problem, most of them sharing the iterative concept of splitting-up the parameter vector estimation optimizer and the parameter vector evaluation analyzer as in Figure 2.4 Fig. 2.4 Iterative Optimization Scheme of Optimizer Analyzer