A unified approach for a posteriori high-order curved mesh generation using solid mechanics

Int. J. Numer. Meth. Engng

et al. 2017

143

SUMMARYWe propose a new computational framework for the treatment of acousto-magneto-mechanical coupling that arises in low-frequency electro-magneto-mechanical systems such as magnetic resonance imaging scanners. Our transient Newton-Raphson strategy involves the solution of a monolithic system obtained from the linearisation of the coupled system of equations. Moreover, this framework, in the case of excitation from static and harmonic current sources, allows us to propose a simple linearised system and rigorously motivate a single-step strategy for understanding the response of systems under different frequencies of excitation. Motivated by the need to solve industrial problems rapidly, we restrict ourselves to solving problems consisting of axisymmetric geometries and current sources. Our treatment also discusses in detail the computational requirements for the solution of these coupled problems on unbounded domains and the accurate discretisation of the fields using hp -finite elements. We include a set of academic and industrially relevant examples to benchmark and illustrate our approach.

Section: T] and The Associated Residualsmentioning

confidence: 99%

A linearisedhp-finite element framework for acousto-magneto-mechanical coupling in axisymmetric MRI scanners

Bagwell

Ledger

Int. J. Numer. Meth. Engng

et al. 2017

143

“…The geometry and the three-dimensional curved p " 4 tetrahedral mesh of the flexoelectric structure is shown in Figure 12. To represent the geometry of the problem accurately, we employ the high order curvilinear finite elements recently developed by Poya et al [2018] which uses a posteriori mesh morphing technique presented in Poya et al [2016] to represent the CAD boundaries of the flexoelectric structure accurately (notice the curved elements representing the circle in the top conical frustum) without requiring a change in the mixed finite element functional spaces presented in Appendix C.…”

Section: Nanocompression Of a Flexoelectric Conical Pyramidmentioning

confidence: 99%

On a family of numerical models for couple stress based flexoelectricity for continua and beams

Poya

Journal of the Mechanics and Physics of Solids

Ortigosa

et al. 2019

Self Cite

A family of numerical models for the phenomenological linear flexoelectric theory for continua and their particularisation to the case of three-dimensional beams based on a skew-symmetric couple stress theory is presented. In contrast to the standard strain gradient flexoelectric models which assume coupling between electric polarisation and strain gradients, we postulate an electric enthalpy in terms of linear invariants of curvature and electric field. This is achieved by introducing the axial (mean) curvature vector as a strain gradient measure. The physical implication of this assumption is many-fold. Firstly, analogous to the standard strain gradient models, for isotropic (non-piezoelectric) materials it allows constructing flexoelectric energies without breaking material's centrosymmetry. Secondly, unlike the standard strain gradient models, nonuniform distribution of volumetric part of strains (volumetric strain gradients) do not generate electric polarisation, as also confirmed by experimental evidence to be the case for some important classes of flexoelectric materials. Thirdly, a state of plane strain generates out of plane deformation through strain gradient effects. Finally, under this theory, extension and shear coupling modes cannot be characterised individually as they contribute to the generation of electric polarisation as a whole. As a first step, a detailed comparison of the developed couple stress based flexoelectric model with the standard strain gradient flexoelectric models is performed for the case of Barium Titanate where a myriad of simple analytical solutions are assumed in order to quantitatively describe the similarities and dissimilarities in effective electromechanical coupling under these two theories. From a physical point of view, the most notable insight gained is that, if the same experimental flexoelectric constants are fitted in to both theories, the presented theory in general, reports up to 200% stronger electromechanical conversion efficiency. From the formulation point of a view, the presented flexoelectric model is also competitively simpler as it eliminates the need for high order strain gradient and coupling tensors and can be characterised by a single flexoelectric coefficient. In addition, three distinct mixed flexoelectric variational principles are presented for both continuum and beam models that facilitate incorporation of strain gradient measures in to a standard finite element scheme while maintaining the C 0 continuity. Consequently, a series of low and high order mixed finite element schemes for couple stress based flexoelectricity are presented and thoroughly benchmarked against available closed form solutions in regards to electromechanical coupling efficiency. Finally, nanocompression of a complex flexoelectric conical pyramid for which analytical solution cannot be established is numerically studied where curvature induced necking of the specimen and vorticity around the frustum generate moderate electric polarisation.

“…13 Furthermore, by choosing to linearise the transient nonlinear system of coupled equations about the static fields, the formulation simplifies greatly, resulting in a transient linear system of equations dependent on the transient magnetic field excitation, as shown in our other work. 13 Similar techniques, involving the additive split of a nonlinear problem to a series of linear problems, have been successfully applied to the field of computational mechanics, such as analysis of structural membranes, [38][39][40] high-order mesh generation, 41 and in biomedical applications. 42 This formulation permits a simple linearised approach that can be solved in the frequency domain and, given the relatively small frequency ranges of MRI applications, provides rapid solutions to industrial problems.…”

Section: Introductionmentioning

confidence: 99%

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

Bagwell

Ledger

Numerical Meth Engineering

et al. 2018

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