Arbitrary Lagrangian Eulerian Electromechanics in 3D

Transcription

1 Progress In Electromagnetcs Research Symposum 2006, Cambrdge, USA, March Arbtrary Lagrangan Euleran Electromechancs n 3D R. Reben, B. Walln, and D. Whte Lawrence Lvermore Natonal Laboratory, USA Abstract We present results from an effort to couple the equatons of electromagnetc dffuson wth the equatons of arbtrary Lagrangan-Euleran (ALE) hydrodynamcs. The electromagnetc dffuson equatons are dscretzed usng a novel mxed fnte element method coupled wth a generalzed Crank-Ncholson tme dfferencng scheme. At each dscrete tme step, electromagnetc force and heat terms are calculated and coupled to the hydrodynamc equatons n an operator splt approach. We present prelmnary results from a fully coupled electromechancal smulaton as well as results concernng advecton technques for electromagnetc quanttes. 1. Introducton We are nterested n the smulaton of electromechancal devces and magnetohydrodynamc events n three dmensons. Our prmary goal s a numercal method that solves, n a self-consstent manner, the equatons of electromagnetcs (prmarly statcs and dffuson), heat transfer (prmarly conducton), and non-lnear mechancs (elastc-plastc deformaton, and contact wth frcton). In ths paper, we focus on the numercal dscretzaton of electromagnetc dffuson n an arbtrary Lagrangan-Euleran (ALE) fashon for the purposes of computng J B forces for mechancal (or hydrodynamc) calculatons and J E Joule heatng terms for thermal calculatons. The equatons of electromagnetc dffuson can be derved from the full wave Maxwell equatons by makng the good conductor approxmaton (.e., gnorng dsplacement current), whch s standard practce n magnetohydrodynamc (MHD) formulatons. For conductng materals movng wth a velocty v wth respect to a fxed Euleran (or laboratory) frame, we can derve the so called dynamo equaton (also known as the hydromagnetc equaton) n terms of magnetc flux densty B t = ( 1 σ 1 µ B) + ( v B) (1) In the Euleran descrpton the velocty v s a functon of tme t and poston x. In the Lagrangan (or materal) descrpton (whch we wll desgnate wth a prme symbol), the flow s descrbed by followng the poston x( x,t) of the materal pont that started at poston x at t = 0. In functonal form, we have x = x ( x,t); x = x( x,t) To convert between the two representatons, we defne the Jacoban matrx as J,j = x j x (2) As shown n [1], the followng quanttes are nvarant wth respect to the Lagrangan-Euleran representatons Lagrangan Euleran E d x = ( E + v B) d x B d a = B d a (3) It s well known that dfferental arc length and surface area elements transform accordng to d x = J T d x (4) d a = J J 1 d a (5) As a consequence, the electrc feld ntenstes and magnetc flux denstes must transform nversely to mantan the nvarance property of (3)

2 266 Progress In Electromagnetcs Research Symposum 2006, Cambrdge, USA, March The dynamo equaton n the Lagrangan frame s therefore E = J 1 ( E + v B) (6) B = 1 J JT B (7) d B = ( 1 σ 1 µ B ) (8) In a typcal ALE hydrodynamc calculaton, an operator splt method s employed where all calculatons are performed on a Lagrangan mesh (.e., a mesh that moves wth the materals). When the Lagrange moton of the mesh causes sgnfcant mesh dstorton, that dstorton s corrected wth an equpotental relaxaton of the mesh, followed by a 2nd order monotonc remap of mesh quanttes. Ths remap s equvalent to an advecton of materal through the mesh. In our proposed ALE formulaton of MHD, we wll employ an operator-splt method wth three dstnct steps: Electromagnetc Dffuson Solve the dynamo equaton n the Lagrangan frame at one dscrete tme step for fxed materals. Lagrangan Moton Move mesh nodes accordng to J B forces assumng a d B condton. = 0 frozen flux Euleran Advecton Only requred f mesh s relaxed, advect (or transport) magnetc (vector potental) flux quanttes to new mesh. Note that the second step (effectvely draggng the electromagnetc quanttes along wth the mesh durng Lagrangan moton) wll only work f our dscretzaton of the electromagnetc quanttes satsfes the nvarance relaton of (3) (see also [2]). In the Euleran advecton step of the calculaton, the computed electromagnetc degrees of freedom must be remapped or advected n a way whch preserves a dscrete dvergence-free property of the magnetc flux densty wth mnmal magnetc energy loss. 2. Numercal Formulaton The dvergence-free (or solenodal) nature of the magnetc flux densty, B = 0, mples that B = A where A s a magnetc vector potental. Ths n turn mples that the electrc feld n the Lagrangan frame s gven by E = φ t A, where φ s an electrc scalar potental. Usng the gauge condton σ A = 0, we can reformulate the dynamo equaton (8) n terms of potentals as σ φ = 0 (9) σ d A = 1 µ A σ φ (10) Note that ths formulaton has an addtonal ellptc PDE (9) to solve for the scalar potental. A key advantage of ths formulaton s that voltage, whch s often the only known quantty for electromechancal engneerng applcatons, appears explctly n the equatons as an essental boundary condton for the ellptc soluton of (9). To compute force and heat terms, we defne the secondary varables n terms of the potentals as B = A (11) J = σ E = φ d A (12) Fnally, there are dvergence constrants on both the prmary and secondary felds, namely σ A = 0 (13) B = 0 (14)

3 Progress In Electromagnetcs Research Symposum 2006, Cambrdge, USA, March To dscretze the potental formulaton n the Lagrangan frame, we apply the mxed fnte element methods (FEM) of [3] whch are based on the propertes of dfferental forms and have been shown to preserve dscrete dvergence-free propertes and to mantan accuracy n secondary varables (e.g., J and B) even when computed from potentals. Most mportantly, the dscrete vector felds transform dentcally to (6) and (7), thereby preservng the nvarance property of (3). In our proposed ALE formulaton the scalar potental wll be dscretzed on mesh nodes (.e., a dscrete 0- form feld), the vector potental wll be dscretzed on mesh edges (.e., a dscrete 1-form feld) and the secondary varables B and J wll be dscretzed on mesh faces (.e., dscrete 2-form felds) as follows φ A B J v W 0 (15) a W 1 (16) b W 2 (17) j W 2 (18) where W l denotes a dscrete l-form bass functon. In [3], varous mass, stffness, dervatve and dscrete Hodge matrces are defned. Gven these matrces, the fully dscrete form of the potental dffuson equaton s gven n [3] by applyng a Generalzed Crank-Ncholson method to obtan S 0 v n+α = f 0 n+α (19) (M 1 (σ) + α ts 1 (µ 1 ))a n+1 = (M 1 (σ) (1 α) t S 1 (µ 1 ))a n td 01 v n+α (20) where α [0.1] s weghtng parameter whch determnes the type of ntegraton such that 0 Explct, 1st Order Accurate Forward Euler α = 1/2 Implct, 2nd Order Accurate Crank Ncholson 1 Implct, 1st Order Accurate Backward Euler Once the values for the prmary potentals have been solved for, the dscrete secondary felds can be computed as e n+α = K 01 v n+α 1/ t(a n+1 a n ) (21) b n+1 = K 12 a n+1 (22) M 2 (σ 1 )j n+α = H 12 e n+α (23) These terms are used to compute J B forces whch wll accelerate the mesh nodes durng the Lagrangan moton step. The dscrete dvergence constrants are gven by (D 01 (σ)) T a = 0 (24) (D 01 (σ)) T e = 0 (25) K 23 b = 0 (26) and as shown n [3], these constrants are mplctly satsfed for all tme, assumng the ntal condtons and the source terms are dvergence free. To demonstrate a fully coupled Lagrangan calculaton, we consder a numercal experment n whch a 5 KV capactor bank s dscharged nto a can shaped alumnum structure (see Fg. 1). The voltage through the can (effectvely an nductve and resstve load) s computed va a smple SPICE model. The resultng voltage vs. tme profle s then used as an essental boundary condton for the dscrete scalar potental solve of (19) whch

4 268 Progress In Electromagnetcs Research Symposum 2006, Cambrdge, USA, March drves the problem. An essental boundary condton of the form ˆn A = 0 s appled to the front sde of the mesh whle the remander of the surface s subject to the natural boundary condton ˆn 1 µ A = 0. A peak current of roughly 0.8 MA s generated n the can, creatng a J B force whch causes the can to ntally compresses (or mplode). However, the force s not strong enough to cause the alumnum can to yeld, and so the can effectvely rngs over tme n an elastc response as shown n Fgs. 1 and 2. Fgure 1: Snapshot of the fully coupled electromechancal smulaton. In ths mage the alumnum can s elastcally expandng after ntally beng compressed. The dsplacement has been exaggerated by a factor of 300 for vsual clarty. Fgure 2: Measured pressure response n the alumnum can due to an electromagnetc force. 3. Constraned Transport Methods on Unstructured Grds Durng the optonal Euleran advecton phase of our operator splt method, the computed electromagnetc values must be remapped (or advected). Remappng refers to the process of updatng the representaton of the feld gven a new grd. We consder only new grds whch are nearby n the sense that only small perturbatons of the grd are allowed (.e., the mesh nodes should not travel farther than one mesh element n any one tme step). Ths s known as the contnuous remap approxmaton (CRA). We propose to use the so called constraned transport method orgnally developed by [4] and later expanded by [5]. Suppose we have calculated the magnetc flux densty B n a Lagrangan tme step va (22), we then have a local element representaton of B The degrees of freedom (DOF) b old B old b old W 2,old (27) n ths expanson carry the unts of magnetc flux. For the specal case of lowest order (p = 1) bass functons (.e., sx DOF per element), ths mples that we know the magnetc flux through every face n the Lagrangan mesh (or the old mesh). Now n a standard ALE step, the old mesh s relaxed under the CRA to a new mesh. Therefore, our goal s to compute new values of the magnetc flux b new whch wll allow us to represent the magnetc flux densty on the new mesh. For the specal case of lowest order (p = 1) bass functons, the dscrete dvergence free property s smply a statement that the 6 fluxes n the face sum to zero. The goal of constraned transport s to preserve ths property on the new mesh. For unstructured hexahedral grds, we can update the magnetc flux (or vector potental flux A d x) by effectvely solvng Faraday s law for a movng conductor (equvalent to magnetc transport under the frozenflux condton) Φ new Φ old C ( u B) d l (28) where u s the mesh dsplacement. Our goal now s to apply (28) n an algorthmc fashon to update the fluxes on the faces of a new mesh. A schematc representaton of ths process s shown n Fg. 3. It s clear from

5 Progress In Electromagnetcs Research Symposum 2006, Cambrdge, USA, March the depcton of Fg. 3 that we can approxmate the flux through a new face gven the flux through the old face and a measurement of the tme rate of change of flux (an effectve voltage) along the closed crcut path C depcted n green. For a gven face n the new mesh, the algorthm of (28) can be used to update the edge flux contrbutons Fgure 3: Schematc dagram depctng the relatonshp between magnetc flux through old and new mesh faces and the most accurate locaton for measurng the update EMF. A dx for each edge n the face (thereby updatng the vector potental) or the total magnetc flux B d a through the face. By constructon, the new flux values wll sum to zero, provded the old fluxes do so as well. In order for ths algorthm to work, B must be evaluated at the dsplacement vector mdponts for the dscrete path ntegral; however, ths s problematc for faced based representatons of B, snce they are by constructon, dscontnuous along element edges. To overcome ths, a smooth B feld must be patch recovered usng a contnuous vector nodal approxmaton. 4. Conclusons We have presented and dscussed an operator splt approach for solvng the coupled equatons of electromechancs and magnetohydrodynamcs usng the novel mxed fnte element methods of [3] to dscretze the equatons of electromagnetc dffuson. We have presented prelmnary results for a fully coupled Lagrangan calculaton and have dscussed methods for advectng magnetc flux for ALE calculatons. Acknowledge Ths work was performed under the auspces of the U.S. Department of Energy by the Unversty of Calforna, Lawrence Lvermore Natonal Laboratory under contract No. W-7405-Eng-48, UCRL-ABS REFERENCES 1. Lax, M. and D. F. Nelson., Maxwell equatons n materal form, Phys. Rev. B, Vol. 13, No. 14, , Hu, P. J., A. Robnson, and R. Tumnaro., Towards robust z-pnch smulatons: dscretzaton and fast solvers for magnetc dffuson n heterogenous conductors, Electronc Trans. Num. Analyss, Vol. 15, , Reben, R. and D. Whte., Verfcaton of hgh-order mxed fem soluton of transent magnetc dffuson problems., IEEE Trans. Mag., artcle n press, October Evans, C. R. and J. F. Hawley, Smulaton of magnetohydrodynamc flows: a constraned transport method, The Astrophyscal Journal, Vol. 332, , Bochev, P. and M. Shaskov., Constraned nterpolaton (remap) of dvergence-free felds, Comput. Methods Appl. Mech. Engrg., Vol. 194, , 2005.