Uncertainty quantification in linear magnetostatic problems with correlated random reluctivities

Abstract

An approach to quantify uncertainty in linear three-dimensional magnetostatic problems with correlated random reluctivities is proposed. Such strategy is based on a reduced-order model and a spectral approximation of the deterministic parametric magnetostatic problem for accurately and efficiently estimating the statistics of the quantities of interest.

Keywords

Uncertainty quantification magnetostatics finite integration technique

1. Introduction

Various techniques can be used to overcome the limitations of Monte Carlo methods for uncertainty quantification, in particular those based on Polynomial Chaos Expansion [1]. Unfortunately, for correlated random variables, these techniques become cumbersome [2] and are generally not applied in practice. On the other hand, in many practical electromagnetic applications the input random variables cannot be assumed to be statistically independent.

In this paper the following approach to uncertainty quantification is proposed for dealing with correlated random variables: a parametric reduced-order model is extracted by a Model Order Reduction (MOR) technique from the deterministic magnetostatic problem; the deterministic relationship between the material parameters and the solution of the magnetostatic problem is represented by a spectral approximation; this approximation is used for estimating any statistics of the solution, assuming the actual multivariate probability density function of the magnetic reluctivities.

2. The test case

As a test case, a typical geometry of a magnetic circuit (Fig. 1) is considered [3]. A total DC current is imposed on surface $\sigma$ of a rectangular coil $\Omega_{C}$ and the aim is to statistically characterize the magnetic flux through section $\Sigma$ .

Regions $\Omega_{k}$ ( $k=1,\ldots,4$ ) are assumed to have reluctivities that are uncertain [4], modelled as correlated random variables, with uniform marginal probability density functions $\nu_{1}=$ 0.1 $\pm$ 0.01 mm/H, $\nu_{2}=$ 0.05 $\pm$ 0.005 mm/H, $\nu_{3}=$ 0.033 $\pm$ 0.0033 mm/H, and $\nu_{4}=$ 0.025 $\pm$ 0.0025 mm/H, and with a Gaussian copula of a 4 $\times$ 4 correlation matrix of entries $\rho_{ii}=$ 1, $\rho_{12}=\rho_{21}=$ $-$ 0.4; $\rho_{13}=\rho_{31}=\rho_{24}=\rho_{42}=$ $-$ 0.2; $\rho_{14}=\rho_{41}=$ $-$ 0.1; $\rho_{23}=\rho_{32}=$ $-$ 0.5.

Figure 1.

Geometry of the problem: a $=$ 50 mm, b $=$ c $=$ 200 mm, d $=$ 10 mm, e $=$ 30 mm, f $=$ 70 mm, g $=$ 10 mm.

3. Deterministic FIT formulation and model order reduction approach

The stochastic Finite Integration Technique (FIT) formulation here proposed starts from a deterministic FIT formulation, reformulated by the augmented dual grids introduced in [5, 6, 7] and constituted by an arbitrary polynomial primal grid ${\cal G}$ , a barycentric dual grid $\tilde{\cal G}$ and a barycentric dual grid of the boundary $\tilde{\cal G}_{b}$ .

Deterministic magnetostatic problems can be discretized by FIT in various ways. In a vector potential formulation, hereinafter considered, Ampère’s law is discretized, in exact form, as follows

${\rm{\bf\tilde{C}\tilde{h}}}+{\rm{\bf\tilde{C}}}_{b}{\rm{\bf\tilde{h}}}_{b}={% \rm{\bf\tilde{j}}},$ (1)

in which ${\rm{\bf\tilde{h}}}$ and ${\rm{\bf\tilde{h}}}_{b}$ are the arrays of the circulations of the magnetic field along the edges of the dual grids $\tilde{\cal G}$ and $\tilde{\cal G}_{b}$ [7], ${\rm{\bf\tilde{j}}}$ is the array of the fluxes of current density through the faces of $\tilde{\cal G}$ , ${\rm{\bf\tilde{C}}}$ is the face-edge incidence matrix of $\tilde{\cal G}$ , ${\rm{\bf\tilde{C}}}_{b}$ is the incidence matrix between the faces of $\tilde{\cal G}$ and the edges of $\tilde{\cal G}_{b}$ . The solenoidality law of magnetic induction is written in terms of the vector potential in the form

${\rm{\bf b}}={\rm{\bf Ca}},$ (2)

in which ${\rm{\bf b}}$ is the array of the fluxes of magnetic induction through the faces of ${\cal G}$ , ${\rm{\bf a}}$ is the array of circulations of the vector potential along the edges of ${\cal G}$ , and ${\rm{\bf C}}={\rm{\bf\tilde{C}}}^{T}$ is the face-edge incidence matrix of ${\cal G}$ .

The magnetic constitutive law is written as follows

${\rm{\bf\tilde{h}}}={\rm{\bf M}}_{\nu}{\rm{\bf b}},$ (3)

where ${\rm{\bf M}}_{\nu}$ is a symmetric positive definite matrix derived using the energetic approach introduced in [7]. The generic element at the $i$ -th row and $j$ -th column of ${\rm{\bf M}}_{\nu}$ is given by

$m_{\nu}^{ij}=\int_{\Omega}\nu({\bf r}){\bf w}^{f}_{i}({\bf r})\cdot{\bf w}^{f}% _{j}({\bf r})d\Omega$ (4)

in which $\nu({\rm{\bf r}})$ is the magnetic reluctivity function and ${\rm{\bf w}}_{i}^{f}({\rm{\bf r}})$ are the face functions introduced in [8]. In usual magnetostatic problems it can be assumed that the magnetic reluctivity is uniform and equal to $\nu_{k}$ in each subregion $\Omega_{k}$ composed of a distinct material. As a result, the discrete constitutive matrix takes the form

${\rm{\bf M}}_{\nu}=\sum_{k=1}^{K}\nu_{k}{\bf M}_{k}.$ (5)

As far as boundary conditions are concerned, by assuming as usual that the tangential component of the vector potential is zero on the boundaries, such conditions can be written in terms of the face-edge incidence matrix ${\rm{\bf\tilde{C}}}_{b}$ of $\tilde{\cal G}_{b}$ as follows

${\rm{\bf\tilde{C}}}_{b}^{T}{\rm{\bf a}}={\rm{\bf 0}}.$ (6)

Grouping the unknown circulations of the vector potential along the edges of ${\cal G}$ not belonging to the boundary in the column array ${\rm{\bf a}}_{\overline{b}}$ , it can be written

${\rm{\bf a}}={\rm{\bf\tilde{C}}}_{\overline{b}}{\rm{\bf a}}_{\overline{b}},$ (7)

in which ${\rm{\bf\tilde{C}}}_{\overline{b}}$ is the matrix mapping ${\rm{\bf a}}_{\overline{b}}$ onto ${\rm{\bf a}}$ . Substituting Eq. (2) into Eq. (3) and Eq. (3) into Eq. (1), multiplying on the left by ${\rm{\bf\tilde{C}}}_{\overline{b}}$ , using Eq. (7), and eliminating all variables different from the vector potential ${\rm{\bf a}}_{\overline{b}}$ , previous equations can be reduced to the linear system of $N$ equations in $N$ unknowns

${\rm{\bf Na}}_{\overline{b}}={\rm{\bf y}}$ (8)

in which

$\displaystyle{\rm{\bf N}}={\rm{\bf\tilde{C}}}_{\overline{b}}^{T}{\rm{\bf C}}^{% T}{\rm{\bf M}}_{\nu}{\rm{\bf C\tilde{C}}}_{\overline{b}},$ (9) $\displaystyle{\rm{\bf y}}={\rm{\bf\tilde{C}}}_{\overline{b}}^{T}{\rm{\bf\tilde% {j}}}.$ (10)

System of Eq. (8) is underdetermined since no gauging is introduced. It is the discretization of curl-curl equation for the magnetic vector potential. Even if undetermined, the system can be efficiently solved by Krylov iterative methods, in particular by the conjugate gradient method [9]. In case Eq. (5) holds, Eq. (8) takes the form

$\sum^{K}_{k=1}\nu_{k}{\bf N}_{k}{\bf a}_{\bar{b}}={\bf y},$ (11)

in which

${\rm{\bf N}}_{k}={{\tilde{\bf C}}}_{\bar{b}}^{T}{\bf C}^{T}{\bf M}_{k}\tilde{% \bf C}_{\bar{b}}.$ (12)

A reduced model can be obtained assuming that [10]

${\rm{\bf a}}_{\overline{b}}={\rm{\bf V\hat{a}}}_{\overline{b}}$ (13)

and projecting Eq. (11) onto the reduced space, so that

$\sum^{K}_{k=1}\nu_{k}\hat{\bf{N}}_{k}\hat{\bf{a}}_{\bar{b}}=\hat{\bf{y}},$ (14)

with ${\rm{\bf\hat{y}}}={\rm{\bf V}}^{T}{\rm{\bf y}}$ and ${\rm{\bf\hat{N}}}_{k}={\rm{\bf V}}^{T}{\rm{\bf N}}_{k}{\rm{\bf V}}$ .

4. Deterministic FIT formulation and model order reduction approach

The magnetic reluctivity is now assumed to depend on a small number $Q$ of random variables $\xi_{1},\ldots,\xi_{Q}$ , which can always be assumed to be statistically independent. Such variables form a vector ${\rm{\bf\xi}}=[\xi_{q}]$ and we can write $\nu=\nu({\rm{\bf r}},{\rm{\bf\xi}})$ .

A Polynomial Chaos Expansion (PCE) can be introduced for all fields and from these expansions analogous expansions follow for the discrete variables of FIT. In this way, for instance, the array $\hat{\bf a}_{\bar{b}}({\bf\xi})$ of the circulations of the vector potential in the reduced order model is approximated in the form

$\hat{\bf a}_{\bar{b}}({\bm{\xi}})=\sum_{|{\bm{\alpha}}|\leqslant P}\hat{\bf a}% _{\bar{b},{\bm{\alpha}}}\psi_{\alpha}(\xi),$ (15)

in which ${\bm{\alpha}}=(\alpha_{1},\cdots,\alpha_{Q})$ is a multi-index of $Q$ elements, $|{{\bm{\alpha}}}|=\alpha_{1}+\alpha_{2}+\cdots\alpha_{Q}$ and

$\psi_{\rm{\bf\alpha}}({\rm{\bf\xi}})=\psi_{\alpha_{1}}^{1}(\xi_{1})\psi_{% \alpha_{1}}^{2}(\xi_{2})\cdots\psi_{\alpha_{Q}}^{Q}(\xi_{Q}),$

in which functions $\psi_{\rm{\bf\alpha}}({\rm{\bf\xi}})$ are polynomials of degrees not greater than $P$ , forming a basis of dimension

$M=\left({{\begin{array}[]{*{20}c}{P+Q}\hfill\\ Q\hfill\\ \end{array}}}\right)$ (16)

satisfying the orthonormality property ${\rm{\bf E}}[\psi_{\rm{\bf\alpha}}({\rm{\bf\xi}})\psi_{\rm{\bf\beta}}({\rm{\bf% \xi}})]=\delta_{{\rm{\bf\alpha\beta}}}$ , in which ${\rm{\bf E}}[\cdot]$ indicates the statistical mean value. As a result ${\rm{\bf\hat{a}}}_{\bar{b},{\rm{\bf\alpha}}}$ denotes the projection of the stochastic variable $\hat{\bf a}_{\bar{b}}({\bf\xi})$ onto $\psi_{\bf\alpha}({\bf\xi})$ , or equivalently

$\hat{\bf a}_{\bar{b},{\bf\alpha}}={\bf E}[{\hat{\bf a}}_{\bar{b}}({{\bf\xi}})% \psi_{\bf\alpha}({\bf\xi})].$

All these projections, with $|{{\bm{\alpha}}}|\leqslant P$ , can be grouped in the vector

$\hat{\bf A}_{\bar{b}}=[\hat{\bf a}_{\bar{b}},{\bf\alpha}],$

in which sub-vectors of multi-indices ${\rm{\bm{\alpha}}}$ , with $|{\rm{\bm{\alpha}}}|\leqslant P$ , are sorted in lexicographical order [11].

An intrusive stochastic approach is obtained by substituting the PCEs of discrete variables into discrete equations, multiplying all members of such equations by polynomials $\psi_{\rm{\bf\alpha}}({\rm{\bf\xi}})$ and taking the expected values, so that

$\sum_{k=1}^{K}\left({\bf R}_{k}\otimes\hat{\bf{N}}_{k}\right)\hat{\bf A}_{\bar% {b}}=\hat{\bf Y},$ (17)

with ${\rm{\bf\hat{Y}}}={\rm{\bf V}}^{T}{\rm{\bf Y}}$ , ${\rm{\bf Y}}=({\rm{\bf 1}}_{M}\otimes{\rm{\bf\tilde{C}}}_{\overline{b}}^{T}){% \rm{\bf\tilde{J}}}$ , and in which $\otimes$ indicates the tensor product.

5. Numerical results

A spectral approximation of degree 6 for the magnetic flux through section and its probability density function (PDF) is computed in 38 seconds on a 2.3 Ghz Intel Core i7 (Fig. 2). A very accurate match is obtained with the results of Monte Carlo analysis (106 tries, 2074 s) using a reduced-order model of dimension 15. An analogous Monte Carlo analysis performed directly on the discrete FIT problem would require about 2220 hours. As can be noted, the correlation between magnetic reluctivities have a relevant impact on the PDF under analysis. As a result, the proposed approach exhibits a speed up of about 3.85 $\cdot$ 105 with respect to brute force Monte Carlo analysis of the discrete FIT problem.

Figure 2.

Probability density functions of the magnetic flux.

Figure 3 shows that, despite the random variables are correlated, the rate of convergence is still exponential (and so optimal) as in the uncorrelated case.

Figure 3.

Error in the mean and in the standard deviation vs. the PCE order.

6. Conclusion

For the chosen magnetostatic problem the proposed model order reduction approach is proved to be effective not only when random variables are uncorrelated, as in the case of [10] in an electrokinetic problem, but also when the random material parameters are correlated. The same convergence rate is observed in both cases. The case of correlated random variables is very common in many applications and the benefits of the proposed approach are the reduced computational burden together with the use of a polynomial spectral expansion as for the uncorrelated case.

References

Beddek

Clénet

Moreau

and Le Menach

, Spectral stochastic finite element method for solving 3D stochastic eddy current problems, International Journal of Applied Electromagnetics and Mechanics 39 (2012), 753–760.

Feinberga

and Langtangen

H.P.

, Chaospy: An open source tool for designing methods of uncertainty quantification, Journal of Computational Science 11 (2015), 46–57.

Beddek

Le Menach

Clenet

and Moreau

, 3-D stochastic finite-element method in static electromagnetism using vector potential formulation, IEEE Trans. Magn. 47 (2011), 1250–1253.

Ramarotafika

Benabou

and Clenet

, Stochastic modeling of anhysteretic magnetic curve using random inter-dependant coefficients, International Journal of Applied Electromagnetics and Mechanics 43 (2013), 151–159.

Weiland

, Discretization method for the solution of Maxwell’s equations for six-component fields, AEU-Archiv fur Elektronik und Ubertragungstechnik 31 (1977), 116–120.

Schreiber

Clemens

and Van Rienen

, Conformal FIT formulation for simulations of electro-quasistatic fields, International Journal of Applied Electromagnetics and Mechanics 19 (2004), 193–197.

Codecasa

, Refoundation of the cell method by means of augmented dual grids, IEEE Transactions on Magnetics 50 (2014).

Codecasa

Specogna

and Trevisan

, A new set of basis functions for the discrete geometric approach, Journal of Computational Physics 229 (2010), 7401–7410.

Ren

, Influence of the r.h.s. on the convergence behaviour of the curl-curl equation, IEEE Trans. Magn. 32 (1996), 655–658.

10.

Codecasa

and Di Rienzo

, Fast model order reduction approach to uncertainty quantification in electrokinetics, Proc. of Progress in Electromagnetics Research Symposium (PIERS) 2015, Prague, Czech Republic, 6–9 July, 2015, pp. 369–373.

11.

Xiu

, Numerical methods for stochastic computations: a spectral method approach, Princeton University Press, 2010.