Transient solutions to nonlinear acousto‐magneto‐mechanical coupling for axisymmetric MRI scanner design

Abstract

Summary In this work, we simulate the coupled physics describing a magnetic resonance imaging (MRI) scanner by using a higher‐order finite element discretisation and a Newton‐Raphson algorithm. To apply the latter, a linearisation of the nonlinear system of equations is necessary, and we consider two alternative approaches. In the first approach, ie, the nonlinear approach, there is no approximation from a physical standpoint, and the linearisation is performed about the current solution. In the second approach, ie, the linearised approach, we realise that the MRI problem can be described by small dynamic fluctuations about a dominant static solution and linearise about the latter. The linearised approach permits solutions in the frequency domain and provides a computationally efficient way to solve this challenging problem, as it allows the tangent stiffness matrix to be inverted independently of time or frequency. We focus on transient solutions to the coupled system of equations and address the following two important questions: (i) how good is the agreement between the computationally efficient linearised approach compared with the intensive nonlinear approach and (ii) over what range of MRI operating conditions can the linearised approach be expected to provide acceptable results for outputs of interest in an industrial context for MRI scanner design? We include a set of academic and industrially relevant examples to benchmark and illustrate our approach.

A High Order Finite Element Coupled Multi-Physics Approach To MRI Scanner Design

“…More recent works [26,150,93,120,121] have utilised an alternative approach, which avoids the direct computation of electromotive forces and instead works directly with a (physically motivated) Maxwell stress tensor. This work permits a more rigorous treatment of the coupling through a complete linearisation of the coupled system [26,25].…”

“…These displacements are typically of the order of microns [25] and the conductors can therefore be assumed to behave elastically, given the small strains developed.…”

“…This linearisation technique, known as the linearised approach, in the context of MRI scanners, is motivated by the knowledge that the static DC current source J DC is several orders of magnitude stronger than the weaker AC time varying source J AC , leading to a strong DC field and a weaker time varying AC field [25].…”

“…and V 3 = (−1, 0) defined in the referential coordinate system, given as 25) in the reference domain Ω ξ . The hierarchic set of low (vertex) and high order (edge, interior) modal shape functions, generated on the reference triangular element, are defined in [217,256,119] and Appendix E. …”

“…Thus, to describe the acoustic behaviour of an MRI scanner the acoustic wave equation under sourcing must be developed to form part of the fully coupled system. The equation describing the acoustic behaviour is obtained by first considering the relationship between the pressure P and displacement field in the non-conducting medium by 25) where κ defines the bulk modulus of the acoustic (free space) medium. This definition corresponds to the hydrostatic mechanical part of the Cauchy stress tensor From this relationship the free space region surrounding the scanner can now be modelled as a solid, governed by the elasticity equation …”