Michal Maciejewski

The design specification of future high-field accelerator magnets require innovation in and integration of multiple disciplines. Numerical models underpin each step of the design. In this paper, we present a methodology for collaborative modeling based on relevant concepts of model-based systems engineering. The methodology is composed of three pillars: encapsulated computing environments with service query interface, model notebooks with auto-generated model views, and model query interface with results caching. The methodology aims at a de-centralized approach to multi-model and multi-scale collaboration that maintains the cornerstones of traceability and reproducibility. It is demonstrated with a multi-model coil design optimization of a high-field superconducting magnet.

The cross-sectiondesign of $\cos (\theta)$ superconducting magnets is historically developed in a two-step process: initially, the coil geometry is defined on the basis of magnetic optimizations; then, the structure is designed around the coil. The first step searches for the best coil cross-section maximizing the magnetic field, margin, field quality and conductor efficiency. The latter step aims at limiting the coil stresses and deformations. However, the coil design, defined with the initial magnetic optimization, can influence the mechanical behaviour of the magnet, altering, for example, the peak stress during operation. As the critical current is a function of the applied strain, the mechanical implications of the coil cross-section design can limit the achievable performance. In this paper we propose an integrated optimization process that targets the peak stress on the conductor in addition to the magnetic objectives. The results are presented for a sample high-field Nb $_{3}$ Sn dipole: a 4-layer design aiming at ultimate conductor performance.

The application of high-temperature superconductors to accelerator magnets for future particle colliders is under study. Numerical methods are crucial for an accurate evaluation of the complex dynamical behavior of the magnets, especially concerning the magnetic field quality and thermal behavior. We present a coupled A-H field formulation for the analysis of magneto-thermal transients in accelerator magnets. The magnetic field strength H accounts for the eddy current problem in the source regions containing the superconducting domains, while the magnetic vector potential A represents the magnetoquasistatic problem in the normal and non-conducting domains. Furthermore, we include a thin shell approximation for the source regions, making the formulation suitable for large scale models composed of thousands of tapes. In this work, the relevant equations are derived and discussed, with emphasis on the coupling conditions. The weak formulation is derived, and numerical results are provided in order to both, verify the formulation and scale it to the size of an accelerator magnet.

In this paper we present an algorithm for the coupling of magneto-thermal and mechanical finite element models representing superconducting accelerator magnets. The mechanical models are used during the design of the mechanical structure as well as the optimization of the magnetic field quality under nominal conditions. The magneto-thermal models allow for the analysis of transient phenomena occurring during quench initiation, propagation, and protection. Mechanical analysis of quenching magnets is of high importance considering the design of new protection systems and the study of new superconductor types. We use field/circuit coupling to determine temperature and electromagnetic force evolution during the magnet discharge. These quantities are provided as a load to existing mechanical models. The models are discretized with different meshes and, therefore, we employ a mesh-based interpolation method to exchange coupled quantities. The coupling algorithm is illustrated with a simulation of a mechanical response of a standalone high-field dipole magnet protected with CLIQ (Coupling-Loss Induced Quench) technology.

Simulating the transient effects occurring in superconducting accelerator magnet circuits requires including the mutual electro-thermo-dynamic interaction among the circuit elements, such as power converters, magnets, and protection systems. Nevertheless, the numerical analysis is traditionally done separately for each element in the circuit, leading to possible inconsistent results. We present STEAM, a hierarchical cosimulation framework featuring the waveform relaxation method. The framework simulates a complex system as a composition of simpler, independent models that exchange information. The convergence of the coupling algorithm ensures the consistency of the solution. The modularity of the framework allows integrating models developed with both proprietary and in-house tools. The framework implements a user-customizable hierarchical algorithm to schedule how models participate to the cosimulation, for the purpose of using computational resources efficiently. As a case study, a quench scenario is cosimulated for the inner triplet circuit for the high luminosity upgrade of the Large Hadron Collider at CERN.

In this paper we present the co-simulation of a PID class power converter controller and an electrical circuit by means of the waveform relaxation technique. The simulation of the controller model is characterized by a fixed-time stepping scheme reflecting its digital implementation, whereas a circuit simulation usually employs an adaptive time stepping scheme in order to account for a wide range of time constants within the circuit model. In order to maintain the characteristic of both models as well as to facilitate model replacement, we treat them separately by means of input/output relations and propose an application of a waveform relaxation algorithm. Furthermore, the maximum and minimum number of iterations of the proposed algorithm are mathematically analyzed. The concept of controller/circuit coupling is illustrated by an example of the co-simulation of a PI power converter controller and a model of the main dipole circuit of the Large Hadron Collider.

In this paper we present the co-simulation of a PID class power converter controller and an electrical circuit by means of the waveform relaxation technique. The simulation of the controller model is characterized by a fixed-time stepping scheme reflecting its digital implementation, whereas a circuit simulation usually employs an adaptive time stepping scheme in order to account for a wide range of time constants within the circuit model. In order to maintain the characteristic of both models as well as to facilitate model replacement, we treat them separately by means of input/output relations and propose an application of a waveform relaxation algorithm. Furthermore, the maximum and minimum number of iterations of the proposed algorithm are mathematically analyzed. The concept of controller/circuit coupling is illustrated by an example of the co-simulation of a PI power converter controller and a model of the main dipole circuit of the Large Hadron Collider.