INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2017; 112:1323 1352 Published online 13 July 2017 in Wiley Online Library wileyonlinelibrary.om)..5559 A linearised hp finite element framework for aousto-magneto-mehanial oupling in axisymmetri MRI sanners Sott Bagwell 1, Paul D Ledger 1 *,, Antonio J Gil 1, Mike Mallett 2 and Marel Kruip 2 1 Zienkiewiz Centre for Computational Engineering, College of Engineering, Swansea University, Bay Campus, Swansea, SA1 8EN, UK 2 Siemens PLC, Healthare Setor, MR Magnet Tehnology, Wharf Road, Eynsham, Witney, OX29 4BP, Oxon, UK SUMMARY We propose a new omputational framework for the treatment of aousto-magneto-mehanial oupling that arises in low-frequeny eletro-magneto-mehanial systems suh as magneti resonane imaging sanners. Our transient Newton Raphson strategy involves the solution of a monolithi system obtained from the linearisation of the oupled system of equations. Moreover, this framework, in the ase of exitation from stati and harmoni urrent soures, allows us to propose a simple linearised system and rigorously motivate a single-step strategy for understanding the response of systems under different frequenies of exitation. Motivated by the need to solve industrial problems rapidly, we restrit ourselves to solving problems onsisting of axisymmetri geometries and urrent soures. Our treatment also disusses in detail the omputational requirements for the solution of these oupled problems on unbounded domains and the aurate disretisation of the fields using hp finite elements. We inlude a set of aademi and industrially relevant examples to benhmark and illustrate our approah. Copyright 2017 The Authors. International Journal for Numerial Methods in Engineering Published by John Wiley & Sons, Ltd. Reeived 11 August 2016; Aepted 1 April 2017 KEY WORDS: aousto-magneto-mehanial oupling; finite element methods; MRI sanner; multifield systems; newton methods; spetral 1. INTRODUCTION Magneti resonane imaging MRI) has beome a widely used and popular tool in the medial industry apable of diagnosis of many medial ailments, suh as tumours, damaged artilage and internal bleeding as well as its use in neuroimaging. The most popular type of magnet used in these devies are superonduting magnets, onsisting of onduting wire ontained within a superooled vessel of liquid helium known as the ryostat, whih ahieve the field strengths required for high-resolution imaging. Figure 1 shows a typial setup of an MRI sanner, onsisting essentially of four main omponents: the main magnet oils, seondary magneti oils, the ryostat omprised of a set of radiation shields that enapsulate the liquid helium immersed magneti oils) and resistive gradient oils. A set of main magneti oils produe a strong uniform stationary magneti field aross the radial setion of the sanner, required to align the protons of the hydrogen atoms in the patient in the axial diretion. The seondary magneti oils are used to minimise the large stray magneti fields arising outside of the sanner. The ryostat onsists of a set of metalli vessels in vauum to maintain the superooled magnet temperatures and shield the oils from radiation. A set of resistive oils inside the imaging volume, known as gradient oils, produe pulsed magneti field gradients that exite ertain regions of the protons to generate an image of the patient. *Correspondene to: Paul Ledger, Zienkiewiz Centre for Computational Engineering, College of Engineering, Swansea University, Bay Campus, Swansea SA1 8EN, UK. E-mail: p.d.ledger@swansea.a.uk 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 2017 The Authors. International Journal for Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1324 S. BAGWELL ET AL. Figure 1. Primary omponents of a simplified linial magneti resonane imaging sanner. Reent developments in MRI sanner design have inreased the field strength of the main magnet, improved the transient gradient oil urrent signature and onsidered hanges in bore geometry in order to improve the quality of MRI images. However, this also introdues new hallenges beause image quality must also be balaned with the needs of linial effiieny and patient safety [1]. Transient magneti fields produe eddy urrents in onduting omponents, whih, in turn, perturb the magneti field and generate Lorentz fores, exerting eletro-mehanial stresses ausing them to vibrate and deform [2]. The vibrations also ause perturbations of the surrounding air, whih produe an aousti pressure field. These phenomena an have undesired effets ausing imaging artefats ghosting), dereased omponent life and unomfortable onditions for the patient, beause of the noise from mehanial vibrations. Minimising ghosting effets, keeping noise levels to aeptable limits and ensuring patient safety are key riteria for new MRI sanner designs. The onsistent exposure to time-varying magneti fields during a san proedure has resulted in patients experiening tingling sensations [3]. During some sans, when patients have been exposed to high sound levels, adverse effets have also been reported [4,5]. Current guidelines [1] state there is little risk of a permanent threshold shift in hearing in those exposed to noise assoiated with MRI proedures on a one-off or oasional basis. It goes on to say that, in urrent low-frequeny MRI sanners, linially signifiant effets on hearing are unlikely in most subjets for noise levels below 85 dba) of sans lasting less than an hour. Nevertheless, IEC reommend that hearing protetion should be used if equipment is apable of produing more than 99 dba). Many attempts have been made to measure the sound profiles of these harmful noise levels [6,7] as well as proposing methods to redue them, suh as ative noise-anelling tehnologies [8]. Experimental prototyping and testing of new sanner designs is expensive, and to redue this ost, there is onsiderable interest in the development of aurate omputational tools, whih an aid the design proess. However, the development of suh tools is hallenging due to omplexity that results from the oupled physis of eletromagnetism, mehanis and aoustis and the non-trivial task of aurately modelling the omplex field behaviour. Previously, there has been interest in applying ommerial multi-physis finite element method) FEM)) pakages, suh as COMSOL [9], Ansys [10] and NACS [11], to simulate the oupled nature of MRI sanners. Raush et al. have presented an approah based on the FEM-BEM boundary element method) program CAPA [12] for the magneto-mehanial oupling effets of an MRI sanner using a low-order disretisation [13]. This important three-dimensional simulation was extended to inlude aousti effets in [14] and, to the best of the authors' knowledge, is the only fully oupled simulation of a omplete MRI sanner that has been presented to date. To desribe the eletromagneti fores, they adopt the approah of Kaltenbaher, whih employs a layer of elements adjaent to the ondutor for their omputation [15]. In our previous work [2,16 18], we advoated an alternative approah, whih allows us to avoid the diret omputation of eletromotive fores and instead work entirely with a physially motivated Maxwell stress tensor. Others have attempted to model these omplex physial effets to aid the design of the MRI magneti oils [19,20], to analyse the plane strain effets on superonduting solenoids [21] and the effets of the magneti field exposure on the patient [22]. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1325 The simulation of MRI sanners also builds on the expanding literature devoted to magnetomehanis and oupled problems inluding FEM-BEM oupling [23,24], magneto-mehanial damping mahines [25], magneto-mehanial effets on material parameters [26], enhaned basis funtions for magneto-mehanial oupling [27] and strongly oupled systems [28]. When the oupling inludes aousti effets, areful treatment of the far field boundary is also required and tehniques suh as perfetly mathed layers PML), FEM-BEM oupling, infinite elements and absorbing boundary onditions have been developed for this see, e.g. [29] for a reent review). Although ommerial odes provide an effiient environment for many problems, our interest lies in providing a low-ost dediated industrial design tool. This tool should operate over a wide range of frequenies more effetively in order to predit the response from an analysis of general gradient oil time signatures, whih an be deomposed by Fourier expansion. As suh, not only do we require a reliable means of treating the oupled nature of aousto-magneto-mehanial problems, but we also need to be able to aurately resolve the potentially small skin depth in onduting omponents as well as aurately resolve the propagation of aousti waves. The aforementioned ommerial odes are typially designed with low-order FE disretisations in mind, whih require dense meshes for handling the small skin depths and wave propagation at higher frequenies. On the other hand, hp FE disretisations offer possibilities for high auray on loally refined meshes and have been shown to aurately resolve the skin depth in the onduting omponents [30], handle the omplex oupling [16 18] and to resolve the propagation of the aousti waves [31] and we, therefore, hoose to adopt them here. Furthermore, in the omputation of unbounded domains using a PML, they have also been shown to offer superior performane [32]. Building on the established hp FEM methodology, the main novelty of our work is to provide a new rigorous theoretial framework for the simulation of aousto-magneto-mehanial effets in MRI sanners, whih forms the basis of our design tool. We undertake the onsistent linearisation of the transient equations and arrive at a simplified monolithi single-step strategy in the ase of harmoni gradient oil exitations. It greatly improves on our previous work [2], whih required nonphysially motivated simplifying assumptions and resulted in a fixed point strategy with a growth of iterations for inreasing frequeny. We also extend our framework to inorporate the effets of the aousti field and propose a rigorous set of interfae onditions, whih ouple the various physis together. The entire framework is suitable for three-dimensional geometries disretised by hp FEM, but suh simulations would be prohibitively expensive for the industrial design yle. As our interest lies in the development of a rapid design tool, we fous on the simulation of problems on axisymmetri geometries. The presentation of the material proeeds as follows: In Setion 2, we outline the governing equations, the oupling between the fields and present a fully oupled non-linear transient transmission problem. Setion 3 presents the onsistent linearisation of the weak form of the transmission problem, and we derive a simplified monolithi strategy in the ase of harmoni gradient oil exitations. Then, in Setion 4, we briefly disuss the redution to axisymmetri geometry, the omputational far field treatment and the hp FE disretisation. Setion 5 presents a seletion of numerial ase studies to validate eah physial field independently and highlights the omputational hallenges of small skin depth and high-frequeny wave propagation. Then we present a series of results for omplex oupled ase studies of industrial relevane before losing with onluding remarks. 2. COUPLING APPROACH In the following, we desribe the governing equations and oupling methodologies that link the eletromagneti, mehanial and aousti behaviour of an MRI sanner. We begin, in Setion 2.1, with the transient eddy urrent model to desribe the eletromagneti response from a onduting region Ω when illuminated by a low-frequeny bakground magneti field. This field arises from a urrent soure J s with support in an unbounded region of free spae R 3 Ω, as illustrated in Figure 2. Then, in Setion 2.2, we present the orresponding transient mehanial response of Ω resulting from eletromagneti stresses generated in this region. Finally, in Setion 2.3, we present the transient Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1326 S. BAGWELL ET AL. Figure 2. Conduting region Ω exited by oils ontained within the unbounded R 3 Ω spae. aousti response resulting from the vibration of Ω in R 3 Ω. The omplete oupled transmission problem is stated in 2.4. 2.1. Eletromagneti desription For our hosen appliation, the transient eddy urrent approximation of Maxwell's equations is valid [2], where the displaement urrent terms are negleted beause of the high ondutivity of the onduting omponents and the low frequeny of the exiting urrents. A rigorous justifiation involves the topology of the onduting region [33]. Defining E, H, D, B as the eletri, magneti, eletri flux and magneti flux intensity vetors, respetively, and introduing a vetor potential A suh that B = A, this model an be desribed by μ 1 A) =J s + J o + J l = J s γ A u + γ t t A) in R3, 1a) A = 0 in R 3 Ω, 1b) A = O x 1) as x, 1) where we assume x to be measured from the entre of Ω. The previous equation assumes the regions to be homogenous and isotropi, suh that B = μh and J o = γe where γ denotes the eletrial ondutivity and μ the magneti permeability. The solenoidal external urrent soures J s are assumed to lie in free spae, R 3 Ω,whereγ = 0andμ = μ 0 = 4π 10 7 H/m. The term J l = γ u t A denotes the Lorentz urrents where u is the mehanial displaement in the onduting region Ω. The vetor potential A satisfies the transmission onditions on the ondutor boundary Ω n [A] Ω = 0, n [μ 1 A] Ω = 0, where [ ] Ω denotes the jump on this interfae and n is a unit outward normal vetor to Ω. 2.2. Mehanial desription The onduting region Ω is assumed to behave elastially and the mehanial displaements u to satisfy σ m u)+σ e A)) = ρ 2 u in Ω t 2, 2) The temporal gauge has been applied in Ω and the Coulomb gauge in R 3 Ω [30]. Note we set A = O x 1) as x aording to the mathematial model desribed by Ammari, Buffa and Nédéle; [33] here, the big O notation implies that the rate is at least as fast as x 1 and an be faster in pratie; for details, see the aforementioned paper. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1327 where σ m u) =λtrεu))i + 2Gεu), is the mehanial ) ontribution to the Cauhy stress tensor, λ, G denote the Lamé parameters, ε = u + u T 2 the linear strain tensor, I the identity tensor, T the transpose and σ e A) =μ H H 1 ) 2 H 2 I =μ A 1 A 1 ) 2 A 2 I, is the magneti omponent of the Maxwell stress tensor [2]. In Equation 2), we have already used the fat that the magneto-ponderomotive foring term an be expressed as f e = σ e. We write Ω = Ω D Ω N and fix u = u D on Ω D in order to stop the onduting omponent floating away. 2.3. Pressure desription In free spae, σ m has only a volumetri part σ m = I = κ u)i where is some pressure field. vol This means that 2) redues to the salar wave equation with a souring term 2 1 2 2 t 2 = σe ) in R 3 Ω, 3) where = κ ρ is the speed of sound and κ denotes the bulk modulus of the medium. This must be aompanied by the assoiated radiation ondition lim x x + ) = O x 1 ). 4) t The soure term in 3) is only non-zero in the support of J s ; this an be seen by onsidering the alternative form of f e in free spae below f e = σ e = μ 0 H H = A J s = A μ 1 0 A), 5) and thus it follows that in the same region f e = σ e )=μ 0 H J s J s 2 ). 6) Taking this into aount, on the interfae Ω shown in Figure 3, the pressure field, mehanial displaements and stresses are oupled by the transmission onditions σ m + σ e ) Ω n = I + σ e ) + Ω n, ρ + 2 u n = + σ e ) + t 2 Ω n, Ω Note that the latter ondition also further redues to + Ω D n = 0, ρ + 2 u t 2 Ω N n = + σ e ) + Ω n, in light of the known Dirihlet displaement ondition on Ω D for the mehanial problem. Here, we assume Biot Savart oils. If the oils are instead treated as rigid or deformable onduting bodies, then their support instead forms part of Ω We reall here the relationship between the url and gradient operators H)H 1 2 H H) = H H). Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1328 S. BAGWELL ET AL. Figure 3. Interfae onditions at the ondutor boundary. 2.4. Coupled transmission problem Combining the statements from the previous Setions 2.1, 2.2 and 2.3, we arrive at the following transmission problem for desribing our oupled aousto-magneto-mehanial system in a time period [0, T]: Find A, u, )t) R 3 R 3 R)[0, T] suh that μ 1 A)+γ A = J s + γ u t t A) in R3, 7a) A = 0 in R 3 Ω, 7b) σ m u)+σ e A)) = ρ 2 u t 2 in Ω, 7) 2 1 2 2 t 2 = σe A)) in R 3 Ω, 7d) A = O x 1), 7e) lim x x + ) = O x 1 ) as x, 7f) t u = u D, 7g) + n = 0 Ω D on Ω D, 7h) ρ + 2 u n = + σ e A)) + t 2 Ω n Ω N on Ω N, 7i) n [A] Ω = 0, 7j) n [μ 1 A] Ω = 0, 7k) σ e A)+σ m u)) Ω n = I + σ e A)) + Ω n on Ω, 7l) At = 0) =0 in R 3, 7m) ut = 0) = u t t = 0) =0 in Ω, 7n) t = 0) = t t = 0) =0 in R3 Ω, 7o) where we have hosen to set the initial onditions for the fields to be zero, orresponding to a system at rest at t = 0. An illustration of the fully oupled system is shown in Figure 4. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1329 Figure 4. Physial representation of the oupling effets in an magneti resonane imaging environment. For our desired appliation, the system 7) is exited through the urrent soure J s t). In pratie, the appliation allows for the deomposition J s t) =J DC +J AC t),wherej DC orresponds to the stati urrent soure of the main magneti oils and J AC t) the transient urrent soure of the gradient oils [2]. This deomposition, illustrated in Figure 5, allows us to introdue the following stati problem: find A DC, u DC, DC R 3 R 3 R suh that μ 1 A DC )=J DC in R 3, 8a) A DC = 0 in R 3, 8b) σ m u DC )+σ e A DC )) = 0 in Ω, 8) 2 DC = σ e A DC )) in R 3 Ω, 8d) A DC = O x 1), 8e) DC = O x 1 ) as x, 8f) u DC = u D DC on Ω D, 8g) n [A DC ] Ω = 0, 8h) n [μ 1 A DC ] Ω = 0, 8i) DC + σ e A DC ) ) + n = 0, Ω 8j) σ m u DC )+σ e A DC ) ) n = DC I + σ e A DC ) ) + n Ω Ω on Ω, 8k) where we have assumed a similar deomposition of the Dirihlet displaement ondition u D = u D DC + ud AC t). 3. LINEARISATION With developing a FE framework for the approximate solution of 7) and 8) in mind, we linearise weighted residual statements of the transmission problems. To this end, it is onvenient to introdue the following X ={A Hurl, R 3 ) A = 0 in R 3 Ω }, Yg) ={u H 1 Ω ) 3 u = g on Ω D }, Z ={ H 1 R 3 Ω )}, whih will be used to desribe the weak solutions to the dynami and stati transmission problems, where Hurl, R 3 ) and H 1 R 3 ) have their usual definitions e.g. [34]). We start with the treatment of the simpler stati problem 8) and then ontinue to our approah for the transient system 7). Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1330 S. BAGWELL ET AL. Figure 5. Current soure deomposition. 3.1. Linearisation of the stati problem Consider possible weak solutions A DC, u DC, DC ) X Yu D ) Z and the assoiated residuals DC R DC A Aδ ; A DC ) = R 3μ 1 A DC A δ )dω R 3J DC A δ dω, 9a) R DC u u δ ; A DC, u DC, DC ) = σ m u DC )+σ e A DC )) u δ dω Ω I + σ e ) + n u δ ds, 9b) Ω N R DC δ ; A DC, DC ) = DC δ + σ e ) δ )dω, 9) R3 Ω for all A δ, u δ, δ ) X Y0) Z. The diretional derivatives of these residuals are u u u A Aδ ; A DC )[δ DC A ]= μ 1 δ DC A Aδ )dω, 10a) R 3 u δ ; A DC, u DC, DC )[δ DC A ]= μ 1 SA DC, δ DC A ) uδ dω Ω μ 1 Ω 0 SADC, δ DC A ) + n u δ ds, 10b) N u δ ; A DC, u DC, DC )[δ DC u ]= σ m δ DC u ) u δ dω, 10) Ω u δ ; A DC, u DC, DC )[δ DC ]= δ DC + n u δ ds, 10d) Ω N δ ; A DC, DC, )[δ DC A ]= δ DC A ) JDC ) δ dω, 10e) suppj DC ) δ ; A DC DC )[δ DC ]= δ DC δ dω, 10f) R3 Ω where n = n + = n and we have introdued the linearised eletromagneti stress tensor Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1331 SA DC, δ DC A ) = ADC δ DC A + δdc A ADC A DC δ DC ) A I. At a ontinuous level, equations 9,10) an be used to introdue the Newton Raphson iteration: Find, δ DC[m],δ DC[m] ) X Y0) Z suh that δ DC[m] A u u u u A Aδ ; A DC[m] )[δ DC[m] ]= R DC A ]+ A u δ ; A DC[m], u DC[m], DC[m] )[δ DC[m] u δ ; A DC[m], u DC[m], DC[m] )[δ DC[m] u ]+ A Aδ ; A DC[m] ), 11a) u δ ; A DC[m], u DC[m], DC[m] )[δ DC[m] ]= R DC u u δ ; A DC[m], u DC[m], DC[m] ), 11b) δ ; A DC[m], DC[m] )[δ DC[m] ]+ A for all A δ, u δ, δ ) X Y0) Z where δ ; A DC, DC[m] )[δ DC[m] ]= R DC δ ; A DC[m], DC[m] ), 11) A DC[m+1] =A DC[m] + δ DC[m] A, u DC[m+1] =u DC[m] + δ DC[m] u, DC[m+1] = DC[m] + δ DC[m]. The initial guesses are suh that A DC[0], u DC[0], DC[0] ) X Yu D ) Z). To permit the omputational solution of 11), a spatial FE disretisation is required, whih we will disuss in Setion 4. DC However, at this stage, it is already useful to note that, beause of the speifi nature of the equations, 11a) an be solved independently, followed by 11) and then 11b) without iteration. Moreover, if the system is solved monolithially, the solution will onverge to A DC, u DC, DC ) X Yu D DC ) Z) in a single iteration. In our previous approah [2], we negleted the effets of the stati displaement, driven by the stati magneti field, as we were primarily interested in omputing the output power of the system that depends only upon the transient displaements, whih were assumed to be harmoni. However, in this new formulation, we also inlude the stati effets of the displaements to allow for a rigorous treatment of the linearised transient sheme and maintain onsisteny of the physial fields. 3.2. Linearisation of the dynami problem Consider possible transient weak solutions A, u, )t) X Yu D ) Z)[0, T] and the assoiated AC residuals R A A δ ; A, u) = μ 1 A A δ + γ A R 3 t Aδ) dω J s A δ dω R 3 γ u Ω t A) Aδ dω, 12a) R u u δ ; A, u, ) = Ω σ m u)+σ e A)) u δ dω+ Ω ρ 2 u t 2 uδ dω I + σ e A)) + n u δ ds, Ω N R δ ; A, u, ) = δ + 1 2 ) R3 Ω 2 t 2 pδ + σ e δ dω 12b) ρ + 2 u n + p δ ds, 12) Ω N t 2 Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1332 S. BAGWELL ET AL. for all A δ, u δ, δ ) X Y0) Z). The diretional derivatives of these residuals are DR A A δ ; A, u)[δ A ]= μ 1 δ A A δ + γ δ ) A A δ dω R 3 t γ u ) Ω t δ A) A δ dω, 13a) DR A A δ ; A, u)[δ u ]= γ δ ) u A) A δ dω, 13b) Ω t DR u u δ ; A, u, )[δ A ]= Ω μ 1 SA, δ A ) u δ dω μ 1 Ω 0 SA, δ A) + n u δ ds, 13) N ) DR u u δ ; A, u, )[δ u ]= σ m δ u ) u δ + ρ 2 δ u u δ dω, 13d) Ω t 2 DR u u δ ; A, u, )[δ ]= δ + n u δ ds, Ω N DR δ ; A, u, )[δ A ]= δa μ 1 suppj s 0 A)) ) 13e) + A μ 1 0 δ A)) ) δ dω, 13f) DR δ ; A, u, )[δ u ]= ρ + 2 δ u n + δ ds, 13g) Ω N t 2 DR δ ; A, u, )[δ ]= R3 Ω δ δ + 1 2 2 δ t 2 δ ) dω. 13h) In the preeding equations, we have found it onvenient to use the alternative form of σ e introdued in 5) when linearising R. One strategy for solving the temporal system, after spatial disretisation, would be to adopt a disrete time integration sheme and then apply the Newton Raphson algorithm at eah time-step to solve the non-linear equations using the diretional derivatives omputed previously. However, suh an approah is omputationally expensive and, in the interests of developing a fast omputational tehnique, we adopt a different strategy. Rather than integrating the equations in time and solving the non-linear equations at eah time step, we hoose instead to linearise the full time dependant equations about the stati solution. This linearisation in the ontext of MRI sanners is motivated by the knowledge that the stati DC urrent soure is several orders of magnitude stronger than the weaker AC time varying soure, leading to a strong DC field and a weaker time varying field. Similar tehniques, involving the additive split of a non-linear problem to a series of linear problems, have been suessfully applied to the field of omputational mehanis suh as analysis of strutural membranes [35 37], high-order mesh generation [38] and in biomedial appliations [39]. In this ase, the residuals of the dynami problem beome R AC A Aδ ) =R A A δ ; A DC, u DC )= suppj )J AC A δ dω, AC 14a) R AC A uδ ) =R u u δ ; A DC, u DC, DC )=0, 14b) R AC δ ) =R δ ; A DC, u DC, DC )= A suppj ) DC ) J AC ) δ dω, AC 14) Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1333 and the assoiated diretional derivatives take the form A Aδ )[δ A ] =DR A A δ ; A DC, u DC )[δ A ] = μ 1 δ A A δ + γ δ ) A A δ dω, 15a) R 3 t A Aδ ; A DC )[δ u ] =DR A A δ ; A DC, u DC )[δ u ]= γ δ u A DC ) A δ dω, 15b) Ω t u u δ ; A DC )[δ A ] =DR u u δ ; A DC, u DC, DC )[δ A ]= μ 1 SA DC, δ A ) u δ dω Ω μ 1 Ω 0 SADC, δ A ) + n u δ ds, 15) N u u δ )[δ u ] =DR u u δ ; A DC, u DC, DC )[δ u ] ) = σ m δ u ) u δ + ρ 2 δ u u δ dω, 15d) Ω t 2 u u δ )[δ ] =DR u u δ ; A DC, u DC, DC )[δ ]= Ω N δ + n u δ ds, 15e) δ ; A DC )[δ A ] =DR δ ; A DC, DC )[δ A ]= suppj DC ) δa ) J DC) δ dω A DC μ 1 suppj DC ) suppj AC 0 δ A)) ) δ dω, ) 15f) δ )[δ u ] =DR δ ; A DC, u DC, DC )[δ u ]= ρ + 2 δ u n + δ ds, Ω N t 2 15g) δ )[δ ] =DR δ ; A DC, DC )[δ ]= δ δ + 1 2 δ R3 Ω 2 t 2 δ ) dω. 15h) Finally, realling that A DC, u DC, DC are all time invariant, we see that the residuals and the diretional derivatives in 14) and 15), respetively, are linear in the time dependent terms δ A, δ u,δ and J AC. This motivates the time harmoni representations δ A δ A e iωt, δ u δ u e iωt, δ δ e iωt, J AC J AC e iωt, where ω denotes the angular frequeny of the driving urrent in the gradient oils in the ase of a harmoni exitation. In reality, the gradient oils are driven using non-harmoni exitations, but their time signals an be deomposed into its different frequeny modes and the same approah applied. The solution of the linear harmoni problem beomes as follows: find δ A, δ u,δ ) X Yu D AC ) Z suh that A Aδ )[δ A ]+D R AC A Aδ ; A DC )[δ u ]= R AC A Aδ ) u u δ ; A DC )[δ A ]+ u δ ; A DC )[δ A ]+ = suppj AC )J AC A δ dω, 16a) u δ )[δ u ]+D R AC u u δ )[δ ]= R AC u u δ ) = 0, 16b) δ )[δ u ]+D R AC δ )[δ ]= R AC δ ) = suppj AC ) A DC ) J AC δ dω. 16) Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1334 S. BAGWELL ET AL. for all A δ, u δ, δ ) X Y0) Z where it assumed that u D AC ud AC eiωt. We may then deompose the full temporal solution into its stati and time varying omponents, whih in the ase of a single frequeny exitation are given by At) =A DC + Reδ A e iωt ), ut) =u DC + Reδ u e iωt ), t) = DC + Reδ e iωt ). The diretional derivatives in Equation 16) expliitly beome A Aδ )[δ A ]= μ 1 δ A A δ + iωγδ A A δ) dω, 17a) R 3 A Aδ ; A DC )[δ u ]= Ω iωγδ u A DC ) A δ dω, 17b) u u δ ; A DC )[δ A ]= μ 1 SA DC, δ A ) u δ dω Ω μ 1 Ω 0 SADC, δ A ) + n u δ ds, 17) N u u δ )[δ u ]= σ m δ u ) u δ ρω 2 δ u u δ) dω, 17d) Ω u u δ )[δ ]= Ω N δ + n u δ ds, 17e) δ ; A DC )[δ A ]= suppj DC ) δa ) J DC) δ dω suppj DC ) suppj AC ) A DC μ 1 0 δ A)) ) δ dω, 17f) δ )[δ u ]= ω 2 ρ + δ u n + δ ds, 17g) Ω N ) δ )[δ ]= δ δ ω2 R3 Ω 2 δ δ dω. 17h) In the absene of pressure, the formulation previously proposed in [2] for the dynami ase is idential to that obtained in the preeding equations; however, previously, a number of assumptions had to be made in order to drop terms to arrive at the onsidered system, whih is now no longer the ase. Moreover, in the preeding formulation, the equations are rigorously established through a linearisation of the dynami system and the solutions an be obtained by a single monolithi solve rather than an iterative fixed point sheme, previously proposed. 4. COMPUTATIONAL TREATMENT In this setion, we disuss the key omponents for the omputational solution to Equations 16) and 11). This inludes the redution for rotationally symmetri geometries in Setion 4.1, the far field treatment in Setion 4.2 and the hp FE disretisation in Setion 4.3. 4.1. Redution for rotationally symmetri geometries The formulation proposed in Setion 3 is valid for general three-dimensional domains involving a onduting region surrounded by an unbounded region of free spae ontaining the urrent soures. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1335 To a first approximation, the geometry of an MRI sanner is lose to ylindrial and the strong DC urrent soure J DC has only the omponent J DC in ylindrial oordinates r,φ,z). However,ofthe φ three sets of AC gradient oils, it is only the z-gradient oil that exhibits rotational symmetry, and in our quest for a rapid industrial design tool, we must neglet the x and y gradient oils. Under these assumptions, our simplified MRI sanner is rotationally symmetri with respet to the azimuth, the problem redues to solving for A = A φ r, z)e φ, u = u r r, z)e r + u z r, z)e z, = r, z) [2]. Here e r, e φ and e z denote the standard basis of the ylindrial oordinate system. The redution of the full threedimensional problem to the axisymmetri meridian r, z) plane Ω m isshowninfigure6.however, by projetion of this plane, full three-dimensional results are still ahieved. When transformed to the axisymmetri domain, the spaes in whih the weak solutions are sought in the variational statements 16) and 11) must also be adapted. In general, this leads to neessity to seek for solutions in weighted spaes to ensure the fields are well behaved at the radial axis [40]. To avoid the omplexity of weighted spaes, we transform the fields A φ = râ φ and u r = r u r aording to that proposed in [2], and note that  φ H 1 Ω m ), ũ = [ u r, u z ] H 1 Ω m ). The aousti pressure presents no diffiulty beause r, z) H 1 Ω m ). The treatment of the bilinear and linear forms assoiated with the terms in 11) and 16) follow similar steps to that presented in [2], and hene, we only give as an example ) δ )[δ ]=2π m δ m δ ω2 R2 Ω m 2 δ δ r dω m, 18) where Ω m is the projetion of Ω on to the meridian plane, m = [ r, z] T and the fator of 2π results from the integration of the azimuthal diretion. This fator similarly appears in the treatment of all of the other terms in the linear system and therefore anels. 4.2. Far field treatment The strong forms of the dynami and stati problems inlude radiation and deay onditions, whih desribe the behaviour of A, A DC, and DC as x. To allow the omputational treatment of the problem, the unbounded free spae region R 3 Ω is trunated at a finite distane from Ω and the region Ω n is reated, whih ontains all the urrent soures. The three-dimensional omputational domain Ω =Ω n Ω R 3 beomes Ω m = Ω m n Ω m R 2 for axisymmetri problems. This means that integrals over R 3 Ω in 11) and 16) beome integrals over Ω n and then, in turn, Ω m n. On Ω m, the stati deay of A, A DC, DC is approximated by fixing Ω m to be loated suffiiently far from the region of interest and setting A = A DC = A φ e φ = A DC φ e φ = 0, DC = 0 on Ω m. Naturally, the quality of the approximation improves as the size of Ω n Ω m n ) is inreased. However, the harmoni pressure field, and in partiular the treatment of δ )[δ ], annot merely be approximated by trunating the domain and fixing = 0at Ω m, beause this would result in refletions that Figure 6. Transformation from three-dimensional full sanner to simplified 2D axisymmetri ase. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1336 S. BAGWELL ET AL. would pollute the omputational domain beause of its wave behaviour. The radiation ondition 4) in the ontinuous problem desribes the orret deay of this field, whih must also be approximated omputationally. Therefore, we add a PML [32] Ω pml Ω m pml ) to the exterior of Ω n Ω m n ) so that the omputational domain now beomes Ω=Ω Ω n Ω pml Ω m =Ω m Ω m n Ω m ). For terms other pml than δ )[δ ] in 16), Ω pml an be merely thought of as a free spae extension of Ω n.however, the aforementioned term 18) is treated differently as δ )[δ ]= Ω m n + Ω m pml ) δ δp 2 m δ m δ ω2 Λ 1 m δ m δ ω2 2 r dω m ) Λ 2δ p δ r dω m, where Λ 1, Λ 2 are both omplex funtions of position in the layer and redue to identity on Ω m n Ω m. The oeffiients of these funtions an be established through a omplex oordinate strething pml of the domain Ω m n following the approah in [32]. For the axisymmetri ase, this omplex strething is equivalent to introduing the omplex position-dependent funtions Λ 1 = 1 r z z z r z r 0 0 z r z r z z, Λ 2 = z rz zz r. r In the preeding equation, the omplex oordinate transform z s is desribed as a power law in terms of the distane to the layer d s and thikness of the layer t s, as illustrated in Figure 7b. The prime indiates differentiation with respet to the argument, and expliitly, we hoose { s 0 s < dk z s s) = ) s ds 5s, s i t s dk s where s =[r, z]. The hoie of a power law of degree 5 and user-defined thikness t s is somewhat arbitrary, provided that the resulting omplex field behaviour in Ω m is properly resolved. If this is pml aomplished, the aousti pressure field is absorbed without refletion and we an set = 0on Ω m. 4.3. hp FE disretisation Building on the previous suess in [2], we hoose to adopt the hp FE H 1 onforming basis funtions, proposed by Zaglmayr and Shöberl [41,42], for aurately disretising the fields  φ, ũ 19) Figure 7. Representation of the perfetly mathed layer. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1337 and. Denoting their two-dimensional basis funtion set for triangular grids by X hp the disrete stati problem beomes δ DC[m] A φ hp e φ, δ DC[m],δ DC[m] ) = rδ DC[m] ũhp hp e φ, rδ DC[m] e  φ hp û r hp r + δ DC[m] e u z hp z,δ DC[m] ) hp X0) rx hp e φ Y0) rx hp e r + X hp e z ) Z0) X hp suh that u u A ũ δ hp ; ADC[m] φhp u ũ δ hp ; ADC[m] φhp ũ δ hp ; ADC[m] φhp Aδ φhp e φ; A DC[m] φhp e φ, ũ DC[m] hp e φ, ũ DC[m] δ hp ; ADC[m] φhp hp, DC[m] hp e φ, ũ DC[m] hp e φ )[δ DC[m] e φ]= R DC A φ hp )[δ DC[m] A φ hp e φ]+, DC[m] )[δ DC[m] ]+ hp ũhp, DC[m] )[δ DC[m] ] hp hp e φ, DC[m] hp δ hp ; ADC φhp e φ, DC[m] hp = R DC u )[δ DC[m] A φ hp e φ]+ )[δ DC[m] hp A ũ δ hp ; ADC[m] φhp ]= R DC Aδ φhp e φ; A DC[m] φhp e φ ), 20a) e φ, ũ DC[m] hp, DC[m] hp ), 20b) δ hp ; ADC[m] e φhp φ, DC[m] ), 20) hp for all A δ φhp e φ, ũ δ hp,δ hp )=râδ φhpe φ, r u δ rhp e r + u δ zhp e z, δ hp ) X0) rx hp e φ Y0) rx hp e r, X hp e z ) Z0) X hp.intheaboveweuse Xg) ={A Hurl, Ω) A = 0 in Ω n Ω pml, A = g on Ω}, Zg) ={ H 1 Ω n Ω pml ), = g on Ω}, to aount for the domain trunation introdued in Setion 4.2. The orresponding disrete linear harmoni problem for δ Aφ hpe φ, δũhp,δhp )=rδ Âφ hpe φ, rδ ur hpe r + δ uz hpe z,δhp ) X0) rx hp e φ Yu D AC ) rx hpe r + X hp e z ) Z0) X hp an be developed analogously. We hoose not to give the expliit formulae for the terms beause they an be found by following the approah for axisymmetri problems, whih we have previously desribed in [2]. The omplete algorithm is summarised in Algorithm 1. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1338 S. BAGWELL ET AL. 5. NUMERICAL RESULTS In this setion, we present a series of aademi and industrial numerial examples in order to demonstrate the apabilities of the presented framework. Firstly, we inlude examples with analytial solutions to demonstrate the independent validation of the eletromagneti, mehanial and aousti fields for unoupled problems in Setions 5.1. Seondly, we inlude validation of oupled physis problems and appliation to hallenging industrial benhmarks in Setion 5.2. In order to analyse the quality of the solutions, we set e L 2 = e, e) 1 2, e H 1 = e 2 L 2 + e 2 L 2 ) 1 2 for appropriate salar and vetorial errors respetively, where e, e) = Ω e e dω is the standard L 2 inner produt where the overbar denotes the omplex onjugate. For mehanial problems, we also onsider σe) SNSΩ) = trσe)) L 2 assoiated with the sum of normal stresses to examine the extent to whih our formulation an overome mehanial loking [16]. 5.1. Validation of single-physis problems 5.1.1. Conduting sphere in uniform alternating magneti field. A losely related problem to the solution of 1), with J s = J l = 0, is that of a onduting objet loated in free spae exited by a uniform harmoni bakground magneti field of amplitude H 0 and frequeny ω. In this ase, μ 1 0 A H 0 as x. For a spherial ondutor Ω ={x x 2 R 2 } of radius R, permeability μ s and ondutivity γ s, as illustrated in Figure 8, an analytial axisymmetri solution is presented in [43] for A φ. We onsider the ase of R = 1m, γ s = 10 7 S m, μ s = μ 0, H 0 = μ 1 0 e zwb and angular frequenies ω = [5, 50, 500]rad/s. To simulate this problem, the omputational domain is hosen as Ω m = [0, 4] [ 4, 4])m 2 and we solve a suitably simplified version of Algorithm 1 where δ Aφ hp Ωn = A φ Ωn and we set A hp φ = δ A φ hp. We generate a oarse mesh of 578 unstrutured quasiuniform triangular elements of maximum size h = 0.5m and use here, and subsequently, a blending funtion approah to represent the exat geometry of the sphere's surfae [44]. This funtion avoids any geometrial error in the solution beause of oarse approximation of the boundary. In light of the smooth nature of the solution, we onsider uniform polynomial enrihment orresponding to p = 1, 2, 10 on this mesh and plot in Figure 9 the relative error measures A φ A hp φ L 2 A φ L 2, A φ A hp φ H 1 A φ H 1 against the number of degrees of freedom for varying frequenies of the alternating magneti field. Figure 9a shows the onvergene of the error against the number of degrees of freedom on a logarithmi sale, where eah point represents a polynomial refinement and the different urves orrespond to different frequenies and error measures. Eah line indiates a downward sloping Figure 8. Conduting sphere in a uniform alternating magneti field: problem setup. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1339 Figure 9. Conduting sphere in a uniform alternating magneti field: onvergene of A φ A hp φ L 2 A φ L 2 and A φ A hp φ H 1 A φ H 1 under p-refinement. Figure 10. Conduting sphere in a uniform alternating magneti field: ontours of B A hp φ e φ) around the onduting sphere at different frequenies. urve suggesting that the onvergene is exponential. This is onfirmed by plotting the error on a logarithmi sale against the number of degrees of freedom raised to the power 1 2onanalgebrai sale in Figure 9b. After a pre-asymptoti region, eah urve beomes a straight line indiating that onvergene of the numerial to the analytial solution is exponential with respet to the square root number of degrees of freedom for p-refinement of this problem. This orresponds to the expeted rate for smooth solutions, as reported by Babuška and Guo [45]. As the frequeny of the alternating magneti field inreases, the gradient of the lines in Figure 9b redues, indiating that, although still exponential, the rate of onvergene is lower. Physially, this is due to the smaller skin depths s = 2 ωγμ), [43] whih haraterises the depth to whih the eddy urrents J o = γe iωγa hp φ e φ deay to 1 e of their surfae value, assoiated with higher ω. It is possible to improve the gradient in the plots in Figure 9b by using a graded mesh towards the sphere's surfae. We have also onduted h-refinement studies, whih ahieve the expeted and slower) algebrai rates of onvergene for this problem. [45] Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1340 S. BAGWELL ET AL. To illustrate the different fields and skin depth effets for different frequenies, Figure 10 shows the ontours of B A hp φ e φ) for the various frequenies of the onverged solutions. This figure illustrates the smaller s for higher ω and the need to use higher fidelity disretisations to apture the solution with the same degree of auray. 5.1.2. Mehanial shell. To verify the treatment of the elastiity system, we onsider the solution of equation 2) for σ e = 0 and stati displaements u ut) for the ase of a spherial mehanial shell Ω=Ω ={x r 2 x 2 r 2 i o} of inner and outer radii r i, r o, respetively. The inner and outer surfaes of the shell are subjet to tration onditions i n and o n resulting from internal and external pressures i, o respetively, on different parts of Ω N as illustrated in Figure 11. This problem is axisymmetri and has the analytial solution [46], whih an be expressed in terms of the ylindrial displaement omponents u r, u z ). Speifially, we solve the problem orresponding to r i = 0.5m, r o = 1m, E = 210 10 9 Pa, ν = 0.49, i = o = 10 4 Pa so that the shell is nearly inompressible. As desribed in Setion 2.2, we must fix part of the boundary of the shell to avoid it floating away. We solve a suitably simplified Figure 11. Mehanial shell subjet to interior and exterior pressure: problem setup. Figure 12. Mehanial shell subjet to interior and exterior pressure: onvergene of u u hp L 2 u L 2, u u hp SNS u SNS under p-refinement. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1341 version of 20) for a single iteration where we hoose to fix the displaements δ DC[0] = u ũhp r e r +u z e z on a small boundary segment Ω D aording to the analytial solution. The region of omputation orresponds to Ω m =Ω m ={r, z) r 2 r 2 +z 2 ) r 2 i o}, whih we disretise by a quasi-uniform mesh of 68 unstrutured triangular elements of maximum size h = 0.5m. We perform the same p-refinement study that was desribed in Setion 5.1.1 and now measure onvergene using u u hp L 2 u L 2, u u hp SNS u SNS where u hp = δũhp. The results shown in Figure 12a and 12b illustrate similar trends to those shown previously indiating that exponential onvergene with respet to the number of degrees of freedom raised to the power 1 2 is also ahieved through p-refinement for the mehanial problem. In partiular, p-refinement serves as a method for overoming volumetri loking that is known to be assoiated with the displaement formulation of elastiity for nearly inompressible material [47,48] and leads to exponential rates of onvergene of the error measured in the SNS norm. Although we do see stagnation of onvergene when the norms of the error reah 10 13,whih oinides with the numerial preision of our omputation. The displaements in the radial and axial diretions of the shell, obtained using p = 10, are illustrated in Figure 13. 5.1.3. Aousti sattering of a sphere. Finally, to verify the aousti system 3, 4), we onsider the problem of a sound hard sphere of radius R, illustrated in Figure 14. The sphere is illuminated by a harmoni inident wave in = 0 e ikz of amplitude 0 and wavenumber k = ω. The omplete solution is of the form = in + s, subjet to the boundary ondition n = 0on Ω and admits an axisymmetri analytial solution for s, [49] Speifially, we hoose to solve the problem orresponding to R = 1m, 0 = 1Pa, k = [4π 3, 10, 30]m 1. We solve a suitably simplified version of Algorithm 1 on the omputational domain Ω m =[0, 5.6] [ 5.6, 5.6]) {r, z) r 2 + z 2 R 2 }m 2 of whih Ω m pml =Ω m [0, 4] [ 4, 4])m 2, with thikness parameters [t r, t z ]=[1.6, 1.6]m, distane parameters [d r, d z ]=[4, 4]m and δhp = 0on Ω m. We fix a quasi-uniform mesh of 408 unstrutured triangular elements of maximum size h = 0.5m and perform the same refinement study of p = 1, 2, 10 as previously and measure onvergene using s shp L 2 s L 2, s shp H 1 s H 1,where shp = δhp. Figure 15a and 15b illustrates similar downward sloping trends, as shown in the previous examples. However, in this ase the pre-asymptoti region is now affeted by the inrease in wave number. For higher wave numbers there is an initial stage of inrease in error, whih results from wave dispersion effets, disussed in [31] and [50]. This effet is overome by further inreasing p and eventually Figure 13. Mehanial shell subjet to interior and exterior pressure: ontours of u hp r and u hp z. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1342 S. BAGWELL ET AL. Figure 14. Sound-hard sphere subjet to an inident aousti pressure field: problem setup. Figure 15. Sound-hard sphere subjet to an inident aousti pressure field: onvergene of s shp L 2 s L 2, s shp H 1 s H 1 under p-refinement. results in the same expeted exponential rates of onvergene as before, onfirmed in Figure 15b. This again indiates that exponential onvergene with respet to the number of degrees of freedom raised to the power 1 2 is also ahieved through p-refinement for the aousti problem, provided suffiiently high refinement is used to eliminate numerial dispersion. For the ase of k = 4π 3m -1 for p > 7, the onvergene behaviour is suboptimal beause of the effet of the PML, whih is an approximate absorbing boundary ondition, but nevertheless, aurate solutions are still obtained. The finest solution, using p = 10, of the sattered pressure field arising from the inident pressure field for wave numbers k =[4π 3, 10, 30] m -1 are illustrated in Figure 16. 5.2. Coupled multi-physis problems 5.2.1. Aousti wave sattering of thin elasti shell. We onsider a oupled aousto-mehanial problem onsisting of a thin elasti shell of thikness t, mid surfae radius R and material parameters ρ s, ν and E. The shell is plaed in a bakground medium desribed by ρ f and f and is illuminated by harmoni inident pressure field in. The onfiguration is illustrated in Figure 17. This problem requires the solution of Equation 7) in absene of eletromagneti oupling and naturally lends itself to a harmoni treatment. For thin shells, the solution to this problem an be approximated by Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1343 Figure 16. Sound-hard sphere subjet to an inident aousti pressure field: ontours of Re s hp ). Figure 17. Elasti shell subjet to an inident aousti pressure field: problem setup. the Kirhhoff shell theory [49,51] and the total pressure exterior beomes = in + s + r. For the inident field in = 0 e ikz, the omponent s orresponds to the hard sattering by the sphere and r to the radiated pressure [49,51]. We onsider the partiular ase of t = 0.05m, R = 1m, 0 = 1Pa,k = ω f =[4π 3, 10, 30] m 1, ρ f = 1000 kg/m 3, f = 1460m s, ρ s = 7800 kg/m 3, ν = 0.3, E = 210GPa. We treat this problem omputationally by applying a suitably simplified version of Algorithm 1 with shp + rhp = δhp on the omputational domain Ω m =[0, 5.6] [ 5.6, 5.6]) {r, z) r 2 + z 2 R t 2) 2 }m 2 with the same PML settings as in Setion 5.1.3. As in Setion 5.1.2, to avoid the shell from floating away, we fix a small boundary segment Ω D to have displaements u = 0. The problem is driven by the inident pressure field in the form of a Neumann ondition set on the external boundary of the shell {r, z) r 2 + z 2 =R + t 2) 2 } as n δhp = n in and the oupling aording to the interfae onditions in equation 7). The inside boundary of the shell is left free and the aousti effets inside of the shell are ignored. Given that the onvergene rates of the mehanial and aousti fields have already been verified in Setions 5.1.2 and 5.1.3, we now instead diretly ompare the effetiveness of both h-refinement and Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1344 S. BAGWELL ET AL. p-refinement with the analytial solution for k = 4π 3m 1. In Figure 18, we plot various hp-enrihed solutions for the line segment 1.025 m r 5.6 m,z = 0, taken from the outer surfae of the shell to the trunated boundary. For both h-refinement and p-refinement, the omputed solution tends to the analytial for r 4m.However,forh-refinement, a mesh of h 0.1m with 15 752 elements, with 7779 unknowns, is required to obtain good agreement with the analytial solution, with a level of auray of O10 2 ). On the other hand, using p = 2 on a mesh with 530 elements requires only 1021 unknowns for omparable auray in the solution. If we further refine p to p = 4, then the number of unknowns inreases to 4163, but with improvement in the relative auray of two orders of magnitude. The PML is defined by the grey area, in whih the omputed solution is non-physial and absorbed. Figure 19 shows the omparisons in omputed and analytial solutions for higher wave numbers of k =[10, 30]m 1. For both ases, the order of p = 4, used to obtain the finest solution in Figure 18, offers reasonable agreement with the analytial solution and is able to apture the higher frequenies of the waves. However, for inreasing wave numbers, our omputed solution requires even further p- refinement in order to aurately apture the solution in these regions beause of the aforementioned dispersion effets. The solution ase for p = 4 and a suitably refined solution of eah ase is plotted Figure 18. Elasti shell subjet to an inident aousti pressure field: Effets of h-refinementand p-refinement on the aousti pressure field. Figure 19. Elasti shell subjet to an inident aousti pressure field: effets of p-refinement for high wave number k on the aousti pressure field. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1345 against the analytial solution in Figure 19a and b. The mesh density required, for p = 1 elements, to apture the high-frequeny wave effets at k = 30m 1 beomes impratial for suh geometries. The interation between the pressure field and the displaements of the shell is illustrated in Figure 20, where the omputed deformed shape of the shell is plotted in the surrounding aousti field. The suess of high-order hp disretisations for the preeding ase studies motivates our strategy for the following industrial example, whih does not have an analytial solution. 5.2.2. Siemens benhmark problem. We now onsider an industrially relevant benhmark problem, proposed by Siemens Magnet Tehnology, in whih a simplified quarter-size representation of an MRI sanner was modelled and previously presented in [2]. The problem setup omprises of the same main omponents illustrated in Figure 1, with a redued omplexity in the oil onfiguration. The setup omprises of three metalli shields known as the outer vauum hamber OVC) Ω OVC, 77 K radiation shield Ω 77K and 4 K helium vessel Ω 4K, whih make up Ω and eah with different material parameters γ, μ, ν, E, ρ). A pair of main oils, with stati urrent soure J DC, are loated on the outside of the three shields and a pair of gradient oils, with alternating urrent soure J AC t), are loated within the imaging bore. Both are assumed as Biot Savart oils and are illustrated in Figure 21. We treat this problem omputationally for two ases; in the first ase, we apply a suitably simplified version of Algorithm 1 in whih we neglet the aousti effets and fous on the purely magneto-mehanial oupling mehanisms, as in [2]. In the seond ase, we onsider the fully oupled aousto-magneto-mehanial systems in Algorithm 1. We trunate the non-onduting Figure 20. Elasti shell subjet to an inident aousti pressure field: deformed shell interating with surrounding aousti pressure field. Figure 21. Simplified magneti resonane imaging sanner subjet to alternating and stati urrent driven oils: problem setup. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1346 S. BAGWELL ET AL. region, omprised of air, and reate the domain Ω m =[0, 1.26] [ 1.68, 1.68])m 2, with the PML Ω m pml =Ωm [0, 0.9] [ 1.2, 1.2])m 2. As in Setion 5.1.2, to avoid the shell from floating away, we fix a small boundary segment Ω D of the ondutors to have displaements u = 0. The exat geometries and material parameters of the onduting omponents are onfidential and so are not to be displayed in this paper. We analyse the solution for an unstrutured mesh of 8464 triangles of maximum size h = 0.25m, but with substantial refinement in Ω m. Of these elements, 1700 are loated within Ω m. This example serves to show the preditive apability of our approah, and to validate our approah, we ompare our results with industrial data supplied by Siemens Magnet Tehnology. Dissipated power and eddy urrents A quantity of industrial interest is the power dissipated in and Ω 4K. This measure is used to quantify the resonane behaviour of the MRI system and to determine the frequenies at whih operation is undesirable. This measure is given in terms of the Ohmi urrents, and for the harmoni omponent of the magneti vetor potential δ Aφhp, beomes Ω OVC, Ω 77K P o Ω ω, δ A)= 1 2 Ω γ 1 J o 2 dω= 1 2 Ω γ E 2 dω πω 2 Ωm γ δ Aφhp 2 rdω m. 21) In, [2] we previously ompared P o Ω OVC ω, A AC ), P o Ω 77K ω, A AC ) and P o Ω 4K ω, A AC ),wherea AC A hp φ e φ is the equivalent to δ Aφhp e φ but obtained by the fixed point sheme, to a set of industrial results using the NACS software [11]. We revisit this by inluding the results of the new monolithi formulation, both with and without aousti effets. The onverged results are plotted in Figure 22 where we Figure 22. Simplified magneti resonane imaging sanner subjet to alternating and stati urrent driven oils: Ohmi power dissipation as a funtion of alternating urrent frequeny Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1347 Figure 23. Simplified magneti resonane imaging sanner subjet to alternating and stati urrent driven oils: ontours of the eddy urrents ReJ o ) for p = 1, 5andf =[160, 4100]Hz. φ Figure 24. Simplified magneti resonane imaging sanner subjet to alternating and stati urrent driven oils: effets of p-enrihment on the eddy urrent resolution ReJ o ) in Ω4K φ for f =[160, 4100]Hz. perform a sweep over the alternating urrent driving frequenies in the range ω = 2πf = 2π10 f 5000) rad/s. In the figure, the blak line represents the results obtained by Siemens using NACS, the red line the results of the previous fixed point sheme, the blue line our new monolithi magneto-mehanial formulation and the yellow line the fully oupled monolithi system with aousti effets. In the absene of aousti effets, and for f 3000 Hz, the fixed point, monolithi and NACS ω, A AC ) and P o Ω 77K ω, A AC ) are in lose agreement with eah other beause in this results for P o Ω OVC ase the problem is well approximated by the pure eddy urrent model. The fixed point and monolithi results for P o Ω 4K ω, A AC ) also give good agreement in this region; however, the results obtained by NACS offer very small differenes of O10 1 ) for f > 1000Hz. We onjeture that this is due to the limitations of the low-order elements in aurately resolving the skin depth effets with inreasing frequeny, illustrated in Figures 23 and 24. These methodologies also give a similar predition of the resonane region oupied by 3500 Hz f 4500 Hz with the NACS model being damped beause of the artifiial Rayleigh damping [52]. The effet of this damping on the response of the system results in a hange in the amplitude and frequeny range of the resonane region [52,53]. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
1348 S. BAGWELL ET AL. Our new monolithi framework offers omputational advantages over our previous fixed point strategy beause the solution is obtained in a single iteration, as opposed to multiple iterations, whih grows in the resonane region. This enables us to perform rapid and robust solutions at eah frequeny, whih offers trivial parallelism and greater resolution in the resonane region for the same omputational ost. Notably, the resonant frequenies omputed using the fixed point sheme exatly math those obtained by the monolithi sheme albeit with differing magnitude of the peaks beause of the effets of matrix equation onditioning. The prediative apability of our approah is further demonstrated by the inlusion of aousti effets, whih has substantial effets in P o Ω 77K ω, A AC ) and P o ω, A Ω 4K AC ) for f 500 Hz, not inluded in the NACS software or our previous fixed point sheme. Negligible effets for P o ω, A Ω OVC AC ) are obtained beause the OVC is loated losest to the inner bore tube and, therefore, the gradient oils, and dissipated power is dominated by the eletromagneti effets. In ontrast, the other two shields are loated to the outside of the OVC and hene aousti propagation effets ause these shields to further perturb the output power. Note that repeating the results with oils treated as deformable ondutors leads to only negligible hanges in the output power, and therefore, these results are not shown. Figure 25. Simplified magneti resonane imaging sanner subjet to alternating and stati urrent driven oils: T Ω OVC ω, δ u ) and resonant mode shapes. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.
ACOUSTO-MAGNETO-MECHANICAL COUPLING IN MRI SCANNERS 1349 To illustrate the different skin effets at different frequenies, we show in Figure 23 the eddy urrent distributions at f = 160Hz and f = 4100Hz for both p = 1andp = 5. At the lower frequeny, the skin effets are already well resolved by p = 1 elements, but the higher frequeny p 4 elements are required to resolve the small skin depth. This is further illustrated in Figure 24, whih shows the onvergene of J o in Ω 4K along the line z = 0.04m for p = 1, 2,, 5. 5.2.2.1. Kineti energy and mode shapes. The kineti energy of Ω OVC, Ω 77K and Ω 4K is of industrial interest for understanding the motion of the ondutors, highlighting the resonane frequenies and the orresponding mode shapes of the sanner's struture. In terms of the omputed displaements δũ hp,thisis T Ω ω, δ u )= 1 4 Ω m v 2 dω= 1 4 Ω ρω 2 δ u 2 dω π 2 Ω mρω 2 δũhp 2 rdω m. 22) Figure 26. Simplified magneti resonane imaging sanner subjet to alternating and stati urrent driven oils: magneti flux lines red), aousti ontour lines yellow) and displaed shields Ω. Numerial Methods in Engineering Published by John Wiley & Sons, Ltd.