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

Transcription

1 Reeived: 17 November 2017 Revised: 10 February 2018 Aepted: 9 Marh 2018 DOI: /nme.5802 RESEARCH ARTICLE Transient solutions to nonlinear aousto-magneto-mehanial oupling for axisymmetri MRI sanner design S. Bagwell 1 P.D. Ledger 1 A.J. Gil 1 M. Mallett 2 1 Zienkiewiz Centre for Computational Engineering, College of Engineering, Swansea University Bay Campus, Swansea, UK 2 Siemens Healthare Ltd., MR Magnet Tehnology, Witney, Oxon, UK Correspondene P.D. Ledger, Zienkiewiz Centre for Computational Engineering, College of Engineering, Swansea University Bay Campus, Swansea SA1 8EN, UK. p.d.ledger@swansea.a.uk Funding information Engineering and Physial Sienes Researh Counil EPSRC and Siemens Magnet Tehnology, Grant/Award Number: EP/L505699/1; EPSRC, Grant/Award Number: EP/R002134/1; Sêr Cymru National Researh Network in Advaned Engineering and Materials; European Training Network AdMoRe, Grant/Award Number: Summary In this work, we simulate the oupled physis desribing a magneti resonane imaging MRI sanner by using a higher-order finite element disretisation and a Newton-Raphson algorithm. To apply the latter, a linearisation of the nonlinear system of equations is neessary, and we onsider two alternative approahes. In the first approah, ie, the nonlinear approah, there is no approximation from a physial standpoint, and the linearisation is performed about the urrent solution.inthe seondapproah,ie,thelinearised approah, we realise that the MRI problem an be desribed by small dynami flutuations about a dominant stati solution and linearise about the latter. The linearised approah permits solutions in the frequeny domain and provides a omputationally effiient way to solve this hallenging problem, as it allows the tangent stiffness matrix to be inverted independently of time or frequeny. We fous on transient solutions to the oupled system of equations and address the following two important questions: i how good is the agreement between the omputationally effiient linearised approah ompared with the intensive nonlinear approah and ii over what range of MRI operating onditions an the linearised approah be expeted to provide aeptable results for outputs of interest in an industrial ontext for MRI sanner design? We inlude a set of aademi and industrially relevant examples to benhmark and illustrate our approah. KEYWORDS aousto-magneto-mehanial oupling, linearisation, MRI sanner, multifield systems, Newton methods, time integration impliit 1 INTRODUCTION In reent years, magneti resonane imaging MRI sanners, illustrated in Figure 1, have beome an indispensable tool for use in medial imaging. Their apability to nonintrusively diagnose a range of medial ailments, suh as tumours, 1 damaged artilage, fratures, 2 internal bleeding, and even detetion of multiple slerosis 3 make them a desirable imaging... This is an open aess artile under the terms of the Creative Commons Attribution Liense, whih permits use, distribution and reprodution in any medium, provided the original work is properly ited. Copyright 2018 The Authors. International Journal for Numerial Methods in Engineering Published by John Wiley & Sons, Ltd. Int J Numer Methods Eng. 2018;115: wileyonlinelibrary.om/journal/nme 209

2 210 BAGWELL ET AL. FIGURE 1 Patient on magneti resonane imaging MRI sanner bed with loalised torso reeiver oils, ourtesy of Siemens [Colour figure an be viewed at wileyonlinelibrary.om] tehnique for linial use. The inreasing auray in MRI has beome a topi of partiular importane in reent years, due to the need for more aurate diagnosis of medial onditions, suh as aner. 4,5 Magneti resonane imaging sanner resolution is determined by the strength of the stati magneti field, ie, H DC, produed by the main magnet. The magnitude of magneti flux density of suh fields, ie, B DC, is typially in the region of 1.5 to 3 T approximately to times the strength of that of the Earth for linial operation, 6-8 with some 7-T units in use for medial researh appliations. 9,10 Reently, the Siemens 7-T magnet, the MAGNETOM Terra, 11 has also been leared for linial use and researh use. The magnitude of the magneti flux density, quoted by manufaturers, is defined as the maximum value of the flux density magnitude on the imaging bore axis,* in the entre where the patient lies, as shown in Figure 1. Reent advanes in MRI design have resulted in magnets of flux densities of up to 12 T oming into prodution, whih will allow for very high resolution images to be obtained, ompared with the urrent systems. 12 These sanners all typially utilise superonduting magnets onsisting of wound onduting wires resembling solenoids that are superooled by being immersed in liquid helium to temperatures of approximately 4 K. Some open C-shaped MRI sanners, whih utilise permanent magnets to generate the stati field, are still available, but these are less ommon in urrent imaging units due to their relatively low flux densities of approximately 0.3 T. In addition to the stati field generated by the main magnet, MRI sanners use pulsed time-varying magneti field gradients, generated through sets of resistive oils, whih exite the tissues and generate images of the patient. The gradient in the magneti flux density of these fields is muh smaller than the flux density of the main stati field, typially with amplitudes in the region of 30 to T/m. 6-8, In the presene of onduting bodies, these time-varying fields give rise to eddy urrents, whih exert Lorentz fores in the ondutors. These fores generate magneti stresses in the onduting omponents, whih, in turn, ause them to deform and vibrate. Furthermore, these vibrations generate Lorentz urrents, whih ause further perturbations in the magneti field. They also ause the surrounding free-spae region of air to perturb, generating aousti pressure waves, resulting in audible noise that an ause disomfort for the patient. These generated pressure waves an also rebound off other onduting omponents ausing further deformations to the ondutors and more aousti waves to form. This, in turn, reates further perturbations of the magneti field resulting in a fully oupled transient system of nonlinear transient PDEs omprising the Maxwell, elastiity, and aousti equations. 13 With inreasing interest in the design of more eonomial redued amount of material, dereased power onsumption and more advaned MRI systems apable of produing stronger magneti fields, 14 of late, the predition of the physial behaviours of suh sanners has beome an important area of researh, in partiular, the omputational modelling of these systems. As suh, an emphasis on the redution of the omputational ost of simulations is of great importane to the medial imaging industry. Many single field attempts to analyse the magneti field of MRI sanners have been published; finite-differene time-domain methods for the alulation of eddy urrents arising from transient gradient oils in MRI sanners, with analysis also of the effets on the human body, and methods for fast analysis and design of the MRI oils. 21,22 A number of works have also been published, whih fous on the analysis of superonduting solenoids, 23 as well as the on full MRI sanners 24,25 and onsider the strutural design of higher field sanners. 26 Aousti effets in MRI sanners have also been investigated, 27 with attempts to design noise redution systems 28,29 andevenanalysethe *The imaging bore is loated in free spae and as suh B DC = μ 0 H DC,whereμ 0 = 4π 10 7 H/m is the permeability of free spae. The gradient field is typially measured in terms of spatial rate of hange in the magneti flux density along the imaging bore axis in teslas per metre T/m. The magnitude of the magneti flux density arising from these oils is orders of magnitude smaller than that of the main oils.

3 BAGWELL ET AL. 211 aousti effets in the human head. 30 More reently, a few attempts to analyse the magneto-mehanial oupling in MRI systems have been onsidered; with the modelling of axisymmetri superonduting solenoids in self magneti fields 31 and effiient low-order FE solvers for magneto-mehanial oupling in the work of Raush et al, 32 whih utilised alulation of body fores to generate a weakly oupled algorithm. This work was later extended to inlude aousti effets in the aforementioned authors' other work. 33 Our previous work 34 foused on the solution of oupled magneto-mehanial behaviour of geometrially idealised axisymmetri MRI sanners, with arbitrarily high-order finite elements, 35,36 through a stress tensor approah to avoid diret alulation of the eletromagneti body fores. Given the relatively small-sale phenomena that emerge in suh magneti and aousti appliations, suh as small skin depths in onduting omponents known as skin effet 37 and high frequeny aousti waves, high-order elements are neessary to aurately resolve suh effets and ensure aurate solutions, as shown in our other work. 13 Furthermore, by hoosing to linearise the transient nonlinear system of oupled equations about the stati fields, the formulation simplifies greatly, resulting in a transient linear system of equations dependent on the transient magneti field exitation, as shown in our other work. 13 Similar tehniques, involving the additive split of a nonlinear problem to a series of linear problems, have been suessfully applied to the field of omputational mehanis, suh as analysis of strutural membranes, high-order mesh generation, 41 and in biomedial appliations. 42 This formulation permits a simple linearised approah that an be solved in the frequeny domain and, given the relatively small frequeny ranges of MRI appliations, provides rapid solutions to industrial problems. However, the appliability of this linearisation will depend on the strength of the oupling between the aousti, mehanial, and eletromagneti fields. In the ontext of MRI sanners, it is thus imperative to answer two important questions: i how good is the agreement between the omputationally effiient linearised approah ompared with the intensive treatment of the fully nonlinear system, denoted the nonlinear approah and ii over what range of MRI operating onditions an the linearised approah be expeted to provide aeptable results for outputs of interest in an industrial ontext for MRI sanner design? The nonlinear approah is trusted as there is no approximation from a physial standpoint. The aim of this paper is to address these two questions. This is ahieved through the following novelties. 1. We revisit the linearisation of the nonlinear equations presented in our other work 13 and provide an alternative formulation, suitable for transient simulation of the full nonlinear problem, using a Newton-Raphson and alpha-type time integration shemes. We refer to this as the nonlinear approah. 2. We demonstrate numerially that the linearised approah is in exellent agreement with the nonlinear approah and derive estimates of the energy present in the nonlinear approah, whih is not aptured by the linearised approah. 3. We undertake a rigourous omparison of the linearised and nonlinear approahes in terms of outputs of interest for a set of hallenging industrially motivated examples. In this paper, we begin by presenting the oupled transient transmission problem and its assoiated initial onditions in Setion 2. Then, in Setion 3, we present the ontinuous weak treatment of the nonlinear and linearised approahes,derive estimates of the energy not aptured by the linearised approah and present a transient Newton-Raphson proedure for the nonlinear iteration of the ontinuous fields in the two approahes. In Setion 4, we briefly reall the spatial disretisation of the system by means of high-order finite elements. We inlude the temporal disretisation, where we have implemented a seond order generalised α-sheme, to allow for inlusion of numerial dissipation into the model. The system is then solved through an iterative monolithi Newton-Raphson solution proedure, whih is summarised in an algorithm. We then onlude with numerial results in Setion 5, whih have been obtained by using the in-house software written by the group for this work. 2 COUPLED SYSTEM Previously, in our other work, 13 we presented the fully oupled system of nonlinear equations desribing the magneti, mehani, and aousti behaviours of an MRI sanner. The work inluded the transmission onditions present at the interfae between the onduting and nononduting regions, as well as initial and far-field onditions of the system. From this oupled transmission problem, a simpler linearised sheme was introdued by a suitable additive split of the exiting urrent soure J s t as J s t =J DC +J AC t. In this setion, we briefly reall the fully oupled transmission problem derived in our earlier paper, desribe in detail the physial motivation for the additive split of J s t, and provide a set of realisti initial onditions.

4 212 BAGWELL ET AL. FIGURE 2 Physial representation of the oupling effets in an magneti resonane imaging MRI environment [Colour figure an be viewed at wileyonlinelibrary.om] 2.1 Transient nonlinear system The governing oupled transmission problem, illustrated in Figure 2, desribing the aousto-magneto-mehanial behaviour of MRI sanners is given by the following. Find A, u, t R 3 R 3 R0, T] suh that μ 1 A+J e A =J s + J l A, u in R 3, 1a A = 0 in R 3 Ω, 1b σ m u+σ e A = ρu tt in Ω, tt = σ e A in R 3 Ω, 1d A = O x 1, 1e lim x x + t = O x 1 as x, 1f u = u D, 1g + n = 0 on Ω D Ω D, 1h ρ + u tt Ω n = + σ e A + n on Ω N N, 1i Ω n [A] Ω = 0, 1j n [μ 1 A] Ω = 0, 1k σ e A+σ m u n = I + σ e A + n on Ω, 1l Ω Ω subjet to an appropriate set of initial onditions. In 1, the magneti flux density B is related to the magneti field H and the magneti vetor potential A by B = μh = A, whereμ is the magneti permeability of the medium, is the aousti pressure field in free-spae region of air R 3 Ω,andu is the displaement field of the onduting bodies Ω. The eddy urrents J e A =γa t and Lorentz urrents J l A, u =γu t A are expressed in terms of the field variables, respetively. The notation desribing the system of equations in 1 is the same as in our other work, 13 where the magneti vetor potential A, mehanial displaements u, and aoustipressure have been introdued above. Here, we use u t = u t and u tt = 2 u t 2 and note that J s are the magneti soure urrents, x is the position vetor, γ is the eletrial ondutivity, and σ m u =λtr εu I + 2Gεu, is the Cauhy stress tensor. In the above, λ and G denote the Lamé parameters, ρ the material density, εu = u + u T 2 the linear strain tensor, I the identity tensor, T the transpose, n the unit normal vetor, and σ e A =μ 1 A A 1 2 A 2 I, 2 the magneti omponent of the Maxwell stress tensor.

5 BAGWELL ET AL. 213 The urrent soure J s t in 1 an be deomposed through the additive split J s t =J DC + J AC t,wherej DC is a stati urrent soure provided by the main magnet and J AC t is a time-varying urrent soure through a series of onduting gradient oils. Thus, the supports of J DC and J AC t are different and their maximum magnitudes and orientations are defined during the MRI design proess. In all urrent MRI designs, they are also hosen so that J DC J AC by several orders of magnitude to provide a strong bakground stati magneti field, whih is perturbed by weaker time-varying magneti fields generated from the gradient oils and their interation with the onduting omponents. Furthermore, J DC is always present and J AC t is only nonzero during imaging sequenes. 43 In absene of J AC t, the system in 1 simplifies onsiderably as there is no time variation and it is instead desribed by stati solutions A DC, u DC, DC. 13 If μ = μ 0 = 4π 10 7 H/m in R 3,thenu DC = 0 and DC = 0; however, in general, this is not the ase. Taking advantage of the fat that MRI sanners are typially maintained at full field strength, whilst in linial use, as they are maintained at superonduting temperatures, and that the gradient time-varying fields are then only applied during imaging sequenes, the initial onditions for 1 are set as the solution of the stati field omponents At = 0 =A DC in R 3, 3a ut = 0 =u DC, u t t = 0 =0 in Ω, 3b t = 0 = DC, t t = 0 =0 in R 3 Ω. 3 3 WEAK TREATMENT OF THE CONTINUOUS PROBLEM The transient problem desribed by 1 represents a nonlinear system of oupled equations inluding eletromagneti, mehani and aousti effets in an MRI environment. We first establish the ontinuous weak form of this oupled nonlinear approah and present the ontinuous weak form of the linearised approah, previously derived in our other work. 13 Based on estimates of the energy present in the nonlinear approah, whih is not aptured by the linearised approah, presented in the Appendix, we summarise onditions under whih we expet the linearised approah to provide an aurate representation of the solution of the nonlinear approah. A Newton-Raphsoniteration is then presented to resolve the nonlinearity in the ontinuous weak forms. 3.1 Weak forms of the nonlinear and linearised approahes Nonlinear approah In our nonlinear approah, we assume that there exists a fixed point weak ontinuous solution A, u, t XA DC Ỹu DC, u D Z DC 0, T] of the oupled system 1 satisfying R A A δ ; A, u, J s = R 3 μ 1 A A δ dω+ Ω γat A δ dω Ω γut A A δ dω suppj s J s A δ dω =0, 4a R u u δ ; A, u, = Ω σ m u+σ e A u δ dω+ Ω ρutt u δ dω Ω N u δ I + σ e A + n ds = 0, Ω R δ ; A, u, = R3 Ω + σ e A δ dω+ R3 Ω δ 1 Ω N δ ρ + u tt n ds = 0, Ω N 2 tt dω 4b 4

6 214 BAGWELL ET AL. for all weighting funtions A δ, u δ, δ X Y0 Z, denoted by a supersript δ,where Xg = { A X At = 0 =ginr 3}, X = { A Hurl, R 3 A = 0inR 3 } Ω, Ỹg, h = { u Yh ut = 0 =g, u t t = 0 =0inR 3}, Yh = { u H 1 Ω 3 u = h on Ω D }, Zq = { Z t = 0 =q, t t = 0 =0inR 3 } Ω, Z = { H 1 R 3 Ω }, and Hurl, R 3 and H 1 R 3 have their usual definitions, eg, the work of Monk Linearised approah The ontinuous version of the linearised approah, previously derived in our other work, 13 orresponds to seeking weak solutions A DC + A AC t, u DC + u AC t, DC + AC t, whih are obtained from a diret urrent DC and an alternating urrent AC stage. DC-stage The stati solutions A DC, u DC, DC X Yu D Z are the fixed point solution of R DC A Aδ ; A DC, u DC =R A A δ ; A DC, u DC, J DC = μ 1 A DC A δ dω J DC A δ dω =0, 5a R 3 suppj DC R DC u u δ ; A DC, u DC, DC =R u u δ ; A DC, u DC, DC = Ω σ m u DC +σ e A DC u δ dω Ω N u δ DC I + σ e A DC + n ds = 0, Ω 5b R DC δ ; A DC, u DC, DC =R δ ; A DC, u DC, DC = R3 Ω DC + σ e A DC δ dω=0, 5 for all A δ, u δ, δ X Y0 Z. AC-stage The transient weak solutions A AC, u AC, AC t X0 Ỹ0, 0 Z00, T] are the solution of the linearised problem DR A A δ ; A DC, u DC [A AC ]+DR A A δ ; A DC, u DC [u AC ]= R A A δ ; A DC, u DC, J s, 6a DR u u δ ; A DC, u DC, DC [A AC ]+DR u A δ ; A DC, u DC, DC [u AC ]+ DR u A δ ; A DC, u DC, DC [ AC ]= R u A δ ; A DC, u DC, DC, 6b DR δ ; A DC, u DC, DC [A AC ]+DR A δ ; A DC, u DC, DC [u AC ]+ DR A δ ; A DC, u DC, DC [ AC ]= R A δ ; A DC, u DC, DC, 6 for all A δ, u δ, δ X Y0 Z under the assumption A AC, u AC, AC t are small perturbations from A DC, u DC, DC. In the above, the notation DR A A δ ; A DC, u DC [A AC ] denotes the diretional derivative of R A about the stati solutions

7 BAGWELL ET AL. 215 A DC, u DC in the diretion A AC with similar meanings for the other terms. 45 By deriving the required terms, this an be expliitly written as: Find A AC, u AC, AC t X0 Ỹ0, 0 Z00, T] suh that μ 1 A AC A δ dω+ γa AC t A δ dω R 3 Ω Ω γu AC t A DC A δ dω suppj AC J AC A δ dω =0, 7a σ m u AC +μ 1 A DC, A AC u δ dω+ ρu AC tt u δ dω Ω Ω u δ AC I + μ 1 A DC, A AC + Ω N Ω n R3 Ω AC + σ e A AC δ dω+ R3 Ω 1 2 AC tt δ dω Ω N δ ρ + u AC tt Ω N ds = 0, n ds = 0, for all A δ, u δ, δ X Y0 Z. In the above, the linearised eletromagneti stress tensor, introdued previously in our other work, 13 is a, b = a b+ a b a b I. 7b Energy not aptured by the linearised approah In the Appendix, we estimate the energy present in the nonlinear approah, whih is not aptured by the linearised approah. We find that this will be small, at a ontinuous level, provided the following riteria are met for all time t: 1. γu AC t t L Ω B DC 2 L Ω 1, 2. B AC t 2 L Ω udc L Ω + u AC L Ω 1, B AC t 2 L Ω N udc L Ω N + uac L Ω N 1, 3. μ 0 H AC t L suppj AC JAC L suppj AC DC L suppj AC + AC L suppj AC 1, 4. B AC t L Ω B DC L Ω 1. Provided thattheseonditions hold, we onjeture thatthelinearised approah will be in good agreement with the nonlinear approah. We argue that these onditions will be met on physial grounds for MRI sanners as follows: 1 requires that the veloities are small in omparison to γ and B DC 2, whih is typially the ase for MRI sanners; 2 requires L Ω that the displaements and strains displaement gradients are small in omparison to B AC 2 L Ω and BAC 2 L Ω N, whih is again typially the ase for MRI sanners; 3 requires the pressure gradients in the oils are small, whih is typially the ase for MRI sanners under the Biot-Savart oil assumption; and 4 requires the B DC field obtained from the strong main magnet results in a B DC L Ω that is orders of magnitude larger than B AC L Ω, obtained from the weaker AC oils and smaller field perturbations aused by eddy and Lorentz urrents in the ondutors. In Setion 5, we will demonstrate numerially that the 2 approahes are in exellent agreement for suh problems. We inlude a simple model below to illustrate the appliability of these riteria Simple model relating field and urrent strengths Whilst the true field strength B = μh an only be found after solving 1 by means of the solution of the ontinuous weak forms for the nonlinear or linearised approahes, desribed in Setions and 3.1.2, there is merit in also onsidering a simple model to relate B to the applied urrent density in the oils. This is beause manufaturer's data, available for MRI sanners, is typially only quoted in terms of the maximum apable main and gradient field strengths at the entral axis of the imaging bore. 6-8 Therefore, to determine the operating ranges of the urrent densities on in-use sanners, the relationships given by this simplified model offer useful insight. In the simplified model, as shown in Figure 3, all transient, eddy urrent, and oupling effets are negleted, as is the mutual indutane between oils. Coil i has a onstant ross-setional area a i, is irular, and hene, rotationally symmetri and so is best expressed in terms of ylindrial oordinates r,,z and arries a uniform urrent density

8 216 BAGWELL ET AL. A B FIGURE 3 Single urrent loop, representing a lumped mass of oils. A, Magneti field around single urrent loop; B, Multiple oil onfiguration [Colour figure an be viewed at wileyonlinelibrary.om] J s = J s e. Then, by an additive appliation of the Biot-Savart law for a single oil, 46 the following relationship an be derived for the field strength along the axis r = 0forNoils plaed at different loations R i, Z i N z Zi 2 + R J s i = B0,,z, 8 R i a i i where B0,,z = B z r = 0, ze z along this axis, R i is the radius, and Z i the axial position of the ith urrent soure relative to the sanner's entral axes. Provided the aforementioned assumptions are enfored, this relationship an be applied to obtain the urrent densities that are assoiated only with the main magnet J s = J DC or with the gradient oil J s = J AC from their produed field strengths. It also provides a guide as to the ranges of ratios of J AC J DC and B AC L Ω B DC L Ω over whih the linearised approah is appliable. 3.3 Nonlinear iteration We now present a nonlinear iteration to resolve the nonlinear terms present in 4 and 5, resulting in linearised problems not to be onfused with linearised approah being solved at eah iteration. The unknowns in the linearised problems are ontinuous update fields, whih we relate to the ontinuous solution of 4 and 5. We make expliit those terms, whih are time dependent Newton-Raphson for the nonlinear approah To resolve the nonlinearity in 4 a nonlinear iteration, whih requires the solution of a linearised problem at eah iteration, is performed. This is expressed in the form of the following iterative Newton-Raphson proedure: Find δ [k] A, δ [k] u,δ [k] t X0 Ỹ0, 0 Z00, T] suh that DR A A δ ; A [k], u [k][ [k] δ ] A + DR A A δ ; A [k], u [k][ [k] δ ] u = R A A δ ; A [k], u [k] 9a, DR u u δ ; A [k], u [k], [k][ [k] δ ] A + DR u u δ ; A [k], u [k], [k][ [k] δ ] u + DR u u δ ; A [k], u [k], [k] [ ] δ [k] = R u u δ ; A [k], u [k], [k] 9b, DR δ ; A [k], u [k], [k][ [k] δ ] A + DR δ ; A [k], u [k], [k][ [k] δ ] u + DR δ ; A [k], u [k], [k] [ ] δ [k] = R δ ; A [k], u [k], [k] 9, for all A δ, u δ, δ X Y0 Z for a partiular iteration [k],where A [k+1] t =A [k] t+δ A [k] t, u [k+1] t =u [k] t+δ u [k] t, 10a 10b [k+1] t = [k] t+δ [k] t. 10

9 BAGWELL ET AL. 217 As standard, this linearisation is established under the assumption that the updates δ [k] A, δ [k] u,δ [k] t are small.45 Given an initial guess A [0], u [0], [0] t XA DC Ỹu DC, u D Z DC, and performing iterations of this kind, leads to a sequene of ontinuous iterates A [0], u [0], [0] t, A [1], u [1], [1] t, A [2], u [2], [2] t, XA DC Ỹu DC, u D Z DC 0, T] with the goal of onverging to the ontinuous solution A, u, t XA DC Ỹu DC, u D Z DC 0, T] of 4. Provided the initial guess A [0], u [0], [0] t is hosen to be suffiiently lose to A, u, t this is expeted to be the ase. The requirements for the onvergene of Newton-Raphson shemes are well doumented and are onsequently not disussed further, see e.g. 45 It is instrutive to introdue the notation qt =q A, q u, q t =A, u, t and to rewrite 9 in the form of the Newton-Raphson iteration: Find δ q [k] t W0, 00, T] suh that M [k] δ q, qδ + C [k] δ tt q, t qδ ; q [k] + K δ [k] q, q δ ; q [k], q [k] t = R q q δ ; q [k], q [k] t, q [k] tt, Js, 11 for all q δ W0,where q [k+1] t =q [k] t+δ [k] q t, 12 and Wq DC, u D = XA DC Ỹu DC, u D Z DC and Wu D =X Yu D Z. As before, if q [0] Wq DC, u D is hosen suffiiently lose to q Wq DC, u D then the iterates q [0] t, q [1] t, q [2] t, Wq DC, u D are expeted to onverge to qt Wq DC, u D. This formulation has the benefit that the forms of M, C,andK, are assoiated with the mass, damping, and stiffness ontributions to the system, respetively, and an be found to be M [k] δ q, qδ = tt ρ [k] δu Ω tt qδ u q δ ρ + [k] δ u n dω ΩN tt + ds 1 + dω, 13a R3 Ω 2 C δ q [k] t, qδ ; q [k] = R 3 K δ [k] q, q δ ; q [k], q [k] t = R 3 Ω N C δ [k] γ δa [k] t qδ A ttq δ μ 1 δ A [k] q δ A dω Ω γ δ u [k] t q [k] A dω Ω γ q [k] u q δ A dω, dω [k] δ A q δ A t + μ 1 q [k] Ω A, δ [k] A q δ u dω q δ u μ 1 Ω 0 q [k] A, δ [k] A + n ds N + σ m [k] δ u q δ u Ω dω ΩN suppj s + q [k] A δ [k] A μ 1 0 q[k] A + δ [k] q δ R3 Ω μ 1 0 δ A [k] q δ dω. δ [k] + n q δ u dω The system residual R q easily follows from 4 and is defined as R q q δ ; q [k], q [k] t, P q [k] tt, Js = M q [k] tt, qδ + C q [k] t, qδ ; q [k] + μ 1 q [k] q δ R A A dω+ σ q m [k] u + σ e q [k] q δ 3 Ω A u dω q δ u q [k] I + σe q [k] + n ds + q [k] q δ A R3 + σ e q [k] A Ω ds q δ dω 13b 13 R 3 J s t q δ A dω. 14 The diretional derivatives in 9a are already stated in our other work 13 and onsequently not repeated here.

10 218 BAGWELL ET AL Newton-Raphson for the linearised approah DC-Stage A nonlinear iteration, with a linearised problem being solved at eah iteration, is also required to resolve the nonlinearity in 7. For this, we employ the Newton-Raphson proedure: Find δ DC[k], δ DC[k] A u,δ DC[k] X Y0 Z suh that DR DC A A δ ; A DC[k] [ ] δ DC[k] = R DC A A A δ ; A DC[k], 15a DR DC u u δ ; A DC[k], u DC[k], DC[k] [ ] δ DC[k] A u δ ; A DC[k], u DC[k], DC[k] [ δ DC[k] + DR DC u DR DC δ ; A DC[k], u DC[k], DC[k] [ δ DC[k] A + DR DC + DR DC u ] ] + DR DC ] δ ; A DC[k], u DC[k], DC[k] [ δ DC[k] u δ ; A DC[k], u DC[k], DC[k] [ ] δ DC[k] u u δ ; A DC[k], u DC[k], DC[k], 15b = R DC u δ ; A DC[k], u DC[k], DC[k] [ δ DC[k] u δ ; A DC[k], u DC[k], DC[k], 15 = R DC for all A δ, u δ, δ X Y0 Z for a partiular iteration [k], whih follows from the assumption that δ DC[k], δ DC[k] A u,δ DC[k] are small. The solution updates δ DC[k], δ DC[k] A u,δ DC[k] are used to obtain A DC[k+1], u DC[k+1] and DC[k+1] in a similar way to 10. One again, if the initial guess A DC[0], u DC[0], DC[0] X Y0 Z is hosen to be suffiiently lose to A DC, u DC, DC X Y0 Z, the iterates A DC[0], u DC[0], DC[0], A DC[1], u DC[1], DC[1], A DC[2], u DC[2], DC[2], X Y0 Z are expeted to onverge to the ontinuous solutions A DC, u DC, DC X Y0 Z of 5. This problem an also be formulated in terms of q DC = A DC, u DC, DC.Findδ DC[k] q W0 suh that K δ DC[k] q for all q δ W0 with q DC[k+1] = q DC[k] + δ DC[k] q., q δ ; q DC[k], 0 = R q q δ ; q DC[k], 0, 0, J DC, 16 AC-stage For ompleteness, we also reast 7 to: Find δ q t q AC t W0, 00, T],suhthat M δ q tt, q δ + C δ q t, q δ + K δ q, q δ = R q q δ, 17 for all q δ W0, whih does not require iteration as it is linear in the unknown fields; hene, the reason we all it the linearised approah. The bilinear forms M, C and K are assoiated with the mass, damping and stiffness ontributions in the linearised approah, and an be expressed in terms of the definitions in 13 as ] M δ q tt, q δ = M δ q tt, q δ, C δ q t, q δ = C δ q t, q δ ; q DC, K δ q, q δ = K δ q, q δ ; q DC, 0, R q q δ =R q q δ ; q DC, 0, 0, J s. 18a 18b 18 18d In this ase the solution of 17 will ontain only the transient alternating urrent AC omponent of the fields, and thus the omplete desription is qt =q DC + δ q t. The linear nature of this system also allows for it to be transformed to the frequeny domain by the appliation of a Fourier transform, whih was the approah followed in our other work DISCRETISATION OF THE SYSTEM In this setion, we desribe the spatial and temporal disretisation of 11 for the nonlinear approah and 16,17 for the linearised approah. The eventual aim of this formulation is to handle fully oupled nonlinear problems, whih, for the nonlinear approah, will employ an iterative proedure at eah time step to fully resolve the solution while, for the The diretional derivatives in 15 have been previously stated in our other work 13 and, in the following, we relate them to the expliit expressions in 13 to avoid repetition.

11 BAGWELL ET AL. 219 linearised approah, will involve a stati iterative solve followed by a linear time integration without iteration. The presentation inludes a brief desription of the spatial disretisation using hp-finite elements, whih result in a semidisrete system of oupled seond order ODEs. We then present the temporal disretisation of the system of equations to allow for solutions in the time domain and lose the setion by presenting a solution algorithm for the nonlinear and linearised approahes, whih summarises the steps required one the problems are fully disretised. 4.1 Spatial disretisation The spatial disretisation of the linearised system of oupled aousto-magneto-mehanial equations in 16 and 17 is extensively overed in our other work 13 and the work of Bagwell 47 and the spatial disretisation of 11 is analogous. Hene, we only briefly summarise the approah. We limit ourselves to the treatment of idealised MRI sanners that are assumed to be rotationally symmetri in the azimuthal diretion so that the problem is axisymmetri. Our domain then beomes the meridian plane Ω m ={r, z 0 < r <, < z < }, the urrent soures have the form J s = J s r, ze, and the unknown fields are A = A r, ze, u = u r r, ze r + u z r, ze z and = r, z, wheree r, e,ande z are the standard bases for the ylindrial oordinate system r, z,. However, by projetion of Ω m the full 3-dimensional results are still ahieved. When transformed to the axisymmetri domain, the spaes in whih the weak solutions are sought in the variational statements 11,16,17 must be adapted following the approah desribed in our other work 13 and the work of Bagwell. 47 To overome the need for the solution in weighted spaes, to ensure the fields are well behaved at the radial axis, we transform the fields as A = râ, u r = rû r and note that  H 1 Ω m and ũ = [û r, u z ] H 1 Ω m 2 and reall from our other work 13 that the aousti pressure r, z H 1 Ω m presents no diffiulty. The ontinuous update fields and test funtions in the Newton-Raphson iterations are also treated in an analogous manner. To ahieve a finite omputational domain, Ω m is trunated by the introdution of infinite elements 48 for the far-field treatment of  and a perfetly mathed layer approah 49 for the far-field treatment of. The far-field deay of the update fields is also treated analogously. For the details, we refer again to our other work 13 and the work of Bagwell. 47 Then, a nonoverlapping partition,, of Ω m is ahieved through the generation of a hybrid unstrutured triangular-quadrilateral mesh, where denotes the set of verties, the set of edges, and the set of ells. In eah element, we employ the H 1 onforming hierarhi finite element basis funtions proposed by Shöberl and Zaglmayr, 50,51 whih allow for arbitrary inreases in element order p and loal refinement of the mesh spaing h, for the spatial disretisation of the fields A, ũ,and and the assoiated ontinuous update fields. For a disretisation onsisting of uniform order p elements, and a mesh onsisting of N verties, N edges, N T triangular ells and N Q quadrilateral ells, there will be N = N +p 1N + 1 p 2p 1N 2 T +p 12 N Q degrees of freedom for eah omponent of eah field Nonlinear approah Following a spatial disretisation using a uniform order approximation, the semidisrete vetor of updates and unknowns for problem 11 beome { } [k] { } [k] δa qa δ [k] q t = δũ t R 4N, q [k] t = qũ t R 4N, 19 δ for eah time t and the orresponding semidisrete solution to the nonlinear approah redues. Finding δ [k] q t at eah time t 0, T] satisfying the following initial value problem: [k] Mδ q t+c [k] [k] δ q t+k [k] δ [k] q t = R [k] q, 20a q q [k+1] t =q [k] t+δ q [k] t, q0 =0, 20b 20 q0 =q DC, 20d [k] where δ q = [k] [k] dδq dt and δ q = d 2 δ [k] q dt 2. Equation 20 involves iteratively solving a seond-order system of ODEs until R [k] q < TOL for some user-defined tolerane TOL. The onvergene of R [k] q to 0 will be quadrati. 45 In the above,

12 220 BAGWELL ET AL. M, C [k], K [k],andr [k] q are the disrete linearised mass, damping and stiffness matries, and residual vetor, respetively, whih are obtained by disretising the terms in 13 and depend, with the exeption of the mass matrix, on the solution at iteration [k]. Expliit expressions for the matries an be found in the work of Bagwell. 47 The vetor q DC R 4N is the onverged solution of 21 below Linearised approah DC-stage In a similar manner, the linearised approah takes the form of a time-independent Newton-Raphson iteration. Find δ q DC[k] R 4N suh that Kδ q DC[k] = R DC[k] q, 21a q DC[k+1] = q DC[k] + δ q DC[k], q DC[0] = 0, 21 with the iteration performed until R DC[k] q < TOL, whih follows from the disretisation of 16. The onvergene of R DC[k] q to 0 will again be quadrati. AC-stage The linearised approah also requires the solution of the semidisrete initial value problem. Find δ [k] q t R 4N at eah time t 0, T] suh that M δ q + C δ q + Kδ q = R q, 22a δ q 0 = δ q 0 =0, 22b whih follows from the disretisation of 17, where M, C, K,and R q are the disrete mass, damping and stiffness matries, and the residual vetor, respetively, obtained by disretising the terms in 18. Expliit expressions for the matries an be found in the work of Bagwell. 47 The semidisrete solution for the linearised approah is qt =q DC + δ q t. 21b 4.2 Temporal disretisation There are a onsiderable number of alternative time integration tehniques that ould be employed for the temporal integration of 20 and 22, suh as Euler shemes, 52 trapezoidal shemes, 53 and the Newmark method. 54 In this paper, we hoose to adopt a seond-order generalised-α sheme see the work of Chung and Hulbert 55 to disretise our system of equations. This sheme is designed to allow for tailor-made numerial dissipation in the solution and is of suffiient degree, given the seond-order nature of 20 and 22. Given the omplexity of the oupled system, it is diffiult to know the true initial onditions for the full transient problem, when applying a fored exitation. The numerial dissipation of the generalised-α sheme is therefore benefiial as it an be used to damp out any artifiial frequenies that pollute the solution. There exists also a number of high-order integration shemes, suh as bakward differentiation formulas, 56 expliit Runge-Kutta and leap-frog type methods, 57,58 that allow for higher-order auraies in time and larger time steps, although with inreased temporal auray omes a signifiant inrease in omputational ost. However, for our problem, the required time step size is ditated by the need to apture the exitation frequeny and resonant frequenies of the system typially in the region of 5000 Hz. For this reason, the need for higher-order auray is signifiantly outweighed by the requirement of smaller time step sizes anyway, and thus, the generalised-α sheme appears a sensible hoie Generalised-α time integration sheme seond order In this setion, we fous on the development of the temporal disretisation of 20, whih an similarly be developed for 22. The impliit form of this sheme evaluates the system of nonlinear equations in 20 at an intermediate time step t n+1 αf as [k] Mδ q n+1 α + C [k] [k] m t δ q n+1 αf n+1 α + K [k] [k] f t δ q n+1 αf n+1 α = R q [k] f t, 23 n+1 αf where C [k] t, K [k] n+1 αf t,andr q [k] n+1 αf t are evaluated at the intermediate time step t n+1 αf and the values of the fields at n+1 αf this time step are given by q [k] n+1 α =1 α m q [k] m n+1 + α m q n, 24a

13 BAGWELL ET AL. 221 q [k] n+1 α f =1 α f q [k] n+1 + α f q n, 24b q [k] n+1 α f =1 α f q [k] n+1 + α f q n, 24 t n+1 αf =1 α f t n+1 + α f t n. 24d The generalised-α sheme, desribed in the aforementioned work, 55 expresses q [k] n+1 and q[k] n+1 in terms of q[k] n+1, known as an aeleration-based formulation. This an be manipulated into a displaement based formulation, where q [k] n+1 and q [k] n+1 are expressed in terms of q[k] n+1,as q [k] n+1 = 1 β q [k] n+1 = ς β q [k] n+1 q n q n 1 Δt 2 Δt 2 β q n, 25a β β ς 1 q n + ς 1 Δt q n, 25b 2 q [k] n+1 q n + Δt where ς = 1 α 2 m + α f, β = 1 1 α 4 m + α f 2, α m = 2ρ 1, α ρ +1 f = ρ, Δt is the time step size and ρ ρ +1 denotes the user-speified value of the spetral radius in the high-frequeny limit Preditor-multi-orretor step A preditor-orretor approah has been followed from the implementation standpoint. 59 The predition step [k = 0], based on a initial guess of the displaement field q [0] n+1, is defined as q [0] n+1 = 1 q [0] n+1 q n q n 1 β Δt 2 Δt 2 β q n, 26a q [0] n+1 = ς q [0] n+1 q n β β + β Δt ς 1 q n + ς 1 Δt q n, 26b 2 where 26a and 26b are onsistent with 25a and 25b, respetively. In other words both the veloity and aelerations preditors preserve seond-order auray, as disussed in the work of Jansen et al. 60 If we return our attention to 25b and 25a and substitute the fields at iteration step [k+1], we obtain that the update variables of the first- and seond-order fields are [k] δ q n+1 = 1 βδt δ 2 q [k] n+1, 27a [k] δ q n+1 = ς βδt δ q [k] n+1. 27b Fully disrete nonlinear approah Using the relations between the updates, in 27, we may rewrite the disrete system, in 23, in terms of the update in [k] the zeroth-order solution vetor δ q n+1 at time t [k] n+1 as the following Newton-Raphson proedure. Find δ q n+1 R4N for n = 0, 1, 2,, N Δt = T Δt suh that 1 αm M + ς1 α f C [k] βδt 2 t +1 α f K [k] [k] βδt n+1 αf t δ q n+1 αf n+1 = R q [k] t, 28a n+1 αf q [k+1] n+1 = q [k] n+1 + δ q [k] n+1, 28b q 0 = 0, q 0 = q DC, 28d with the preditors 26 used to provide the initial guesses q [0] n+1 and q[0] n+1, whih are used to start the iteration over [k] until R [k] q < TOL. 28 Traditionally with the generalised α method, γ is used to refer to the first stability parameter; however, given our formulation reserves γ for the material ondutivity, we use instead the alternative symbol ς.

14 222 BAGWELL ET AL Fully disrete linearised approah AC-stage The fully disrete version of 23 for the linearised approah is similarly expressed. Find δ qn+1 R 4N for n = 0, 1, 2,, N Δt = T Δt suh that 1 αm M + ς1 α f C +1 α βδt 2 f K δ q βδt n+1 = R q tn+1 αf, 29a δ q0 = δ q0 = 0, 29b where the system matries M, C,and K are independent of the previous temporal solutions, and R q depends only on the evaluation of J AC at time level t n+1 αf and so an be solved in a single step, whih is a simplified version of Solution algorithm A general algorithm for omputing the transient variation in the fields for both the nonlinear and linearised approahes, proposed above, under a generalised-α sheme is summarised in Algorithm 1. In the nonlinear approah, the algorithm involves 2 nested loops, the first over n performs the integration of the oupled system over time and, the seond, over [k] resolves the nonlinearity through the appliation of a Newton-Raphson iteration. In the linearised approah, this simplifies to a single loop over n to perform the time integration. Furthermore, Figure 4 summarises the steps required to onstrut the disretised Newton-Rapshon iterative sheme in 28 from the transient nonlinear transmission problem 1. 5 NUMERICAL EXAMPLES In this setion, we disuss two benhmark MRI test problems to justify that the onditions presented in Setion 3.2 do hold in pratie. We study the linearity of the fields for a range of different urrent density values in the oils to determine the validity of the linearised approah.

15 BAGWELL ET AL. 223 A B C D FIGURE 4 Summary of steps to solving transient nonlinear system. A, Residual weak form of oupled non-linear system; B, Linearisation of the nonlinear system; C, Disretisation of the diretional derivatives; D, Temporal disretisation and building system matrix [Colour figure an be viewed at wileyonlinelibrary.om] FIGURE 5 Test magnet problem: omponents of the simplified geometry. OVC, outer vauum hamber [Colour figure an be viewed at wileyonlinelibrary.om] 5.1 Test magnet problem First, we onsider solutions to the industrially relevant test magnet problem, whih was previously presented in our other work 13 as the Siemens benhmark problem. The onduting region Ω of the test magnet onsists of three metalli shields, known as the outer vauum hamber OVC Ω OVC,77 K radiation shield Ω 77K,and4 K helium vessel Ω 4K,eah with different material parameters γ, μ, λ, G, ρ. The exat geometries and material parameters of the onduting omponents are ommerially sensitive and as suh are not displayed in this paper. By hoosing to study only the Z-gradient oils and noting that the urrents in the main oils and geometry of the onduting omponents are rotationally symmetri, the problem may be redued to an axisymmetri desription and solved in ylindrial oordinates r,,z on the meridian plane Ω m. The full 3-dimensional representation of this simplified MRI sanner is depited in Figure 5. A pair of main magnet oils, eah with a stati urrent soure J DC = J DC r, ze, are loated on the outside of the three shields and a pair of Z-gradient magnet oils, eah with alternating urrent soure J AC t =J AC t, r, ze, are loated within the imaging bore, both of whih are assumed as Biot-Savart ondutors and are loated in free spae. Realisti exitations of the gradient oil are nonsinusoidal in nature; however, for the purposes of omparison between the linearised andnonlinear approahes, we restrit ourselves to sinusoidal exitations, where the urrent density is desribed by t =Re JAC eiωt. We onsider several frequenies of exitation of ω = 2π[1000, 1500, 2000]rad/s, whih lie outside J AC of the resonane region of ω 2π[3500], predited in our other works. 13,34 The magnitude of the stati J DC and gradient urrent soures J AC to be onsidered are obtained from manufaturer's data.6-8 This data quotes the maximum apable flux density on the entral axis of the imaging bore r = 0 and to obtain the orresponding urrent densities we have used the model desribed in Setion 3.2.1, the results of whih are summarised, for key linial field strengths, in Table 1. Given riteria 4 in Setion 3.2, and hene the impliation of the urrent density ratios, the greatest nonlinearity should appear for a magnet with the weakest stati field of 1.5 T and the strongest gradient field of T/m. This would result in a ratio between the stati and gradient urrent density values of J AC JDC 7.2% for this problem. We therefore study the two approahes aross a range of urrent density ratios of J AC JDC =[5, 10, 15, 20]% to provide a rigourous test of the linearised approah for appliations of higher levels of nonlinearity than urrent MRI sanners are apable of.

16 224 BAGWELL ET AL. TABLE 1 Test magnet problem: typial values of the urrent densities in stati and gradient oils and ranges of stati and gradient field strengths from manufaturers data 6-8 Main Coil Gradient Coil max B DC z=0 [T] J DC [ 10 A/m 2] max B AC [ z=0 10 T/m ] J AC [ 10 A/m 2] The problem is subjet to the following boundary onditions: we fix the Dirihlet boundaries of the ondutors to u D = 0 to fix the ondutors in spae, as in Setion 2, and we set the value of the magneti vetor potential on the outer boundary to A = 0 due to the eddy urrent deay. The initial onditions of the problem are defined by 3. We treat this problem omputationally for both the linearised and nonlinear approahes. We trunate the nononduting free-spae region, omprised of air, and reate the domain Ω m, whih is the same as in our other work. 13 In terms of spatial disretisation, we analyse the solution for a single unstrutured mesh of 2842 triangles of maximum size h = 0.25 m, but with substantial refinement in the ondutors Ω m, resulting in 570 elements in Ω m. The mesh parameters used are the same as in our other work 13 ; however, due to improvements in the mesh generator used, 61 resulted in a better distribution of elements and hene fewer elements. We apply a single layer of 18 infinite quadrilateral elements on the outer boundary of Ω m to resolve the stati deay of the magneti field, so that the boundary ondition A = 0 is effetively imposed at infinity. We onsider p-refinement for elements of order p =[1, 2, 3, 4, 5]. The temporal disretisation used to resolve these waves are studied for a time step size of Δt = 2π ωn Δt, where we vary the number of points per wavelength of the exitation frequeny N Δt =[10, 20, 30, 40]. The spetral radius of the α-sheme time integrator ρ allows for damping of ertain frequeny regimes. For ρ = 1 the amplitude of the wave is fully preserved and no damping of any high frequenies is introdued. For ρ = 0, the sheme is fully dissipative and higher frequeny waves are ompletely damped; however, the damping of physial modes also ours. Given that we wish to reover only physial modes in our problem, we therefore hose the value of ρ = 0.8, to allow for numerial damping of any nonphysial high frequenies indued through the foring, whilst still preserving the physial lower frequeny waves. Solutions to this problem are obtained by applying Algorithm 1 for both the nonlinear and linearised approahes aross the range of urrent densities and exitation frequenies disussed Convergene study in outputs of interest We onsider quantities of industrial interest for both the mehanial and eletromagneti fields. For the eletromagneti field, we are onerned with the output power dissipation in the onduting omponents P o and for the mehanial field Ω we are onerned with the kineti energy of the onduting omponents E k. The formal definition of these quantities in Ω a full time-domain desription, where the quantities are averaged over the time period T of exitation frequeny, are t+t P o t, A = 1 γ A Ω T t t 2 dωdt, E k t, u = 1 Ω Ω T whih redue, in a time harmoni representation, to t t+t 1 2 Ω ρ u t 2 dωdt, 30 P o Ω ω, A = 1 2 Ω γω 2 Aω 2 dω, E k Ω ω, u = 1 4 Ω ρω 2 uω 2 dω, 31 where Aω and uω are the omplex amplitudes of their respetive fields as At =Re Aωe iωt and ut =Re uωe iωt, as desribed in our other work. 13 To ompute these quantities for the test magnet problem, using either the transient solutions obtained by the linearised or nonlinear approahes, Algorithm 1 is run at speifi N Δt, Δt, ρ for a suffiiently long time until the field responses reah steady state periodi solution. Equation 30 is approximated by Gaussian quadrature for the spatial integration with suffiient quadrature points orresponding to the degree of the integrand and a Trapezoidal rule for the temporal integration so that no further approximations are made. In order to determine suitable spatial resolution in the solution, we perform a p-refinement study. We apply Algorithm 1 for N Δt = 30, ρ = 0.8 using the linearised approah to ompute the output power and kineti energy aross the frequeny

17 BAGWELL ET AL. 225 Output Power [W] p=1 p=2 p=3 p=4 p=5 Kineti Energy [J] p=1 p=2 p=3 p=4 p=5 A Frequeny [Hz] B Frequeny [Hz] FIGURE 6 Test magnet problem. Results for element order p =[1, 2, 3, 4, 5]. A, Output power of the OVC outer vauum hamber; B, Kineti energy of the OVC spetrum for ω 2π[5000], the results of whih are illustrated in Figure 6. For p 3 the omputed urves, illustrated in Figure 6, do not math one another and so the results have not yet reahed onvergene. The urves for p = 4andp = 5, however, are pratially indistinguishable for both the output power and kineti energy and suggests that for elements of order p = 4 the results are suffiiently onverged. We hene deide to adopt elements of p = 4 for all subsequent omputations of this problem on the mesh speified previously. A similar study to determine the required time step size was also arried out, where a range of number of time steps per wavelength N Δt =[10, 20, 30, 40] for p = 4 elements were studied. The results suggest that N Δt = 30 offers suffiient temporal resolution to apture the amplitudes of the dominant frequeny as well as frequenies twie the dominant frequeny 2ω. The importane of this frequeny doubling will be explained later in Setion Thus, for an exitation frequeny of ω = 2π[1000], the time step size Δt = s,forω = 2π[1500] the Δt = s and for ω = 2π[2000] the Δt = s. For the full set of results, see our other work. 47 Using these parameters, in Setion 5.1.4, we ompute the quantities in 30 by applying Algorithm 1 for the linearised and nonlinear approahes in the time domain and ompare them with our previous frequeny domain solver. 13 However, first we onsider the results for the linearised and nonlinear approahes for transient eletromagneti and mehanial fields Eletromagneti field To perform omparisons between the linearised and nonlinear approahes, we ompare the transient response of the magneti field for both approahes. Given that the output power of the ondutors P o, desribed above, is driven by the Ω temporal derivative of the magneti vetor potential A t, we measure and ompare the response of this field. Figure 7 summarises the transient results of both approahes for the ω = 2π[2000]rad/s sinusoidal exitation. The graphs in the left-hand olumn plot the time signal obtained from both the linearised approah in red and nonlinear approah in blak. The right hand olumn plots the orresponding frequeny spetrum of the signal, obtained by performing fast Fourier transforms FFTs. We an see from this figure that the linearised approah provides an aurate approximation of the magneti vetor potential aross the full range of urrent density ratios. The relative norm of the differene in the time signals is for J AC JDC = 5% and for J AC JDC = 20%. Themagnitude of the frequenies aross the spetrum are almost idential and the linearised approah is even apable of apturing all the fundamental frequenies, around ω = 2π[ ]rad/s. Figure 8 plots the transient response and orresponding frequeny spetrum for a urrent density ratio of J AC JDC = 20% and a range of exitation frequenies. These plots again illustrate the agreement between the linearised and nonlinear approahes for different exitation frequenies.

18 226 BAGWELL ET AL. Time signal Frequeny spetrum 0.4 linearised nonlinear linearised t nonlinear t 5% da/dt t da/dtf time [s] f [Hz] 0.4 linearised nonlinear 10 0 linearised t nonlinear t % da/dt t 0 da/dtf time [s] f [Hz] FIGURE 7 Test magnet problem. Time signals and orresponding fast Fourier transforms FFTs of A t for both the linearised red line and nonlinear approahes blak line, for various values of J AC JDC subjet to a ω = 2π[2000]rad/s sinusoidal exitation [Colour figure an be viewed at wileyonlinelibrary.om] Exitation Freq. Time signal Frequeny spetrum linearised nonlinear linearised t nonlinear t 1000 Hz da/dt t da/dtf time [s] f [Hz] linearised nonlinear linearised t nonlinear t 1500 Hz da/dt t da/dtf time [s] f [Hz] FIGURE 8 Test magnet problem. Time signals and orresponding fast Fourier transforms FFTs of A t for both the linearised and nonlinear approahes,for J AC JDC = 20% subjet to various frequenies of exitation [Colour figure an be viewed at wileyonlinelibrary.om]

19 BAGWELL ET AL. 227 Time signal Frequeny spetrum linearised nonlinear linearised t nonlinear t 5% du r /dt t du r /dtf time [s] linearised nonlinear f [Hz] linearised t nonlinear t 20 % du r /dt t du r /dtf time [s] f [Hz] FIGURE 9 Test magnet problem. Time signals and orresponding fast Fourier transforms FFTs of u r t for both the linearised and nonlinear approahes, for various values of J AC JDC subjet to a ω = 2π[1000]rad/s sinusoidal exitation [Colour figure an be viewed at wileyonlinelibrary.om] Mehanial field We now ompare the response of the mehanial field for both the linearised and nonlinear approahes. Given that the kineti energy of the ondutors E k, desribed in Setion 5.1.1, is driven by their mehanial veloity u Ω t,wemeasureand ompare the response of this field. Figure 9 summarises the transient results of both approahes for the ω = 2π[1000]rad/s sinusoidal exitation. The graphs in the left-hand olumn plot the time signal obtained from both the linearised approah in red and nonlinear approah in blak. From the plots it appears as though the two approahes offer good agreement, espeially when looking at the time signals. The relative norm of the differenes between signals being for J AC JDC = 5% and for J AC JDC = 20%. However, in the frequeny spetrum there appears to be an extra frequeny at ω = 2π[2000]rad/s that is piked up in the nonlinear approah, butnotinthelinearised approah. This term appears due to the nonlinearity in the Maxwell stress tensor 2, whih we an observe is quadrati in the magneti field. From a deomposition of the magneti field into stati and dynami omponents, as shown in Setion 3.1.2, it an be shown that this nonlinear term omprises of a produt of the dynami omponent of the field with itself, whih disappears in the linearised approah. This term auses a frequeny doubling effet, in other words it results in a omponent of exitation of the mehanial field that is double the frequeny of the AC urrents. So for a ω = 2π[1000]rad/s wave, as in Figure 10, an exitation at ω = 2π[2000]rad/s would also appear, whih mathes exatly with the results obtained. However, from Figure 10, it is lear that the magnitude of this term is far smaller than that of the amplitude assoiated with the exiting frequeny and thus has little effet on the solution. As the ratio of the urrent densities inreases so too does the magnitude of this term. However, even for a ratio of 20% the magnitude is still several orders smaller than the main exitation and smaller also than the most dominant resonant frequenies. Thus the magnitude of this doubled frequeny omponent provides a useful measure in determining the nonlinearity of the problem. Figure 10 plots the transient response and orresponding frequeny spetrum for a urrent density ratio of J AC JDC = 20% and a range of exitation frequenies. We see from these plots that for the AC urrent frequenies of 1500 Hz, the agreement between the two approahes results in almost indistinguishable time signals. The relative norm of the differenes between signals being for ω = 2π[1500]rad/s and for ω = 2π[2000]rad/s. However, for the

20 228 BAGWELL ET AL. Exitation Freq. Time signal Frequeny spetrum linearised nonlinear linearised t nonlinear t 1500 Hz du r /dt t du r /dtf time [s] 10-6 linearised nonlinear f [Hz] linearised t nonlinear t 2000 Hz du r /dt t 0 du r /dtf time [s] f [Hz] FIGURE 10 Test magnet problem. Time signals and orresponding fast Fourier transforms FFTs of u r t for both the linearised and nonlinear approahes,for J AC JDC = 20% subjet to various frequenies of exitation [Colour figure an be viewed at wileyonlinelibrary.om] ω = 2π[2000]rad/s AC urrents' differenes in the time signal beome more visible. This is beause the doubled frequeny exitation omponent of 4000 Hz lies within the resonane region. When exiting lose to the resonane region, the problem results in matries of high ondition numbers, whih are lose to singular. Consequently, the tangent stiffness matrix inversion beomes more hallenging and less reliable and, as a result, an lead to differenes in the amplitudes aross the frequeny spetrum. Despite this effet however, the differenes in the time signal are still very small and the harateristis of the system and predition of the resonane region remain well aptured by the linearised approah Comparisonoflinearised and nonlinearapproahes To benhmark the auray of the solution from the linearised approah with the nonlinear approah, weomparethe omputation of the outputs of interest for the two approahes, presented in Setion 5.1.1, aross the range of frequenies ω 2π[5000]rad/s. Figure 11 illustrates the outputs of interest omputed by the linearised approah in both frequeny and time domain and the nonlinear approah in time domain, using the definitions in 30 and 31 for a urrent density ratio of J AC JDC = 10%. Using the frequeny domain approah, desribed in our other work, 13 the outputs of interest P o and E k an be diretly Ω Ω omputed for a given exitation frequeny. Whereas, in the time domain, we must run the time solver until steady state is obtained and then apply the definition in 30 aross a time period. Due to the inreased omputational ost of omputing the outputs of interest for transient solutions, we have hosen to ompute them for a oarser frequeny sweep ompared with the frequeny domain results. Figure 11A plots the output power in the OVC P o ω, A AC and Figure 11B plots Ω OVC the kineti energy in the OVC E k ω, A AC. For the results of the other shields, whih show similar agreement, see our Ω OVC other work. 47 From the plots, the urves produed by the linearised and nonlinear approahes are almost indistinguishable aross the frequeny spetrum whih suggests that thelinearised approah provides a very aurate approximation to the full nonlinear approah aross the full spetrum for J AC JDC = 10% for the two outputs of interest. In fat, given that the individual fields, analysed in Setions and 5.1.3, also show very good agreement for J AC JDC = 20%,wean hypothesise this to be the ase also for the outputs of interest as they are diretly related.

21 BAGWELL ET AL linearised linearisedt nonlinear 10 0 linearised linearisedt nonlinear Output Power [W] Kineti Energy [J] A Frequeny [Hz] B Frequeny [Hz] FIGURE 11 Test magnet problem. Results for the linearised approah in time and frequeny domain as well as the nonlinear approah. A, Output power of the OVC outer vauum hamber; B, Kineti energy of the OVC linearised nonlinear linearised nonlinear t = s t = s linearised nonlinear 2 du r /dt t time [s] t = s t = s linearised nonlinear linearised nonlinear FIGURE 12 Test magnet problem. Comparison between the linearised and nonlinear approahes for displaements of the outer vauum hamber OVC at different times for J AC JDC = 20% and ω = 2π2000rad/s

22 230 BAGWELL ET AL. TABLE 2 Test magnet problem: average omputational times per time step of the linearised and nonlinear approahes for speifi element order p =[1, 2, 3, 4, 5] Linearised Approah Nonlinear Approah p Computational time [s] p Computational time [s] Speed-up We now ompare the displaements of the mehanial shields by plotting the displaements of the OVC, in three dimensions, at interesting instanes of the time signal for the radial veloity u r t omputed for the ase of J AC JDC = 20% and ω = 2π2000rad/s in Figure 12. The hosen time instanes aross the time signal are plotted, where the differene in the mehanial veloities between the linearised and nonlinear approahes is notieable. The ontour plots of the displaement fields are all saled suh that the olour maps between the linearised and nonlinear approahes are the same. The displaements in Figure 12 are saled by several orders of magnitude to show visually the displaement shapes of the shield. We an see that for all snapshots the differenes between the two approahes are almost indistinguishable. This suggests that even for J AC JDC = 20% and an exitation frequeny of ω = 2π2000rad/s, where the doubled frequeny omponent of ω = 2π4000rad/s resides in the resonane region, see Setion 5.1.3, that the linearised approah provides aurate and omparable results to the full nonlinear approah. For the results of the other shields, see our other work. 47 The average omputational timings, per time step, for the two approahes aross a range of different element orders p are summarised in Table 2. The omparison between the omputational timings of the two approahes suggests that the linearised approah is orders of magnitude more effiient in terms of omputational ost than the nonlinear approah. For low-order p = 1 elements the linearised approah requires 383 times less omputational effort than the nonlinear approah. Whereas, for higher-order p = 5 elements thelinearised approah requires 164 times less omputational effort than the nonlinear approah. This speed-up fator appears to offer an inverse exponential behaviour with p, whih is due to the higher requirement on the solver for higher-order elements. Nevertheless, the omputational timings displayed in the table suggest that the linearised approah offers orders of magnitude inrease in omputational effiieny over the nonlinear approah. 5.2 Realisti magnet problem We now onsider a more realisti problem that represents, very aurately, the sorts of MRI sanner designs urrently used in linial operation. The geometry is illustrated in Figure 13A. This problem onsists of a similar onstrution to the previous problem, where the onduting region is omprised of the three radiation shields Ω =Ω OVC Ω 77K Ω 4K, Grad Coils Main Coils Rad Shields Bore Axis r=0 A FIGURE 13 B Realisti magnet problem: omponents of the simplified geometry. A, R 3 domain, Ω; B, Positive Half Meridian domain, Ω m

23 BAGWELL ET AL. 231 t = s t = s t = s t = s linearised nonlinear 0.5 du r /dt t time [s] 10-3 t = s t = s t = s t = s FIGURE 14 Full magnet problem. Displaements of the outer vauum hamber OVC at different times for J AC JDC = 15% and ω = 2π1500rad/s [Colour figure an be viewed at wileyonlinelibrary.om] eah with different material parameters γ, μ, λ, G, ρ. The geometry of the radiation shields, however, is more omplex and their topology represents that of losed ylindrial shells of trapezoidal ross setion, with urved fae end setions. However, despite the inreased omplexity in topology, the geometry is still ylindrial and an be treated as axisymmetri. Again, the exat geometries and material parameters of the onduting omponents are ommerially sensitive and as suh are not displayed in this paper. The onfiguration of the stati main oil onsists of the same blok ross setion as FIGURE 15 Full magnet problem. Distorted outer vauum hamber OVC, main oils, and orresponding stati magneti field lines for the stati problem where J s = J DC and t = 0 s [Colour figure an be viewed at wileyonlinelibrary.om]

24 232 BAGWELL ET AL. the test magnet problem, but ontains more sets of oils inluding also a set of seondary oils, whih at to minimise the magneti stray field by reversing the polarity of the magneti field. The gradient oils of this problem represent a far more realisti Z-gradient oil struture, whih also ontains a set of primary and seondary oils for shielding. The oils A B C D E F G H FIGURE 16 Full magnet problem. Snapshots of the distorted outer vauum hamber OVC, gradient oils and orresponding gradient magneti field lines at various time intervals for J AC JDC = 15% and ω = 2π1500rad/s. A, t = s; B, t = s; C, t = s; D, t = s; E, t = s; F, t = s; G, t = s; H, t = s [Colour figure an be viewed at wileyonlinelibrary.om]