• Very high electromagnetic fields - maximum magnetic field on the inner cavity surface up to B_pk =100mT, maximum electric field on the inner cavity surface up to E_pk =50MV/m. These high field result in the strong Lorenz forces which cause the cavity wall deformation; the cavity wall deformation;
• Low temperature - down to 2K, that again causes wall displacements and inner volume change after cool down;
• The pulse regime of operation that results in the addition requirements on cavity rigidity;
• High vacuum conditions (10^-9-10^-10) and extra pressure on cavity walls from the helium tank also deform the cavity shape;
• High tolerances and quality surface requirements.

Generally there is a clear separation between electromagnetic and the structural cavity design. Separate tasks often lead to designs that conflict and do not take into account each others advantages and strong points. Considering cavity design as one integrated task, where the electromagnetic fields, heat transfer, strong structure cool-down and external pressures are all combined into one multi-physics problem and where iterations and optimisations are integrated into the design, is not only possible today but can be extended into areas yet unexplored. In this paper we describe the fundamentals and procedure of SC cavity design and the integration between different cavity analyses.

The simulation model for electrodynamics is the cavity inner volume. For the system structural analysis the simulation model is the cavity walls together with cryo-module environments that could be simulated with shell and, or volume elements. The most important issue during such kind of calculations is the ability to exchange results from one type of modelling to another using them in this case as the input data and back with so-called coupled analysis [1]. The full cavity shape parameterization is highly required for both electromagnetic and mechanics.

The accelerating field, E_acc, is proportional to the peak electric (E_pk) as well as magnetic field (H_pk) on the surface of the cavity. In some analyses the peak electromagnetic fields on the internal cavity surface are used as the reference data for further normalizations. It means the proper peak field definitions define the accuracy of the results and require strong attention to the model meshing.

An acceptable mesh creation is an iterative process. All simulation results including electromagnetic, surface heat flux, etc. are highly dependent on the mesh density. That is why the usual procedure is to create a fine mesh in critical areas on the surfaces, while retaining a larger mesh in not so important places of the body in order to reduce run time and memory usage. A simple way to achieve this mesh variation is to divide the vacuum volume into sub-volumes depending on the required local mesh size. In this way, not only the surface mesh can be controlled by sizing areas and lines but the "global" mesh size can be set on the local basis for each sub-volume, resulting in better mesh control. The final mesh adjustment should be provided taking into account the real field distributions in the cavity.

All SC RF resonators are niobium cavities that are enclosed within helium vessels. These vessels are filled with liquid helium that floods the cavities and maintain the 2K operating temperature. Mechanical analysis consists of design calculations for all critical cavity assembly components, cavity tuning sensitivity analysis, active tuner and bench tuner load determination, cavity assembly cool-down analysis, natural frequency and random response analysis, inertia friction weld analysis and critical flaw size calculations.

The high repetition rate like the 50 Hz of an accelerator will require a closer look to the mechanical resonances of the cavities. Mechanical resonances can influence the phase behavior of the cavity during a pulse, which can hardly be compensated by a good control system, even if a lot of additional power is available. Additionally, the cavity RF resonance is sensitive to vibrations of sub-μm amplitudes. These microphonic effects cause low frequency noise in the accelerating fields. Therefore, a careful mechanic eigen-mode analysis of the cavity together with its environments should be conducted.

Lorentz force cavity detuning is a function of the RF field, which is forcing term, mechanical mode frequency, modal mass and mode's damping degree. However, findings of mode frequencies, corresponding stiffness and especially damping degrees are quite difficult for the real situation, since these dynamic properties are very sensitive to the boundary conditions such as connection scheme, strength, equivalent masses and equivalent stiffness of surroundings that is attached to the cavity. Only the relative comparisons are available before having experimental measurements of mechanical properties with actual cryo-module. Even after having measured values about dynamic mechanical properties of cavity, the predictions sometimes are not accurate with a conventional RF modeling, since RF fields and mechanical vibrations are strongly coupled and both are dynamic.

1

E. Zaplatin, "FZJ SC Cavity coupled analyses", Workshop on RF Superconductivity SRF06, 2005, Ithaca, USA.

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