JP2010160782A - Magnetic field analysis device and magnetic field analysis method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a magnetic field analysis device for accurately and efficiently performing magnetic field analysis with respect to a system with a magnetic body. <P>SOLUTION: The magnetic field analysis device 1 includes a storage means for storing an analysis condition and a motion equation of a magnetic field and a calculation means for calculating the initial values and constant numbers of the coefficient and variable number of the motion equation of the magnetic field concerning particles constituting the magnetic body based on the analysis condition, calculating the solution of the motion equation of the magnetic field, and calculating the magnetic field at an arbitrary point in a space based on the solution of the motion equation of the magnetic field. The magnetic field analysis device 1 calculates the arbitrary magnetic field in the space through the use of the calculation means based on the motion equation of the magnetic field and input information. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a magnetic field analysis apparatus and a magnetic field analysis method.

  In recent years, electrical devices such as motors that use magnetic energy as a drive source have been required to have high precision and high efficiency, and at the same time, have been required to increase development speed.

  In order to satisfy such requirements, verification by magnetic field analysis of electrical equipment is indispensable.

  Conventionally, the finite element method has been widely used as a magnetic field analysis method.

  For example, Patent Document 1 describes an analysis method using a finite element method as an example of magnetic field analysis when a magnetic core is present in air.

Japanese Patent Laid-Open No. 10-188042

  However, when the magnetic field analysis is performed by the finite element method, it is necessary to divide the entire space including the target electric device into meshes.

  Therefore, when analyzing large-scale, complex-shaped electrical equipment or dealing with high aspect ratio problems such as laminated iron cores, the analysis requires a tremendous amount of time and memory of the analytical equipment, and performs magnetic field analysis efficiently with sufficient accuracy. There was a problem that it was difficult.

  The present invention has been made in view of such problems, and an object thereof is to provide a magnetic field analysis apparatus capable of performing magnetic field analysis with sufficient accuracy and efficiency for a system having a magnetic material. .

  In order to achieve the above-mentioned object, the present inventor, as a result of intensive studies, uses an information processing device to calculate the solution of the motion equation of the magnetic field of the particles constituting the magnetic material, and based on the solution of the motion equation of the magnetic field. It has been found that by calculating the magnetic field at any place in the space where the magnetic material is arranged, the magnetic field analysis can be efficiently performed with sufficient accuracy for the system having the magnetic material, and the present invention has been achieved. .

  That is, the first invention is a magnetic field analysis device for analyzing a magnetic field at an arbitrary point in a space in which a magnetic material is arranged by using an information processing device, and storage means for storing analysis conditions and a motion equation of the magnetic field; From the analysis conditions, the coefficient of the motion equation of the magnetic field of the particles constituting the magnetic body, the initial value and the constant of the variable are calculated, the solution of the motion equation of the magnetic field is calculated, and based on the solution, A magnetic field analysis apparatus comprising: a calculation means for calculating a magnetic field at an arbitrary point.

  The second invention is a program for causing a computer to function as the magnetic field analysis apparatus described in the first invention.

  A third invention is a magnetic field analysis method for analyzing a magnetic field at an arbitrary point in a space in which a magnetic body is arranged using an information processing apparatus, the step (a) storing an analysis condition and a motion equation of the magnetic field; Based on the analysis conditions, the coefficient of the magnetic field equation of motion of the particles constituting the magnetic body, the initial value of the variable and the constant are calculated, the solution of the magnetic field equation of motion is calculated, and the solution of the magnetic field equation of motion is calculated. And (b) calculating a magnetic field at an arbitrary point in the space.

  According to the present invention, it is possible to provide a magnetic field analysis apparatus capable of performing a magnetic field analysis with sufficient accuracy without the need to mesh the entire space including the analysis target for a system having a magnetic body. .

  Further, if the solution of the motion equation of the magnetic field derived from the Lagrangian of the particles constituting the magnetic material is calculated, the total energy of the system is preserved when the external input does not change.

2 is a diagram illustrating a hardware configuration of the magnetic field analysis apparatus 1. FIG. 3 is a diagram showing a storage device 5. FIG. 2 is a diagram showing an outline of the configuration of an SPM motor 31. FIG. FIG. 4 is an enlarged perspective view around one of the stator teeth 43 of FIG. 3. 4 is a flowchart showing a procedure for analyzing a magnetic field using the magnetic field analysis apparatus 1. FIG. 6A is an enlarged perspective view of the coil 45, and FIG. 6B is a diagram showing an example in which the coil 45 is divided into local conductors. It is a figure which shows the relationship between the rectangular parallelepiped conductor 45a and a local coordinate. It is a figure which shows the relationship between the arc-shaped columnar conductor 45c and a local coordinate. It is explanatory drawing of the element used for the magnetic field calculation of a magnetic body.

  DESCRIPTION OF EMBODIMENTS Hereinafter, preferred embodiments of the present invention will be described in detail based on the drawings.

  First, a hardware configuration of the magnetic field analysis apparatus 1 according to the present embodiment will be described with reference to FIG.

  It should be noted that the hardware configuration in FIG. 1 is an example, and is not limited to this.

  As shown in FIG. 1, the magnetic field analysis apparatus 1 includes a control unit 3, a storage device 5, a media input / output unit 6, an input unit 7, a display unit 9, a printer port 11, and the like connected to each other via a bus 13.

  The control unit 3 includes a CPU (Central Processing Unit), a ROM (Read Only Memory), a RAM (Random Access Memory), and the like, and is connected via the bus 13 according to a program stored in the storage device 5 as a storage unit. Each of the devices is driven and controlled.

  As shown in FIG. 2, the storage device 5 stores a control program 15 for driving and controlling each component of the magnetic field analysis device 1, and a magnetic field analysis program 17 for carrying out the present invention.

  The magnetic field analysis program 17 includes input information 21 that is information having analysis conditions, and an arithmetic program 19 that calculates a magnetic field based on the motion equation of the magnetic field based on the input information 21.

  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.

  Next, a procedure for analyzing a magnetic field using the magnetic field analyzer 1 will be described with reference to FIGS.

  Here, a magnetic field analysis of an SPM motor 31 (Surface Permanent Magnet Motor), which is a kind of PM motor (Permanent Magnet Motor), will be described as an example.

  First, an outline of the configuration of the SPM motor 31 will be described with reference to FIGS. 3 and 4.

  As shown in FIG. 3, the SPM motor 31 has a rotor 33 as a rotor (moving element) and a stator 35 as a stator.

  The rotor 33 has 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 46 that is provided outside the core back 44. Has been.

  As shown in FIGS. 3 and 4, a coil 45 that is a conductor such as 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 specifications to form a bundle, but in this embodiment, as illustrated in FIGS. 3 and 4. The 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 perform the magnetic field analysis of 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 body constituting the rotor 33 and the stator 35, and to analyze the magnetization phenomenon of these magnetic bodies. is there.

  Next, the analysis procedure will be described with reference to FIGS.

  In the following procedure, the SPM motor 31 is divided into a plurality of elements to form an aggregate of particles for each element, and is handled as a rigid body model. However, the present invention is not limited to this, and particles are not used. May be treated as a rigid model for each element.

  In the following procedure, the magnetic material indicates 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.

  First, the control unit 3 of the magnetic field analysis apparatus 1 activates the magnetic field analysis program 17 and functions representing the three-dimensional structure (shape, coordinate point), mass density, and magnetization curve of the magnetic material as analysis conditions of the SPM motor 31 to be analyzed. The current density vector of the conductor is stored as input information 21 of the storage device 5 (step 101 in FIG. 5).

  These physical quantities may be read from a recording medium such as a CD-ROM via the media input / output unit 6, for example.

  The information of the three-dimensional structure of the SPM motor 31 is data such as CAD.

  Further, when the physical quantity is stored in advance as the input information 21, the above step is not necessary.

  The current density vector is necessary when calculating the magnetic field vector formed by the coil 45.

  Furthermore, a function representing the magnetization curve of the magnetic material is required when calculating the magnetization vector (bold) M.

  The above is the details of step 101 in FIG.

  Next, the control unit 3 of the magnetic field analysis apparatus 1 determines that the magnetic material is N particles (a polyhedral element whose center is the position vector of the particle (cubic element in this embodiment) from the information of the three-dimensional structure included in the input information 21. ), And the particle position vector is calculated and stored in the storage device 5 (step 102 in FIG. 5).

  Here, N is an arbitrary integer, and the number N of particles and the position vector are calculated based on the three-dimensional structure of the input information 21 and the shape of the polyhedral element stored in the storage device 5 in advance.

  The above is the details of step 102 in FIG.

  Next, the control unit 3 determines that the conductor (coil 45 shown in FIG. 6A) is a local conductor (rectangular conductors 45a and 45b) as shown in FIG. And arc-shaped columnar conductors 45c and 45d), the coefficient for calculating the magnetic field vector created by each conductor is calculated and stored in the storage device 5 (step 103 in FIG. 5).

  Here, step 103 in FIG. 5 will be specifically described.

  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 control unit 3 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 control unit 3 reads the three-dimensional structure of the conductor and the current density vector stored in the storage device 5 as the input information 21 from step 101, and a, b, c that are the dimensions of the local conductor (cuboid conductor 45a) and the particles The position vector is calculated and stored in the storage device 5.

  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 control unit 3 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. 8) 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.

The control unit 3 reads the three-dimensional structure of the conductor and the current density vector stored in the storage device 5 as the input information 21 from step 101, and the dimensions r _a and z _{b of the} local conductor (arc-shaped columnar conductor 45c). , Θ, R _c and the particle position vector are calculated and stored in the storage device 5.

  Further, when the physical quantity is stored in advance as the input information 21, the above step is not necessary.

  The above is the details of step 103 in FIG.

  Next, the control unit 3 of the magnetic field analysis device 1 stores the position vector of N particles constituting the magnetic body calculated up to this step and stored in the storage device 5 and stored in the storage device 5 in advance. The coefficient of the equation of motion of the magnetic field of the particles constituting the magnetic body is calculated according to the shape of the polyhedral element and stored in the storage device 5 (step 104 in FIG. 5).

  Here, step 104 in FIG. 5 will be specifically described.

  When the divided polyhedral elements (cubic elements in this embodiment) are displayed two-dimensionally, the shape shown in FIG. 9 is obtained.

Here, r _{g is} the particle position vector (bold), q is the midpoint of the element boundary surface of the polyhedral element centered on the particle position vector, and p is the midpoint between the particle position vector and q point. Define points.

  The control unit 3 reads the position vector of the particles (polyhedral elements (cubic elements in the present embodiment) having the center of gravity of the particles) constituting the magnetic body calculated in step 102 and stored in the storage device 5.

  If the particle position vector is updated and stored in the storage device 5 in step 113, the control unit 3 reads the value.

  From the particle position vector and the shape of the polyhedron element stored in the storage device 5 in advance, the control unit 3 calculates the points p and q in the polyhedron element whose center of gravity is the position vector of all particles constituting the magnetic body. The position vector and the normal vector (bold) n to the element boundary surface are calculated and stored in the storage device 5.

  Next, the control unit 3 determines the boundary area ΔS of the polyhedral element having the particle constituting the magnetic body as the center of gravity and the polyhedral element volume having the vertex of the polyhedral element boundary surface having the particle position vector as the center of gravity and the particle position as the vertex. ΔV is calculated and stored in the storage device 5.

  The above is the details of step 104 in FIG.

  Next, the control unit 3 of the magnetic field analysis apparatus 1 uses the size of the coil stored in the storage device 5 in step 103 and the current density vector stored in the storage device 5 up to this step to perform bio-savart. Based on the analytical solution obtained by integrating the above law, the energized coil 45 calculates the magnetic field vector formed on the particles constituting the magnetic material and stores it in the storage device 5 (step 105 in FIG. 5).

  Here, step 105 in FIG. 5 will be specifically described.

  Here, when it is assumed that an arbitrary position vector is a point P where the stator teeth 43 as a magnetic body in FIG. 4 are present, a procedure for calculating a magnetic field vector generated by the energized coil 45 on the position vector of the P point. Will be described as an example.

  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 ) is applied as shown in FIG.

  This local coordinate system is described in step 103.

Next, the position vector of the point P 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 (bold) is the position vector of the point P in the local coordinate system (x _s , y _s , z _s ), and x _ps , y _ps , z _ps are x _s , y _s , z _s. Direction value.
π is the circumference ratio.

H _xs , H _ys , and H _zs 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 , and z _k represent the upper and lower limits of integration in the x _s , y _s , and z _s directions, and the relationship shown in Expression (4) is established.

  Here, a, b, and c are dimensions of a 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.

When the local conductor is the arc-shaped columnar conductor 45c, the local coordinate system (x _c , y _c , z _c ) is applied as shown in FIG.

  This local coordinate system is described in step 103.

The position vector of the P point after being converted to the local coordinate system (x _c , y _c , z _c ) is expressed by the 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 the _{Z k,} R _{j, Z k} is the upper limit of the integration represents the lower limit, the relationship shown in equation (12) holds.

  The above is the calculation procedure of the magnetic field vector created by the coil at an arbitrary point P.

  The control unit 3 of the magnetic field analysis apparatus 1 reads the position vector of the p point inside the polyhedron element having the center of gravity as the position vector of all particles constituting the magnetic body already calculated in step 104 and stored in the storage device 5. .

Next, the control unit 3 converts the position vector of the point p into the local coordinate system (x _s , y _s , z _s ), and calculates the dimensions of the rectangular parallelepiped conductor 45 a calculated in step 103 and stored in the storage device 5. From step 101, the current density vector stored in the storage device 5 as the input information 21 is read, and the magnetic field vector is calculated based on the equations (1) to (4).

  When the current density vector is updated in step 116 and stored in the storage device 5, the control unit 3 reads the value.

  Next, the control unit 3 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.

  Thereby, the magnetic field vector which the energized rectangular parallelepiped conductor 45a produces in p point is calculated | required.

Further, the control unit 3 converts the position vector of the point p into 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 ). Convert to

Next, the control unit 3 reads the dimensions r _a , θ, z _b and R _c of the arc-shaped columnar conductor 45 _c calculated in step 103 and stored in the storage device 5, and the control unit 3 inputs in step 101. A current density vector stored in the storage device 5 as information 21 is read.

  When the current density vector is updated in step 116 and stored in the storage device 5, the control unit 3 reads the value.

  The controller 3 calculates a magnetic field vector based on the equations (6) to (12).

  Next, the calculated magnetic field vector is converted into an orthogonal coordinate system, further converted into a global coordinate system (x, y, z), and stored 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.

  The above is the details of step 105 in FIG.

  Next, the control unit 3 of the magnetic field analysis apparatus 1 uses the calculation program 19 to calculate the magnetic field vector after the virtual time step δt without considering the constraint from the motion equation of the magnetic field of the particles constituting the magnetic body, and stores it. It memorize | stores in the apparatus 5 (step 106 of FIG. 5).

  Here, step 106 in FIG. 5 will be specifically described.

  The Lagrangian of the N particles constituting the magnetic material is assumed to have a shape represented by the equations (13) to (15).

Here, in Expression (13), α is a virtual mass, bold r is a position vector, bold H is a magnetic field vector, bold bold H is a time derivative of the magnetic field vector, bold M is a magnetization vector, and bold n Is the normal vector of the polyhedral element boundary surface with the particle position vector as the center of gravity, bold Hex is the externally applied magnetic field vector, ΔS is the area of the element boundary surface of the polyhedral element with the particle position vector as the center of gravity, and ΔV is The vertex of the polyhedral element boundary surface with the particle position vector as the center of gravity and the polyhedral element volume with the particle position as the vertex, χ is the magnetic susceptibility, μ ₀ is the permeability of vacuum, λ is the Lagrange's undetermined constant, and π is the circumference The rate, s, is the number of element boundary faces of a polyhedral element whose center of gravity is the particle position vector.

  The subscript ip of each physical quantity is the physical quantity of the p point in the polyhedron element whose center is the position vector of the i-th particle, and the subscript jq is the polyhedron element whose center is the position vector of the j-th particle. The physical quantity of q point is shown.

  The points p and q are described in step 104.

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,} and (bold with _{neighbors) H ip,} and substituting Lagrangian of formula (13) to (15) to the Lagrange equation of motion, the motion equation of the magnetic field Can be described as in equation (16).

  Here, the second term on the right side of Equation (16) imposes a constraint (the divergence of the magnetization vector is 0) through Lagrange's undetermined constant.

  The third 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 shown in the second term on the right side of Expression (16), the present embodiment employs the SHAKE method, which is a generalized constraint introduction method.

  When Expression (16) is discretized by the jumping method without considering the constraint, the following Expression (17), Expression (18), and (19) are obtained.

  Here, δt is a virtual time step 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 (n + 1). This corresponds to the physical quantity at δt.

  The control unit 3 of the magnetic field analysis apparatus 1 reads the position vectors of the p point and the q point in the polyhedron element whose center of gravity is the particle position vector already calculated in step 104 and stored in the storage device 5.

  Next, the control unit 3 generates a magnetic field vector generated at the point p inside the polyhedron element whose center of gravity is the particle constituting the magnetic body, which is already calculated in step 105 and stored in the storage device 5. Read as a vector.

  Further, the control unit 3 calculates the polyhedral element boundary surface having the centroid of the element boundary area, normal vector, and particle position vector of the particles constituting the magnetic body, which are already calculated in step 104 and stored in the storage device 5. The volume of a polyhedron element whose vertex is the particle position is read.

  Further, the control unit 3 reads the attenuation constant, virtual mass, and virtual time increment stored in advance in the storage device 5.

  Next, the control unit 3 reads the magnetic field vector stored in advance in the storage device 5 and the initial value of the time differentiation of the magnetic field vector.

  In addition, the control part 3 reads the value, when the magnetic field vector and the time differentiation of the magnetic field vector are updated in step 107 mentioned later.

  Note that the magnetization vector is calculated by substituting the magnetic field vector into a function representing the magnetization curve stored in the storage device 5 as the input information 21 from step 101.

  The control unit 3 calculates the equations (17), (18), and (19) with respect to all p points in the polyhedron element having the position vector of the particles constituting the magnetic body as the center of gravity. The magnetic field vector is stored in the storage device 5.

  The details of step 106 in FIG. 5 have been described above.

  Next, the control unit 3 of the magnetic field analysis device 1 is bound to the magnetic field vector at the point p in the polyhedron element having the center of gravity as the position vector of the particles constituting the magnetic material, which is calculated in step 106 and stored in the storage device 5. A force is applied, and the calculated magnetic field vector is stored in the storage device 5 (step 107 in FIG. 5).

  Here, step 107 in FIG. 5 will be specifically described.

  A binding force is applied to the magnetic field vector at the point p in the polyhedron element with the position vector of the particle constituting the magnetic body as the center of gravity according to the following equations (20) and (21).

  Here, bold H is a magnetic field vector, α is a virtual mass, δt is a virtual time step, γ is an attenuation constant, N is the number of particles constituting the magnetic body, s is a position vector of the particles constituting the magnetic body, and the center of gravity. The subscript i is the physical quantity on the position vector of the i-th particle among the particles constituting the magnetic body, and ip is the i-th particle among the particles constituting the magnetic body. The physical quantity of point p in the polyhedron element whose center is the position vector of, iq is the physical quantity of point q in the polyhedron element whose center of gravity is the position vector of the i-th particle among the particles constituting the magnetic body, n is an arbitrary integer, and corresponds to a physical quantity at nδt, and n + 1 corresponds to a physical quantity at (n + 1) δt.

  The control unit 3 of the magnetic field analysis apparatus 1 uses the calculation program 19 to calculate the magnetic field vector at the point p in the polyhedron element with the particle constituting the magnetic body calculated in step 106 and stored in the storage device 5 as the center of gravity. Read.

  The control unit 3 reads the virtual mass, virtual time step, and attenuation constant stored in advance in the storage device 5.

  The control unit 3 performs a calculation based on Expression (20) and Expression (21) for each particle constituting the magnetic body, and stores the calculated magnetic field vector in the storage device 5.

  The details of step 107 in FIG. 5 have been described above.

  Next, the control unit 3 of the magnetic field analysis apparatus 1 determines whether or not the magnetic field vector obtained in step 107 satisfies the constraint condition, and if satisfied, proceeds to the next step, and if not satisfied, returns to step 107 (FIG. 5 step 108).

  Specifically, the control unit 3 performs the calculation based on the equation (22) using the magnetization vector calculated from the magnetic field vector of each of the particles constituting the magnetic body calculated in step 107 and stored in the storage device 5. .

  Here, the subscript n + 1 corresponds to the physical quantity at (n + 1) δt.

err _i is an error value for the binding condition of the i-th particle among the particles constituting the magnetic material.

  The magnetization vector (in bold) M is calculated in step 107 and the magnetic field vector stored in the storage device 5 is read by the control unit 3 and substituted in the function representing the magnetization curve stored in the input information 21 from step 101. Calculated by

  For the normal vector (bold) n, the control unit 3 reads and substitutes the one already calculated in step 104 and stored in the storage device 5.

  If the error value does not satisfy the equation (23) for all particles, the control unit 3 returns to Step 107.

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

The above is the details of step 108 in FIG.

Next, the control unit 3 of the magnetic field analysis apparatus 1 determines whether the magnetization phenomenon of the magnetic material has reached a steady state, and proceeds to the next step if the condition is satisfied. (Step 109 in FIG. 5)
Here, step 109 in FIG. 5 will be specifically described.

Whether the particles constituting the magnetic material have reached a steady state is determined by the following equation (24).

Here, the bold letter H is the magnetic field vector, μ ₀ is the magnetic permeability of the vacuum, N is the number of particles constituting the magnetic body, and s is the number of faces of the polyhedral element whose center of gravity is the position vector of the particles constituting the magnetic body. The subscript ip is a physical quantity at the point p in the polyhedron element whose center of gravity is the position vector of the i-th particle among the particles constituting the magnetic body.

  Further, the subscript n is an arbitrary integer, which corresponds to a physical quantity at nδt, n−1 corresponds to (n−1) δt, and n + 1 corresponds to a physical quantity at (n + 1) δt.

A ₂ is an arbitrary error determination value for determining whether the magnetic field vector of the magnetic material has reached a steady state.

The control unit 3 of the magnetic field analysis apparatus 1 reads A ₂ and μ ₀ stored in advance in the storage device 5.

  Further, the control unit 3 reads the magnetic field vector at the point p in the polyhedron element having the center of gravity of the particles constituting the magnetic body, which is calculated in step 107 and stored in the storage device 5.

  Next, the control unit 3 calculates Expression (24), and proceeds to the next step if Expression (24) is satisfied, and returns to Step 106 if not satisfied.

The above is the details of step 109 in FIG.

  Next, the control unit 3 of the magnetic field analysis apparatus 1 performs the magnetic field vector, the magnetization vector, the magnetic flux density vector on the position vector of all particles constituting the magnetic body, and the magnetic field on the position vector of all particles constituting the coil. The vector and the magnetic flux density vector are calculated and stored in the storage device 5 (step 110 in FIG. 5).

  Step 110 will be specifically described below.

(Particles constituting magnetic material)
The magnetic field vector on the position vector of the particles constituting the magnetic body is expressed by the following equation (25).

Here, (bold) e _x = (1, 0, 0), (bold) e _y = (0, 1, 0), (bold) e _z = (0, 0, 1).

  The bold letter H is a magnetic field vector, the bold letter n is a normal vector, the suffix i represents the physical quantity on the position vector of the i-th particle of the particles constituting the magnetic body, and the suffix ip is the magnetic body. Is a physical quantity of p point in the polyhedron element whose center of gravity is the position vector of the i-th particle.

  The magnetization vector of the particles constituting the magnetic material can be obtained by substituting the magnetic field vector into a function representing the magnetization curve.

  The magnetic flux density vector on the position vector of the particles constituting the magnetic body is expressed by Expression (26).

  Here, the bold letter B is the magnetic flux density vector, the bold letter H is the magnetic field vector, the bold letter M is the magnetization vector, and the suffix i is the position vector of the i-th particle among the particles constituting the magnetic body. Indicates the physical quantity.

μ ₀ is the vacuum permeability.

  The control unit 3 of the magnetic field analysis apparatus 1 reads the magnetic field vector at the point p in the polyhedron element with the particle constituting the magnetic body as the center of gravity calculated up to step 110 from the storage device 5.

  Next, the control unit 3 reads the normal vector already calculated in step 104 and stored in the storage device 5, and calculates the magnetic field vector on the position vector of the particles constituting the magnetic body based on the equation (25). Calculate and store in the storage device 5.

  Further, the control unit 3 substitutes the magnetic field vector calculated in this step into a function representing the magnetization curve stored in the storage device 5 as the input information 21 from step 101, and the magnetization vector of the particles constituting the magnetic body. Is calculated and stored in the storage device 5.

  Next, the control unit 3 substitutes the magnetic field vector and the magnetization vector calculated in this step into Expression (26), calculates the magnetic flux density vector on the position vector of the particles constituting the magnetic body, and stores it in the storage device 5. .

  The control unit 3 reads the vacuum permeability stored in the storage device 5 in advance.

(Particulate particles)
The magnetic field vector on the position vector of the particles constituting the coil is described by Expression (27) and Expression (28).

  Here, the bold letter r is a position vector, the bold letter H is a magnetic field vector, N is the number of particles constituting the magnetic body, s and ΔS are the number of boundary faces and boundaries of the polyhedron element having the particle constituting the magnetic body as the center of gravity. Area, bold M is a magnetization vector, bold n is a normal vector, subscript i is a physical quantity on the position vector of the i-th particle among the particles constituting the coil, and j is a magnetic material Jq is the physical quantity on the position vector of the j-th particle, and jq is the physical quantity at the point q of the polyhedron element whose center of gravity is the position vector of the j-th particle among the particles constituting the magnetic body. Point to.

  The first term on the right side of Equation (27) is a term describing a magnetic field vector created on the position vector of the particles constituting the coil by the particles constituting the magnetic material, and is represented by Equation (28).

  The second term on the right side of the equation (27) is a magnetic field vector formed on the position vector of the particles constituting the coil by the particles constituting the coil, and is represented by equations (1) to (12).

  The control unit 3 of the magnetic field analysis apparatus 1 reads the position vector of the particles constituting the coil calculated in step 103 and stored in the storage device 5 from the storage device 5.

  In addition, when the position vector of the particle | grains which comprise a coil is updated in step 113, the control part 3 reads the value.

  Next, the control unit 3 calculates the number of particles constituting the magnetic body and the number of faces of the polyhedron element calculated from step 102 and stored in the storage device 5, and q points in the polyhedron element having the particle position vector as the center of gravity. Read the position vector and polyhedral element boundary area.

  Further, the control unit 3 reads the magnetization vector of the particles constituting the magnetic body, which is calculated in this step and stored in the storage device 5.

  Further, the control unit 3 reads the coil size calculated in step 103 and stored in the storage device 5 and the current density vector stored in the storage device 5 as the input information 21 in step 101.

  If the current density vector is updated in step 116, the control unit 3 reads the value.

  Thereafter, the control unit 3 calculates the magnetic field vector on the position vector of the particles constituting the coil in accordance with the equations (27), (28), and (1) to (12), and stores them in the storage device 5.

The control unit 3 converts the magnetic flux density vector (bold) B _i on the position vector of the particles constituting the coil into the magnetic field vector on the position vector of the particles constituting the coil that has already been calculated, and the vacuum permeability. Multiply and store in the storage device 5.

  The magnetic permeability of the vacuum is stored in the storage device 5 in advance.

  The details of step 110 have been described above.

  Thus, the control unit 3 analyzes the magnetization phenomenon of the magnetic material by solving the motion equation of the magnetic field.

  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.

  Next, the control unit 3 of the magnetic field analysis apparatus 1 calculates a force vector acting on all the particles constituting the SPM motor 31 and stores it in the storage device 5 (step 111 in FIG. 5).

  The step 111 in FIG. 5 will be specifically described below.

(Force vector acting on particles constituting magnetic material)
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 force acting on the particles constituting the magnetic material is described as shown in Expression (29).

Here, μ ₀ is the permeability of the vacuum, bold H is the magnetic field vector, bold M is the magnetization vector, bold r is the position vector, and ΔV is the vertex of the polyhedral element boundary surface and the particle centered on the particle position vector. The polyhedral element volume with the position at the apex, N is the number of particles constituting the magnetic body, and the subscripts i and j indicate the physical quantities of the i-th and j-th particles among the particles constituting the magnetic body. The letter ip indicates the physical quantity of the point p in the polyhedron element whose center of gravity is the position vector of the i-th particle among the particles constituting the magnetic body.

Φ (r _i −r _j ) is an interaction potential energy between the i-th particle and the j-th particle.

  In this embodiment, the SPM motor is a rigid body (relative distance between particles is unchanged).

  Assuming that the model is a rigid model, the force acting on the particles constituting the magnetic material is described as in equation (30).

  The control unit 3 of the magnetic field analysis apparatus 1 reads the magnetic field vector at the point p in the polyhedron element with the particle constituting the magnetic body calculated up to step 110 and stored in the storage device 5 as the center of gravity.

  The control unit 3 calculates the magnetization vector at the p point by substituting the magnetic field vector at the p point into a function representing the magnetization curve stored in the storage device 5 as the input information 21 from step 101 to calculate the storage device 5. To remember.

  In addition, the control unit 3 calculates the polyhedral element volume having the vertex of the polyhedral element boundary surface having the center of gravity as the position vector of the particle constituting the magnetic body calculated in step 104 and stored in the storage device 5 and the vertex of the particle position. Read.

  Next, the control unit 3 calculates the partial differentiation of the magnetic field vector at the point p in the polyhedral element with the particle constituting the magnetic body as the center of gravity, and calculates the force vector from the equation (30) based on the value. Store in the storage device 5.

(Force vector acting on the particles constituting the coil)
Of the particles constituting the coil, the position vector of the i-th particle is bold r _i , the current unit vector at that position vector is bold t _i , the magnetic flux density vector is bold B _i , and the current flow direction When the length of the coil and L _i, of the particles constituting the coil, the force vector acting on the i-th particle is described by the following equation (31).

  The control unit 3 of the magnetic field analysis apparatus 1 reads the position vector of the particles constituting the coil calculated in step 103 and stored in the storage device 5.

  If the position vector of the particles constituting the coil has been updated in step 113, the control unit 3 reads the value.

  Further, the control unit 3 reads the dimensions of the coil calculated in step 103 and stored in the storage device 5.

  Next, the control unit 3 reads the current density vector of the coil stored in the storage device 5 as the input information 21 from step 101.

  Note that if the current density vector of the coil is updated in step 116, the control unit 3 uses the value.

  The control unit 3 calculates a force vector acting on the particles constituting the coil based on the equation (31) and stores the force vector in the storage device 5.

  The above is the details of step 111 in FIG.

  Next, the control unit 3 of the magnetic field analyzer 1 calculates the rotation amount, angular velocity, translation amount, translation of the movable unit (moving element) from the force vector acting on the particles constituting the moving part (moving element) calculated in step 111. The speed is calculated and stored in the storage device 5 (step 112 in FIG. 5).

  The step 112 in FIG. 5 will be specifically described below.

  The equation of motion of the translational motion of the rigid body is expressed by equation (32).

Here, m _tot constitutes the mass of the movable portion (movable element), the position vector of the representative point arbitrarily determined in bold r _c is the movable portion (movable element), bold F movable portion (slider) The force vector acting on the particles, N, is the number of particles constituting the movable part (moving element).

  Next, the equation of motion of the rotational motion around an arbitrary fixed point is expressed by Equation (33-1), Equation (33-2), and Equation (33-3).

_{Here, I 1, I 2, I} 3 are principal moments of inertia of a rigid _{body, ω 1, ω 2, ω} 3 is an angular velocity in an inertial main axis coordinate _system, in N _1, N 2, _{N 3} is an inertial main axis coordinate system Torque.

  The rotor 31 of the SPM motor 31 of this embodiment is fixed to a certain rotating shaft through a rotor shaft 41.

  If the fixed rotation axis is the z axis, the motion of the rotor 31 is calculated as follows.

  Since the rotor 31 is fixed to the z-axis through the rotor shaft 41, the motion of the rotor 31 is calculated by solving the motion equation of the rotational motion around the z-axis.

  The equation of motion of the rotational motion of the rotor 31 around the z-axis is expressed by equation (34).

Here, I _z is the moment of inertia of the rotor 31, ω _z is the angular velocity in the z direction, and N _z is the torque in the z direction.

  When the equation (34) is discretized by the flying method, the following equations (35) and (36) are obtained.

Here, θ _z is a rotation amount around the z axis, and dt is a minute time step.

  The subscripts n−1 / 2, n, n + 1/2, and n + 1 correspond to physical quantities in (n−1 / 2) dt, ndt, (n + 1/2) dt, and (n + 1) dt, respectively. Yes.

  The control unit 3 reads the time step width in which an arbitrary value is stored in the storage device 5 in advance.

  Initial values of the rotation amount and the angular velocity are stored in the storage device 5 in advance in the storage device 5.

  If the rotation amount and angular velocity of the rotor 31 have already been updated in step 112, those values are used.

  The controller 3 calculates the moment of inertia from the three-dimensional structure and mass density of the rotor 31 stored in the storage device 5 as the input information 21 from step 101.

  Further, the control unit 3 calculates the equation (35) based on the position vector of the particles constituting the rotor 31 and the force vector acting on the particles constituting the rotor 31 calculated in step 111 and stored in the storage device 5. Equation (36) is calculated, and the rotation amount and angular velocity of the rotor 31 after dt seconds are calculated and stored in the storage device 5.

  The above is the details of step 112 in FIG.

  Next, the control unit 3 of the magnetic field analysis apparatus 1 calculates and stores the position vector of the particles constituting the moving element based on the rotation amount after a minute time and the movement amount of the center of gravity calculated in step 112 ( Step 113 in FIG.

  Next, the control unit 3 of the magnetic field analyzer 1 outputs the position vector, force vector, magnetic field vector, magnetic flux density vector, and magnetization vector of the particles constituting the SPM motor 31 from the printer 12 via the printer port 11 as necessary. (Step 114 in FIG. 5).

  Next, the control unit 3 of the magnetic field analysis apparatus 1 determines whether or not a predetermined end condition (time, amount of movement, etc.) is satisfied. If it is satisfied, the analysis ends, and if not, the process proceeds to step 116. (Step 115 in FIG. 5).

  Next, the control unit 3 of the magnetic field analysis apparatus 1 updates the current density vector flowing in the coil 45 as necessary, and returns to Step 104 (Step 116 in FIG. 5).

  The above is the magnetic field analysis procedure according to the present embodiment.

  Thus, according to the present embodiment, the magnetic field analysis device 1 calculates the magnetization phenomenon of the magnetic material by solving the equation of motion derived from the Lagrangian of the 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 matrix is not handled, the memory required for the calculation is proportional to the number of particles (elements). Compared with computations that handle asymmetric dense matrices, the memory can be significantly reduced.

  Further, according to the present embodiment, when the solution of the motion equation of the magnetic field derived from the Lagrangian of the particles constituting the magnetic material is calculated, the total energy of the system is preserved when the external input does not change.

  In the above-described embodiment, the case where the magnetization phenomenon of the magnetic material is calculated by solving the equation of motion derived from the Lagrangian of the particles constituting the magnetic material has been described, but the present invention is not limited to this. Instead, any equation other than that derived from the Lagrangian may be used as long as it is an equation of motion.

  In the above-described embodiment, the present invention is used for the magnetic field analysis of the SPM motor 31. However, the present invention is not limited to this, and the present invention analyzes an arbitrary magnetic field in the space where the 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 magnetic field analysis in the space shielded by the magnetic material.

DESCRIPTION OF SYMBOLS 1 ………… Magnetic field analyzer 3 ………… Control unit 5 ………… Storage device 7 ………… Input unit 9 ………… Display unit 11 ………… Printer port 12 ……… Printer 13 …… … Bus 31 ……… SPM motor 33 ……… Rotor 35 ……… Stator 37 ……… Rotor core 39 ……… Permanent magnet 41 ……… Rotor shaft 43 ……… Stator teeth 44 ……… Core back 45 …… ... coil 46 ... …… frame

Claims (24)

A magnetic field analysis device that analyzes a magnetic field at an arbitrary point in a space where a magnetic material is arranged using an information processing device,
Storage means for storing the analysis conditions and the equation of motion of the magnetic field;
From the analysis conditions, the coefficient of the equation of motion of the magnetic field of the particles constituting the magnetic body, the initial value of the variable and the constant are calculated, the solution of the equation of motion of the magnetic field is calculated, and based on the solution, A computing means for computing the magnetic field at an arbitrary point;
A magnetic field analysis apparatus comprising: The analysis conditions are:
The shape of the magnetic body;
A function representing a magnetization curve of the magnetic material;
The magnetic field analysis apparatus according to claim 1, comprising:   The magnetic field analysis apparatus according to claim 1, wherein the variable of the equation of motion of the magnetic field is a magnetic field vector of particles constituting the magnetic body. The coefficient of the equation of motion of the magnetic field is
A position vector of particles constituting the magnetic body;
A function representing a magnetization curve of the magnetic material;
The magnetic field analysis apparatus according to claim 1, wherein the magnetic field analysis apparatus is calculated from: The equation of motion of the magnetic field is
5. The method according to claim 1, wherein the Lagrangian of a particle including a magnetic field term is obtained by substituting the Lagrangian equation of motion with a magnetic field vector and a time derivative of the magnetic field vector as a canonical variable. Magnetic field analyzer. A conductor is further arranged in the space,
The magnetic field analysis apparatus according to claim 1, wherein the analysis condition includes a shape of a conductor and a current. 6. The magnetic field analysis apparatus according to claim 5, wherein the Lagrangian of the particles is a function represented by the following formulas (A) to (C) when N particles are included in the magnetic material.

α: virtual mass (bold) r: position vector (bold) H: magnetic field vector (with bold side points) H: time derivative of magnetic field vector (bold) M: magnetization vector (bold) n: particle position Normal vector (in bold) of polyhedral element boundary surface with vector as centroid: Externally applied magnetic field vector ΔS: Area of polyhedral element boundary surface with particle position vector as centroid ΔV: Particle position vector as centroid Polyhedral element volume χ: Magnetic susceptibility μ ₀ : Vacuum permeability λ: Lagrange's undetermined constant π: Circumferential ratio s: Grain position vector is the center of gravity Polyhedral element boundary surface number subscript ip: physical quantity subscript at point p in the polyhedral element centered on the particle position vector jq: physical quantity at point q in the polyhedral element centered on the particle position vector m: mass v: velocity φ (r _i −r _j ): subscript i of interaction potential energy between i-th particle and j-th particle, j: physical quantity of i-th particle and j-th particle The magnetic field analysis apparatus according to claim 7, wherein the equation of motion of the magnetic field is an expression represented by the following expression (D).

The equation of motion of the magnetic field is obtained by adding a discretized equation represented by the following equations (E), (F), and (G) and a binding force represented by the following equations (H) and (I). The magnetic field analysis apparatus according to claim 7, further comprising:

γ: Decay constant δt: Virtual time step subscript n: Physical quantity subscript n−1 / 2: (n−1 / 2) Physical quantity subscript n + 1/2: Physical quantity at (n + 1/2) δt Subscript n + 1: Physical quantity at (n + 1) δt
  The magnetic field analysis apparatus according to claim 1, wherein the magnetic body is a rotor and a stator of a motor.   The magnetic field analysis apparatus according to claim 6, wherein the conductor is a coil of a motor.   The magnetic field analysis apparatus according to claim 1, wherein the magnetic body is a rotor and a stator of a PM motor (Permanent Magnet Motor).   The magnetic field analyzer according to claim 6, wherein the conductor is a coil of a PM motor (Permanent Magnet Motor).   The magnetic field analysis apparatus according to claim 1, wherein the magnetic body is a magnetic body constituting a magnetic shield.   The program for functioning a computer as a magnetic field analyzer in any one of Claims 1-14. A magnetic field analysis method for analyzing a magnetic field at an arbitrary point in a space where a magnetic material is arranged using an information processing device,
Storing the analysis conditions and the equation of motion of the magnetic field (a);
From the analysis conditions, calculate the coefficient and variable initial values and constants of the motion equation of the magnetic field of the particles constituting the magnetic body, calculate the solution of the motion equation of the magnetic field,
A step (b) of calculating a magnetic field at an arbitrary point in the space based on the solution of the equation of motion of the magnetic field;
A magnetic field analysis method characterized by comprising: The analysis conditions are:
A function representing a shape of the magnetic body and a magnetization curve of the magnetic body;
The magnetic field analysis method according to claim 16, further comprising:   The magnetic field analysis method according to claim 16, wherein the variable of the equation of motion of the magnetic field is a magnetic field vector of particles constituting the magnetic body. The coefficient of the equation of motion of the magnetic field is
A position vector of particles constituting the magnetic body;
A function representing a magnetization curve of the magnetic material;
The magnetic field analysis method according to claim 16, wherein the magnetic field analysis method is calculated from: The equation of motion is
20. The particle Lagrangian including a magnetic field term is obtained by substituting the Lagrangian equation of motion with a magnetic field vector and a time derivative of the magnetic field vector as a canonical variable. Magnetic field analysis method. A conductor is further arranged in the space,
The magnetic field analysis method according to claim 16, wherein the analysis condition includes a shape of a conductor and a current. 21. The magnetic field analysis method according to claim 20, wherein the Lagrangian of the particles is a function represented by the following formulas (A) to (C) when N particles constituting the magnetic material are used.

α: virtual mass (bold) r: position vector (bold) H: magnetic field vector (with bold side points) H: time derivative of magnetic field vector (bold) M: magnetization vector (bold) n: particle position Normal vector (in bold) of polyhedral element boundary surface with vector as centroid: Externally applied magnetic field vector ΔS: Area of polyhedral element boundary surface with particle position vector as centroid ΔV: Particle position vector as centroid Polyhedral element volume χ: Magnetic susceptibility μ ₀ : Vacuum permeability λ: Lagrange's undetermined constant π: Circumferential ratio s: Grain position vector is the center of gravity Polyhedral element boundary surface number subscript ip: physical quantity subscript jq in the polyhedral element centered on the particle position vector jq: physical quantity at point q in the polyhedral element centered on the particle position vector,
m: mass v: velocity φ (r _i −r _j ): interaction potential energy index i between the i-th particle and j-th particle, j: physical quantity of the i-th particle and j-th particle The magnetic field analysis method according to claim 22, wherein the equation of motion of the magnetic field is an expression represented by the following expression (D).

The equation of motion of the magnetic field is obtained by adding a discretized equation represented by the following equations (E), (F), and (G) and a binding force represented by the following equations (H) and (I). 24. The magnetic field analysis method according to claim 23, wherein:

γ: damping constant δt: hypothetical time step subscript n: physical quantity at nδt,
Subscript n-1 / 2: Physical quantity subscript n + 1/2 at (n-1 / 2) δt: Physical quantity subscript n + 1 at (n + 1/2) δt: Physical quantity at (n + 1) δt

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