An hp-fem framework for the simulation of electrostrictive and magnetostrictive materials
Introduction
The force exerted at a macroscopic scale on a solid or fluid domain due to the presence of electromagnetic phenomena is described by a ponderomotive force, fEM, which can be expressed as the divergence of a stress tensor [[σEM]] minus the time derivative of the electromagnetic momentum [1] aswhere and are the electric field intensity and magnetic flux density vectors, respectively, and c is the speed of light. In stationary problems or those where the 1/c factor can be considered as negligible, the second term in (1) is neglected.
There are a range of electromagnetic forces that contribute to the first term in (1), which can be grouped into magnetic and electric contributions. Magnetic contributions include the Lorentz, dimagnetophoretic and magnetostrictive forces. Their corresponding electric counterparts are the Coulomb, dielectrophoretic and electrostrictive forces [2]. The Lorentz force is due to magnetic induction whereas the Coulomb force is due to the presence of electric charges. The dimagnetophoretic and dielectrophoretic forces occur due to the spatial gradients of the permeability and permittivity, respectively, of the medium. The magnetostrictive and electrostrictive forces are associated with the deformation of the magnetised and dielectric material, respectively. We remark that there is some variation in the literature as to the explicit definitions and names given to each of these contributions [3], [4], [1], [5]. For instance, in Stratton [3], the dimagnetophoretic and magnetostrictive (dielectrophoretic and electrostrictive) forces are combined and simply called magnetostriction (electrostriction). This was also the approach we followed in [6] and that we will also follow in this paper.
A material is defined as dielectric, magnetic or conducting, depending on whether polarisation, magnetisation or conduction, respectively, is the predominant electromagnetic behaviour of the material [7]. For polarised (magnetised) materials, suitable electromagnetic constitutive laws are established to relate the electric (magnetic ) flux intensity vector and the electric (magnetic ) field intensity vector as followswhere ∊ is the permittivity of the material, which describes the polarisation, and μ is the permeability of the material, which describes the magnetisation. When polarisation is the dominant effect, three categories of dielectrics can be identified, namely ∊/∊0 < 1, the material behaves as vacuum ∊ = ∊0 and ∊/∊0 > 1. Analogously, when magnetisation is the dominant effect, three categories of magnetics can be identified, namely diamagnetic if μ/μ0 < 1, behaving as vacuum when μ = μ0 and paramagnetic if μ/μ0 > 1 [8].
The electromagnetic constitutive relations (2) can be considerably more complicated, with μ and ∊ defined through nonlinear second order tensors, as in the case of ferroelectric or ferromagnetic materials. Well known ferrous metallic objects, which contain significant amounts of iron (i.e. steel), behave differently from a simple magnetic material and exhibit non-linear hysteresis after being magnetised. Although, in an unmagnetised state, their response (2) for small magnetic fields is linear and paramagnetic, beyond a given threshold they have a distinctive nonlinear behaviour [9] where is defined by a hysteresis curve.
The formula provided in (1) may, at first glance, appear quite straightforward. However, closer inspection of the explicit expression of [[σEM]] [1] reveals a very complex coupled expansion dependent upon the deformation of the material and electromagnetic fields. Furthermore, the deformation induced by the force field defined by (1) must be taken into account when evaluating the electromagnetic constitutive relations (2), as μ and ∊ are also deformation dependent. All in all, Eqs. (1), (2) lead to a fully coupled electro–magneto–mechanical problem.
In many applications, simplifying assumptions are usually made [1], [5], [2]. In some mechanical problems involving rigid conductors, the interest lies on the measurement of the Lorentz forces in (1) generated by magnetic induction, such as in MRI scanners, with the purpose of minimising the appearance of vibrations and, subsequently, the deterioration of the resulting imaging technique [10]. In the corresponding case of rigid dielectrics, for small deformations, the effect on the electromagnetic constitutive behaviour (2) can be considered as negligible [11]. In magnetohydrodynamics, of great interest for the analysis of conducting fluids (e.g. molten metals), the Lorentz force is often considered as the dominant term, disregarding other force contributions [4]. However, when considering problems with different fluid phases, dimagnetophoretic contributions become important and still further improvements in accuracy can be accomplished by including magnetostrictive effects. Unlike conducting fluids, Eringen and Maugin describe ferrofluids as essentially non-conducting fluids with high concentrations of magnetic particles suspended in them [5]. Eringen and Maugin also describe potential applications of these fluids, which include as lubricants in chemical and petroleum industries or being employed for medical applications, such as the production of blood flow during surgery. Another important class of fluids are magneto-rheological fluids, which can behave either as a fluid or a solid under the influence of a magnetic field. Such fluids have recently attracted considerable interest as shock absorbers in motor vehicles [12]. Electrostrictive and magnetostrictive effects have not been investigated to the same extent as their counterparts in solids, but new important applications are beginning to emerge [13]. Examples of magnetostrictive effects include dipolar fluids [14], polycrystalline iron films [15], cylindrical type II superconductors [16] and in magneto-rheological elastomers [17]. Magnetostrictive effects in ferrofluids have been studied by [13], [18] and have also found to be important when spark erosion is used to produce them [19]. Further potential applications may exist in the behaviour of magneto-rheological fluids, but, as reported by Dr. Holger Böse, a researcher at the Fraunhofer Material Research Institute [20], “The microscopic mechanisms at work in the fluid are not at all clear. We are not sure what influences the fluid’s strength in the magnetic field” [12].
Knops [11] analysed the problem of two–dimensional electrostriction for rigid dielectrics, where the effect of the deformation on the electric permittivity was disregarded. Under this assumption, the authors presented in [6] a closed form solution to the benchmark problem of electrostriction of an infinite dielectric plate with a rigid circular/elliptical dielectric insert [11], [21], [22]. However, useful as these analytical solutions are, they are limited to simple geometric and loading configurations and do not include how the electrostrictive effect alters the electromagnetic constitutive behaviour of the material (2). The numerical solution algorithm presented in [6] using hp-finite elements shows an approach using a fixed point algorithm applied to the simulation of the fully coupled electrostrictive approach for dielectric materials.
The description of the ponderomotive force (1) and the electromagnetic constitutive behaviour of the material (2) may appear somewhat disjoint. However, for certain classes of conservative materials, an energy formulation offers an informative approach which is capable of describing both the ponderomotive force and the electromagnetic constitutive behaviour of a wide class of dielectrics and magnetics [23], [24], [25], [26], [27]. These have great potential for the description of new novel smart materials. An energy formulation also provides a natural starting point for the numerical simulation of such materials, by performing consistent linearisation and using a quadratically converging Newton–Raphson algorithm. However, in the case of fluid problems, this energy formulation approach can no longer be followed due to the non-self-adjoint nature of the advection operator.
This work is part of ongoing research to bridge the gap between existing techniques which often omit dielectrophoretic/dimagnetophoretic and electrostrictive/magnetostrictive effects in the simulation of coupled solid and fluid problems involving dielectrics and magnetics. Our focus here is to provide a framework to show how these effects can be effectively included. Firstly, our work extends our earlier fixed point strategy and energy functional approach to include the effects of electrostrictive and magnetostrictive effects in solid dielectrics and magnetics. The work is immediately applicable to non–ferroelectric dielectrics and non–ferromagnetic magnetics. For these materials, we present a new consistent Newton–Raphson scheme for including electrostrictive and magnetostrictive effects in these conservative materials. Secondly, we describe a new consistent Newton–Raphson scheme for the simulation of Newtonian viscous fluids. In particular, our development offer a useful platform towards introducing magnetostrictive effects into non–conducting ferromagnetic fluids and conducting magnetohydrodynamic fluids.
From the spatial discretisation point of view, an hp–fem framework is preferred, which ensures the efficient, accurate and robust solution of the coupled electro–magneto–mechanical–fluid problems including electrostrictive and magnetostrictive effects. Other works [6], [28], [29] (and references therein) have previously outlined the advantages of an hp–fem approach and, therefore, we do not repeat them here. When dealing with an incompressibility constraint, suitable mixed formulations and functional spaces, needed to satisfy the well known LBB condition, are followed [30], [31].
The article is broken down into the following sections. Section 2 presents the basic equations governing the electrostatic and magnetostatic fields, where their boundary value problems and equivalent weak forms are introduced. In Section 3, the strong form of the mechanical problem is presented. For the case of conservative elastic solids, the weak form is obtained after computation of the stationary points of a total potential energy functional, whereas for the case of non–conservative incompressible viscous fluids, the weak form is presented directly. Sections 4 Conservative systems and electro/magnetostrictive materials, 5 Non-conservative systems and electro/magnetrostrictive materials describe the coupled formulation followed for electrostrictive and magnetostrictive materials, presenting explicitly the employed Newton–Raphson strategy and the consistent linearisation. Section 6 describes briefly the main features of the hp-finite element framework employed for the spatial discretisation of the weak form. Section 7 includes a series of numerical examples, well known within the solid and fluid mechanics communities, used to demonstrate the accuracy and robustness of the hp–implementation. Finally, some concluding remarks are summarised in Section 8.
Section snippets
Electromagnetic description
For a contractible domain , with boundary , the static Maxwell’s equations decouple aswhere and are the electric and magnetic field intensity vectors, respectively, and are the electric and magnetic flux intensity vectors, respectively, denotes the electric current density and the volume charge density. For linearised materials the constitutive relationshold where and are the second–order permittivity and
Compressible linearised elasticity
For a contractible domain with boundary ∂CE the conservation of linear momentum iswhere is the symmetric Cauchy stress tensor and f is a body force per unit of volume. In the case of linearised elasticity with isotropic homogeneous materials, the constitutive law takes the well known formwhere is the small strain tensor, is the displacement vector and
Conservative systems and electro/magnetostrictive materials
By a conservative system we mean a system for which work done by a force or set of forces is 1) independent of path; 2) equal to the difference between final and initial values of an energy functional and 3) completely reversible [37]. The systems described by (6), (11) represent conservative systems and as such, their weak forms can be obtained by computing the stationary points of a corresponding total potential energy functional [33], [23].
We now wish to consider a class of conservative
Non-conservative systems and electro/magnetrostrictive materials
For non–conservative systems it is not possible to express the weak form as the stationary point of a total potential energy functional. Instead, we set in (13a), (13b) and define for the solutions the residuals and :
Numerical discretisation
In a similar manner to [6], we employ the hp-finite element discretisation of Schöberl and Zaglmayr [39], [40]. For this we consider a regular simplicial triangulation of denoted by , with the set of vertices and the set of edges and recall the low–order vertex, high–order edge–cell based splitting of the hierarchic scalar H1–conforming finite element spacewhere Xh,1 is the classical space of continuous piecewise linear hat functions and
Conservative systems
To benchmark the Newton–Raphson scheme proposed in (37a), (37b) for conservative systems we apply the approach to a series of benchmark problems. For this purpose we let ∥ · ∥ denote the standard Euclidean vector norm and define for a displacement field u the normsassociated with the magnitude of the displacements, energy and sums of normal stress. For all examples, the Newton Raphson iterations are
Conclusions
In this paper, we extend our previous work [6] for the numerical simulation of two-dimensional electrostrictive and magnetostrictive materials. Plane stress and plane strain conservative (elastic solids) and non–conservative (Newtonian viscous fluids) materials are all analysed under a unified framework. In the case of conservative materials, use is made of energy principles when deriving the variational weak form of the problem [23], [24], [25]. The coupled formulation is implemented in a
Acknowledgements
The third author acknowledges the financial support received through “The Leverhulme Prize” awarded by The Leverhulme Trust, United Kingdom.
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