Patent Publication Number: US-2012046888-A1

Title: Analysis apparatus and analysis method

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is based upon and claims the benefit of priority of the prior Japanese Patent Application No. 2010-183284, filed on Aug. 18, 2010, the entire contents of which are incorporated herein by reference. 
     FIELD 
     An embodiment of the invention discussed herein is directed to an analysis apparatus and an analysis method. 
     BACKGROUND 
     Conventional analysis apparatuses simulate magnetic fields generated in magnetic materials by modeling the properties of the magnetic materials and by analyzing the properties of the modeled magnetic materials. For example, the analysis apparatuses model the magnetic hysteresis of a magnetic material in accordance with both an M-H curve, which indicates the relation between magnetization M of the magnetic material and an external magnetic field H of the magnetic material, and the coercive force Hc of the magnetic material. Such analysis apparatuses simulate the property of the magnetic material by analyzing the modeled magnetic hysteresis. 
     As illustrated in  FIG. 17 , in some cases, the magnetization M of a magnetic material may vary in such a manner that curves called minor loops are made within the M-H curves. However, the analysis apparatuses do not simulate the minor loops if the analysis apparatuses, in accordance with the previously determined M-H curve and the coercive force Hc, model the magnetic hysteresis of the magnetic material, thus reducing the accuracy of the modeling of the magnetic material.  FIG. 17  is a schematic diagram illustrating minor loops. 
     In such a case, to accurately model the properties of the magnetic materials, there is a known technology, called micro-magnetization analysis, for modeling the properties of the magnetic materials by dividing the magnetic materials into multiple regions and calculating the magnetization state in each divided region. 
     Specifically, an analysis apparatus that performs a micro-magnetization analysis uses a micro-magnetization vector m in each region of the divided magnetic material. Then, the analysis apparatus solves, as simultaneous equations, both the Landau Lifshitz Gilbert (LLG) equation corresponding to the equation of motion of a micro-magnetization vector m and the equation of a magnetic field representing an external magnetic field H given by a micro-magnetization vector m. More specifically, by solving both the LLG equation and the equation of the magnetic field as the micro-magnetization vector m and the external magnetic field H evolve over time, the analysis apparatus simulates, with high accuracy, the properties of the magnetic materials, including the minor loops. 
     For example, by using the equation of the magnetic field, the analysis apparatus calculates an external magnetic field H t  given by a micro-magnetization vector m t  at a time t. Then, by using both the calculated external magnetic field H t  and the LLG equation, the analysis apparatus calculates the value of a micro-magnetization vector m t+Δt  at a time t+Δt for which the time has evolved by At from the time t. By repeatedly performing this calculation, the analysis apparatus simulates the properties of the magnetic materials that temporally vary. 
     Furthermore, if simultaneous equations include an equation that indicates a non-stationary value, by using the shortest time scale in the time scale used with each equation, the analysis apparatus solves equations by evolving, over time, the non-stationary value. Here, the micro-magnetization vector m has a non-stationary value in terms of the external magnetic field H, and the time scale that is used with the LLG equation is shorter than that used with the equation of the magnetic field. 
     Accordingly, by using the time scale used with the LLG equation, the analysis apparatus evolves, over time, both the micro-magnetization vector m and the external magnetic field H. For example, if the time scale used with the LLG equation is approximately 10 −12  seconds, the analysis apparatuses calculates, from the external magnetic field H t  at the time t, a micro-magnetization vector at a time t+10 −12  and then calculates, from the calculated micro-magnetization vector, an external magnetic field at a time t+10 −12 . 
     However, with the above-described technology that is used for the micro-magnetization analysis, using the time scale used with the LLG equation, both the micro-magnetization vector m and the external magnetic field H are made to evolve over time; therefore, there is a problem in that the period of time for simulating the properties of the magnetic materials is limited to a short period of time. 
     For example, if the time scale used with the LLG equation is 10 −12  seconds, the analysis apparatus evolves both the micro-magnetization vector m and the external magnetic field H over time at approximate time intervals of 10 −12  seconds. Accordingly, even when the analysis apparatus calculates for a long period of time, it only simulates the properties of the magnetic materials for approximately 10 −6  seconds. Furthermore, if the analysis apparatus simulates the properties of the magnetic materials for several seconds, it cannot complete the calculation in a predetermined period of time; therefore, the analysis apparatus cannot simulate the properties of the magnetic materials for several seconds. 
     To use a micro-magnetization analysis for a large magnetic material, it is conceivable to use a technology for dividing the magnetic materials into a plurality of meshes and using, as a magnetization vector, the average value of a plurality of micro-magnetization vectors m that are included in the meshes. However, analysis apparatuses that use such a technology evolve, over time, both the micro-magnetization vector m and the external magnetic field H using the time scale used with the LLG equation. Accordingly, the period of time for simulating the properties is limited to a short period of time; therefore, it is impossible to simulate the properties of the magnetic materials for several seconds. 
     According to an aspect of the disclosed technology, it is possible to simulate the properties of magnetic materials for a long period of time. 
     Patent Document: Japanese Laid-open Patent Publication No. 2008-275403 
     SUMMARY 
     According to an aspect of an embodiment of the invention, an analysis apparatus includes a receiving unit that receives a property information on an object to be analyzed; and a vector potential calculating unit that calculates an average value of a plurality of magnetization vectors allocated to each region of the object that is obtained by dividing the object to be analyzed in accordance with the received property information of the object to be analyzed, and calculates a vector potential obtained after a predetermined time has elapsed by using the calculated average value of the magnetization vectors and an equation of a magnetic field that is a governing equation of a vector potential. 
     The object and advantages of the embodiment will be realized and attained by means of the elements and combinations particularly pointed out in the claims. 
     It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the embodiment, as claimed. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a schematic diagram illustrating an information processing apparatus according to a first embodiment; 
         FIG. 2  is a schematic diagram illustrating an example of a read process performed on a mesh; 
         FIG. 3  is a schematic diagram illustrating an example of a process for setting analysis conditions; 
         FIG. 4  is a schematic diagram illustrating an example of a process for setting the properties of an object to be analyzed; 
         FIG. 5  is a schematic diagram illustrating a calculation in which both the LLG equation and an equation of a stationary magnetic field are evolved over time; 
         FIG. 6  is a schematic diagram illustrating calculation in which both the LLG equation and an equation of a non-stationary magnetic field are evolved over time; 
         FIG. 7  is a schematic diagram illustrating an example of simulating minor loops using a micro-magnetization analysis; 
         FIG. 8  is a schematic diagram illustrating a magnetic field gradient; 
         FIG. 9  is a schematic diagram illustrating a saturation magnetic field; 
         FIG. 10  is a schematic diagram illustrating an example of output results; 
         FIG. 11  is a schematic diagram illustrating an example of a process for evolving time by using an explicit expression M; 
         FIG. 12  is a schematic diagram illustrating an example of a process for evolving time by using an implicit expression M; 
         FIG. 13  is a flowchart illustrating the flow of a process for evolving time by using the explicit expression M; 
         FIG. 14  is a flowchart illustrating the flow of a process for evolving time by using the implicit expression M; 
         FIG. 15  is a flowchart illustrating the flow of a process that uses a magnetic field gradient to perform time evolution; 
         FIG. 16  is a functional block diagram illustrating a computer that executes an analysis program; and 
         FIG. 17  is a flowchart illustrating minor loops. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Preferred embodiments of the present invention will be explained with reference to accompanying drawings. In the following, each embodiment will be described as an information processing apparatus that includes the analysis apparatus disclosed in the present invention. Furthermore, in the following description, a micro-magnetization vector m, an average value M of the micro-magnetization vectors m, an external magnetic field H, and a vector potential A are assumed to be vectors. 
     [a] First Embodiment 
     In a first embodiment, an example of an information processing apparatus that includes an analysis apparatus will be described with reference to  FIG. 1 .  FIG. 1  is a schematic diagram illustrating the information processing apparatus according to the first embodiment. The information processing apparatus according to the first embodiment is a computer that simulates, in accordance with an initial value of a micro-magnetization vectors that is input from outside, the properties of a magnetic material. 
     In the example illustrated in  FIG. 1 , an information processing apparatus  1  includes a memory  2 , a hard disk drive (HDD)  3 , a control unit  4 , and an analysis apparatus  10 . Furthermore, the information processing apparatus  1  is connected to a keyboard  5  and a monitor  6 . The memory  2  is a storing unit that temporarily stores therein arbitrary information. The HDD  3  is a storing unit that stores therein arbitrary information. The keyboard  5  is an input device used for inputting property information on an object to be analyzed or inputting settings of the analysis method. The monitor  6  is a display that displays a screen for inputting the property information on the object to be analyzed or inputting analysis results. 
     The control unit  4  controls the memory  2 , the HDD  3 , and the analysis apparatus  10  that are included in the information processing apparatus  1 . Furthermore, the control unit  4  displays, on the monitor  6 , setting screens for dividing the magnetic material into meshes, for analysis conditions, and for the properties of an object to be analyzed. The control unit  4  transmits, to the analysis apparatus  10 , various kinds of information that is set by a user. Furthermore, if the control unit  4  obtains an analysis result from the analysis apparatus  10 , the control unit  4  displays the obtained analysis result on the monitor  6 . 
     For example, as in the example illustrated in  FIG. 2 , the control unit  4  displays a setting screen for dividing the magnetic material into meshes, which is an object to be analyzed, on the monitor  6 .  FIG. 2  is a schematic diagram illustrating an example of a read process performed on a mesh. A user of the information processing apparatus  1  divides, via the setting screen illustrated in  FIG. 2 , the magnetic material into meshes and sets micro-magnetization vectors m that are allocated to each of the meshes. 
     Furthermore, as in the example illustrated in  FIG. 3 , the control unit  4  displays the setting screen for the analysis conditions on the monitor  6 .  FIG. 3  is a schematic diagram illustrating an example of a process for setting analysis conditions. In the example illustrated in  FIG. 3 , the control unit  4  displays, the setting screen for the selection of a basic calculation, a static magnetic field calculating method, a calculating method, and the selection of stationary or non-stationary state; the model scale of the magnetic material; a time integration method; various parameters; and the like on the monitor  6 . Furthermore, in the example illustrated in  FIG. 3 , the control unit  4  displays, the setting screen for convergence criterion, the CVODE (solves initial value problems for an ordinary differential equation) parameter; and the like on the monitor  6 . 
     Furthermore, as in the example illustrated in  FIG. 4 , the control unit  4  displays, the setting screen for property information on an object to be analyzed on the monitor  6 .  FIG. 4  is a schematic diagram illustrating an example of a process for setting the properties of an object to be analyzed. In the example illustrated in  FIG. 4 , the control unit  4  displays, the setting screen for a basic property value, a conductive property value, a moving element, storing of the magnetic field, material properties, basic material characteristics, settings of the anisotropy, initial magnetization vectors, settings of the lead analysis, settings of the internal magnetic field, additional items, and the like on the monitor  6 . 
     Then, if a user sets micro-magnetization vectors m, analysis conditions, and property information on an object to be analyzed, the control unit  4  transmits each piece of set information to the analysis apparatus  10 . 
     Referring back to  FIG. 1 , the analysis apparatus  10  receives an input of property information on an object to be analyzed. Furthermore, the analysis apparatus  10  calculates, in accordance with the received property information on the object to be analyzed, the average value M (hereinafter, referred to as an “average value M”) of a plurality of micro-magnetization vectors m that are allocated to the object to be analyzed. Then, using both the calculated average value M and an equation of the magnetic field that is a governing equation of a vector potential A, the analysis apparatus  10  calculates a vector potential A obtained after a predetermined time has elapsed. 
     In the following, a calculating process performed by the analysis apparatus  10  will be described in detail. The first thing described in the following description will be a process performed by a conventional information processing apparatus in which the Landau Lifshitz Gilbert (LLG) equation, which is an equation of motion of a micro-magnetization vector m, and an equation of the magnetic field representing the external magnetic field H are simultaneously solved. Then, a process performed by the analysis apparatus  10  will be described. 
     The LLG equation, which is the equation of motion of a micro-magnetization vector m, can be given by Equation (1) below: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       m 
                     
                     
                       ∂ 
                       t 
                     
                   
                   = 
                   
                     
                       
                         - 
                         γ 
                       
                        
                       
                           
                       
                        
                       m 
                       × 
                       
                         H 
                         eff 
                       
                     
                     + 
                     
                       α 
                        
                       
                         ( 
                         
                           m 
                           × 
                           
                             
                               ∂ 
                               m 
                             
                             
                               ∂ 
                               t 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where m is the micro-magnetization vector and γ is the gyromagnetic ratio. Furthermore, in Equation (1), H eff  is the effective magnetic field given by the micro-magnetization vector and is specifically given by Equation (2) below: 
         H   eff   =H   mag   +H   ex   +H   aniso   (2)
 
     where H mag  is the magnetic field formed by the micro-magnetization vector m. Furthermore, H ex  is the switched connection magnetic field. Furthermore, H aniso  is the anisotropic magnetic field. 
     In the following, an equation of the magnetic field representing the external magnetic field H will be described. The equation of the magnetic field representing the external magnetic field H includes both the equation of the stationary magnetic field and the equation of the non-stationary magnetic field. The equation of the stationary magnetic field includes both a governing equation of a scalar potential and a governing equation of a vector potential. Specifically, the governing equation of the scalar potential can be given by Equation (3) and Equation (4) below. In Equation (3), Δ is the Laplace operator, φ is the scalar potential, and ∇ is the vector differential operator. 
       Δφ=∇· m   (3)
 
         H   mag =−∇φ  (4)
 
     Furthermore, the governing equation of the vector potential can be given by Equation (5) and Equation (6) below. In Equation (5), A is the vector potential, μ 0  is the vacuum permeability, and J is the vector representing the excitation current. 
     
       
         
           
             
               
                 
                   
                     ∇ 
                     
                       × 
                       
                         ∇ 
                         
                           × 
                           A 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         μ 
                         0 
                       
                        
                       J 
                     
                     + 
                     
                       ∇ 
                       
                         × 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     H 
                     mag 
                   
                   = 
                   
                     
                       1 
                       
                         μ 
                         0 
                       
                     
                      
                     
                       ∇ 
                       
                         × 
                         A 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The governing equation of the scalar potential can take into consideration the effect of the magnetic field generated by the excitation current, whereas the governing equation of the vector potential can take into consideration the effect of the magnetic field generated by the excitation current. 
     Furthermore, the equation of the non-stationary magnetic field can be given by Equation (7) below. In Equation (7), σ is the conductivity. The equation of the non-stationary magnetic field is an equation taking into consideration the contribution of eddy currents and is represented as the governing equation of the vector potential A. 
     
       
         
           
             
               
                 
                   
                     
                       
                         μ 
                         0 
                       
                        
                       σ 
                        
                       
                         
                           ∂ 
                           A 
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                     + 
                     
                       ∇ 
                       
                         × 
                         
                           ∇ 
                           
                             × 
                             A 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         μ 
                         0 
                       
                        
                       J 
                     
                     + 
                     
                       ∇ 
                       
                         × 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     When simultaneously solving the LLG equation and the governing equation of the stationary scalar potential, the conventional information processing apparatus solves the LLG equation and the governing equation of the stationary scalar potential by treating Equation (1) to Equation (4) as simultaneous equations and then solving the LLG equation and the governing equation of the stationary scalar potential as the micro-magnetization vector m and the external magnetic field H evolve over time. Furthermore, when simultaneously solving the LLG equation and the governing equation of the stationary vector potential, the conventional information processing apparatus solves them by treating Equation (1), Equation (2), Equation (5), and Equation (6) as simultaneous equations and then solving them as the micro-magnetization vector m and the external magnetic field H evolve over time. 
     Furthermore, when simultaneously solving the LLG equation and governing equation of the non-stationary vector potential, the conventional information processing apparatus solves them by treating Equation (1), Equation (2), Equation (6), and Equation (7) as simultaneous equations and then solving them as the micro-magnetization vector m and the external magnetic field H evolve over time. 
     In the following, there will be a description with reference to  FIG. 5  of a process, performed by the conventional information processing apparatus, in which the LLG equation and the equation of the stationary magnetic field are solved by treating them as simultaneous equations as the micro-magnetization vector m and the external magnetic field H evolve over time.  FIG. 5  is a schematic diagram illustrating a calculation in which both the LLG equation and an equation of a stationary magnetic field are evolved over time. In the example illustrated in  FIG. 5 , it is assumed that the conventional information processing apparatus solves the equations, using an explicit method, as the micro-magnetization vector m and the external magnetic field H evolve over time. 
     First, the conventional information processing apparatus obtains, as an initial value, a micro-magnetization vector m 0  at a time 0 (Step  1  in  FIG. 5 ). Then, the conventional information processing apparatus calculates, using the micro-magnetization vector m 0  at the time 0, an external magnetic field H 0  at the time 0 (Step  2  in  FIG. 5 ). Subsequently, by using the calculated external magnetic field H 0  as an effective magnetic field H eff  given by the micro-magnetization vector m, the conventional information processing apparatus calculates a micro-magnetization vector m 1  at the subsequent time step, (Step  3  in  FIG. 5 ). 
     Specifically, the conventional information processing apparatus calculates the external magnetic field H n  from the micro-magnetization vector m n  and then calculates, from the calculated external magnetic field H n , the micro-magnetization vector m n+1  obtained after a predetermined time has elapsed. By repeatedly performing such a series of calculations, the conventional information processing apparatus calculates as the micro-magnetization vector m and the external magnetic field H evolve over time (Step  4  in  FIG. 5 ). 
     In the following, there will be a description with reference to  FIG. 6  of a process, performed by the conventional information processing apparatus, in which the LLG equation and the equation of the non-stationary magnetic field both are solved by treating them as simultaneous equations as the micro-magnetization vector m and the external magnetic field H evolve over time.  FIG. 6  is a schematic diagram illustrating calculation in which both the LLG equation and an equation of a non-stationary magnetic field are evolved over time. In the example illustrated in  FIG. 6 , it is assumed that the conventional information processing apparatus uses an implicit method to solve the equations as the micro-magnetization vector m and the external magnetic field H evolve over time. 
     First, the conventional information processing apparatus obtains, as an initial value, a micro-magnetization vector m 0  at a time 0 (Step  1  in  FIG. 6 ). Then, the conventional information processing apparatus calculates, using the micro-magnetization vector m 0 , an external magnetic field H 1/2  in which the time step is advanced by a half step (Step  2  in  FIG. 6 ). Subsequently, the conventional information processing apparatus calculates, using the external magnetic field H 1/2 , a micro-magnetization vector m 1  in which a time step is further advanced by a half step (Step  3  in  FIG. 6 ). By repeatedly performing such a series of calculations, the conventional information processing apparatus sequentially calculates as the micro-magnetization vector m and the external magnetic field H evolve over time (Step  4  in  FIG. 6 ). 
     In this way, the conventional information processing apparatus treats the LLG equation and the equation of the magnetic field as simultaneous equations and simultaneously solves the equations by evolving them over time so that the apparatus simulates the properties of the magnetic material. For example, as illustrated in  FIG. 7 , the conventional information processing apparatus simulates minor loops produced within the M-H curves.  FIG. 7  is a schematic diagram illustrating an example of simulating minor loops using a micro-magnetization analysis. 
     However, as described above, the conventional information processing apparatus solves the LLG equation and the equation of the magnetic field by making both the micro-magnetization vector m and the external magnetic field H evolve over time using the time scale used with the LLG equation. Accordingly, the conventional information processing apparatus limits the period of time for simulating the properties of the magnetic material to a short period of time. For example, in the examples illustrated in  FIGS. 5 and 6 , the conventional information processing apparatus evolves time only by 10 −12  seconds for each single time step. Accordingly, the conventional information processing apparatus cannot simulate the properties of the magnetic material for several seconds. 
     In the following, a calculating process performed by the analysis apparatus  10  according to the first embodiment will be described. First, an explicit method will be described in which equations are simultaneously solved by explicitly representing the average value M of the micro-magnetization vectors m. The analysis apparatus  10  calculates the average value M of the micro-magnetization vectors m allocated to the object to be analyzed (hereinafter, referred to as “average value M”) using Equation (8) below: 
     
       
         
           
             
               
                 
                   M 
                   = 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         m 
                         i 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     At this stage, if the average value M of the micro-magnetization vectors m is explicitly represented by using the vector potential A, it is possible to obtain Equation (9) below. In Equation (9), A n+1  is the vector potential A at the n+1 th  time step. Furthermore, M n  is the average value M at the n th  time step. 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               μ 
                               0 
                             
                              
                             σ 
                           
                           
                             Δ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         - 
                         
                           ∇ 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       A 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           μ 
                           0 
                         
                          
                         σ 
                       
                       
                         Δ 
                          
                         
                             
                         
                          
                         t 
                       
                     
                     + 
                     
                       A 
                       n 
                     
                     + 
                     
                       ∇ 
                       
                         × 
                         
                           M 
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Then, if Equation (9) is transformed using an inverse matrix, it is possible to obtain Equation (10) below: 
     
       
         
           
             
               
                 
                   
                     A 
                     
                       n 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 μ 
                                 0 
                               
                                
                               σ 
                             
                             
                               Δ 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                           - 
                           
                             ∇ 
                             2 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               
                                 μ 
                                 0 
                               
                                
                               σ 
                             
                             
                               Δ 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                            
                           
                             A 
                             n 
                           
                         
                         + 
                         
                           ∇ 
                           
                             × 
                             
                               M 
                               n 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Specifically, if the average value M is explicitly represented, the analysis apparatus  10  solves Equation (9) from the average value M n  at the n th  time step and calculates, in a single calculation, the vector potential A n+1  at the n+1 th  time step. 
     Because the time scale in which the average value M varies is much shorter than the time scale in which the vector potential A varies, the average value M can be assumed to be stationary during a period of time in which the vector potential A n  changes to A n+1 . Specifically, the average value M n  can be assumed to be in the stationary state with respect to an external magnetic field H n+1  calculated from A n+1 . 
     As a result, because, in the calculation performed as the vector potential A n  evolves over time by A n+1 , the average value M can be assumed to be in the stationary state, the analysis apparatus  10  can evolve the LLG equation over time for a long time scale over which the equation of the magnetic field is made to evolve. Specifically, because the analysis apparatus  10  calculates, from the average value M n , the vector potential A n+1  obtained after a predetermined time has elapsed, the analysis apparatus  10  evolves both the vector potential A and the average value M over time for a long time scale over which the vector potential A evolve. 
     Here, the time scale over which the vector potential A is made to evolve is, for example, approximately 10 −6  seconds. Furthermore, the time scale over which the micro-magnetization vector m is made to evolve over time is, for example, approximately  10   −12  seconds. Specifically, because the analysis apparatus  10  evolves the micro-magnetization vector m and the vector potential A over time for a long time scale, the analysis apparatus  10  can simulate the properties of the magnetic material over a long period of time. 
     In the following, if the analysis apparatus  10  calculates the vector potential A n+1  using the average value M n , the analysis apparatus  10  calculates the external magnetic field H n+1  using the calculated vector potential A n+1 , Equation (11), and Equation (12) below: 
     
       
         
           
             
               
                 
                   
                     B 
                     
                       n 
                       + 
                       1 
                     
                   
                   = 
                   
                     ∇ 
                     
                       × 
                       
                         A 
                         
                           n 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     H 
                     
                       n 
                       + 
                       1 
                     
                   
                   = 
                   
                     
                       1 
                       
                         μ 
                         0 
                       
                     
                      
                     
                       B 
                       
                         n 
                         + 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Then, the analysis apparatus  10  uses Equation (1) with the calculated external magnetic field H 1+1  and then calculates the average value M n+1 . If the analysis apparatus  10  further evolves the average value M n+1  over time, the analysis apparatus  10  repeats the process for sequentially solving Equation (10) to Equation (12) and calculates the average value M n+2  at the subsequent time step. 
     At this stage, if the time intervals for evolving the simultaneous equations over time are large, to avoid the degradation of the calculation accuracy, the analysis apparatus  10  performs a calculation using an implicit method, in which the average value M is implicitly represented and both the average value M and the vector potential A are made to evolve over time. For example, if the vector potential A and the average value M are represented using a central difference with respect to time, Equation (13) below is given: 
     
       
         
           
             
               
                 
                   
                     
                       
                         μ 
                         0 
                       
                        
                       σ 
                        
                       
                         
                           
                             A 
                             
                               n 
                               + 
                               1 
                             
                           
                           - 
                           
                             A 
                             n 
                           
                         
                         
                           Δ 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                     - 
                     
                       
                         
                           ∇ 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               A 
                               n 
                             
                             + 
                             
                               A 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                   
                   = 
                   
                     
                       ∇ 
                       
                         × 
                         
                           ( 
                           
                             
                               M 
                               n 
                             
                             + 
                             
                               M 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     If the average value M is implicitly treated, a calculation is repeatedly performed, using A n+1   1 =A n+1   0 +δA, until δA becomes “ 0 ” or until δA converges to a predetermined threshold ε or below. Specifically, the analysis apparatus  10  treats a value obtained by adding A n+1   0  to an increment δA of the vector potential A as a new vector potential A n+1   1 ; calculates, using the vector potential A n+1   1 , an average value M 1 ; and then calculates a new δA using the calculated M 1 . Then, the analysis apparatus  10  repeatedly performs a series of calculations until the calculated new δA becomes “0” or it converges to a predetermined threshold ε or below. 
     Here, it is assumed that A n+1   0  represents the vector potential A n+1  that has not been subjected to the repeated calculations for converging δA. In the following description, it is assumed that A n+1   i  represents the calculated vector potential A n+1  that is obtained after the repeated calculations for converging δA are performed “i” times. 
     If A n+1 =A n+1 +δA is used with Equation (13) and Equation (13) and is represented in the nonlinear and non-stationary incremental modes, Equation (14) below can be obtained: 
     
       
         
           
             
               
                 
                   
                     
                       
                         μ 
                         0 
                       
                        
                       σ 
                        
                       
                         
                           
                             ( 
                             
                               
                                 A 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                               + 
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 A 
                               
                             
                             ) 
                           
                           - 
                           
                             A 
                             n 
                           
                         
                         
                           Δ 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                     - 
                     
                       
                         
                           ∇ 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               A 
                               n 
                             
                             + 
                             
                               ( 
                               
                                 
                                   A 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                                 + 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   A 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                       2 
                     
                   
                   = 
                   
                     
                       ∇ 
                       
                         × 
                         
                           ( 
                           
                             
                               M 
                               n 
                             
                             + 
                             
                               M 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Furthermore, if Equation (14) is transformed to a linear equation of δA, Equation (15) below can be obtained: 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               μ 
                               0 
                             
                              
                             σ 
                           
                           
                             Δ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         - 
                         
                           
                             1 
                             2 
                           
                            
                           
                             ∇ 
                             2 
                           
                         
                       
                       ) 
                     
                      
                     δ 
                      
                     
                         
                     
                      
                     A 
                   
                   = 
                   
                     
                       
                         
                           μ 
                           0 
                         
                          
                         
                           σ 
                            
                           
                             ( 
                             
                               
                                 A 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                               - 
                               
                                 A 
                                 n 
                               
                             
                             ) 
                           
                         
                       
                       
                         Δ 
                          
                         
                             
                         
                          
                         t 
                       
                     
                     + 
                     
                       
                         
                           ∇ 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               A 
                               n 
                             
                             + 
                             
                               A 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                     + 
                     
                       
                         ∇ 
                         
                           × 
                           
                             ( 
                             
                               
                                 M 
                                 n 
                               
                               + 
                               
                                 M 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Accordingly, the increment δA of the vector potential A can be given by Equation (16) below: 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     A 
                   
                   = 
                   
                     
                       - 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   μ 
                                   0 
                                 
                                  
                                 σ 
                               
                               
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 t 
                               
                             
                             - 
                             
                               
                                 1 
                                 2 
                               
                                
                               
                                 ∇ 
                                 2 
                               
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                     
                      
                     
                         
                       
                         [ 
                         
                           
                             
                               
                                 μ 
                                 0 
                               
                                
                               
                                 σ 
                                  
                                 
                                   ( 
                                   
                                     
                                       A 
                                       
                                         n 
                                         + 
                                         1 
                                       
                                     
                                     - 
                                     
                                       A 
                                       n 
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               Δ 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                           + 
                           
                             
                               
                                 ∇ 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     A 
                                     n 
                                   
                                   + 
                                   
                                     A 
                                     
                                       n 
                                       + 
                                       1 
                                     
                                   
                                 
                                 ) 
                               
                             
                             2 
                           
                           + 
                           
                             
                               ∇ 
                               
                                 × 
                                 
                                   ( 
                                   
                                     
                                       M 
                                       n 
                                     
                                     + 
                                     
                                       M 
                                       
                                         n 
                                         + 
                                         1 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             2 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     By using the average value M n  in Equation (16), the analysis apparatus  10  calculates δA and calculates, from the calculated δA, new A n+1   1 =A n+1   0 +δA. Then, the analysis apparatus  10  calculates, from the new A n+1   1  using Equation (13), the average value M 1  and calculates new δA from the calculated average value M 1 . Furthermore, the analysis apparatus  10  calculates, from the calculated δA, A n+1   2 =A n+1   1 +δA. By repeatedly performing the calculations until δA becomes “0” or δA converges to a predetermined threshold ε or below, the analysis apparatus  10  can evolve both the average value M and the vector potential A over time. 
     As described above, because the time scale in which the average value M varies is much shorter than the time scale in which the vector potential A varies, the average value M can be assumed to be stationary during a period of time in which the vector potential A n  changes to A n+1 . Specifically, in the repeated calculations from A n+1   0  to A n+1   n , the average value M n  can be assumed to be in the stationary state with respect to the external magnetic field H n+1 . 
     As a result, in the repeated calculations performed as the vector potential A n  evolves over time by A n+1 , the analysis apparatus  10  can evolve both the vector potential A and the micro-magnetization vector m over time for a long time scale over which the vector potential A evolves. 
     In the following, a calculating method will be described in which, when calculations are repeatedly performed using Equation (15), a gradient of a governing equation is used to speed up the convergence of the increment δA. For example, if a term taking into consideration the gradient of the average value M is added to Equation (15), Equation (17) below is obtained. 
     
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               μ 
                               0 
                             
                              
                             σ 
                           
                           
                             Δ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         - 
                         
                           
                             1 
                             2 
                           
                            
                           
                             
                               ∇ 
                               2 
                             
                              
                             
                               + 
                               
                                 1 
                                 2 
                               
                             
                           
                            
                           
                             ∂ 
                             
                               ∂ 
                               A 
                             
                           
                            
                           
                             ∇ 
                             
                               × 
                               
                                 M 
                                 n 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     δ 
                      
                     
                         
                     
                      
                     A 
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                             μ 
                             0 
                           
                            
                           
                             σ 
                              
                             
                               ( 
                               
                                 
                                   A 
                                   
                                     n 
                                     + 
                                     1 
                                   
                                 
                                 - 
                                 
                                   A 
                                   n 
                                 
                               
                               ) 
                             
                           
                         
                         
                           Δ 
                            
                           
                               
                           
                            
                           t 
                         
                       
                     
                     + 
                     
                       
                         
                           ∇ 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               A 
                               n 
                             
                             + 
                             
                               A 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       2 
                     
                     + 
                     
                       
                         ∇ 
                         
                           × 
                           
                             ( 
                             
                               
                                 M 
                                 n 
                               
                               + 
                               
                                 M 
                                 
                                   n 
                                   + 
                                   1 
                                 
                               
                             
                             ) 
                           
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     At this stage, if variable transformation is performed on the differentiation of the average value M, Equation (18) below is obtained. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∂ 
                             
                               ∂ 
                               A 
                             
                           
                            
                           
                             ∇ 
                             
                               × 
                               M 
                             
                           
                         
                         = 
                         
                           
                             
                               ∂ 
                               B 
                             
                             
                               ∂ 
                               A 
                             
                           
                            
                           
                             ∂ 
                             
                               ∂ 
                               B 
                             
                           
                            
                           
                             ∇ 
                             
                               × 
                               M 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               
                                 ∂ 
                                 B 
                               
                               
                                 ∂ 
                                 A 
                               
                             
                              
                             
                               ∇ 
                               
                                 × 
                                 
                                   
                                     ∂ 
                                     M 
                                   
                                   
                                     ∂ 
                                     B 
                                   
                                 
                               
                             
                           
                           = 
                           
                             
                               1 
                               
                                 μ 
                                 0 
                               
                             
                              
                             
                               
                                 ∂ 
                                 B 
                               
                               
                                 ∂ 
                                 A 
                               
                             
                              
                             
                               ∇ 
                               
                                 × 
                                 
                                   
                                     ∂ 
                                     M 
                                   
                                   
                                     ∂ 
                                     H 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Because the analysis apparatus  10  cannot analytically calculate δM/δH, the analysis apparatus  10  calculates ΔM/ΔH by calculating an average value M* in which the external magnetic field is set as H=H+ΔH and using an equation ΔM=M*−M. In the calculation of ΔM/ΔH, a value ΔH obtained at the previous time step is used for ΔH. Specifically, ΔM/ΔH is calculated using Equation (19) below: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       M 
                     
                     
                       ∂ 
                       H 
                     
                   
                   = 
                   
                     
                       
                         Δ 
                          
                         
                             
                         
                          
                         M 
                       
                       
                         Δ 
                          
                         
                             
                         
                          
                         H 
                       
                     
                     = 
                     
                       
                         
                           M 
                           
                             n 
                             + 
                             1 
                           
                         
                         - 
                         
                           M 
                           n 
                         
                       
                       
                         Δ 
                          
                         
                             
                         
                          
                         
                           H 
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     If the analysis apparatus  10  performs the calculation using Equation (17) that takes into consideration the gradient term of such an average value M, the convergence speed of δA becomes fast; therefore, it is possible to evolve, over time at a faster pace, the vector potential A and the average value M. For example, in the example illustrated in  FIG. 8 , the analysis apparatus  10  sets a value, which is obtained by dividing the difference between M n+1  and M n  by ΔH n  obtained at the previous time step, as the gradient of the average value M.  FIG. 8  is a schematic diagram illustrating the magnetic field gradient. 
     Because the number of calculated average values M is finite, the M-H curves are not smooth and have minute irregularities. Because the gradient of the average values M is not accurate if the irregularities on the M-H curve are large, in some cases, the analysis apparatus  10  cannot converge δA at a high speed. 
     Accordingly, the analysis apparatus  10  calculates δH by dividing a saturation magnetic field of the object to be analyzed by the number of average values M. For example, the analysis apparatus  10  obtains the saturation magnetic field H sat  illustrated in  FIG. 9 .  FIG. 9  is a schematic diagram illustrating the saturation magnetic field. 
     Then, if the sum of δH and a predetermined coefficient α is smaller than ΔH, the analysis apparatus  10  estimates that the irregularities on the M-H curves are within the variation of ΔH and calculates an increment δA using Equation (17). In contrast, if the sum of δH and a predetermined coefficient α is greater than ΔH, the analysis apparatus  10  determines that the analysis apparatus  10  picks up the irregularities on the M-H curves and thus calculates the increment δA using Equation (16). 
     Specifically, by appropriately using Equation (8) to Equation (19) above, the analysis apparatus  10  calculates the vector potential A that is made to evolve over time using the average value M. 
     In the following, processes performed by each units included in the analysis apparatus  10  will be described. In the following description, it is assumed that the analysis apparatus  10  obtains, from the control device  4 , m 0  as initial values of the plurality of micro-magnetization vectors m. 
     Referring back to  FIG. 1 , the analysis apparatus  10  includes a receiving unit  11 , a magnetization vector average value calculating unit  12 , a vector potential calculating unit  13 , an external magnetic field calculating unit  14 , a micro-magnetization vector calculating unit  15 , a magnetic field gradient calculating unit  16 , and an output unit  17 . 
     The receiving unit  11  receives an input of property information on an object to be analyzed. For example, if the receiving unit  11  receives, from the control unit  4 , a plurality of micro-magnetization vectors m 0  that are set by a user, the receiving unit  11  transmits, to the magnetization vector average value calculating unit  12 , the obtained micro-magnetization vectors m 0 , the analysis conditions, and property information on the object to be analyzed. 
     The magnetization vector average value calculating unit  12  calculates, in accordance with the property information on the object to be analyzed received by the receiving unit  11 , the average value M 0  of the micro-magnetization vectors m 0  allocated to the object to be analyzed. Specifically, the magnetization vector average value calculating unit  12  obtains, from the receiving unit  11 , the micro-magnetization vectors m 0 , the analysis conditions, and the property information on the object to be analyzed. Then, using Equation (8), the magnetization vector average value calculating unit  12  calculates the average value M 0  from the obtained micro-magnetization vectors m 0 . Thereafter, the magnetization vector average value calculating unit  12  transmits, to the vector potential calculating unit  13 , the calculated average value M 0 , the analysis conditions, and the property information on the object to be analyzed. 
     Furthermore, if the magnetization vector average value calculating unit  12  obtains a plurality of micro-magnetization vectors m n  calculated by the micro-magnetization vector calculating unit  15 , which will be described later, the magnetization vector average value calculating unit  12  calculates an average value M n  of the obtained micro-magnetization vectors m. Then, the magnetization vector average value calculating unit  12  transmits the calculated average value M n  to the vector potential calculating unit  13 . 
     By using the average value M calculated by the magnetization vector average value calculating unit  12  and using an equation of the magnetic field that is a governing equation of the vector potential A, the vector potential calculating unit  13  calculates a vector potential A obtained after a predetermined time has elapsed. 
     In the following, a process performed by the vector potential calculating unit  13  will be specifically described. The first thing described in the following description will be a process in which the vector potential calculating unit  13  determines whether it treats the average value M explicitly or implicitly. Then, a process performed when the average value M is explicitly treated will be described, and then a process performed when the average value M is implicitly treated will be described. 
     First, a process in which the vector potential calculating unit  13  determines whether it treats the average value M explicitly or implicitly will be described. For example, the vector potential calculating unit  13  obtains, from the micro-magnetization vector calculating unit  15 , the average value M 0 , the analysis conditions, and the property information on the object to be analyzed. 
     Then, when the vector potential calculating unit  13  obtains both the analysis conditions and the property information on the object to be analyzed, the vector potential calculating unit  13  determines, in accordance with the obtained analysis conditions and the property information on the object to be analyzed, the time scale over which both the LLG equation and the equation of the magnetic field evolve. Then, if the time scale over which both the LLG equation and the equation of the magnetic field evolve is shorter than a predetermined threshold, the vector potential calculating unit  13  determines to explicitly treat the average value M. In contrast, if the time scale over which both the LLG equation and the equation of the magnetic field evolve is greater than a predetermined threshold, the vector potential calculating unit  13  determines to implicitly treat the average value M. 
     In the following, a process in which the vector potential calculating unit  13  explicitly treats the average value M will be specifically described. If the vector potential calculating unit  13  explicitly treats the average value M, the vector potential calculating unit  13  solves, using the average value M 0  obtained from the magnetization vector average value calculating unit  12 , Equation (10) and calculates the vector potential A 1  obtained after a predetermined time has elapsed. Then, the vector potential calculating unit  13  transmits the calculated vector potential A 1  to the external magnetic field calculating unit  14 . 
     Furthermore, the vector potential calculating unit  13  obtains a new average value M (for example, an average value M n  at the n th  time step) calculated by the magnetization vector average value calculating unit  12 . Then, the vector potential calculating unit  13  solves Equation (10) using the average value M n  and calculates the vector potential A n+1  obtained after a predetermined time has elapsed. Then, the vector potential calculating unit  13  transmits the calculated vector potential A n+1  to the external magnetic field calculating unit  14 . 
     In the following, a process in which the vector potential calculating unit  13  implicitly treats the average value M will be specifically described. First, if the vector potential calculating unit  13  determines to implicitly treat the average value M, the vector potential calculating unit  13  calculates the above-described δH and ΔH and determines whether the sum of the calculated δH and a predetermined coefficient α (for example, α=“3”) is greater than ΔH. 
     If the vector potential calculating unit  13  determines that the sum of δH and the predetermined coefficient α is greater than ΔH, the vector potential calculating unit  13  solves Equation (16) using the average value M 0  and calculates an increment δA of the vector potential. Then, the vector potential calculating unit  13  transmits the calculated δA to the magnetic field gradient calculating unit  16 . 
     In contrast, if the vector potential calculating unit  13  determines that the sum of δH and the predetermined coefficient is smaller than ΔH, the vector potential calculating unit  13  obtains the magnetic field gradient from the magnetic field gradient calculating unit  16 , which will be described later. Then, by solving Equation (17) using the obtained magnetic field gradient and the average value M 0 , the vector potential calculating unit  13  calculates the increment δA of the vector potential. Then, the vector potential calculating unit  13  transmits the calculated δA to the magnetic field gradient calculating unit  16 . 
     Furthermore, when the vector potential calculating unit  13  calculates the increment δA of the vector potential, the vector potential calculating unit  13  determines whether the absolute value of the calculated increment δA is smaller than the predetermined threshold ε. If the vector potential calculating unit  13  determines that the absolute value of the calculated increment ΔA is greater than the predetermined threshold ε, the vector potential calculating unit  13  calculates a new vector potential A n+1   1  by adding the vector potential A n+1   0  to the increment δA. 
     Furthermore, if the vector potential calculating unit  13  calculates the new vector potential A n+1   1 , the vector potential calculating unit  13  calculates, by solving Equation (1), Equation (6), and Equation (8) using the calculated vector potential A n+1   1 , a new average value M 1 . Then, the vector potential calculating unit  13  solves, using the average value M 1 , Equation (16) so that it calculates an increment δA of the new vector potential. The vector potential calculating unit  13  repeatedly performs a series of calculations until the absolute value of the increment δA of the vector potential becomes smaller than the predetermined threshold ε. 
     In contrast, if the vector potential calculating unit  13  determines that the absolute value of the increment δA is smaller than the predetermined threshold ε, the vector potential calculating unit  13  transmits, as the vector potential A n+1  to the external magnetic field calculating unit  14 , A n+1   i  calculated in the previous repeated calculation (for example, calculated at the i th  time). Furthermore, the vector potential calculating unit  13  transmits, as the vector potential A n+1  to the magnetic field gradient calculating unit  16 , the calculated A n+1   i . 
     In this way, the vector potential calculating unit  13  calculates, using the average value M, a vector potential A at the subsequent time step. Furthermore, as described above, the average value M can be assumed to be stationary with respect to the vector potential A. Accordingly, the vector potential calculating unit  13  can calculate the vector potential A using a long time step. 
     The external magnetic field calculating unit  14  calculates, in accordance with the vector potential A calculated by the vector potential calculating unit  13 , the external magnetic field H formed around the magnetic material. For example, the external magnetic field calculating unit  14  obtains the vector potential A 1  that is transmitted by the vector potential calculating unit  13 . Then, the external magnetic field calculating unit  14  solves Equation (6) using the obtained vector potential A 1  so that it calculates the external magnetic field H 1 . Then, the external magnetic field calculating unit  14  transmits the calculated external magnetic field H 1  to the micro-magnetization vector calculating unit  15 . 
     The micro-magnetization vector calculating unit  15  calculates, in accordance with the external magnetic field H calculated by the external magnetic field calculating unit  14 , new micro-magnetization vectors m used for the magnetic material. For example, the micro-magnetization vector calculating unit  15  obtains the external magnetic field H 1  transmitted by the external magnetic field calculating unit  14 . Then, by using the obtained external magnetic field H 1 , the micro-magnetization vector calculating unit  15  solves Equation (1) for each mesh of the magnetic material so that it calculates new micro-magnetization vectors m 1 . Then, the micro-magnetization vector calculating unit  15  transmits the calculated micro-magnetization vectors m 1  to the output unit  17  and the magnetization vector average value calculating unit  12 . 
     Specifically, using each unit  12  to  15 , the analysis apparatus  10  calculates the average value M n  of the micro-magnetization vectors m n  and calculates, from the calculated average value M n , a vector potential A n+1  that is made to evolve, to the subsequent time step, over time. Then, the analysis apparatus  10  calculates, from the calculated vector potential A n+1 , an external magnetic field H n+1  and calculates, from the calculated external magnetic field H n+1 , new micro-magnetization vectors m n+1 . By repeatedly performing this process, the analysis apparatus  10  can solve the LLG equation and the equation of the magnetic field as both the micro-magnetization vector m and the vector potential A evolve over time. 
     The magnetic field gradient calculating unit  16  calculates the magnetic field gradient by dividing the increment of the average value M by the increment of the external magnetic field H. For example, the magnetic field gradient calculating unit  16  obtains both the increment δA and the vector potential A n+1  calculated by the vector potential calculating unit  13 . Then, if repeated calculations according to the implicit method are not performed, the magnetic field gradient calculating unit  16  calculates ΔH using Equation (20) below: 
     
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     H 
                   
                   = 
                   
                     
                       1 
                       
                         μ 
                         0 
                       
                     
                      
                     
                       ∇ 
                       
                         × 
                         δ 
                          
                         
                             
                         
                          
                         A 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Furthermore, if repeated calculations according to the implicit method are performed multiple times (for example, n times), the magnetic field gradient calculating unit  16  calculates ΔH by solving Equation (21) using both the obtained A n+1  and the previously obtained A n . 
     
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     H 
                   
                   = 
                   
                     
                       1 
                       
                         μ 
                         0 
                       
                     
                      
                     
                       ∇ 
                       
                         × 
                         
                           ( 
                           
                             
                               A 
                               
                                 n 
                                 + 
                                 1 
                               
                             
                             - 
                             
                               A 
                               n 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Furthermore, the magnetic field gradient calculating unit  16  calculates, from the previously obtained A n , an external magnetic field H n  and calculates H* that is the sum of the calculated H n  and ΔH. Furthermore, the magnetic field gradient calculating unit  16  solves Equation (1) and Equation (8) using the calculated H* and H n  and calculates the average value M* and the average value M n . Thereafter, the magnetic field gradient calculating unit  16  calculates the difference δM between the calculated average value M* and the average value M n . Then, the magnetic field gradient calculating unit  16  transmits, to the vector potential calculating unit  13 , ΔM/ΔH as the magnetic field gradient. 
     The output unit  17  outputs the new micro-magnetization vectors m calculated by the micro-magnetization vector calculating unit  15 . For example, the output unit  17  obtains, from the micro-magnetization vector calculating unit  15 , the micro-magnetization vectors m 1  to m n+1 . Then, as illustrated in  FIG. 10 , the output unit  17  creates, in accordance with the obtained micro-magnetization vectors m 1  to m n+1 , an image in which the properties of the magnetic material are simulated. Then, the output unit  17  displays, on the monitor  6  by using the control unit  4 , the created image.  FIG. 10  is a schematic diagram illustrating an example of output results. 
     For example, the control unit  4  and the analysis apparatus  10  are electronic circuits. Furthermore, the receiving unit  11 , the magnetization vector average value calculating unit  12 , the vector potential calculating unit  13 , the external magnetic field calculating unit  14 , the micro-magnetization vector calculating unit  15 , the magnetic field gradient calculating unit  16 , and the output unit  17  are electronic circuits. In the embodiments, examples of the electronic circuit include an integrated circuit, such as an application specific integrated circuit (ASIC) or a field programmable gate array (FPGA), a central processing unit (CPU), and a micro processing unit (MPU). 
     The memory  2  is a semiconductor memory device, such as a random access memory (RAM), a read only memory (ROM), and a flash memory, or it is a storing unit, such as a hard disc drive and an optical disk. 
     In the following, a process, performed by the analysis apparatus  10 , for evolving the external magnetic field H over time by using an explicit expression M will be described with reference to  FIG. 11 .  FIG. 11  is a schematic diagram illustrating an example of a process for evolving time by using the explicit expression M. In the example illustrated in  FIG. 11 , the analysis apparatus  10  calculates, from a stationary average value M 0 , a non-stationary external magnetic field H 1  obtained after 10 −6  seconds and calculates, from the calculated external magnetic field H 1 , a stationary average value M 1 . Accordingly, because the analysis apparatus  10  calculates the external magnetic field H that is made to evolve over time by using the average value M, the analysis apparatus  10  can evolve the time for a longer time scale than the conventional information processing apparatus that calculates a micro-magnetization vector m that is made to evolve over time by using the external magnetic field H. 
     In the following, a process, performed by the analysis apparatus  10 , for evolving the vector potential A over time using an implicit expression M will be described with reference to  FIG. 12 .  FIG. 12  is a schematic diagram illustrating an example of a process for evolving time by using the implicit expression M. In the example illustrated in  FIG. 12 , the analysis apparatus  10  calculates, using A n+1   1 =A n +δA, a stationary state M, and calculates, from the calculated M 1 , A n+1   2 =A n   1 +δA. By repeatedly performing such calculations until δA becomes equal to or less than a predetermined threshold ε, the analysis apparatus  10  can calculate, from A n , A n+1  after 10 −6  seconds. 
     In the following, there will be a description, with reference to  FIG. 13 , of the flow of a process, performed by the analysis apparatus  10 , in which time evolves by using the explicit expression M.  FIG. 13  is a flowchart illustrating the flow of a process for evolving time by using the explicit expression M. 
     First, the analysis apparatus  10  initializes, to “0”, n that represents the number of times that time is made to evolve (Step S 101 ). Then, the analysis apparatus  10  solves Equation (10) using an average value M n  and calculates a vector potential A n+1  (Step S 102 ). Subsequently, the analysis apparatus  10  solves Equation (6) using the calculated vector potential A n+1  and calculates an external magnetic field H n+1  (Step S 103 ). Furthermore, the analysis apparatus  10  solves, using the calculated external magnetic field H n+1 , Equation (1) and Equation (8) and calculates an average value M n+1  of the micro-magnetization vectors with respect to the external magnetic field H n+1  (Step S 104 ). 
     Then, the analysis apparatus  10  determines whether the number of times n that time is made to evolve is greater than a predetermined value “Nall” (Step S 105 ). Here, Nall is, for example, the time scale for evolving both the micro-magnetization vector m and the vector potential A over time and is a value divided by the period of time for simulating the properties of the magnetic material set by a user. If the analysis apparatus  10  determines that the number of times n that time is made to evolve is smaller than a predetermined value “Nall” (No at Step S 105 ), the analysis apparatus  10  adds “1” to n (Step S 106 ) and calculates a new vector potential A n+1  (Step S 102 ). 
     In contrast, if the analysis apparatus  10  determines that the number of times n that time is made to evolve is greater than the predetermined value “Nall” (Yes at Step S 105 ), the analysis apparatus  10  ends the calculation (Step  5107 ) and then ends the process. 
     In the following, there will be a description, with reference to  FIG. 14 , of the flow of a process, performed by the analysis apparatus  10 , in which time evolves by using an implicit expression M.  FIG. 14  is a flowchart illustrating the flow of a process for evolving time by using the implicit expression M. In the example illustrated in  FIG. 14 , it is assumed that the analysis apparatus  10  repeatedly performs calculations without using a magnetic field gradient. 
     First, the analysis apparatus  10  initializes, to “0”, both i that is the number of repeated calculations and n that represents the number of times that time is made to evolve (Step S 201 ). Then, the analysis apparatus  10  solves Equation (16) using the average value M n  and calculates δA (Step S 202 ). The analysis apparatus  10  then calculates A n+1   i+1 =A n+1   i +δA (Step S 203 ). Furthermore, the analysis apparatus  10  determines whether the absolute value of δA is greater than the predetermined threshold ε (Step S 204 ). 
     If the analysis apparatus  10  determines that the absolute value of δA is greater than the predetermined threshold ε (Yes at Step S 204 ), the analysis apparatus  10  solves Equation (6) using A n+1   i+1  and calculates an external magnetic field H n+1   1+1  (Step S 205 ). Then, the analysis apparatus  10  solves, using the external magnetic field H n+1   i+1 , Equation (1) and Equation (8) and calculates an average value M n+1   1+1  of the micro-magnetization vectors with respect to the external magnetic field H n+1   i+1  (Step S 206 ). By adding “1” to i (Step S 207 ), the analysis apparatus  10  solves Equation (16) using M n+1   i+1  and again calculates δA (Step S 202 ). 
     In contrast, if the analysis apparatus  10  determines that the absolute value of δA is smaller than the predetermined threshold ε (No at Step S 204 ), the analysis apparatus  10  sets A n+1   i+1  as A n+1  (Step S 208 ). Specifically, if the analysis apparatus  10  determines that the absolute value of δA is smaller than the predetermined threshold ε, the analysis apparatus  10  sets, as the result of repeated calculations with respect to δ, the calculated A n+1   i+1  as a new vector potential A n+1 . Then, the analysis apparatus  10  determines whether the number of times n that time is made to evolve is greater than the predetermined value “Nall” (Step S 209 ). 
     If the analysis apparatus  10  determines that the number of times n that time is made to evolve is smaller than the predetermined value “Nall” (No at Step S 209 ), the analysis apparatus  10  adds “1” to n and initializes i to “0” (Step S 210 ). Thereafter, the analysis apparatus  10  again calculates δA (Step S 202 ). Furthermore, if the analysis apparatus  10  determines that the number of times n that time is made to evolve is greater than the predetermined value “Nall” (Yes at Step S 209 ), the analysis apparatus  10  ends the calculation (Step S 211 ) and then ends the process. 
     In the following, there will be a description, with reference to  FIG. 15 , of a process, performed by the analysis apparatus  10 , in which time evolves by using both the magnetic field gradient and the implicit expression M.  FIG. 15  is a flowchart illustrating the flow of a process, in which time evolves, using a magnetic field gradient. In the process illustrated in  FIG. 15 , Steps S 302  to S 306  and S 308  are the same as Steps S 202  to S 206  and S 207  illustrated in  FIG. 14 ; therefore, descriptions thereof will be omitted. Furthermore, Steps S 309  to S 312  are the same as Steps S 208  to S 211  illustrated in  FIG. 14 ; therefore, descriptions thereof will be omitted. 
     First, the analysis apparatus  10  initializes i, which is the number of repeated calculations, to “0”; initializes n, which represents the number of times that time is made to evolve, to “0”; and initializes AM/AH, which is the magnetic field gradient, to “0”; and initializes the predetermined coefficient α to “3” (Step S 301 ). Furthermore, if i is “0”, the analysis apparatus  10  calculates ΔH using Equation (20), whereas if i is not “0”, the analysis apparatus  10  calculates ΔH using Equation (21). Then, the analysis apparatus  10  calculates the magnetic field gradient ΔM/ΔH using the calculated ΔH (Step S 307 ). 
     Advantage of First Embodiment 
     As described above, the information processing apparatus  1  according to the first embodiment calculates an average value M n  of a plurality of micro-magnetization vectors m allocated to the magnetic material that corresponds to the object to be analyzed. Then, by using the calculated average value M n  and the equation of the magnetic field that is the governing equation of the vector potential, the information processing apparatus  1  calculates a vector potential A n+1  at the subsequent time step. Accordingly, the information processing apparatus  1  can evolve both the LLG equation and the equation of the magnetic field over time for a long time scale, thus simulating the properties of the magnetic material over a long period of time. 
     Furthermore, by using Equation (10) that explicitly represents the average value M of the magnetization vectors that is derived from both LLG Equation (1) and from Equation (7) of the magnetic field and by using the average value M n  of the magnetization vector, the information processing apparatus  1  calculates a vector potential A n+1  obtained after a predetermined time has elapsed. Accordingly, the information processing apparatus  1  can calculate, in a single calculation, the vector potential A n+1  at the subsequent time step. As a result, because the information processing apparatus  1  reduces the amount of calculation in which both the LLG equation and the equation of the magnetic field are made to evolve over time, the information processing apparatus  1  can simulate the properties of the magnetic material over a further longer period of time. 
     Furthermore, by using Equation (13) that implicitly represents the average value M of the magnetization vectors derived from LLG Equation (1) and from Equation (7) of the magnetic field and by using the average value M n  of the micro-magnetization vectors, the information processing apparatus  1  calculates the increment δA of the vector potential. Then, in accordance with the calculated increment δA of the vector potential, the information processing apparatus  1  calculates the vector potential A n+1  at the subsequent time step. Accordingly, even when the implicit method is used, the information processing apparatus  1  can evolve the LLG equation and the equation of the magnetic field over time for a long time scale. As a result, it is possible to accurately simulate the properties of the magnetic material for a long period of time. 
     Furthermore, during the process for evolving time from A n  to A n+1 , when repeatedly performing calculations according to the implicit method, the information processing apparatus  1  calculates, from the average value M n   i  of the micro-magnetization vectors, the increment δA of the vector potential. Then, the information processing apparatus  1  calculates, in accordance with the calculated increment δA of the vector potential, a new vector potential A n+1   i+1 . 
     Specifically, in the repeated calculations of the vector potential from A n  to A n+1 , the information processing apparatus  1  calculates, from the vector potential A n+1   0 , A n+1   i+1  by assuming each of the average values M n   0  to M n   i+1  of the micro-magnetization vectors to be stationary. Accordingly, in each Step of the repeated calculations according to the implicit method, the information processing apparatus  1  can evolve both the LLG equation and the equation of the magnetic field for a longer time scale, thus further extending the period of time for simulating the properties of the magnetic material. 
     Furthermore, the information processing apparatus  1  speeds up, using the magnetic field gradient ΔM/ΔH, the convergence of δA in the repeated calculations. Accordingly, the information processing apparatus  1  can reduce the number of repeated calculations according to the implicit method; therefore, it is possible for the information processing apparatus  1  to reduce the amount of calculation and to further extend the period of time for simulating the properties of the magnetic material. 
     Furthermore, if the sum of a predetermined coefficient α and δH, where δH is obtained by dividing the saturation magnetic field H sat  by the number of micro-magnetization vectors m allocated to the magnetic material, is greater than the increment ΔH of the external magnetic field, the information processing apparatus  1  performs a calculation using the magnetic field gradient ΔM/ΔH. Specifically, if the sum of δH and α is greater than the increment ΔH of the external magnetic field, irregularities produced on the M-H curve are large; therefore, the gradient of the average values M is not accurate. Accordingly, if the sum of ΔH and the predetermined coefficient α is smaller than the increment ΔH of the external magnetic field, the information processing apparatus  1  can use the appropriate magnetic field gradient ΔM/ΔH. As a result, the information processing apparatus  1  can appropriately speed up the convergence of δA in the repeated calculations. 
     [b] Second Embodiment 
     The embodiments of the present invention have been described; however, the present invention is not limited to the embodiments described above and can be implemented with various kinds of embodiments other than the embodiments described above. Accordingly, another embodiment included in the present invention will be described as a second embodiment. 
     (1) Predetermined Coefficient α 
     If the sum of δH and the predetermined coefficient α is smaller than ΔH, the analysis apparatus  10  according to the first embodiment described above estimates that the irregularities on the M-H curves are within the variation of ΔH. When evolving time using the magnetic field gradient and the implicit expression M, the analysis apparatus  10  initializes a predetermined coefficient α to “3”; however, the embodiment is not limited thereto. The predetermined coefficient α can be another value. For example, predetermined coefficient α can be a value that can appropriately be changed in accordance with the properties of the magnetic material that is the object to be analyzed or changed in accordance with the number of micro-magnetization vectors m allocated to the magnetic material. 
     (2) Time Scale Over Which Time Evolves 
     The analysis apparatus  10  according to the first embodiment described above evolves both the LLG equation and the equation of the magnetic field over time at time intervals of 10 −6 ; however, the embodiment is not limited thereto. Any given time step can be used so long as the vector potential A can be accurately calculated at the subsequent time step. Furthermore, for example, the time scale of the time step can be changed for each calculation. For example, the analysis apparatus  10  can evolve both the LLG equation and the equation of the magnetic field over time for a given time scale between approximately 10 −6  and 10 −12  seconds. 
     (3) Information processing apparatus 
     The information processing apparatus  1  according to the first embodiment has a single analysis apparatus  10 ; however, the embodiment is not limited thereto. For example, an information processing apparatus  1   a  according to the second embodiment has a plurality of analysis apparatuses  10 A to  10 Z that have the same function as that performed by the analysis apparatus  10 . If the size of the magnetic material that is the object to be analyzed is large, the information processing apparatus la divides the magnetic material into regions A to Z and allows the analysis apparatuses  10 A to  10 Z to perform a parallel calculation of the properties of each of the divided regions A to Z. 
     Calculating the average value M from the micro-magnetization vectors m can be independently performed for each of the regions A to Z. Accordingly, the information processing apparatus  1   a  can accurately calculate, in a short period of time, the properties of the magnetic material that is the object to be analyzed over a long period of time. 
     (4) Program 
     With the analysis apparatus  10  according to the first embodiment, a case in which various processes are implemented using hardware has been described; however, the embodiments are not limited thereto. For example, it is possible to implement an analysis program prepared in advance and executed by a computer. Accordingly, in the following, a computer that executes the analysis program having the same functions as those performed by the analysis apparatus  10  described in the first embodiment will be described as an example with reference to  FIG. 16 .  FIG. 16  is a functional block diagram illustrating a computer that executes the analysis program. 
     In a computer  100  illustrated in  FIG. 16 , a random access memory (RAM)  120 , a read only memory (ROM)  130 , and a hard disk drive (HDD)  150  are connected via a bus  170 . Furthermore, in the computer  100  illustrated in  FIG. 16 , a central processing unit (CPU)  140  is connected via the bus  170 . Furthermore, an input/output (I/O)  160  that receives an input from a user is connected to the bus  170 . 
     The ROM  130  stores therein, in advance, a receiving program  131  and a vector potential calculating program  132 . In the example illustrated in  FIG. 16 , the CPU  140  reads each of the programs  131  and  132  from the ROM  130  and executes them so that the programs  131  and  132  functions as a receiving process  141  and a vector potential calculating process  142 , respectively. Furthermore, each of the processes  141  and  142  has the same function, respectively, as that performed by the units  11  and  12  illustrated in  FIG. 1 . Each of the processes  141  and  142  can also have the same function as that performed by each unit according to the second or third embodiment. 
     The analysis program in the embodiments can be implemented by a program prepared in advance and executed by a computer, such as a personal computer or a workstation. The program can be sent using a network such as the Internet. Furthermore, the program can be stored in a computer-readable recording medium, such as a hard disc drive, a flexible disk (FD), a compact disc read only memory (CD-ROM), a magneto optical disc (MO), and a digital versatile disc (DVD). Furthermore, the program can also be implemented by the computer reading it from the recording medium. 
     According to an embodiment, it is advantageously possible to simulate the properties of magnetic materials for a long period of time. 
     All examples and conditional language recited herein are intended for pedagogical purposes to aid the reader in understanding the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions, nor does the organization of such examples in the specification relate to a showing of the superiority and inferiority of the invention. Although the embodiments of the present invention have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention.