Solution of Static Field Problems With Random Domains

Transcription

1 Soluion of Saic Field Problems Wih Random Domains Duy Hung Mac, Séphane lene, Jean-laude Mipo, Olivier Moreau To cie his version: Duy Hung Mac, Séphane lene, Jean-laude Mipo, Olivier Moreau. Soluion of Saic Field Problems Wih Random Domains. I Transacions on Magneics, Insiue of lecrical and lecronics ngineers, 00, 46 (8), pp <0.09/TMAG >. <hal > HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied on 0 Sep 03 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Science Ars & Méiers (SAM) is an open access reposiory ha collecs he work of Ars e Méiers ParisTech researchers and makes i freely available over he web where possible. This is an auhor-deposied version published in: hp://sam.ensam.eu Handle ID:.hp://hdl.handle.ne/0985/775 To cie his version : Duy Hung MA, Séphane LNT, Jean-laude MIPO, Olivier MORAU - Soluion of Saic Field Problems Wih Random Domains - I Transacions on Magneics - Vol. 46, n 8, p Any correspondence concerning his service should be sen o he reposiory Adminisraor : archiveouvere@ensam.eu

3 9-NUMRIAL MTHODS Soluion of Saic Field Problems wih Random Domains D.H. Mac,3, S. léne,, J.. Mipo 3, and O. Moreau 4 LP/Ar e Méiers ParisTech, Lille, Lille cedex-france LP/LAML/Ar e Méiers ParisTech, Lille cedex-france 3 VALO-Sysèmes lecriques, réeil-france 4 LAML/dF R&D, 94 lamar edex-france Absrac A mehod o solve sochasic parial differenial equaions on random domains consiss in using a one o one random mapping funcion which ransforms he random domain ino a deerminisic domain. Wih his mehod, he randomness is hen suppored by he consiuive relaionship of he maerial. In his paper, his mehod is applied in elecrokineics in he case of scalar poenial and vecor poenial formulaions. An example is reaed and he proposed mehod is compared o a Non Inrusive Mehod based on he remeshing of he random domains. I. INTRODUTION The finie elemen mehod (FM) has been widely used o solve he Maxwell equaions leading o valuable ools for undersanding and predicing he feaures of elecromagneic devices. In several cases, he available inpu daa are known wih a finie level of confidence. These uncerainies can arise for insance from he aging of he maerials or from imperfecions of he manufacuring processes. Since he numerical models are more and more accurae due o he improvemen of numerical mehods (in 3D for example) and also due o he increasing of compuer performances, some of hese uncerainies can no be considered negligible any more. In several works, a probabilisic approach using random variables is used in order o ake ino accoun hese uncerainies []. In [], mehods o accoun for uncerainies on he maerial behavior were used o solve saic field problems. However, he case of uncerainies on he geomery is much less sudied. In [3], a mehod o solve differenial equaions in random domains based on a one o one random mapping funcion which ransforms he random domain ino a deerminisic one is proposed. In his paper, we propose o use such approach o solve a saic field problem wih random dimensions. Firs, we presen he problem o solve in he case of random linear behavior laws. Second, we show how a random domain problem can be ransformed ino a random behavior law problem using a one o one random mapping. Then, mehods, proposed in he lieraure o rea he case of random maerial behavior laws, can be applied o solve he problem wih random dimension such as he projecion mehod ha is shorly presened. Finally, o illusrae he mehod, a numerical example is presened and a comparison is done wih a Non Inrusive Mehod based on a remeshing procedure. II. PROLM WITH UNRTAINTIS ON TH HAVIOR LAW In his par, we will recall shorly how uncerainies on he behavior law can be aken ino accoun. The elecrokineics problem wih uncerainies on he behavior law defined in deerminisic domain D can be wrien: div J( x, ) = 0 curl ( x, ) = 0 () J( x, ) = σ ( x, ) ( x, ) The uncerainy on he behavior law is represened by he random field σ ( x, ), where is he oucome belonging o he space Ω. The curren densiy J and he elecric field are hen also random fields. We assume ha he domain D is bounded by he surface Γ = Γ Γ Γ where he boundary condiions are given by: 3 J( x, ) n = 0 on Γ () ( x, ) n = 0 on Γ and Γ3 where n is normal uni vecor and he elecromoive force beween and 3 is imposed o V. quaion() can be solved by eiher he scalar poenial formulaion or by he vecor poenial formulaion. If we denoe ϕ( x, θ ) he scalar poenial ha is a random field such ha: ( x, θ ) = grad ϕ( x, θ ) (3) quaion () can be wrien: div( σ ( x, θ ) grad ϕ( x, θ )) = 0 (4) The weak formulaion becomes: D grad ϕ( x, θ ) σ ( x, θ ) grad λ( x).d Ω ( x) = 0 (5) where λ( x) is a scalar es funcion ha is equal o zero on Γ and Γ 3. In FM, o approximae he scalar poenial, nodal shape funcions are commonly used. This problem can be sudied using Mone arlo Simulaion Mehod (MSM) [4] ha is a very reliable mehod bu very ime consuming. Alernaive mehods can also be used ha were sudied in [], [] and [5]: Specral Sochasic Finie lemen Mehod (SSFM) and Non-Inrusive Mehod (NIM). SSFM and NIM consis in projecing he field ϕ( x, θ ) on space K( D, Ω ) = S( D) H ( Ω ) where S( D ) is he space spanned by

4 9-NUMRIAL MTHODS he se of nodal shape funcions (x) (see secion II) and H ( Ω ) H ( θ ) (he is spanned by a se of orhogonal polynomials { i } polynomial chaos) [7]. The main difference beween SSFM and NIM is ha wih SSFM, he projecion ϕ( x, θ ) is underaken direcly on K( D, Ω ) = S( D) H ( Ω ) while wih NIM, ϕ( x, θ ) is firsly projeced on S( D) and afer on H ( Ω ). III. PROLM WITH UNRTAINTIS ON GOMTRY Wih SSFM, MSM and NIM, one difficuly in he case of random domains compared o he case of random behavior law is ha, a priori, geomeric variaion leads o a modificaion of he mesh. Since he boundaries of he domain are random so does he posiion of he nodes locaed on ha boundary. The shape funcions (x,θ) associaed o hese nodes become random fields. The space S(D) is no longer independen of H(), herefore wih SSFM, he random scalar poenial ϕ( x, θ ) can no be direcly approximaed by he projecion in K( D, Ω ) = S( D) H ( Ω ). To overcome ha difficuly, an idea based on a one o one random mapping funcion ha ransforms he random domain o a deerminisic domain is proposed in [3]. We will ranspose ha approach in elecrokineics. Le consider a domain D(θ) wih random boundaries and le suppose ha i exiss a one o one random mapping X = X ( θ, x) which ransforms he random domain D(θ) o a reference domain for each oucome θ (see fig.). Fig.. Transformaion mehod Thus, applying he random mapping, he weak formulaion (5) wrien on D can be wrien on and becomes: M ( X,). σ ( X ).M( X,) grad ϕ( X,) grad λ ( X )dx = 0 (6) de(m( X,) where M is he Jacobian marix of he mapping. If we denoe he conduciviy ensor: M ( X,) σ ( X ) M( X,) σ ( X,) = (7) de(m( X,) he problem wih uncerain dimensions on he domain D can be considered equivalen o a problem wih uncerainies on a modified behavior law wih a conduciviy σ (X,θ) on he reference domain. In a similar way, he problem () can be solved using a vecor poenial formulaion wriing J = curl T wih T he vecor poenial. The weak formulaion of he problem is hen: url ( T( X,)) σ ( X,) url( ψ ( X )) dx = 0 (8) D() X=X (x,) wih (X) is es funcion and - (X,) is inverse marix of conduciviy ensor (X,). The vecor poenial can be approximaed in T(D) H ( Ω) wih T(D) he edge elemen space. To solve he new problem given by (6) or (8), mehods presened in secion II can be used. One can noe ha o use one of hese mehods, a one o one random mapping funcion has o be defined. IV. NUMRIAL APPLIATION We focus now on he elecrokineic problem defined in a random domain D(θ) presened in Fig.. I is a cubic domain D wih a conduciviy σ = (.m) - wih an edge lengh (a = 4m). This domain holds anoher cube D wih random dimensions (l, l, l 3 ) wih a conduciviy σ = 0 (.m) -.The dimensions l (θ), l (θ) and l 3 (θ) are independen uniform random variables in he inerval [;.5](m). On wo opposie sides of he domain D an elecromoive force V = 4 (Vol) is prescribed. Since he dimensions of D are random so does he power. The aim is o calculae he power W() dissipaed in he domain D(). The power is approximaed by: N W( θ ) = w H ( θ ) (9) i= where H i () is he muli-dimensional orhogonal Legendre polynomials and w i a real coefficien. To calculae hese coefficiens, we use a NIM ha is based on a projecion mehod []. The coefficiens are given by: [ W( θ ) Hi ( θ )] wi = (0) Hi ( θ ) A where [X()] is he expecaion of he random variable X(). The calculaion of he denominaor can be done analyically whereas he calculaion of he numeraor can jus be done numerically using a quadraure mehod. For he quadraure mehod, we consider several specific realizaions of he random dimensions l (θ), l (θ) and l 3 (θ). For each realizaion we solve he problem (6) on he reference domain applying random mapping ha will be describe below. The conduciviies on each sub-domain D and D are reevaluaed using (7) for each poin of quadraure. The calculaion is underaken on a unique mesh of he reference domain, only he conduciviy disribuion changes from a quadraure poin o anoher. In an opposie way, wih a remeshing mehod (no mapping is required), o each quadraure poin which corresponds o a new geomery, he problem (5) is solved wih a new mesh. a : = 0 a l 3 l D l D i i a : = V Fig.. lecro-kineic sysem

5 9-NUMRIAL MTHODS In he following, we will deail he definiion of he random mapping. ecause of he symmery of he device, we can divide he domain D(θ) ino 8 idenical sub-domains D (θ) where he problem will be sudied. We define he random mapping X = X(x,θ) ha ransforms he domain D ino a domain wih l = l = l 3 = m. Fig. 3. The ransformaion X We divide he domain D and ino several sub-domains D i and i (8i) (Fig. 4). The mapping will be defined for each sub-domain D i by a linear ransformaion (dilaion) which is presened in Fig. 5. x 3 D () x x Fig. 4. Division of he random domain a l l 3 a 3 l a X=X(x, ) Fig.5. The linear dilaion The linear dilaion which ransforms a cube of dimensions a x a x a 3 ino a cube of dimensions b x b x b 3 is: b 0 0 a X x b X 0 0 x = a + () X 3A x 3 b A a3a which is a consan vecor linking he posiions of he wo ceners of he cubes. We obain he Jacobian marix of his ransformaion: 4 3 X =X(x,) b 0 0 a b = a b a3a M 0 0 b 3 b 4 a/ a/ b a/ X 3 X X () For example, for he ransformaion of he subdomain D 4 ino he sub-domain 4 wih dimensions 0.5a x 0.5a x 0.5a we have he following Jacobian marix (cf Fig. 3): a 0 0 l a M4 = 0 0 (3) ( a l ) a 0 0 ( a l3) A Therefore, he equivalen conduciviy of his sub-domain can be wrien under he form: ( a l )( a l3 ) 0 0 a l l ( a l3 ) σ = σ (4) a ( a l ) l ( a l ) 0 0 a ( a l3 ) A Finally, he iniial weak formulaion which was wrien on D(θ) under he form: 8 i= Di grad ϕ( x,) σ ( x) grad λ( x,) dx = 0 (5) D can be wrien using he mapping on : 8 i= i i grad ϕ( X,) σ ( X,) gradλ( X ) dx = 0 (6) i The same approach can be used o solve he vecor poenial formulaion. For he expression(9), we use an expansion of order 4 of he mulivariae Legendre polynomials. A ensorial Legendre Gauss quadraure mehod is used o calculae he coefficiens of he polynomial expansions (4 3 = 64 poins are calculaed). The quadraure poins are hen given by he roos of he Legendre polynomial of order 4. The domain has been meshed wih 36 erahedra (Fig. 6). Fig. 6. Mesh of 36 elemens The MSM has been also used. In ha case, a dimensions sample of size 0000 (l, l, l 3 ) is deermined using a random number generaor. For each realizaion, he problem is solved using he FM coupling wih he random mapping. We ge a he end a sample of power values of size The saisical momens are hen esimaed.

6 9-NUMRIAL MTHODS The saisical momens obained wih he NIM and he MSM by using he scalar poenial formulaion are repored in he Table I. We noice ha he NIM gives saisical momens ha belong o he 95% confidence inerval obained wih he MSM. The expansion (9) enables o obain good resuls and much faser han he MSM. TAL MONTARLO SIMULATION MTHOD OMPARD WITH NON- INTRUSIV MTHOD Mone arlo mehod Informaion (95% confidence inerval) Non-inrusive mehod Power (W) Power (W) Mean [8.86 : 8.99] 8.9 Sandard deviaion [3.34 : 3.4] 3.38 Skewness [0.43 : 0.56] 0.49 Kurosis [.40 :.69].55 A remeshing mehod can also be used o approximae he expansion (9) of he power W(θ). In fac, for each poin of quadraure, a new mesh is generaed direcly on he domain D and he problem is solved using boh poenial formulaions. Wih he mapping mehod, he geomery and he mesh don change whereas wih he remeshing mehod, he conduciviies in he subdomains don change. Wih he mapping mehod, we use four meshes wih M=8, M=79, M3=95, and M4=685 elemens. Wih he remeshing mehod, i almos impossible o keep he number of elemens consan since he geomery changes from a quadraure poin o anoher bu we ry o keep his number approximaely consan. Finally, solving he problem wih he wo mehods using boh poenial formulaions leads o four expansions of he power. From each expansion, we can easily deermine he mean and he sandard deviaion. In he Fig. 7 and Fig. 8, we give for he differen meshes (Mi, i = o 4), he evoluion of he mean and of he sandard deviaion. Firs we can see ha he scalar poenial formulaion gives a mean of he power greaer han he one given by he vecor poenial formulaion. This propery can be deduced from he energy bound propery observed in he saic elecromagneism in he deerminisic case [6]. The exac soluion is in beween he resuls given by boh formulaions. The smaller he gap is beween boh poenial formulaions, he more accurae he numerical model. We can see ha he gap is a decreasing funcion of he number of elemens and also ha he gap is smaller wih he mapping mehod han wih he remeshing mehod. This remark can also be done in he case of he sandard deviaion. Since he mesh is modified from a quadraure poin o anoher wih he remeshing mehod, i appears an addiional noise on he oupu of he model (he power W(θ)). This addiional noise increases he numerical error. Moreover, he mapping mehod is less ime consuming since here is no need o remesh. V. ONLUSION We have presened a mehod o solve a saic field problem wih uncerainies on he geomeric dimensions. This mehod consiss in using a random mapping ha enables o ransfer he randomness on he behavior law. We compare his mehod on an academic example wih a mehod ha consiss in remeshing he domain. The resuls obained show ha he mapping mehod seems o be more accurae and less ime consuming. The key poin of he mapping mehod is he deerminaion of he ransformaion ha can always be deermined easily. Mehods o deermine hese mappings are proposed in he lieraure bu need o be compared. M e a n v a lu e S a n d a rd d e v ia io n Transformaion-Scalar poenial Transformaion-Vecor poenial Remesh-Scalar poenial Remesh-Vecor poenial Mesh Fig. 7. Mean value of he Power W() obained by differen mehods Transformaion-Scalar poenial Transformaion-Vecor poenial Remeshing-Scalar poenial Remeshing-Vecor poenial Mesh Fig. 8. Sandard deviaion of he Power W() obained by differen mehods VI. RFRNS [] R. Ghanem, P. D. Spanos, Sochasic Finie lemens: A specral approach. Mineola, NY: Dover 003 [] R. Gaignaire, S. lene, O. Moreau, and. Sudre. 3D specral sochasic finie elemen mehod in elecromagneism. I Trans.Magn. vol.43, no.4, pp. 09-, 007. [3] D. Xiu e D. M. Tarakovsky. Numerical mehods for differenial equaions in random domains. SIAM J.SI OMPUT. No 3, pp.67-85, 006. [4] Has Minsky and Ibragimov, Saisical esimaion asympoic heory. Springer Verlag, 98. [5] R.Gaignaire, S.lene, O.Moreau,.Sudre. urren alculaion in lecrokineics Using a Specral Sochasic Finie lemen Mehod. Magneics, I Transacions Vol. 44, p.p , June 008 [6] W. Prager, J. L. Synge. Approximaion in elasiciy based on he concep of funcions space, Quar. Appl. Mah. 5, p.p. 6-69, 947 [7] D. Xiu and G. Karniadakis, The Wiener Askey polynomial chaos for saic differenial equaions, SIAM J. Sci. ompu. Vol.4, No., pp