CN108416107B - Particle motion pushing finite element algorithm applied to PIC - Google Patents
Particle motion pushing finite element algorithm applied to PIC Download PDFInfo
- Publication number
- CN108416107B CN108416107B CN201810114140.XA CN201810114140A CN108416107B CN 108416107 B CN108416107 B CN 108416107B CN 201810114140 A CN201810114140 A CN 201810114140A CN 108416107 B CN108416107 B CN 108416107B
- Authority
- CN
- China
- Prior art keywords
- grid
- particles
- particle
- pic
- solving
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
Abstract
The invention belongs to the field of numerical simulation of particle simulation, and particularly relates to a particle motion propelling finite element algorithm applied to PIC. The invention uses the unstructured grid which can better fit the shape of the model boundary, so that the PIC (positive-impedance converter) pushes the particle motion algorithm to have higher calculation precision under the condition of complex boundary; the method that the FEM pushes the particles to move is combined into the typical PIC method, so that the FEM is utilized to obtain higher finite element calculation accuracy while the excellent characteristics of simple and rapid calculation of the typical PIC method are maintained; the FEM method can be well matched with a complex boundary, can use non-uniform grids according to simulation requirements, is not limited by numerical stability conditions, and can optimize space grids and time step length under the condition of keeping calculation accuracy, so that the simulation efficiency is greatly improved.
Description
Technical Field
The invention belongs to the field of numerical simulation of Particle-in-cell (PIC), and particularly relates to a Particle motion pushing finite element algorithm applied to PIC.
Background
The PIC method is a numerical simulation method widely applied to the interaction physics problem of charged particles and electromagnetic field, and obtains macroscopic characteristics and motion rules by tracking the motion of a large number of charged particles in an external and self-consistent electromagnetic field and counting and averaging. After decades of developments, the PIC simulation method has become a powerful numerical means for studying the physical problem of the interaction between the charged particles and the electromagnetic field, and is widely applied to many fields related to the interaction between the charged particles and the electromagnetic field, such as magnetic confinement fusion plasma, inertial confinement fusion plasma, nuclear explosion, space plasma, artificial plasma (including electron gun, ion source, etc.), electric propulsion, free electron laser, and electric vacuum devices.
The core steps for PIC solution are as follows:
1. solving the electromagnetic field, namely solving a Maxwell equation set (degraded into Poisson equation under an electrostatic model) satisfied by the electromagnetic field to obtain the electromagnetic field on all grid points;
2. solving the stress of the particles, namely obtaining the potential distribution in the grid through the potential values of the related grid points, solving the negative gradient of the potential distribution to obtain the electric field at the position of the particles, and then solving the stress;
3. promoting the particle motion, namely updating the motion information such as the speed and the position of the particle by solving the motion equation of the discrete particle, and further updating the grid to which the particle belongs;
4. the distribution of the source, namely, the contribution of the particles to the charge and the current of the surrounding grid points is obtained according to the positions of the particles, and then the charge and the current contributions of all the particles to the grid points are accumulated to obtain the charge density and the current density of the grid points;
the above process is continuously circulated until the calculation result converges or the time is artificially set.
Wherein step 3 drives particle motion is one of the core steps of PIC, and the precise and efficient solution of this step is very important for controlling the overall solution accuracy and efficiency of PIC. To date, there are two main approaches to driving particle motion in PIC, the Finite Difference (FD) method and the embedded finite element (IFE) method.
FD method: in the application of PIC to drive the particles to move, the FD method discretely solves the area by adopting the structured grid, so that the algorithm forms of solving the discrete particle motion equation and judging the grid to which the particles belong are simple and easy to understand, but the PIC has the following defects in the application of driving the particles to move:
1. the FD method adopts a structured grid divided by orthogonal lines, and has poor fitting on a complex curved boundary, so that the solving precision of the movement of the pushing particles near the complex curved boundary is low;
2. the FD method has higher requirement on the size uniformity of the grids, so the method is limited by the limitation of fine physical structures in a simulation system, and the grid which is small enough must be divided to meet the requirement on the calculation precision, so that the total grid number is huge, and the number of simulated particles is in direct proportion to the total grid number, thereby causing that the calculation amount for pushing the particle motion solution is huge;
3. the FD method is strictly limited by the numerical stability condition, i.e. in the numerical simulation of the motion of the PIC push particles, if the spatial grid size is small, the time step is also small, which further increases the FD numerical simulation burden of the time-loop solution of the push particles.
In response to the shortcomings of the FD method in PIC particle motion-driven applications, Kafafy and Wang in 2003 proposed IFE methods applicable to PIC particle motion-driven applications.
An IFE method: in applications where PIC forces particle motion, the IFE method discretely solves for regions by using an invasive unstructured grid. The intrusive unstructured meshing situation is shown in fig. 1, and it can be seen that the meshing is equivalent to 2-fold meshes, where 1-fold mesh is a structured mesh and 2-fold mesh is an intrusive unstructured mesh that further divides each structured mesh in 1-fold mesh into five tetrahedrons. The IFE method in PIC application uses a 2 nd re-invasive non-structural grid for field solution, while the pushing particle motion is performed in the 1 st re-structured grid, so the IFE method does not solve the disadvantages of the FD method in PIC pushing particle motion application.
Disclosure of Invention
Aiming at the problems and the defects, the invention provides a Finite Element (FEM) algorithm applied to PIC (positive influence modeling) for pushing particle motion, which aims to solve the problems of low boundary matching degree, low solving precision and large numerical simulation load in the solution of pushing particle motion by FD and IFE methods. The specific technical scheme is as follows:
step 1, solving an electromagnetic field.
Obtaining electromagnetic fields on all grid points by solving a discrete Maxwell equation set (degraded into an electrostatic discrete Poisson equation under an electrostatic model) satisfied by an electromagnetic field;
and 2, solving the stress of the particles, obtaining the potential distribution in the grid through the potential values of the related grid points, solving the negative gradient of the potential distribution to obtain the electric field at the position of the particles, and then solving the stress.
Step 3, pushing the particles to move
A global unstructured grid is used as shown in fig. 2. And (3) by solving a discrete particle motion equation, motion information such as the speed and the position of the particle is updated, and the grid to which the particle belongs is further updated. The detailed solving process is divided into the following two steps:
step 3.1, update the speed and position of the particle
Under a rectangular coordinate system, the motion equation of the particle obtained by considering the relativistic effect is shown as the formula (1):
wherein: one point in the superscript represents the first derivative of the variable with respect to time, and two points represent the second derivative of the variable with respect to time; eta is the charge-to-mass ratio of the particles; gamma is a relativistic factor; ex、Ey、Ez、Bx、By、BzThe electromagnetic field component of the position of the particle is obtained in the step 2;
the velocity and position of the particle can be updated by solving equation (1) using the method of Boris or Runge-Kutta, etc.
Step 3.2, updating the grids to which the particles belong
Updating the grid to which the particle belongs after updating the speed and position of the particle each time is an indispensable link in the PIC, and only updating the grid to which the particle belongs can perform source calculation (namely step 4), module addition of characteristic physical processes occurring in the grid, relevant numerical diagnosis and the like.
Because the number of analog particles in the PIC is large and an unstructured grid is used at this time (the grid to which the particles belong is not directly obtained as in the case of a structured grid), the calculation of the grid to which all the particles belong is a key step that affects the overall PIC calculation accuracy and efficiency. Therefore, the invention provides a rapid particle positioning algorithm based on an unstructured grid, which is used for accurately and efficiently calculating the grid to which the particle belongs, and the specific implementation mode is as follows:
1) and initializing grid numbers where the particles are positioned according to the particle appearance position or emission surface setting when the first step time is long.
2) Starting from the second step time step, it is first determined whether the particle is still within the current grid. If so, ending the search;
if not, go to 3).
3) Center of gravity P of grid PreId where particle is located at moment before connection1And the position P of the particle at the current moment2To obtain a line segment P1P2And find the line segment P1P2The intersection FaceId with the pred, and a new mesh sharing the intersection FaceId with the pred, set the new mesh to pred, and go back to 2).
There are three main reasons for always selecting the center of gravity of a grid as a calculation point rather than other grid points: 1. the center of gravity is positioned inside the grid; 2. the gravity center is convenient to calculate, and the components in the directions of x, y and z of the gravity center coordinate can be obtained only by respectively summing the components of the coordinates of x, y and z of each vertex of the grid where the gravity center is located and dividing the sum by the total number of the vertices of the grid; 3. avoiding the appearance of dead circulation when the program judges that the particles are at special positions, if the vertex of a certain grid is just in the line segment P1P2The above time.
The particle rapid positioning algorithm based on the unstructured grid combines the grid where a time step is located on the particle, searches according to a depth-first algorithm, starts from an initial position according to a definite direction, gradually approaches the grid where the particle is located, and finally achieves accurate and rapid positioning.
The specific schematic diagram of the algorithm is shown in fig. 3. In order to better describe the particle fast localization algorithm, the schematic diagram adopts a two-dimensional representation form, in the figure, a two-dimensional triangular mesh corresponds to a three-dimensional tetrahedral mesh, and a two-dimensional line corresponds to a three-dimensional plane.
The specific flow chart of the algorithm is shown in fig. 4.
And 4, allocating the source.
Obtaining the contributions of the particles to the charges and currents of the surrounding grid points according to the positions of the particles, and then accumulating the contributions of all the particles to the charges and the currents on the grid points to obtain the charge density and the current density on the grid points;
the solution of steps 1, 3 to 4 uses structured, immersive unstructured or completely unstructured grids.
And (5) circulating the steps 1 to 4 until a convergence condition or a simulation termination condition is reached, and finally performing numerical diagnosis.
The invention is suitable for two-dimensional and three-dimensional structures, and is suitable for changing the grid division from a tetrahedral grid to a triangular grid in two dimensions.
Compared with the FD method and the IFE method for driving the particles to move by PIC, the method has the advantages that:
1. the unstructured grid is used, and the grid can better fit the shape of the model boundary, so that the PIC (positive-impedance converter) push particle motion algorithm has higher calculation accuracy under the condition of complex boundary;
2. the method that the FEM pushes the particles to move is combined into the typical PIC method, so that the FEM is utilized to obtain higher finite element calculation accuracy while the excellent characteristics of simple and rapid calculation of the typical PIC method are maintained;
3. the FEM method can be well matched with a complex boundary, can use non-uniform grids according to simulation requirements, is not limited by numerical stability conditions, and can optimize space grids and time step length under the condition of keeping calculation accuracy, so that the simulation efficiency is greatly improved.
Drawings
FIG. 1 is a schematic diagram of an IFE grid for PIC solution;
FIG. 2 is a schematic diagram of a FEM grid for PIC solution;
FIG. 3 is a schematic diagram of a particle fast localization algorithm;
FIG. 4 is a flow chart of a particle fast localization algorithm;
FIG. 5 is a schematic diagram of an example of PIC electrostatic model calculation for a seven-aperture dual-gate ion optical system;
fig. 6 is a schematic diagram of an example grid division of a PIC electrostatic model calculation for a seven-aperture double-gate ion optical system.
Detailed Description
The present invention will be described in further detail below by way of examples.
Taking the ion thruster seven-aperture double-gate ion optical system as an example, a schematic diagram thereof is shown in fig. 5. The specific implementation steps of the PIC electrostatic simulation for this example using the algorithm of the present invention are as follows:
step 1, solving an electric field.
Obtaining electric fields on all grid points by solving an electrostatic discrete Poisson equation met by the electric fields;
step 2, solving the stress of the ions, obtaining the potential distribution in the grid through the potential values of the related grid points, solving the negative gradient of the potential distribution to obtain the electric field at the position of the particles, and then solving the stress;
and 3, driving the ions to move.
And (3) updating the motion information such as the speed and the position of the ions by solving a discrete ion motion equation by adopting a global unstructured grid so as to update the grid to which the ions belong. The detailed solving process is divided into the following two steps:
step 3.1, update of ion velocity and position
Here, regardless of the magnetic field, the ion motion equation in the rectangular coordinate system is shown by the equation (2):
wherein: one point in the superscript represents the first derivative of the variable with respect to time, and two points represent the second derivative of the variable with respect to time; eta is the charge-to-mass ratio of the ions; gamma is a relativistic factor; ex、Ey、EzThe electric field component of the ion position is obtained by the step 2;
solving the formula (2) by using a Runge-Kutta method to obtain the new speed and position of the ions after a time step.
Step 3.2, updating the grid to which the ions belong
And updating the grid to which the ions belong after updating the speed and the position of the ions every time so as to prepare for the subsequent step 4.
The method adopts an unstructured grid-based rapid ion positioning algorithm for accurately and efficiently calculating the grids to which ions belong, and the specific implementation mode is as follows:
1) and when the time step of the first step is long, initializing the grid number where the ions are located according to the ion appearance position or emission surface setting.
2) Starting with the second step time step, it is first determined whether the ions are still within the current grid. If so, ending the search;
if not, go to 3).
3) Center of gravity P of grid PreId where ion is located at moment before connection1And the position P of the ion at the current moment2To obtain a line segment P1P2And find the line segment P1P2The intersection FaceId with the pred, and a new mesh sharing the intersection FaceId with the pred, set the new mesh to pred, and go back to 2).
And 4, distributing the charges.
The contribution of the ions to the charges of the surrounding grid points is obtained according to the positions of the ions, and then the charge contributions of all the ions to the grid points are accumulated to obtain the charge density on the grid points;
the solving of steps 1, 3 to 4 can adopt structured, immersion unstructured and completely unstructured grids.
And (5) circulating the steps 1 to 4 until a convergence condition or a simulation termination condition is reached, and finally performing numerical diagnosis. The results of PIC electrostatic simulation of this example using the algorithm of the present invention are shown in fig. 5.
Claims (3)
1. A finite element solution method for pushing particle motion applied to PIC comprises the following steps:
step 1, solving an electromagnetic field; obtaining electromagnetic fields on all grid points by solving a discrete Maxwell equation set met by an electromagnetic field or an electrostatic discrete Poisson equation met by an electrostatic model;
step 2, solving the stress of the particles, obtaining the potential distribution in the grid through the potential values of the related grid points, solving the negative gradient of the potential distribution to obtain the electric field at the position of the particles, and then solving the stress;
step 3, promoting the particles to move, adopting a global unstructured grid, and updating the speed and position movement information of the particles by solving a discrete particle motion equation so as to update the grid to which the particles belong;
step 3.1, update the speed and position of the particle
Under a rectangular coordinate system, the motion equation of the particle obtained by considering the relativistic effect is shown as the formula (1):
wherein one point in the superscript represents the first derivative of the variable with respect to time and two points represent the second derivative of the variable with respect to time; eta is the charge-to-mass ratio of the particles; gamma is a relativistic factor; ex、Ey、Ez、Bx、By、BzThe electromagnetic field component of the position of the particle is obtained in the step 2;
step 3.2, updating the grids to which the particles belong;
1) when the first step time step is long, initializing grid numbers where the particles are located according to the setting of the appearance positions or emission surfaces of the particles;
2) starting from the time step of the second step, firstly judging whether the particles are still in the current grid; if so, ending the search; if not, go to the next step 3);
3) center of gravity P of grid PreId where particle is located at moment before connection1And the position P of the particle at the current moment2To obtain a line segment P1P2And find the line segment P1P2Obtaining a new grid sharing the intersection face FaceId with the PreId at the same time, setting the new grid as the PreId, and returning to 2);
the grid algorithm to which the updated particles belong is combined with the grid where a time step length of the particles is located, searching is carried out according to a depth-first algorithm, the grid where the particles are located is gradually approached from the initial position according to a definite direction, and finally accurate and rapid positioning is achieved;
step 4, source allocation;
obtaining the contributions of the particles to the charges and currents of the surrounding grid points according to the positions of the particles, and then accumulating the contributions of all the particles to the charges and the currents on the grid points to obtain the charge density and the current density on the grid points;
and (5) circulating the steps 1 to 4 until a convergence condition or a simulation termination condition is reached, and finally performing numerical diagnosis.
2. A method of push particle motion finite element solution as claimed in claim 1 applied to a PIC, wherein: in the step 3, the equation (1) is solved by using a Boris or Runge-Kutta method, and the velocity and the position of the particle are updated.
3. A method of push particle motion finite element solution as claimed in claim 1 applied to a PIC, wherein: the solving of the steps 1, 3 to 4 adopts structured, immersed unstructured or completely unstructured grids.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201810114140.XA CN108416107B (en) | 2018-02-05 | 2018-02-05 | Particle motion pushing finite element algorithm applied to PIC |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201810114140.XA CN108416107B (en) | 2018-02-05 | 2018-02-05 | Particle motion pushing finite element algorithm applied to PIC |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN108416107A CN108416107A (en) | 2018-08-17 |
| CN108416107B true CN108416107B (en) | 2021-07-06 |
Family
ID=63126937
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201810114140.XA Active CN108416107B (en) | 2018-02-05 | 2018-02-05 | Particle motion pushing finite element algorithm applied to PIC |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN108416107B (en) |
Families Citing this family (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110955955B (en) * | 2019-11-05 | 2022-11-29 | 中国电波传播研究所（中国电子科技集团公司第二十二研究所） | Dust plasma numerical calculation speed-up method based on logarithmic fitting |
| CN111695281B (en) * | 2020-06-01 | 2023-04-25 | 电子科技大学 | Particle rapid positioning method for tetrahedral mesh division finite element particle simulation |
| CN111737894B (en) * | 2020-06-01 | 2023-04-07 | 电子科技大学 | Particle rapid positioning method for triangular surface gridding finite element particle simulation |
| CN111967148B (en) * | 2020-07-31 | 2023-07-07 | 电子科技大学 | Particle simulation Voronoi diagram particle merging algorithm |
| CN111967149B (en) * | 2020-08-03 | 2022-11-04 | 电子科技大学 | Particle motion semi-interpolation solving method for particle simulation algorithm |
Citations (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104239623A (en) * | 2014-09-05 | 2014-12-24 | 兰州空间技术物理研究所 | Method for obtaining satellite surface potential based on multi-time-scale particle push |
| CN105593747A (en) * | 2013-06-09 | 2016-05-18 | 光电网股份有限公司 | Broad-modulation-bandwidth low-optical-loss modulators |
| CN106155306A (en) * | 2015-05-13 | 2016-11-23 | 意美森公司 | The system and method for the sense of touch feedback of modularized equipment |
| CN106503850A (en) * | 2016-11-01 | 2017-03-15 | 上海电机学院 | A kind of SVM forecast models for wind power prediction |
| CN106991237A (en) * | 2017-04-05 | 2017-07-28 | 东北大学 | A kind of electromagnetic agitation analysis construction method based on electric melting magnesium furnace |
| CN107425527A (en) * | 2017-09-15 | 2017-12-01 | 国网江苏省电力公司经济技术研究院 | A kind of THE UPFC static security prevention and control method |
| CN107492591A (en) * | 2016-06-13 | 2017-12-19 | 隆达电子股份有限公司 | Chip scale package structure of light emitting diode |
| CN107526887A (en) * | 2017-08-22 | 2017-12-29 | 电子科技大学 | A kind of LeapfrogADI FDTD method parallel based on GPU |
Family Cites Families (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US20130083389A1 (en) * | 2011-09-30 | 2013-04-04 | Optical Air Data Systems, L.L.C. | Laser Doppler Velocimeter Optical Electrical Integrated Circuits |
| US9557486B2 (en) * | 2015-06-29 | 2017-01-31 | Elenion Technologies, Llc | Optimized 2×2 3dB multi-mode interference coupler |
| CN107577639B (en) * | 2017-08-19 | 2020-08-11 | 电子科技大学 | MPM (Multi-Point modulation) hybrid model simulation method applied to ECR (Electron cyclotron resonance) ion source numerical simulation |
| CN107611965B (en) * | 2017-09-15 | 2021-11-19 | 东南大学 | UPFC-containing power system economic and static safety comprehensive optimization method |
-
2018
- 2018-02-05 CN CN201810114140.XA patent/CN108416107B/en active Active
Patent Citations (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105593747A (en) * | 2013-06-09 | 2016-05-18 | 光电网股份有限公司 | Broad-modulation-bandwidth low-optical-loss modulators |
| CN104239623A (en) * | 2014-09-05 | 2014-12-24 | 兰州空间技术物理研究所 | Method for obtaining satellite surface potential based on multi-time-scale particle push |
| CN106155306A (en) * | 2015-05-13 | 2016-11-23 | 意美森公司 | The system and method for the sense of touch feedback of modularized equipment |
| CN107492591A (en) * | 2016-06-13 | 2017-12-19 | 隆达电子股份有限公司 | Chip scale package structure of light emitting diode |
| CN106503850A (en) * | 2016-11-01 | 2017-03-15 | 上海电机学院 | A kind of SVM forecast models for wind power prediction |
| CN106991237A (en) * | 2017-04-05 | 2017-07-28 | 东北大学 | A kind of electromagnetic agitation analysis construction method based on electric melting magnesium furnace |
| CN107526887A (en) * | 2017-08-22 | 2017-12-29 | 电子科技大学 | A kind of LeapfrogADI FDTD method parallel based on GPU |
| CN107425527A (en) * | 2017-09-15 | 2017-12-01 | 国网江苏省电力公司经济技术研究院 | A kind of THE UPFC static security prevention and control method |
Non-Patent Citations (1)
| Title |
|---|
| 包含碰撞效应的离子引出三维FEM-PIC模拟研究;黄桃 等;《第十八届全国等离子体科学技术会议摘要集》;20170726;第204页 * |
Also Published As
| Publication number | Publication date |
|---|---|
| CN108416107A (en) | 2018-08-17 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN108416107B (en) | Particle motion pushing finite element algorithm applied to PIC | |
| Grote et al. | The warp code: modeling high intensity ion beams | |
| WO2018176596A1 (en) | Unmanned bicycle path planning method based on weight-improved particle swarm optimization algorithm | |
| CN111259599B (en) | Test particle simulation method for analyzing constraint performance of high-energy particles in tokamak | |
| CN107577639B (en) | MPM (Multi-Point modulation) hybrid model simulation method applied to ECR (Electron cyclotron resonance) ion source numerical simulation | |
| Hu et al. | Recent developments on EOS 2-D/3-D electron gun and collector modeling code | |
| CN109357678A (en) | A kind of multiple no-manned plane paths planning method based on heterogeneousization dove colony optimization algorithm | |
| CN107705217B (en) | Method for calculating electric field below cross spanning area of alternating-current and direct-current transmission line | |
| CN111709150B (en) | Simulation method for magnetic field spatial distribution of magnet in any shape | |
| CN108460188B (en) | Charge distribution finite element FEM solving algorithm applied to PIC electrostatic model | |
| Chu et al. | AN IMMERSED-FINITE-ELEMENT PARTICLE-IN-CELL SIMULATION TOOL FOR PLASMA SURFACE INTERACTION. | |
| CN107944113A (en) | A kind of method for calculating three-dimensional high-speed translation Electromagnetic Scattering of Target field | |
| CN111695281A (en) | Particle rapid positioning method for tetrahedral mesh division finite element particle simulation | |
| CN108280309B (en) | Potential finite element solving method applied to PIC (peripheral interface controller) electrostatic model | |
| CN108446429B (en) | Particle stress finite element solving algorithm applied to PIC (positive-impedance converter) electrostatic model | |
| CN114237282A (en) | Intelligent unmanned aerial vehicle flight path planning method for intelligent industrial park monitoring | |
| Frenod et al. | An exponential integrator for a highly oscillatory Vlasov equation | |
| CN117473756A (en) | Cable laying optimization design method based on complex terrain | |
| CN111581876A (en) | Particle source solving method for particle simulation of cylindrical coordinate system | |
| CN107705310B (en) | Method and system for dividing land blocks | |
| Kovaleski | Calculation of the ion extraction boundary of a plasma ion source | |
| Qi et al. | PIC simulation of plasma properties in the discharge channel of a pulsed plasma thruster with flared electrodes | |
| CN112595333B (en) | Road navigation data processing method and device, electronic equipment and storage medium | |
| Brandt et al. | Magnetohydrodynamics and particle-in-cell codes simulation of plasma processes in micro HEMP-Thrusters | |
| Xue et al. | An Efficient iterative MoM-PO Hybrid Method for Calculating Scattered Fields of the Multiscale and Multiphysics Scatterers |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |