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3D Spectral Stochastic Finite Element Method in Electromagnetism R. Gaignaire, S. Clénet, B. Sudret, O. Moreau Abstract—Spectral Stochastic Finite Element Method (SSFEM), used in mechanics to take into account random aspects of input data, has been implemented, as an extended version, in the 3D FEM software CARMEL dedicated to electromagnetic field computation. As a test-case, this approach has been applied to a 3D electrostatic problem and successfully validated by comparing with the Monte Carlo Simulation Method involving usual “deterministic” CARMEL resolutions. Index Terms—Spectral Stochastic Finite Element Method. Random problem I.

I

INTRODUCTION

n electromagnetism, most models are based on the numerical solution of Maxwell Equations, assuming all input data are perfectly known. Unfortunately, geometry and material characteristics for instance would rather present uncertainties. Under those conditions, the output data (magnetic field distributions, global quantities like torque, flux, current…) become also uncertain. To deal with propagation of the input data uncertainties to the output data, probabilistic models would rather be more appropriate than deterministic models. Whereas many methods have already been proposed in mechanics, very few have been applied in electromagnetism [1]. Among all the available techniques, the Monte Carlo Simulation Method (MCSM) [2] is probably the most wellknown. It is already widely used in different scientific domains (like financial mathematics, biostatistics, or mechanics) because of its is simplicity and robustness but remains very time consuming. Besides, the “perturbation method” consists in expanding the unknown field around its mean [4]. This approach is very useful to determine the moment of the first and second orders (mean and variance) of the unknown field. However, the extension to moments of higher orders is very difficult and time consuming also.

Manuscript received April 24, 2006. This work is involved in the CNRT “réseaux and machines électriques du futur” and is supported by the region Nord Pas de Calais, the French government and the company EDF. R. Gaignaire is with the “Ecole Nationale des Arts et Métiers” , 8 Boulevard LOUIS XIV, Lille, 59046 CEDEX, France and with Electricité de France, R&D Division, 1 Avenue du Général de Gaulle, 92141 Clamart Cedex (email : [email protected]). S. Clenet is with the “Ecole Nationale des Arts et Métiers” , 8 Boulevard LOUIS XIV, Lille, 59046 CEDEX, France (phone: 303-555-5555; fax: 303555-3333; e-mail: [email protected]). B. Sudret is with Electricité de France, R&D Division, Renardières, 77818 Moret-sur-Loing Cedex (e-mail: [email protected]). O. Moreau is with Electricité de France, R&D Division, 1 Avenue du Général de Gaulle, 92141 Clamart Cedex (e-mail: [email protected]).

Moreover, Neumann Expansion Method, where operators are expanded into Neumann series, seems to present a very low convergence rate [2]. Finally, risk and reliability studies in mechanics have been using for fifteen years the Spectral Stochastic Finite Element Method (SSFEM) [3] based on discretizing simultaneously onto both spatial and random domains. The present work intends to apply this method to a 3D electrostatic problem. First, we will define the functions which span a dense subset of the space of the random variables of finite variance. The next section will deal about the space where the output fields are approximated. Then, he matrix system to be solved will be described and some properties given. Finally, the validation has been carried out by comparing, on a test-case, results obtained with Monte Carlo Simulation Method (MCSM) involving the “deterministic” 3D FEM software CARMEL, developed at the LAMEL-L2EP, with SSFEM implemented in its “Stochastic” extended version. II. DESCRIPTION OF THE PROBLEM Let us consider a spatial domain D (Fig.1). In the following, x represents the spatial dependence, and θ the random dependence. The domain D is divided into M sub domains Di on which the permittivity iε(θ) is assumed to be uniform but random. Thus, the permittivity ε(x,θ) can be written: M

ε ( x,θ ) = ∑ i ε (θ )1Di ( x) . i =1

(1)

where 1Di(x) is the indicator function (this function is equal to 1 if x belongs to Di and else zero). Since the permittivity is random, the electric field E and the electrical flux density D are also random fields. Both fields satisfy the electrostatic charge free problem which can be written as:  curl x ( E ( x,θ )) = 0  (2)  divx ( D( x,θ )) = 0  D ( x , θ ) = ε ( x, θ ) E ( x , θ )  curlx, divx represent the standard spatial differential operator (derivative versus the position x). Since E(x,θ) is curl free, the scalar potential ϕ(x, θ)) can be introduced : E = grad x (ϕ ( x,θ )) + grad x (ϕC ( x)) (3) Where φC is chosen in order to have homogeneous Dirichlet boundary conditions on φ. So, by using (2) and (3), the equation to be solved is: divx (ε ( x,θ ) grad x (ϕ ( x,θ )) . (4) = −divx (ε ( x,θ ) grad x (ϕC ( x)))

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The problem is the same as in deterministic case and is to determine the scalar potential distribution in both spatial and random dimension knowing ε(x,θ) and the boundary conditions on D. Generally, the equation (4) has no analytical solution. To solve it numerically a weak form of (4) is used: E ( ∫ div ε ( x,θ ) grad xϕ ( x,θ ) ω ( x,θ ) dD) = D

(5)

− E ( ∫ div ε ( x,θ ) grad xϕC ( x)) ω ( x,θ )dD)

B. Multi-dimensional case: As in [2]-[3], multi-dimensional Hermite Polynomials can be written as a product of one dimensional Hermite polynomials of independent standard normal variables. Considering for instance M independent standard normal variables (ξi)i=1…M , an integer sequence α =( α1,…, αM), and M

an integer p = | α | = ∑ α i , let us define: i =1

D

Where ω is a test function as defined in [8] and E(.) denotes the mathematical expectation: +∞

E[X(θ )] = ∫ xf X (x )dx

(6)

-∞

With X a random variable and fX(x) its random distribution. By using the Green Formula [8] it can be written: E ( ∫ [ε ( x,θ ) grad x (ϕ ( x,θ ) grad x (ω ( x,θ ))dD ]) D

= − E ( ∫ ε ( x,θ ) grad xϕC ( x) grad x ( ω ( x,θ ))dD  )

(7)

D

Equations (7) require a finite dimension space of functions to be solved numerically. That space is spanned by the tensor product of n spatial shape functions (λi(x))1≤i≤n related to the n nodes of a spatial mesh, and P Hermite Polynomials (jΨ((ξ1… ξM))1≤j≤P with (ξi)i=1…M where M are independent standard normal random variables (described in the section III). That leads to an approximated scalar potential (also denoted φ(x,θ) ) that can be written: n P

ϕ ( x,θ ) = ∑ ∑ ijϕ λi ( x) j Ψ (ξ1 (θ ),..., ξ M (θ )) , i =1 j =1

(8)

The scalars ( ijϕ )11≤≤ij≤≤nP will be calculated in order to have the approximation ϕ so close as possible of the exact solution of (7) (see section IV). In the following, the dependance to θ of the random variables (ξi)i=1…M will be omitted for clarity. III. POLYNOMIAL CHAOS OF HERMITE This section will be dedicated to present the Hermite Polynomial Chaos in one dimension first and then in dimension M. A. One-dimensional case: The distribution Φ ( s ) of the Gaussian measure is defined as: s2 ). (9) 2 2π The Hermite polynomials (denoted hn(s)) can be expressed by the recurrence relation as follow [2]-[3]:  s2  d n  exp(− )  2  s2   n (10) exp( )  . h n( s ) = (−1)  2  d sn  All Hermite polynomials hn(ξ) are orthogonal to each other for the Gaussian measure (E(hn(ξ) hm(ξ)) = 0 if n is different of m) and the polynomial hn(ξ) is of order n. Φ( s) =

1

exp(−

α

M

Ψ (ξ1...ξ M ) = Π hα i (ξ i ) i =1

(11)

For M=2, the polynomial of degree 0 is 1Ψ(ξ1, ξ2)=h0(ξ1)h0(ξ2) (α=(0,0)) , the polynomials of order 1 are 2 Ψ(ξ1,ξ2)=h1(ξ1)h0(ξ2) (α=(1,0)) and 3Ψ(ξ1,ξ2)=h0(ξ1)h1(ξ2) (α=(0,1))… Those polynomials are a set of dense functions in the space of the random variable that have a finite variance (the expectation of the square of the variable is finite). In order to have a finite sequence of interpolation function, equation (8) has been truncated with P multi dimensional polynomials. For M normal standard variables and mono dimensional Hermite polynomials of order less than p, the number P of polynomials j Ψ is equal to [7]: ( M + p)! P= . (12) M ! p! It is worth noticing that the integer P increases dramatically with M and p. To simplify the notation hereafter, the polynomial chaos jΨ(ξ1,ξ2,…,ξM) will be denoted jΨ(ξ). IV. SSFEM To be able to deal with any random permittivity of finite variance with SSFEM, it is more convenient to expand the permittivity given by (1) onto polynomial chaos: M P

ε (x,θ ) = ∑ ∑ ijε j Ψ (ξ )1Di ( x) . i =1 j =1

(13)

Where ijε are scalars depending on the input laws. The approximated scalar potential is a linear combination of the functions [λi(x)jΨ(ξ)]1≤i≤n, 1≤j≤P. Then, as usual in FEM, these functions are used as test functions (except for the functions corresponding to nodes located on the boundary where homogeneous Dirichlet conditions are set on the scalar potential). By using (8) and (13) in (7), with the functions [λi(x)jΨ(ξ)]1≤i≤n, 1≤j≤P as test functions it comes: n P k P j j m m g ∑ ∑ ∑ ∑ ε i ϕl E ( ∫ 1Di Ψgrad x (λl Ψ )grad x (λ f Ψ ))

l =1 m =0 i =1 j =1

D

(14)

= − E ( ∫ ε ( x,θ ) grad x (ϕC )grad x ( λ f g Ψ )  )   D

Where lmφ are the n×P unknowns. That is equivalent to: n

P

∑ ∑

l =1 m =1

m l ϕ

M P

M P

j jmg ∑ ∑ {i ε D ∫ ( grad x (λl ) grad x (λ f )}

i =1 j =1

Di

= - ∑ ∑ {ij ε E ( g Ψ j Ψ ) ∫ ( grad x (λl ) grad x (ϕC ))} i =1 j =1 Di

(15)

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Where Djmg is the mathematical expectation of the product of the multi dimensional Hermite polynomials jΨ, mΨ and gΨ (see Appendix). Applying (17) for each interpolating functions [λi(x)jΨ(ξ(θ))]1≤i≤n, 1≤j≤P leads to n×P linear equations with n×P unknowns which are the ( ijϕ )11≤≤ij≤≤nP and the right hand side of those equations forms a n×P vector B. A matrix system is then obtained: Aϕ = B (16)

example 1, the values of each relative permittivity are quite close to realistic values. For example 2, those values have TABLE I MEAN AND VARIANCE OF RELATIVE PERMITTIVITY ON EACH SUBDOMAIN Example Relative permittivity on each sub Mean Variance Domain iε 1 5.1 1.04 1ε~LN(1.6,0.04) 1

2

ε~U (]5,7[)

Where A is a n×P symmetric matrix, B is a n×P vector including the boundary conditions.

1

3

ε~LN(2.3,0.0035)

2

1

ε~LN(1.5,0.5)

5.75

21.5

2

2ε~LN(2,0.2)

8.17

14.8

V. MATRIX SYSTEM

2

3

ε~LN(2,0.5)

9.48

58

The matrix A is assembled from local matrices, so called elementary matrices, calculated on each element using the same procedure as for deterministic FEM. If we consider an element e with ne nodes located in the sub domain Di, all of them are assumed to be associated with an unknown scalar potential leading to en nexP unknowns on that element e. Denoting Aelement the matrix obtained in the deterministic case on the element e Aelement is a nexne matrix which terms (aij) (1≤i≤ne and 1≤j≤ne, i and j are local index on the element e) are given by: aij = ∫ grad x λi ( x ) grad x λ j ( x ) de (17) e

Denoting Aelemntsto the nexP elementary matrix defined on the element e located in sub domain Di (1≤i≤M), and according to (15), Aelementsto can be written as: P j  P j  j11 jP1 ∑ i ε D Aelement   ∑ i ε D Aelement j =0  j =0   (18) Aelementsto =    P j  P j ε D j1P A  jPP ∑ i ε D Aelement  element ∑ i j =0  j =0  jgm The term D can be calculated analytically (A.2), and then the calculation Aelemensto of the elementary matrix is done easily from the calculation of the matrix Aelement already calculated in the deterministic case. In the same way, the procedure of assembling is very close to the one used in the deterministic case. VI. NUMERICAL RESULTS In this section, we will test the SSFEM by comparing results obtained on a 3D example with results obtained with Monte-Carlo Simulation Method (MCSM). A. Description of the test-case The geometry and the mesh of the studied problem are given in Fig.1. Each straight part is assumed to be homogeneous with a constant but random permittivity. Consequently, for that problem, M is equal to 3. Table I indicates the law, the mean and the variance of each three relative permittivity which are assumed to be independent in two different examples (Example 1 and Example 2). In Table I, LN(m,v) represents a lognormal of parameter m and v, U([a,b]) represents an uniform law on the segment [a,b]. For

6

0.33

10.1

3.6

been chosen with large variance in order to test the robustness of the method. The potential is fixed on two parts of boundary (see fig 1). The mesh of the domain D holds 290 tetrahedra with 131 nodes. There are only 114 unknowns associated to nodes, prescribed boundary conditions on the scalar potential have been imposed on the remaining nodes. B. SSFEM The maximum value of p (order of hn see (10)) is equal to 6, as M is equal to 3, therefore, using (12) that leads to a maximum of P=84 degree of freedom onto the random aspect at each nodes. Globally, with the SSFEM the problem to be solved has at much 9576 unknowns, the matrix A has at much 3751524 non zero terms.

Fig. 1 Mesh of the domain D and surfaces where the scalar potential is fixed.

C. MCSM In the following, we will describe in the example under consideration how the MCSM enables to propagate the randomness of the permittivity ε(x,θ) onto the output fields φ(x,θ). For more details, see [5]-[7]. At first, a sample of size R of the three relative permittivities (1iε,2iε, 3iε)1≤i≤R have been generated following the law given in Table.1. To generate those values, we use the RANDOM’s FORTRAN module (given in [9]) which makes it possible to generate some samples of different laws. Then, we solve R finite element deterministic problems using the mesh given in Fig.1 with the R realizations (1iε,2iε, 3iε)1≤i≤R as input data leading to R distributions of the scalar potential (iφMCSM) 1≤i≤R. Finally, statistical treatments can be undertaken using this set of samples to calculate for example the moments at a selected order. The MCSM is very robust and accurate for a sample of sufficient size R.

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D. Numerical results The validation has been performed by comparing the moments Mk (Mk=E(Xk) with k=1 to 5) predicted both by MCSM and SSFEM. The moments are significant values of a random variable. To apply MCSM, R has been chosen equal to 700000. Thus, a sample of (1iε,2iε, 3iε) of size 700 000 is generated leading to 700000 realizations of the scalar potential at each nodes that leads to the solution of 700000 deterministic systems of 114 unknowns. Then, for each node i, the sample moments of order k, denoted iRMkMCSM, have been calculated using the following formula: 1 R R MCSM = ∑ ilϕ k MCSM . (19) i Mk R l =1 With ijϕ the value of the scalar potential at the node i of the jth sample. SSFEM , defined as the moment of order k of the scalar iMk potential φ obtained by the SSFEM for the node i can be calculated analytically. So, to compare MCSM and SSFEM, we denote the error Err(i,k) between both moments of order k related to the node i: Err (i, k ) =

| Ri M kMCSM − i M kSSFEM | R MCSM i Mk

(20)

*100

Where |.| is the absolute value. Note that this error is in percent. Let us define a maximum error Emk onto the moment of order k related to the whole mesh (ie onto the entire spatial mesh) following: Emk = max ( Err (i, k )) (21) i∈node

Table II give the maximum error (given by (21)) for both examples for two values of P. The results obtained by SSFEM are very close to the ones given by MCSM. We can notice that the error increases with the order of the moment considered. TABLE II ERROR MAXIMUM ON THE MESH FOR DIFFERENT P IN %

TABLE III NUMERICAL CONSIDERATION AROUND SSFEM

P

Em1

Em2

Em3

Em4

Em5

1 1 2 2

84 20 84 20

0.0035 0.036 0.0231 0.55

0.006 0.007 0.043 0.7

0.01 0.011 0.062 0.65

0.01 0.015 0.08 0.7

0.015 0.019 0.095 1.05

For both examples, the convergence is very good even for P = 20 (p=3). We can see more clearly on the second example the convergence with the increasing of P. Table III gives the number of non zero terms in A (nz), the CPU time (in s) and the number of unknowns. The number of unknowns is a linear function of P, whereas the number of non zero terms increases with a superior order of P (for a given mesh). The CPU time of SSFEM increases highly with P. It has been observed that the time to solve the linear system (16) is negligible versus the time to construct the matrix (the ratio is greater than 100 for the considered examples). In our case, the CPU time for MCSM is much greater with about 90,480 s.

P

nz

Unknown

Time (s.)

1 1 2 2

84 20 84 20

3,751,524 213,540 3,751,524 213,540

9,576 2,280 9,576 2,280

10,538 163 10,480 172

VII. CONCLUSION SSFEM has been tested successfully in electrostatics with random permittivity. This method can be also used to take into account random boundary conditions on the scalar potential, or random electric charge distribution. Nevertheless, taking into account random geometric parameters is not so straightforward with this kind of method and requires further investigations. APPENDIX Let us consider the scalar dijk as the expectation of the product of three polynomials hi(ξ), hj(ξ) and hk(ξ) where ξ is a standard normal variable. Then dijk is equal to zero except if i+j+k is even and |i-j|