CC BY
477
2014 VESTNIK OF SAINT PETERSBURG UNIVERSITY Ser. 10 Issue4
APPLIED MATHEMATICS
UDC 629.439:4.027.3:621.313.282:538.945
V. M. Amoskov1, D. N. Arslanova1, A. M. Bazarov1, A. V. Belov1, V. A. Belyakov1, T. F. Belyakova1, A. A. Firsov1, E. I. Gapionok1, M. V. Kaparkova1, V. P. Kukhtin1, E.A. Lamzin1, M.S. Larionov1, N.A. Maximenkova1, V.M. Mikhailov1, A. N. Nezhentzev1, D. A. Ovsyannikov2, A. D. Ovsyannikov2, I. Yu. Rodin1, N. A. Shatil1, S. E. Sychevsky1, V. N. Vasiliev1, A. A. Zaitzev3
SIMULATION OF ELECTRODYNAMIC SUSPENSION SYSTEMS FOR LEVITATING VEHICLES. I. MODELLING OF ELECTROMAGNETIC BEHAVIOUR OF MAGLEV VEHICLES WITH ELECTRODYNAMIC SUSPENSION
1 D. V. Efremov Scientific Research Institute of Electrophysical Apparatus, 3, Doroga na Metallostroy, St. Petersburg, 196641, Russian Federation
2 St. Petersburg State University, 7/9, Universitetskaya embankment, St. Petersburg, 199034, Russian Federation
3 St. Petersburg State Transport University, 9, Moskovskii pr., St. Petersburg, 190031, Russian Federation
An original computational technique, aimed at electromagnetic analysis of devices for particle physics and fusion applications, has been adapted to simulation of the magnetic levitation (maglev) transport. The paper describes modelling approaches and their implementation. A description of electrodynamic suspension systems is presented employing both superconducting and permanent magnets. The technique is oriented to parallel computations on supercomputers to take advantage of improved computational efficiency. Bibliogr. 22. Il. 5. Table 1.
Keywords: magnetic levitation, vehicle, electromagnetic suspension, finite elements, simulation, magnetic field, eddy current, lifting and drag forces, superconducting coil, permanent magnet.
Amoskov Victor Mikhailovich — candidate of physical and mathematical sciences, senior scientists; e-mail: avm@sintez.niiefa.spb.su
Arslanova Daria Nikolaevna — mathematician; e-mail: sytch@sintez.niiefa.spb.su Bazarov Alexandr Mikhailovich — mathematician; e-mail: sytch@sintez.niiefa.spb.su Belov Alexander Vyacheslavovich — leading of group; e-mail: sytch@sintez.niiefa.spb.su Belyakov Valery Arkadievich — doctor of physical and mathematical sciences, professor, deputy director general — Director of SRC "Sintez"; e-mail: belyakov@sintez.niiefa.spb.su
Belyakova Tatiana Fedorovna — senior programmer; e-mail: sytch@sintez.niiefa.spb.su Firsov Alexey Anatolievich — head of laboratory; e-mail: sytch@sintez.niiefa.spb.su Амосков Виктор Михайлович — кандидат физико-математических наук, ведущий научный сотрудник; e-mail: avm@sintez.niiefa.spb.su
Арсланова Дарья Николаевна — математик; e-mail: sytch@sintez.niiefa.spb.su Базаров Александр Михайлович — математик; e-mail: sytch@sintez.niiefa.spb.su Белов Александр Вячеславович — начальник группы; e-mail: sytch@sintez.niiefa.spb.su Беляков Валерий Аркадьевич — доктор физико-математических наук, профессор, заместитель генерального директора — директор НТЦ «Синтез»; e-mail: belyakov@sintez.niiefa.spb.su
Белякова Татьяна Федоровна — ведущий программист; e-mail: sytch@sintez.niiefa.spb.su Фирсов Алексей Анатольевич — начальник лаборатории; e-mail: sytch@sintez.niiefa.spb.su
B. М. Амосков1, Д. Н. Арсланова1, А. М. Базаров1, А. В. Белов1, В. А. Беляков1, Т. Ф. Белякова1, А. А. Фирсов1, Е. И. Гапионок1, М. В. Капаркова1, В. П. Кухтин1, Е. А. Ламзин1, М. С. Ларионов1, Н. А. Максименкова1, В. М. Михайлов1, А. Н. Не-женцев , Д. А. Овсянников2, А. Д. Овсянников2, И. Ю. Родин1, Н. А. Шатиль1,
C. Е. Сычевский1, В. Н. Васильев1, А. А. Зайцев3
ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ЭЛЕКТРОДИНАМИЧЕСКИХ ПОДВЕСОВ ЛЕВИТАЦИОННЫХ ТРАНСПОРТНЫХ СИСТЕМ. I. ВЫЧИСЛИТЕЛЬНАЯ ТЕХНОЛОГИЯ МОДЕЛИРОВАНИЯ ЭЛЕКТРОМАГНИТНЫХ ПРОЦЕССОВ В ЭЛЕКТРОДИНАМИЧЕСКИХ ПОДВЕСАХ МАГНИТО-ЛЕВИТАЦИОННЫХ ТРАНСПОРТНЫХ СИСТЕМ
1 НИИ электрофизической аппаратуры им. Д. В. Ефремова, Российская Федерация, 196641, Санкт-Петербург, Дорога на Металлострой, 3
2 Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7/9
3 Санкт-Петербургский государственный университет путей сообщения, Российская Федерация, 190031, Санкт-Петербург, Московский пр., 9
Разработанная вычислительная технология моделирования электромагнитных систем установок для исследований по проблеме управляемого термоядерного синтеза была адаптирована для целей анализа и оптимизации магнито-левитационных систем. Вычислительные модели базируются на разработанных авторами пакетах вычислительных программ. Обеспечена возможность математического моделирования всех значимых для практики видов магнитных систем электродинамических подвесов, включая сверхпроводящие катушки и постоянные магниты. Технология ориентирована на использование отечественных супер-ЭВМ с применением параллельных вычислений, что позволяет эффективно выполнить значительный объем вычислений. Библиогр. 22 назв. Ил. 5. Табл. 1.
Ключевые слова: магнитная левитация, транспортное средство, электромагнитный подвес, конечные элементы, численное моделирование, магнитное поле, вихревой ток, подъемная и тормозящая силы, сверхпроводящая катушка, постоянный магнит.
Gapionok Elena Igorevna — senior mathematician; e-mail: sytch@sintez.niiefa.spb.su Kaparkova Marina Viktorovna — senior researcher; e-mail: sytch@sintez.niiefa.spb.su Kukhtin Vladimir Petrovich — candidate of physical and mathematical sciences, senior scientist; e-mail: sytch@sintez.niiefa.spb.su
Lamzin Evgeni Anatolievich — doctor of physical and mathematical sciences, head of laboratory; e-mail: sytch@sintez.niiefa.spb.su
Larionov Mikhail Sergeevich — head of test stand; e-mail: sytch@sintez.niiefa.spb.su Maximenkova Nina Alexandrovna — senior programmer; e-mail: sytch@sintez.niiefa.spb.su Mikhailov Valery Mikhailovich — senior design engineer; e-mail: sytch@sintez.niiefa.spb.su Nezhentzev Andrey Nikolaevich — senior design engineer; e-mail: sytch@sintez.niiefa.spb.su Ovsyannikov Dmitri Alexandrovich — doctor of physical and mathematical sciences, professor, department chairman; e-mail: dovs45@mail.ru
Гапионок Елена Игоревна — ведущий математик; e-mail: sytch@sintez.niiefa.spb.su Капаркова Марина Викторовна — ведущий исследователь; e-mail: sytch@sintez.niiefa.spb.su Кухтин Владимир Петрович — кандидат физико-математических наук, старший научный сотрудник; e-mail: sytch@sintez.niiefa.spb.su
Ламзин Евгений Анатольевич — доктор физико-математических наук, начальник лаборатории; e-mail: sytch@sintez.niiefa.spb.su
Ларионов Михаил Сергеевич — начальник стенда; e-mail: sytch@sintez.niiefa.spb.su Максименкова Нина Александровна — ведущий программист; e-mail: sytch@sintez.niiefa.spb.su Михайлов Валерий Михайлович — ведущий конструктор; e-mail: sytch@sintez.niiefa.spb.su Неженцев Андрей Николаевич — ведущий конструктор; e-mail: sytch@sintez.niiefa.spb.su Овсянников Дмитрий Александрович — доктор физико-математических наук, профессор, заведующий кафедрой; e-mail: dovs45@mail.ru
Introduction. A computational technique has been developed at D. V. Efremov Scientific Research Institute for electromagnetic (EM) simulations of various devices related to accelerators, particle physics, and fusion [1]. Particular efforts have been made to adapt the technique to modelling and optimization of maglev transportation systems.
Magnetic levitation is a non-contact type of transportation. A vehicle is levitated and stabilised above a guideway by electromagnetic forces. Propulsion is driven with a linear actuator.
There are four principal types of maglev systems [2-6]:
1) electromagnetic suspension (EMS);
2) electrodynamic suspension (EDS);
3) INDUCTRACK (permanent magnet EDS);
4) magnetodynamic suspension (MDS).
The first two types are applied in commercial transportation [5].
The EDS system, the object of our interest, uses electromagnets on a vehicle. When moving, they induce eddy currents in conducting structures of the guideway. The electromagnetic interaction results in the repulsive force that lifts the vehicle. The repulsion depends on the control current, speed and levitation gap. A promising candidate for onboard magnets is superconducting coils [2-6].
The INDUCTRACK system [7-11] is in fact an EDS modification which utilises permanent magnets on board of the vehicle.
Background for simulation of maglev systems. The basic tool to study and optimise electromagnetic issues for the maglev systems is a set of original computer codes KLONDIKE [12], TYPHOON [13, 14], TORNADO [1], and KOMPOT [15, 16]. These codes intended for electromagnetic simulations of complex systems have been validated in the course of activities supporting development of the International Thermonuclear Experimental Reactor [1, 17].
The starting point is a system of the Maxwell equations for a macroscopic field that gives a differential formulation to relate electric field strength E, magnetic intensity H, electric-flux density D, and magnetic induction B as well as free charge density p and
Ovsyannikov Alexcmdr Dmitrievich — assistant professor; e-mail: ovc74@mail.ru
Rodin Igor Yurievich — candidate of technical sciences, head of department; e-mail: sytch@sintez.niiefa.spb.su
Shatil Nikolay Alexandrovich — candidate of technical sciences, leading scientist; e-mail: sytch@sintez.niiefa.spb.su
Sychevsky Sergey Evgenievich — doctor of physical and mathematical sciences, head of department; e-mail: sytch@sintez.niiefa.spb.su
Vasiliev Vyacheslav Nikolaevich — deputy head of department; e-mail: sytch@sintez.niiefa.spb.su
Zaitzev Anatoly Alexandrovich — doctor of economical sciences, professor, head of Scientific and educational centre for innovations in railway passenger transport; e-mail: nozpgups@mail.com
Овсянников Александр Дмитриевич — доцент; e-mail: ovc74@mail.ru
Родин Игорь Юрьевич — кандидат технических наук, начальник отдела; e-mail: sytch@sintez. niiefa.spb.su
Шатиль Николай Александрович — кандидат технических наук, ведущий научный сотрудник; e-mail: sytch@sintez.niiefa.spb.su
Сычевский Сергей Евгеньевич — доктор физико-математических наук, начальник отдела НИВО; e-mail: sytch@sintez.niiefa.spb.su
Васильев Вячеслав Николаевич — заместитель начальника отдела НИВО; e-mail: sytch@sintez. niiefa.spb.su
Зайцев Анатолий Александрович — доктор экономических наук, профессор, руководитель Научно-образовательного центра инновационного развития пассажирских железнодорожных перевозок; e-mail: nozpgups@mail.com
conduction current density j. From these relations the continuity equation Vj+dp/dt = 0, is derived which expresses the charge conservation law. Here V is the Hamilton operator, t is time, d/dt is the time derivative at a given point.
The system of the Maxwell equations is complemented with the constitutive equations D = e0eE, B = ^H, where e0 = 8.854 ■ 10~12 F/m, = 4n ■ 10~7 H/m. Many homogeneous and isotropic media can be described via a model with pre-determined time and space variations of relative dielectric and magnetic permeabilities e, n that are independent on E or H. The current density is approximated by Ohm's law in the differential form. Then the current density at a given point in a stationary medium is expressed as j = a E, where a is the specific conductivity. For linear anisotropic media tensorial forms of e, n and a are applicable.
In many cases [18] a pre-determined field of an extraneous current je can be introduced in addition to the conduction current j. Then the vector j in the Maxwell equations is replaced by the sum of j + je.
All problems of macroscopic electrodynamics can therefore be unambiguously described by the above combination of the Maxwell equations with the constitutive equations and relevant boundary and initial conditions. In special cases these equations can be simplified.
In a quasi-stationary approximation, the field exhibits quite slow variations so that polarization accommodates the field changes. Thus, the relations between the vectors D, E, B, H, j are independent on their time derivatives [18, 19]. By the virtue of a finite velocity of propagation, the quasi-stationary approximation for an ac field is valid only within a limited spatial domain [19]. In the metals the displacement current density tends to be negligibly low as compared with the conduction current density at any rate of field change. So, the first Maxwell equation can be rewritten as Vx H = j + je, the second Maxwell equation for a quasi-stationary state has a form: Vx E = -dB/dt.
A comprehensive description of complex magnet configurations results in calculation models that require extensive meshing. An efficient way to reduce the mesh dimension is to apply a reduced magnetic scalar potential. This approach is known as the T-Q method, or the method of the electric vector potential. In this case, the number of unknowns is in fact minimized to the number of mesh nodes. This makes the magnetic scalar potential attractive for the use in magnetostatic computations.
In the context of the approach developed by Ya. Frenkel, V. Greshnyakov, C. J. Carpenter, K. Demirchan, N. Doinikov, V. Chechurin et al., we can formally introduce vector P such that Vx P = j.
The relation Vx H = j leads to the expression: VxH = VxP or Vx (H — P) = 0. Then the field H — P can be determined in terms of the magnetic scalar potential p as H — P = —Vp or H = —Vp + P. With known P, a distribution of the scalar magnetic potential p is found by solving a boundary-value problem for the equation V ■ ¡i0^(—Vp + P) = 0 using boundary conditions for p.
Physically, P corresponds to a specific magnetic moment of a double magnetic layer that is analogous to closed currents under certain conditions from the viewpoint of the magnetic field [19]. To construct P, the domain of interest is partitioned with conditional magnetic shells corresponding to double magnetic layers. The partitions are arbitrary shaped and bounded by closed current loops. Vector P is co-directed with the surface vector element of the partitions. In practice, P is constructed reasoning from a specific problem formulation, an optimization technique for the calculation algorithm, and software functionality.
Such approximation is most efficient in EM calculations of magnet systems modelled with thin conducting shells [14]. The approximation is accurate if the field decay time t is much less than the characteristic time At of the observed EM process and satisfies the condition t « p0ah? c At, where h is the shell thickness, a = l/p is the shell conductivity. In this case the problem is limited to finding a single unknown: the normal component of vector P = Pnen, assuming j • h = Vx P. Here en is the unit normal vector to the shell, j is the density of induced eddy current.
For a singly connected shell S located in a variable field Bext, the normal component Pn is defined as [14]
Vr x Ç- [VrPn(t, r) x e„] h
Г [VrPn(t, r) x en] x (г - г7) _ dBn^{t, г)
4тг dt J |r _ r'|3 S ~~ dt
Uniquely defined (up to the constant) P is applicable only to singly connected surfaces. In order to transform multi-connected or branched surfaces into singly connected shells, conditional cuts and non-conducting areas are introduced. Value Pn jumps at the boundary of conducting and non-conducting surfaces.
If surface S is branched so that more than two shells are connected over a contour C, the first Kirchhoff law is applied in the differential form Ел (ec • Ien,A x eTjAj) dPn/dC\X G = 0.
Here Л is a shell joining the contour C; eC is the unit vector directed along C, eT is the unit vector directed from C tangentially to Л.
Using discretization, we obtain a system of ordinary differential equations resulted in the matrix equation Ej (MijdPj/dt + RijPj) = —d^i/dt.
Numerical solution gives a distribution of Pn and time variations of eddy currents induced in shells. Also, distributed fields, ponderomotive forces, Joule's heat and other parameters are calculated. This formulation of the problem was implemented in the computer code TYPHOON [13, 14].
A 3D model is applied when the shell approximation is inaccurate or needs validation. The shell approximation is adequate for thin walls. At the initial stage of an EM transient, when the field penetration depth is much less than the thickness of a solid structure, the perfect conductor model is applied to evaluate the external field.
A 3D representation to study EM transients in massive solids has been implemented in the computer code TORNADO [1]. TORNADO uses 4 types of finite elements: 8-node hexahedra, 6-node trilinear prisms, 5-node wedges, and 4-node tetrahedra. Equations are solved in terms of three-component electric vector potential P and magnetic scalar potential y. As P is determined accurate to an arbitrary scalar gradient, a gauge condition P • u = 0 is applied, where u = u(r) is an arbitrary stationary vector field with unclosed field lines. u(r) = 0 over the subdomains of R3 where P is nonzero.
The total field strength Htot is a superposition of unknown field H from the eddy current and external field Hext: Htot = H + Hext. The electric and magnetic potentials are found by solving the following system of equations assuming the absence of magnetic materials (^ = 1):
Vx(p •Vx P)+ Mod (P + Vy)/3t = — Mod HeXt/dt, V- (P + Vy)= 0, H = P + Vy, P • u = 0.
The first equation is defined only over simply connected conducting subdomain with resistivity p. The second equation refers to the entire domain R3. To determine the eddy
e
n
current with the current density VxP = j we need to know space and time distributions of the external field Hext over conducting areas of R3. Such distribution may be obtained from simulations with dedicated codes KLONDIKE [12] or KOMPOT [15, 16].
Computational model for EDS. In the study two coordinate systems are used: a global coordinate system related to the immovable guideway and a local coordinate system related to the moving vehicle.
1. The global coordinates are taken as a right-handed coordinate system X, Y, Z. The plane XY is horizontal, the axis X is directed towards propulsion. The axis Z is normal to the plane XY and directed upward.
2. The origin of local coordinates is associated with the onboard magnet of the moving vehicle.
The magnetic suspension is calculated as the EM force density integrated over the volume of current-carrying coils or permanent magnets fixed on the vehicle. The lifting force is directed vertically upward, the magnetic drag is opposite to the velocity, the sideward force acts in the direction normal to the lift.
The guideway structures involve active windings of a linear induction motor and passive rails. When the vehicle moves, the onboard magnets induce currents in the rails, the induced currents interact with the onboard magnet, and the resulting repulsion makes the vehicle to levitate. EDS systems with permanent magnets often utilize the Halbach array [7]. The Halbach array is a special arrangement of identical permanent magnets (PM) uniformly magnetized and shaped as square bars. Magnetization vectors of all magnets in the array lie in the same plane. The magnetic orientation of each magnet is rotated at a fixed angle a to the orientation of adjacent magnets. The Halbach array with a = 90° consists of four magnets (M = 4). The rotation pattern with a = 45° involves eight magnets (M = 8). The length of the array LM is determined as a sum of sizes of M magnets in the direction of propagation. The field, which varies periodically in space along the array, is concentrated on one side of the array and cancelled to near zero on the opposite side.
For EDS systems with electromagnets the field periodicity and magnetic period are determined in a similar way.
When the vehicle moves, the frequency of the first harmonic of the periodical field is expressed as f = V/LM, where V is the vehicle speed. Assuming the maximal speed to be V ~ 333 m/s and minimal wavelength LM ~ 4 ■ 5 ■ 10~2 m = 0.2 m, the frequency of the first harmonic is estimated as 2 ■ 103 Hz.
Reasoning from sizes, speed and field periodicities typically employed in maglev trains, a quasi-stationary field approximation is applicable for a numerical model [19, 20]. This means that at any time point the field and electromagnetic coupling of time-varying currents are assumed to be the same as those of direct currents of the magnitudes equal to respective instantaneous values of the variable currents [19].
The model of a levitated vehicle comprises onboard field sources and track structures where eddy currents are induced. A special attention is paid to possible electromagnetic interaction of the components.
Predictive simulations with the codes TYPHOON and TORNADO give distributed eddy currents, fields, heat release, ponderomotive forces and other parameters for all components of the vehicle-track system. The levitation performance is estimated with regard to nonlinear magnetic properties of materials. The outputs form an extensive database suitable as inputs for subsequent thermal-hydraulic analysis, particularly, for the case of superconducting onboard magnets [21], and strain computations to study various
motion scenarios. A combination of multivariant analytical studies and complementary numerical modelling make it possible to efficiently solve coupled electromagnetic-mechanical problems within a single computational algorithm. Such strategy enables iterative optimization of the maglev system design and performance at all R&D stages.
Model of permanent magnets. As an example, a model of an INDUTRACK EDS system is presented [8-11]. The system is configured as a Halbach array of 5 permanent magnets with ¡i0Mr = 1 T and a solid aluminium track, as shown in fig. 1.
Fig. 1. Layout of 5 permanent magnets arranged in Halbach array for INDUTRACK-type EDS system, side and top views; Vx is speed
Omitting PM demagnetization, the field from a permanent magnet can be calculated in the approximation of a uniformly magnetized polyhedron or equivalent surface currents [12]. These analytical descriptions are implemented in the code KLONDIKE. Specifically, the number of equivalent current loops is determined from a comparison of EM forces evaluated with 2 models:
• model 1 reflects a couple of identical permanent magnets. Each magnet has sizes 50 mm x 500 mm x 50 mm related to the axes X, Y, Z. The magnets are located symmetrically with respect to the plane Z = 0 with a gap of 2hz between them. The magnetic orientation is directed along the axis Z and is positive for the upper magnet and negative for the lower magnet;
• model 2 describes two rectangular solenoids with N infinitely thin turns that are magnetic equivalent to the permanent magnets (see fig. 2).
EM force acting on each magnet is calculated using numerical integration of components of the Maxwell tension tensor [19]. The repulsive forces evaluated for both models are listed in table. The table demonstrates that with the number of equivalent current loops as low as N = 4 — 8 the results differ only by several percents. This proves
Fig. 2. Equivalent circuit with N = 4 current loops for M = 5 Halbach array, side and top views
an acceptable accuracy and, therefore, adequacy of the numerical model used for an EDS with permanent magnets.
Repulsive force and relative deviation 5 vs air gap hz and number of turns N
Gap hz, mm Repulsive force, N
Model 1 Model 2
N = 4 N = 8 N = 16 N = 32
5 3337 3206, (5 = 3.9% 3301, (5 = 1.1% 3328, (5 = 0.28% 3335, (5 = 0.07%
10 2163 2105, (5 = 2.7% 2147, (5 = 0.7% 2159, (5 = 0.17% 2162, (5 = 0.04%
30 590 583, (5 = 1.2% 588, (5 = 0.3% 589.7, (5 = 0.08%
50 235 233, (5 = 0.68% 234, (5 = 0.17%
100 50 49.94, Ô = 0.25%
200 8 7.92, Ô = 0.09%
400 0.8978 0.8976, ô = 0.03%
The model gives near sinusoidal field in the air gap [7]. At speeds above 10 m/s, the skin effect appears, and the induced eddy currents tend to concentrate at a thin surface layer of the track. From theoretical estimates [19, 20], about 86% of the current generated by magnetic flux oscillating with a frequency u = 2nf are expected to concentrate in a non-ferromagnetic conductor with resistivity po in the skin layer with the depth defined as: 2Ao = y^Spo/¡j^ouj. For aluminium structures this depth is about 5 mm at the frequency of 103 Hz. In this case, an adequate description of electromagnetic behaviour of the track structures implies construction of FE models with a high level of details and accuracy.
Model of superconducting magnet. Fig. 3 illustrates a layout of the EDS system employing a pair of superconducting (SC) coils fixed on a vehicle levitated along a solid aluminium track. The coil cross-sectional size is assumed to be very small if compared
with the distance to an observation point, so each SC coil is modelled as an equivalent rectangular current loop located in the horizontal plane [12, 22]. The total current of a coil is taken as 800 kA-turns. The maximal lifting force achievable at the perfect skin effect (that means infinitely high speed or perfect conduction of the track) is evaluated with the use of the code KLONDIKE for a range of air gap values. The results are plotted in fig. 4. The lifting force should exhibit increase with the speed to the asymptotic value obtained in the KLONDIKE simulation.
Fig. 3. Layout of EDS with SC coils, side and top views
Fig. 4. SC EDS lifting force vs air gap
\ , "u • i j i Fig. 5. Lift and drag of SC EDS
between vehicle and track _ . r.
for two options of initial coil current 1 - TYPHOON simulation with v = 50 m/s; ( , 7 r .
(v = 50 m/s, h =15 mm) 2 - KLONDIKE simulation for maximal ^ _ . „ ^ ,
1, 3 - lifting force; 2, 4 - drag force.
achievable lifting force.
Stationary electromagnetic forces in the EDS system were calculated through solving a non-stationary problem via the stationary solution. Two options with different initial
conditions were analysed. For option 1 the coil current at t = 0 was assumed to have its rated value, eddy currents taken zero, the vehicle started moving with constant speed at t = 0 + 0. For option 2, a current jump from zero to the rated value was taken assuming infinite conductivity, the vehicle started moving with constant speed at t = 0 + 0. The stationary solutions for both options are plotted in fig. 5 and demonstrate a good match. This proves the reliability of the model.
As shown in fig. 4, the curve obtained in TYPHOON simulations is limited within the range related to the asymptotic behaviour of the maximal achievable lifting force evaluated with the use of the code KLONDIKE. A discrepancy of the TYPHOON and KLONDIKE results is caused by different models involved. The TYPHOON model implies a track with a finite width, while the KLONDIKE model treats the track as an infinite plane. The discrepancy of the simulated results is most notable if the air gap size is close to the track width.
Conclusions.
• An original computational technique has been adapted to simulation of the electromagnetic behaviour of maglev transport.
• To reach reliable prediction of the EDS performance at high frequencies of electromagnetic field, complex enhanced FE models were generated. Multi-variant computations have enabled selection of the appropriate design solutions. In the course of modelling and comparative simulations, the optimal discretization of the models with sufficient accuracy has been assessed.
• The computational technique is based on vector algorithms and oriented to parallel computations on supercomputers to take advantage of improved computational efficiency and shorter time cycle of the step by step design and optimization.
• The proposed technique is applicable for simulation of all basic types of EDS systems employing both superconducting and permanent magnets. A realistic description of geometry, media and magnetic properties is implemented.
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The article is received by the editorial office on June 26, 2014. Статья поступила в редакцию 26 июня 2014 г.
