## 2 Robust Design Optimization Methods and Challenges
### Robustness in Machine Design and Manufacturing  
Robustness is a general, desirable property in many engineering fields [42]. The design of an induction brazing process may well illustrate the need to consider various uncertainties (Figure 2). The optimized shape of the inductor has to be designed together with its control, considering the manufacturing tolerances to minimize the amount of waste products during the mass manufacturing process.  
Figure 1: A one- and two-dimensional explanations of the robustness if the goal function(s) is (are) maximized. In (**a**), the yellow dots illustrate the problem of (under-)sampling during the gradient calculation (sensitivity analysis) of a numerically expensive optimization task. (**b**) shows the best solutions in the different tolerance classes in a robust optimization test problem (with Artap [35]).  
Due to its fast, clean, and contactless properties, induction brazing is a preferred method of many assembly processes [43; 44]. The adjacent surfaces need to be heated, and the soldering material is melted on this surfaces. However, in the case of aluminum brazing, the difference between the temperature of the melting point of the soldering material and the aluminum is relatively small. To make a good joint, the temperature of the adjacent surfaces cannot differ by more than 10-20 K from optimal (Figure 2c,d), while the absolute value of the temperature reaches about 600 K from the initial room temperature within some tens of seconds. However, the temperature hot-spot is very sensitive to different kinds of positioning, manufacturing errors, the shape of the inductor, and the control of the excitation current [31; 32; 45]. To reduce the amount of waste produced during the mass manufacturing process, the tolerances and the control have to be designed and optimized together with the shape of the inductor.  
In the case of electrical machines, the design is robust if it is insensitive to small parameter changes and other uncertainties. These uncertainties can come from four different sources, and they are usually handled by a stochastic design approach [46; 47]. Firstly, the most apparent and widely investigated source of these errors is the effect of the manufacturing tolerances on the electrical machine. Secondly--at the system or application level--the imbalances or non-linear properties of the electrical drives or changes in the operational or environmental conditions affect the performance of the machine [46; 48; 49; 50]. For example, the proper handling of multi-harmonic excitations requires novel design methodologies. This issue concerns not only rotating machines, but also the insulation system design of an inverter fed by power transformer [51; 52]. In some cases, it is necessary to consider the temperature of the environment to accurately calculate the demagnetization of the magnets [53]. Thirdly, the modeling or approximation errors can represent a different class of uncertainties. This kind of modeling error is unavoidable in the first, preliminary design stage, which deals with simplified machine models [9; 54]. Another example is the parallel calculation of the time-dependent transient eddy current based quantities in electrical machines (multigrid-reduction-in-time algorithms [55]), like the parallel calculation of the steady-state parameter in the case of asynchronous [56] or permanent magnet machines [54] or the lightning impulse response calculation for large power transformers. In addition to the three already mentioned, [PERSON] [46] distinguished another source of uncertainty, the feasibility error. It considers what constraints of the design variables must be fulfilled. This kind of robustness is not independent of the first two cases. In the case of an electrical machine design, we also have limited knowledge about the exact material properties and boundary conditions. A wide literature on the numerical modeling of the B-H curves and demagnetization for electrical machine design exits [57; 58; 59].