JP2017194884A - Analysis apparatus and analysis method - Google Patents

Abstract

The present invention provides an analysis technique that simultaneously improves the accuracy and speed of arithmetic processing by a magnetic bead method. An analysis apparatus includes a magnetic moment applying unit that applies a magnetic moment to particles of a particle system defined in a virtual space, and a numerical solution of a magnetic field generated by a reference particle to which the magnetic moment is applied. Reference data that is calculated in advance for each of a plurality of lattice points set around and holds the correspondence between the vector amount of the magnetic moment applied to the reference particle and the vector amount of the numerical solution of the magnetic field for each of the plurality of lattice points. Using the holding unit 230 and the correspondence relationship held in the reference data holding unit 230, the magnetic field calculation unit 121 that calculates the magnetic field generated by each particle to which the magnetic moment is applied by the magnetic moment applying unit 110, and the magnetic field calculation unit 121 And a particle state calculation unit 124 that numerically calculates a governing equation that governs the motion of each particle using the calculation result in. [Selection] Figure 19

Description

  The present invention relates to an analysis apparatus and an analysis method for analyzing a particle system.

  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. Other techniques for magnetic field analysis include the difference method and the magnetic moment method.

  Simulation based on the renormalization group molecular dynamics method developed to handle macro-scale systems as a method for exploring phenomena in general materials science using computers based on classical mechanics and quantum mechanics. Is known (see, for example, Patent Document 2). Since the particle method can handle not only a static phenomenon but also a dynamic phenomenon such as a flow, it is attracting attention as a simulation method replacing the above-mentioned finite element method which mainly analyzes a static phenomenon. For example, by applying a magnetic moment to each particle and calculating a magnetic physical quantity using an exact solution based on the spherical symmetry of each particle, it is possible to obtain a relatively accurate simulation result at high speed. A bead method has been proposed (see, for example, Patent Document 3).

JP-A-11-146688 JP 2009-37334 A Japanese Patent Laid-Open No. 2015-111401

Erik Bitzek et al., “Structural Relaxation Made Simple”, Physical Review Letters, October 27, 2006, 97 L.D.Landau, E.M.Lifshitz, L.P.Litaevskii, "Electrodynamics of Continuous Media 2nd ed", January 1, 1984, p. 205 Toshihiko Kuwahara, Takeshi Takeda, “Boundary Element Method Using Analytical Integration for Three-Dimensional Laplace Problems”, IEEJ Transactions Vol. 106, No. 11, 1986, p. 34

  In the magnetic bead method described above, an object is divided into a plurality of elements, each element is regarded as a spherical particle, and calculation is performed using an exact solution based on spherical symmetry. At this time, if an object is filled with a plurality of particles, gaps are formed between the particles, and errors due to the gaps or spherical particles may occur. In order to increase the calculation accuracy, it is necessary to divide the target object with a voxel such as a cube without gaps and execute the area based on the exact solution according to the shape of the voxel, but trying to calculate the area with high accuracy Then, the calculation time becomes enormous.

  The present invention has been made in view of these problems, and an object of the present invention is to provide an analysis technique that achieves both improvement in accuracy and speed of arithmetic processing by the magnetic bead method.

  In order to solve the above-described problem, an analyzer according to an aspect of the present invention includes a magnetic moment applying unit that applies a magnetic moment to particles of a particle system defined in a virtual space, and a reference particle to which the magnetic moment is applied. The numerical solution of the magnetic field to be created is calculated in advance for each of the plurality of lattice points set around the reference particle, and the correspondence between the vector amount of the magnetic moment applied to the reference particle and the vector amount of the numerical solution of the magnetic field A reference data holding unit for each lattice point, and a magnetic field calculation unit for calculating a magnetic field generated by each particle to which a magnetic moment is applied by the magnetic moment applying unit, using a correspondence relationship held in the reference data holding unit; A particle state calculation unit that numerically calculates a governing equation that governs the motion of each particle using a calculation result in the magnetic field calculation unit.

  Another aspect of the present invention is an analysis method. This method assigns a magnetic moment to the particles of the particle system defined in the virtual space, and the numerical solution of the magnetic field created by the reference particles to which the magnetic moment is applied is set for each of a plurality of lattice points set around the reference particles. The correspondence between the vector quantity of the magnetic moment applied to the reference particle and the vector quantity of the numerical solution of the magnetic field is held for each of the plurality of lattice points, and the correspondence relation calculated and held in advance is used. The magnetic field generated by each particle to which a magnetic moment is applied is calculated, and the governing equation governing the motion of each particle is numerically calculated using the calculation result of the magnetic field generated by each particle.

  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, a magnetic phenomenon can be suitably handled in a simulation.

It is a block diagram which shows the function and structure of the analyzer which concerns on embodiment. It is a data structure figure which shows an example of the particle | grain data holding part of FIG. It is a flowchart which shows an example of a series of processes in the analyzer of FIG. It is a figure which shows the calculation result corresponding to expansion | deployment to n = 1st order. It is a figure which shows the calculation result corresponding to expansion | deployment to n = 2nd order. It is a figure which shows the calculation result corresponding to expansion | deployment to n = 3rd order. It is a figure which shows the calculation result corresponding to expansion | deployment to n = 4th order. It is a figure which shows the relationship between the order of a spherical harmonic function, and a calculated value. It is a figure which shows a hysteresis curve. It is a figure which shows the analysis model of the conductor sphere comprised with a bead. It is a graph which shows the calculated value of the magnetic field which changes with time on the surface of a conductor sphere. It is a graph which shows the phase error and amplitude error of a calculation value. It is a graph which shows the calculated value of the magnetic field on the surface of a conductor sphere, an electric current, and an electromagnetic force. It is a figure which shows typically the near field and the far field set around the reference particle. It is a figure which shows typically the method of calculating the magnetic field of an observation point from the magnetic field of the some lattice point which adjoins. It is a figure which shows typically the lattice point required in order to calculate a magnetic field in the arbitrary coordinates of a near region. FIGS. 17A, 17 </ b> B, and 17 </ b> C are diagrams schematically illustrating an example of a method for executing the area of the cube surface. It is a figure which shows typically the method of performing the area part of a micro triangle. It is a block diagram which shows the function and structure of the analyzer which concerns on embodiment. It is a flowchart which shows an example of a series of processes in the analyzer of FIG. It is a flowchart which shows the detail of the process of S36 of FIG. It is the graph which compared the exact solution of the magnetic field by an area part, and the numerical solution of the magnetic field by a dipole approximation. FIGS. 23A and 23B are diagrams showing calculation results according to the present embodiment. It is a graph which shows the calculation result concerning this embodiment.

  Hereinafter, the same or equivalent components, members, and processes shown in the drawings are denoted by the same reference numerals, and repeated description is appropriately omitted.

(First embodiment)
Molecular dynamics method (MD method) and quantum molecular dynamics method (Quantum Molecular) are methods for exploring phenomena in general material science using computers based on classical mechanics and quantum mechanics. Dynamics Method) and simulation based on Renormalized Molecular Dynamics (hereinafter referred to as RMD method), which is an MD method developed to handle a macro-scale system, is known (for example, Patent Document 2). reference). To be precise, the MD method and the RMD method are kinetic methods (using statistical mechanics to calculate physical quantities), and the particle method is a method for discretizing differential equations that describe a continuum. However, the MD method and the RMD method are also called particle methods here.

  Since the particle method can handle not only a static phenomenon but also a dynamic phenomenon such as a flow, it is attracting attention as a simulation method replacing the above-mentioned finite element method which mainly analyzes a static phenomenon.

  The particle method has a differential view of obtaining a particle system to be analyzed by discretizing a continuum with particles. For example, when dealing with fluids in the particle method, the Navier-Stokes equation is often discretized with particles.

  On the other hand, as another view of the particle method, there is also an integral view that many particles are collected to form a continuum. This is, for example, the view that small iron grains are collected and hardened to form a large iron ball.

  In general, when a magnetic field at a certain point in a space where many magnetic moments exist is obtained, the magnetic fields generated by the magnetic moments at the point are added over the magnetic moments by the principle of superposition. The present inventor uniquely found the affinity between the method for obtaining the magnetic field from such a collection of magnetic moments and the particle method in the case of an integral view, and imparts a magnetic moment to the particles in the particle method. I came up with it. This makes it possible to extend the application range of the particle method to magnetic field analysis while maintaining the advantage of the particle method that is strong in convection and large deformation analysis.

  Furthermore, the present inventor considers the induced magnetization induced in each particle in addition to the interaction term between the magnetic moments applied to each particle in the particle method, thereby reducing the application range of the particle method to the electromagnetic induction phenomenon. It was thought that it can be extended to dynamic magnetic field analysis including. As a result, it is possible to realize a dynamic magnetic field analysis by the particle method for a physical phenomenon that involves a large deformation and requires consideration of electromagnetic induction, such as an interaction between a stator and a rotor in an induction motor.

  FIG. 1 is a block diagram illustrating functions and configurations of an analysis apparatus 100 according to an 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.

  In this embodiment, a case where a particle system is analyzed following the RMD method will be described. However, an MD method without renormalization, DEM (Distinct Element Method), SPH (Smoothed Particle Hydrodynamics), or MPS (Moving Particle Semi-implicit). It will be apparent to those skilled in the art who have touched the present specification that the technical idea according to the present embodiment can be applied to the case where a particle system is analyzed following other particle methods.

  The analysis device 100 is connected to the input device 102 and the display 104. The input device 102 may be a keyboard, a mouse, or the like for receiving user input related to processing executed by the analysis device 100. The input device 102 may be configured to receive input from a network such as the Internet or a recording medium such as a CD or a DVD.

  The analysis apparatus 100 includes a particle system acquisition unit 108, a magnetic moment application unit 110, a numerical calculation unit 120, a display control unit 118, and a particle data holding unit 114.

  The particle system acquisition unit 108 is based on input information acquired from the user via the input device 102, and is a particle system composed of N (N is a natural number) particles defined in one, two, or three-dimensional virtual space. Get the data. The particle system is a particle system that has been brought in using the RMD method.

  The particle system acquisition unit 108 arranges N particles in the virtual space based on the input information, and gives a speed to each of the arranged particles. The particle system acquisition unit 108 registers the particle ID that identifies the arranged particle, the position of the particle, and the velocity of the particle in the particle data holding unit 114 in association with each other.

  The magnetic moment imparting unit 110 imparts a magnetic moment to the particle-based particles acquired by the particle-based acquiring unit 108 based on input information acquired from the user via the input device 102. For example, the magnetic moment applying unit 110 requests the user to input the magnetic moment of the particle-based particles via the display 104. The magnetic moment applying unit 110 registers the input magnetic moment in the particle data holding unit 114 in association with the particle ID.

  In the following, a case will be described in which all particles of the particle system are set to be the same or equivalent, and the potential energy function is set to have two potentials and have the same shape regardless of the particles. However, it will be apparent to those skilled in the art who have touched this specification that the technical idea according to the present embodiment can be applied to other cases.

The numerical calculation unit 120 numerically calculates a governing equation that governs the motion of each particle in the particle system represented by the data held by the particle data holding unit 114. In particular, the numerical calculation unit 120 repeatedly performs calculations according to the discretized equation of motion of particles. The equation of motion of particles has a term that depends on the magnetic moment.
Numerical calculation unit 120 includes a magnetic field calculation unit 121, a force calculation unit 122, a particle state calculation unit 124, a state update unit 126, and an end condition determination unit 128.

  The magnetic field calculation unit 121 calculates the magnetic field generated by the particle system represented by the data held by the particle data holding unit 114. The magnetic field calculation unit 121 calculates a magnetic field based on the magnetic moment of each particle held by the particle data holding unit 114. A magnetic field is a magnetic physical quantity associated with a particle system. The magnetic field calculation unit 121 may calculate the magnetic flux density and magnetization instead of or in addition to the magnetic field.

  The force calculation unit 122 refers to the particle data held by the particle data holding unit 114, and calculates the force acting on the particles based on the distance between the particles for each particle in the particle system. The force acting on the particle includes a force based on an interaction between magnetic moments. The force calculation unit 122 determines, for the i-th particle (1 ≦ i ≦ N) in the particle system, a particle whose distance from the i-th particle is smaller than a predetermined cutoff distance (hereinafter referred to as a close particle). To do.

  For each neighboring particle, the force calculation unit 122 determines that the neighboring particle is the i-th particle based on the potential energy function between the neighboring particle and the i-th particle and the distance between the neighboring particle and the i-th particle. Calculate the force exerted on. In particular, the force calculator 122 calculates the force from the gradient value of the potential energy function at the distance value between the adjacent particle and the i-th particle. The force calculation unit 122 calculates the force acting on the i-th particle by adding the force exerted by the adjacent particle on the i-th particle for all the adjacent particles.

  The particle state calculation unit 124 refers to the particle system data stored in the particle data storage unit 114, and applies the force calculated by the force calculation unit 122 to the discrete particle motion equation for each particle in the particle system. To calculate at least one of the position and velocity of the particles. In the present embodiment, the particle state calculation unit 124 calculates both the position and velocity of the particle.

  The particle state calculation unit 124 calculates the velocity of the particle from the discretized particle motion equation including the force calculated by the force calculation unit 122. The particle state calculation unit 124 uses the predetermined minute time step Δt based on a predetermined numerical analysis method such as the jumping method or the Euler method for the i-th particle in the particle system to move the particle. By substituting the force calculated by the force calculation unit 122 into the equation, the velocity of the particle is calculated. For this calculation, the velocity of the particle calculated in the previous cycle of repetition is used.

  The particle state calculation unit 124 calculates the position of the particle based on the calculated particle velocity. The particle state calculation unit 124 calculates, for the i-th particle in the particle system, a relational expression between the position and velocity of the particle that is discretized using the time step Δt based on a predetermined numerical analysis technique. The particle position is calculated by applying the velocity. For this calculation, the position of the particle calculated in the previous cycle of repetition calculation is used.

  The state update unit 126 updates the position and velocity of each particle of the particle system held in the particle data holding unit 114 with the position and velocity calculated by the particle state calculation unit 124.

  The end condition determination unit 128 determines whether or not the repetitive calculation in the numerical value calculation unit 120 should be ended. Termination conditions for ending the repetitive calculation are, for example, that the repetitive calculation has been performed a predetermined number of times, that an end instruction has been received from the outside, or that the particle system has reached a steady state. When the end condition is satisfied, the end condition determination unit 128 ends the repetitive calculation in the numerical value calculation unit 120. The end condition determination unit 128 returns the process to the force calculation unit 122 when the end condition is not satisfied. Then, the force calculation unit 122 calculates the force again based on the particle position and speed updated by the state update unit 126.

  The display control unit 118 displays the state of time evolution of the particle system and the state at a certain time on the display 104 based on the position, velocity, and magnetic moment of each particle of the particle system represented by the data held in the particle data holding unit 114. Let This display may be performed in the form of a still image or a moving image.

  FIG. 2 is a data structure diagram illustrating an example of the particle data holding unit 114. The particle data holding unit 114 holds the particle ID, the position of the particle, the velocity of the particle, and the magnetic moment of the particle in association with each other.

  In the above-described embodiment, examples of the holding unit are a hard disk and a memory. Based on the description in this specification, each unit is realized by a CPU (not shown), an installed application program module, a system program module, a memory that temporarily stores the contents of data read from the hard disk, and the like. It is understood by those skilled in the art who have touched this specification that they can do this.

The operation of the analysis apparatus 100 configured as above will be described.
FIG. 3 is a flowchart illustrating an example of a series of processes in the analysis apparatus 100. The particle system acquisition unit 108 acquires a particle system that has been transferred in accordance with the RMD method (S12). The magnetic moment imparting unit 110 imparts a magnetic moment to the acquired particle-based particles (S14). The magnetic field calculation unit 121 calculates a magnetic field generated by the particle system (S16). The force calculator 122 calculates the force acting on the particles from the distance between the particles and the magnetic moment of each particle (S18). The particle state calculation unit 124 calculates the velocity and position of the particle from the equation of motion of the particle including the calculated force (S20). The state update unit 126 updates the position and speed of the particles held in the particle data holding unit 114 with the calculated position and speed (S22). The termination condition determination unit 128 determines whether the termination condition is satisfied (S24). If the end condition is not satisfied (N in S24), the process repeats steps S16 to S22. When the end condition is satisfied (Y in S24), the display control unit 118 outputs the calculation result (S26).

In general, the particle method can more appropriately handle dynamic analysis objects such as fluid, separation phenomenon, and cutting.
And according to the analysis apparatus 100 which concerns on this Embodiment, the analysis by a particle method and the magnetic field analysis based on a magnetic moment can be coupled. Therefore, it is possible to analyze a dynamic analysis target with higher accuracy and in a short time including magnetic properties such as a magnetic field.

  For example, when the object to be analyzed is iron sand, meshes cannot be suitably set by a mesh-based analysis method such as a conventional finite element method, so that it is difficult to simulate well. Therefore, according to the analysis apparatus 100 according to the present embodiment, the iron sand can be suitably analyzed by the particle method in a form that takes into account the magnetism of the iron sand.

  For example, when the analysis target is a motor, the rotating body rotates with respect to the fixed body, so that the mesh-based analysis method is not suitable and the static analysis is stopped. Therefore, according to the analysis apparatus 100 according to the present embodiment, the magnetic interaction between the rotating body and the fixed body can be dynamically handled while simulating the rotation of the rotating body with high accuracy by the particle method. Furthermore, even for an analysis object including an electromagnetic induction phenomenon, such as an induction motor, the time variation of the magnetic field can be taken into account by applying an induction magnetization that takes the electromagnetic induction phenomenon into account and a magnetic moment based on the induction magnetization. Can handle magnetic interactions.

  Hereinafter, the principle of the analysis method used in the analysis apparatus 100 will be described.

Magnetic field is a macro vector potential:
Is obtained as follows. The direction of spin is taken on the z axis, and a is the atomic radius. The magnetic field generated by one spin at lattice point j generates a circular current that generates spin S _jz (magnetic moment m _jz ).
age,
here,
Uniform magnetization when leaving only n = 0:
Is equivalent to the magnetic field created by the sphere.
When this method is applied to a true spherical bulk material, if only n = 0 is adopted, uniform magnetization cannot be obtained, and the same result as the magnetic moment method is obtained. It can be improved by adding higher order terms. The FEM becomes uniform.

Convert to Cartesian coordinate system (
)
Note that when sin θ _j = 0, B _θ = 0 to B _x = B _y = 0. Field when the magnetic moments _m x, _{m y} is x-axis, parallel respectively to the y-axis
Is obtained in the same way.

Magnetic field in
(Contribution from N spins) is
here,
Means all outside markets. By simultaneous with P-14 and P-15, the magnetic moment _{m p} is obtained. The magnetic field generated outside the magnetic body is P-15, and the magnetic field generated inside the magnetic body is
P-14 takes into account the demagnetizing field due to the magnetization generated on the sphere surface
Satisfied.

P-13 converges slowly because the matrix is not a diagonal (j = p is excluded). Therefore, a solution is obtained from the following equation of motion.
Here, m _v = 5 × 10 ⁻² dt ² is a virtual mass, and γ = m _v / (10 dt) is an attenuation coefficient.

If FIRE (see Non-Patent Document 1) is added, the speed can be further increased. The FIRE code is shown below. In addition, it is 10.05 [s] by calculation in the number of particles 802 and residual < ^10-8 . Without FIRE, the residual did not ^fall below ^10-5 .

The special cases of the Legendre function and the Double Legendre function are listed below. (X: = cos θ _j )

Below, the magnetization of a magnetic sphere (susceptibility 999) in a uniform external field with B _ox = 0.1 [T] in the x-axis direction
The result of calculating is shown. The number of particles (atoms) constituting the sphere was 4009 and arranged in an fcc structure. The exact solution is
It is. 4, 5, 6, and 7 show the results of the first to fourth orders of multipole expansion. Both cross sections passing through the center were displayed. It can be seen that increasing the order of the multipole expansion works well with the exact solution. If the order is low, the convergence is poor. Most of the calculation time is a spherical harmonic function call.

FIG. 4 is a diagram showing calculation results corresponding to expansions up to n = 1. For the exact solution μ _o M _x = 0.2991 [T], the calculated value is an average of 0.346231 [T]. It can be confirmed that the magnetization is parallel and uniform to the external field B _x = 0.1 [T]. The residual was less than 1e-13 and the calculation time was about 80 minutes.

FIG. 5 is a diagram showing a calculation result corresponding to the expansion up to n = 2 order. For the exact solution μ _o M _x = 0.2991 [T], the calculated value is an average of 0.321891 [T]. It can be confirmed that the magnetization is parallel and uniform to the external field B _x = 0.1 [T]. The residual was less than 1e-8 and the calculation time was about 3 hours. The residual did not fall below 1e-9.

FIG. 6 is a diagram showing calculation results corresponding to expansions up to n = third order. For the exact solution μ _o M _x = 0.2991 [T], the calculated value is an average of 0.312444 [T]. It can be confirmed that the magnetization is parallel and uniform to the external field B _x = 0.1 [T].

FIG. 7 is a diagram illustrating calculation results corresponding to expansions up to n = 4th order. For the exact solution μ _o M _x = 0.2991 [T], the calculated value is an average of 0.312222 [T]. It can be confirmed that the magnetization is parallel and uniform to the external field B _x = 0.1 [T]. After about 5 hours, the residual did not fall below 1e-9.

  FIG. 8 is a diagram showing the relationship between the order of the spherical harmonic function and the calculated value. It can be seen that increasing the order approaches the exact solution.

  The principle of the analysis method used in the analysis apparatus 100 can also be described as follows.

M _p is the mass of the nucleus, and _me is the mass of the electron. e is the charge (absolute value) of the nucleus and the electron. Eventually, the electric charge is replaced by electric current, and the mass of the electron is added to the mass of the nucleus to become the mass M of the atom, so that it does not appear positively. δV is the volume of the magnetic sphere, m _v is the virtual mass, and λ _i is the unsteady number of Lagrange. In order to satisfy the constraint strictly, a convergence calculation represented by the SHAKE method is necessary, but in this method, λ _i = 1 is assumed, and a minute vibration around the correct answer is treated as an error.

The direction of spin is taken on the z axis, and a is the atomic radius. The circular current that generates the spin S _jz (magnetic moment m _jz ) at the lattice point j is
So the magnetic field is a macro vector potential:
Is obtained as follows.
here,
Uniform magnetization when leaving only n = 0:
Is equivalent to the magnetic field created by the sphere.

Convert to Cartesian coordinate system (
If you use)
Note that when sin θ _j = 0, H _θ = 0 to H _x = H _y = 0. Field when the magnetic moments _m x, _{m y} is x-axis, parallel respectively to the y-axis
Is obtained in the same way.

Magnetic field felt by moments
(Contribution from N spins) is
Induction magnetic field
If the p point is inside the conductor,
If point p is outside the conductor,
C ₁ and C ₂ are, for example, in the formula shown in Non-Patent Document 2.
It is a compensation coefficient when compared with the limit to be, and takes values of C ₁ = 1/3 and C ₂ = 3/5.

Here, the first term (or Q-21-2) on the right side of Q-21 is a term representing the induced magnetization M _ant induced in each particle by an external magnetic field that changes over time. Further, the second term on the right side of Q-21 and the right side of Q-22 represent the magnetic field obtained by the interaction between magnetic moments _mant based on the induced magnetization of each particle represented by the formula Q-21-4. Term. By expressing the induced magnetic field in this way, it is possible to realize a dynamic magnetic field analysis that takes into account the influence of a time-varying external magnetic field.

At the time of retraction, the electrical conductivity σ is
Now, δ = 2.

Lagrange differentiation on the right side of Q-21-2
Is a particle method
Can be replaced. However, the translational movement of the magnetization is considered by the movement of the particles, but the rotation of the magnetization vector needs to be considered separately (Landau-Lifschitz, Electromagnetism, §51).

The magnetic field generated inside the magnetic material is
And the relationship between the total external field H 2 _O (x _p ) and the magnetization M (H) is

It is. Magnetic moment by combining Q-26 and Q-27
Is obtained. The magnetic field outside the magnetic body is
It becomes.

Non-linear magnetization and hysteresis Magnetic field from Q-27 and Q-24
Erase:
By solving Q-28
(A case of B = f (H) is shown in FIG. 9).

FIG. 9 is a diagram illustrating a hysteresis curve. M is obtained from the intersection of the hysteresis curve and B + 2 μ _o H = 3B _o . now
BH curve with permanent magnetization:
Is given.
Coalesced with
Get. The general B = f (H) is solved by the Newton-Laphson method.

Magnetic field particleization method Q-19 has a slow convergence because the matrix is not a super diagonal (j = p is excluded). Therefore, a solution is obtained from the following equation of motion.
Here, H 2 _O (x _p ) is represented by Q-26. M _v is the virtual mass. The recommended value is 250 × dt × dt. If m _v is sufficiently small, minute damped oscillation around the correct answer occurs, and the error can be reduced.

Consider a model:
The solution of forced oscillation with a damping term (Landau and Lifshitz, Mechanics, Sec. 26) is
Given that
It becomes. That is if you take sufficiently small m _v, the original equations to be solved
Matches the solution of

Discretization If discretization is performed according to the skipping method (speed is evaluated at the midpoint between n and n + 1), the number of convergence is written as s,
Here, in discretization of the induced magnetic field H _ind (x _i ),
Take it. At this time, if only the diagonal elements are used in the future, an explicit solution can be obtained, and the efficiency is improved by about 30%. Most of the calculations are special function calls.

Speeding up with FIRE: Effective only for static magnetic field analysis. Further speeding up is possible by adding FIRE (see Non-Patent Document 1). The FIRE code is shown below. In addition, it is 10.05 [s] by the calculation in the number of particles 802 and residual <10 < ^-8 >. Without FIRE, the residual did not ^fall below ^10-5 .

Accuracy Verification: Uniform Magnetization of Sphere FIG. 10 is a diagram showing an analysis model of a conductor sphere composed of beads. In the accuracy verification, a spatially uniform external magnetic field satisfying H _{ext, x} = H _O sin (2πft) was applied in the x-axis direction. Here, H 2 _O = 7.958 × 10 ⁴ [A / m], f = 200 [Hz], radius of the conductor sphere A = 10 [mm], electrical conductivity σ = 59 × 10 ⁶ [S / m] It was. The number of particles (atoms) constituting the conductor sphere was 6099, and the conductor sphere was arranged in an fcc structure.

  FIG. 11 is a graph showing the calculated value of the time-varying magnetic field on the surface of the conductor sphere, on the barycentric position (rx = 9.89 [mm], ry = rz = 0) of the particles located near the conductor surface. The time change of the magnetic field Hx is shown. In addition, in this figure, the value by an external magnetic field and exact solution is also shown with the calculation result by simulation. It was shown that the magnetic field inside the conductor changed with a phase lag with respect to the external magnetic field due to the influence of the induced magnetic field, and the tendency of the exact solution was in good agreement with the calculation result.

FIG. 12 is a graph showing the phase error and the amplitude error of the calculated values, and shows the phase error ε _phase and the amplitude error ε _amplitude of the magnetic field at the gravity center position of the particle located on the x-axis. The phase error ε _phase and the amplitude error ε _amplitide are
Defined. Here, θ _exact is the phase of the exact solution, and θ _MBM is the phase of the analysis result. _Hexact is the magnetic field obtained by exact solution, and _HMBM is the magnetic field of the analysis result. Incidentally phase _error, compared with the phase theta _{exact the H exact} becomes _maximum, and a phase theta _{MBM that H MBM} is maximized. From FIG. 12, the phase error of the magnetic field of the analysis result on the x-axis and the exact solution is 5.86% at the maximum and the amplitude error is 6.38%, and an analysis result with higher accuracy than the exact solution can be obtained. It has been shown.

Series display of Legendre function The special cases of Legendre function and Double Legendre function are listed below. (X: = cos θ _j )

Deriving the forces acting on particles Lagrangean, which describes the interaction between particles (electrons and nuclei) and particle magnetic fields,
It is. Vector potential
And the electromagnetic field is
It is.
To obtain the following equation of motion.
The equation of motion for the atom _{M = M} p + _{m e} is,
It becomes. Right side: Lorentz force average current density
The electrical moment
Replace with
From Maxwell's equation and Q-24
The force that finally acts on the atom ignores the electric polarization,
It becomes.

The force acting on the particles can also be obtained directly from Maxwell's stress tensor.
Moment created by current localized in a sphere
The force acting on is at the lowest order
It is. Therefore, the potential energy is combined with the potential energy by electric polarization,
However, it does not include eddy currents or true currents.

After all,
Can be obtained magnetic force acting on the magnetic sphere.

Magnetization current
And eddy current
Can't be completely carved out,

Derivative calculation Calculate the necessary differential coefficients below.
If only r _pj > a is enumerated from (
Is proportional to sin θ, so there is no divergence at θ = 0),

Calculation of the magnetizing current generated by the magnetic moment When magnetized parallel to the z axis Once again, write the magnetic field when magnetized to the z axis
Magnetizing current
The magnetic force generated by magnetization
Is
It is. First,
Ask for.

When magnetized parallel to the x axis Write the magnetic field when magnetized to the x axis:
Magnetization current
Ask for.

When magnetized parallel to the y-axis Write the magnetic field when magnetized to the y-axis:
Magnetization current
Ask for.

Calculation of the eddy current generated by the induced magnetic field
It becomes. Eddy current is
The above equation R-52-2 is the same as the case where the expansion order n = 0 of the spherical harmonic function in R-12.

Calculated m x of the force acting on the _{particle, m y,} be added the contribution by m _z and the eddy currents, the total current is obtained.
Coordinate
The force acting on the particles
Finally
It is necessary to add a term derived from the derivative of. This is a magnetic field error
It is the apparent force that arises from the movement and preserves the total energy of motion and magnetic field.

(Second Embodiment)
In the first embodiment described above, the induced magnetic field equation (Q-21) expressed by the induced magnetization shown in the first term on the right side of Q-21 and the interaction of the magnetic moment shown in the second term on the right side. ) Was used to perform the dynamic magnetic field analysis. In the present embodiment, instead of using the equation of Q-21, the induced magnetic field is formulated using the exact solution of the vector potential derived from the differential equation of the vector potential. Hereinafter, the difference from the first embodiment will be mainly described.

For the vector potential A, the following differential equation holds.
The solution of the above differential equation is
It becomes. here,
If divided into each particle, the vector potential at the position r _i of each particle is
It becomes. Here, v _j is the volume of each particle. It should be noted that the vector
If the crosses the particle surface, the sum is not taken. Moreover, the conditions that the particle diameter a should satisfy are:
It is.

At this time, by dividing each object into particles, the translational symmetry of the vector potential A may be lost. Therefore, the present inventor has found that the translational symmetry of the vector potential A can be recovered by introducing the gauge transformation shown in S-4 below. This gauge conversion is also referred to as “bead gauge” in this specification.
Here, the vector g _i is a local centroid vector relating to a particle group (number of particles: N _i ) to be calculated,
It is represented by Thus, the vector g _i represents the position of the center of gravity of the particles. Also, beads gauge shown in S-4 is said to be described by using a cross product of the magnetic flux density B _j and a local centroid vector g _i.

For the above bead gauge, the relational expression between magnetic flux density B and vector potential A
If you ask for
It becomes. Therefore, it is understood that the number of particles N _i is if sufficiently large, translational symmetry is satisfied.

Here, the time differentiation of the vector potential contained in the right side of S-3 from the gauge transformation shown in S-4
Can be expressed as follows using the time derivative of the magnetic flux density B.
Furthermore, the vector d _ji is
Then the time derivative of the vector potential is
It is expressed. Therefore, if S-9 is substituted for S-3 and integration is performed, the exact solution of the vector potential A can be described using the time derivative of the magnetic flux density B.

In order to derive an exact solution of the vector potential A, first, it is a time derivative of the z component of the magnetic flux density B.
Perform integration when only works. Thus, the vector potential A is
It becomes. It is a time derivative of the x component and the y component of the magnetic flux density B.
If you add the contribution of
Is obtained.

The induction magnetic field H describing the inside of the conductor to be analyzed is
From the relational expression of
It becomes. here,
With the induction magnetic field H as
Is obtained.

  The magnetic field calculation unit 121 calculates the induction magnetic field generated by the particle system using the expression of the induction magnetic field shown in S-13. The calculation for obtaining the time change value of the induced magnetic field may be performed using the same method as that shown in the first embodiment. The magnetic field calculation unit 121 may calculate the current density from the obtained induced magnetic field. Moreover, the force calculation part 122 may calculate the electromagnetic force which acts on each particle | grain from the value of the obtained magnetic field and electric current.

In this embodiment, the local centroid vector g _i shown in S-5 is introduced to the particle group to be the same potential, for multiple particle groups electrically insulated from one another, the separate A local centroid vector is introduced. For example, when the particle system to be analyzed has a plurality of portions that are electrically insulated from each other, local centroid vectors having different centroid positions are applied to the particle groups constituting each portion. In addition, when the object to be analyzed deforms over time and splits into a plurality of objects, and the split parts are insulated from each other, the center of gravity position is set to the particle group constituting each part. Apply different local centroid vectors.

Therefore, when a plurality of particle groups including the first particle group and the second particle group that are electrically insulated from each other are arranged in the particle system to be analyzed, the magnetic field calculation unit 121 corresponds to each particle group. Analyze using the induction magnetic field formula. For example, the induced magnetic field in the first particle group is calculated using the first induced magnetic field equation derived by gauge conversion using the first local centroid vector representing the centroid position of the first particle group. The induced magnetic field in the second particle group is calculated using the second induced magnetic field expression derived by gauge conversion using the second local centroid vector representing the centroid position of the second particle group. More specifically, in the induction magnetic field equation shown in S-13, _dij included in the first term on the right side takes a different value in the calculation for each particle group.

The accuracy of the analysis method using the induction magnetic field formula (S-13) according to the present embodiment was verified. Similar to the first embodiment, a spatially uniform external magnetic field of H _{ext, x} = H _O sin (2πft) is applied in the _x -axis direction using the conductor sphere analysis model shown in FIG. The calculation conditions are: H 2 _O = 7.948 × 10 ⁴ [A / m], f = 100 [Hz], radius of the conductor sphere A = 10 [mm], electric conductivity σ = 59 × 10 ⁶ [S / m ]. The number of particles (atoms) constituting the conductor sphere was 6099, and the conductor sphere was arranged in an fcc structure.

  FIG. 13 is a graph showing calculated values of the magnetic field, current density, and electromagnetic force on the surface of the conductor sphere. This figure shows the time change of (a) magnetic field Hx, (b) current density jz, and (c) electromagnetic force Fy at the position of the center of gravity of particles located near the conductor surface. In the figure, the exact solution value is indicated by a solid line, and the calculated value is indicated by a broken line. As shown in the figure, it has been found that when the analysis method according to the present embodiment is used, the calculation result and the exact solution agree well.

(Third embodiment)
In the above-described first embodiment, an object is divided into particles, and an exact solution based on the spherical symmetry of each particle is used to obtain a calculation result with high speed and excellent accuracy. However, when an object that is a continuum is divided by particles, gaps are generated between the particles, and calculation errors due to the gaps are inevitably generated. In particular, when calculating the magnetic field near the particle when the distance between the particles is short, the influence of the error becomes large. In order to increase the calculation accuracy, it is necessary to divide the object into voxels such as a cube and a Voronoi polyhedron, and to divide the object without gaps by these voxels. In this case, since it is necessary to execute the area based on the exact solution according to the shape of the voxel, the calculation time becomes enormous.

  In the present embodiment, an exact solution corresponding to the voxel shape is used when calculating the magnetic field in the vicinity of the particle, and dipole approximation is used when calculating the magnetic field far from the particle where the error due to the approximation can be almost ignored. At this time, instead of executing the exact solution calculation according to the voxel shape each time, the correspondence between the vector amount of the magnetic moment given to the voxel in advance and the vector amount of the magnetic field created by the voxel is based on the exact solution. The magnetic field in the vicinity of each particle is calculated by calculating and referring to the correspondence. Therefore, according to the present embodiment, it is possible to calculate the magnetic field with high accuracy and at high speed using the correspondence relationship based on the exact solution calculated in advance while avoiding the calculation for each area with a high load. Hereinafter, the present embodiment will be described focusing on differences from the above-described embodiment.

Field observation point r _i defined in the virtual space, a magnetic field 'magnetization M (r' of) make each minute volume dv obtained by integrating over all space, the table by the following formula (T-1) Is done. Formula (T-1) can be transformed into Formula (T-2) by dividing it into a volume term and an area term. V is a region volume, S is a surface area, and n _S is a normal vector of the surface.

In the magnetic bead method, the target space is divided into N particles (beads), the magnetization vector M _j inside each bead is constant, and the following formula (T-1) obtained by dipole approximation is obtained ( The magnetic field vector H (r _i ) at the coordinate r _i is calculated by T-3). Here, r _ij = r _i −r _j , n _ij = r _ij / | r _ij |, ΔV _j is the bead volume, and γ _j is the bead filling rate.

On the other hand, in the magnetic moment method, the object is divided into N elements (voxels), the magnetization vector M _j inside the element is constant, and the following expression (T−) obtained by modifying Expression (T-2) 4) Calculate the magnetic field vector H (r _i ) at the coordinate r _i . Here, ds _j means a minute surface of each element, and if the voxel is a regular hexahedron (cube), it is necessary to execute the area for six surfaces.

  Comparing the magnetic bead method and the magnetic moment method, the magnetic moment method is superior in calculation accuracy, and the magnetic bead method is excellent in calculation speed. In the magnetic bead method, the continuum is divided into N spherical particles, and the magnetic field generated by each particle is obtained by dipole approximation, so the calculation accuracy of the magnetic field in the vicinity of each particle is low. On the other hand, for the magnetic field in a distant region away from each particle, the error due to the dipole approximation can be ignored, so the calculation accuracy is high. In the magnetic moment method, the continuum is divided into polyhedral elements so that no gap is generated between the elements, so that there is no approximation error that occurs in the magnetic bead method, and the calculation accuracy is high.

  In the magnetic moment method, it is necessary to execute the area shown in the above formula (T-4) with high accuracy, and calculation with a numerical integration method or an exact solution is essential. At this time, the area needs to be executed by the number of times n (n = 6 in the case of a cube) that the polyhedral element has, and the calculation time becomes enormous as the number of integration increases. On the other hand, in the magnetic bead method, if an exact solution based on the spherical shape of the bead is used, the area is not included as shown in the above formula (T-3), so that the calculation time can be reduced.

Therefore, in this embodiment, in order to achieve both the calculation accuracy and the calculation speed, the calculation of the magnetic field in the far region where the dipole approximation is established uses the formula (T-3) based on the magnetic bead method, In the calculation of the magnetic field in the vicinity region where the child approximation is not established, the formula (T-4) based on the magnetic moment method is used. Further, in order to speed up the calculation processing based on the equation (T-4), the calculation result for the area is calculated in advance, and the correspondence between the magnetization vector M _j and the magnetic field H _{j formed} around the magnetization vector M _j Prepare a magnetic field table that defines the relationship. In the calculation processing at the time of executing the simulation, the magnetic field is calculated by referring to a magnetic field table prepared in advance instead of executing the area for each time.

FIG. 14 is a diagram schematically showing a near region C _A and a far region C _B set around the reference particle V _j . FIG. 14 shows a case where the object is divided by cubic elements (voxels), and lattice points _Gk are set at the vertices of each element. The length of one side of each cubic element is twice the particle radius a (2a). An orthogonal coordinate system (ξ, η, ζ) is set around the reference particle (reference element) V _j, and the coordinates r _{j of} each element are set at the center of the cube. Around the reference particles V _j determined that virtual sphere of radius a _s, the interior of the sphere becomes neighboring region C _A, outside the far region C _B. The radius a _s of the sphere that defines the neighborhood region C _A is determined according to the distance at which the dipole approximation is established. For example, the radius is 4 times or more of the particle diameter a, preferably 6 times or more of the particle diameter a. It is set as a _s. In the illustrated example, 6 times the radius _{a s} a particle diameter a sphere _(i.e., a s = 6a) is set to.

In FIG. 14, the magnetic field H _B (r _i ) at an arbitrary coordinate r _i in the far region C _B among the magnetic fields _generated by the reference particle V _j around is calculated based on the equation (T-3). On the other hand, among the magnetic fields _generated by the reference particle V _j around, the magnetic field H _A (r _i ) at an arbitrary coordinate r _i inside the neighboring region C _A is a plurality of lattice points G _{1 to} G that are close to the coordinate r _{i. 4} (actually, it is calculated by averaging the magnetic field of 8 lattice points shown in FIG. 15 described later). The magnetic field at each lattice point is calculated based on the equation (T-4), and is calculated using the value of the magnetic field table calculated in advance for each lattice point.

FIG. 15 is a diagram schematically illustrating a method of calculating the magnetic field _HA at the observation point r _i from the magnetic fields H _{1 to} H _{8 at} a plurality of adjacent lattice points. The magnetic field _{H A} in a neighborhood area _{C A,} the magnetic field _{H A} operation subject to the coordinates of (ξ, η, ζ) averaging the magnetic field _H k (k = _1~8) at eight grid points surrounding the And is represented by the following formula (T-5). Note that N _k (ξ, η, ζ) is expressed by Expression (T-6).

Here, the magnetic field H _k that the reference particle V _{j creates} at each lattice point G _k (position r _i ) can be expressed by the following formula (T-7) from the formula (T-4).

The magnetic field H _k represented by the equation (T-7) is an exact solution using the area. Here, if the equation (T-7) is modified and can be expressed as the following equation (T-8) using a tensor T _k that defines the correspondence between the magnetic field H _k and the magnetization vector M _j , any magnetization can be obtained. It becomes possible to easily calculate the magnetic field H _k that the particle having the vector M _{j creates} at each lattice point G _k . From equation (T-7), each component of the tensor _{T k,} can be represented by the following formula (T-9). _{Incidentally,} n _x, n y, _{n z,} respectively the x-direction, y-direction, the unit vector in the _z-direction, x _ij, y _ij, each _{z ij} is the reference particle _{V j} to the observation point (grid point) _These are the x component, y component, and z component of the distance r _ij .

Therefore, in the present embodiment, a numerical solution of each component of the tensor T _k represented by Expression (T-9) is calculated in advance, and the magnetic field H _{k at} each lattice point is calculated from the product of the numerical solution and the magnetization vector M _j. It calculates, further, to be able to calculate the magnetic field H _a in a neighborhood area C _a. In other words, it calculates the magnetic field _{H A} in a neighborhood area _{C A} using the following equation (T-10). If the numerical values of the respective components of the tensor T _k are obtained in advance, the area of the tensor Tk is not included in the calculation of the equation (T-10), so the calculation is highly accurate in a much shorter time than when the area is executed each time. The result can be obtained.

Figure 16 is a diagram schematically showing the grid points G _k which is necessary for calculating the magnetic field H _A in any coordinate r _i proximity area C _A. To be able to calculate the magnetic field H _A in all coordinates r _i of the internal sphere of the virtual radius a _s is in the range of voxels that can surround a neighboring region C _A (within the heavy line) The tensor T _k that defines the correspondence between the magnetic field _HA and the magnetization vector M _j may be calculated for all the lattice points G _{k in} FIG. For example, if the range defining the neighborhood region C _A is 6 times the bead diameter a, the number of grid points that need to be pre-calculated is less than 500. This number is less than the total number of particles contained in the particle system defined in the virtual space. Therefore, it is possible to derive the tensor T _k that defines the magnetic correspondence in a shorter time than calculating the magnetic field generated by all particles once using the area. That is, the calculation load for calculating the magnetic table is not so high, and the ratio of the calculation time of the magnetic table to the total calculation time is small.

Next, an example of a specific method for calculating the tensor T _k that defines the magnetic correspondence will be described. The calculation method of the numerical solution of each component of the tensor T _k is not particularly limited as long as the calculation of Expression (T-9) can be performed. For example, the method disclosed in Non-Patent Document 3 can be used. FIGS. 17A, 17 </ b> B, and 17 </ b> C are diagrams schematically illustrating an example of a method for executing the area of the cube surface. First, take out one of the six faces that are the target of the area, and consider a small triangle that is obtained by dividing the square that was taken out into two. In this way, the six surfaces are divided into 12 minute triangles, and the area is executed for each minute triangle.

FIG. 18 is a diagram schematically illustrating a method of executing the area of a minute triangle. In the local coordinate system (x ′, y ′, z ′) shown in FIG. 18, the minute triangle Δ123 that is the target for the area is arranged in the x′y ′ plane, and the magnetic field ΔH _k created by the minute triangle Δ123. The coordinate transformation is performed so that the coordinates r _{i of} are arranged on the z ′ axis. At this time, if the magnetic field ΔH _k created by the minute triangle Δ123 is expressed as ΔH _k = ΔT _k M _j , the tensor ΔT _k resulting from the minute triangle Δ123 is expressed by the following formula (T-11). Each variable included in Expression (T-11) is defined by Expression (T-12).

Thus, by calculating and adding up the tensors ΔT _k resulting from the minute triangle Δ123 for all the surfaces of the reference particle V _j , it is possible to derive the tensors T _k that define the magnetic correspondence. The tensor T _k that defines the magnetic correspondence is calculated for all lattice points G _k that are necessary to calculate the magnetic field H _A at an arbitrary coordinate r _i of the neighboring region C _A , and the numerical solution is used as a magnetic table. Retained.

FIG. 19 is a block diagram illustrating functions and configurations of the analysis apparatus 200 according to the embodiment. The analysis apparatus 200 according to the present embodiment is different from the analysis apparatus 100 shown in FIG. 1 in that it further includes a reference data holding unit 230. The reference data holding unit 230 holds a correspondence relationship between the vector amount of the magnetic moment applied to the reference particle and the vector amount of the exact solution of the magnetic field generated by the reference particle. That is, the reference data holding unit 230 holds a magnetic table in which a numerical solution of a tensor T _k that defines a magnetic correspondence is determined for each lattice point G _k .

The magnetic field calculation unit 121 executes calculation processing for deriving the above magnetic table according to the particle system defined by the particle system acquisition unit 108 and the magnetic moment application unit 110. Field calculation unit 121 sets the neighboring region C _A in accordance with the definition of the particle system, identifying a plurality of grid points G _k the calculated required tensor T _k defining a magnetic relationship. Field arithmetic unit 121 performs operations based particles V _j arranged at the origin is related to the magnetic field H _k to make the positions of the grid points G _k, showing the placement of the reference particles V _j and each grid point G _k The numerical solution of the tensor T _k is stored in the reference data holding unit 230 together with the value.

When calculating the magnetic field H _j (r _i ) that the particle j creates at the coordinate r _i , the magnetic field calculation unit 121 calculates the magnetic field with reference to the magnetic table if the coordinate r _i is in the vicinity region of the particle j, If the coordinate r _i is a far region (outside the near region) of the particle j, the magnetic field is calculated using an equation based on dipole approximation. When the distance between the particle j and the coordinate r _i exceeds a predetermined value, that is, the particle j at a position far away from the coordinate r _i is ignored without being subject to the calculation of the magnetic field. Good. For example, a magnetic field calculation by dipole approximation may be performed for an intermediate region close to a nearby region among distant regions, and a magnetic field calculation may not be performed for a region farther than the intermediate region. The range of the intermediate region is, for example, about 10 times the particle diameter a.

When calculating the magnetic field with reference to the magnetic table, the magnetic field calculation unit 121 calculates the magnetic field by converting the position of the coordinate r _{i into} the orthogonal coordinate system (ξ, η, ζ) shown in FIG. calculates the magnetic field of the coordinate r _i by returning to the coordinate system. At this time, the coordinate transformation between the coordinate system of the particle system and the orthogonal coordinate system (ξ, η, ζ) of the magnetic table only needs to be translated, so the processing load related to the coordinate transformation can be reduced. In the modification, coordinate conversion processing such as enlargement / reduction or rotation may be performed during coordinate conversion.

  FIG. 20 is a flowchart illustrating an example of a series of processes in the analysis apparatus 200. The particle system acquisition unit 108 acquires a particle system that has been transferred according to the RMD method (S30). The magnetic moment applying unit 110 applies a magnetic moment to the acquired particle-based particles (S32). The magnetic field calculation unit 121 creates a magnetic table according to the defined particle system and stores the magnetic table in the reference data holding unit 230 (S34). The magnetic field calculation unit 121 calculates a magnetic field generated by the particle system (S36). The force calculator 122 calculates the force acting on the particle from the distance between the particles and the magnetic moment of each particle (S38). The particle state calculation unit 124 calculates the velocity and position of the particle from the equation of motion of the particle including the calculated force (S40). The state update unit 126 updates the position and speed of the particles held in the particle data holding unit 114 with the calculated position and speed (S42). The end condition determination unit 128 determines whether or not the end condition is satisfied (S44). If the end condition is not satisfied (N in S44), the steps S36 to S42 are repeated. When the end condition is satisfied (Y in S44), the display control unit 118 outputs the calculation result (S46).

FIG. 21 is a flowchart showing details of the process of S36 of FIG. The magnetic field calculation unit 121 sets an initial value i = 1 for the variable i (S50). Field calculation unit 121, neighboring particles with reference to the magnetic table held in the reference data holding unit 230 calculates the magnetic field generated by the coordinate r _i (S52), distal particles using a dipole approximation equation coordinates r _i The magnetic field to be generated is calculated (S54). The magnetic field calculation unit 121 adds the calculated magnetic fields, and calculates the magnetization vector M _i (magnetic moment) of the particle at the coordinate r _i using the relational expression of M _i = χH _i (S56). 1 is added to the variable i (S58). If the variable i is equal to or less than the number of particles N (N in S60), the steps S52 to S58 are repeated. If the variable i is greater than the number of particles N (Y in S60), S36 Terminate the process.

FIG. 22 is a graph comparing an exact magnetic field solution B1 by area and a magnetic field solution B2 by dipole approximation. In FIG. 22, the horizontal axis represents a value obtained by normalizing the distance r _ij between the source particle j and the observation point i by the particle diameter a, and the vertical axis represents the value of the magnetic flux density Bx in the x-axis direction. As shown in the figure, when the distance r _ij between the particle j and the observation point i is doubled, the difference between the exact solution B1 (solid line) and the approximate solution B2 (broken line) increases, and the difference increases as the distance r _ij increases. It can be seen that the difference hardly occurs when the distance r _ij becomes 6 times or more of the particle diameter a. Therefore, it can be said that it is sufficient to set the range of about 6 times the particle diameter a as neighboring region C _A, error almost negligible by dipole approximation for a range of more than 6 times.

  FIGS. 23A and 23B are diagrams showing calculation results according to the present embodiment. This figure shows a calculation example of the magnetization phenomenon of a rectangular parallelepiped magnetic body in which 10 mm is set in the x direction and 5 mm is set in the y direction and the z direction. The external magnetic field was constant 0.1 [T], and the magnetization characteristics were linear (χ = 999). FIG. 23A is a comparative example, and shows the result of calculating the magnetic field by executing the area based on the above equation (T-4) for all particles. FIG. 23B shows the result of calculating the magnetic field using the magnetic table according to the present embodiment. When both were compared, it was found that the calculation results agreed well. In the example of FIG. 23A, the calculation time per step is 2.15 seconds, whereas in the example of FIG. 23B, the calculation time per step is 0.03 seconds. Therefore, by the method according to the present embodiment, the calculation speed can be increased by about 70 times while maintaining high calculation accuracy.

  FIG. 24 shows the calculation result according to the present embodiment, and shows the absolute value | B | of the magnetic flux density distribution on the y-axis in the calculation example shown in FIG. A magnetic flux density distribution B3 indicated by a solid line corresponds to the calculation example according to the embodiment of FIG. 23B, and a magnetic flux density distribution B4 indicated by a broken line corresponds to the calculation example according to the comparative example of FIG. As shown in the figure, it can be seen that the calculation results of both agree well.

  According to the present embodiment, it is possible to increase the calculation speed while maintaining high calculation accuracy by making good use of the feature of the magnetic bead method that the voxel shape does not change in the calculation process. In the finite element method, the voxel shape changes due to the change of the node position, and the shape of the surface element on the surface of the voxel changes. Therefore, it is necessary to execute an area corresponding to the changed shape of the surface element. On the other hand, in the magnetic bead method, since the shape of the voxel does not change over the calculation process, the result for the area using the common reference particles can be used. Thus, in the present embodiment, the calculation speed can be improved by preparing in advance a magnetic table based on common reference particles by using the characteristics of the magnetic bead method.

In the modification, a plurality of types of particles having different voxel shapes may be defined in the particle system. In this case, a magnetic table is prepared in advance for each of a plurality of types of reference particles having different voxel shapes, and different types of magnetic tables are prepared according to the voxel shape of the particle j that creates a magnetic field at the position of an arbitrary observation point r _i. The magnetic field may be calculated by referring to it. Thereby, even if it is a case where multiple types of voxel shapes are utilized, there can exist an effect similar to the above-mentioned embodiment. The voxel shape may be any shape as long as it is a polyhedron, and may be a regular tetrahedron, a regular octahedron, a regular dodecahedron, or the like. Further, it may be a Voronoi polyhedron obtained by Voronoi division of the target space.

In the above-described embodiment, the case where the mathematical expression used for the near particle and the far particle is switched is shown. In a modified example, the magnetic field calculation may not be performed for particles that are far away (for example, particles that are more than 10 times the particle diameter a), and all the particles that are magnetic field calculations may be calculated with reference to the magnetic table. That is, the magnetic correspondence T _k may be calculated for a range where the particle diameter a is 10 times, and the magnetic field calculation may be performed based on the correspondence T _k .

In the above-described embodiment, the case where the tensor T _k is held in the magnetic table has been shown. However, other types of numerical solutions can be used as long as the form is necessary for executing the calculation represented by the equation (T-10). It may be held as a magnetic table. For example, in a modified example, a numerical solution of a tensor S _k described as H _A (ξ, η, ζ) = S _k (ξ, η, ζ) M _j may be held as a magnetic table.

  The configuration and operation of the analysis apparatus according to the embodiment have been described above. These embodiments are exemplifications, 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 embodiment, the case where both the position and the speed of the particle are calculated in the numerical value calculation unit 120 has been described. However, the present invention is not limited to this. For example, there is a numerical analysis method such as the Verlet method in which the particle position is directly calculated from the force acting on the particle when calculating the particle position, and the velocity of the particle does not have to be calculated explicitly. The technical idea according to the present embodiment may be applied to such a method.

  DESCRIPTION OF SYMBOLS 200 Analysis apparatus, 102 Input device, 104 Display, 108 Particle system acquisition part, 110 Magnetic moment provision part, 114 Particle data holding part, 118 Display control part, 120 Numerical calculation part, 121 Magnetic field calculation part, 122 Force calculation part, 124 Particle state calculation unit, 126 state update unit, 128 end condition determination unit, 230 reference data holding unit.

Claims (9)

A magnetic moment applying unit for applying a magnetic moment to particles of a particle system defined in a virtual space;
A numerical solution of a magnetic field created by a reference particle to which a magnetic moment is applied is calculated in advance for each of a plurality of lattice points set around the reference particle, and a vector amount of the magnetic moment applied to the reference particle and the magnetic field A reference data holding unit that holds the correspondence with the vector quantity of the numerical solution for each of the plurality of grid points;
A magnetic field calculation unit that calculates a magnetic field generated by each particle to which a magnetic moment is applied by the magnetic moment applying unit, using the correspondence relationship held in the reference data holding unit;
An analysis apparatus comprising: a particle state calculation unit that numerically calculates a governing equation governing the motion of each particle using a calculation result in the magnetic field calculation unit.   The reference data holding unit holds a correspondence relationship between a vector amount of a magnetic moment applied to the reference particle and a vector amount of an exact solution of a magnetic field generated by the reference particle for each of the plurality of lattice points. The analysis device according to claim 1.   The reference data holding unit holds the correspondences at a plurality of lattice points set in the vicinity of the reference particle, and the number of lattice points is smaller than the number of particles to which a magnetic moment is applied by the magnetic moment applying unit. The analysis apparatus according to claim 1, wherein:   The magnetic field calculation unit uses the correspondence relationship in which the magnetic field in the vicinity region of each particle among the magnetic fields generated by each particle to which the magnetic moment is applied by the magnetic moment applying unit is held in the reference data holding unit. The analyzer according to any one of claims 1 to 3, wherein the analyzer calculates and calculates a magnetic field outside a region near each particle by approximating the magnetic moment of each particle to a dipole.   The analysis apparatus according to claim 4, wherein the neighboring region is set such that a distance from the center of each particle is in a range larger than twice the particle diameter and smaller than ten times.   When the magnetic field calculation unit calculates the magnetic field of the observation point in the neighboring region of each particle, the magnetic field calculation unit calculates the magnetic field of a plurality of lattice points close to the observation point from the correspondence relationship, and 6. The analysis apparatus according to claim 4, wherein the magnetic field at the observation point is calculated from the magnetic field calculated for the lattice point. The magnetic moment imparting unit imparts a magnetic moment to a plurality of types of particles having different voxel shapes defined in the particle system,
The reference data holding unit holds a plurality of types of correspondences calculated in advance for a plurality of types of reference particles having different voxel shapes,
The magnetic field calculation unit calculates a magnetic field generated by each particle using the correspondence relationship according to the voxel shape of each particle to which a magnetic moment is applied by the magnetic moment applying unit. The analysis device according to any one of 1 to 6. Giving a magnetic moment to particles of a particle system defined in virtual space,
A numerical solution of a magnetic field created by a reference particle to which a magnetic moment is applied is calculated in advance for each of a plurality of lattice points set around the reference particle, and a vector amount of the magnetic moment applied to the reference particle and the magnetic field Holding the correspondence with the vector quantity of the numerical solution for each of the plurality of grid points;
Using the correspondence relationship calculated and held in advance, calculate the magnetic field created by each particle to which a magnetic moment is applied,
An analysis method characterized by numerically calculating a governing equation governing the motion of each particle, using a calculation result of a magnetic field generated by each particle. A function of giving a magnetic moment to particles of a particle system defined in a virtual space;
A numerical solution of a magnetic field created by a reference particle to which a magnetic moment is applied is calculated in advance for each of a plurality of lattice points set around the reference particle, and a vector amount of the magnetic moment applied to the reference particle and the magnetic field A function of maintaining the correspondence with the vector amount of the numerical solution for each of the plurality of grid points;
A function of calculating a magnetic field created by each particle to which a magnetic moment is applied, using the correspondence relationship calculated and held in advance;
A computer program for causing a computer to realize a function for numerically calculating a governing equation governing the motion of each particle using a calculation result of a magnetic field generated by each particle.