Combined RKDG and LDG Method for the Simulation of the Bipolar Charge Transport in Solid Dielectrics

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1 PIERS ONLINE, VOL. 5, NO. 1, Combined RKDG nd LDG Method for the Simution of the Bipor Chrge Trnsport in Soid Dieectrics Jihun Tin, Jun Zou, nd Jinsheng Yun Deprtment of Eectric Engineering, Tsinghu University, Beijing , Chin Abstrct The spce chrge trnsport described by the convection-rection equtions is simuted by using the Runge-Kutt discontinuous Gerkin RKDG method. Compred to the trdition finite voume method, the RKDG method hs higher ccurcy nd better performnce of cpturing the steep front in the chrge profie. Combined with the RKDG method, the oc discontinuous Gerkin LDG method ws dopted to sove the Poisson s eqution insted of the previous boundry eement method BEM, in order tht the numeric qudrture for the soution of the convection-rection equtions is repced by nytic formus. The bipor chrge trnsport under dc votge is simuted by the RKDG+LDG method, which is more efficient nd produces identic resuts with those of the RKDG+BEM method. 1. INTRODUCTION The spce chrge dynmics pys n importnt roe on the degrdtion nd the brekdown of soid dieectrics under the high votge due to the intensifiction of the oc eectric fied. The trnsport of the different species of spce chrges under the eectric fied cn be described by set of convection-rection equtions couped with the Poisson s eqution. In our previous work, the RKDG method ws utiized to sove the convection-rection equtions [1]. Unike the trdition QUICKEST method with the ULTIMATE fux imiter [, 3], the RKDG method cn hnde the convection nd the rection terms simutneousy without incorporting the spit procedure [4], nd the high-order ccurcy cn be obtined without incorporting wide stenci, which is more suitbe for the shock cpturing thn the QUICKEST method [5]. For the soution of the Poisson s eqution, the oc discontinuous Gerkin LDG method proposed by B. Cockburn et. [5, 6] is dopted in this pper insted of the genery used boundry eement method BEM [7], in order to repce the numeric qudrture with nytic formus in the RKDG wek formution of the convection-rection equtions. This combined RKDG+LDG method is verified by the simution of the spce chrge trnsport in ow-density poyethyene LDPE smpe under high dc votge. The resuts show tht the proposed method is more efficient nd produces identic spce chrge profies with those of the RKDG+BEM method.. MATHEMATICAL MODEL DESCRIPTIONS The fuid mode for the spce chrge trnsport in this pper ws initiy proposed by Aison nd Hi [8]. It hs redy been widey dopted by Le Roy nd Begroui et. in [, 3, 7]. The mode is given s foows: n t + f x = S convection-rection eqution, 1 E x = ρ ε 0 ε r Poisson s eqution, f = ±µ n E trnsport eqution, 3 where n is the number of crriers per unit voume nd the subscript represents different crrier types. Usuy, four species of crriers re tken into ccount to describe the chrge dynmics in soid dieectrics, i.e., free eectrons nd hoes, which represent the crriers in the show trps; trpped eectrons nd hoes, which represent the immobie crriers residing in the deep trps. The source term S in 1 comprises the effects of trpping nd recombintion. Both the source term S nd the grdient of the crrier convection fux f contribute to the vrition of the crrier concentrtion ccording to 1. At the eectrodes, the inet boundry condition, i.e., crrier injection, for the convection-rection eqution is usuy prescribed s the Schottky injection w, whie the Ohm s w is dopted s the outet boundry condition, i.e., crrier extrction. For the definitions nd the physic menings of the mode prmeters, reders my refer to [1, 3, 7, 8] for detis.

2 PIERS ONLINE, VOL. 5, NO. 1, RKDG METHOD FOR CONVECTION-REACTION EQUATION The uthors of [3, 7] used the QUICKEST method with the ULTIMATE fux imiter to sove the convection-rection Eqution 1. However, it cnnot ccurtey simute the spce chrge profie with very steep wvefront, i.e., shock wve, due to tht wide stenci is dopted to interpote the numeric fux. Moreover, the spit tretment of the originy couped convection nd rection terms brings ddition errors [1]. In our previous work [1], the RKDG method ws ppied to the convection-rection eqution, in which the high-order piecewise continuous Legendre poynomis were chosen s the bsis functions to expnd the spce chrge profie. A speci Runge-Kutt method combined with sope imiter ws used for the time discretiztion, which ensures the numeric stbiity. The source term S nd the convection term f / x were hnded simutneousy by the RKDG method without the spit procedure. High-order ccurcy cn be obtined without incorporting wide stenci, which is more suitbe for the shock cpturing thn the QUICKEST method. The bsic steps of the RKDG method re s foows: 1. Discontinuous Gerkin DG spce discretiztion: The crrier concentrtion n of the jth eement = x j 1, x j+ 1 cn be pproximted by the Legendre poynomis {P x} =0,...,k s: n = k =0 P x xj, 4 where is the order of the poynomi nd is the ength of. By tking {P x} =0,...,k s the weighting functions, the convection-rection Eqution 1 cn be weky enforced. After the simpifiction by utiizing the orthogonity of the Legendre poynomis, the semi-discrete form of the convection-rection eqution cn be obtined: d dt α t = + 1 where ˆf is the numeric fux. P x xj f P x xj ˆf x j+1/ x j 1/ + S P x xj, 5. Numeric fux: The Lx-Friedrichs numeric fux ws dopted in the RKDG method ccording to [9]. For the right boundry of the jth eement, it is given s: where ˆf j+1/ n j+1/, n + j+1/ = 1 [ ] f j n j+1/ + f j n + j+1/ Cn + j+1/ n j+1/, 6 C = mx v t = 0, x, 7 x n j+ nd n + 1 j+ re the eft nd right imit vues of n 1 t x j+ 1, v t = 0 is the initi crrier speed. 3. RK time discretiztion: According to [9], the RK time discretiztion shoud be performed in k + 1 intermedite steps to mtch the ccurcy degree of the spce domin. With the chrge distribution n m t the time point m t, the RK time stepping is s foows: Set n 0 = n m ; b For the s intermedite steps s = 1,..., k + 1: n s = ΛΠ s 1 α s ω s, 8 =0

3 PIERS ONLINE, VOL. 5, NO. 1, where ω s = n + β s α s tl j {n }, 9 L j {n } represents the right-hnd side of 5. ΛΠ is sope imiter proposed by Bisws et., which ensures the numeric stbiity of the RKDG method [10]. The determintion of the coefficients α s nd β s is introduced by Gottieb et. in [11]. c Set n m+1 = n k LDG METHOD COMBINED WITH RKDG METHOD The purpose of dopting the LDG method for the soution of the Poisson s eqution is to promote the efficiency of the RKDG method by nyticy evuting the integrs insted of the numeric qudrture. In the RKDG semi-discrete formution of the convection-rection Eqution 5, there exists the integrnd which is the product of the crrier concentrtion n nd the eectric fied E: f P x xj = x xj. 10 If the eectric fied E is soved by the BEM, this integr shoud be evuted by numeric methods, such s the Guss-Legendre qudrture. However, for the combined RKDG+LDG method, the Legendre poynomis re seected s the bsis functions for both the crrier concentrtion n nd the eectric fied E. This fct enbes us to nyticy ccute the bove integr LDG Method for Poisson s Eqution For the one-dimension Poisson s eqution defined in domin Ω with the first kind condition on the domin boundry Γ D : dq = f in Ω, 11 q = E = du, u = g D on Γ D, its wek soution stisfies the foowing two equtions: x j+ 1 qr = rû u dr, 1 x j 1 q dv = x j+ 1 fv + vˆq, 13 x j 1 where r nd v beongs to the spce of Legendre poynomis defined on with the degree t most k, i.e., r, v P k = {P x : = 0,..., k}. The numeric fux û in 1 nd ˆq in 13 re defined s: ˆq {q} C11 C = + 1 [u] û {u} C 1 C [q] ˆq q = û + C 11 u + g D n on Γ g D, n = D in Ω, 14 { 1 eft boundry 1 right boundry, 15 where C 11 = C = 1, C 1 = C 1 = 0, {q} = 1 q+ + q, [q] = q + q. {u} nd [u] dopt j+ 1 j+ 1 j+ 1 j+ 1 the simir definitions s {q} nd [q]. By expnding u nd its grdient q with Legendre poynomis, the coefficients of the bsis functions cn be obtined by soving 1 nd 13.

4 PIERS ONLINE, VOL. 5, NO. 1, Anytic Evution of the Integr In the RKDG+LDG method, both the chrge density n nd the eectric fied E re expnded by the Legendre poynomis with the form s in 4. Therefore, the integr in 10 cn be directy given by nytic formus. Tke the Legendre poynomi spce P k where k = 1 nd k = s exmpe, the nytic soution of the integr cn be given s: x xj = { 0 = 0 ±µ 0 b b 1 = 1 x xj 0 =0 = ±µ 0 b b b =1 ±µ 0 b b b b 1 = for k = 1, 16 for k =, 17 where {b } 0 represent the coefficients of the Legendre poynomis for E. The bove nytic formus re fexibe nd cn be ppied to vrious situtions, due to tht they re derived from the integrtion with respect to the product of the Legendre poynomi series which re used to pproximte the rbitrry distribution of the eectric fied nd the crrier density. 5. RESULTS AND COMPARISONS A 118 µm LDPE smpe under extern dc votge 1 kv is used s test mode for the bipor chrge trnsport. The dopted prmeter vues re the sme s those in [3]. To verify the combined RKDG+LDG method with different bsis function orders k = 1 nd k =, its simution resuts t t = 0, 30, 50, 100 s re compred with those obtined from the RKDG+BEM method. In the impementtion of the RKDG+BEM method, five-point Guss-Legendre numeric qudrture ws dopted in ech spci eement in. The retive errors of the eectric fied nd the net chrge density re given in Tbe 1. The retive errors re defined s foows: ε E = E BEM E LDG E BEM, ε ρ = ρ /BEM ρ /LDG ρ /BEM 18 It cn be seen from these dt tht for the cse k = 1, the RKDG+LDG method produces most identic resuts s those of the RKDG+BEM method. For k =, the retive error of the eectric fied is sti very sm. Athough the retive error of the net chrge density increses to bout 1%, the simuted chrge profies of the two methods re sti very cose to ech other see Figure 1. Moreover, by using the RKDG+LDG method, of the numeric qudrtures impemented in the RKDG+BEM method re repced by nytic soutions, which sves 40% of the computtion time. From our numeric tests, the difference in the computtion time between the BEM nd the LDG method for soving the Poisson s eqution is negigibe nd the incresed efficiency of the RKDG+LDG method is miny due to repcing the numeric qudrtures by nytic formus in the soution of the convection-rection eqution. Therefore, the combined RKDG+LDG method is verified nd shows to be more efficient thn the RKDG+BEM method. 8 Net Chrge Density C/m s 50s 30s 0s RKDG+LDG k= RKDG+BEM k= Xm x 10-4 Figure 1: Comprison of the simuted spce chrge profies by using the rd order RKDG+LDG nd RKDG+BEM methods.

5 PIERS ONLINE, VOL. 5, NO. 1, Tbe 1: Retive errors of the eectric fied nd the net chrge density by using the RKDG+BEM nd the RKDG+LDG methods. Timess ε E k = 1 ε E k = ε ρ k = 1 ε ρ k = e e e e e e e e e e e e CONCLUSIONS The combined RKDG nd LDG method proposed in this pper hs high-order ccurcy nd is suitbe for simuting the spce chrge trnsport with steep fronts. Both the RKDG nd the LDG methods re bsed upon the discontinuous Gerkin spce discretiztion with the Legendre poynomis s the bsis functions, which enbes us to repce the numeric qudrtures in the RKDG+BEM method with nytic formus. The RKDG+LDG method is verified by the simution of the bipor chrge trnsport in n LDPE smpe, which produces identic resuts with those of the RKDG+BEM method. Due to the empoyment of the nytic integr, the RKDG+LDG method cn promote the efficiency of the numeric scheme by sving 40% of the computtion time tht needed by the RKDG+BEM method. ACKNOWLEDGMENT This work is supported by the Ntion Ntur Science Foundtion of Chin REFERENCES 1. Tin, J., J. Zou, Y. Wng, J. Liu, J. Yun, nd Y. Zhou, Simution of bipor chrge trnsport with trpping nd recombintion in poymeric insutors using Runge-Kutt discontinuous gerkin method, Journ of Physics D: Appied Physics, Vo. 41, No. 19, 10, October Le Roy, S., Numeric methods in the simution of chrge trnsport in soid dieectrics, IEEE Trnsctions on Dieectrics nd Eectric Insution, Vo. 13, No., 39 46, Begroui, E., I. Boukhris, A. Ke, G. Teyssedre, nd C. Lurent, A new numeric mode ppied to bipor chrge trnsport, trpping nd recombintion under ow nd high dc votges, Journ of Physics D: Appied Physics, Vo. 40, No. 1, , Toro, E. F., Riemnn Sovers nd Numeric Methods for Fuid Dynmics, Chpter 15, Spitting Schemes for PDEs with Source Terms, Springer, Cockburn, B. nd C. W. Shu, Runge-kutt discontinuous gerkin methods for convectiondominted probems, J. Sci. Comput, Vo. 16, No. 3, , Cockburn, B. nd C. W. Shu, The oc discontinuous gerkin method for time-dependent convection-diffusion systems, SIAM Journ on Numeric Anysis, Vo. 35, No. 6, , Le Roy, S., P. Segur, G. Teyssedre, nd C. Lurent, Description of bipor chrge trnsport in poyethyene using fuid mode with constnt mobiity: Mode prediction, Journ of Physics D: Appied Physics, Vo. 37, No., , Aison, J. M. nd R. M. Hi, A mode for bipor chrge trnsport, trpping nd recombintion in degssed crossinked poyethene, Journ of Physics D: Appied Physics, Vo. 7, , Cockburn, B. nd C.-W. Shu, TVB Runge-Kutt oc projection discontinuous Gerkin finite eement method for conservtion ws II: Gener frmework, Mthemtics of Computtion, Vo. 5, No. 186, , Bisws, R., K. D. Devine, nd J. E. Fherty, Pre, dptive finite eement methods for conservtion ws, App. Numer. Mth., Vo. 14, No. 1 3, 55 83, Gottieb, S. nd C.-W. Shu, Tot Vrition diminishing Runge-Kutt schemes, Mthemtics of Computtion, Vo. 67, No. 1, 73 85, 1998.