The Simulation Code XPDP1

Transcription

1 Chapter 3 The Smulaton Code XPDP1 3.1 Introducton The smulaton program used n ths research s XPDP1, an open source software developed by the Plasma Theory and Smulaton Group PTSG at Unversty of Calforna, Berkeley. It s runnng on Unx workstatons wth X-Wndows, and PC wth an X-Wndows emulator. XPDP1 s the X-Wndows verson of PDP1, whch has the same physcs kernel as PDP1. PDP1 s among the PDx1 codes from UC Berkeley, where x stands for P (Planar), C (Cylndrcal), and S (Sphercal) electrodes. The codes are wrtten n object-orented style, and n standard C (C language). PDP1 (Plasma Devce Planar) smulates a plasma wthn planar electrodes, wth or wthout a unform appled DC magnetc feld n an arbtrary drecton. The two electrodes n PDP1 are symmetrc, as llustrated n Fgure 3.1. For other versons n the same seres, there are PDC1 and PDS1. PDC1 (Plasma Devce Cylndrcal) smulates a plasma wthn concentrc (coaxal) cylndrcal electrodes and allows an axal DC magnetc feld. PDS1 (Plasma Devce Sphercal) smulates a plasma wthn concentrc sphercal electrodes, wthout a magnetc feld. The nner electrode of both PDC1 and PDS1 s of fnte sze. They are useful for smulatng dscharges wth 9

2 dfferent electrode areas [5]. In ths work, we wll only focus on the planar verson, PDP1. Fgure 3.1: Schematc dagram of XPDP1 code [5]. PDP1 s the modfed verson of the W. S. Lawson s PDW1 code (1983). It s a bounded electrostatc code, smulatng one-dmensonal plasma devces. The code smulates a bounded plasma wth external crcut, whch nclude R, L, C elements, and AC, DC, ramped current/voltage sources. These characterstcs (ncludng partcles and electrostatc felds) are specfed by the user at run tme usng an nput fle. The code uses Partcle-n-Cell (PIC) technque to smulate the electrons and ons, leap-frog method for the ntegraton of the equaton of moton, and Monte-Carlo collsonal (MCC) model for electron-neutral and on-neutral collsons [15]. The PIC technque, leap-frog method and MCC model are descrbed n sectons 3., 3.3 and 3.4 respectvely. The smulaton proceeds n real tme, and the user may vew the output as the code s runnng, n the form of varous user-specfed dagnostc wndows. The dagnostc wndows are updated at each tme step (anmaton). The applcatons of XPDP1 code range from collsonal capactve RF dscharges, used n materals processng to collsonless fuson problems [5]. 3

3 3.3 Partcle-n-Cell (PIC) Method In realty, plasma s a collecton of partcles whch consst of electrons, ons wth varous charge states, neutral atoms and molecules. In the Partcle-n-Cell method, there are computer partcles (superpartcles). Each computer partcle s a homogeneous collecton of a large number of real partcles (commonly 1 4 to 1 6 partcles), whch s havng the same mass-to-charge rato as the real partcles, thus mnmzng the amount of partcles to be smulated. In the PIC scheme, the physcal volume s dvded nto cells by lnes whch run parallel to the boundares. The ntersectons of these lnes are called mesh ponts or grd ponts. An example of the grd s llustrated n Fgure 3.. Fgure 3.: A mathematcal grd of PIC scheme [16]. 31

4 In Fgure 3., A mathematcal grd s set nto the plasma regon, to measure the charge densty ρ and current densty J. From the measured charge and current denstes, we wll obtan the electrc feld E and magnetc feld B on the grd. The partcle quanttes such as velocty and poston of a charged partcle q at (x, y) locaton wll be counted n terms of ρ at the nearby grd ponts (, ), (1, ), (1, 1), (, 1) and n terms of J at the faces between these ponts [16]. The partcle quanttes, such as veloctes, v and poston, x are known at the partcle, and may take on all values n the phase space. For each tme step, the charge and current denstes on the grd are calculated. The process to produce the charge and current denstes ( ρ, J) on the grd from the partcle postons x and veloctes v mples some weghtng (lnear weghtng) to the grd ponts that s dependent on partcle poston. Once the denstes are establshed on the grd, the ρ and J are used to obtaned the electrc and magnetc felds (E, B) by solvng the feld equatons. Posson s equaton s used to solve for E n electrostatc smulaton. For electromagnetc smulatons, the full set of Maxwell s equatons s used to solve for E and B. Wth the felds on the grd, but partcles scattered around wthn the grd, the force at the partcle F s obtaned through nterpolaton of the feld from the grd to the partcles by agan performng a weghtng. Newton-Lorentz equaton of moton s used for the calculaton, and the partcles are advanced to new postons and veloctes. Next, partcle boundary condtons such as absorpton and emsson are appled. If the model s collsonal, the Monte Carlo Collson (MCC) scheme s appled. The flow of the PIC-MCC scheme wthn one tmestep s shown schematcally n Fgure 3.3 [16, 17, 18]. 3

5 Fgure 3.3: Flow chart for an explct PIC-MCC scheme [16]. 3.3 Integraton of the Partcle Equatons of Moton The ntegraton method used n the code s called the leap-frog method. The two frst-order dfferental equatons to be ntegrated are dv m = F,...(3.1) dt dx = v.... (3.) dt These equatons are replaced by the fnte-dfference equatons v v new old m = t F old, (3.3) x new x t old = v new...(3.4) 33

6 In the leap-frog method, as shown n Fgure 3.4, v() s pushed back to v(- t / ) usng the force F calculated at t =. Consequently, the fnte dfference equatons of the leap-frog method are v m v t t+ t / t t / = F,...(3.5) x t+ t x t t = v....(3.6) Fgure 3.4: Scheme of the leap-frog ntegraton method [4]. Therefore, v F = t + v,....(3.7) m t+ t / t t / t+ t x = v t + x t...(3.8) where force, F = q( E + V B). The leap-frog method s explct snce t t / v + and t t x + are determned only from values at earler tme levels. The stablty and 34

7 accuracy of the leap-frog method can be tested by applyng t to the smple harmonc oscllator model d x = ω x...(3.9) dt Equaton (3.9) s substtuted nto the fnte dfference scheme to obtan t + t+ t t t t x x x = ω x t..(3.1) The solutons of the equaton are of the form x t = C exp( ωt),.(3.11) x t+ t = C exp[ ω ( t + t)]....(3.1) Usng Euler s formula, the fnte dfference becomes ω t sn ω = ± t..(3.13) ω t For 1 <<, ω ω as desred. If ω t >, the real soluton for ω becomes complex wth growng and decayng roots, ndcatng numercal nstablty. For smulatons whch use the leap-frog mover, typcally ω t. s taken for stablty [16, 17, 18]. More detals on the accuracy and stablty requrements for PIC are stated n secton

8 3.4 Monte Carlo Collson Model In the Monte Carlo Collson (MCC) model, random numbers are beng used to decde whether or not a partcle s subjected to a collson, and what type of collson s to occur. The MCC model statstcally descrbes the collson processes, usng cross sectons for each type of collson. Consder a set of partcles ncdent on another set of partcles (targets). The probablty, P of a collson event for the th ncdent partcle 1 of energy ε = mv can be wrtten as, P ( x) ( ) v ] = 1 exp[ n σ ε t,...(3.14) g T where the total cross secton s the sum over all processes, σ T ( ε ) σ ( ε ) =..(3.15) j j Here n g (x) s the spatally varyng target densty, v s the ncdent speed, and tme nterval. t s the For example, assume the partcle spece s has N types of collsons wth the target spece, the knetc energy of the th partcle of the ncdent s spece s gven by 1 ε = msv,.(3.16) 36

9 where ε s needed n calculatng the collson cross sectons. The total cross secton s σ ( ε ) σ ( ε ) T =,..(3.17) j j where ( ) j ε σ for 1 j N, s the cross secton of the jth type of collson between the s spece and the target spece. The collson probablty for the th partcle s calculated based on the dstance s = v t traveled n each tme step, t. Consequently, P exp[ n ( x) ( ) v t] = 1 exp[ n ( x) ( ) s ] = σ ε σ ε. If a 1 g T g T unformly dstrbuted random number on the related tme step s less than P, a collson wll take place. Then another random number s chosen to determne the type of collson. Every computer partcle must be evaluated n every tme step of the smulaton. Another approach s the null collson method, whch s computatonally more effcent. In ths method, only one collson probablty, whch s energy ndependent, s used to model all the partcles. The smulaton program wll randomly select a number of partcles based on the collson probablty, and concentrate on evaluatng ths selected fracton partcles, then decdng whch collson they are to undergo. The null collson model has a sgnfcant speed advantage over other collson models whch query each of the fracton of partcles for collsons at every tme step. 37

10 3.5 Accuracy and Stablty Requrements for PIC The basc dscretzaton parameters for explct PIC method are mesh spacng, tme step, x, t, and number of partcles per cell, PPC. Tryng to ncrease the mesh spacng, x and tme step, t s often done for speedng-up the smulatons. However, there are certan constrants on and stablty. For electrostatc PIC, the x and x and t n order to mantan accuracy t are determned by λ D ω t p. and 1, x 1 e n ε T where the plasma frequency, ω = p, the Debye length, λ D = and ε m en 1 the mesh spacng, nterelectrode gap length x =. number of spatal cells, nc In these equatons, T s the plasma temperature n electron volts, n s the plasma densty, ε s the permttvty of free space ( F m -1 ), e s the electronc charge, and m s the mass of the lghtest speces nvolved n the collson [18, 19]. 38

11 3.6 Boundary Condtons The boundary condtons of the code nclude surface charge on the electrodes, whch are connected to a seres RLC crcut wth drvng voltage, V(t) or current, I(t). The general form of the appled source s S t) = DC + Ramp t + AC sn(π f t + )..(3.18) ( θ An external crcut has ts own ntrnsc tme scales whch are not related to the plasma tme scales. The smulaton tme step must be small enough to resolve these tme scales, or the crcut smulaton wll produce naccurate results regardless of the plasma parameters []. Referrng to Fgure 3.1, the current n the external crcut nteracts wth the plasma va the surface charge on the electrodes. The potental wthn the plasma regon s affected by the dstrbuton and moton of space charge, the electrode surface charge, and the current n the external crcut [1]. By applyng Gauss law to the system, the boundary condtons for the potental equaton are obtaned [1]. S E ds = V ρ dv + ε A σ A σ ε =,...(3.19) where the surface S encloses the plasma and electrodes. A + and A - refer to the left and rght electrode respectvely, and σ s the surface charge n the respectve 39

12 electrode. ρ has unts of charge/volume and σ has unts of charge/area. Usng the defnton of potental, we obtan Φ j + 1 Φ x j + Φ j 1 = ρ j ε,.(3.) for planar electrodes [1]. For one dmensonal system, the boundary condtons can be wrtten as [1] Φ nc = (3.1) and σ + E =... (3.) ε Equaton (3.1) can be wrtten at one-half grd cell from the boundary, n conjuncton wth a central dfference appled to the defnton of potental, whch gves E 1/ Φ Φ1 x 1 = σ ε = + x + ρ,.(3.3) for planar electrodes [4]. The charge conservaton equaton at each wall, A σ = Qconv + Q...(3.4) becomes t t t t t t t Qconv + Q Q σ = σ +....(3.5) A where Q s the charge on one plate of the external crcut capactor [1]. 4

13 In the code, there are four cases whch cover the full range of external crcut parameters [1]: (1) General Seres RLC Crcut (Voltage-Drven Crcut) For the general voltage-drven seres RLC crcut, the capactor charge Q s advanced usng Krchhoff s voltage law, d Q dq Q L + R + = V ( t) + Φ Φ nc. (3.6) dt dt c where Φ nc s the reference potental fxed at zero, whch to be the potental of the grounded electrode, and Φ s the potental at tme zero. One second order dfference s Q t t t t V ( t) + Φ nc Φ K =, (3.7) α where K t = t t t t t 3 t t 4 t α 1Q + α Q + α 3Q + α 4Q...(3.8) 9 L 3 R 1 α = t t C α 1 L R = 6 t t α = 11 L t + 1 R t α 3 = L t α = L t 41

14 () Open Crcut When C. the mpedance approaches nfnty, therefore the external crcut becomng an open crcut. The potentals on the boundares are floatng, and the surface charges on the electrodes nfluence the potental as always, but the electrodes cannot exchange charge va external current. The charge conservaton equaton s σ = + Q. (3.9) t t + σ + t conv (3) Short-Crcut When R=L= and C, the external crcut s a short crcut, wth Φ Φ nc = V ( )...(3.3) t The short-crcut case s appled when C εa / l > (3.31) (4) Current-Drven Crcut In ths case, an deal current source s assumed to be drvng the specfed tmevaryng current I(t). The external crcut elements R, L, and C are gnored snce an deal current source s an open crcut. The stablty of the crcut wth L s gven by the characterstc equaton ξ (3 + t / RC) 4ξ + 1 =,..(3.3) 4

15 where ξ 1 s requred. The roots are ± 1 t / RC ξ =..(3.33) 3 + t / RC 3.7 Partcle Condtons - Loadng Intal Dstrbutons In order to calculate the partcle energy dstrbuton, the followng equatons are used [17]: The Maxwell-Boltzmann dstrbuton, ε f ( ε ) = f exp, kt (3.34) 1 3 ε = mυ t = kt.... (3.35) In one dmenson, equatons (3.34) and (3.35) becomes, υ = f ( υ ) f exp,...(3.36) υt 1 1 mυ t = kt (3.37) Invert the cumulatve dstrbuton functon (3.38), F( υ ) = υ exp( υ / υt ) dυ. (3.38) exp( υ / υ ) dυ t 43

16 By usng Box-Muller method [35], we choose two pseudo-random numbers ( < v, v 1) dscardng them f R = v + v > 1. Then ln( R ) υ 1 = v1,...(3.39) R n( R ) υ = v...(3.4) R Equaton (3.38) must be nverted numercally for general cutoffs. 3.8 General Structure of the Code XPDP1 code s mplemented wth an object-orented structure n standard C n order to facltate enhancements. The code s separated nto a physcs applcaton and the wndowng core. Therefore, new physcs and dagnostcs can be added wthout alterng the wndowng core. The nteracton between wngraphcs manager and the physcs kernel s llustrated n Fgure

17 Fgure 3.5: Schematc representaton of the nteracton between WnGraphcs and the physcs kernel [15]. 45

18 Usng the wndowng core, all dagnostcs are updated dynamcally n tme. The dagnostcs can also update n ndvdual tme-steps, and pausng for keystroke before contnung the smulaton [15]. PDP1 can run ndefntely wthout a tme step lmtaton. Tme hstores are combed perodcally n such a way that there are never more than HISTMAX (a constant) value stored [15]. PDP1 employs four crcut solvers to handle the full range of external crcut parameters. () The general voltage-drven seres RLC case. () The open crcut case, where C. () The short crcut case, when C and R=L=. (v) The deal current source case. More detals have been descrbed prevously n secton 3.6. A Monte Carlo Collsonal model for electron-neutral and on-neutral collsons s used n PDP1. All components of velocty can be specfed ndependently n the nput fle. The man flow of the code s shown n Fgure 3.6. Table 3.1 s the descrpton of the XPDP1 code man flow. 46

19 Fgure 3.6: Man flow of XPDP1 code. 47

20 Table 3.1: Descrpton of XPDP1 code man flow. The descrpton of the XPDP1 program man flow dsplay_ttle The code starts wth the dsplay of the program ttle and the general nformaton of the program developer, etc.. XGInt Intalze the wndow graphcs (XGrafx) stuff. Here, the wndow manager wll get the code and the nput fle names, read, then ntalze the wndow array. Start The start functon opens fles and read the general nput parameters n the nput fle, then checks for errors. The program wll ext, when there s an error. If no error, the program wll proceed to allocate feld parameters and dagnostcs arrays. DagArray Allocate dagnostc arrays. Ths functon wll allocate space for all the dagnostcs. The dagnostcs are also group nto tme hstory dagnostcs and average dagnostcs. Then the program wll setup the parameters for each speces, e.g. on, electron, etc.. speces Read speces parameters from the nput fle, and assgn nput parameters to arguments. Scalng factors to convert from physcal unts to code unt and vs versa. Next, allocate partcle arrays. SpecesDagArray Ths s the dagnostc arrays of the speces. The program wll allocate 48

21 space for veloctes dstrbuton, and allocate space for the dstrbuton functons arrays. Calculate the coeffcent for the mover n the presence of a magnetc feld. Then, the program wll load n all speces partcles, whch s properly dstrbuted. Ths wll start wth a dump fle wth all the record of a dstrbuton, or f a dump fle s not gven, t wll load the system wth the ntal dstrbuton. Restore load Runnng the dump fle f gven. Load one speces and one drecton at a tme. There are loader for a cold beam and loader for thermal dstrbuton. Maxwellan dstrbuton functon s used n the code. (More detals n secton 3.7 and Chapter - secton.6) mccdag_nt Intalze Monte Carlo Collson rate dagnostc. Allocate arrays for dagnostc rates. IntWndows setrho Intalze dagnostc wndows. Setup each wndow structure. Set ntal charge densty, ncludes smoothng the charge densty of each speces. felds Intalze feld arrays. feld_nt Intalzng the arrays and parameters for the feld solve. Here, the program wll setup the arrays for the posson solve. Then, decde whch crcut solver to use. 49

22 There are four optons of crcut solver: (More detals n secton 3.6). when C, open crcut. when current source s appled. when L=R=, C nfnty, short crcut v. the general case wth external voltage source Calculate the external crcut. Hstory Intalze hstory arrays. Ths s the tme hstory accumulator. It calculates and stores all hstory values, and performs combng on hstory values when low on memory. XGStart Ths wll start the XGrafx man loop, whch s the wngraphcs manager. It wll then lnk to the man physcs loop, whch s the XGManLoop. XGManLoop For a no subcyclng loop, ntally the program wll proceed to advance tme for the speces sp (on). Then advance poston and velocty n (*moveptr). adjust Remove partcles that cross boundares. Ths s the routne to adjust (ntalze, re-pack and nject) partcles to the 5

23 desred condtons. boundary There are four parts of smulaton process. Frst, ntalze array for computng postons of njected partcles. Then, t wll go to left hand sde (LHS) wall dagnostcs, md system dagnostcs, and njecton of new partcles at walls (one speces at a tme). (*mccptr) e.g.: argonmcc Monte Carlo Collsons for speces sp (on). Ths s the part of Monte Carlo Collson for electron-neutrals and on-neutrals. Here, the program wll be calculatng the null collson probablty, then the electron collsons wth argon. For the collsons, there 51

24 are fve types of collsons: collson, onzaton, elastc exctaton, scatterng and charge exchange collson. The program wll determne the type of collson, and follow wth the calculaton. (More detals n secton 3.4 and Chapter - secton.7) gather Assgn charge denstes to the grd. Ths functon s n separate loop, because new partcles mght be created n mcc and adjust that mght not have been weghted. Then the program wll run the felds, and hstory. Lastly back to the flow n XGStart n the wngraphcs manager. 5

25 XPDP1 s avalable n the zpped form from the PTSG webste at []. The wngraphcs manager (XGrafx) s avalable separately at the same webste. The nstallaton gude s avalable n the webste at [3]. XPDP1 fully support a mouse for selecton of tems, buttons, etc.. The buttons on the man menu nclude RUN, STOP, STEP, SAVE, and QUIT. The dagnostc wndow has four buttons, whch are RESCALE, TRACE, PRINT, and CROSS-HAIR. The nput fle s used to specfy the parameters for the smulaton. The descrpton of the nput fle s provded n the user manual [15], whch comes together wth the XPDP1 dstrbuton. There are mplct and explct schemes n XPDP1. In ths work, explct scheme s used for all the smulatons due to ts smplcty as we are dealng wth DC dscharges. For the RF dscharges we contnue to use explct scheme for comparson. 53