TEMF, TU Darmstadt, Darmstadt, Germany
The design and optimization of particle accelerator components are fundamentally based on beam dynamics simulations. The knowledge of the interaction of moving charged particles with the surrounding materials and fields enables to optimize individual devices and consequently to take the best advantage of the entire machine. Among the essential accelerator components are radio-frequency cavities which are utilized for acceleration as well as for beam diagnostics. In these applications, precise beam dynamics simulations urgently require high-precision data of the electromagnetic fields. Numerical simulations based on Maxwell’s equations have to represent the resulting fields on an acceptable level of quality even with limited amount of degrees of freedom. On the other hand, the particle beam itself gives rise to the excitation of undesired modes which have to be extracted from the cavities. In the current work, some of the challenges faced in high precision cavity simulations are summarized. Based on high-performance eigenvalue calculations, important features like "low-noise" field evaluations or port-mode boundary approximations to enable traveling-wave transport are addressed.
Rostock University, Faculty of Computer Science and Electrical Engineering, Rostock, Germany
Geometry perturbations affect the eigenmodes of a resonant cavity and thereby can improve but also impair the performance characteristics of the cavity. To investigate the effects of both, intentional and inevitable geometry variations parameter studies are to be undertaken. Using common eigenmode solvers involves to perform a full eigenmode computation for each variation step, even if the geometry is only slightly altered. Therefore, such investigations tend to be computationally extensive and inefficient. Yet, the computational effort for parameter studies may be significantly reduced by using perturbative computation methods. Knowing a set of initial eigenmodes of the unperturbed geometry these allow for the expansion of the eigenmodes of the perturbed geometry in terms of the unperturbed modes. In this paper, we study the complexity of a numerical implementation of perturbative methods. An essential aspect is the computation and analysis of the unperturbed modes since the number and order of these modes determine the accuracy of the results.
Rostock University, Faculty of Computer Science and Electrical Engineering, Rostock, Germany
TUAAI1
TEMF, TU Darmstadt, Darmstadt, Germany
RF sources represent a key component for FEL radiation sources. The most important tool for their analysis is numerical simulation. However, the broad spectrum of phenomena involved, including space-charge interaction, retardation effects as well as cavity wakefields, make this type of simulations extremely demanding. Beam dynamics simulation codes are faced with the challenge of reproducing all these effects occurring at different temporal and spatial scales while using a possibly small amount of computational resources and simulation times. According to the physical assumptions employed, these codes may be categorized as
- space-charge codes in the boosted frame,
- codes based on the Lienard-Wiechert interaction of individual particles,
- Particle-In-Cell codes and
- wakefield codes. In this work, a comparative study of these simulation approaches will be presented.
DESY, Hamburg, Germany
HZB, Berlin, Germany
LBNL, Berkeley, California, USA
An efficient numerical method for computing wakefields due to coherent synchrotron radiation (CSR) has been implemented using a one-dimensional integrated Green function approach. The contribution from CSR that is generated upstream and propagates across one or more lattice elements before interacting with the bunch is included. This method does not require computing the derivative of the longitudinal charge density, and accurately includes the short-range behavior of the CSR interaction. As an application of this method, we examine the importance of upstream transient wakefields within several bending elements of a proposed Next Generation Light Source.
UNM, Albuquerque, New Mexico, USA
BNL, Upton, Long Island, New York, USA
In our self-consistent algorithm for calculating 2D CSR effects we reduce the field calculation to a 2D integral over the 2D charge and current densities of the bunch and their time history. Our code VM3@A (Vlasov-Maxwell Monte-carlo Method @ Albuquerque) implements this in a time stepping algorithm as discussed in PRST-AB 12, 080704 (2009). A major expense is the integration over history at each time step. By going to Fourier space the 2D integral is reduced to a 1D convolution over history. This may on its own have a computational advantage, however, using the kernel compression technique of Alpert, Greengard and Hagstrom [1, 2], we approximate the convolution kernel by a sum of exponentials. This allows a time step to be taken using information only from the previous time step, thus eliminating the integral over history. Of course 2D Fourier transforms must be calculated at each step, these can be done with an FFT (or NFFT). We discuss the flop count for the two approaches. In addition we implement this as an option in VM3@A and compare efficiencies of our new and old approaches in the context of a bunch compressor system for the LCLS.
[1] SIAM J. Numer. Anal. 37(2000) 1138. See also PhD thesis at http://web.njit.edu/~jiang/pub.html
[2] S. R. Lau, J. Math. Phys. 46, 102503, (2005). Supported by DE-FG02-99ER41104
WESAI1
CERN, Geneva, Switzerland
IS, Beijing, People's Republic of China
IMP, Lanzhou, People's Republic of China
ANL, Argonne, USA
Electromagnetic simulations are fundamental for accelerator modeling. In this paper two high-order numerical methods will be studied. These include continuous Galerkin (CG) method with vector bases, and discontinuous Galerkin (DG) method with nodal bases. Both methods apply domain decomposition method for the parallelization. Due to the difference in the numerical methods, these methods have different performance in speed and accuracy. DG method on unstructured grid has the advantages of easy parallelization, good scalability, and strong capability to handle complex geometries. Benchmarks of these methods will be shown on simple geometries in detail first. Then they will be applied for simulation in accelerator devices, and the results will be compared and discussed.
KU Leuven, Kortrijk, Belgium
The dependence of the eigenfrequencies of a superconductive cavity on its geometry are represented by a stochastic response surface model. The model is constructed on the basis of both information on the eigenfrequencies as on their sensitivities with respect to the geometry. The eigenmodes are calculated using the 2D or 3D finite element method or finite integration technique. The stochastic representation does not only model uncertainties on the geometrical parameters but also inaccuracies of the eigenmode solvers, e.g. due to remeshing. Variations or optimisations of the geometry are carried out on the surrogate model. The model allows an efficient evaluation of microphoning and Lorentz detuning of accelerator cavities.
PSI, Villigen, Switzerland
ETH, Zurich, Switzerland
The new X-ray Free Electron Laser (SwissFEL) at the Paul Scherrer Institute (PSI) employs, among many other radio frequency elements, a transverse deflecting cavity for beam diagnostics. Since the fabrication process is expensive, an accurate 3-D eigenmodal analysis is indispensable. The software package Femaxx has been developed for solving large scale eigenvalue problems on distributed memory parallel computers. Usually, it is sufficient to assume that the tangential electric field vanishes on the cavity wall. To better approximate reality, we consider the cavity wall conductivity is large but finite, and thus the tangential electrical field on the wall is nonzero. We use the surface impedance boundary conditions (SIBC) arising from the skin-effect model. The resulting nonlinear eigenvalue problem is solved with a nonlinear Jacobi–Davidson method. We demonstrate the performance of the method. First, we investigate the fundamental mode of a pillbox cavity. We study resonance, skin depth and quality factor as a function of the cavity wall conductivity. Second, we analyze the transverse deflecting cavity to assess the capability of the method for technologically relevant problems.
CERN, Geneva, Switzerland
EPFL, Lausanne, Switzerland
Rostock University, Faculty of Computer Science and Electrical Engineering, Rostock, Germany
The prediction of RF properties of complex accelerating structures is an important issue in computational accelerator physics. This paper describes the derivation of state space equations for complex structures based on real eigenmodes of sections of the decomposed complex structure. The state space equations enable the calculation of system responses due to port excitations by means of standard ordinary differential equation solvers. Therefore, the state space equations are referred to as lumped equivalent models of such complex RF structures. Besides fast computation of system responses, the equivalent models enable the calculation of secondary quantities such as external quality factors. The present contribution discusses theoretical aspects and illustrates an application example.
TEMF, TU Darmstadt, Darmstadt, Germany
For acceleration of charged particles at the heavy-ion synchrotron at the GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt two ferrite-loaded cavity resonators are installed within the ring. Their eigenfrequency can be tuned by properly choosing a bias current and thereby modifying the differential permeability of the ferrite material. The goal of the presented work is to numerically determine the lowest eigensolutions of accelerating ferrite-loaded cavities based on the Finite Integration Technique. The newly developed solver includes two subcomponents: Firstly, a magnetostatic solver supporting nonlinear material for the computation of the magnetic field which is excited by the specified bias current. This enables to linearize the constitutive equation for the ferrite material at the current working point, at which also the differential permeability tensor is evaluated. Secondly, a Jacobi-Davidson type eigensolver for the subsequent solution of the nonlinear eigenvalue problem. Particular emphasis is put on the implementation to enable efficient distributed parallel computing. First numerical results for biased ferrite-filled cavity resonators are presented.
Rostock University, Faculty of Computer Science and Electrical Engineering, Rostock, Germany
For the design and optimization of Higher-Order-Mode Coupler, used in RF accelerator structures, numerical computations of electromagnetic fields as well as scattering parameter are essential. These computations can be carried out in time domain. In this work the implementation and investigation of a time integration scheme, using the Arbitrary high-order DERivatives (ADER) approach, applied on the Discontinuous Galerkin finite-element method (DG-FEM) is demonstrated for solving 3-D electromagnetic problems in time domain. This scheme combines the advantage of high accuracy with the possibility of an efficient implementation as local time stepping scheme, which reduces the calculation time for special applications considerable. It is implemented in NUDG++*, a framework written in C++ that deals with the DG-FEM for spatial discretization of the Maxwell equations. Accuracy and performance is analyzed by a suitable benchmark.
* Nodal Unstructured Discontinuous Galerkin in C++