JP2012128492A - Analyzer and analysis method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To more firmly conserving energy in a coupled analysis of a magnetic field and a motion.SOLUTION: An analyzer 1 analyzes a behavior of an object to be analyzed after the object to be analyzed has been represented thereby as a particle system comprising a plurality of particles. The analyzer 1 comprises: a magnetic field calculating section 66 for calculating a magnetic field by numerically solving an equation of motion, which represents a relation to be satisfied by the magnetic field in a form of the equation of motion, of a virtual particle; and a mechanism and elasticity calculating section for calculating a motion of each particle based on the magnetic field calculated by the magnetic field calculating section 66 and errors caused by calculations performed by the magnetic field calculating section 66 so as to update a position of the each particle.

Description

  The present invention relates to an analysis apparatus and an analysis method.

  In recent years, simulations incorporating magnetic field analysis are often used in the field of design and development of electric devices such as motors as computer computing power increases. When simulation is used, it is possible to evaluate to some extent without actually creating a prototype, so that the speed of design development can be improved.

  For example, Patent Document 1 describes a motor analysis device including an arithmetic processing device that performs magnetic field analysis. The arithmetic processing unit executes a static magnetic field analysis by a finite element method and a torque calculation by Maxwell's stress method according to an external command based on a user operation. Mesh division is performed for static magnetic field analysis by the finite element method. The mesh division is also performed on the core region and the housing region, and also on the outer air layer region.

  Further, a simulation method using renormalization group molecular dynamics is known (see Patent Document 2).

JP-A-11-146688 JP 2006-285866 A

In the Japanese Patent Application No. 2010-128664, the applicant of the present application has proposed, for example, a method of analyzing magnetic field, mechanism, elasticity, and control in combination. In this approach, the canonical variables that are derived from one of the Lagrangian (bold H _{ip '',} bold neighbors with H _{ip '')} equation of motion and the canonical variables of virtual particles (bold r _{i '',} bold The motion equations of the particles with r _{i ′} ′) with side points are numerically integrated by the jumping method, which is one of the general solutions of ordinary differential equations, and coupled analysis is performed.
(In Japanese Patent Application No. 2010-128664, the equation of motion of the virtual particle is named and described as the equation of motion of the magnetic field. The method shown in Japanese Patent Application No. 2010-128664 originally uses a virtual particle and has a magnetic material. In this specification, the virtual particle is described as a virtual particle so as to limit the name and make it more physically clear).
By numerically integrating the equation of motion of the virtual particle, the behavior of the virtual particle in the phase space shown in FIG. 11A is calculated (magnetic field analysis), and by numerically integrating the equation of motion of the particle, FIG. The behavior of the particles in the phase space shown in Fig. 2) is calculated (mechanism / elastic analysis). Magnetic field / mechanism / elastic coupled analysis is performed by calculating the behavior of virtual particles and the behavior of particles alternately. By adopting the jumping method in the form of the equation of motion, the energy of the system can be preserved in the numerical calculation of the virtual particle equation of motion and the particle equation of motion.

  However, when iterative calculation is performed until the magnetic field state of the system having the magnetic material reaches a steady state by the equation of motion of the virtual particles, it is necessary to terminate the calculation repeatedly with an arbitrary error value. Note that the error decreases as the iterative calculation is performed, but it is virtually impossible that the error is zero.

  Due to this truncation error, the energy of the entire system can fluctuate each time the cycle from the calculation of the virtual particle behavior (magnetic field analysis) to the calculation of the particle behavior (mechanism / elastic analysis) is repeated. As the number of cycles increases, that is, as the simulation becomes more complex and larger, the effects of this variation accumulate and cannot be ignored.

  Further, not only the truncation error, but errors generated by the calculation of the magnetic field are accumulated as the cycles are repeated, and the influence on the energy conservation cannot be ignored.

  The present invention has been made in view of these problems, and an object thereof is to provide an analysis technique that further enhances energy conservation in a coupled analysis of a magnetic field, a mechanism, and elasticity.

  One embodiment of the present invention relates to an analysis apparatus. This analysis device is an analysis device that analyzes the behavior of the object after describing the object to be analyzed as a particle system composed of a plurality of particles. Numerically solve the equation of motion of the virtual particle describing the relationship that the particles should satisfy in the form of the equation of motion, and calculate the magnetic field state of the system from the behavior of the virtual particle and the equation of motion of the particle numerically A mechanism / elasticity calculation unit that calculates the behavior of the particles by solving.

According to this aspect, the equation of motion of the particle includes the numerical error generated by the magnetic field analysis as a restoring force, and the numerical error generated by the magnetic field analysis is automatically reflected in the behavior of the particle, and the energy of the entire system is Preservation becomes strong.
Further, according to this aspect, it is possible to incorporate numerical errors caused by the magnetic field calculation into the particle behavior (mechanism / elasticity calculation).

  It should be noted that any combination of the above-described constituent elements, or those obtained by replacing the constituent elements and expressions of the present invention with each other between apparatuses, methods, systems, computer programs, recording media storing computer programs, and the like are also included in the present invention. It is effective as an embodiment of

  According to the present invention, energy conservation can be further strengthened in a coupled analysis of a magnetic field and a mechanism / elasticity.

It is a block diagram which shows the function and structure of the analyzer which concerns on embodiment. It is a figure which shows the outline of a structure of a SPM motor. It is an expansion perspective view of one circumference | surroundings of the stator teeth of an SPM motor. It is a flowchart which shows a series of processes in the analyzer of FIG. It is explanatory drawing of a coil. It is a figure which shows the relationship between a rectangular parallelepiped conductor and a local coordinate. It is a figure which shows the relationship between an arc-shaped columnar conductor and a local coordinate. 5 is a flowchart showing details of a part of the processing shown in FIG. 4. It is explanatory drawing of the particle | grains (cubic element) corresponding to a magnetic body, and the virtual particle inside it. It is explanatory drawing which shows transition of the energy of the whole system with the case where a restoring force is introduce | transduced and the case where it is not. FIGS. 11A and 11B are explanatory diagrams for explaining a phase space in which virtual particles and particles move respectively.

  The present invention will be described below based on preferred embodiments with reference to the drawings. The same code | symbol may be attached | subjected to the same or equivalent component, member, and process which are shown by each drawing. Moreover, the description which overlaps suitably is abbreviate | omitted.

  In the analysis apparatus according to the embodiment, a coupled simulation of a magnetic field, a mechanism, and elasticity is performed on an actual object such as a motor. The object of analysis is an aggregate of particles. Especially, for the mechanism / elastic analysis, the equation of motion of the particle (molecular dynamics) is used, and for the magnetic field analysis of the magnetic material, using the equation of motion of the virtual particles in the particles constituting the magnetic material. Coupled calculations are performed. In this coupled calculation, the analysis apparatus restores the numerical error generated in the magnetic field analysis by the virtual particle motion equation to the particle motion equation. As a result, the energy conservation of the entire system becomes stronger, and a highly accurate coupled analysis becomes possible.

  The numerical error that occurs in the magnetic field analysis using the virtual particle equation of motion is at least a truncation error that is the difference between the approximate solution and the exact solution that occur when the iterative solution is terminated indefinitely. And a discretization error corresponding to the difference between the solution of the problem obtained by performing the above and the solution of the original continuous problem. In the following, let us consider a case in which a truncation error is handled among numerical errors generated in a magnetic field analysis using a virtual particle equation of motion. In other embodiments, numerical errors other than the truncation error may be handled.

  FIG. 1 is a block diagram showing functions and configuration of an analysis apparatus 1 according to the embodiment. Each block shown here can be realized by hardware such as a computer (CPU) (central processing unit) and other elements and mechanical devices, and software can be realized by a computer program or the like. Here, The functional block realized by those cooperation is drawn. Therefore, it is understood by those skilled in the art who have touched this specification that these functional blocks can be realized in various forms by a combination of hardware and software.

The analysis device 1 analyzes the behavior of an object to be analyzed as a particle system composed of a plurality of particles. As the particle system here, a particularly brought-in particle system is used. The renormalization is detailed in Patent Document 2.
The analysis device 1 includes a control unit 3, a storage device 5, a media input / output unit 6, an input unit 7, a display unit 9, and a printer port 11, and these members are connected to each other via a bus 13. It is connected.

  The control unit 3 includes a CPU, a ROM (Read Only Memory), a RAM (Random Access Memory), and the like, and controls each device connected via the bus 13 in accordance with a program stored in the storage device 5 as a storage unit. Drive control.

  The storage device 5 has input information that is information having initial conditions. The media input / output unit 6 is a device for inputting / outputting information to / from a medium such as a floppy (registered trademark) disk, CD, or DVD. The input unit 7 is an input device such as a keyboard and a mouse, and the display unit 9 is a display device such as a display. A printer 12 as an output device is connected to the printer port 11.

  The control unit 3 includes a particle model generation unit 60, a retraction unit 62, an interaction calculation unit 64, a magnetic field calculation unit 66, a mechanism / elasticity calculation unit 68, a control parameter update unit 70, and an end determination unit 72. ,including. Details of each part will be described later.

  Below, the case where the motor, especially the SPM motor 31 (Surface Permanent Magnet Motor) which is a kind of permanent magnet motor is employed as the object to be analyzed by the analysis apparatus 1 will be described.

An outline of the configuration of the SPM motor 31 will be described with reference to FIGS.
As shown in FIG. 2, the SPM motor 31 includes a rotor 33 that is a rotor (moving element) and a stator 35 that is a stator. The rotor 33 includes a cylindrical rotor core 37 that is a magnetic body such as iron, and a permanent magnet 39 is provided on the surface of the rotor core 37. A rod-shaped rotor shaft 41 is provided at the axial center of the rotor core 37.

  The stator 35 includes a tooth-like stator teeth 43 that is a magnetic body, a core back 44 that is a cylindrical magnetic body provided outside the stator teeth 43, and a cylindrical frame that is provided outside the core back 44. 46.

  As shown in FIGS. 2 and 3, a coil 45 that is a conductor such as a metal is wound around the stator teeth 43. In the actual SPM motor 31, the coil 45 is wound around the stator teeth 43 with the number of turns according to the specification to form a bundle, but in this embodiment, it is depicted in FIGS. 2 and 3. In addition, a coil bundle is handled as one conductor without modeling each coil.

  In the SPM motor 31 having such a structure, the rotor 33 and the stator 35 are magnetized by the magnetic field of the permanent magnet 39 and the magnetic field generated by passing a current through the coil 45. The SPM motor 31 is driven by the deviation of the magnetic energy of the magnetic material.

  Therefore, in order to analyze the SPM motor 31, it is necessary to calculate the magnetic field vector that the coil 45 and the permanent magnet 39 form on the magnetic bodies constituting the rotor 33 and the stator 35, and to analyze the magnetization phenomenon of these magnetic bodies. .

  FIG. 4 is a flowchart showing a series of processes in the analysis apparatus 1 according to the embodiment. In the following procedure, the magnetic material refers to the magnetic material constituting the rotor 33 and the stator 35 and the permanent magnet 39, and these are distinguished by a function representing a magnetization curve. The conductor is, for example, a coil 45.

(Step S202)
The analysis device 1 acquires initial conditions including information on the shape of the SPM motor 31 and characteristics of the material (such as material constants) and stores them in the storage device 5 as input information. This initial condition may be read from a recording medium such as a CD-ROM via the media input / output unit 6, for example. Furthermore, when the initial conditions are stored as input information in advance, step S202 is not necessary.

  The initial conditions include the three-dimensional structure (shape, coordinate point) of the SPM motor 31, the mass density, the function representing the magnetization curve of the magnetic material, and the current density vector of the conductor. The information on the three-dimensional structure of the SPM motor 31 is data such as CAD. The current density vector is necessary when calculating the magnetic field vector formed by the coil 45. A function representing the magnetization curve of the magnetic material is required when calculating the magnetization vector.

(Step S204)
The particle model generation unit 60 generates a particle model of the SPM motor 31 from the initial conditions stored in the storage device 5. First, the particle model generation unit 60 divides the SPM motor 31 into N particles (N is an integer of 2 or more) from the information of the three-dimensional structure included in the initial condition, and calculates the position vector of each particle. Store in storage device 5. Next, the particle model generation unit 60 calculates the position vector of the virtual particle in each particle constituting the magnetic body and stores it in the storage device 5. N particles constitute the particle system S. The particle may be an atomic or molecular unit, or may be a polyhedral element having the position of the particle as the center of gravity. Details of the virtual particles will be described later.

  The number N of particles and the position vector are calculated based on the three-dimensional structure included in the initial condition and the shape of the polyhedral element stored in the storage device 5 in advance. Further, a polyhedral element constituted by the center of gravity and the surface of each particle constituting the magnetic body is assumed to be a virtual particle. In the following, it is assumed that the particles corresponding to the magnetic body of the SPM motor 31 are cubic elements whose center of gravity is the position of the particles, and the virtual particles are pentahedrons composed of the center of gravity of the particles and each surface. That is, one particle contains six virtual particles. Details of the calculation of the position of the virtual particle will be described in the explanation of step S102.

Moreover, the particle | grains corresponding to the coil 45 which is a conductor are determined as follows.
FIG. 5 is an explanatory diagram of the coil 45. FIG. 5A is an enlarged perspective view of the coil 45, and FIG. 5B is a diagram illustrating an example in which the coil 45 is divided into local conductors. FIG. 6 is a diagram illustrating a relationship between a rectangular parallelepiped conductor and local coordinates. FIG. 7 is a diagram illustrating a relationship between the arc-shaped columnar conductor and the local coordinates.

  From the three-dimensional structure information included in the initial conditions, the particle model generation unit 60 converts the coil 45 shown in FIG. 5A to a local conductor (rectangular conductors 45a and 45b and an arc shape as shown in FIG. 5B). Dividing into columnar conductors 45c and 45d), a coefficient for calculating a magnetic field vector created by each conductor is calculated and stored in the storage device 5.

First, the case where the local conductor is a rectangular parallelepiped conductor 45a will be described.
In addition, since it is the same as that of the case where a local conductor is the rectangular parallelepiped conductor 45a when a local conductor is the rectangular parallelepiped conductor 45b, description is abbreviate | omitted.

When the local conductor is a rectangular parallelepiped conductor 45a, the particle model generation unit 60 applies a local coordinate system (x _s , y _s , z _s ) as shown in FIG. What this in the local coordinate system to the center of gravity of the local conductor (rectangular conductor 45a) and the origin _{O s,} dimensions of rectangular conductor 45a is having a length of _{x s} direction 2a, _{y s} direction 2b, _{z s} directions 2c And And shall particles is located at the origin O _s.

  The particle model generation unit 60 reads the three-dimensional structure of the coil 45 included in the initial conditions, calculates the dimensions a, b, and c of the local conductor (cuboid conductor 45a) and the particle position vector, and stores them in the storage device 5. To do.

Next, a case where the local conductor is an arc-shaped columnar conductor 45c will be described.
Note that the case where the local conductor is the arc-shaped columnar conductor 45d is the same as the case where the local conductor is the arc-shaped columnar conductor 45c, and a description thereof will be omitted.

When the local conductor is the arc-shaped columnar conductor 45c, the particle model generation unit 60 applies the local coordinate system (x _c , y _c , z _c ) as shown in FIG. Origin O _c are present on an arc of the central axis, and that the arcuate columnar conductor 45c are symmetrical with respect to the height direction of the arcuate columnar conductor 45c (z _c direction in FIG. 7) in this local coordinate system Shall exist.

_{Moreover, x c, y c, z} c _, when viewed in x c -y _c plane, + _{x c-axis} as a base point, and so the arc of the arcuate columnar conductor 45c is + when viewed from the _{z c-axis} counterclockwise decide. The average value of the inner diameter and the outer diameter of the arcuate columnar conductor 45c and _{R c, 2r} the thickness in the radial direction a, the _{z c} direction height and 2z _b.

Current as a base point a + x _c-axis, + when viewed from the z _c the angle a θ of the counter-clockwise direction, current is assumed to flow in a uniform current density j in this direction. It is assumed that the particles are located at (R _c , θ / 2, 0) in the cylindrical coordinate system.

Particle model generation unit 60 reads the three-dimensional structure of a coil 45 contained in the initial condition, the dimension of the local conductor (arcuate columnar conductor _{45c) r a, z b,} θ, calculate R _c and particle position vector And stored in the storage device 5.
The particle model generation unit 60 may set particles in the same manner regardless of the magnetic material and the conductor.

(Step S206)
The renormalization unit 62 performs a renormalization process on the particle system S using the method of renormalization group molecular dynamics described in Patent Document 2, and generates a regenerated particle system S ′. In the following, it is assumed that the first renormalization factor is α, the second renormalization factor is γ = 0, the third renormalization factor δ = 2, and the dimension number d of the space is d = 3. According to Patent Document 2, between the parameters of the particle system S and the regenerated particle system S ′,
The relationship holds. Moreover, according to the original consideration by the inventor (described later),
The relationship holds. The physical quantities described in the following explanation are all the quantities in the brought-in particle system S ′.

(Step S208)
The interaction calculation unit 64 calculates the interaction potential energy φ ′ due to the interaction between particles. The calculation of the interaction potential energy φ ′ is detailed in Patent Document 2. In particular, in the present embodiment, the interaction potential energy φ ′ has a shape that takes elasticity into consideration.

(Step S210)
The magnetic field calculation unit 66 calculates the magnetic field on the particle position vector by numerically solving the equation of motion of the virtual particle describing the relationship to be satisfied by the virtual particle in the particles constituting the magnetic material in the form of the equation of motion. . At that time, the magnetic field calculation unit 66 repeatedly performs a calculation based on the equation of motion of the virtual particles until a predetermined steady state is reached. Details will be described later.

(Step S212)
The mechanism / elasticity calculation unit 68 numerically integrates the equation of motion of each particle based on the interaction potential energy φ ′ calculated by the interaction calculation unit 64 and the magnetic field on the particle position vector calculated by the magnetic field calculation unit 66. Update the position and speed of each particle. Next, the position of the virtual particle is updated based on the updated position of the particle. The particle equation of motion has a term of restoring force due to an error in the magnetic field solution calculation. Details will be described later.

(Step S214)
The control unit 3 confirms whether there is an output instruction from the user.
(Step S216)
When there is an output instruction from the user (Y in Step S214), the control unit 3 performs rescaling corresponding to Step S206 on the particle system S ′ that has been brought forward.
(Step S218)
The control unit 3 outputs the particle position vector, force vector, magnetization vector, magnetic field vector on the particle position vector, magnetic flux density vector, and the like obtained from the rescaling from the printer 12 via the printer port 11.

(Step S220)
When there is no output instruction from the user (N in Step S214), the end determination unit 72 determines whether a predetermined end condition (time, movement amount, etc.) is satisfied. When the termination condition is satisfied (Y in step S220), the termination determination unit 72 terminates the analysis.
When the termination condition is not satisfied (N in Step S220), the control unit 3 returns the process to Step S208. That is, the control unit 3 calculates the interaction potential energy φ ′ and the magnetic field again using the updated position and velocity of each particle.

  Note that the control parameter update unit 70 is a case where the termination condition is not satisfied (N in Step S220), and when there is a request from the user, the control parameter given from the outside of the transferred particle system S ′. Then, control parameters that affect the magnetic field acting on each particle, such as a current density vector, are updated. The magnetic field calculation unit 66 calculates a magnetic field using the updated current density vector.

Steps S210 and S212 will be described in detail below.
FIG. 8 is a flowchart showing details of step S210 and step S212 of FIG.

(Step S102)
The magnetic field calculation unit 66 uses a coil size and a current density vector to calculate a virtual particle position vector in a particle in which the energized coil corresponds to the magnetic substance by an analytical solution obtained by integrating Bio-Savart's law. The magnetic field vector created above is calculated and stored in the storage device 5.

FIG. 9 is an explanatory diagram of particles corresponding to a magnetic body and virtual particles. The shape of the particles is a cube. FIG. 9 shows an example of a shape when a cubic element is displayed two-dimensionally.
Here, the position vector of the particle (the center of gravity of the cube element) is (bold) r _g ′, and the midpoint of the particle (cube element) surface is the q ′ point. A pentahedral element composed of a particle position and a particle surface is defined as a virtual particle, and an intermediate point p ′ between the particle position and the q ′ point is defined as a virtual particle position. Bold n ′ _{q ′} is the normal vector of the element boundary surface to which the middle point q ′ belongs, and ΔS ′ _{q ′} is the surface area of the virtual particle to which the middle point q ′ belongs.

Returning to FIG.
First, the case where the local conductor is a rectangular parallelepiped conductor 45a will be described.
In addition, since it is the same as that of the case where a local conductor is the rectangular parallelepiped conductor 45a when a local conductor is the rectangular parallelepiped conductor 45b, description is abbreviate | omitted.

When the local conductor is a rectangular parallelepiped conductor 45a, the local coordinate system (x _s ′, y _s ′, z _s ′) defined in FIG. 6 is applied.
Next, the position vector of point P, which is an arbitrary point, is converted into the local coordinate system (x _s ′, y _s ′, z _s ).

  Magnetic field vectors created at the point P by the energized rectangular parallelepiped conductor are described by the following equations (1) to (3).

Here, r _ps ′ (in bold) is a position vector of the P point in the local coordinate system (x _s ′, y _s ′, z _s ′), and x _ps ′, y _ps ′, z _ps ′ are x _{s', y s',} is the value of _{z s'} direction. π is the circumference ratio. H _{x ′s} ′, H _{y ′s} ′, and H _z ′ _s ′ are each component of the magnetic field vector in the local coordinate system (x _s ′, y _s ′, z _s ′). j ′ is the current density.

X _i ′, y _j ′, z _k ′ represent the upper and lower limits of integration in the x _s ′, y _s ′, z _s ′ direction, and the relationship shown in Expression (4) is established.

  Here, a ', b', and c 'are dimensions of the rectangular parallelepiped conductor.

Next, a case where the local conductor is an arc-shaped columnar conductor 45c will be described.
Note that the case where the local conductor is the arc-shaped columnar conductor 45d is the same as the case where the local conductor is the arc-shaped columnar conductor 45c, and a description thereof will be omitted.

Local conductors when arcuate columnar conductors 45 c, to apply the local coordinate system defined in Figure _{7 (x c ', y c} ', z c ').
The position vector of the point P after conversion to the local coordinate system (x _c ′, y _c ′, z _c ′) is expressed as a cylindrical coordinate system (bold) r _pc ′ = (R _pc ', Φ _pc ', Z _pc ').
A magnetic field vector formed at the point P by the energized circular columnar conductor 45c is expressed by the following equations (6) to (11).

Here, H _rc ′, H _tc ′, and H _zc ′ are each component of the magnetic field vector in the cylindrical coordinate system. j ′ is the current density. r _a ′, θ ′, and z _b ′ are dimensions of the arc-shaped columnar conductor 45c, and R _c ′ is an average value of the inner diameter and the outer diameter of the arc. sgn is the sign of Z _k ′, R _j ′ and Z _k ′ represent the upper and lower limits of integration, and the relationship shown in Expression (12) is established.

The above is the calculation procedure of the magnetic field vector created by the coil 45 at an arbitrary point P.
The magnetic field calculation unit 66 reads the position vectors of the positions p ′ of all the virtual particles corresponding to the magnetic substance in the renormalized particle system S ′.
If the position vectors of the particles and virtual particles are updated and stored in the storage device 5 in step S212, the magnetic field calculation unit 66 reads the values.

Next, the magnetic field calculation unit 66 converts the position vector of the point p ′ in FIG. 9 into the local coordinate system (x _s ′, y _s ′, z _s ′), and the dimensions of the rectangular parallelepiped conductor 45a, the current density vector, And calculate the magnetic field vector based on the equations (1) to (4).

Next, the magnetic field calculation unit 66 converts the magnetic field vector calculated by the equations (1) to (4) into the global coordinate system (x ′, y ′, z ′), and stores the converted magnetic field vector in the storage device 5. To remember.
As a result, a magnetic field vector generated at the point p ′ by the energized rectangular parallelepiped conductor 45a is obtained.

Further, the magnetic field calculation unit 66 converts the position vector of the point p ′ to the local coordinate system (x _c ′, y _c ′, z _c ′), and further, the cylindrical coordinate system (bold) r _pc ′ = (R _pc ′). , Φ _pc ′, Z _pc ′).

Next, the magnetic field calculation unit 66 reads the dimensions r _a ′, θ ′, z _b ′, and R _c ′ of the arc-shaped columnar conductor 45c, and the magnetic field calculation unit 66 reads the current density vector.

The magnetic field calculation unit 66 calculates a magnetic field vector based on the equations (6) to (12).
Next, the magnetic field calculation unit 66 converts the calculated magnetic field vector into an orthogonal coordinate system, further converts it into a global coordinate system (x ′, y ′, z ′), and stores it in the storage device 5.
As a result, a magnetic field vector created at the point p ′ by the energized circular columnar conductor 45c is obtained.

(Step S104)
The magnetic field calculation unit 66 calculates a magnetic field vector created at the position of the virtual particle in the particle corresponding to the magnetic material by the permanent magnet 39 and stores it in the storage device 5.

(Step S106)
The magnetic field calculation unit 66 calculates the sum of the magnetic field vector calculated in step S102 and the magnetic field vector calculated in step S104, and stores it in the storage device 5.

(Step S108)
The magnetic field calculation unit 66 calculates the coefficient of the equation of motion of the virtual particle corresponding to the magnetic body based on the virtual particle position vector in the particle corresponding to the magnetic body and the position and shape of the particle (cube element), and stores them in the storage device 5. Remember.

The magnetic field calculation unit 66 reads the position vectors of particles and virtual particles corresponding to the magnetic material.
If the position vectors of the particles and virtual particles are updated and stored in the storage device 5 in step S212, the magnetic field calculation unit 66 reads the values.

  The magnetic field calculation unit 66 calculates the position vector of the q ′ point on the virtual particle surface and the normal vector (in bold) n ′ of the virtual particle surface from the particle position vector and the shape of the particle (cubic element), and the storage device Store in 5.

  Next, the magnetic field calculation unit 66 calculates the virtual particle surface area ΔS ′ and the virtual particle volume ΔV ′ in the particle corresponding to the magnetic material, and stores them in the storage device 5.

  Next, the magnetic field calculation unit 66 calculates the behavior (value of the canonical variable) of the virtual particle after the time step δt ′ without considering the constraint condition from the equation of motion of the virtual particle in the particle corresponding to the magnetic substance. And stored in the storage device 5.

  The Lagrangian of (N′−w) particles corresponding to the magnetic material is assumed to have a shape represented by the formulas (13) to (15). w is a natural number, and is the number of particles corresponding to the coil 45 in the transferred particle system S ′.

Here, in Expression (13), α ′ is a virtual mass, bold r ′ is a position vector, bold H ′ and bold H ′ with a cantilever are canonical variables, bold M ′ is a magnetization vector, bold Hext 'Is an externally applied magnetic field vector, χ' is magnetic susceptibility, μ ₀ 'is vacuum permeability, λ' is Lagrange's undetermined constant, π is pi, s' is the number of virtual particles in the particle .

  The subscript ip ′ of each physical quantity is the p′-th virtual particle in the i-th particle in the renormalized particle system, and the subscript jq ′ is q ′ in the j-th particle in the renormalized particle system. Indicates the physical quantity of the second virtual particle.

In equation (13), m ′ is the mass of the particle, v ′ is the velocity, and φ ′ (r _i ′ −r _j ′) is the interaction potential energy between the i th particle and the j th particle. Subscripts i and j indicate physical quantities of the i and j-th particles, respectively.

Next, the canonical variables (in bold) _{H ip 'when',} (with bold neighbor) _{H ip 'and'} substitutes the Lagrangian of the formula (13) to (15) to the Lagrange equation of motion, The equation of motion of the virtual particle can be described as in equation (16).

Here, the third term on the right side of Equation (16) imposes a constraint condition (the divergence of the magnetization vector is 0) through Lagrange's undetermined constant.
The fourth term on the right side of Equation (16) is an attenuation term for quickly following when the externally applied magnetic field vector changes, and γ ′ is an attenuation constant.

In solving the equation of motion including the constraint condition shown in the third term on the right side of Equation (16), the present embodiment adopts the SHAKE method, which is a generalized constraint condition introduction method.
If equation (16) is discretized by the jumping method without considering the constraint condition, the following equations (17), (18), and (19) are obtained.

Here, δt ′ is a time step size used for calculating the convergence of the magnetization phenomenon.
The subscript n is an arbitrary integer, a physical quantity at nδt ′, n−1 / 2 is a physical quantity at (n−1 / 2) δt ′, n + 1/2 is a physical quantity at (n + 1/2) δt ′, and n + 1 is This corresponds to the physical quantity at (n + 1) δt ′.

  The magnetic field calculation unit 66 reads the position vectors of the points p ′ and q ′ that have already been calculated and stored in the storage device 5.

Next, the magnetic field calculation unit 66 generates a magnetic field vector generated at the virtual particle position p ′ point in the particle corresponding to the magnetic material by the coil 45 and the permanent magnet 39 that are already calculated in step S106 and stored in the storage device 5. Read as an externally applied magnetic field vector.
Further, the magnetic field calculation unit 66 reads the surface area, normal vector, and volume of the virtual particles corresponding to the magnetic material, which are already calculated and stored in the storage device 5.
Further, the magnetic field calculation unit 66 reads the attenuation constant, the virtual mass, and the time step size stored in advance in the storage device 5.

Next, the magnetic field calculation unit 66 reads the initial values of the canonical variables of virtual particles ((bold) _{Hip ′} ′, (with bold side points) _{Hip ′} ′) stored in the storage device 5 in advance.
In addition, when the canonical variable of the virtual particle is updated in step S110 described later, the magnetic field calculation unit 66 reads the value.
The magnetization vector is calculated using a function representing a magnetization curve included in the initial condition and (in bold) _{Hip ′} ′.

The magnetic field calculation unit 66 calculates Equation (17), Equation (18), and Equation (19) for the virtual particles in the particles corresponding to the magnetic material, and calculates the canonical variable (in bold) H _{ip 'stores} the value in the storage device 5', (with bold neighbor) _{H ip '').}

(Step S110)
Field calculating unit 66 is stored in the calculation and the storage device 5 in step S108, (bold) H _{ip ''} to the constraints added, were calculated (bold) H _{ip 'a',} the storage device 5 To remember.

  A constraint condition is added to the virtual particles according to the following formulas (20) and (21).

The magnetic field calculation unit 66 reads the value of the canonical variable of the virtual particle in the particle corresponding to the magnetic body calculated in step S108 and stored in the storage device 5.
The magnetic field calculation unit 66 reads the virtual mass, the time step size, and the attenuation constant that are stored in advance in the storage device 5.
The magnetic field calculation unit 66 performs calculations based on the equations (20) and (21) for the virtual particles in each particle corresponding to the magnetic material, and stores the calculated (bold) H _{ip ′} ′ in the storage device 5. To remember.

(Step S112)
Next, the magnetic field calculation unit 66 determines whether or not the (bold) _{Hip ′} ′ obtained in step S110 satisfies the constraint condition, and if satisfied, proceeds to the next step, and if not satisfied, returns to step S110. .

Specifically, the magnetic field calculation unit 66 calculates the magnetization vector calculated from _{Hip ′} ′ of each virtual particle in the particle corresponding to the magnetic body calculated in step S110 and stored in the storage device 5 (bold). The calculation based on the equation (22) is performed.

err _i ′ is an error value with respect to the constraint condition of the i-th particle among the particles corresponding to the magnetic material.

The magnetization vector (bold) M ′ is calculated in step S110 and stored in the storage device 5 (bold) _{Hip ′} ′ by the magnetic field calculation unit 66 and represents the magnetization curve included in the initial condition. It is calculated by using the function.

  For the normal vector (bold) n ′, the magnetic field calculation unit 66 reads and substitutes what has already been calculated and stored in the storage device 5.

  If the error value does not satisfy Expression (23) for all particles, the magnetic field calculation unit 66 returns to Step S110.

In Expression (23), A ₁ ′ is an error determination value, and an arbitrary value is stored in the storage device 5 in advance.

(Step S114)
The magnetic field calculation unit 66 determines whether or not a predetermined truncation condition for determining whether or not to repeat the repetitive calculation based on the motion equation of the virtual particle is satisfied. Here, the criterion of whether or not the magnetization phenomenon of the magnetic material has reached a steady state is used as the truncation condition.
Whether or not the virtual particle in the particle corresponding to the magnetic body has reached a steady state is determined by the following equation (24).

A ₂ ′ is an error determination value for determining whether the virtual particle has reached a steady state, and can be set arbitrarily.

The magnetic field calculation unit 66 reads A ₂ ′ and μ ₀ ′ stored in advance in the storage device 5.
Further, the magnetic field calculation unit 66 reads (in bold) _{Hip ′} ′ calculated in step S110 and stored in the storage device 5.

  Next, the magnetic field calculation unit 66 calculates Expression (24). If Expression (24) is satisfied, the process proceeds to the next step, and if not satisfied, the process returns to Step S108.

It should be noted that the expression (24) is satisfied and the process proceeds to the next step, that is, when the iterative calculation in the magnetic field calculation unit 66 is terminated.
Is a truncation error under the above truncation conditions, and is often a non-zero finite value.
Step S210 includes step S102, step S104, step S106, step S108, step S110, step S112, and step S114.

(Step S116)
The mechanism / elasticity calculation unit 68 calculates the magnetic field vector, the magnetization vector, and the magnetic flux density vector at the positions of all the particles corresponding to the SPM motor 31 and stores them in the storage device 5. That is, the mechanism / elasticity calculation unit 68 performs the magnetic field vector, magnetization vector, and magnetic flux density vector at all particle positions corresponding to the magnetic material, and the magnetic field vector and magnetic flux density at all particle positions corresponding to the coil 45. The vector is calculated and stored in the storage device 5.

(Particles corresponding to magnetic materials)
The magnetic field vector at the position of the particle corresponding to the magnetic body is expressed by the following equation (25) from the virtual particle.

Where (bold) e _{x '} ' = (1, 0, 0), (bold) e _{y '} ' = (0, 1, 0), (bold) e _{z '} ' = (0, 0 , 1).
The magnetization vector of the particle corresponding to the magnetic material can be obtained using a magnetic field vector on the particle position vector and a function representing a magnetization curve.

  The magnetic flux density vector at the position of the particle corresponding to the magnetic body is expressed by Expression (26).

  Here, B ′ in bold is a magnetic flux density vector.

The mechanism / elasticity calculation unit 68 reads, from the storage device 5, the already calculated _{Hip ′} ′ (bold) in the particle corresponding to the magnetic material.

  Next, the mechanism / elasticity calculation unit 68 reads the normal vector already calculated and stored in the storage device 5, and calculates the magnetic field vector at the position of the particle corresponding to the magnetic body based on the equation (25). Store in the storage device 5.

  Further, the mechanism / elasticity calculation unit 68 calculates the magnetization vector of the particle corresponding to the magnetic body by using the magnetic field vector calculated in this step and the function representing the magnetization curve included in the initial condition, and the storage device 5. To remember.

Next, the mechanism / elasticity calculation unit 68 substitutes the magnetic field vector and the magnetization vector calculated in this step into the equation (26), calculates the magnetic flux density vector at the position of the particle corresponding to the magnetic material, and stores it in the storage device 5. Remember.
The control unit 3 reads the vacuum permeability stored in the storage device 5 in advance.

(Particle corresponding to coil)
The magnetic field vector at the position of the particle corresponding to the coil 45 is described by equations (27) and (28).

The first term on the right side of the equation (27) is a term describing a magnetic field vector created at the position of the particle corresponding to the coil 45 by the virtual particle in the particle corresponding to the magnetic body, and is represented by the equation (28).
The second term on the right side of the equation (27) is a magnetic field vector generated at the position of the particle corresponding to the coil 45 by the particle corresponding to the coil 45, and is represented by equations (1) to (12).

The mechanism / elasticity calculation unit 68 reads, from the storage device 5, the particle position vector corresponding to the coil 45 in the retreated particle system S ′.
If the position vector of the particle corresponding to the coil 45 has been updated in step S120, the mechanism / elasticity calculation unit 68 reads the value.

The mechanism / elasticity calculation unit 68 reads the number of particles and the number of virtual particles corresponding to the magnetic material calculated and stored in the storage device 5, and the position vector and surface area of the q ′ point on the surface of the virtual particle.
Further, the mechanism / elasticity calculation unit 68 reads the magnetization vector of the virtual particle in the particle corresponding to the magnetic material, which has been calculated up to this step and stored in the storage device 5.
Further, the mechanism / elasticity calculation unit 68 reads the dimensions of the coil 45 and the current density vector.

  Thereafter, the mechanism / elasticity calculation unit 68 calculates the magnetic field vector at the position of the particle corresponding to the coil 45 in accordance with the equations (27), (28), and (1) to (12), and stores them in the storage device 5. To do.

The mechanism / elasticity calculation unit 68 converts the magnetic flux density vector (bold) B _i ′ at the position of the particle corresponding to the coil 45 into a magnetic field vector at the position of the particle corresponding to the coil 45 that has already been calculated. Multiply the permeability and calculate and store in the storage device 5. The vacuum permeability is stored in the storage device 5 in advance.

In this way, the control unit 3 analyzes the magnetization phenomenon of the magnetic material by solving the motion equation of the virtual particles. Therefore, it is not necessary to divide the entire space into meshes unlike the finite element method, and the magnetic field analysis can be efficiently performed with sufficient accuracy for a system having a magnetic body.
In addition, since the equation of motion is solved, a matrix is not handled and the amount of memory necessary for the calculation is proportional to the number of particles.

(Step S118)
The mechanism / elasticity calculation unit 68 calculates a force vector acting on all particles corresponding to the SPM motor 31 and stores it in the storage device 5. The mechanism / elasticity calculating unit 68 is an elastic force acting on the particles derived from the interaction potential energy φ ′ calculated by the interaction calculating unit 64 and a magnetic force acting on the particles derived from the magnetic field calculated by the magnetic field calculating unit 66. The force and the restoring force based on the repetitive calculation truncation error in the magnetic field calculation unit 66 are added to obtain a total force acting on the particle.

(Particles corresponding to magnetic materials)
Let the canonical variables be bold r _i ′ and bold r _i ′ with side points, and the Lagrangian of equation (13) is substituted into the Lagrangian equation of motion. Then, the equation of motion of the particle corresponding to the magnetic material can be described as in Expression (29).

The first term on the right side of Equation (29) is the force acting on the particle derived from the interaction potential energy φ ′ calculated by the interaction calculation unit 64. In particular, when the interaction potential energy φ ′ is determined in consideration of elasticity, this force corresponds to an elastic force. The second term on the right side of Equation (29) is the magnetic force acting on the particles derived from the magnetic field calculated by the magnetic field calculation unit 66. The third term on the right side of Expression (29) is a restoring force based on the truncation error of the repetitive calculation in the magnetic field calculation unit 66. If the truncation error is zero, in the third term on the right side
Therefore, the restoring force is zero.
Equation (29) shows that the force corresponding to the elastic force, magnetic force and restoring force acts on the particles corresponding to the magnetic substance.

(Particle corresponding to coil)
Of the particles corresponding to the coil 45, i-th position vector of a particle bold the r _i ', t _i the unit vector bold current at that position vector', the magnetic flux density vector bold B _i ', current Assuming that the length of the coil in the flowing direction is L _i ′, the force vector acting on the i-th particle among the particles corresponding to the coil 45 is expressed by the following equation (30).

(Step S120)
The mechanism / elasticity calculation unit 68 updates the position vector and velocity vector of each particle by numerically solving the equation of motion of the particle.
Step S212 includes step S116, step S118, and step S120.

  According to the analysis apparatus 1 according to the present embodiment, errors generated in the magnetic field analysis by the virtual particle motion equation are included in the particle motion equation as a restoring force. Specifically, this is caused by the second term in the outer side [] of the first term on the right side of the Lagrangian of Expression (13). Thereby, the influence on the energy of the entire system due to an error generated in the magnetic field analysis can be compensated on the mechanism / elasticity analysis side in the form of restoring force. As a result, the energy conservation of the entire system can be made stronger in the coupled analysis of the magnetic field, mechanism, and elasticity.

  FIG. 10 is an explanatory diagram showing the transition of the energy of the entire system when the restoring force is introduced and when it is not introduced. The horizontal axis in FIG. 10 indicates the number of steps when the steps S208, S210, and S212 in FIG. 4 are regarded as one step, and the simulation progresses as the number of steps increases. The vertical axis in FIG. 10 indicates the energy of the entire system to be analyzed.

An alternate long and short dash line 90 in FIG. 10 shows the transition of energy of the entire system when no restoring force is introduced. In this case, as a result of simply truncating the error generated in the magnetic field analysis, the energy of the entire system moves away from the true value E ₀ as the number of steps increases.

A solid line 96 in FIG. 10 shows the transition of energy of the entire system in the case of the present embodiment in which the restoring force is introduced. In this case, the energy of the entire system exists around the true value E ₀ even if the number of steps increases.

  In the analysis apparatus 1 according to the present embodiment, the magnetic field calculation unit 66 repeatedly performs a calculation based on the motion equation of virtual particles until a steady state is reached, and the mechanism / elasticity calculation unit 68 includes a magnetic field calculation unit 66. The motion of the particle is computed using the restoring force based on the truncation error of the iterative computation at. Therefore, when the magnetic field analysis is performed by repeatedly performing the calculation based on the equation of motion of the virtual particle, the energy drift of the entire system due to the truncation error can be suppressed.

  Further, according to the analysis apparatus 1 according to the present embodiment, the elastic analysis in the interaction calculation unit 64, the magnetic field analysis in the magnetic field calculation unit 66, the mechanism analysis in the mechanism / elasticity calculation unit 68, the control analysis in the control parameter update unit 70, Are coupled. Therefore, it is possible to perform a simulation with high accuracy and enhanced energy conservation for the object to be analyzed.

  The configuration and operation of the analysis apparatus 1 according to the embodiment have been described above. This embodiment is an exemplification, and it is understood by those skilled in the art that various modifications can be made to each component and combination of processes, and such modifications are within the scope of the present invention.

  In the above description, the case where the magnetization phenomenon of the magnetic material is calculated by solving the equation of motion of the virtual particle derived from the Lagrangian of the particle corresponding to the magnetic material is described. However, the present invention is not limited to this. Instead, it may be an equation other than that derived from Lagrangian.

  In the above description, the present embodiment is used for the analysis of the SPM motor 31. However, the present invention is not limited to this, and the present invention analyzes a magnetic field at an arbitrary point in a space where a magnetic body is arranged. If so, for example, a motor other than the SPM motor 31 such as a linear motor or a magnetic shield surrounding the space with a magnetic material may be used for analysis in the space shielded with the magnetic material.

In the following, considerations made independently by the present inventor regarding the relationship between the parameters of the particle system S and the renormalized parameters of the particle system S ′ will be described.
When the first renormalization factor is α, the second renormalization factor is γ, the third renormalization factor is δ, and the number of dimensions of the space is d, according to Patent Document 2,
It becomes.

(Magnetic field renormalization)
Consider a bulk with spin (nuclear magnetic moment) μ _i (i = 1, 2,..., N) on N atoms. Magnetic induction produced by the nuclear spin μ _i inside the bulk is
It is expressed. When this is coarse-grained,
It becomes.

Retract the magnetic moment as shown below.
The magnetic flux density brought in is
It is expressed.

As with the magnetic flux density, when a magnetic field is applied,
The renormalized magnetization is
It is. The force acting on the coarse-grained magnetic moment m = Σμ _i is
And scaled as shown below.

The magnetic flux density produced by the coarse-grained charge flow j is
It is. Interaction between the magnetic flux density to make the coarse-grained and magnetic moments m _i and the current density j is
Thus, the current density is scaled as shown below.

In summary,
It becomes.

  DESCRIPTION OF SYMBOLS 1 Analysis apparatus, 3 Control part, 5 Storage device, 6 Media input / output part, 7 Input part, 9 Display part, 11 Printer port, 12 Printer, 13 Bus | bath, 60 Particle model production | generation part, 62 Renormalization part, 64 Interaction calculation Unit, 66 magnetic field calculation unit, 68 mechanism / elasticity calculation unit, 70 control parameter update unit, 72 end determination unit.

Claims (5)

An analysis device that analyzes the behavior of an object after describing the object to be analyzed as a particle system composed of a plurality of particles,
A magnetic field calculation unit for calculating a magnetic field by numerically solving a motion equation of a virtual particle describing a relationship to be satisfied by the magnetic field in the form of a motion equation;
A mechanism / elasticity calculation unit that updates the position and velocity of each particle by numerically solving the equation of motion of the particle including a force caused by a numerical error generated in the magnetic field calculation unit, Analysis device to do.   The analysis apparatus according to claim 1, wherein the mechanism / elasticity calculation unit calculates a motion of each particle including a truncation error of repetitive calculation in the magnetic field calculation unit. It further comprises an interaction calculator that calculates the interaction potential energy due to the interaction between the particles,
The mechanism / elasticity calculation unit includes a first force acting on the particle derived from the interaction potential energy calculated by the interaction calculation unit and a first force acting on the particle derived from the magnetic field calculated by the magnetic field calculation unit. 3. The analysis according to claim 2, wherein the force of 2 and the third force based on the truncation error of the repetitive calculation in the magnetic field calculation unit are added to obtain a total force acting on the particle. apparatus. An analysis method for analyzing the behavior of the object after describing the object to be analyzed as a particle system composed of a plurality of particles,
Calculating the magnetic field by numerically solving the equation of motion of the virtual particle describing the relationship to be satisfied by the magnetic field in the form of the equation of motion;
Calculating the motion of each particle based on the calculated magnetic field and the error caused by the calculation of the magnetic field, and updating the position and velocity of each particle. A computer program that allows a computer to realize a function of analyzing the behavior of an object after describing the object to be analyzed as a particle system composed of a plurality of particles,
A function to calculate the magnetic field by numerically solving the equation of motion of the virtual particle describing the relationship to be satisfied by the magnetic field in the form of the equation of motion,
A computer program for causing the computer to realize a function of calculating the motion of each particle based on the calculated magnetic field and an error caused by the calculation of the magnetic field, and updating the position and velocity of each particle.