| Περίληψη: | The topic of this thesis is the development of two hybrid numerical methods for the solution of nonlinear problems dealing with magnetic field analysis in superconducting accelerator magnets. The first method is a combination of Finite Element Method (FEM) and Boundary Element Method (BEM). FEM is an ideal method for modeling the material nonlinearity of the iron yoke, while BEM is convenient in treating the linear exterior air domain with the superconducting coils. The source magnetic field produced by the superconducting coils is calculated with high accuracy via a Biot–Savart integral, thus, a very dense volume discretization between the coil blocks is avoided. The other hybrid numerical method combines the Fragile Points Method (FPM) and the Boundary Element Method. The FPM is a Galerkin-type meshless method employing for field approximation simple, local, and discontinuous point–based interpolation functions. Because of the discontinuity of these functions, the FPM can be considered as a meshless discontinuous Galerkin formulation where numerical flux corrections are employed for the treatment of local discontinuity inconsistencies. The FPM possesses the advantages of a mesh-free method, evaluates with high accuracy field gradients, and treats nonlinear magnetostatic problems by utilizing, for the same problem, fewer nodal points than the Finite Element Method (FEM). Therefore, in the work of the present thesis, the nonlinear ferromagnetic material of an accelerator magnet is treated by the FPM, while the BEM is employed for the infinitely extended, surrounding air space. Both proposed schemes are based on the scalar potential formulation of \cite{Mayergoyz}. This formulation overcomes the main issues of the interfacial jump condition difficulty and cancellation errors appearing in the total and reduced scalar potential formulations, respectively. Furthermore, in comparison with nodal vector potential formulations, the computational cost is reduced in 3D problems, as only one unknown per node is required. The applicability of both methods is demonstrated with the solution of representative 2-D and 3-D nonlinear magnetostatic problems dealing with different types of accelerator magnets and the obtained numerical results are compared to those provided by the commercial FEM package ANSYS. Finally, the elastic deformation of an accelerator magnet, due to the body force produced by the superconducting coils, is estimated for 2D cases. For this purpose, the FPM is demonstrated and developed for the solution of 2D elasticity problems for accelerator magnets.
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