Method for estimating dielectric electromagnetic effects based on sigmoid variable step FETD

Abstract

The application discloses a dielectric electromagnetic effect estimation method based on a sigmoid variable step FETD. The method comprises the following steps: constructing a simulation model of a dielectric and an estimation region under the excitation of a preset plane electromagnetic wave and performing grid division, so as to establish a time domain iteration matrix of the estimation region; performing time step iteration on the time domain iteration matrix within a preset total time, to obtain the error mean of the single-step space electric field intensity of the estimation region; using a variable step adjustment method based on a sigmoid function to update the preset time step until the iteration is stopped after the preset simulation time is reached, to obtain the electric field distribution of the estimation region of the dielectric within the preset total time, so as to obtain the electromagnetic effect parameters of the estimation region. The application can realize the dynamic self-adaptation of the time step in the time domain finite element method, improve the estimation efficiency, reduce the estimation time, more accurately control the estimation precision of the electric field, suppress the numerical oscillation of the estimation precision and improve the stability, and has a strong practical engineering application value.

Description

TECHNICAL FIELD

[0001] The application relates to a dielectric electromagnetic effect estimation method, in particular to a dielectric electromagnetic effect estimation method based on Sigmoid variable step FETD. BACKGROUND

[0002] When solving some structures with high precision requirements, there are two choices for the time domain finite element simulation of electromagnetic fields: 1. reducing the mesh size, using denser meshes to discretize the target object or increasing the order of the basis function to control the solving precision; 2. using a step-adaptive time domain finite element FETD (finite element time domain) algorithm in the time domain to control the solving precision. However, the former method will bring more unknowns, and the selection of the time step will be limited by the minimum mesh size. Moreover, for large-size electric structures, the calculation time is very long by using the high-order basis function method. Compared with the former method, controlling the time step in the time domain is more stable and flexible in manipulating the solving precision, and can also significantly improve the calculation efficiency. The existing variable step accelerating time domain finite element FETD method mainly includes the bisection method and the variable step method based on the exponential function, both of which can improve the solving efficiency to a certain extent, but the stability of the solving precision needs to be improved. SUMMARY

[0003] In order to solve the problems in the background art, the application provides a dielectric electromagnetic effect estimation method based on Sigmoid variable step FETD. Based on the fixed step fourth-order single diagonal implicit Runge-Kutta SDIRK4 (Single Diagonal Implicit Runge-Kutta 4th order) difference format and its bisection variable step acceleration strategy, the estimation precision and step-adaptive method of the Sigmoid-based variable step strategy are researched, which can more quickly and efficiently and stably estimate the electromagnetic field target problem.

[0004] The technical scheme adopted by the application is:

[0005] The dielectric electromagnetic effect estimation method based on Sigmoid variable step FETD of the application comprises:

[0006] In the first step, a simulation model of a dielectric body and an estimation region thereof under the excitation of a preset plane electromagnetic wave is constructed and meshed, so as to establish a time domain iteration matrix of the estimation region of the dielectric body in the simulation model.

[0007] In the second step, the time domain iteration matrix is iterated by time steps within a preset total time, and the error mean of the single-step space electric field intensity of the estimation region is obtained every preset time step.

[0008] Thirdly, according to the error mean value of the single-step space electric field intensity of the estimated region, the preset time step is updated using a variable step adjustment method based on a Sigmoid function improvement, and the next time step is iteratively estimated using the updated time step until the preset simulation time is reached to stop iteration, thereby obtaining the electric field distribution of the estimated region of the dielectric body in the preset total time, and thus obtaining the electromagnetic effect parameter of the estimated region.

[0009] In the first step, the estimated region of the dielectric body is the outer edge region of the dielectric body; the simulation model is meshed using a tetrahedral mesh, and the structural information of the meshed simulation model is obtained, including node information and element information after meshing, the node information including node serial number and node coordinates, and the element information including element serial number and node serial number contained in each element; simulation parameters are set, and the structural information of the model can be read.

[0010] In the first step, the time-domain iterative matrix of the estimated region of the dielectric body in the simulation model is as follows:

[0011]

[0012]

[0013]

[0014]

[0015]

[0016]

[0017] wherein, , and respectively represent the first, second and third coefficient matrices obtained in the construction of the time-domain iterative matrix, i and j respectively represent the first and second dimension lengths of the first, second and third coefficient matrices; E represents the electric field intensity in the estimated region of the dielectric body under the excitation of a preset plane electromagnetic wave; t is the time; represents the external electromagnetic wave excitation on the i-th grid node; and respectively represent the dielectric constant and the magnetic permeability; and respectively represent the first-order basis function on the i-th grid node and the weighted shape function on the j-th grid node; represents the overall range of the estimated region; represents the electrical conductivity; and respectively represent the boundary of the estimated region and the Neumann boundary part therein; Y represents an intermediate parameter; denotes the normal vector of the unit grid surface; denotes the Hamiltonian operator; denotes the preset current density excitation in the estimation region; denotes the weighted shape function at the i-th Neumann boundary; is a known function on the Neumann boundary; denotes the total number of grid nodes on the Dirichlet boundary; denotes the weighted shape function at the j-th Dirichlet boundary grid node; denotes the electric field intensity at the j-th Dirichlet boundary grid node; denotes the wave number of the electromagnetic wave, denotes the normalized surface impedance on the Neumann boundary.

[0018] In the second step, according to the electric field intensity of the preset plane electromagnetic wave excitation of the estimation region at the initial time, the time step iteration of the finite element time domain (FETD) is performed on the time domain iteration matrix using the fourth-order single-diagonal implicit Runge-Kutta (SDIRK4) difference format within the preset total time, and two similar Butcher matrices are randomly used during the time step iteration; the error mean of the single-step space electric field intensity of the estimation region obtained is as follows:

[0019]

[0020] wherein n denotes the total number of nodes of the estimation region divided by the tetrahedral grid; and respectively denote the electric field intensity of the k-th node in the estimation region at the time obtained by the fourth-order single-diagonal implicit Runge-Kutta (SDIRK4) difference format using two similar Butcher matrices.

[0021] In the third step, the error mean of the single-step space electric field intensity of the estimation region is taken as the estimation accuracy at the current time, and the time step is updated using a variable step adjustment method based on the improved Sigmoid function, as follows:

[0022]

[0023]

[0024] wherein and respectively denote the updated time step at the current time t and the next time t+1; k denotes the step adjustment factor based on the improved Sigmoid function.​​ and respectively represent the first and second adjustment parameters; represents the estimated accuracy at the current time t, represents the preset estimated accuracy.

[0025] In the third step, the adjustment of the time step is based on the preset deviation threshold The adjustment is as follows:

[0026] If , the current time step is not adjusted;

[0027] If , the time step is updated using a variable step adjustment method based on a Sigmoid function, and the time step is reduced;

[0028] If , the time step is updated using a variable step adjustment method based on a Sigmoid function, and the time step is increased.

[0029] In the third step, in the iteration process, the electric field intensity at each node in the simulation model is updated by the time domain iteration matrix of the estimated region of the dielectric body until the iteration ends, and the final electric field intensity at each node is obtained, and then the electric field distribution of the estimated region of the dielectric body is obtained.

[0030] According to the size relationship between the current estimated accuracy and the preset estimated accuracy, the size of the time step at the next time is adjusted, so that the time step length of the finite element time domain (FETD) method is self-adaptive, and then the iteration matrix equation of the fourth-order single diagonal implicit Runge-Kutta (SDIRK4) difference format is used to update the electric field intensity at the time, and each point in the space can obtain the correct electric field intensity value, and the process of time step updating is completed; repeat the steps until the time iteration ends.

[0031] The electronic device of the application comprises a memory and a processor coupled to each other, wherein the memory stores program data, and the processor calls the program data to execute the method as described above.

[0032] The computer readable storage medium of the application has program data stored thereon, and the program data is executed by the processor to implement the method as described above.

[0033] The beneficial effects of the application are:

[0034] The application can realize dynamic self-adaption of time step in time domain finite element method, compared with current variable step length methods such as time domain finite element FETD fixed step length algorithm and dichotomy, improves the estimation efficiency of time domain finite element method, reduces the estimation time, can more accurately control the estimation precision of electric field, suppresses the numerical oscillation of estimation precision, improves the stability of estimation precision, and has strong practical engineering application value. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 is a schematic diagram of the effect of a metal cylinder on incident plane electromagnetic waves in the simulation method of the application;

[0036] Figure 2 is a schematic diagram of the effect of a metal cylinder on incident plane electromagnetic waves at the same time in the dichotomy variable step length time domain finite element FETD method simulation;

[0037] Figure 3 is a time step change diagram obtained by using the method of the application;

[0038] Figure 4 is a schematic diagram of the Sigmoid improved step length adjustment function of the application;

[0039] Figure 5 is a comparison diagram of estimation precision change with time step obtained by using the method of the application and the traditional double-length variable step length method in the estimation region. DETAILED DESCRIPTION

[0040] The application will be further described in detail below in combination with the drawings and specific embodiments.

[0041] The dielectric body electromagnetic effect estimation method of the Sigmoid-based variable step length FETD of the application is as follows:

[0042] Firstly, a simulation model of a dielectric body and its estimation region under preset plane electromagnetic wave excitation is constructed and meshed, so as to establish a time domain iteration matrix of the estimation region of the dielectric body in the simulation model; the estimation region of the dielectric body is the outer edge region of the dielectric body; the simulation model is meshed by using a tetrahedral mesh, the structure information of the simulation model is obtained after meshing, including node information and element information after meshing, the node information includes node serial number and node coordinates, and the element information includes element serial number and node serial number contained in each element; simulation parameters are set, and the structure information of the model can be read.

[0043] The time domain iteration matrix of the estimation region of the dielectric body in the simulation model is as follows:

[0044]

[0045]

[0046]

[0047]

[0048]

[0049]

[0050] wherein, , and denote the first, second and third coefficient matrices obtained in the time-domain iterative matrix construction, i and j denote the first and second dimension length of the first, second and third coefficient matrices, respectively; E denotes the electric field intensity in the estimation region of the dielectric body under the preset plane electromagnetic wave excitation; t is the time; denotes the applied electromagnetic wave excitation on the i-th grid node; and denote the permittivity and permeability, respectively; and denote the first-order basis function on the i-th grid node and the weighted shape function on the j-th grid node, respectively; denotes the overall range of the estimation region; denotes the conductivity; and denote the boundary of the estimation region and the Neumann boundary part therein, respectively; Y denotes the intermediate parameter; denotes the normal vector of the unit grid surface; denotes the Hamiltonian operator; denotes the preset current density excitation in the estimation region; denotes the weighted shape function at the i-th Neumann boundary; is the known function on the Neumann boundary; denotes the total number of grid nodes on the Dirichlet boundary; denotes the weighted shape function at the j-th Dirichlet boundary grid node; denotes the electric field intensity at the j-th Dirichlet boundary grid node; denotes the wave number of the electromagnetic wave, denotes the normalized surface impedance on the Neumann boundary .

[0051] When establishing the time-domain iteration matrix, the wave equation containing only the electric field vector is first derived based on Maxwell's equations. Then, based on the known plane electromagnetic wave excitation set in the estimation region, the Galerkin method is used to test both sides of the equation, and the basis functions are expanded to obtain the final time-domain iteration formula. The time-domain iteration matrix is ​​then filled and calculated according to the iteration formula.

[0052] The first two equations of Maxwell's equations in the time domain, with electric field strength E and magnetic field strength H as unknowns in the estimation region, are constructed as follows:

[0053]

[0054]

[0055] Eliminating the magnetic field strength H from the above equation yields a wave equation containing only the electric field vector. Performing a Galerkin test on the equation and then expanding it using basis functions, we get:

[0056]

[0057] The equation was subjected to Galerkin testing, and the electric field strength E and magnetic field strength H were expanded using basis functions, as follows:

[0058] Eliminating the magnetic field from the first two equations of Maxwell's equations in the time domain, we obtain the wave equation containing only the electric field vector as follows:

[0059]

[0060] Substituting the typical homogeneous Dirichlet boundary condition and the mixed boundary condition of the impedance surface:

[0061] ,exist superior

[0062] ,exist superior

[0063] in, Represents the Dirichlet boundary portion of the estimation region. The tangential electric field value on; This represents the relative permeability.

[0064] Substituting the formula, the Galerkin test is applied to the wave equation containing Dirichlet and mixed boundary conditions. Through vector identities and divergence theorem, the weak solution expression for this boundary value problem is obtained as follows:

[0065]

[0066] Use basis functions to define the electric field vector The base function in the expansion is selected in the same form as the weighted shape function, and the time-domain iterative matrix is obtained by arrangement.

[0067] When filling , , and , the order is filled according to the encoding of the tetrahedral unit, and the matrix obtained has the block diagonal characteristic because each unit has no interaction with other units.

[0068] Secondly, the time-domain iterative matrix is iterated in time steps within a preset total time, and the error mean of the single-step spatial electric field intensity of the estimated region is obtained every preset time step; the fourth-order single-diagonal implicit Runge-Kutta (SDIRK4) difference format is used to perform time-step iteration of the finite element time-domain (FETD) within the preset total time according to the electric field intensity of the preset plane electromagnetic wave excitation of the estimated region at the initial time, and two similar Butcher matrices are randomly used during the time-step iteration; the error mean of the single-step spatial electric field intensity of the estimated region obtained is as follows:

[0069]

[0070] wherein n represents the total node number of the estimated region divided by the tetrahedral grid; and respectively represent the electric field intensity of the kth node in the estimated region at the time obtained by using two similar Butcher matrices of the fourth-order single-diagonal implicit Runge-Kutta (SDIRK4) difference format.

[0071] The matrix equation is numerically approximated according to the difference format of the fourth-order single-diagonal implicit Runge-Kutta (SDIRK4), and the following equation is obtained:

[0072]

[0073]

[0074]

[0075] wherein , and respectively represent the estimated electric field intensity within the single time step at the time , and when the iteration is performed according to the difference format of the fourth-order single-diagonal implicit Runge-Kutta (SDIRK4), and​​​ denote the elements of the c vector in the Butcher matrix, denotes the single time step; denotes denotes the initial electric field intensity at time denotes the element of the a matrix in the Butcher matrix; denotes the n-order unit matrix; f() denotes an intermediate variable; denotes the element of the b vector in the Butcher matrix; denotes the T matrix in the time-domain iterative matrix equation; denotes the electric field intensity denotes the external electromagnetic wave excitation at time and denote the S matrix and the R matrix in the time-domain iterative matrix equation, respectively; denotes the derivative of the electric field intensity .

[0076] In the estimation region, the variable step strategy calculates the average single-step error of the electric field intensity at time in the fourth-order single-diagonal implicit Runge-Kutta SDIRK4 difference format under two different pre-given estimation Butcher matrices.

[0077] The selection examples of the two pre-set estimation Butcher matrices are as follows:

[0078]

[0079]

[0080]

[0081]

[0082]

[0083]

[0084] Thirdly, according to the average single-step error of the spatial electric field intensity in the estimation region, the pre-set time step is updated using the variable step adjustment method based on the improved Sigmoid function, and the next time step is iteratively estimated using the updated time step, until the pre-set simulation time is reached to stop the iteration, and the electric field distribution of the estimation region of the dielectric body in the pre-set total time is obtained, thereby obtaining the electromagnetic effect parameters of the estimation region.

[0085] The mean error of the single-step spatial electric field strength of the estimated region is taken as the estimation accuracy at the current time, and a variable step size adjustment method based on a Sigmoid function is used to update the time step, as follows:

[0086]

[0087]

[0088] wherein, and respectively represent the updated time step at the current time t and the next time t+1; k represents a step size adjustment factor based on a Sigmoid function improvement; and respectively represent first and second adjustment parameters; represents the estimation accuracy at the current time t, represents a preset estimation accuracy.

[0089] The first parameter A of the step size adjustment factor is used to control the maximum adjustment range of the step size of the next time step, and the second parameter B is used to adjust the stretching of the Sigmoid improvement function, thereby controlling the sensitivity of the step size adjustment amplitude and the deviation between the current estimation accuracy and the preset numerical accuracy .

[0090] The adjustment of the time step is performed according to a preset deviation threshold , as follows:

[0091] If , the current time step is not adjusted.

[0092] If , the variable step size adjustment method based on the Sigmoid function improvement is used to update the time step, and the time step is reduced.

[0093] If , the variable step size adjustment method based on the Sigmoid function improvement is used to update the time step, and the time step is increased.

[0094] In the iteration process, the time-domain iteration matrix of the estimated region of the dielectric body is used to update the electric field strength at each node in the simulation model until the iteration ends, and the final electric field strength at each node is obtained, and then the electric field distribution of the estimated region of the dielectric body is obtained.

[0095] According to the size relationship between the current estimation accuracy and the preset estimation accuracy, the size of the time step at the next time is adjusted, thereby realizing the time step length adaptation of the finite element time domain (FETD) method, and then the iteration matrix equation of the fourth-order single diagonal implicit Runge-Kutta (SDIRK4) difference format is used The electric field intensity is updated at each time step, and the correct electric field intensity value can be obtained at every point in space. At this point, the entire time step update process is completed; the steps are repeated until the time iteration ends.

[0096] The electromagnetic effect parameters such as scattering in the estimated region are obtained by using the electric field distribution of the estimated region. The post-processing is the same as in the time-domain finite element FETD method, so it will not be described in detail here.

[0097] To verify the correctness and effectiveness of this invention, the scattering effect of a metal cylinder on a uniformly incident plane electromagnetic wave is analyzed below. The change in spatial electric field intensity during this process is calculated using the step-adaptive time-domain finite element method (FETD) based on the Sigmoid function and the fixed-step time-domain finite element method (FETD) with the same numerical accuracy.

[0098] The calculation example uses a metal cylinder with a radius of 0.25 m. The origin of the coordinate system is fixed at the center of the cylinder, and a Cartesian coordinate system is established. A plane electromagnetic wave is incident along the positive X-axis, with the electric field intensity vector linearly polarized in the Z-direction. The frequency of the electromagnetic wave is 300 MHz. The electric field intensity distribution within the solution domain at a certain moment when the plane wave passes through the metal cylinder, obtained by the step-size adaptive FETD method based on the Sigmoid function, is shown below. Figure 1 As shown, the electric field intensity in the solution domain estimated by the bisection variable step size FETD method at the same instant is as follows: Figure 2 The time step variation of this problem is estimated by the step-size adaptive FETD method based on the Sigmoid function, as follows: Figure 3 As shown, using this method, the change in step size can quickly correspond to the preset estimation accuracy and tend to stabilize. In this embodiment, the mean single-step error of the next long-space electric field intensity of the fourth-order single-diagonal implicit Runge-Kutta SDIRK4 difference scheme is calculated once in the first 6 steps and after every 4 steps of the time step iteration, under two different preset estimation Butcher matrix configurations. In this embodiment, the iteration step size is adjusted based on the improved Sigmoid function. The improved Sigmoid function used to adjust the step size is as follows: Figure 4 As shown, A is the first adjustment parameter, used to set the range of change of the coefficient to be multiplied in the next time step. In this embodiment, the required estimation accuracy of the electric field strength is preset. 2×10 -22 The estimation accuracy of the step-size adaptive FETD method varies with the number of iterations, as shown in the comparison with the bisection method variable step-size FETD method. Figure 5 As shown, compared with the traditional variable step-size estimation method using the bisection method, this method has higher estimation accuracy and stability, and can stabilize the estimation accuracy more quickly.

[0099] In addition, for those skilled in the art, the steps in the embodiments described in the present application, whether in whole or in part, can be realized by corresponding hardware devices through programming instructions. The corresponding control program can be stored in various computer readable media, such as but not limited to read-only memory, hard disk or optical disk and the like.

[0100] The above embodiments are only examples for implementing the present application, and the present application is not limited to the above embodiments. Any non-essential addition or replacement made by those skilled in the art according to the technical features of the technical solutions of the present application shall fall within the protection scope of the present application.

Claims

1. A method for estimating dielectric electromagnetic effects based on Sigmoid- based variable step FETD, characterized in that, The method comprises the following steps: Firstly, a simulation model of a dielectric body and an estimated area thereof under a preset plane electromagnetic wave excitation is constructed and meshed, so as to establish a time-domain iteration matrix of the estimated area of the dielectric body in the simulation model; Secondly, the time-domain iteration matrix is iterated at a preset time step, and an error mean of a single-step spatial electric field intensity of the estimated area is obtained every preset time step; Thirdly, the preset time step is updated according to the error mean of the single-step spatial electric field intensity of the estimated area by using a variable step adjustment method based on an improved Sigmoid function, and the updated time step is used for iteration estimation of a next time step until the iteration is stopped after a preset simulation time is reached, so that an electric field distribution of the estimated area of the dielectric body in the preset total time is obtained, and electromagnetic effect parameters of the estimated area are obtained. In the first step, the time-domain iteration matrix of the estimated area of the dielectric body in the simulation model is as follows: in, , and represents the first, second, and third coefficient matrices obtained in the construction of the time-domain iterative matrix, i and j represent the first and second dimension lengths of the first, second, and third coefficient matrices, respectively; E represents the electric field intensity in the estimated region of the dielectric under the preset plane electromagnetic wave excitation; t is the time. This represents the external electromagnetic wave excitation on the i-th grid node; and They represent the dielectric constant and magnetic permeability, respectively. and Let represent the first-order basis function on the i-th grid node and the weighted shape function on the j-th grid node, respectively; Indicates the overall extent of the estimated region; Indicates electrical conductivity; and represents the boundary of the estimation region and its Neumann boundary portion, respectively; Y represents the intermediate parameters; Represents the normal vector of a unit mesh surface; Represents the Hamiltonian operator; This indicates a preset current density excitation within the estimated region; Denotes the weighted shape function at the i-th Neumann boundary; A known function on the Neumann boundary; This represents the total number of grid nodes on the Dirichlet boundary; This represents the weighted shape function at the j-th Dirichlet boundary grid node; This represents the electric field intensity at the j-th Dirichlet boundary grid node; The wave number represents the electromagnetic wave. Represents the Neumann boundary Normalized surface impedance; In the third step, the error mean of the single-step spatial electric field intensity of the estimated area is used as an estimation accuracy at a current time, and the time step is updated by using the variable step adjustment method based on the improved Sigmoid function, as follows: wherein, and respectively represent the current t time and the updated time step of next t+1 time; k represents a step adjustment factor; and respectively represent the first and second adjustment parameters; represents the estimation accuracy of the current t time, represents the preset estimation accuracy.

2. The Sigmoid-based variable step-size FETD-based dielectric electromagnetic effect estimation method of claim 1, wherein: In the first step, the estimated area of the dielectric body is an outer edge area of the dielectric body; the simulation model is meshed by using a tetrahedral mesh, and structure information of the simulation model is obtained after the meshing, including node information and element information after the meshing, the node information including node serial numbers and node coordinates, and the element information including element serial numbers and node serial numbers contained in each element.

3. The Sigmoid-based variable step-size FETD-based dielectric electromagnetic effect estimation method of claim 2, wherein: In the second step, according to the preset plane electromagnetic wave excitation electric field intensity of the estimation region at the initial moment, a fourth-order single-diagonal implicit Runge-Kutta SDIRK4 difference format is used for time domain iteration matrix in a preset total time, time step iteration of the finite element time domain FETD is carried out, and two similar Butcher matrices are randomly used in time step iteration; and an error mean value of the obtained single-step space electric field intensity of the estimation region is obtained As follows: wherein n represents the total number of nodes of the estimation region of the tetrahedral mesh division; and respectively represent the electric field intensity of the kth node in the estimation region at the time moment when the fourth-order single-diagonal implicit Runge-Kutta SDIRK4 difference format adopts two similar Butcher matrices. respectively represent the electric field intensity of the kth node in the estimation region at the time moment when the fourth-order single-diagonal implicit Runge-Kutta SDIRK4 difference format adopts two similar Butcher matrices.​ 4. The Sigmoid-based variable step-size FETD-based dielectric electromagnetic effect estimation method of claim 1, wherein: In the third step, the adjustment of the time step is based on a preset deviation threshold The adjustment is made as follows: If then the current time step is not adjusted; If , the time step is reduced using the variable step size adjustment method based on the improved Sigmoid function to update the time step. If then the time step is increased using the modified variable step size adjustment method based on the Sigmoid function.

5. The Sigmoid-based variable step-size FETD method for dielectric body electromagnetic effect estimation according to claim 2, characterized in that: In the third step, in the iteration process, the electric field intensity at each node in the simulation model is updated by using the time-domain iteration matrix of the estimated area of the dielectric body until the iteration is ended, so that the electric field intensity at each node is obtained, and the electric field distribution of the estimated area of the dielectric body is obtained.

6. An electronic device, comprising: The program data is executed by the processor to implement the method of any one of claims 1-5. The program data is executed by the processor to implement the method of any one of claims 1-5.

7. A computer readable storage medium having stored thereon program data, wherein, ​

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