CN108416107A - A kind of promotion Particles Moving finite element algorithm applied to PIC - Google Patents
A kind of promotion Particles Moving finite element algorithm applied to PIC Download PDFInfo
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Abstract
The invention belongs to the numerical simulation fields of particle simulation, and in particular to a kind of promotion Particles Moving finite element algorithm applied to PIC.The present invention uses unstructured grid, the grid to be capable of the shape on better model of fit boundary so that pushes Particles Moving algorithm to have higher computational accuracy in complex boundary PIC；The FEM methods for pushing Particles Moving are attached in typical PIC methods, while keeping the simple calculating of typical case's PIC methods, quick good characteristic, higher FEM calculation precision is obtained using FEM；Since FEM methods can both be well matched with complex boundary, it can be needed to use non-uniform grid according to simulation again, and not limited by the numerical stability conditions, therefore can be under conditions of keeping computational accuracy, optimize space lattice and time step, to which simulation precision be significantly increased.
Description
Technical field
The invention belongs to the numerical simulation fields of particle simulation (Particle-in-cell is abbreviated as PIC), and in particular to
A kind of promotion Particles Moving finite element algorithm applied to PIC.
Background technology
PIC methods are a kind of numerical simulations being widely used in charged particle and electromagnetic field interaction physics problem
Method, it is adding by tracking a large amount of charged particles and is obtaining macroscopic properties from the movement and statistical average be in harmony in electromagnetic field outside
And the characteristics of motion.After decades of development, PIC analogy methods have become research charged particle and electromagnetic field interactant
A kind of strong numerical value means of reason problem are widely used in charged particle and many necks involved by electromagnetism field interactions
Domain, such as magnetic confinement fusion plasma, inertial confinement fusion plasma, nuclear blast, space plasma, artificial plasma
(including electron gun, ion source etc.), electric propulsion, free-electron laser and electron tube etc..
The core procedure that PIC is solved is as follows：
1, electromagnetic field solves, i.e., (degenerates under static electric charge model by solving the maxwell equation group that electromagnetism place meets and be
Poisson's equation), obtain the electromagnetic field on all mesh points；
2, particle stress solves, i.e., is worth to the Potential distribution in grid by the current potential on related grid point, and solve
Its negative gradient obtains particle position electric field, then solves stress；
3, Particles Moving is pushed, i.e., by solving the discrete particle equation of motion, the movements such as the speed of more new particle and position
Information, and then the affiliated grid of more new particle；
4, the distribution in source acquires its contribution to around mesh point charge and electric current, so according to the position where particle
All particles add up to the charge on mesh point with current contribution afterwards and obtain the charge density on mesh point and current density；
Process as above is constantly recycled, until the time until numerical convergence or being artificially arranged.
It is one of core procedure of PIC that wherein step 3, which pushes Particles Moving, the accurate and efficient solution of the step for
The control of the integrated solution precision and efficiency of PIC is particularly significant.Until up to now, Particles Moving is pushed mainly to have two in PIC
Kind method, is finite difference (FD) method and embedded type finite element (IFE) method respectively.
FD methods：In the application that PIC pushes Particles Moving, FD methods are by using the discrete solution area of structured grid
Domain, thus solve the discrete particle equation of motion and judge the affiliated grid scheduling algorithm form of particle simply, it can be readily appreciated that still existing
It is had the disadvantage that in the application of PIC promotion Particles Movings：
1, FD methods are using structured grid made of being divided by cross line, for complicated curved boundary fitting compared with
Difference, so that the solving precision of the promotion Particles Moving of complicated curved near border is relatively low；
2, the requirement due to FD methods to size of mesh opening uniformity is relatively high, is limited to tiny physics in simulation system
The limitation of structure, it is necessary to computational accuracy requirement could be met by dividing sufficiently small grid, so that total grid number is huge, and mould
Quasi- number of particles is proportional to total grid number, this results in the calculation amount for pushing Particles Moving to solve very huge；
3, FD methods are severely limited to the limitation of the numerical stability conditions, i.e., in the Numerical-Mode for pushing Particles Moving to PIC
In quasi-, if space lattice is small-sized, the very little that time step can also take therewith, this can be further increased to pushing particle
The FD numerical simulations burden that run duration cycle solves.
The deficiency occurred in Particles Moving application is pushed in PIC for FD methods, Kafafy and Wang were proposed in 2003
It can be applied to PIC and push IFE methods in Particles Moving.
IFE methods：PIC push Particles Moving application in, IFE methods by using intrusive unstructured grid from
Dissipate domain.Intrusive unstructured grid dividing condition is as shown in Figure 1, it can be seen that this mesh generation has been equivalent to 2 weights
Grid, wherein 1 weight grid is structured grid, 2 weight grids are further to divide each structured grid in 1 weight grid
At five tetrahedral intrusive unstructured grids.In the application of PIC, field solves intrusive non-using the 2nd weight IFE methods
Structured grid, and Particles Moving is pushed to be carried out in the 1st weight structure grid, therefore there is no solve the side FD for IFE methods
Disadvantage of the method in PIC pushes Particles Moving application.
Invention content
It is in view of the above problems and insufficient, to solve FD and IFE methods in pushing Particles Moving to solve to boundary
Matching degree is not high, solving precision is not high and numerical simulation bears big problem, and the present invention provides a kind of pushing away applied to PIC
Dynamic Particles Moving finite element (FEM) algorithm.Specific technical solution is as follows：
Step 1, electromagnetic field solve.
By solving the discrete maxwell equation group that electromagnetism place meets, (degenerate under static electric charge model is the discrete Poisson of electrostatic
Equation), obtain the electromagnetic field on all mesh points；
Step 2, particle stress solve, and are worth to the Potential distribution in grid by the current potential on related grid point, and ask
It solves its negative gradient and obtains particle position electric field, then solve stress.
Step 3 pushes Particles Moving
Using global unstructured grid, as shown in Figure 2.By solving the discrete particle equation of motion, the speed of more new particle
The movable informations such as degree and position, and then the affiliated grid of more new particle.Detailed solution procedure is divided into following two steps：
The speed of step 3.1, more new particle and position
Under rectangular coordinate system, consider the equation of motion for the particle that relativistic effect obtains as shown in (1) formula：
Wherein：One point of subscript indicates that the first order derivative of the variables versus time, two points indicate the two of the variables versus time
Subderivative；η is the charge-mass ratio of particle；γ is relativistic factor；Ex、Ey、Ez、Bx、By、BzFor the electromagnetic field of particle position
Component is acquired by step 2；
Formula (1), the speed of more new particle and position are solved using the methods of Boris or Runge-Kutta.
The affiliated grid of step 3.2, more new particle
The affiliated grid of more new particle is indispensable link in PIC later for the speed of each more new particle and position, only
There is the affiliated grid of more new particle, could carrying out the calculating in source, (module of the feature physical process occurred i.e. in step 4), grid adds
Be subject to and correlation values diagnosis etc..
Due to the simulation particle huge amount in PIC and at this time using unstructured grid (not as structuring net
The affiliated grid of particle can be immediately arrived under lattice situation), therefore be to influence entirety PIC to calculate on the calculating of the affiliated grid of all particles
The committed step of precision and efficiency.For this purpose, the present invention proposes a kind of particle fast locating algorithm based on unstructured grid, use
In accurately and efficiently calculating the affiliated grid of particle, specific implementation is as follows：
1) in first step time step, position is occurred according to particle or the surface of emission is set, initializes grid residing for particle
Number.
2) since second step time step, first determine whether particle whether still in current grid.If terminated
Search；
If it was not then into 3).
3) the center of gravity P of grid PreId where line previous moment particle1With current time particle position P2, obtain line
Section P1P2, and find out line segment P1P2With the intersection FaceId of PreId, while obtaining sharing intersection FaceId's with PreId
New grid is set as PreId, returned to 2) by new grid.
For always select the center of gravity of grid rather than other mesh points as calculate point the main reason for there are three：1, again
The heart one is positioned inside grid；2, center of gravity calculation is convenient, it is only necessary to by the x on each vertex of grid where center of gravity, y, z coordinate point
Measure the x that summation divided by grid vertex sum respectively can be obtained the barycentric coodinates, y, z durection components；3, avoid particle special
There is endless loop when program judges when position, such as a certain grid vertex is exactly in line segment P1P2When upper.
The above-mentioned particle fast locating algorithm based on unstructured grid, the grid in conjunction with residing for a time step on particle,
It is scanned for by depth-priority-searching method, according to specific direction since initial position, grid where Step wise approximation particle, finally
Realize accurate quickly positioning.
The specific schematic diagram of algorithm is as shown in Figure 3.In order to preferably describe the particle fast locating algorithm, schematic diagram uses two
The form of expression is tieed up, two-dimensional triangular mesh is equivalent to three-dimensional tetrahedral grid in the figure, and two-dimensional line is equivalent to three-dimensional
Face.
Algorithm particular flow sheet is as shown in Figure 4.
The distribution of step 4, source.
Its contribution to around mesh point charge and electric current is acquired according to the position where particle, then by all particles pair
Charge on mesh point adds up with current contribution obtains the charge density on mesh point and current density；
Step 1,3 to 4 solution using structuring, immersion is unstructured or complete unstructured grid.
Circulation step 1 to 4 finally carries out numerical value diagnosis until reaching the condition of convergence or simulation end condition.
The present invention is suitable for two dimension and three-dimensional structure, and when being suitable for two dimension, mesh generation becomes triangle from tetrahedral grid
Shape grid.
The FD methods of Particles Moving and IFE methods, beneficial effects of the present invention is pushed to be embodied in relative to PIC：
1, using unstructured grid, which is capable of the shape on better model of fit boundary so that in complex boundary
In the case of PIC push Particles Moving algorithm have higher computational accuracy；
2, the FEM methods for pushing Particles Moving are attached in typical PIC methods, in the meter for keeping typical case's PIC methods
While simple, quick good characteristic, higher FEM calculation precision is obtained using FEM；
3, it since FEM methods can not only be well matched with complex boundary, but also can be needed to use non-homogeneous net according to simulation
Lattice, and not limited by the numerical stability conditions, thus can under conditions of keeping computational accuracy, optimize space lattice and when
Between step-length, to which simulation precision be significantly increased.
Description of the drawings
Fig. 1 is the IFE grid schematic diagrames that PIC is solved；
Fig. 2 is the FEM grid schematic diagrames that PIC is solved；
Fig. 3 is particle fast locating algorithm schematic diagram；
Fig. 4 is particle fast locating algorithm flow chart；
Fig. 5 is the PIC static electric charge model calculated examples schematic diagrames of seven apertures in the human head double grid ion-optic system；
Fig. 6 is the PIC static electric charge model calculated examples mesh generation schematic diagrames of seven apertures in the human head double grid ion-optic system.
Specific implementation mode
Below by embodiment, invention is further described in detail.
By taking ion propeller seven apertures in the human head double grid ion-optic system as an example, schematic diagram is as shown in Figure 5.Using being calculated in the present invention
The specific implementation step that method carries out this example PIC electrostatic simulations is as follows：
Step 1, electric field solve.
The discrete Poisson's equation of electrostatic met by solving electric field, obtains the electric field on all mesh points；
Step 2, ion stress solve, and are worth to the Potential distribution in grid by the current potential on related grid point, and ask
It solves its negative gradient and obtains particle position electric field, then solve stress；
Step 3 pushes ion motion.
Speed and position of ion etc. are updated by solving discrete ion motion equation using global unstructured grid
Movable information, and then update the affiliated grid of ion.Detailed solution procedure is divided into following two steps：
Step 3.1, the speed for updating ion and position
Magnetic field is not considered herein, and the ion motion equation under rectangular coordinate system is as shown in (2) formula：
Wherein：One point of subscript indicates that the first order derivative of the variables versus time, two points indicate the two of the variables versus time
Subderivative；η is the charge-mass ratio of ion；γ is relativistic factor；Ex、Ey、EzFor the electric field component of ion position, by step
Rapid 2 acquire；
Formula (2) is solved using Runge-Kutta methods, obtains ion after a time step new speed and position.
Step 3.2, the update affiliated grid of ion
The speed of update ion and position update the affiliated grid of ion later every time, prepare for follow-up step 4.
Here the ion fast locating algorithm based on unstructured grid is used, for accurately and efficiently calculating belonging to ion
Grid, specific implementation are as follows：
1) in first step time step, position is occurred according to ion or the surface of emission is set, initializes grid residing for ion
Number.
2) since second step time step, first determine whether ion whether still in current grid.If terminated
Search；
If it was not then into 3).
3) the center of gravity P of grid PreId where line previous moment ion1With current time ion position P2, obtain line
Section P1P2, and find out line segment P1P2With the intersection FaceId of PreId, while obtaining sharing intersection FaceId's with PreId
New grid is set as PreId, returned to 2) by new grid.
Step 4, charge distribution.
Its contribution to mesh point charge around is acquired according to the position where ion, then by all ion pair mesh points
On charge contribution cumulative obtain the charge density on mesh point；
Step 1,3 to 4 solution can be used that structuring, immersion be unstructured and completely unstructured etc. grids.
Circulation step 1 to 4 finally carries out numerical value diagnosis until reaching the condition of convergence or simulation end condition.Using this hair
Bright middle algorithm carries out PIC electrostatic simulations to this example, and the results are shown in Figure 5.
Claims (3)
1. a kind of promotion Particles Moving finite element algorithm applied to PIC, specific as follows：
Step 1, electromagnetic field solve；By solving discrete maxwell equation group that electromagnetism place meets or static electric charge model is lower expires
The discrete Poisson's equation of electrostatic of foot, obtains the electromagnetic field on all mesh points；
Step 2, particle stress solve, and are worth to the Potential distribution in grid by the current potential on related grid point, and solve it
Negative gradient obtains particle position electric field, then solves stress；
Step 3 pushes Particles Moving, using global unstructured grid, by solving the discrete particle equation of motion, more new particle
Speed and the movable informations such as position, and then the affiliated grid of more new particle；
The speed of step 3.1, more new particle and position
Under rectangular coordinate system, consider the equation of motion for the particle that relativistic effect obtains as shown in (1) formula：
Wherein one point of subscript indicates that the first order derivative of the variables versus time, two points indicate that the secondary of the variables versus time is led
Number；η is the charge-mass ratio of particle；γ is relativistic factor；Ex、Ey、Ez、Bx、By、BzFor the electromagnetic field point of particle position
Amount, is acquired by step 2；
The affiliated grid of step 3.2, more new particle；
1), in first step time step, position is occurred according to particle or the surface of emission is set, grid residing for particle is initialized and compiles
Number；
2), since second step time step, first determine whether particle whether still in current grid；If terminating to search
Rope；If it was not then entering in next step 3)；
3) the center of gravity P of grid PreId where line previous moment particle1With current time particle position P2, obtain line segment
P1P2, and find out line segment P1P2With the intersection FaceId of PreId, while obtaining sharing the new of intersection FaceId with PreId
New grid is set as PreId, returned to 2) by grid；
The above-mentioned particle fast locating algorithm based on unstructured grid, the grid in conjunction with residing for a time step on particle, by deep
Degree priority algorithm scans for, and according to specific direction since initial position, grid where Step wise approximation particle is final to realize
Accurate quickly positioning；
The distribution of step 4, source；
Its contribution to around mesh point charge and electric current is acquired according to the position where particle, then by all particles to grid
Charge on point adds up with current contribution obtains the charge density on mesh point and current density；
Circulation step 1 to 4 finally carries out numerical value diagnosis until reaching the condition of convergence or simulation end condition.
2. being applied to the promotion Particles Moving finite element algorithm of PIC as described in claim 1, it is characterised in that：In the step 3
Formula (1), the speed of more new particle and position are solved using Boris or Runge-Kutta methods.
3. being applied to the promotion Particles Moving finite element algorithm of PIC as described in claim 1, it is characterised in that：The step 1,3
Solution to 4 uses structuring, immersion unstructured or complete unstructured grid.
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| CN111967148B (en) * | 2020-07-31 | 2023-07-07 | 电子科技大学 | Particle simulation Voronoi diagram particle merging algorithm |
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