material and shape derivative method for quasi

c 2012 Society for Industrial and Applied Mathematics 

SIAM J. NUMER. ANAL. Vol. 50, No. 3, pp. 1086–1110

MATERIAL AND SHAPE DERIVATIVE METHOD FOR QUASI-LINEAR ELLIPTIC SYSTEMS WITH APPLICATIONS IN INVERSE ELECTROMAGNETIC INTERFACE PROBLEMS∗ ´ † IVAN CIMRAK Abstract. We study a shape optimization problem for quasi-linear elliptic systems. The state equations describe an interface problem and the ultimate goal of our research is to determine the interface between two materials with different physical properties. The interface is identified by the minimization of the shape (or the cost) functional representing the misfit between the data and the simulations. For shape sensitivity of the shape functional we elaborate the material and the shape derivative method. In this concept a vector field is introduced that deforms the unknown shape toward the optimum. We characterize the elliptic interface problems whose solutions give the material and the shape derivatives. In particular, we show the existence of weak as well as strong material derivatives. Further, we employ the adjoint variable method to obtain an explicit expression for the gradient of the shape functional. This gradient is then used for the actual implementation of the minimization algorithm. In simulations we use the level set method for the representation of the interface. We present the simulation results showing the reconstructed voids in the nonlinear ferromagnetic material from the near-boundary measurements of magnetic induction. Key words. sensitivity analysis, shape derivative, material derivative, adjoint problem, shape optimization, level set method AMS subject classifications. 65M32, 49Q12 DOI. 10.1137/100810800

1. Introduction. Second order elliptic partial differential equations with discontinuous coefficients are often encountered in material sciences and fluid dynamics. This is the case when two distinct materials or fluids with different conductivities, permeabilities, densities, or diffusion coefficients are involved. Such problems are known as interface, transmission, or diffraction problems. The two materials or fluids are modeled by a discontinuous piecewise smooth coefficient function in a linear or quasi-linear equation. We aim at reconstruction of the unknown interface between the two materials. Let D be a bounded domain in R2 with C 2 boundary and Ω ⊂ D its proper subdomain with C 2 boundary. Assume that the function β : D × R → R is defined piecewise by  β1 (s) for x ∈ Ω, (1.1) β(x, s) = β2 (s) for x ∈ D \ Ω, where β1 , β2 are smooth nonlinear functions. We study inverse interface problems for the quasi-linear elliptic equation   u = 0 on ∂D, (1.2) −∇ · β(x, |∇u|2 )∇u = f (x) in D, with the interface conditions   (1.3) u  = 0, ∂Ω

  β(x, |∇u|2 )∇u · n 

= 0, ∂Ω

∗ Received by the editors October 5, 2010; accepted for publication (in revised form) January 5, 2012; published electronically May 10, 2012. http://www.siam.org/journals/sinum/50-3/81080.html † Department of Software Technologies, Faculty of Management Science and Informatics, University of Zilina, Univerzitna 8215/1, 01026 Zilina, Slovakia ([email protected]).

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  where v is the jump of a quantity v across the interface ∂Ω and n the unit outward normal to the boundary ∂Ω. The ultimate goal of this work is the reconstruction of Ω if we possess the values of ∇u on a specific part ω ⊂ D. We point out that such an inverse problem for quasilinear state equations has not yet been studied, to the author’s knowledge. Its linear version has been studied in [14]. In section 2 we motivate this theoretical setting by a real-world application in the detection of voids in nonlinear electromagnetic materials. Direct problem. The direct problem can be formulated in a weak sense. For a given Ω find u ∈ W01,2 (D) such that     (1.4) β(|∇u|2 )∇u, ∇ϕ = f, ϕ is satisfied for all ϕ ∈ W01,2 (D). In section 4 we study this direct problem. We review known existence and regularity results. We list the monotonicity, boundedness, and differentiability properties that must be possessed by function β. Further, we discuss the existence result for (1.4) from [25] and we nontrivially verify its assumptions. Subsequently we positively answer the question of the uniqueness for the solution of (1.4). We are interested in the reconstruction of Ω for given β1 , β2 and given data on ω ⊂ D. The data corresponds to ∇u. The inverse problem is to find Ω such that the gradient of the state variable ∇u fits the data represented by a vector field M. We construct a cost functional measuring the fidelity of u to the measurements  1 |∇u(Ω) − M|2 , (1.5) J(Ω) = 2 ω where u(Ω) is the solution of (1.4) for given Ω. Inverse problem. The inverse problem is defined as follows. Find the shape Ω for which the cost function J(Ω) is minimal, i.e., find Ωopt : (1.6)

Ωopt = arg min J(Ω). Ω

To this end we use a gradient-type minimization method to minimize the cost function. For this we need to compute the gradient DJ of J. We employ the shape sensitivity analysis using the material and shape derivative as tools for computing DJ. The shape and material derivative has been widely used in shape optimization; see [5, 13, 29, 32] and the references therein. This concept has been applied in shape sensitivity for unilateral problems describing such physical phenomena as contact problems in elasticity, elastoplastic torsion problems, and obstacle problems. We derive basic identities and developments, including the form of the equation that is satisfied by the material derivative. We also show the existence of the weak as well as the strong material derivative. We point out that the transition from the linear elliptic state equation treated in [15] to the quasi-linear case is not trivial. In section 6 we determine the form of the equation that must be satisfied by the shape derivative. In section 7 we use the adjoint method to explicitly express the derivative of the cost functional DJ. In the following section we elaborate the level set method to represent the interface ∂Ω and we show how DJ is evaluated in practice. We describe the optimization algorithm that eventually finds Ω by minimization of the cost function J(Ω). Finally, in section 9 we show the implementation of the minimization algorithm and we present the numerical results.

´ IVAN CIMRAK

1088 D

ω material

d th

Ω void

measurements

Fig. 2.1. Model setup with piecewise smooth reluctivity. The ferromagnetic material occupies a square domain D\Ω without a hole Ω inside it. Measured data is available on a strip, denoted by ω, located around the boundary ∂D. The boundary of the strip is indicated by a dashed line. The reluctivity has a discontinuity across ∂Ω.

2. Motivation. As a concrete application we present the defect characterization in magnetic materials using local magnetic measurements. Consider a hard magnetic workpiece made of a nonlinear magnetic material; see Figure 2.1. Such a material is characterized by a strongly nonlinear magnetic reluctivity β(x, |B|2 ) (often denoted by ν) that depends not only on the position vector x but also on the strength of the magnetic induction. During the production process small air gaps or cracks can appear inside such a workpiece. The magnetic reluctivity of the air is significantly different from that of the nonlinear magnetic material. Therefore the magnetic reluctivity β(x, |B|2 ) defined over the whole D has discontinuities over the borders of the air gaps or cracks. Theory implies that D has a C 2 boundary, which is not satisfied by our choice of a rectangle. However, this condition probably could be relaxed to the piecewise C 2 boundary, although we do not prove it here rigorously. We model the case where D contains one or multiple air gaps. The union of these air gaps is represented by an open set Ω, on which the reluctivity has significantly different values than on the rest of the domain. The static distribution of the magnetic field under the induced current with the current density f is governed by the magnetic vector potential formulation of the static Maxwell equations. We consider the planar symmetry with the symmetry plane xy. The only nonzero component of the vector potential A is denoted by u, so A = (0, 0, u)T . In this formulation with planar symmetry, the magnetic induction has two nonzero components and can be expressed as

T ∂u ∂u ,− ,0 (2.1) B= ∇×A = . ∂y ∂x This reduces the three-dimensional case to the two-dimensional case. Suppose that the magnetic potential vanishes on the boundary. Then u satisfies the quasi-linear problem (1.2), (1.3), where f (x) represents the current density. For linear materials, the reluctivity is a scalar function leading to a simpler linear elliptic PDE. In such a case, the Dirichlet problem (1.4) has a unique weak solution for any strictly positive β(x) ∈ L∞ (Ω). Such linear system describes the problem of electrical impedence tomography, which has been thoroughly studied by many authors; see [4] and the references therein.

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Note, that space dependence is expressed only by the splitting of the domain. To be consistent with (1.5), we represent the three-dimensional measurements vector of magnetic induction B = (B 1 , B 2 , 0)T by rotated vector M = (−B 2 , B 1 ), so that (2.1) is fulfilled. In [10] a similar case has been studied. Simulations were performed to detect the defects of the rectangular shape from real data. The authors determined two parameters, the position and the width of the defect. 3. Notations. We use classical Sobolev spaces W 1,2 (D), W01,2 (D), the space with square integrable functions L2 (D), and the space L∞ (D) of bounded functions. The scalar product in L2 (D) is denoted by (·, ·). The norm in L2 (D), L∞ (D) is denoted by  · 2 ,  · ∞ and the norm in general space X by  · X . The vectors in Rd are denoted by bold symbols, e.g., x, or by pairs (in two dimensions) or triples (in three dimensions), e.g., x = (x, y, z)T or x = (x1 , x2 , x3 )T . The scalar product of two vectors u, v in Rd will be denoted by u · v. The partial derivative of f (x, y) with respect to x is denoted either by ∂f ∂x or by fx . We frequently use the restriction of a function. Therefore to simplify the notation we use the expression f ∈ L2 (Ω) even for f : D → R, instead of the longer notation f |Ω ∈ L2 (Ω). 4. Analysis of the direct problem. When the interface ∂Ω is smooth enough, the solution of the interface problem is also smooth in individual regions separated by the discontinuities. The global regularity, however, is very low. For regularity studies in the case of linear equations, see [19, 21]. These results have been used in finite element (FE) approximations to show the convergence and error estimates for FE methods [1, 6]. The literature concerning the case of quasi-linear equations is very rich. See [12, 17, 25], where the authors consider smooth domains, and see [3] for conical domains. The FE method approximation of nonlinear interface problems has been studied in [28, 30]. For the quasi-linear case with the special structure (1.2), the authors of [16] suggested a simple substitution fixed-point iterative method. The iterations are designed in such a way that the nonlinearity is taken from the previous iteration. The convergence analysis of this method, however, strongly relies on the fact that the nonlinearity function β(x, s) is decreasing in s. We consider the case when the nonlinearity is increasing so that the approach from [16] cannot be used. We solve this problem using the Newton–Raphson algorithm. We formulate the properties of β1 , β2 appearing in (1.1). These properties are consistent with the electromagnetic application described in section 2. For i = 1, 2, the following assumptions hold: A1. The function s → βi (s) is nondecreasing. A2. There exists positive βmin such that βi (s) ≥ βmin . A3. There exists positive βmax such that βi (s) ≤ βmax .  A4. βi is differentiable with well-defined derivative βi satisfying βi ≤ βmax . The first assumption creates the monotonicity structure of the differential operator. The second assumption guarantees that (1.4) does not degenerate. The third and fourth assumptions are needed for further analysis. Further, we assume the following: (4.1)

f, ∇f ∈ L∞ (D).

We make use of theoretical results from [25] concerning the strong solution of quasi-linear diffraction problems. The theorem was originally formulated for general

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quasi-linear operators in divergence form; however, for the sake of simplicity we state the following theorem, adapted for the operator of the form (1.2). Theorem 4.1 (Theorem 2 in [25]). A classical solution of (1.2), (1.3) satisfying ∂u ∂u ∈ C 0,α (Ω), ∈ C 0,α (D \ Ω), ∂xi ∂xi ∂2u ∂2u ∈ C 0,α (Ω), ∈ C 0,α (D \ Ω), i, j = 1, 2, ∂xi ∂xj ∂xi ∂xj

u ∈ C 0,α (D),

for 1 > α > 0 exists if the following conditions are satisfied: 1. Any strong solution—if one exists—to the auxiliary problem

  |∇u| 2 −∇ · β(x, |∇u| )∇u = f ξ (4.2) in D, u = 0 on ∂D, R1      (4.3) u  = 0, β(x, |∇u(x)|2 )∇u · n  = g(u), ∂Ω

∂Ω

ξ( |u| R )),

must be bounded independently of R1 and R, i.e., where g(u) := −u(1 − maxx∈D |u| < M , and M does not depend on R1 and R. The function ξ(t) is a smooth nonincreasing cut-off function equal to 1 for t < 1 and equal to 0 for t > 2. 2. There exist positive k, M such that for all η, p ∈ Rd and for x ∈ Ω or x ∈ D\Ω the following inequalities hold: ⎫   ∂(β(|p|2 )pi ) ⎪ ⎬ k|η|2 ≤ ηi ηj ≤ M |η|2 ∂p . (4.4) j i j ⎪ ⎭ |β(x, |p|2 )|pi (1 + |p|) + |f (x)| + |∇f (x)| ≤ M (1 + |p|2 ) 3. For 1 > α > 0 the following smoothness conditions for i = 1, . . . , d must be valid: a1i (p) := β1 (|p|2 )pi ∈ C 1,α (Ω × Rd ),

a2i (p) := β2 (|p|2 )pi ∈ C 1,α (D \ Ω × Rd ).

To verify condition 1 we reformulate (4.2), (4.3) in the weak sense,     |∇u|  2 β(x, |∇u| )∇u, ∇ϕ − g(u)ϕ = f ξ ,ϕ in D, u = 0 on ∂D. R1 ∂Ω We have to show that the strong solution to (4.2), (4.3) is bounded. Let us denote this solution by u ˜. It is also a weak solution to the same problem. Set b(x) = β(x, |∇˜ u|2 ), u| ˜(x) = g(˜ u). Then u ˜ is also a weak solution to the following b(x) = f ξ( |∇˜ R1 ), and g linear problem:      b(x)∇u, ∇ϕ = b(x), ϕ + g˜(x)ϕ. ∂Ω

Realize that since f is bounded (from (4.1)), then also b(x), b(x) are bounded independently of R and R1 ; see assumptions A1–A4. Since ξ is bounded we have that |g(u)| ≤ |u|. The trace theorem guarantees that uL2 (∂Ω) ≤ CuW 1,2 (D) , which together with the previous estimate gives that g(u)L2 (∂Ω) ≤ CuW 1,2 (D) with C independent of R1 and R. Therefore, ∂Ω g˜(x)ϕ forms a bounded functional on W 1,2 (D). Summarizing the previous statements we conclude from [12, Theorem 8.15] that u ˜ is bounded independently of R1 and R.

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To verify condition 2 we just need to do some simple manipulation and realize that assumptions A1–A4 and (4.1) are valid, and to verify condition 3 we realize that inside Ω and inside D \ Ω functions β1 and β2 are not dependent on x. Remark 1. In the proof of the previous theorem from [25] the authors derive the following a priori bounds: max |∇u(x)| + max |∇u(x)| ≤ N, x∈Ω

∇uC 0,α (Ω) + ∇uC 0,α(D\Ω) ≤ C

x∈D\Ω

with N and C dependent on k, M from (4.4). However, k and M are not dependent on the domain Ω. This is crucial because it gives us the bounds for ∇u on Ω and on D \ Ω independently of Ω. Consequently we are able to perform the sensitivity analysis on perturbations in Ω in the following section. For general quasi-linear equations as studied in [19, 25] we cannot expect the uniqueness result, as already pointed out in [19]. Therefore, for general interface problems we are not able to prove the uniqueness. However, in our case we can use results from [31] to conclude the uniqueness of the weak solutions. 5. Material derivative. The material derivative method has been addressed by many authors; see [29] and references therein for an overview. In this section we first introduce basic concepts and notation, including the notion of the solution to the direct problem in a time instance t denoted by ut . Then we derive the equation that is satisfied by wt = (ut − u)/t. Next we show that wt weakly converges to the weak material derivative denoted by u, ˙ and we formulate the equation for u. ˙ Further we show the strong convergence. 5.1. Basic concepts. We introduce an artificial time variable denoted by t and we let the domain Ω evolve in time for t > 0. We denote by Ωt the evolved Ω in the time instance t. The direct problem in the time instance t can be written in the weak formulation (5.1)

(βt (|∇ut |2 )∇ut , ∇ϕt ) = (f, ϕt ).

The nonlinearity is defined as (5.2)

βt (x, s) =

 β1 (s) β2 (s)

The cost functional is then written as (5.3)

1 J(Ωt ) = 2



for x ∈ Ωt , for x ∈ D \ Ωt .

|∇ut − M|2 .

ω

We use the symbol h(x) for the velocity field. We look for the response of J onto a small movement of Ω in the direction of this velocity field. Performing such a sensitivity analysis will allow us to determine the correct h under which Ω must be moved in order to decrease the value of J. For nonnegative t ∈ R define the mapping Ft : R2 → R2 by (5.4)

Ft (X) = X + th(X),

where h(X) = (h1 (X), h2 (X))T ∈ (C 1,1 (R2 ))2 and h = 0 on ∂D. For t sufficiently small let Ωt = Ft (Ω) be the image of the fixed domain Ω. Since Ft |t=0 = Id we have Ω0 = Ω. We use X for the points in R2 where R2 is considered as the definition

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domain of Ft . We use x for points in R2 where R2 is considered as the range of Ft . Ft is considered as the mapping from the fixed frame to the moving frame. The moving frame moves under the velocity field h. A symbol D in front of a vector function f means the matrix ⎞ ⎛ Df = ⎝

∂f1 ∂x1 ∂f2 ∂x1

∂f1 ∂x2 ∂f2 ∂x2

⎠.

Denote Mt = (DFt )−1 , It = det(DFt ), At := Mt MtT It , and A := ∇ · hId − (DhT + Dh). In Appendix A we list several important identities. We distinguish between the functions with domain in the fixed frame from those having domain in the moving frame. The functions depending on X (i.e., those with domain in the fixed frame) are marked with a superscript t, and functions depending on x (i.e., those with domain in moving frame) are marked with a subscript t. Thus ut (X) = ut (x) = ut (Ft (X)),

or ut = ut ◦ Ft .

Since (MtT )−1 = (Mt−1 )T we have the following: (5.5)

∇ut = DF T (∇ut ) ◦ Ft ,

MtT (∇ut ) ◦ Ft−1 = ∇ut .

Remark 2. The mapping Ft defined in (5.4) depends linearly on t. It is possible, however, to extend our considerations to a slightly more general case. In particular, consider h(X, t) in (5.4) such that limt→0 h(X, t) = h(X), limt→0 Dh(X, t) = Dh(X), uniformly for each X ∈ D. Then all the relations (A.1)–(A.4), (5.5) remain valid. t ˙ We define the material derivative u˙ = limt→0 u −u t . To derive the equation for u we first find out what equation is satisfied by wt := (ut − u)/t and then we pass in the limit for t → 0. 5.2. Equation for wt = (ut − u)/t. Consider the direct problem (5.1) for the positive time instance t > 0. We perform the change of variables x = Ft (X). The dependence of βt on t is solely through the integration domain Ωt . After the change of variables, Ωt changes to Ω and thus βt does not depend on t anymore. Since Ω0 = Ω we can write βt |t=0 = β0 = β and we obtain (β(|MtT ∇ut |2 )At ∇ut , ∇ϕt ) = (It f t , ϕt ). We subtract the direct problem for the time instance t = 0 from the previous equation and we divide the resulting equation by t. The test functions will be denoted simply by ϕ. After some manipulation we arrive at



1 1 ∇ut − ∇u T t 2 T t 2 At − Id , ∇ϕ + ∇u, ∇ϕ β(|Mt ∇u | )At β(|Mt ∇u | ) 2 t 2 t

1 β(|MtT ∇ut |2 ) − β(|∇u|2 ) + ∇u, ∇ϕ 2 t



1 1 β(|MtT ∇ut |2 ) − β(|∇u|2 ) T t 2 At − Id t t + β(|Mt ∇u | ) ∇u , ∇ϕ + ∇u , ∇ϕ 2 t 2 t





1 It − 1 t ∇ut − ∇u ft − f (5.6) + , ∇ϕ − f ,ϕ − , ϕ = 0. β(|∇u|2 ) 2 t t t

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Using this notation we regroup some terms and obtain

β(|MtT ∇ut |2 )At + β(|∇u|2 ) At − Id ∇ut + ∇u ∇wt , ∇ϕ + β(|MtT ∇ut |2 ) , ∇ϕ 2 t 2       at1 (w t ,ϕ)



+ (5.7)

bt1 (ϕ)

− β(|∇u| ) ∇u + ∇u , ∇ϕ t 2

t It − 1 t f −f − f ,ϕ − , ϕ = 0. t t     

β(|MtT ∇ut |2 )

2

t

bt4 (ϕ)

bt5 (ϕ)

The first term on the left-hand side of the previous equality can be considered as a bilinear form and we denote this form by at1 (wt , ϕ). The second, fourth, and fifth terms can be considered as linear functionals and we denote them by bt1 (ϕ), bt4 (ϕ), and bt5 (ϕ), respectively. From Theorem 4.1 we know that ∇uL∞ (D) ≤ C. Since ||MtT ||L∞ (D) ≤ C, from Remark 1 we have the same estimate for MtT ∇ut . Therefore the pointwise values |MtT ∇ut | and |∇u| are bounded. From the differentiability of β we consequently conclude that for each t and X there exists η(X) satisfying     min |MtT ∇ut (X)|2 , |∇u(X)|2 ≤ η(X) ≤ max |MtT ∇ut (X)|2 , |∇u(X)|2 such that β(|MtT ∇ut (X)|2 ) − β(|∇u(X)|2 ) = β  (η(X))(|MtT ∇ut (X)|2 − |∇u(X)|2 ), where β  (η(X)) is understood as β1 (η(X)) for X ∈ Ω and as β2 (η(X)) for X ∈ D \ Ω. We plug the previous expression into the remaining integral on the left-hand side of (5.7) and we split the result into several terms: 1 2

β(|MtT ∇ut |2 ) − β(|∇u|2 ) (∇ut + ∇u), ∇ϕ t

1 M T ∇ut − ∇u · (MtT ∇ut + ∇u)(∇ut + ∇u), ∇ϕ = β  (η) t 2 t

T 1 M − Id ∇ut · (MtT − Id)∇ut (∇ut + ∇u), ∇ϕ = β  (η) t 2 t   



bt2 (ϕ)

1 MtT − Id  t t t + β (η) ∇u · (∇u + ∇u)(∇u + ∇u), ∇ϕ 2 t    bt3 (ϕ)

+

 1  β (η)∇wt · (MtT − Id)∇ut (∇ut + ∇u), ∇ϕ   2 at2 (w t ,ϕ)

 1  β (η)∇wt · (∇ut + ∇u)(∇ut + ∇u), ∇ϕ . +   2 at3 (w t ,ϕ)

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The first and second terms on the right-hand side of the previous equality can be considered as linear functionals and we denote them by bt2 (ϕ) and bt3 (ϕ), respectively. The third term and the fourth term can be considered as bilinear forms and we denote them by at2 (wt , ϕ), at3 (wt , ϕ), respectively. The sum of at1 , at2 , at3 is denoted by at and the sum of bt1 , . . . , bt5 is denoted by bt . Using the notations introduced above we have (5.8)

at (wt , ϕ) + bt (ϕ) = 0,

which is the equation for wt . In Appendix B we show the continuity and coercivity of at (wt , ϕ) and continuity of bt (ϕ). Consequently, from the Lax–Milgram theorem we conclude that for every positive t there exists a unique solution wt of (5.8). 5.3. Weak and strong material derivative and its equation. We set ϕ = (ut − u)/t, and using the coercivity and continuity of at we can show that for t sufficiently small the following inequality holds: ∇wt 2 ≤ C.

(5.9)

From this we directly have that ∇ut − ∇u ≤ Ct and therefore we obtain that ut → u

strongly in W 1,2 (D).

From (A.3) we have MtT → Id in L∞ and therefore we conclude that MtT ∇ut → ∇u strongly in L2 (D). If we now consider a sequence of functions defined as wn = wtn , where tn → 0, then we have the boundedness of this sequence and thus a weak convergence of a subsequence still denoted wn in W 1,2 (D) to some element from W 1,2 (D) that will be denoted as u. ˙ We are going to derive an equation which is satisfied by u. ˙ In Appendix C we bound the following expressions and compute their limits for n → ∞: ˙ ∇ϕ)|, A1 := |at1n (wn , ϕ) − (β(|∇u|2 )∇u, A3 := |at3n (wn , ϕ) − 2(β  (|∇u|2 )∇u˙ · ∇u∇u, ∇ϕ)|, B1 := |bt1n (ϕ) − (β(|∇u|2 )A∇u, ∇ϕ)|, B3 := |bt3n (ϕ) − 2(β  (|∇u|2 )(−DhT )∇u · ∇u∇u, ∇ϕ)|, B4 := |bt4n (ϕ) − (−f ∇ · h, ϕ)|, B5 := |bt5n (ϕ) − (−∇f · h, ϕ)|. Remark 3. The limits of A1 and A3 for n → ∞ are zero only under the assumption that ϕ ∈ C0∞ (D). Using the density argument in further developments leading to Theorem 5.1 we see that this assumption is not restrictive. On the other hand, the limits of B1 , B3 , B4 , and B5 for n → ∞ are zero for a broader class of test functions, namely, for all ϕ ∈ W 1,2 (D). That means that, e.g., taking ϕ = wtn we can obtain that lim |btn (wtn ) − b(wtn )| = 0.

n→∞

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This observation will be crucial for later considerations about the strong convergence of wt . Realize that from (B.1) and (B.2) we have that limn→∞ |at2n (wn , ϕ)| = 0 and limn→∞ |bt2n (ϕ)| = 0. We have prepared all the ingredients to derive the equation for u. ˙ We introduce the bilinear form a(v, ϕ) = (β(|∇u|2 )∇v, ∇ϕ) + 2(β  (|∇u|2 )∇v · ∇u∇u, ∇ϕ) and the functional b(ϕ) = (β(|∇u|2 )A∇u, ∇ϕ) + 2(β  (|∇u|2 )(−DhT )∇u · ∇u∇u, ∇ϕ) + (−∇ · (f h), ϕ). Similar to what been shown for at and bt , we can prove the continuity and coercivity of a and continuity of b. Using the density argument we can prove that if the identity a(w, ϕ) + b(ϕ) = 0 is satisfied for all ϕ ∈ C0∞ (D), then it is also satisfied for all ϕ ∈ W 1,2 (D). From the limits computed before we know that if wn u˙ in W 1,2 (D), then u˙ satisfies a(u, ˙ ϕ) + b(ϕ) = 0 for all ϕ ∈ W 1,2 (D). However, from the Lax– Milgram theorem we know that the solution of a(v, ϕ) + b(ϕ) = 0 is unique and therefore we obtain that not only wn u˙ but also wt u˙ in W 1,2 (D). To formalize this result we state the following theorem. Theorem 5.1. There exists a weak material derivative u˙ for which wt u˙ in 1,2 W (D). Moreover, the weak material derivative satisfies the equation (β(|∇u|2 )∇u, ˙ ∇ϕ) + 2(β  (|∇u|2 )∇u˙ · ∇u∇u, ∇ϕ) (5.10)

+ (β(|∇u|2 )A∇u, ∇ϕ) + 2(β  (|∇u|2 )(−DhT )∇u · ∇u∇u, ∇ϕ) + (−∇ · (f h), ϕ) = 0

for all ϕ ∈ W 1,2 (D). Further, the solution u˙ satisfies (5.11)

∂ u˙ ∂ u˙ ∈ C 0,α (Ω), ∈ C 0,α (D \ Ω). ∂xi ∂xi

Proof. The first part of the theorem has been proved before. The second part is a direct consequence of [19, Theorem 16.2]. To fulfill the assumptions of the theorem one needs to guarantee that the coefficients of the linear problem (5.10) belong to C 0,α (Ω) and to C 0,α (D \ Ω). Those coefficients, however, are the solutions of (1.2), (1.3) and the required regularity is verified by Theorem 4.1. Further, we would like to show the existence of strong material derivative. We show several ingredients of the proof in Appendix D. Application of Lemmas D.1 and D.2 gives ˙ u)) ˙ = lim (at (wt , wt ) − a(u, ˙ u)) ˙ + lim (a(u, ˙ u) ˙ − at (u, ˙ u)) ˙ = 0. lim (at (wt , wt ) − at (u,

t→0

t→0

t→0

Using this result we can write lim at (wt − u, ˙ wt − u) ˙ = lim (at (wt , wt ) − at (u, ˙ u)) ˙ − 2 lim at (wt − u, ˙ u) ˙ =0

t→0

t→0

t→0

with Lemma D.3 applied at the end. From the coercivity of at we have that at (wt − u, ˙ wt − u) ˙ ≥ Cwt − u ˙ 2W 1,2 (D) , t 1,2 which together with the previous result gives that w → u˙ strongly in W (D).

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6. Shape derivative. Let us define a function u by u = u˙ − h · ∇u; we call it a shape derivative of u. The shape derivative is used for the direct computation of DJ. For further developments we need that u ∈ W 2,2 (Ω) and u ∈ W 2,2 (D \ Ω). However, this is not guaranteed from Theorem 4.1. Therefore, for the moment we assume this regularity of u. Next we would like to find out what equation describes u . Let us denote   E := a(u , ϕ) + (∇ · β(|∇u|2 )∇u , h · ∇ϕ). We compute   E = (β(|∇u|2 )∇u , ∇ϕ) + 2(β  (|∇u|2 )∇u · ∇u∇u, ∇ϕ) + (∇ · β(|∇u|2 )∇u , h · ∇ϕ) = (β(|∇u|2 )∇u, ˙ ∇ϕ) − (β(|∇u|2 )∇(h · ∇u), ∇ϕ) + 2(β  (|∇u|2 )∇u˙ · ∇u∇u, ∇ϕ) − 2(β  (|∇u|2 )∇(h · ∇u) · ∇u∇u, ∇ϕ) + (β(|∇u|2 )Δu, h · ∇ϕ) + (∇(β(|∇u|2 )) · ∇u, h · ∇ϕ). The sum of the underlined terms in the previous expression is equal to a(u, ˙ ϕ) and thus we can replace it by −b(ϕ) from Theorem 5.1. Further we use ∇(h · ∇u) = DhT ∇u + D2 uh to obtain E = −(β(|∇u|2 )A∇u, ∇ϕ) + 2(β  (|∇u|2 )DhT ∇u · ∇u∇u, ∇ϕ) + (∇ · (f h), ϕ) −(β(|∇u|2 )∇(h · ∇u), ∇ϕ) − 2(β  (|∇u|2 )(DhT ∇u + D2 uh) · ∇u∇u, ∇ϕ) + (β(|∇u|2 )Δu, h · ∇ϕ) + (∇(β(|∇u|2 )) · ∇u, h · ∇ϕ) = (β(|∇u|2 ), −A∇u · ∇ϕ − ∇(h · ∇u) · ∇ϕ + Δuh · ∇ϕ) + (∇ · (f h), ϕ) − 2(β  (|∇u|2 )D2 uh · ∇u∇u, ∇ϕ) + (∇(β(|∇u|2 )) · ∇u, h · ∇ϕ). Next using ∇(β(|∇u|2 )) = 2β  (|∇u|2 )D2 u∇u we end up with E = (β(|∇u|2 ), −A∇u · ∇ϕ − ∇(h · ∇u) · ∇ϕ + Δuh · ∇ϕ) + (∇ · (f h), ϕ) − 2(β  (|∇u|2 )D2 uh · ∇u∇u, ∇ϕ) + (2β  (|∇u|2 )D2 u∇u · ∇u, h · ∇ϕ) = (β(|∇u|2 )[−A∇u − ∇(h · ∇u) + Δuh], ∇ϕ) + (∇ · (f h), ϕ) 2(β  (|∇u|2 )[−D2 uh · ∇u∇u + D2 u∇u · ∇uh], ∇ϕ). In the appendix of [15] it was verified by simple calculations that [−A∇u − ∇(h · ∇u) + Δuh] · ∇ϕ = −∇ × (h2 ux − h1 uy ) · ∇ϕ, where the curl operator acting on a scalar function is defined as ∇ × f = (fy , −fx ). It is also easy to verify that −D2 uh · ∇u∇u + D2 u∇u · ∇uh = −(h2 ux − h1 uy )



uyy uy + uxy ux −uxy uy − uxx ux

1 = − (h2 ux − h1 uy )∇ × (|∇u|2 ). 2



1097

MATERIAL AND SHAPE DERIVATIVE METHOD

We can therefore use the previous findings to go on in the computation of E; E = −(β(|∇u|2 )∇ × (h2 ux − h1 uy ), ∇ϕ) + (∇ · (f h), ϕ) − (β  (|∇u|2 )∇ × (|∇u|2 )(h2 ux − h1 uy ), ∇ϕ) (6.1)

= −(∇ × [β(|∇u|2 )(h2 ux − h1 uy )], ∇ϕ) + (∇ · (f h), ϕ).

  We use the Green theorem for a two-dimensional region S: ∂S F · t = S Fx2 − Fy1 . Here, t is the tangential unit vector in the counterclockwise direction defined as t = (t1 , t2 )T = (−n2 , n1 ) with n = (n1 , n2 )T being the outward normal unit vector. Thus the two vectors n and t in this order form the counterclockwise direction. Taking F = r∇ϕ one obtains 

 r∇ϕ · t = −

(6.2) ∂S

∇ × r · ∇ϕ. S

We are going to compute the first of the two integrals in (6.1), IN T := (∇ × [β(|∇u|2 )(h2 ux − h1 uy )], ∇ϕ). We split the integration domain D into two subdomains Ω and D \ Ω. We denote by r the expression r := h2 ux − h1 uy . Notice that the space dependent function r and the function β(|∇u|2 ) both have a discontinuity across ∂Ω. We use the superscripts + and − to indicate the limit values when approaching the boundary ∂Ω from outside Ω and from inside Ω, respectively, that is, f + (x) = lim f (xn ) for xn →x

xn

∈ D \ Ω,

f − (x) = lim f (xn ) for xn →x

xn

∈ Ω.

We perform integration by parts for two domains separately using (6.2), IN T = (β(|∇u|2 )r, ∇ϕ·tD )∂D −(β(|∇u|2 )+ r+ , ∇ϕ·tΩ )∂Ω +(β(|∇u|2 )− r− , ∇ϕ·tΩ )∂Ω , where tΩ is defined as counterclockwise unit tangential vector to Ω. Since ϕ = 0 on ∂D we have that ∇ϕ is perpendicular to tD so the first integral vanishes. We write h as a sum of its projections onto the orthonormal system (nΩ , tΩ ) h = hn nΩ + ht tΩ . We know that tΩ = (t1 , t2 ) = (−n2 , n1 ) and thus r = h2 ux − h1 uy = ht nΩ · ∇u − hn tΩ · ∇u. Therefore we obtain IN T = −(β(|∇u|2 )+ (ht nΩ · ∇u+ − hn tΩ · ∇u+ ), ∇ϕ · tΩ )∂Ω + (β(|∇u|2 )− (ht nΩ · ∇u− − hn tΩ · ∇u− ), ∇ϕ · tΩ )∂Ω .

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If we introduce another notation P u = β(|∇u|2 )∇u we can use the interface condition (1.3) that can be written as (P u+ − P u− ) · nΩ = 0 and proceed IN T = −(ht (P u+ − P u− ) · nΩ , ∇ϕ · tΩ )∂Ω + (hn (P u+ − P u− ) · tΩ , ∇ϕ · tΩ )∂Ω = (hn (P u+ − P u− ) · tΩ tΩ , ∇ϕ)∂Ω . Now realize that (P u+ −P u− )·tΩ tΩ is nothing else than the projection of P u+ −P u− onto tΩ . But from the interface condition we know that P u+ − P u− is perpendicular to nΩ and therefore (P u+ − P u− ) · tΩ tΩ = P u+ − P u− . We put the previous result and the expression for IN T into (6.1) to obtain   a(u , ϕ) + (∇ · β(|∇u|2 )∇u , h · ∇ϕ) = (hn (P u+ − P u− ), ∇ϕ)∂Ω + (∇ · (f h), ϕ).   From (1.2) we have that ∇ · β(|∇u|2 )∇u = −f and thus a(u , ϕ) − (f, h · ∇ϕ) = (hn (P u+ − P u− ), ∇ϕ)∂Ω + (∇ · (f h), ϕ)a(u , ϕ) − (f h · nD , ϕ)∂D + (f h · nΩ , ϕ)∂Ω − (f h · nΩ , ϕ)∂Ω = (hn (P u+ − P u− ), ∇ϕ)∂Ω . Since ϕ = 0 on ∂D, three terms from the previous expression vanish. We can successfully conclude this section with the following theorem. Theorem 6.1. Assume the solutions to the direct problem (1.2), (1.3) satisfy u ∈ W 2,2 (Ω),

u ∈ W 2,2 (D \ Ω).

Then the shape derivative u satisfies the following elliptic interface problem: (6.3)

a(u , ϕ) = (h · nΩ (P u+ − P u− ), ∇ϕ)∂Ω .

7. Adjoint problem. To know the response of the cost functional on the small changes of Ω under the velocity field induced by Ft we differentiate the cost (5.3) (7.1)    J(Ωt ) − J(Ω) 1 = lim DJ := lim |∇ut − M|2 − |∇u − M|2 = (∇u−M)·∇u t→0 t→0 2t t ω ω using results from [15, 29]. We introduce an adjoint problem in order to explicitly compute the derivative of the cost function J(Ω). This reduces computational costs tremendously in comparison with the conventional method of perturbations or with the method of sensitivity equation. This speed up is caused by the fact that the direct problem is nonlinear and therefore it must be solved iteratively. A similar approach of an adjoint variable has been used in many applications [8, 7, 11, 23, 27]. Denote by p a W02,2 (D) function such that for all ψ ∈ W02,2 (D)  (7.2) a(p, ψ) = (∇u − M) · ∇ψ. ω

Take the following test functions ϕ = p in (6.3) and ψ = u in (7.2). The left-hand sides of the resulting equalities are equal and therefore we obtain  (7.3) DJ = (∇u − M) · ∇u = (h · nΩ (P u+ − P u− ), ∇p)∂Ω . ω

MATERIAL AND SHAPE DERIVATIVE METHOD

1099

Therefore, the steepest descent direction (denoted by hsd ) for the gradient-type algorithms minimizing J is given by (7.4)

hsd = −(P u+ − P u− ) · ∇pnΩ

on ∂Ω.

8. Implementation. We adopt the level set method for the description of the geometry. For an overview see [15, 22] and the references therein. The pioneering work about the level set approach for inverse problems involving obstacles is [26]. This technique has been used in the determination of electromagnetic inclusion [24]; however, magnetic materials have been considered linear in [24]. We represent the boundary of Ω as a zero level set of a function φ. We set φ in such a manner that Ω = {x ∈ D | φ(x) > 0}, D \ Ω = {x ∈ D | φ(x) < 0}. We define the Heaviside function H in a classical way by ⎧ ⎨0, φ < 0, H(φ) = 1/2, φ = 0, ⎩ 1, φ > 0. The derivative H  (φ) of the Heaviside function is the Dirac delta function. To minimize the shape functional J we would like to move the interface ∂Ω in the steepest descent direction hsd . The level set method allows us to do this by solving the Hamilton–Jacobi equation φt + hsd · ∇φ = 0. Since ∂Ω = {x ∈ D|φ(x) = 0} we can write nΩ = ∇φ/|∇φ| and thus the previous equality becomes (8.1)

φt + hsd |∇φ| = 0,

where hsd := |hsd |. Given data on the domain ω, the following algorithm identifies the unknown Ω inside the domain D. Algorithm 1. (a) Set an initial level set function φ as an initial guess. By j we indicate the quantities in the jth step of this algorithm. For j = 0, j = 1, . . . , do the following until the algorithm converges. (b) Solve (1.4) with Ωj instead of Ω to obtain the solution uj of the direct problem. (c) Solve (7.2) with Ωj and uj instead of Ω and u to obtain the solution pj of the adjoint problem. (d) Evaluate the normal steepest descent direction hjsd from (7.4). (e) Update the level set function by solving φjt + hjsd |∇φj | = 0. (f) If the convergence is reached then stop otherwise shift the index j with the corresponding quantities and go to the (b) part of this algorithm. Some parts of Algorithm 1 need more detailed discussion. We will use a finite element method for the finite dimensional approximation of φ. For the approximation of W 1,2 (D) we choose Lagrange finite elements of the first order. In part (b) we need to compute a nonlinear elliptic equation (1.4). This equation can be considered as an operator equation G(u) = f , where G:u∈    G is a mapping  W01,2 (D) → G(u) ∈ L2 (D) such that β(|∇u|2 ∇u, ∇ϕ = G(u), ϕ , This operator equation is nonlinear and therefore it will be solved for all the numerical examples by

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the same iterative algorithm. Starting from the initial guess u0 , we use the Newton– Raphson algorithm based on the update ui+1 = ui − [DG(ui )]−1 (G(ui ) − f ). Weak formulation of the previous equation will generate a symmetric coercive bilinear form guaranteeing the invertibility of DG(ui ). Notice that for each iteration one linear PDE has to be solved. In part (c) we need to solve just a linear PDE, which is straightforward. For the evaluation of hsd we need to project (P u+ − P u− ) · ∇p onto space of Lagrange finite elements. This is done by solving the simple linear equation 

+

−



(P u − P u ) · ∇pϕdx =

(8.2) ∂Ω

hsd ϕdx. D

In section 9 we discuss how we tackle the line integral on the left-hand side. Part (e) involves the solution of the Hamilton–Jacobi equation. We use the scheme φj+1 − φj + hjsd |∇φj | = 0. Δt The step size Δt is chosen dynamically. It is doubled if the shape functional decreases; otherwise it is divided by 2 until we obtain the decrease in functional. For evaluation of |∇φj | different approaches can be used. For an overview of upwind schemes on triangular meshes see [2] and [20] and the references therein. The widely used ENO and WENO schemes have been used in numerous applications. We do not use any upwinding and still we obtain satisfactory results without oscillations. The convergence in part (f) is controlled by checking if the shape functional J sufficiently drops. If |(J(Ωj ) − J(Ωj+1 ))/J(Ωj )| < eth , where eth is some small threshold, we stop the algorithm. We picked the threshold value eth by handsetting eth = 3 × 10−5 in all our simulations. 9. Numerics. In computations, to achieve numerical robustness, the use of a smeared out Heaviside function is recommended [22]. We use the following smooth approximation of the Heaviside function: (9.1)

Hk (φ) =

1 1 + arctan(kφ) 2 π

with k being a real parameter influencing how steep the approximation around zero is. For k → ∞, Hk (φ) converges pointwise to H(φ). The derivative of Hk (φ) is Hk (φ) = k/(π(1 + k 2 φ2 )). This smooth approximation is crucial mainly for the computation of the line integrals. Thus, instead of (8.2) we have  D

Hk (φ)(β2 (|∇u|2 )∇u − β1 (|∇u|2 )∇u) · ∇pϕdx =

 hsd ϕdx. D

To quantify the convergence of the method we introduce a distance between two shapes. Since any shape is represented by a zero level set of a level set function, we define the distance dist(φ1 , φ2 ) between two shapes using its level set representations φ1 , φ2 as dist(φ1 , φ2 ) = Hk (φ1 ) − Hk (φ2 )L2 (D) and we say that the sequence of shapes represented by φn converges to a shape represented by φ iff dist(φn , φ) → 0.

1101

MATERIAL AND SHAPE DERIVATIVE METHOD 7000

nonlinearity β2

6000 5000 4000 3000 2000 1000 0 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 9.1. Nonlinear function β2 (s).

Notice that we can manipulate the distance by changing the value of k. Indeed, regardless what value of k has been used in determining the shape, for measuring the obtained shape we can use a different k. The higher k means that the interface is sharper and therefore two different shapes are better distinguished. Throughout this section we consider Ω ∈ R2 to be a square (−0.5, 0.5)×(−0.5, 0.5). The material parameter functions β1 , β2 are set to conform to the real physical quantities as described in the introduction. β1 = 7.961 × 105 , which is a constant equal to the inverse of the magnetic permeability of the air. β2 is chosen to be β2 (s) = d1 +

c1 sb1 . ab11 + sb1

This function approximates the inverse of magnetic permeability of 4% Si steel. From the graph of β2 (s) in Figure 9.1 one can see that assumptions A1–A4 are satisfied. The concrete values are set to be a1 = 1.78, b1 = 14, c1 = 6000, d1 = 245. Initially, all simulations featured oscillations of the zero level set. To stabilize the optimization process we introduce the regularization and we add a Tikhonov stabilizing term. We choose the squared norm of the gradient of the level set function. We use the coefficient α to control the trade-off between the fidelity term and the regularizing term. The cost function J from (1.5) thus obtains a new term resulting in   1 |∇u(Ω) − ∇m|2 + α |∇φ|2 dx. J(D) = 2 ω D The expression (8.2) for the evaluation of the normal steepest descent direction hsd changes by adding the corresponding derivative of the regularization term to (9.2)    Hk (φ)(β2 (|∇u|2 )∇u − β1 (|∇u|2 )∇u) · ∇pϕdx + 2α ∇φ · ∇ϕdx = hsd ϕdx. D

D

D

All the linear problems are solved on the regular triangular mesh with 2dim2 triangles constructed by splitting the square into dim2 small squares and next splitting each of them into two triangles. 9.1. Single void. We design the following numerical example. We generate synthetic data using the model described in section 2. This is to replace the real measurements with nonlinear material. The synthetic data will be available on a

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(a)

(b)

(c)

(d)

Fig. 9.2. The computational domain D is a square (−0.5, 0.5) × (−0.5, 0.5). (a) The exact Dex consists of an egg-like shape plotted with black dashed line. Measurements are available on a strip bounded by the boundary ∂D and the dashed square. The initial guess is an ellipse plotted with the red solid line. The approximation represented by the zero level set of φ after eight iterations is plotted with the green dotted line. (b) Part of a picture. Evolution of the zero level set after 34 iterations is plotted with the red solid line and after 94 iterations is plotted with the green dotted line. (c) Part of a picture. Evolution of the zero level set after 752 iterations is plotted with the red solid line and after 1812 iterations is plotted with the green dotted line. (d) Evolution of the cost value (blue star marker), value of the cost drop (red plus marker), and the convergence curve (green cross marker).

strip ω of a specific thickness dth around the boundary ∂D. We set the parameters f = 5 × 105 , k = 40, dim = 30. The simulations for dth = 0.5, i.e., when ω coincides with D, have been carried out in [9]. This case is computationally easier because one possesses the data above the air gap and can directly see the approximate location of the voids in the electromagnetic material. Our case when the data is available around the boundary is more realistic. For comparison we present both cases. The current density function f is a constant over the whole D. This setting generates a magnetic field with strength between 0 and 2.7T. In this example we choose the exact domain to be of an egg-like shape located around the boundary of the measurements. The initial guess φ0 is set to be an ellipse shifted away from the exact shape; see Figure 9.2(a). We have not used any specialized approaches to choose the regularization weight α automatically. We always picked the value by hand. The use of the L-curve or other methods is not the prime aim of this work. We focus on demonstrating the usefulness of the algorithm. We run two simulations, one with thickness of ω 0.3 and one with 0.5. From Figure 9.2(a)–(c) we can see how the approximated void evolved for dth = 0.3. The shape has moved to the right position and simultaneously has filled the exact shape.

MATERIAL AND SHAPE DERIVATIVE METHOD

(a)

(b)

(c)

(d)

1103

Fig. 9.3. See also Figure 9.2. Measurements are available on the whole D. (a) The approximation represented by the zero level set of φ after two iterations is plotted with the green dotted line. (b) Part of a picture. Evolution of the zero level set after 12 iterations is plotted with the red solid line and after 28 iterations is plotted with the green dotted line. (c) Part of a picture. Evolution of the zero level set after 101 iterations is plotted with the red solid line and after 257 iterations is plotted with the green dotted line. (d) Evolution of the cost value, value of the cost drop, and the convergence curve.

To reach the final state it was necessary to run 1812 iterations. The regularization weight has been set α = 0.005. In Figure 9.2(d) we presented a qualitative study of the algorithm behavior. A blue star plots cost value J(φn ) against the number of iterations. We see that this value decreases monotonically. A green times symbol plots dist(φn , φex ). For a sharper interface we used high k = 500 for the evaluation of the distance. We see that in the beginning the distance monotonically decreases. Then it reaches a flat valley beginning around the 1000th iteration. Its minimum is reached at around the 1812th iteration, where the last snapshot has been taken in Figure 9.2(c). Afterward it starts to increase. At this time the regularization term takes over the fidelity term in the determination of the gradient direction. The last curve, plotted with  a red plus, showsthe relative decrease in the value of the cost functional defined by J(Ωn ) − J(Ωn+1 ) /J(Ωn ). This value was oscillating so we averaged the value to see the tendency. Our hope was that this curve can somehow detect when we have to stop iterations; however, it did not help. We picked the threshold value eth by handsetting eth = 3 × 10−5 . The next simulation was run with measurements available over the whole D, that is, for dth = 0.5 (see Figure 9.3). The regularization term α equals 0.01 and eth = 3 × 10−5 . The evolution of the curves is similar; however, the significant difference is in the number of iterations. While with dth = 0.3 we needed 1800 iterations to reach the optimal shape, with dth = 0.5 we need only around 350 iterations. This is due to the amount of the available data.

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(a)

(b)

(c)

(d)

Fig. 9.4. (a) The exact Dex consists of two circles plotted with dashed line. Measurements are available on a strip bounded by the boundary ∂D and dashed square. The initial guess is a large ellipse encircling Dex plotted with the red solid line. The approximation represented by the zero level set of φ after six iterations is plotted with the green dotted line. (b) Part of a picture. Evolution of the zero level set after 33 iterations is plotted with the red solid line and after 85 iterations is plotted with the green dotted line. (c) Part of a picture. Evolution of the zero level set after 220 iterations is plotted with the red solid line and after 955 iterations is plotted with the green dotted line. (d) Evolution of the cost value, value of the cost drop, and the convergence curve.

9.2. Multiple voids. The next example demonstrates the ability of the algorithm to cope with multiple voids. We choose the exact shape to consist of two circles; see Figure 9.4(a). The initial shape will be one ellipse encircling both parts and thus we test the ability of the algorithm to detect topological changes. The parameters have been set as in the previous example. We again run two simulations, one with a thickness of ω = 0.3 and one with 0.5. From Figure 9.4(a)–(c) we can see how the approximated void evolved for dth = 0.3. The initial ellipse shrinks around two circles and eventually it divides into two parts. Consequently it approaches the exact shape. To reach the final state it was necessary to run 955 iterations. The regularization weight has been set as α = 0.01 and eth = 3 × 10−5 . In Figure 9.4(d) we again see curves representing the cost value, convergence, and relative cost drop. We see that the distance monotonically decreases; however, around the 900th iteration it starts to increase. This point determines where the approximated shape was the closest to the exact solution. Afterward, the regularization term takes over the fidelity term in the determination of the gradient direction. Consequently, the approximated shape and the exact shape draw apart. In Figure 9.5 we depict simulation results with measurements available over the whole D, that is, for dth = 0.5. The regularization term α equals to 0.01 and eth = 3 × 10−5 . The evolution of the curves is similar; however, the significant difference is

MATERIAL AND SHAPE DERIVATIVE METHOD

(a)

(b)

(c)

(d)

1105

Fig. 9.5. See also Figure 9.4. Measurements are available on the whole D. (a) The approximation represented by the zero level set of φ after four iterations is plotted with the green dotted line. (b) Part of a picture. Evolution of the zero level set after 19 iterations is plotted with the red solid line and after 37 iterations is plotted with the green dotted line. (c) Part of a picture. Evolution of the zero level set after 88 iterations is plotted with the red solid line and after 235 iterations is plotted with the green dotted line. (d) Evolution of the cost value, value of the cost drop, and the convergence curve.

in the number of iterations. While with dth = 0.3 we needed 955 iterations to reach the optimal shape, with dth = 0.5 we need only around 235 iterations. 10. Conclusions. The main contribution of this work is twofold. First we theoretically studied the material and shape derivative method for nonlinear—more specifically, quasi-linear—elliptic problems. The obtained results for interface problems are easily adaptable to the case when Neumann or Dirichlet boundary conditions are imposed on ∂Ω. Then, the elliptic problem is considered only on D \ Ω. Second, we used the aforementioned shape derivative method and we solved the inverse interface problems. We emphasize that this kind of quasilinear problem has not been studied before. Further, we showed in two examples the performance of the optimization algorithm based on the gradients computed from shape derivatives. We first implemented a level set method to compute the quasi-linear elliptic equation describing the magnetic processes inside a nonlinear magnetic material that contains air gaps or cracks. In this way we obtained an efficient solver for the nonlinear direct problem. Second, we used an iterative procedure based on the gradient-type minimization algorithms to determine the optimal shape. Optimality was controlled by the shape functional. For evaluation of the shape gradients we implemented the adjoint variable method. This method brings a tremendous reduction of the computational costs in the computation of the gradient direction compared to the classical perturbation methods.

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Appendix A. Identities for material derivative method. We list several identities that hold: " # 1 1 1 + t ∂h t ∂h ∂x1 ∂x2 (A.1) , It = det(DFt ), DFt = 2 2 t ∂h 1 + t ∂h ∂x1 ∂x2 Mt = DFt−1 = (DFt )−1 , At = Mt MtT It = DFt−1 (DFt−1 )T It , A := ∇ · hId − (DhT + Dh), " ∂h #  ∂h1 1 It − 1 DFt − Id  ∂x1 ∂x2 |t=0 = ∇ · h, (A.2) = ∂h2 ∂h2 = Dh, At |t=0 = Id,  t t t=0 ∂x ∂x " ∂h 1 ∂h2 #   1 2 −1 T T − −   Mt − Id  (DFt ) − Id  ∂x1 ∂x1 (A.3) = = = −DhT ,   ∂h1 ∂h2 t t − − t=0 t=0 ∂x2 ∂x2 " ∂h #  ∂h1 ∂h1 ∂h2 2 − ∂x2 − ∂x1 At − Id  ∂x2 − ∂x1 (A.4) = = ∇ · hId − (DhT + Dh) = A.  ∂h ∂h ∂h1 1 2 2 t − ∂x2 − ∂x1 − ∂h + t=0 ∂x2 ∂x1 Appendix B. Coercivity and continuity of at and bt . From the L∞ estimates of MtT ∇ut and ∇u we get the continuity of at1 . Together with boundedness of β  we get the continuity of at3 . Finally, from (A.3) we have that (MtT −Id)L∞ (D) ≤ tC, which gives (B.1)

|at2 (wt , ϕ)| ≤ tC∇wt 2 ∇ϕ2 .

We conclude that at is a continuous bilinear form in W 1,2 (D). Since βi are nondecreasing functions we have that βi ≥ 0 for i = 1, 2 and thus t a3 (wt , wt ) ≥ 0. Since 0 < βmin ≤ β(s) we have that at1 (wt , wt ) ≥ βmin /2∇wt 22 . Finally, from (B.1) we have at2 (wt , wt ) ≥ −Ct∇wt 22 . Altogether we obtain at (wt , wt ) =

3 

ati (wt , wt ) ≥

i=1

βmin − Ct ∇wt 22 , 2

which for sufficiently small t gives the coercivity of at . From (A.4) we have that t−1 (At − Id)L∞ (D) ≤ C and together with L∞ boundedness of ∇ut and ∇u we obtain the boundedness (and thus continuity) of bt1 . From (A.2) we have that t−1 (It − 1)L∞ (D) ≤ C and thus bt4 is bounded and continuous. From the smoothness properties of ft we can conclude that bt5 is bounded and continuous. From (A.3) we get that (MtT − Id)/t is bounded in L∞ and thus together with boundedness of β  and ∇ut , ∇u we get (B.2)

|bt2 (ϕ)| ≤ Ct∇ϕ2 ,

|bt3 (ϕ)| ≤ C∇ϕ2 ,

which confirms the continuity of bt (ϕ). Appendix C. Limits Ai , Bi . To find out what equation is satisfied by the weak material derivative, we need to pass in the limit n → ∞ for expressions Ai , Bi . First we show an auxiliary lemma.

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Lemma C.1. β(|MtT ∇utn |2 )Atn → β(|∇u|2 ) strongly in L2 (D) for n → ∞. Proof. We have that     β(|MtT ∇utn |2 )Atn − β(|∇u|2 ) ≤ β(|MtT ∇utn |2 )(Atn − Id)   + β(|MtT ∇utn |2 ) − β(|∇utn |2 )   + β(|∇utn |2 ) − β(|∇u|2 ) . Using that β1 , β2 are Lipschitz continuous we obtain $ $ $β(|M T ∇utn |2 )Atn − β(|∇u|2 )$ t 2 % $ $ T tn 2 ≤ C β(|Mt ∇u | )(Atn − Id)2 + $(MtT − Id)∇utn (MtT + Id)∇utn $2 $ $ & + $(∇utn − ∇u)(∇utn + ∇u)$2 % ≤ C β(|MtT ∇utn |2 )∞ Atn − Id)∞ + MtT − Id∞ ∇utn ∞ MtT & + Id∞ ∇utn ∞ + ∇utn − ∇u2 ∇utn + ∇u∞ . From (A.3) we know that MtT → Id in L∞ and from (A.4) we have that Atn → Id in L∞ . Also utn → u in W 1,2 (D). This together with boundedness of all the other terms confirms the statement of the lemma. Now we are ready to compute the limit for A1 ,    ˙ ∇ϕ) lim |2A1 | = lim  [β(|MtT ∇utn |2 )Atn + β(|∇u|2 )]∇wn , ∇ϕ − 2(β(|∇u|2 )∇u, n→∞ n→∞ %   ≤ lim  [β(|MtT ∇utn |2 )Atn − β(|∇u|2 )]∇wn , ∇ϕ  n→∞   & + 2  β(|∇u|2 )(∇wn − ∇u), ˙ ∇ϕ  . The first limit is zero. This is true because ∇wn is L2 bounded and we suppose that ϕ ∈ C0∞ (D). From Lemma C.1 we have strong convergence of the rest. The second limit is zero since wn u˙ and β(|∇u|2 )∇ϕ is fixed and bounded in L2 . The conclusion is that if ϕ ∈ C0∞ (D), then limn→∞ |2A1 | = 0. The next auxiliary lemma will help us to compute the limit of A3 . Lemma C.2. β  (η tn ) → β  (|∇u|2 ) strongly in L2 (D) for n → ∞. Proof. The function β  should be understood as β1 for arguments from Ω and as β2 for arguments from D \ Ω. A diagram on p. 88 of [18] describes the mutual relations between types of convergence. Of interest to us is the relation between the convergence in Lp (D) and the existence of a subsequence that converges almost everywhere. From this relation we can state that since MtT ∇un converges strongly to ∇u in L2 (D), then there exists a subsequence still denoted by un such that for almost all X ∈ D the sequence MtT ∇un (X) → ∇u(X). But we know that     min |MtT ∇utn (X)|2 , |∇u(X)|2 ≤ η tn (X) ≤ max |MtT ∇utn (X)|2 , |∇u(X)|2 , which means that also η tn (X) → |∇u(X)|2 for almost all X ∈ D. Using this and the continuity of βi , i = 1, 2, we obtain that βi (η tn (X)) → βi (|∇u(X)|2 ), i = 1, 2. Finally, from convergence almost everywhere we conclude that β  (η tn ) → β  (|∇u|2 ) strongly in W 1,2 (D).

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We go on and compute the limit of A3 for n → ∞. Again, the function β  should be understood as β1 for arguments from Ω and as β2 for arguments from D \ Ω. lim |2A3 | = lim |(β  (η tn )∇wn · (∇utn + ∇u)(∇utn + ∇u)

n→∞

n→∞

− 4β  (|∇u|2 )∇u˙ · ∇u∇u, ∇ϕ)| %   ≤ lim  (β  (η tn ) − β  (|∇u|2 ))∇wn · (∇utn + ∇u)(∇utn + ∇u), ∇ϕ  n→∞   +  β  (|∇u|2 )∇wn · (∇utn − ∇u)(∇utn + ∇u), ∇ϕ    +  β  (|∇u|2 )∇wn · 2∇u(∇utn − ∇u), ∇ϕ    & +  β  (|∇u|2 )(∇wn − ∇u) ˙ · 2∇u2∇u, ∇ϕ  . The last limit is zero since ∇wn ∇u˙ in L2 (D) and β  (|∇u|2 )∇ϕ · ∇u∇u is fixed and bounded in L2 . Next, for ϕ ∈ C0∞ (D) we have boundedness of ∇ϕ, ∇utn , ∇u in L∞ (D). From (5.9) and from Lemma C.2 we conclude that the first limit is zero too. Next we use the boundedness of β  and ∇utn , ∇u, ∇ϕ to end up with lim |2A3 | ≤ C lim ∇wn 2 ∇utn − ∇u2 ,

n→∞

n→∞

and since ∇wn is bounded in L2 (D) and utn → u in W 1,2 (D) we strongly we conclude that for ϕ ∈ C0∞ (D) we have limn→∞ |A3 | = 0. Appendix D. Strong convergence of the material derivative. In this section we show some auxiliary lemmas needed for the strong convergence of the material derivative. Lemma D.1. at (wt , wt ) → a(u, ˙ u) ˙ for t → 0. Proof. Setting ϕ = wt in (5.8) we have at (wt , wt ) = bt (wt ). From Remark 3 we ˙ = limt→0 |bt (wt ) − b(wt )| + limt→0 |b(wt ) − b(u)| ˙ = 0, know that limt→0 |bt (wt ) − b(u)| t t which results in b (w ) → b(u). ˙ From (5.10) we have that a(u, ˙ u) ˙ = b(u). ˙ Lemma D.2. at (u, ˙ u) ˙ → a(u, ˙ u) ˙ for t → 0. Proof. Note that |at2 (u, ˙ u)| ˙ ≤ tC since ∇ut and ∇u are bounded and MtT → Id. Then ˙ u) ˙ − a(u, ˙ u)| ˙ |at (u,    ˙ ∇u˙ − 2(β(|∇u|2 )∇u, ˙ ∇u) ˙  =  [β(|MtT ∇ut |2 )At + β(|∇u|2 )]∇u, + |(β  (η)∇u˙ · (∇ut + ∇u)(∇ut + ∇u), ∇u) ˙  2 ˙ + tC − 4(β (|∇u |)∇u˙ · ∇u∇u, ∇u)|

≤ β(|MtT ∇ut |2 )At − β(|∇u|2 )2 ∇u ˙ 2∞ + β  (η) − β  (∇u|2 )2 ∇u ˙ ∞ ∇ut + ∇u∞ ∇ut + ∇u∞ ∇u ˙ ∞ + β  (∇u|2 )∞ ∇u ˙ ∞ ∇ut − ∇u2 ∇ut + ∇u∞ ∇u ˙ ∞  2 t + 2β (∇u| )∞ ∇u ˙ ∞ ∇u∞ ∇u + ∇u∞ ∇u ˙ ∞ + tC. From Theorem 5.1 we know that ∇u˙ is L∞ bounded and we also know from Lemma C.1 that β(|MtT ∇ut |2 )At → β(|∇u|2 ) in L2 . Further, we know that ∇ut is L∞ bounded and that β  (η) → β  (∇u|2 ) in L2 . This confirms the statement of the lemma. Lemma D.3. at (u, ˙ wt − u) ˙ → 0 and at (wt − u, ˙ u) ˙ → 0 for t → 0.

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Proof. First we show that at (u, ˙ wt − u) ˙ → 0. We have lim |at (u, ˙ wt − u)| ˙ = lim |at (u, ˙ wt ) − at (u, ˙ u)| ˙

t→0

t→0

≤ lim |at (u, ˙ wt ) − a(u, ˙ wt )| + lim |a(u, ˙ wt − u)| ˙ t→0

t→0

˙ u) ˙ − at (u, ˙ u)|. ˙ + lim |a(u, t→0

The second limit is zero since wt u˙ in W 1,2 (D) and the third limit is zero from Lemma D.2. For the first limit realize that |at2 (u, ˙ u)| ˙ ≤ tC since ∇ut and ∇u are T bounded and Mt → Id. We estimate |at (u, ˙ wt ) − a(u, ˙ wt )|    =  [β(|MtT ∇ut |2 )At + β(|∇u|2 )]∇u, ˙ ∇wt − 2(β(|∇u|2 )∇u, ˙ ∇wt ) + |(β  (η)∇u˙ · (∇ut + ∇u)(∇ut + ∇u), ∇wt )

− 4(β  (|∇u2 |)∇u˙ · ∇u∇u, ∇wt )| + tC ˙ ∞ ∇wt 2 ≤ β(|MtT ∇ut |2 )At − β(|∇u|2 )2 ∇u

+ β  (η) − β  (∇u|2 )2 ∇u ˙ ∞ ∇ut + ∇u∞ ∇ut + ∇u∞ ∇wt 2 + β  (∇u|2 )∞ ∇u ˙ ∞ ∇ut − ∇u2 ∇ut + ∇u∞ ∇wt 2

(D.1)

+ 2β  (∇u|2 )∞ ∇u ˙ ∞ ∇u∞ ∇ut + ∇u∞ ∇wt 2 + tC.

From Theorem 5.1 we know that ∇u˙ is L∞ bounded and we also know from Lemma C.1 that β(|MtT ∇ut |2 )At → β(|∇u|2 ) in L2 . We also have other bounded terms, which proves that the limit of at (u, ˙ wt ) − a(u, ˙ wt ) is zero too. Therefore the first part of the lemma is valid. ˙ u) ˙ →0 Although the bilinear form at is not symmetric, the proof of at (wt − u, can be repeated in a very similar way. The only unsymmetric expression at (wt , u) ˙ − ˙ becomes symmetric after taking the absolute value and applying the estimaa(wt , u) tions as was done in (D.1). This concludes the proof of this lemma. REFERENCES [1] I. Babuˇ ska, The finite element method for elliptic equations with discontinuous coefficients, Computing, 5 (1970), pp. 207–213. [2] T. J. Barth and J. A. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comput. Phys., 145 (1998), pp. 1–40. [3] M. Borsuk, The transmission problem for quasi-linear elliptic second order equations in a conical domain. I, II, Nonlinear Anal., 71 (2009), pp. 5032–5083. ¨ hl and M. Hanke, Recent progress in electrical impedance tomography, Inverse Prob[4] M. Bru lems, 19 (2003), pp. 65–90. [5] J. C´ ea, Conception optimale ou identification de formes: Calcul rapide de la d´ eriv´ ee directionnelle de la fonction coˆ ut, RAIRO Mod´ el. Math. Anal. Num´ er., 20 (1986), pp. 371–402. [6] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79 (1998), pp. 175–202. ´ k and V. Melicher, Sensitivity analysis framework for micromagnetism with appli[7] I. Cimra cation to optimal shape design of magnetic random access memories, Inverse Problems, 23 (2007), pp. 563–588. ´ k and V. Melicher, Determination of precession and dissipation parameters in the [8] I. Cimra micromagnetism, J. Comput. Appl. Math., 234 (2010), pp. 2239–2249. ´ k and R. Van Keer, Level set method for the inverse elliptic problem in nonlinear [9] I. Cimra electromagnetism, J. Comput. Phys., 229 (2010), pp. 9269–9283. ´ k, P. Sergeant, and A. Abdallh, Analysis of a non-destructive eval[10] S. Durand, I. Cimra uation technique for defect characterization in magnetic materials using local magnetic measurements, Math. Probl. Eng., 2010 (2010), 574153.

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