I Introduction
Accelerator devices require advanced, simulation based, design approaches due to demanding performance requirements and a considerable level of technical complexity. This is particularly true for superconducting accelerator cavities, which are a key element of many modern accelerator facilities. A typical design process involves 2D as well as 3D numerical solutions of the Maxwell eigenvalue problem, followed by optimization and uncertainty analysis and quantification studies. The latter have been conducted within the accelerator community from the 1970s, Halbach_1976aa ; Weiland_1977 ; vanRienen_1987 . However, these studies have been mainly based on (local) sensitivity analysis which should be applied with care to quantify uncertainties in the cavities’ geometry. Indeed, the eigenmodes and other measures of interest depend strongly on the shape of the cavity and local measures may not yield reliable results.
The topic of simulation based uncertainty quantification has seen tremendous developments in recent years, also in computational electromagnetics, see e.g. Clenet_2013aa . Nowadays, significant computational resources are available and uncertainty studies, taking into account systematically large parameter variations at all steps of the design process, come into reach. In particular, the concept of global sensitivity analysis Sobol_1990aa
has received much attention, where sensitivities are characterized through the contribution of each parameter (or parameter combination) to the variance of a system output quantity. Sobol sensitivity indices permit not only to analyze the importance of model input parameters, which in turn is useful in guiding modeling efforts, but also to identify important combined highorder parameter variations. Although the concept of global sensitivity analysis is wellestablished, the efficient computation of Sobol coefficients is a difficult task, mainly due to the complexity of the underlying eigenvalue problem
Georg_2019aa . This is addressed in the uncertainty literature by introducing surrogate models which emulate the relation between eigenmodes, or other quantities of interest on the model parameter.In this respect, this paper reports on comprehensive efforts on uncertainty quantification and global sensitivity analysis for the simulation of TESLA shaped, superconducting, cavities. Such cavities have been produced in considerable quantity for the European Xray Free Electron Laser (EXFEL) and measurement data is available to substantiate the approach. Each step of the manufacturing chain is documented together with the involved uncertainties. Some of these steps are mimicked on the simulation side, e.g. by introducing a random eigenvalue problem. The uncertainties are then quantified numerically and in particular the sensitivities give valuable insight into the systems behavior. We also compare these findings to purely statistical studies carried out during the manufacturing. However, the simulation of all manufacturing steps would require the solution of several random inverse problems and is postponed to future work.
Uncertainty studies in an accelerator physics context have been reported before, see Xiao_2007aa ; Schmidt_2014aa ; Corno_2015ac . In this work, we use more advanced, adaptive, surrogate modeling techniques, which are crucial to incorporate a large number of uncertain parameters. The main contribution is the detailed comparison and fusion of real manufacturing data on the one hand and simulation based uncertainty studies on the other hand. The paper also clearly points out important directions of future research, which would allow to further combine measurements and simulation.
The structure of the paper is given as follows: Section II introduces quantities of physical interest for accelerator cavities and the Maxwell eigenvalue problem. In Section III, details on the manufacturing of the EXFEL cavities and measurements are reported. This is followed by uncertainty modeling and quantification studies in Section IV and concluding remarks.
Ii Cavities and Maxwell Eigenvalue Problem
Accelerating cavities are devices used to accelerate particles to higher energies. Elliptical cavities are the accepted geometrical shape for particle velocities close to the speed of light (, where is the ratio of particle velocity to the speed of light in vacuum). The shape of an elliptical cell is defined by two elliptical arcs connected by a tangent straight line as shown in FIG. 1. The fundamental mode of the cavity is the TM mode that is typically used as the operating mode of the cavity.
In order to enhance the accelerating efficiency, multicell cavities are created by connecting several cells together via their irises (see FIG. 2 for the EXFEL cavity described in more detail in Section III). Each mode of the singlecell generally divides into modes of the same type in a multicell cavity with cells and forms the so called passband of that mode. In this work, we are interested in the TESLA cavity shape Aune2000 , which is composed of cells.
Let refer to the inner domain of the multicell cavity with boundary , where
denotes a vector of shape parameters to be specified. The fields in the sourcefree, timeharmonic case, are given by Maxwell’s equations:
(1)  
where and denote the electric and magnetic field strength, and are the permittivity and permeability of vacuum, respectively. The walls are modeled as perfect electric conductor, i.e.,
(2) 
One derives the Maxwell eigenvalue problem from (1) by eliminating . Introduction of the wave number yields
(3)  
for and . One should be aware that, although not explicitly specified, the field and the eigenfrequencies depend on the shape parameters . For each mode, i.e., a solution of (3) in the passband, there is a phase shift between fields of neighboring cells that can vary from 0 to radians. The mode, with frequency , of the TM passband is used in multicell cavities for acceleration.
In order to maximize the voltage across the cavity, the length of the middlecells is fixed as , where refers to the wavelength of the mode, see Padamsee_2015aa . Thus as the particle traverses a cell, the direction of the field reverses such that the particle is exposed to an identically directed electric field along the whole multicell cavity.
The modes in a passband have a small frequency difference. If the frequencies of the modes in the fundamental passband are very close to each other, there is a risk of exciting a mode close to by the RF generator.
The spread of modes in the first passband is reflected in the celltocell coupling coefficient, which is defined as Belomestnykh2005 :
(4) 
where refers to the lowest frequency in the passband. The celltocell coupling coefficient is dimensionless and a sensitive quantity in the design phase. A large aperture radius () typically gives rise to a stronger celltocell coupling.
If the energy of the fundamental mode is evenly distributed in the cells, the accelerating voltage is maximized (Padamsee1998, , p.129). Furthermore, a uniform field distribution allows for higher field magnitudes before reaching the surface electromagnetic (EM) field limit. The field flatness is a central figure of merit that indicates the uniformity of the field distribution of the fundamental mode between the cells. In this paper, the field flatness is defined as
(5) 
where refers to the maximum axial electric field in cell .
The resonant frequencies strongly depend on the geometry parameter in each cell, i.e., the equatorial radii , with , and the iris radii , with . In Section IV we consider perturbations of the form and for the equatorial radii and the iris radius, respectively. Then, the parameter vector is given as
(6) 
These perturbations change the resonant frequency of the respective cell(s) and consequently affect the frequency and the field distribution of the mode in the multicell cavity. It has been observed, that for the mode, the change in the field amplitude of each cell is proportional to the frequency change by a factor of Sekutowicz1999 ; Nagle1967 . Thus, a small celltocell coupling increases the sensitivity of the field profile with respect to geometrical perturbations.
Iii Cavity Manufacturing
The European Xray Free Electron Laser XFEL_report facility is constructed to produce Xray pulses with the properties of laser light and at intensities much brighter than those produced by conventional synchrotron light sources. The superconducting linear accelerator of the EXFEL has a length of almost 2.1 km and brings electrons to an energy of up to . This is achieved by using a total of superconducting cavities installed in the three main linac sections and the injector. The production of cavities Singer2016 , the largest in the history of cavity production, was realized by the two companies Research Instruments GmbH (RI) and Ettore Zanon S.p.A. (EZ). EXFEL uses ninecell TESLA cavities build from solid niobium with a nominal MHz. Each cavity (see FIG. 2) consists of 10 main subcomponents, welded together at the equator area. The subcomponents consist of dumbbells (DB) and endgroups (EGL, EGS), referring to short and long endgroups, respectively. A different shape of the endgroups’ halfcells provides the desired asymmetry of the Higher Order Mode (HOM) field distributions and increases the efficiency of their extraction.
Geometric deviations of the inner cavity shape, the cavity length, and the spectra of frequencies as well as deviations in HOM field distributions occur due to random inaccuracies during manufacturing. These uncertainties have a strong impact on the quantities of interest described in Section II. Hence, dedicated measures are undertaken during production to ensure acceptable tolerances according to the EXFEL cavity specification, in particular to obtain and to keep the deviation of below . These measures, together with sources of uncertainty, are summarized in FIG. 3 and described in detail in the following.

Step: Production
In this step DBs, consisting of half cells each, EGSs and EGLs are produced. 
Step: Trimming
The target of this step is to compensate for shape deviations by trimming the components. It is applied to all components (DB, EGS and EGL) and allows to obtain the necessary cavity length and frequencies with required accuracy. 
Step: Selection and Sorting
The manufacturer selects two end groups and dumbbells. To minimize the influence of shape variations for the eight DBs on the asymmetry of the HOM field distribution, the DBs are sorted during the cavity assembly: a DB with average frequency is installed at the last position, the remaining DBs are installed in order of decreasing frequency (see FIG. 4). 
Step: Welding
All components are welded, which induces shape deformations. The materials from the different suppliers exhibit different shrinkages at the welding joint. The resulting equator diameters may be slightly different depending on the cavity position during welding, see Sulimov2015 . 
Step: Chemical treatment
The chemical treatment removes impurities and spikes, see Singer2016 , Sulimov2016 . The homogeneity of the removed material from the cavity surface depends on many parameters of the process and the facility. The electrochemical polishing treatment is usually less homogeneous and more unstable than the equator welding. 
Step: Tuning
The cavity is tuned, i.e., mechanically stretched or compressed, according to the procedure described by Kreps_1996aa which adjusts with an accuracy of and ensures for the field flatness. 
Step: Final preparation (for operation)
The procedures applied in this step vary for different manufacturers and are shown in detail in FIG. 2 of Singer2016 . Those procedures include, e.g., final buffered chemical polishing etching, the integration of the cavities into the Helium tanks, a pressure test using water under the pressure of bar and the cool down to . 
Step: Cavity
All cavities are operational and the statistics of the fundamental mode spectra are measured under . A description of the measurement procedures is given below.
Quality assurance by mechanical measurements of the inner surface dimensions becomes impossible after cavity welding and polishing. Measurement data can only be obtained by ultrasonic or RF measurements. These methods are used for control of equator welding stabilities Sulimov2015 or homogeneity of the polishing process Sulimov2016 . In the following the fundamental mode spectra are discussed for both manufactures based on RF measurements.
The measurements were carried out at operation temperature, i.e., cryogenic tests at 2 were used (FIG. 5). The measurement results are presented in Table 1. Note, that the mode finally operates at 1.3
, which is ensured by an additional tuning step (not described here). The results show that the standard deviation of the
mode (mode 8) in the relaxed condition after cool down, is about 50 . This value increases with decreasing mode number. A similar behavior can be found for the differencesof the averaged frequencies for each mode between the two manufacturers RI and EZ, which also increases with decreasing mode number, see Tab. 1. In particular, the difference of average frequencies is about 14 for mode 8 and 1098 for mode 0, respectively. These results indicate a high deviation of the celltocell coupling coefficient between the manufacturers with
which are given in the form of average value plus/minus one standard deviation as obtained from database, see also Tab. 4. Preliminary studies Sulimov2013 show that only variations of the iris radius can explain such high deviation in the celltocell coupling coefficient. This will be analysed in more detail numerically in Section IV. Finally, the field flatness distribution for the EXFEL cavities is presented in FIG. 6. It can be observed that at least of the cavities possess a field flatness of more than .
Mode  Frequency, MHz  

Average  St Dev  
RI  EZ  RI  EZ  
0  1275.832  1274.734  0.226  0.268 
1  1277.951  1276.926  0.212  0.245 
2  1281.194  1280.294  0.186  0.208 
3  1285.178  1284.460  0.157  0.168 
4  1289.423  1288.921  0.124  0.129 
5  1293.442  1293.161  0.089  0.090 
6  1296.762  1296.649  0.064  0.066 
7  1298.944  1298.946  0.052  0.054 
8  1299.702  1299.716  0.050  0.052 
Iv Uncertainty quantification
A stochastic setting is adopted here to model manufacturing and measurement uncertainties and assess their influence on the cavity design, i.e. each component of
becomes a random variable
(7) 
where denotes a random outcome.
Uncertainty quantification encompasses various methods for uncertainty propagation, Bayesian inverse problems, optimal experimental design and robust optimization, among others. The reader is referred to ghanem1991 ; xiu2010 ; lemaitre2010
for a detailed background. In this work, uncertainty propagation is of great interest, in particular, propagating distributions of cell deformation parameters to distributions of quantities of interest, such as the celltocell coupling coefficient. This can be achieved by sampling according to the underlying distribution and repetitively solving the cavity eigenvalue problem. Thereby, surrogate modeling is a key step to keep the computational workload manageable. The probability distributions of the cavity geometry parameters are modeled based on descriptions of the manufacturing process, as described in Section
III. A more general approach would consist in inferring these input distributions from measured RF data, which would require the solution of an inverse cavity eigenvalue problem. Such a study is out of the scope of the present work and can only be briefly addressed here, applying significant simplifications. In fact, the present study, should be considered as a step towards a more complete treatment of uncertainties in the cavity eigenvalue problem.In this paper, calculations are carried out with the help of Superlans code SLANS . Superlans is a 2Daxisymmetric finiteelementbased code used for the calculation of the monopole modes of azimuthally symmetric geometries. For each simulation, the data describing the contour of the cavity is created by Matlab Matlab and saved in a format readable by Superlans. Superlans is then called by Matlab to triangulate the geometry, solve the resulting eigenvalue problem and calculate the relevant secondary parameters. The results are finally read by Matlab for the postprocessing. The detailed description of the uncertainty modeling steps, which are the numerical counterpart of the production chain in Section III, are as following:

Step: Production
We generate random (”virtual”) cavities by creating, in turn, seven independent random mid cells and two end cells per cavity. In particular, those virtual cavities are obtained by drawing random numbers for the vector defined in (7) where we assume independence of the random variables. Although, the normal distribution seems to be the right choice, we opt for beta distribution in the range
to . Beta distributions can be used to approximate normal distributions but have bounded support (xiu2010, , Appendix B), which is very important in numerical studies to avoid nonphysical parameter configurations. FIG. 7represents such an approximation with probability density function (PDF)
where and denote the lower and upper bound, respectively. In this particular case the shape parameters of the beta distribution were chosen such that a normal distribution with a interval of to is approximated, as illustrated in FIG. 7.

Step: Trimming
In contrast to real manufacturing, in the simulation approach, we build cavities out of elementary cells instead of dumbbells. In this setting, it is not clear how to explicitly model the trimming, however it is implicitly taken into account by the cavity length limitation (8) which is considered in the next step. 
Step: Selection and Sorting
The virtual 9cell cavity is obtained from the random building blocks of step 1 as follows: We apply a sorting procedure which computes for all middle cells the fundamental resonance frequency (by solving a onecell eigenvalue problem), places the cell closest to the average frequency at the 8 position and orders the remaining cells according to decreasing frequency between positions 2 and 7. Additionally, a total length constraint(8) is enforced. If the constraint is violated after tuning (step 6), the corresponding virtual cavity is disregarded. In real manufacturing, the constraint is already incorporated by trimming and compensation effects.
This selection and sorting influences the probability distribution. Hence, we introduce a new random vector
where the density is estimated with kernel density techniques. In particular, we employ an Epanechnikov kernel
Epanechnikov on the selected (and sorted) sample of size , which complies with the length constraint. The estimated densities are presented in Fig. 8. 
Step: Welding
Again, since we consider cells instead of dumbbells for the simulation, deformations during the welding procedure cannot be modeled properly and are therefore neglected. 
Step: Chemical treatment
An appropriate modeling of chemical treatment would require very fine resolutions or even multiscale analyses which is beyond the scope of the present paper. 
Step: Tuning
The virtual tuning procedure we apply, considers each cell individually and is therefore unable to incorporate field flatness constraints directly. However, we observe that an acceptable field flatness of at least was obtained for all eigenvalue problems that are eventually solved. In particular, each cell is tuned to by changing its length . Computationally, this requires the solution of a nonlinear root finding problem for the objective functionwhere denotes the fundamental eigenfrequency of cell (onecell eigenvalue problem). The root finding problem is solved using fzero in Matlab.
In FIG. 9 the values of the 9cell cavity accelerating frequency are depicted for different choices of the length and equatorial radius of the first cell. Considering the contour (in black in the figure), we observe that the tuning process correctly identifies a length such that a value of is obtained (the magenta dots in the figure lie on the contour line as expected).

Step: Final preparation (for operation)
Due to insufficient data and involved numerical modeling this step has to be neglected. 
Step: Cavity
The Maxwell eigenvalue problem is solved to obtain the first nine modes for each virtual cavity and statistics of the spectra are computed. It shall be noted that, in general, eigenvalue tracking Georg_2019aa should be employed to ensure a consistent matching of the eigenfrequencies. However, it has been observed that, in this case, the fundamental eigenfrequencies do not cross with respect to parameter changes for the considered variations. Hence, mode tracking is not applied here. The flow diagram of simulation steps is given in Fig. 10.
iv.1 Stochastic Computations
In order to avoid the tremendous computational cost of repeatedly solving a large number of eigenvalue problems in step 3 and step 8, surrogate modeling is employed. In this work a dimensionadaptive Leja scheme narayan2014 is used to construct accurate polynomial surrogate models. In particular, the algorithm described in (georg2018, , Chapter 3.2) is adapted to address the case of multiple quantities of interest. In order to obtain high uniform accuracies, we employ the classical unweighted Leja points. This algorithm constructs polynomial models adaptively which yields high computational efficiency. Moreover, we control the associated approximation errors by crossvalidation to be sufficiently small.
The 3variate polynomial surrogate models employed in step 3 are computed by solving 50 onecell eigenvalue problems while 500 evaluations of the 9cell eigenvalue problem are used to construct the 10variate polynomial surrogate model for step 8. For all surrogate models, crossvalidation with a uniformly distributed random sample of size 1000 indicates an error below for all fundamental resonance frequencies which is smaller than the standard expected deviations.
Now, all statistical quantities of interest can be obtained from the sample computed in step 8. In particular, we employ unbiased statistical estimates for the moments. The first two moments are depicted in FIG.
11 and Table 2.In the same way, we estimate the first four statistical moments of the celltocell coupling coefficient , see Table 3. To investigate the influence of the stochastic dependency in the input parameters, we additionally construct a sample of size containing independent realizations. This is achieved by sampling from the kerneldensities illustrated in FIG. 8. By, again, evaluating the surrogate model and using Monte Carlo estimates for this sample, the last column in Tab. 3 is obtained. Although, deviations can be observed, particularly in the even moments, the differences are small enough to resort to an independent analysis. Therefore, in the following, we employ Sobol indices as sensitivity measure, despite the fact that they are defined for independent parameters. Generalizations of Sobol indices for correlated data xu2007 exist but are not considered here. We use the OpenTurns implementation of Saltelli’s algorithm saltelli2002 to estimate Sobol indices of the celltocell coupling coefficient for the input parameters . The results presented in FIG. 12 are obtained by evaluating times the surrogate model. As expected, the celltocell coupling coefficient is heavily influenced by the iris radius while the equatorial radii have significantly less impact. In particular, more than of the variation can be attributed to changes in the iris radius.
Moment  Stoch. dependent  Mut. independent 

Mean  1.82807  1.82807 
Variance  0.00045  0.00058 
Skewness  0.01252  0.01336 
Kurtosis  0.41521  0.33525 
To conduct a preliminary inverse analysis, based on the previous findings, we neglect the deformations in the equatorial radii in the following. In this case, the task consists in estimating the parameter from measurements of the celltocell coupling coefficient . FIG. 13 depicts the associated relation which appears to be monotonic and almost linear in the considered range. We collected measurements of the fundamental mode spectra for cavities from the DESY Database desy_db . Parameter estimation is then carried out by numerically inverting . This is implemented by reformulating the rootfinding problem as an optimization problem and applying the scipy implementation of the LBFGSB algorithm. Sample statistics are presented in Table 4.
The XFEL specification requires deviations in the iris radii before welding to be below with respect to the nominal value. It is expected that the mean values and deviations of the iris radii change during the production chain, for example the chemical treatment is considered to have a significant influence. This offset is estimated by our numerical model to be , cf. last row of Table 4. Despite this offset, the standard deviation from both vendors are still within the limits of the specification () after the full production chain. When considering the vendors separately, then both production processes, standard deviations and , operate approximately at a threesigma level.
Manufacturer  

RI  1.854  0.016  
EZ  1.941  0.021  
RI+EZ  1.897  0.047 
V Conclusion
In this contribution the manufacturing chain of the EXFEL cavity was summarized and translated into a simulation workflow considering uncertainties. To analyze the sensitivities of the uncertain parameters, we propose an efficient adaptive surrogate modeling technique. The numerical study confirms the expert knowledge that the iris radius is the most critical parameter for the celltocell coupling coefficient. Finally, the surrogate model is used to infer the first moments of the iris radius variations from frequency measurements. For both manufacturers the obtained standard deviations are within the specification.
The practically very relevant determination of stochastic moments for all design parameters from given measurements is still ongoing research, as well as the investigation of global sensitivities for HOM.
Acknowledgements.
The authors would like to acknowledge the support by the DFG (German Research Foundation) in the framework of the Scientific Network SCHM 3127/1,2 ”Uncertainty quantification techniques and stochastic models for superconducting radio frequency cavities” that provided the basis for this collaborative work. The work of J. Corno, N. Georg and S. Schöps is supported by the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universitat Darmstadt. N. Georg’s work is also funded by the DFG grant RO4937/11. The work of S. Gorgi Zadeh was/is supported by the German Federal Ministry for Research and Education BMBF under contracts 05H15HRRBA and 05H18HRRB1, respectively. J. Heller’s work was supported by the BMBF under the contracts 05K13HR1 and 05K16HRA.References
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