Finite Element Formulation of the Bloch Equations with Dipolar Field Effects

TL;DR

This paper introduces a finite element method for solving the Bloch equations with dipolar field effects, enabling local real-space solutions and reducing computational errors compared to Fourier-based approaches.

Contribution

It presents a novel FEM formulation for the Bloch equations with dipolar effects that simplifies calculations and improves accuracy by avoiding repeated dipolar field computations.

Findings

Local real-space solutions reduce global truncation errors.

Dipolar field calculated once as initial condition.

Enhanced accuracy in simulating magnetic resonance phenomena.

Abstract

A Galerkin finite element (FEM) formulation for the Bloch equations with dipolar field is presented which makes possible the derivation of weak solutions to the Bloch equations. The FEM formulation has the advantage that the equations of motion are local in real space, eliminating the global truncation errors associated with calculations of the dipolar field in Fourier space. The dipolar field and other geometric parameters are calculated only once, before the simulation, and used as an initial condition rather than re-calculated at every time step of some numerical integration.