Numerical modelling of stationary and static

Transcription

1 Numerical modelling of stationary and static electric and magnetic fields loan E. Lager, International Research Centre for Telecommunications-transmission and Radar, Delft University of Technology, CD Abstract A global analysis of numerical methods aimed at the direct computation of the field quantities in case of stationary and static electromagnetic fields is presented. The examined numerical methods allow similar computer code implementations around general purpose, finite element software kernel. The fundamentally different character of the methods is a direct consequence of the different interpretation given to the field quantities and of the choices made for their discretisation. 1 Introduction The computation of the stationary and static electric and magnetic fields remains one of the most widely addressed subjects in the field of numerical modelling of electromagnetic fields. It can be easily observed that, just like in case of time-harmonic and time-domain field problems, the emphasis inside the scientific community has swung from traditional, (vector) potential based methods to those methods that directly yield the electromagnetic field quantities (see the selected papers from the COMPUMAG '97 Conference in IEEE Transactions on Magnetics Vol. 34, No. 5). From a methodological point of view, such numerical methods share the common feature of using the "finite element paradigm", i.e.: the use a conformal grid;

2 66 Software for Electrical Engineering Analysis and Design discretised electromagneticfieldquantities are expressed asfinitelinear combinations of suitably chosen expansion functions and the solution of the initial field problem is translated into the solution of a system of linear, algebraic equations, the coefficients of which follow from contributions that are fully computable at finite element level. As a consequence, all these methods are amenable to a computer code implementation around a general purpose,finiteelement kernel. However, based on the interpretation given to the electromagnetic field quantities and on the manner to formulate thefieldequations, the following classes of methods can be observed: methods based on a local formulation of thefieldequations, the field quantities being interpreted as vector functions of the position vector; methods based on a local formulation of thefieldequations, the field quantities being interpreted as differential forms and methods based on an integral formulation of thefieldequations on finite volumes, the field quantities being interpreted as incompletely represented, complementary, vector functions. Surprisingly enough, the three approaches are analysed in the literature in a completely disjoint (sometimes antithetical) manner. Equally surprising, the motivation of a given formulation owes too often to certain well established idiosyncrasies (the type of expansion functions employed being just one of the most popular ones). One should never overlook the fact that "a model is a mathematical structure that is able to account, within some reasonably definite limits, for a class of concrete physical phenomena" *(p. 1). If within those limits the model provides acceptable results, the validity of the relevant model is demonstrated. Additionally, one should endeavour to construct a (numerical) model such that it remains as close as possible to the physics of the modelled phenomena. It is unfortunate that these phenomena often tend to be obscured by excessively mathematised formulations. In this paper it will be shown that the three classes of methods mentioned above do not mutually exclude each o^er but, on the contrary, they supplement each other. One of the most important steps will be the identification of the set of expansion functions that are best suited for each given numerical formulation. These expansion functions will be chosen such that they implicitly model as many of the conditions that must be satisfied by the field quantities in the relevant context, as possible. Finally, an attempt will be made towards identifying those classes of problems that are best addressed to by each of the three classes of methods. In this manner, it is expected that unnecessary increases in the computational costs caused by the (forced) application of a certain numerical scheme to a problem which is best accounted for by another one can be avoided.

3 2 Thefieldquantities Software for Electrical Engineering Analysis and Design 67 The electromagnetic field problem under discussion is considered in connection with an open domain V C B^ with a smooth outer boundary dt>. A finite number of smooth, non-intersecting interfaces X can be located inside Z>. The unit vectors along the outward oriented normal on dt> and on interfaces are denoted as n and z/, respectively. The outer boundary is partitioned into two disjoint sub-surfaces dt>v and 8T>p such that dt>v is connected and that dt>v U dvp = dv (either dt>v or dvp may be void). In order to attain a maximum degree of generality, the three types of stationary and static, electric and magnetic fields that are met in practice are mapped on a "generic formulation"^. 2.1 Mathematical modelling of the field quantities Electromagnetic field quantities can be modelled by any of the following mathematical entities: vector functions of the position vector; # differential forms and incompletely represented, complementary, vector functions. The most common interpretation of the field quantities is as vector functions (of the position vector). The following set of quantities applies: twofieldquantities, denoted as V (thefieldstrength) and F (the flux density); one material parameter, denoted as (where, for simplicity, all media inside the domain of computation are taken to be linear and isotropic); two volume source terms, denoted as Q* P (the source density of volume current) and p ** (the volume density of charge); a vector function that accounts for the polarisation of the media, denotedas jp^p %).34); two surface source terms (whose support may coincide with interfaces, only), denoted as Q^ (the impressed source density of surface current, with v - Qt P o) and cr* P (the impressed surface density of charge) and two vector functions that are used for prescribing boundary conditions, denoted as V*** (with support dvv} and F*** (with support dt>p)- Another possible interpretation of the field quantities is in terms of differential forms. By using^ \ the following set of quantities applies: twofieldquantities: a 1-form denoted as V (the elementary motive force) and a 2-form denoted as F (the elementary flux);

4 68 Software for Electrical Engineering Analysis and Design one tensor material parameter, denoted as = /, with I denoting the unit tensor of degree 3 (for ensuring the consistency of the model, the tensor character of the material parameter needs to be preserved even in case of isotropic media); two volume source terms: a 2-form denoted as Q* & (the elementary current) and a 3-form denoted as p v (the elementary charge) and two vector differential forms that are used for prescribing boundary conditions: a 1-form denoted as V ** (with support <9XV) and a 2- form denoted as F ** (with support dt>p}> In this case, it was assumed that no domains with permanent polarisation and no interfaces where surface sources manifest themselves are present inside the domain of computation. 2.2 Three possible mathematical models The structure of the sets of quantities in the previous subsection suggests three possible choices for modelling thefieldquantities and, correspondingly, for constructing mathematical models of the electromagnetic field. 1. A first choice is to interpret the field quantities as vector functions of the position vector. Since both V and F are taken to have the same mathematical properties and since the material parameter f is also a point function, it is possible to establish a one-to-one mapping V < > F that allows expressing one of them (F) as a function of the other (V). In this manner, there is but one single field quantity to be computed and only one type of general conditions (e.g. interface conditions) to be satisfied. From this point of view, this choice seems to be the most economic one. This model will hereafter be referred to as the local, vectorial model. 2. From purely mathematical point of view, the above choice for modelling field quantities is not free from interpretation inconsistencies. On one hand, the continuous dependence of a function of the position vector implies that such a function may show singularities %>. 23), that do not manifest themselves in the physical world. It is also worthwhile observing that measuring a physical quantity at a given point is physically impossible ^(p. 23) and ^. On the other hand, assuming that V and F are vector functions, it is obvious that they are different types of vector functions (they belong to different vector spaces). Consequently, a straightforward mapping of them by means of a scalar ( ) is meaningless ^(p. 24). In fact, some authors dispute the vectorial character of the field quantities altogether^. Having in view the observations above, a second choice for modelling field quantities (while still preserving the "local" character of the mathematical model) is by employing differential forms. However, for

5 Software for Electrical Engineering Analysis and Design 69 being able to fully describe the properties of the field, one single differential forms type field quantity does not suffice anymore and, hence, it is the pair {V, F] *(p. 164) that need being utilised. Consequently, it is but the combination of the two field quantities that allows a full representation of the field at any location inside the domain >. Differential forms models of the electromagnetic field cannot be thought of without accounting for complementarity^. Since, in this case, two independentfieldquantities must be computed, this approach to modelling field quantities is less economic then the previous one. 3. The approaches above employed differential (like) equations in order to compute some local quantities (at least, in some sense). In some cases (the extreme case being that of composite media), the domain of computation consists of a large number of sub-domains that are separated by interfaces, where those field quantities are not differentiate anymore. In such cases, the less restrictive integrability condition appears to be clearly more attractive for the modelling functions. To this end, one of the options is to define the field equations in integral form, on elementary domains. The field quantities are now defined exclusively by means of their components that are of interest on the boundaries of those domains. The resulting set of field equations is denoted as the domain-integrated field equations^. Since, in this case, appropriately selected components of some vector functions on the boundary of the elementary domains are used, only, the complete representation of the field requires, again, employing a combination of complementary field quantities. It can than be concluded that the field quantities are modelled by means of incompletely represented, complementary vector functions. By the choice for defining the field equations in an integral form, this method avoids completely the differentiation of the expansion functions, preventing, in this manner, the appearance of non-physical (spurious) surface sources inside the domain of computation. However, as a consequence of the (natural) choice for discretising field quantities by means of completely linear edge and face expansion functions, from computational point of view, this method is less efficient then any of the previous ones. This model will hereafter be referred to as the integral, vectorial model. 3 Numerical models 3.1 The local, vectorial model In this case, the following set of differential equations apply?: V x y = Q^P for r E D\Z, (1) V F = /"* for r G D\%, (2) F = f V for re T>\I. (3)

6 70 Software for Electrical Engineering Analysis and Design These equations are supplemented by the following interface conditions: i/xv *=Q P forrez, (4) v F\l= <T P for r e I, (5) where ^ denotes the jump in the relevant value when crossing an interface X that separates two sub-domains 1 and 2 (the normal r/ being taken as oriented from 1 towards 2), and the following types of boundary conditions: x y = n x y^ for r G 9Dy, (6) n F = n F^ for r G 9%. (7) Equations (l)-(7) define a problem with a unique solution. As mentioned before, the flux density F can be expressed in this case as a function of the field strength V, by using (3). The following aspects are taken into account when choosing the expansion functions to be employed for the discretisation of thefieldquantities: the expansion functions should ensure the continuity of the discretised counterpart V of thefieldstrength V inside interface free sub-domains and the expansion functions should implicitly satisfy the interface conditions (4) and (5) (with (3) being substituted in (5)). In^ was shown how these desiderata can be effectively attained by employing generalised Cartesian expansion functions defined on a tetrahedral mesh. (Generalised) Cartesian expansion functions cannot be used at points and/or edges where singularities of the field are located. In such cases the use of (generalised) Cartesian expansion functions associated with incomplete simplicial stars or expansion functions that account for the type of singularities was proposed. The use of (generalised) Cartesian expansion functions allows replacing the initial field problem by the following sequence of operations: 1. Constructing an (approximate) vector function V QT> that accounts for non-zero boundary conditions and non-zero surface sources. It turns out that, by employing generalised Cartesian expansion functions of the additive type, this constituent has an explicit form. 2. Constructing an (approximate) vector function V^>\x that solves a field problem with zero boundary conditions and no surface sources, by application of a least-squares minimisation of a suitably chosen functional. The effect of non-zero boundary and/or surface sources is taken into account in this functional by means of equivalent volume source distributions. 3. Computing the (approximate)fieldstrength V by summing up the constituents V ' QX> and

7 Software for Electrical Engineering Analysis and Design 71 This numerical formalism was shown to be characterised by a quadratic convergence with mesh refinement of the numerical solution (at least in case of problems where no singularities of thefieldmanifest themselves). Numerical experiments carried out thus far have shown that the conditioning of the system of linear algebraic equations to be solved is excellent. 3.2 Models using differential forms representations This topic is widely addressed in the literature. Formulations of the field equations that employ differential forms are presented in numerous publications (see*' 5 and the strong advocation of their use in*). For the discretisation of thefieldquantities, Whitney elements must be employed: nodal elements for the discretisation of 0- and 3-forms; edge elements for the discretisation of 1-forms and face elements for the discretisation of 2-forms. Two examples of numerical formulations that yield only one of the field quantities are examined in\ while a novel formulation that directly yields both field quantities is presented in^. These formulations require the computation of a significant number of additional degrees of freedom, due to the use of Lagrange multipliers for solving the constrained minimisation problem *(p. 172). One of the consequences of this fact is that the conditioning of the resulting system of linear, algebraic equations is poor*' ^. 3.3 The integral, vectorial model The application of this method requires the splitting of the domain of computation T> into simply connected, elementary domains % with piecewise smooth boundaries dt>&. The elementary domains are taken to be the smallest sub domains for which the material parameters can be physically measured. Since there is no possibility to further investigate the variation of the material parameter inside elementary domains, the media inside them are taken, conventionally, to be homogeneous. The field equations are now defined in an integral fornfi, in such a manner that the obtained equations uniformly converge towards the usual local field equations inside homogeneous media. The following set of equations holds in this case: x V da = f Q P dv, «/ >E (8) r j r " *"** = ] p»dv. (9)

8 72 Software for Electrical Engineering Analysis and Design These equations are supplemented by the following interface conditions: i/ x V continuous for r Z, (10) i/. F continuous for r Z, (11) where, for simplicity, it was assumed that no surface sources carrying interfaces are present inside V. Note that all (parts of) boundaries <9% that separate adjacent elementary sub-domains represent such interfaces where the condition in Eqs. (10) and (11) must be enforced. Finally, the following types of boundary conditions are employed: xv = nxv^ for r e #ZV, (12) n F = n F^ for r G 9%. (13) From the manner in which the field quantities where introduced, it is obvious that no straightforward mapping F < > V is possible, hence a constitutive equation of the type (3) cannot be established. In this case, the relevant mapping can be thought of as linear operator: (with L2( >E)3 denoting the usual L% norm over % and f ( >E) the material parameter of the homogeneous medium inside %), that uniformly converges inside homogeneous media towards 0. Note that the impressed polarisation P' P was included in the constitutive law, also. Obviously, inside homogeneous media the operator a yields the usual form of the constitutive equation: (15) It should however be observed that the field quantities were defined on the boundary of the elementary domain, only. In order to be able to apply (14) it is, thus, necessary to provide a continuation of these values towards the interior of the elementary domains. There are many possible choices, but from many points of view (simplicity, though of relevance, being not the most significant one) a linear interpolation is preferred. Further, for being able to approximate any curved surface up to order Y? with h being the maximum "diameter" of elementary domains, these domains are taken to be tetrahedra. It can now be concluded that the set of equations that apply in case of the integral, vectorial model are implicitly discretised. For deriving a numerical formulation, the field quantities need being discretised. To this end, it is natural to choose for the discretisation of V and F consistently linear, edge and face expansion functions^, respectively. In this manner, the interface conditions in Eqs. (10) and (11) are automatically satisfied. The field equations (8) and (9) are accounted for in the numerical formulation directly, whereas the constitutive equation (14) is

9 Software for Electrical Engineering Analysis and Design 73 modelled in the least-squares sense. The resulting over-determined system of linear, algebraic equations is solved in the least-squares sensed However, this procedure deteriorates the conditioning of the final form of the system of equations. 3.4 Types of problems that are best addressed by the three models The three classes of numerical methods examined above seem to address best certain, well individualised, classes of problems. On basis of numerical experiments and results that are available in the literature, the following classification can be made: 1. The local vectorial model This method can be employed in case of domains of computation that contain relatively few interfaces where surface sources are located or that separate (highly) contrasting media. The media inside the interface free sub-domains need not being homogeneous or isotropic. It can satisfactorily be employed for the analysis of domains of computation that contain non-linear media. The efficiency of the method diminishes in case of the presence of many interfaces while its behaviour in case of configurations with (highly) complicated geometry is still a subject to be investigated. 2. Models using differential forms representations. These methods can be employed in case of domains of commutation that contain relatively few interfaces that separate (highly) contrasting media. The media inside interface free sub-domains are supposed to be linear, isotropic and homogeneous. These methods proved to be very efficient for the analysis of domains of computation with complicated geometry, in particular for the analysis of disconnected domains. When anisotropic and, especially, non-linear media are present inside the domain of computation, the efficiency of the method diminishes considerably. The accuracy of the method is lower when compared with the local vectorial model, for the same mesh density. 3. The integral vectorial model This method can be employed in case of domains of computation that contain a large number of interfaces that separate (highly) contrasting media, as, for example, in case of composite media. The media inside the interface free sub-domains need not being homogeneous or isotropic. It can be employed with good results for the analysis of domains of computation with complicated geometry. The behaviour of the method in cases of disconnected domains or domains of computation that contain non-linear media is still a subject to be investigated.

10 74 Software for Electrical Engineering Analysis and Design 4 Conclusions Three classes of numerical methods that lend themselves to similar computer code implementations were analysed. The characteristics that individualise them were emphasised. The various choices that can be made for the discretisation offieldquantities and equations were examined. In this manner, comparisons between methods that were previously examined in separate publications, exclusively, are greatly facilitated. References 1. Bossavit, A., Computational Electromagnetism. Variational formulations, Complementarity, Edge Elements, Academic Press, San Diego, Lager, I.E., Finite Element Modelling of Static and Stationary Electric and Magnetic Fields, (dissertation), Delft University Press, "Modelarea numerica a campurilor electrice i magnetice in regimurile sta^ionar i static", (dissertation, in Romanian), Department of Electrical Engineering, Delft University of Technology, Report number Et/EM , Delft, 1998, (available upon request). 4. Baldomir, D., Hammond, P., Geometry of Electromagnetic Systems, Oxford University Press, Bossavit, A., "Differential forms and the computation offieldsand forces in electromagnetism", in European Journal of Mechanics, B: Fluids., Vol. 10, No. 5, pp , de Hoop, A.T., Lager, I.E., "Static magneticfieldcomputation - an approach based on the domain-integrated field equations", in IEEE Transactions on Magnetics Vol. 34, No. 5, pp , Lager, I.E., Mur, G., "Least-squares minimising finite-element formulation for and stationary electric and magnetic fields", in IEEE Transactions on Magnetics Vol. 34, No. 5, pp , Dular, P., Remacle, J.-F., Henrotte, F., Genon, A., Legros, W., "Magnetostatic and magneto dynamic mixed formulations compared with conventional formulations", in IEEE Transactions on Magnetics, Vol. 33, No. 2, pp , Alotto, P., Delfino, F., Molfino, P., Nervi, M., Perugia, I., "A mixed face-edge finite element formulation for 3D magnetostatic problems", in IEEE Transactions on Magnetics Vol. 34, No. 5, pp , 1998.