Multiobjective design optimization of a quadrupole resonator under uncertainties

For a precise determination of the rf properties of superconducting materials, a calorimetric measurement is carried out with the aid of a so-called quadrupole resonator (QPR). This procedure is affected by certain systematic measurement errors with various sources of uncertainties. In this paper, to reduce the impact of geometrical uncertainties on the measurement bias, the modified steepest descent method is used for the multiobjective shape optimization of a QPR in terms of an expectation measure. Thereby, variations of geometrical parameters are modeled by the polynomial chaos expansion technique. Then, the resulting Maxwell’s eigenvalue problem with random input data is solved using the polynomial chaos-based stochastic collocation method. Furthermore, to assess the contribution of the particular geometrical parameters, the variance-based sensitivity analysis is proposed. This allows for modifying the steepest descent algorithm, which results in reducing the computational load needed to find optimal solutions. Finally, optimization results in the form of an efficient approximation of the Pareto front for a three-dimensional model of the QPR are shown.

Figure 1

Illustration of the CERN-QPR [3] (left), technical drawing [2] (right).

Figure 2

According to the calorimetric method, the surface resistance of a superconducting sample is derived from a dc measurement [1].

Figure 3

Cross section of a HZB-QPR (left) and a parameterized model of the pole shoes (right). The nomenclature used in the parameterization of the pole shoes follows [1].

Figure 4

Result of the global sensitivity analysis for $f_{·,1}(\mathbf{p})$.

Figure 5

Result of the global sensitivity analysis for $f_{·,2}(\mathbf{p})$.

Figure 6

Result of the global sensitivity analysis for $f_{·,3}(\mathbf{p})$.

Figure 7

Algorithm for the PC-SCM employing CST STUDIO SUITE^® [44] as “black-box” simulation engine. In the flow indicated above Dakota [68] has been exploited.

Figure 8

Convergence to the Pareto front using the VBS-based approach (23) for several initial points $N=101$.

Figure 9

Comparison between the shape of the HZB design, CERN designs, and the obtained solutions.

Figure 10

Probabilistic density function for the focusing factor.

Figure 11

Probabilistic density function for the homogeneity factor.

Figure 12

Probabilistic density function for the dimensionless factor representing the losses on the normal-conducting flange.

Figure 13

Probabilistic density function for the ratio of the magnetic peak values.

Figure 14

Probabilistic density function for the ratio of the magnetic peak by the electric peak values.

Figure 15

Probabilistic density function for the frequency of the third operating mode. The dashed vertical lines show the location of the next two closest neighboring modes calculated for the mean values given in Table 4.