Enhanced electromagnetic track finite element simulation methods, devices, systems, and storage media

Abstract

This invention discloses a finite element simulation method, apparatus, system, and storage medium for enhanced electromagnetic tracks, comprising: Step S1, generating electromagnetic force through an electromagnetic module when the projectile is in its initial position and current is applied; wherein, the performance enhancement of the track is achieved through the cyclic flow of current in the enhanced track; and the contact resistance is simulated through a weak contribution form of the contact resistance; Step S2, installing the armature into the track through an interference fit using a mechanical module, the electromagnetic force causing the armature to acquire acceleration and velocity; wherein, the armature movement generates displacement, leading to mesh deformation; Step S3, controlling the armature displacement by moving the mesh. Using the technical solution of this invention, there is no need to establish a contact layer, greatly reducing the computational complexity; simultaneously, the enhancement effect of parallel enhanced tracks can be quantitatively evaluated.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic launch technology, and particularly relates to an enhanced electromagnetic orbit finite element simulation method, device, system, and storage medium. Background Technology

[0002] Electromagnetic railguns are devices that use electromagnetic force to accelerate projectiles. They possess advantages such as high initial velocity, long range, great power, low cost, strong strike capability, and continuous firing. They also have multiple missions, including long-range precision strikes against land and sea, medium- and long-range air defense and missile defense, and anti-carrier operations. Because they rely entirely on massive electromagnetic energy for launch and carry kinetic energy for kill, they can achieve instant destruction upon impact, forming an effective deterrent. They are hailed as another revolution in the field of weaponry.

[0003] An electromagnetic railgun consists of two parallel rails and an armature that slides along the rail axis. During firing, the armature generates a considerably high velocity. The electromagnetic field and structural mechanical field within the electromagnetic railgun are coupled together, placing the projectile in an extreme multi-field operating environment. When a large current flows through the rails, the contact resistance between the armature and the rails reduces the performance and efficiency of the electromagnetic railgun propulsion system. Furthermore, during operation, the contact between the armature and the rails is a high-speed sliding contact, which directly affects the stability of the electromagnetic launcher, the muzzle velocity, and the system efficiency. Traditional contact resistance mitigation methods involve adding a contact layer, but this increases computational complexity; additionally, the large current required for electromagnetic rail operation makes it impossible to evaluate the enhancement effect of a planar reinforced electromagnetic rail model. Summary of the Invention

[0004] This invention provides an enhanced electromagnetic track finite element simulation method, apparatus, system, and storage medium.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] An enhanced finite element method for electromagnetic orbit simulation includes:

[0007] Step S1: When the projectile is in its initial position and current is applied, an electromagnetic force is generated by the electromagnetic module; the performance of the track is enhanced by the circulation of current in the reinforcing rail; the contact resistance is simulated by the weak contribution of the contact resistance.

[0008] Step S2: The armature is inserted into the track by interference fit through the mechanical module, and the electromagnetic force gives the armature acceleration and velocity; wherein, the movement of the armature generates displacement, which causes mesh deformation;

[0009] Step S3: The electromagnetic module controls the armature displacement by moving the grid.

[0010] The weak contribution form of the contact resistance in the electromagnetic track model is obtained through the energy variational method as follows:

[0011]

[0012] in, Here, E is the differential operator; B is the magnetic flux density; H is the magnetic field strength; t is time; H t V1 represents the tangential magnetic field strength; δ represents the variational sign; a computational domain is selected that encloses the projectile and the orbit, with V1 enclosed by the orbit, V2 enclosed by the projectile, and V3 representing the contact layer between the orbit and the projectile; the thickness of the contact layer V3 is set to approach 0, and V3 is considered as a surface, using S... t Let Γ represent a closed curve enclosing the computational region; R represents R. ct S is the resistivity of the contact surface; V and S1 are the volume and surface area of ​​the calculation region, respectively. It represents the energy lost when current flows through the contact layer; contact resistance is the energy loss when current passes through the contact surface.

[0013] Preferably, the armature is inserted into the track using an interference fit at a loading speed of 0.1 m / s and an interference fit of 0.001 m.

[0014] The present invention also provides an enhanced electromagnetic track finite element simulation device, comprising:

[0015] The first simulation module is used to generate electromagnetic force when the projectile is in its initial position and current is applied via the electromagnetic module; wherein, the performance of the track is enhanced by the circulation of current in the enhancement track; and the contact resistance is simulated by the weak contribution of the contact resistance.

[0016] The second simulation module is used to load the armature into the track through the interference fit via the mechanical module, and the electromagnetic force gives the armature acceleration and velocity; wherein, the armature motion generates displacement, which causes mesh deformation.

[0017] The third simulation module is used by the electromagnetic module to control the armature displacement by moving the grid.

[0018] The first simulation module obtains the weak contribution form of the contact resistance in the electromagnetic track model using the energy variational method:

[0019]

[0020] in, Here, E is the differential operator; B is the magnetic flux density; H is the magnetic field strength; t is time; H tV1 represents the tangential magnetic field strength; δ represents the variational sign; a computational domain is selected that encloses the projectile and the orbit, with V1 enclosed by the orbit, V2 enclosed by the projectile, and V3 representing the contact layer between the orbit and the projectile; the thickness of the contact layer V3 is set to approach 0, and V3 is considered as a surface, using S... t Let Γ represent a closed curve enclosing the computational region; R represents R. ct S is the resistivity of the contact surface; V and S1 are the volume and surface area of ​​the calculation region, respectively. It represents the energy lost when current flows through the contact layer; contact resistance is the energy loss when current passes through the contact surface.

[0021] Preferably, the second simulation module enables the interference fit to insert the armature into the track at a loading speed of 0.1 m / s and an interference fit of 0.001 m.

[0022] The present invention also provides an enhanced electromagnetic orbit finite element simulation system, comprising: a memory and a processor, wherein the memory stores a computer program executed by the processor, and the computer program executes an enhanced electromagnetic orbit finite element simulation method when run by the processor.

[0023] The present invention also provides a storage medium storing a computer program that executes an enhanced electromagnetic orbit finite element simulation method when running.

[0024] This invention establishes a model that represents dynamic contact resistance through weak contributions, eliminating the need to establish a contact layer and greatly reducing computational complexity. At the same time, it establishes a simulation model for planar reinforced electromagnetic tracks under force-electric-magnetic multi-physics coupling, and quantitatively evaluates the reinforcement effect of parallel reinforced tracks. Attached Figure Description

[0025] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0026] Figure 1 This is a flowchart of the enhanced electromagnetic track finite element simulation method according to an embodiment of the present invention;

[0027] Figure 2 This is a schematic diagram of the connection structure between the track and the projectile, where 1 is the track, 2 is the projectile, and A is a schematic diagram of the contact between track 1 and projectile 2;

[0028] Figure 3 for Figure 2An enlarged schematic diagram of A in the diagram, where V1 is the area surrounded by track 1, V2 is the area surrounded by projectile 2, and V3 is the contact layer between track 1 and projectile 2.

[0029] Figure 4 This is a schematic diagram of interference fit; where (a) is a schematic diagram of the filling process; and (b) is a schematic diagram of the filling process.

[0030] Figure 5 This is a schematic diagram of armature-related parameters;

[0031] Figure 6 The diagram shows the current flow direction of an enhanced electromagnetic railgun; (a) is a diagram showing the current flow direction of a dual-rail electromagnetic railgun, (b) is a diagram showing the current flow direction of a quad-rail electromagnetic railgun, (c) is a diagram showing the current flow direction of a six-rail electromagnetic railgun, and (d) is a diagram showing the current flow direction of an eight-rail electromagnetic railgun. Detailed Implementation

[0032] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0033] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0034] Example 1:

[0035] like Figure 1 , 2 As shown, this embodiment of the invention provides an enhanced electromagnetic orbit finite element simulation method, including:

[0036] Step S1: When the projectile 2 is in its initial position and current is applied, an electromagnetic force is generated by the electromagnetic module; wherein, the performance of the track 1 is enhanced by the circulation of current in the reinforcing track; and the contact resistance is simulated by the weak contribution of the contact resistance.

[0037] Step S2: The armature is inserted into track 1 by interference fit through the mechanical module. The electromagnetic force gives the armature acceleration and velocity; wherein, the movement of the armature generates displacement, which causes mesh deformation.

[0038] Step S3: The electromagnetic module controls the armature displacement by moving the grid.

[0039] As one embodiment of the present invention, in step S1, the electromagnetic field control method of the present invention is the H method. The H method is a method that uses only the H magnetic induction intensity as a variable to solve Maxwell's equations, which can effectively improve the speed and accuracy of the simulation process.

[0040] Ampere's circuital law and Faraday's law of electromagnetic induction from Maxwell's equations are as follows:

[0041]

[0042] in, Let J be the differential operator, E be the electric field strength, B be the magnetic flux density, H be the magnetic field strength, and t be the time.

[0043] The constitutive equation of electromagnetism is:

[0044]

[0045] Where ρ is the resistivity of the conductor, μ0 is the absolute permeability, and μ r Let μ be the relative permeability. r =1; Eliminating other physical quantities and retaining only H yields:

[0046]

[0047] Using the H-method as the governing equation can improve the speed of model calculation and reduce the complexity of model setup.

[0048] Because the electromagnetic orbit model has strong symmetry, it was chosen to establish... Models can reduce computational complexity.

[0049] Due to the adoption The model needs to have its boundary conditions constrained, that is, to ensure the following within the plane of symmetry:

[0050] I z =0 (4)

[0051] lateral current I z Setting it to 0 ensures that the simulation results are closer to reality.

[0052] The weak contribution expression of contact resistance in the electromagnetic track model is derived using the energy variational method; Poynting's theorem provides the energy conservation relationship in the electromagnetic field.

[0053]

[0054] Where w is the electromagnetic field energy density, S is the energy flux density, V and S1 are the volume and surface area of ​​the computational domain, and Γ is a closed curve enclosing the computational domain. Substitute ∫ V -δJ·EdV can be used to obtain:

[0055]

[0056] Where δ is the variational symbol.

[0057] Substituting (2) into the equation, we get:

[0058]

[0059] This yields the weak form of the H-method governing equations, which can be understood as the energy flowing out from the boundary being converted into magnetic field energy and the work done by the conductor in the magnetic field. During the model establishment process, a contact layer of a certain thickness will exist between orbit 1 and projectile 2, such as... Figure 2 and Figure 3 As shown. Its thickness is very small. Due to the presence of the contact layer, there will be contact resistance when the current is transmitted from the rail to the armature. However, if the contact layer is constructed during modeling, the computational complexity will be increased. In order to reduce the complexity of modeling, the form of weak contribution is chosen for simplification. Equation (7) can be simplified to:

[0060]

[0061] In this calculation, a region enclosing the projectile 2 and the track 1 is selected. The region enclosing the track 1 is V1, the region enclosing the projectile 2 is V2, and the contact layer between the track 1 and the projectile 2 is V3. The thickness of the contact layer V3 is set to approach 0, and the contact layer V3 is considered as a surface. S0 is used as the basis for the calculation. t To represent; R ct R is the resistivity of the contact surface. ct =ρ c ·d c d c ρ is the contact layer thickness. c Let represent the contact layer resistivity. In the model construction, contact resistance is initially neglected during calculations. Analysis is then performed based on the results. The contact layer thickness is typically 10. -6 The magnitude is considered, so the contact layer thickness is chosen to be 1 μm. The contact ratio between track 1 and projectile 2 is the contact resistivity ρ of copper and aluminum. c ≈3.6065×10 -7 Ω·m.

[0062] Due to the thickness d of the contact layer c Small enough, in equation (8) If we consider it as 0, the above equation can be further simplified:

[0063]

[0064] At the contact surface, the tangential current and normal magnetic field strength can be ignored. Only the tangential magnetic field strength and normal current are retained, and their relationship can be expressed as:

[0065]

[0066] Among them, J n H is the normal current. t The tangential magnetic field strength;

[0067] It can be further simplified:

[0068]

[0069] in, It represents the energy lost when current flows through the contact layer; the contact resistance can be considered as the energy loss when current passes through the contact surface, and thus the contact resistance is expressed in the form of weak contribution.

[0070] In one embodiment of the present invention, in step S2, during the firing of the electromagnetic railgun, the contact force between the armature and the rail 1 has a significant impact on the performance of the electromagnetic railgun. Maintaining good metal-to-metal contact between the armature and the rail 1 is crucial to prevent transition. The magnitude and distribution of the contact pressure between the two are important factors affecting transition, and both the contact pressure and its distribution are related to the armature's structure. To ensure good contact between the armature and the rail 1, an interference fit is used to guarantee this good contact. Figure 4 As shown in (a) and (b), the filling speed V is 0.1 m / s and the interference is 0.001 m.

[0071] During the firing process of an electromagnetic railgun, electromagnetic force is the main driving force for its movement, and the density of electromagnetic force f can be expressed as:

[0072] f = J × B (12)

[0073] Its mechanical equations are written as follows:

[0074] σ ij,j +f i =ρ0a i (13)

[0075] Where, σ ij，j The stress is expressed in tensor form, ρ0 is the armature density, and f is the tensor form. i Let a be the component of the electromagnetic force f in the x, y, z directions. i Let be the acceleration of the armature. During the armature's motion, it must be controlled to move only along the x-direction. Therefore, constraints need to be applied to the armature's motion, namely:

[0076]

[0077] Among them, u y Let y be the displacement of the armature in the y direction.

[0078] This ensures that it will only move within orbit 1. A similar equation applies to orbit 1:

[0079] σ ij,j +f i =0(15)

[0080] To ensure that orbit 1 does not move during launch, fixed constraints are applied to the boundaries of orbit 1.

[0081] In one embodiment of the present invention, in step S3, during the firing of the electromagnetic gun, a large amount of current is passed through in a short period of time, causing the armature to experience a large acceleration. This distorts the mesh, thus affecting the calculation results. To solve this problem, the ALE method is used during modeling, which can adjust the current configuration t. n The previous moment t n-1 The mesh is reconstructed, and the current configuration is linked to the reference configuration through a set of relations.

[0082] In the reference configuration, the material point is denoted by vector X, and its position in the coordinate system in space is denoted by vector x. The deformation gradient F is... When studying the deformation of a continuous medium, Cauchy stress σ and volume force f are defined in the current configuration. b S and For the second Piola-Kirchoff stress (i.e., nominal stress) and body force, Cauchy stress σ and body force f b It can be written as:

[0083]

[0084] Among them, F T J is the transpose of the deformation gradient F. d Let F be the determinant of F. For the strain components, let the strain under the reference configuration be ε. G The strain component under the current configuration is ε A They can be represented as:

[0085]

[0086] Where I is the unit tensor.

[0087] Establish a reference coordinate system That is, t n-1The configuration at time is used to generate a mesh again on the reference configuration. Let the spatial coordinates be x = x(X, t). The spatial coordinates x are a function of the material coordinates X and time t. Alternatively, the spatial coordinates can also be derived from the mesh coordinates. Determined by time t Let the velocity of the object in the material coordinate system be v, and the velocity of the object in the mesh coordinate system be v. The two velocities can be expressed as:

[0088]

[0089] in, and These represent the derivatives with respect to t when the material coordinates and mesh coordinates remain constant;

[0090] Furthermore, the convection velocity C can be defined. v ,

[0091]

[0092] The momentum conservation formula can be written as:

[0093]

[0094] Where ρ0 is the density of the object.

[0095] Similarly, the electromagnetic equations can be written in the form of the reference configuration and the current configuration. First, define the mesh deformation gradient. Take J d The determinant of F can be used to establish the relationship between electromagnetic physical quantities in the spatial coordinate system and electromagnetic physical quantities in the material coordinate system:

[0096]

[0097] Among them, H X B X E X J X These represent the magnetic field strength, magnetic induction, electric field strength, and current density under the reference configuration, respectively; H x B x E x J x Let be the magnetic field strength, magnetic induction intensity, electric field strength, and current density under the current configuration, respectively. According to Faraday's law of electromagnetic induction and Ampere's law, we can obtain:

[0098]

[0099] in, For the curl operator in the current configuration, This is the curl operator under the reference configuration. Similarly, the expression for the H-method governing equations under the reference configuration can be given:

[0100]

[0101] In one embodiment of the present invention, a copper track 1 and an aluminum projectile 2 are selected, and their material parameters are shown in Table 1 and... Figure 5 As shown, l1, l2, and l3 represent the width, length, and height of the armature, respectively, and R1 and R2 represent the radii of the circles used for rounding the corners of the two parts of the armature. Furthermore, an air domain is filled around track 1 and the armature to ensure that the electromagnetic field distribution is the same as in reality.

[0102] Table 1

[0103] Track length (m) Track width (m) Track height (m) Track resistivity (Ω·m) Orbital elastic modulus (Pa) Orbital Poisson's ratio 1 0.02 0.02 <![CDATA[1.72×10 -8 ]]> <![CDATA[110×10 9 ]]> 0.35 l1(m) l2(m) l3(m) Armature resistivity (Ω·m) Armature elastic modulus (Pa) armature Poisson's ratio 0.02 0.04 0.04 <![CDATA[2.85×10 -8 ]]> <![CDATA[70×10 9 ]]> 0.03 R1(m) R2(m) Spacing between each reinforcing rail (m) 0.01 0.005 0.02

[0104] To demonstrate the enhanced effect of the reinforced electromagnetic railgun, the applied current is controlled to be constant, and a current-guiding device is used to ensure it circulates within the parallel rails, thereby achieving a fixed applied current and effectively improving the performance of the electromagnetic railgun. Furthermore, according to... Figure 6 The arrows indicate the direction of the current. An arrow pointing outwards towards rail 1 represents current flowing into the rail, and an arrow pointing outwards from rail 1 represents current flowing out of the rail. Based on this pattern, the current conduction method can be extended to 2n-rail planar reinforced rails, such as... Figure 6 As shown in (a), (b), (c), and (d), by passing a large current through the armature for a short period of time, a considerable electromagnetic force is provided, enabling the armature to achieve a very high velocity. The displacement of the armature in the mechanical module is then introduced into the electromagnetic module as a factor controlling the motion of the dynamic grid, thereby establishing a force-electromagnetic coupling model.

[0105] Example 2:

[0106] This invention also provides an enhanced electromagnetic track finite element simulation device, comprising:

[0107] The first simulation module is used to generate electromagnetic force when the projectile 2 is in the initial position and current is applied through the electromagnetic module; wherein, the performance of the track 1 is enhanced by the circulation of current in the enhancement track; and the contact resistance is simulated by the weak contribution of the contact resistance.

[0108] The second simulation module is used to install the armature into track 1 through the interference fit via the mechanical module, and the electromagnetic force gives the armature acceleration and velocity; wherein, the armature motion generates displacement, which causes mesh deformation.

[0109] The third simulation module is used by the electromagnetic module to control the armature displacement by moving the grid.

[0110] The first simulation module derives the weak contribution expression of the contact resistance in the electromagnetic track model using the energy variational method, namely,

[0111]

[0112] in, Here, E is the differential operator; B is the magnetic flux density; H is the magnetic field strength; t is time; H t V is the tangential magnetic field strength; δ is the variational symbol; a computational domain is selected that surrounds projectile 2 and orbit 1, the region surrounding orbit 1 is V1, the region surrounding projectile 2 is V2, and the contact layer between orbit 1 and projectile 2 is V3; the thickness of the contact layer V3 is made close to 0, and the contact layer is considered as a surface, using S... t Let Γ represent a closed curve enclosing the computational region; R represents R. ct Let V be the resistivity of the contact surface; V and S1 are the volume and surface area of ​​the computational domain, respectively. It represents the energy lost when current flows through the contact layer; contact resistance is the energy loss when current passes through the contact surface.

[0113] In one embodiment of the present invention, the second simulation module causes the interference fit to load the armature into the track 1 at a loading speed of 0.1 m / s and an interference fit of 0.001 m.

[0114] Example 3:

[0115] This invention also provides an enhanced electromagnetic orbit finite element simulation system, comprising: a memory and a processor, wherein the memory stores a computer program executed by the processor, and the computer program executes an enhanced electromagnetic orbit finite element simulation method when run by the processor.

[0116] Example 4:

[0117] This invention also provides a storage medium storing a computer program that executes an enhanced electromagnetic orbit finite element simulation method during runtime.

[0118] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. An enhanced finite element simulation method for electromagnetic tracks, characterized in that, include: Step S1: When the projectile is in its initial position and current is applied, an electromagnetic force is generated by the electromagnetic module; wherein, the performance of the track is enhanced by the circulation of current in the reinforcing rail; and the contact resistance is simulated by the weak contribution of the contact resistance. Step S2: The armature is inserted into the track by interference fit through the mechanical module, and the electromagnetic force gives the armature acceleration and velocity; wherein, the movement of the armature generates displacement, which causes mesh deformation; Step S3: The electromagnetic module controls the armature displacement by moving the grid. The weak contribution form of the contact resistance in the electromagnetic track model is obtained through the energy variational method as follows: in, Here, E is the differential operator; B is the magnetic flux density; H is the magnetic field strength; t is time; H t V1 represents the tangential magnetic field strength; δ represents the variational sign; a computational domain is selected that encloses the projectile and the orbit, with V1 enclosed by the orbit, V2 enclosed by the projectile, and V3 representing the contact layer between the orbit and the projectile; the thickness of the contact layer V3 is set to approach 0, and V3 is considered as a surface, using S... t Let Γ represent a closed curve enclosing the computational region; R represents R. ct S is the resistivity of the contact surface; V and S1 are the volume and surface area of ​​the calculation region, respectively. It represents the energy lost when current flows through the contact layer; contact resistance is the energy loss when current passes through the contact surface.

2. The enhanced electromagnetic track finite element simulation method as described in claim 1, characterized in that, The armature is inserted into the track using an interference fit at a loading speed of 0.1 m / s and an interference fit of 0.001 m.

3. An enhanced electromagnetic track finite element simulation device, characterized in that, include: The first simulation module is used to generate electromagnetic force when the projectile is in its initial position and current is applied via the electromagnetic module; wherein, the performance of the track is enhanced by the circulation of current in the enhancement track; and the contact resistance is simulated by the weak contribution of the contact resistance. The second simulation module is used to load the armature into the track through the interference fit via the mechanical module, and the electromagnetic force gives the armature acceleration and velocity; wherein, the armature motion generates displacement, which causes mesh deformation. The third simulation module is used by the electromagnetic module to control the armature displacement by moving the grid. The first simulation module obtains the weak contribution form of the contact resistance in the electromagnetic track model using the energy variational method: in, Here, E is the differential operator; B is the magnetic flux density; H is the magnetic field strength; t is time; H t V1 represents the tangential magnetic field strength; δ represents the variational sign; a computational domain is selected that encloses the projectile and the orbit, with V1 enclosed by the orbit, V2 enclosed by the projectile, and V3 representing the contact layer between the orbit and the projectile; the thickness of the contact layer V3 is set to approach 0, and V3 is considered as a surface, using S... t Let Γ represent a closed curve enclosing the computational region; R represents R. ct S is the resistivity of the contact surface; V and S1 are the volume and surface area of ​​the calculation region, respectively. It represents the energy lost when current flows through the contact layer; contact resistance is the energy loss when current passes through the contact surface.

4. The enhanced electromagnetic track finite element simulation device as described in claim 3, characterized in that, The second simulation module uses an interference fit to insert the armature into the track at a loading speed of 0.1 m / s and an interference fit of 0.001 m.

5. An enhanced electromagnetic track finite element simulation system, characterized in that, include: A memory and a processor, wherein the memory stores a computer program executed by the processor, the computer program executing the enhanced electromagnetic orbit finite element simulation method as described in any one of claims 1 to 2 when run by the processor.

6. A storage medium, characterized in that, The storage medium stores a computer program that, when executed, performs the enhanced electromagnetic orbit finite element simulation method as described in any one of claims 1 to 2.

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