Author(s)

ANDRÉ BUCHAU, JENS ANDERS

Abstract

The goal of the presented study is to provide a systematic approach for the efficient characterization of vector fields inside a defined region of interest. That means the vector field is described there with a set of coefficients that can be easily derived from the field values and that contains enough information to characterize the vector field accurately. A possible field of application of this approach is the design of defined distributions of vector fields for specific use cases based on optimization algorithms or machine learning approaches. For instance, the homogeneity of the magnetic B-field is an important measure in the context of nuclear magnetic resonance spectroscopy since it directly limits the achievable spectral resolution and applicability of this method. Here, we present a new combination of established techniques of modern boundary element methods, which are typically used for the solution of the field problem, with automatic analysis of the so-called local expansion of the fast multipole method to characterize a vector field based on a robust approach. The local expansion represents the field inside a defined domain, and the effect of all field sources outside this domain is replaced by a small set of local coefficients. Hence, we first discuss the meaning of these local coefficients and then show how they can be computed directly by a smart use of Green’s theorem. Finally, we show the spectrum of local coefficients, which, in the next step, is the basis for a cost function of an optimization problem of the studied vector field.

Keywords

Laplace’s equation, boundary element method, fast multipole method, homogeneous fields, spherical harmonics, vector field characterization