A Penalization and Regularization Technique in Shape Optimization Problems in Arbitrary Dimension

Transcription

1 A Penalization and Regularization Technique in Shape Optimization Problems in Arbitrary imension Peter Philip and an Tiba Abstract We consider shape optimization problems, where the state is governed by elliptic partial differential equations (PE) Using a regularization technique, unknown shapes are encoded via shape functions, turning the shape optimization into optimal control problems for the unknown functions The method is studied for elliptic PE to be solved in an unknown region (to be optimized), where the regularization technique together with a penalty method extends the PE to a larger fixed domain Additionally, the method is studied for the optimal layout problem, where the unknown regions determine the coefficients of the state equation In both cases and in arbitrary dimension, the existence of optimal shapes is established for the regularized and the original problem, with convergence of optimal shapes if the regularization parameter tends to zero Error estimates are proved for the layout problem In the context of finite element approximations, convergence and differentiability properties are shown A series of numerical experiments demonstrate the method computationally for an industrially relevant elliptic PE with two unknown shapes, one giving the region where the PE is solved, and the other determining the PE s coefficients epartment of Mathematics, Ludwig-Maximilians University (LMU) Munich, Theresienstrasse 39, Munich, Germany, philip@mathlmude Institute of Mathematics (Romanian Academy), POBox 1-764, RO Bucharest and Academy of Romanian Scientists, Splaiul Independentei 54, Bucharest, dantiba@imarro Basque Center for Applied Mathematics, Alameda Mazarredo 14, E Bilbao, Basque Country, Spain 1

2 Keywords shape optimization, optimal control, fixed domain method, elliptic partial differential equation, optimal layout problem, error estimate, numerical simulation 1 Introduction A typical shape optimization problem (P ) for elliptic equations has the form: (11) min j(x, y Ω (x)) dx, Ω (12) Λ y Ω = f in Ω, (13) y Ω = 0 on Ω Here Ω is some (unknown) bounded open set, Ω R d or E Ω R d, where the given sets E are also bounded and open, connected Moreover, Ω O, where O denotes the class of all admissible sets (to be specified) In (11) (13), Λ may be either Ω or E (in the second case), f L 2 (), and natural hypotheses are to be imposed on the integrand j(, ) Another important type of shape optimization problems associated to elliptic operators is the so-called optimal layout problem, looking for the optimal distribution of several materials in the given domain R d, such that the obtained composition satisfies certain prescribed properties characterized by a given mapping y d L 2 () In this paper, the case of three different materials is discussed, but our method may be easily applied to an arbitrary number of different materials, in arbitrary dimension If χ 1, χ 2, χ 3 denote the characteristic functions of the regions occupied by each material in, ie χ i (x) {0, 1}, i = 1, 2, 3, χ 1 + χ 2 + χ 3 = 1, we 2

3 formulate the optimization problem (R): (14) min (y y d ) 2 dx, (15) {(a 1 χ 1 + a 2 χ 2 + a 3 χ 3 ) y v + a 0 yv fv} dx = 0, v H 1 () Here each a i > 0 characterizes the properties of the material i, i = 1, 2, 3 and f L 2 (), a 0 > 0 are given On, Neumann boundary conditions are imposed In both problems (11) (13) and (14), (15) more general elliptic operators or other boundary conditions may be considered (in the second case), etc Supplementary constraints may be added on Ω, χ i, i = 1, 2, 3, the state y Ω, respectively y The mathematical literature on such problems, using various types of assumptions and methods, is vast: Pironneau [21], Sokolowski and Zolesio [26], Allaire [1], Bendsoe [2], Neittaanmäki, Sprekels and Tiba [16], Sverak [27], Chenais and Zuazua [5], Neittaanmäki and Tiba [17], etc A main ingredient of our approach is the use of shape functions defined in and of a technique for the approximation of characteristic functions previously used in [13], [15] For instance, if g is continuous in, then (16) Ω g = int{x : g(x) 0} is an open Caratheodory set, not necessarily connected It may even have an infinity of connected components if g has a strongly oscillatory behavior (example in = (0, 1) R, g(t) = t sin 1) t Notice that Ω g defined in (16) is not necessarily a set of class C (ie even the segment property may fail) Some counterexamples may be found in [10], [28] 3

4 It is possible to assume that g is piecewise continuous in and, then, Ω g has similar properties, [20] Notice as well that (16) is essentially different from the level set method [18], [23] since no time dependence for the functions g and no evolution for the corresponding open sets Ω g, is assumed No Hamilton-Jacobi equation will be used in our subsequent argument and this allows very weak assumptions on g, Ω g In fact, we use the term shape functions instead of level functions to underline the difference between the two methods If Ω n Ω in the complementary Hausdorff-Pompeiu metric, then there are g n, g C(), such that Ω n = Ω gn, Ω = Ω g (in the sense of (16)) and g n g uniformly in, [16] The converse is false, in general: take Ω n = Ω, n N and Ω = Ω g for some g C(), g 0; then we may write (for instance) Ω n = Ω gn, g n = 1 n g and clearly g n 0 uniformly in while g 0 In the case of the problem (14), (15), we shall assume that g Hloc 1 (), with some additional conditions (see (31) (33)) The corresponding level sets Ω g = {x : g(x) > 0} are quasi-open sets in the sense of potential theory Since R d and d N is arbitrary, they are just measurable subsets Let H : R R be the Heaviside function Then H(g) : R is the characteristic function of Ω g in (for the second problem, this holds only up to a null set and under the additional assumption that g satisfies (31) below with p i replaced by g, cf (34)) We denote by H ε : R R a regularization of the Yosida approximation of the maximal monotone extension of H in R R Then H ε (g) : R is an approximation of the characteristic function of Ω g in, in a certain sense In the next section, the regularization of the problem (11) (13) will be studied, while Section 3 is 4

5 devoted to the problem (14), (15) In Section 4, error estimates with respect to the penalization/regularization parameter are discussed in both problems Section 5 is devoted to discretization and further approximation properties The final section provides an algorithm for numerical implementation, and it contains a series of numerical results, where the algorithm has been applied to an industrially relevant shape optimization problem that combines features of (11) (13) and (14), (15) 2 Penalization and Regularization We fix Λ = E in the problem (11) (13) and we define its approximation (P ε ) (ε > 0) by: (21) min g E j(x, y ε (x)) dx, (22) y ε + 1 ε (1 H ε(g))y ε = f in, (23) y ε = 0 on We choose in this section the regularization H ε ( ) such that H ε (r) = 1 for r 0 As mentioned in Section 1, since Λ = E, we have to impose the constraint (24) g 0 in E This ensures E Ω g (see (16)) It is clear that y Ω H0(Ω), 1 the solution of (12), (13), may always be extended by 0 to and the cost functional (11) makes sense even for Ω not containing E (respectively without imposing (24)) Under this convention, one may always take Λ = instead of Λ = Ω 5

6 However, this procedure cannot be applied for Neumann boundary conditions as in (14), (15) and we shall not use it If Λ = Ω g, then the cost functional (21) has to be replaced by (25) H ε (g(x)) j(x, y ε (x)) dx and (24) is no more necessary, but might still be imposed additionally The idea of the approximation (21) (24) or (25) is that the unknown geometry Ω = Ω g is hidden in the parametrization g C() and the shape optimization problem (11) - (13) is replaced with a control by the coefficients problem, defined in the given fixed domain R d enote by = U ad C() the family of admissible controls g, ie satisfying (24) and other conditions, if any Proposition 21 Assume that = U ad C() is compact and the convergence j(x, y n (x)) j(x, y(x)) weakly in L 1 (E) if y n y strongly in H 2 () Then the problem (P ε ) has at least one optimal pair [yε, gε] [H 2 () H0()] 1 U ad for each ε > 0 Sketch of Proof For each fixed ε > 0, we have the continuity property from the coefficients g U ad to the solutions y ε of (22), (23), from the uniform topology in C() to the strong topology of H 2 () H0() 1 Applying this on a minimizing sequence (which exists since U ad ) and using the above assumption on j(, ), the result is obtained Remark An important example of U ad satisfying the hypotheses of Prop 21 is given by the signed distance functions that may be associated to any admissible Ω O, where O is assumed to be compact with respect to the Hausdorff-Pompeiu complementary metric [16, p 461] They are uniformly Lipschitzian and they satisfy (24) for any Ω E It is also possible to relax 6

7 the continuity hypotheses on U ad - this point of view will be stressed in the next section enote Ω ε := Ω g ε O and let U ad be given by the signed distance functions associated with the sets in O, where we assume that O is a compact family of open sets of class C (see [16], [28] for details and examples in this respect) Theorem 22 On a subsequence ε 0, we have yε y weakly in H0(), 1 Ω ε Ω O in the complementary Hausdorff-Pompeiu topology, where [y Ω, Ω ] is an optimal pair for (P ) Here, we assume that j(x, y n (x)) j(x, y(x)) weakly in L 1 (E) if y n y weakly in H0() 1 Proof We have Ω ε Ω O on a subsequence, due to the compactness of O As the shape functions gε are the signed distance functions, we have, on the same subsequence, gε g, where g is the signed distance function associated to Ω, [6] Multiply by yε the equations (22), (23) associated to gε We get that {yε} is bounded in H0() 1 and we may assume that yε y weakly in H0() 1 on a subsequence Moreover, we get (26) (1 H ε (gε))(y ε) 2 dx Cε with C > 0 some constant independent of ε > 0 Take K \ Ω an arbitrary, compactly embedded open subset We may find c > 0 such that g K c Then gε K c for ε small, due to 2 the uniform convergence gε g It yields H ε (gε) = 0 in K, for ε ε 0 By (26), we infer (y ε) 2 dx Cε K 7

8 and, consequently, y K = 0 ae As K is arbitrary in Ω, we get y Ω = 0 ae As Ω is of class C, by the Hedberg-Keldys stability property [16], we obtain that y Ω H0(Ω 1 ) Take ϕ C0 (Ω ) arbitrary By the Γ-property [16, p 465], we have ϕ C0 (Ω ε), for ε ε ϕ Then ϕ may be used as a test function in (22) and we have H ε (gε(x)) = 1 on supp ϕ Ω ε by the property that H ε (r) = 1 for r 0, specific to the definition of (P ε ) in this section Then the penalization term in (22), multiplied by ϕ, vanishes and we obtain (27) yε ϕ = fϕ, ϕ C0 (Ω ), ε ε ϕ One can pass to the limit in (27) to see that y Ω H0(Ω 1 ) is the solution of (12), (13) We now use the obvious inequality (28) j(x, yε(x)) dx j(x, y ε (x)) dx E E for each solution y ε H0() 1 of (22), (23) associated to an arbitrary g U ad Repeating the above argument, we get that, on a subsequence, y ε y g weakly in H0(), 1 and y g Ωg H0(Ω 1 g ) is exactly the solution of (12), (13) in Ω g Then the assumption on j(, ) yields j(x, y ε (x)) j(x, y g (x)) weakly in L 1 (E) One can pass to the limit in (28) to show the optimality of [y, Ω ] as claimed Remark If general shape functions from C() are allowed in the definition of (P ε ), no convergence properties may be established For instance, it is even possible that {g ε } is an unbounded sequence (example: g ε = 1 ε g, 0 g C() given) This is due to the nonuniqueness of the parametrization of Ω O by shape functions 8

9 Remark The above convergence result shows that the optimal control g ε found in problem (P ε ) depends in fact on the original geometric optimization problem (P ), which is a natural property in this setting 3 The Optimal Layout Problem In this section, we develop a similar regularization approach for the problem (14), (15) We consider again representations of the characteristic functions χ i, i = 1, 2, 3 via the Heaviside mapping: χ 1 = H(p 1 ), χ 2 = H(p 2 ), χ 3 = 1 H(p 1 ) H(p 2 ) It is enough to work with two shape functions p 1, p 2 Hloc 1 () and the condition χ 1 + χ 2 + χ 3 = 1 is automatically fulfilled On the pair of shape functions, the following compactness and compatibility hypotheses are imposed: (31) p i (x) + p i (x) R d ν > 0 ae in, i = 1, 2, (32) p i H 1+θ iq (Q) M i Q, θ i Q > 0, i = 1, 2, for each Q compactly embedded open subset, (33) p i (x) 0 p j (x) 0 i, j = 1, 2, i j The corresponding level sets are just measurable as R d and d N is arbitrary An important observation is that, by the classical result of Stampacchia [3, p 195], (31) implies (34) µ ( {x : p i (x) = 0} ) = 0, i = 1, 2, where µ( ) is the Lebesgue measure in R d Then, due to (33); H(p 1 ), H(p 2 ), and 1 H(p 1 ) H(p 2 ) are indeed characteristic functions of disjoint regions in 9

10 One can further regularize H (not necessarily in the same way as in Section 2) Here, we impose the requirement (35) H ε (r) = 0 for r 0 We define the approximation via regularization of the problem (R) given by (14), (15) by the problem (R ε ): (36) min [p 1,p 2 ] U ad y ε y d 2 dx, { [a1 H ε (p 1 ) + a 2 H ε (p 2 ) + a 3 (1 H ε (p 1 ) H ε (p 2 )) ] y ε v (37) } + a 0 y ε v fv dx = 0, v H 1 () We have denoted by U ad Hloc 1 ()2 the set of admissible pairs of shape functions defined by (31) - (33) Remark It is clear that the same approximation procedure may be applied to n materials, by using n 1 shape functions The compatibility condition (33) has to be written in the form p i 0 p j 0, j i We also notice that both (R) and (R ε ) have a fixed domain formulation, although the unknowns are the regions Ω i, i = 1, 2, 3 corresponding to each material As in the previous section, the aim is to replace the geometric unknowns by analytic unknowns [p 1, p 2 ] U ad Theorem 31 Under the above assumptions, the problem (R ε ) has at least one optimal triple [yε, p 1ε, p 2ε] H 1 () U ad On a subsequence, we have (38) (39) (310) y ε y p 1ε p 1 p 2ε p 2 weakly in H 1 (), in H 1 loc(), in H 1 loc(), 10

11 where [y, p 1, p 2] H 1 () U ad is an optimal triple for the problem (R) by identifying p 1, p 2 with the measurable sets Ω 1 = {x : p 1 0}, Ω 2 = {x : p 2 0}, respectively Proof ue to (31) - (33), we may take subsequences such that (39), (310) are fulfilled and [p 1, p 2] U ad Here (34) and the ae convergence of p iε and p iε give the argument Fix v = yε in (37) The ellipticity gives that {yε} is bounded in H 1 () and we may assume yε y weakly in H 1 () on a subsequence The set {x : p i (x) = 0} has zero measure, i = 1, 2 Then, ae, we may assume p i (x) > 0 or p i (x) < 0, i = 1, 2 In the first case, by the ae convergence, we infer p iε(x) > c for each ε < ε i (x), and H ε (p iε(x)) = 1 for each ε < ε i (x), i = 1, 2 Similarly, if p i (x) < 0, then, for ε < ε i (x), we obtain H ε (p iε(x)) = 0 Here, we use (35) and the fact that one may suppose H ε (r) = 1, for r > ε (see (42) as well) Consequently, on a subsequence, H ε (p iε) H(p i ) ae in, i = 1, 2 and H(p i ) are indeed characteristic functions in By the Lebesgue theorem, the convergence is valid in L q (), 1 q < One can pass to the limit in (37): The observation is that [a 1 H ε (p 1ε) + a 2 H ε (p 2ε) + a 3 (1 H ε (p 1ε) H ε (p 2ε))] yε is bounded in L 2 () d and it converges weakly in L 1 () d to [a 1 H(p 1) + a 2 H(p 2) + a 3 (1 H(p 1) H(p 2))] y Then the weak limit in L 2 () d is the same and we get that [y, p 1, p 2] satisfy the state system (15) for (R) We also have yε y d 2 dx y y d 2 dx and the same argument as at the end of Theorem 22 shows that [y, p 1, p 2] is an optimal solution for (R) Remark More general cost functionals, for instance integral functionals de- 11

12 fined on smooth manifolds of codimension one, are possible to be studied in this setting Remark One may add supplementary constraints in the definition of U ad For instance, if a certain subset ω should be made, for various reasons, of material i, then it is enough to impose the condition p i ω 0 If the total quantity of material i is limited, but its distribution is free, then we may ask H(p i ) dx k, where k > 0 is given, etc 4 Estimates with Respect to ε > 0 For any g Hloc 1 (), we denote by (41) ε (g) := {x : H ε (g(x)) H(g(x))} a measurable subset of, defined up to a subset of measure 0 Proposition 41 Under hypotheses (31) for g Hloc 1 (), we have: µ( ε (g)) 0 for ε 0, where, as in (34), µ( ) is the Lebesgue measure in R d Proof For all the regularizations H ε ( ) considered in the previous sections, we may impose that (42) ε (g) E ε (g) := {x : g(x) < ε}, 12

13 without loss of generality We notice that, if ε 1 < ε 2, then E ε1 (g) E ε2 (g) Consequently, E ε (g) is a decreasing set sequence for ε 0, and E(g) := lim E ε (g) = E ε (g), ε 0 ε>0 µ(e(g)) = lim ε 0 µ(e ε (g)) Then the result follows from (34) and (42) by contradiction Assume now that R, ie d = 1 Proposition 42 Let K ad be compact in C 1 () and (31) be satisfied by each p K ad Then there exists some natural number n such that Z(p) n, p K ad, where Z(p) denotes the number of zeros of p K ad Proof By contradiction, assume there is {p n } K ad such that Z(p n ) for n On a subsequence, we have p n p K ad, uniformly in (by assumption) Write the zeros of p n (or some subset of them) in increasing order: {x 1 n, x 2 n,, x i n, }, i Z(p n ) For any fixed i, we consider the sequence {x i n} (which may start from some higher order n) As is compact, on a subsequence, we have x i n n x i n 1 n 0, n, for certain roots of p n In certain intermediary points τ n between the above roots we also have p n(τ n ) = 0 On a subsequence, all these points have the same limit x proof We clearly get p( x) = p ( x) = 0, which contradicts (31) and ends the 13

14 Proposition 43 Under the above hypotheses, we have µ( ε (g)) µ(e ε (g)) Cε, with C > 0 a constant independent of ν > ε > 0 and of g K ad Proof E ε (g) is an open subset of and consists of a union of disjoint open intervals contained in The two extreme intervals (to the left and to the right) may have the same endpoints as We notice that g 0 in E ε (g), otherwise (31) is contradicted (for ε < ν) Since g C 1 (), g has a constant sign in each connected component of E ε (g), ie g is strictly monotone in each subinterval of E ε (g) and its slope satisfies g (x) ν ε, x E ε (g) Each such subinterval has a maximal character in the sense that g(x) = ε in its endpoints (with the possible exception of the outermost subintervals) It yields that each interior subinterval contains one zero of g and only one Therefore, their number is limited by n + 2 (the maximal number of roots from Prop 42 plus, possibly, two untypical intervals that may occur toward the endpoints of ) ue to g (x) ν ε, g C(), the length of each subinterval is majorized by 2ε ν ε The result follows with C = 4( n + 2)/ν for ε < ν 2 Based on Propositions 41-43, we establish error estimates between the solutions of the problems (14), (15) and (36), (37) Theorem 44 Let y ε H 1 () be the solution of (37) and y H 1 () the solution of (15), where χ 1 = H(p 1 ), χ 2 = H(p 2 ), χ 3 = 1 H(p 1 ) H(p 2 ), with [p 1, p 2 ] H 1 loc ()2 satisfying (31), (33) 14

15 Then we have y ε y H 1 () Cµ( ε ) q, with q > 0 depending on the dimension of, ε explained below, and C > 0 independent of ε > 0 Proof We subtract the equations (15), (37) and use the test function v = (y y ε ): { [a1 ] (43) 0 = H(p 1 ) + a 2 H(p 2 ) + (1 H(p 1 ) H(p 2 ))a 3 (y yε ) a 0 y y ε 2 } dx [ (a1 a 3 )(H(p 1 ) H ε (p 1 )) + (a 2 a 3 )(H(p 2 ) H ε (p 2 )) ] y ε (y y ε ) dx The last integral in (43) is in fact over ε (p 1 ) ε (p 2 ) =: ε (new notation) By the assumption on a 0,, a 3, it yields (44) y y ε 2 H 1 () C ε y ε (y ε y) dx, where C is computed from the coefficients and is independent of ε > 0 Applying the Cauchy-Schwarz and the Hölder inequality in (44), we infer 1 2 (45) y ε y H 1 () C y ε 2 dx ε C y ε S dx 1 S µ( ε ) S 2 S C 1 µ( ε ) S 2 S ε In (45), S > 2 is given by the regularity and the boundedness in W 1,S () of the solution of (37), Meyers [14] The proof is finished with q = S 2 > S 0 15

16 We also indicate a direct argument in dimension one (which is of special interest here) of the regularity result for (37) (similarly for (15)) Lemma 45 If dim = 1, then y ε, the solution of (37), satisfies y ε W 1,p () for each 1 p <, and y ε L p () C f L 2 (), C > 0 independent of ε > 0 The estimate in Thm 44 is valid for each 0 < q < 1 Proof By the uniform ellipticity, the weak solution y ε H 1 () of (37) satisfies (46) y ε H 1 () C f L 2 () with C > 0 independent of ε > 0 (and in arbitrary dimension) We write (37) in the form (47) a ε (x) y ε v dx = [fv a 0 y ε v] dx, where a ε L (), m a ε (x) a > 0, is the whole coefficient appearing in the first term in (37) Take in (47) v n H 1 (), v n x w, for some α, converging α strongly in W 1,p () with w L p () arbitrarily fixed, 1 p + 1 = 1 The p right-hand side in (47) has a limit for n and the same is valid for the left-hand side Consequently, y ε defines a linear bounded functional on L p (): [ y εa ε (x)w dx = f f(x)µ() 1 p dx + x α x ] w a 0 y ε w dx α a 0 y ε (x)µ() 1 p dx w L p () 16

17 by the Hölder inequality By the known bounds for a ε ( ) and the boundedness of {y ε } in H 1 () as a weak solution according to (46), we get that y ε L p (), p < Corollary 46 For the minimal values of the problems (R) and (R ε ), we have the estimate with C > 0 independent of ε > 0 min(r) min(r ε ) Cµ( ε ) q Proof By the approximation property proved in Thm 44, we obtain (see (14)): min(r) min(r ε ) Cµ( ε ) q Conversely, if [p 1ε, p 2ε] is an optimal pair for (R ε ) and we introduce it in (15), then the estimate from Thm 44 remains valid for the difference between the two states Therefore, the converse inequality is valid and the proof is finished min(r ε ) min(r) Cµ( ε ) q Remark If we introduce [p 1ε, p 2ε] in (15) and denote the obtained solution by ỹ ε, then, by Thm 44, we get (48) ỹ ε y d L 2 () yε y d L 2 () ỹ ε yε H 1 () Cµ( ε ) q, with C > 0 independent of ε > 0 Combining (48) with Corollary 46, we see that the cost obtained in (R) from [p 1ε, p 2ε], denoted by R(p 1ε, p 2ε), satisfies min(r) R(p 1ε, p 2ε) ȵ( ε) q This shows that [p 1ε, p 2ε] is suboptimal in the original problem (R) The subregions of corresponding to H(p 1ε), H(p 2ε), and (1 H(p 1ε) H(p 2ε)) 17

18 produce a cost at distance at most ȵ( ε) q from the optimal value, where Č > 0 is independent of ε > 0 Remark In dimension 1, by Proposition 43, we obtain explicit estimates with respect to ε > 0 Remark In the case of the problem (11) - (13), we have the estimate (26) By multiplying (22), (23) by sgn(y ε ) (or a regularization sgn λ (y ε )), one may infer (49) (1 H ε (g ε)) y ε dx Cε with C > 0 independent of ε > 0 This is due to the monotonicity of sgn( ) However, both (26) or (49) seem too weak in order to extend the estimates of this section to the problem (11) - (13) 5 iscretization Based on the results from previous sections, we study the discretization of the problems (P ε ) and (R ε ) Notice that (P ε ) is an optimal control problem defined in with the control g acting in the coefficients of the lower order term In this section, we assume (51) g Ũad = {z W 1, () : z E 0, z R d 1} This is a variant of the conditions imposed on g in Section 2 and it is partially justified by Thm 22 In, we consider a uniformly regular finite element partition = T h T h T h, h > 0 We assume that the grid in, restricted to E, provides a finite element mesh in E as well 18

19 Let y ε H 2 () H 1 0() be the unique solution of (22), (23) If dim = d 3, then y ε C(), while (51) ensures Ũad C() as well An example when such properties hold is when is polyhedral and convex The discretization of (P ε ) is the following (Pε h ) min j(x, y ε,h (x)) dx, g h Ũ ad h E (52) y ε,h v h dx + 1 (1 H ε (g h ))y ε,h v h dx = ε v h V h H 1 0(), f h v h dx, (53) g h Ũ ad h = { z h Ṽh H 1 () : z h E 0, z h R d 1 } Here V h, Ṽ h are the finite element spaces in constructed with piecewise linear continuous functions (with 0 trace on for elements of V h ) and f h is some interpolant of f L 2 (), f h f in L 2 () We denote simply by [y h, g h ] V h Ũ ad h (ε is fixed here) an optimal pair for (P ε h ) Its existence follows by the convexity and compactness properties of Ũ ad h and the continuity properties of the mapping g h y ε,h defined in (52) Theorem 51 Assume that (51) is included in the definition of (P ε ) Then, [y h, g h ] [y ε, g ε ] in H0() C(), 1 on a subsequence for h 0, where [y ε, g ε ] is an optimal pair of (P ε ) Proof On a subsequence, we may assume g h g ε Ũad, strongly in C() Fix v h = y h in (52), corresponding to g h We get {y h } bounded in H0(), 1 where we also use f h f in L 2 () On a subsequence, y h y ε weakly in H0() 1 For any v H 2 () H0(), 1 one can find v h V h, with v h v strongly in H0() 1 Then, it is 19

20 possible to pass to the limit h 0 in the discretized state system (52) and to infer (54) y ε v dx + 1 (1 H ε (g ε ))y ε v dx = fv dx, v H 1 ε 0() In (54), we also use the density of H 2 () in H 1 () We obtain as well: (55) j(x, y ε (x)) dx = lim j(x, y h (x)) dx h 0 E by the assumptions on j(, ) (see Thm 22) For any g Ũad, there is g h Ũ h ad such that g h g in C() If y ε,h denotes the solution of (52) corresponding to g h, a similar argument shows that y ε,h ỹ ε weakly in H0() 1 and [ỹ ε, g] satisfy (54) As in (55), we get (56) j(x, ỹ ε (x)) dx = lim j(x, y ε,h (x)) dx h 0 E Taking into account (55), (56) and the optimality of [y h, g h ], we see that [y ε, g ε ] is an optimal pair for (P ε ) and the proof is finished Remark It is possible to prove a similar result for U ad as in Theorem 22 The advantage, here, is that Ũad, Ũ ad h are convex as well Theorem 52 The mapping θ ε,h : Ṽ h V h, given by g h y ε,h, is Gâteaux differentiable and z h = θ ε,h (g h )w h satisfies (57) z h v h dx + 1 (1 H ε (g h ))z h v h dx = 1 ε ε E E H ε(g h )y ε,h w h v h dx, for any v h V h and with w h Ṽh 20

21 Proof Let y λ ε,h = θ ε,h(g h + λw h ), λ R Subtract the equations corresponding to yε,h λ, y ε,h and divide by λ 0: [ yλ ε,h y ε,h v h + 1 λ ε (1 H ε(g h + λw h )) yλ ε,h y ] ε,h (58) v h dx λ = 1 H ε (g h + λw h ) H ε (g h ) y ε,h v h dx, v h V h ε λ Since H ε ( ) is in C 1 () and g h + λw h g h in C() for λ 0, we have H ε (g h + λw h ) H ε (g h ) λ H ε(g h )w h uniformly in Taking v h = yλ ε,h y ε,h in (58), we see that it is bounded in λ H0() 1 and in V h with respect to λ 0 (ε and h are fixed here) enoting by z h its weak limit in H 1 0() and in V h on a subsequence, we obtain (57), for λ 0 The limit is in fact in the strong topology since V h is finite dimensional As the solution of (57) is unique and z h depends linearly and continuously on w h, the proof is finished since the convergence is in fact valid without taking subsequences Under the assumption that j y (x, y h ) exists and belongs to L 2 (E) for each y h V h, we define the so-called adjoint system for the adjoint state p h V h : [ p h v h + 1ε ] (1 H ε(g h ))p h v h dx (59) = E j y (x, y ε,h )v h dx, v h V h Corollary 53 The directional derivative of the cost in (P h ε ), at g h Ṽh and in the direction w h Ṽh, is given by 1 H ε ε(g h )y ε,h w h p h dx, 21

22 where p h V h is given by (59) Proof We compute the limit 1 lim j(x, y λ λ 0 λ ε,h) dx j(x, y ε,h ) dx E E = [ p h z h + 1ε ] (1 H ε(g h ))p h z h dx = 1 ε H ε(g h )y ε,h w h p h dx by using (59), (57), under the notations of Theorem 52 Remark By Corollary 53, we notice the descent directions w h = 1 ε H ε(g h )y ε,h p h and ŵ h = y ε,h p h (since the coefficient is positive) Notice that the support of H ε(g h ) is contained in E ε (g h ) (see (42)) By (16), this is a neighborhood of the boundary of Ω gh Therefore, roughly speaking, w h corresponds to boundary variations if gradient methods are applied in (P ) The descent direction ŵ h is more general and allows simultaneous boundary and topological variations, which is useful in applications Remark For cost functionals of the type j(x, y(x)) dx and their approxi- mation (25) in, one has to put Ω H ε (g)j y (x, y ε,h )v h dx in the right-hand side of (59) For the problem (R ε ), due to (31) - (33) defining U ad, it is necessary to use higher order finite elements for the discretization of [p 1, p 2 ] U ad We define Uad h by the conditions: (510) p h i (x) + p h i (x) R d ν > 0 ae, i = 1, 2, (511) p h i H 2 (K) M i K, i = 1, 2, for each K compactly embedded open subset, 22

23 (512) p h i (x) 0 p h j (x) 0 This is a slight strengthening of (31) - (33) and Uad h H2 loc ()2 For the approximation of the solution of the equation (37), piecewise linear finite elements are used, denoted by V h H 1 () The discretized optimization problem is (Rε h ) (513) min y h y d 2 dx, [p h 1,ph 2 ] U ad h E (514) { [a1 H ε (p h 1) + a 2 H ε (p h 2) + a 3 (1 H ε (p h 1) H ε (p h 2)) ] y h v h + a 0 y h v h fv h } dx = 0, v h V h For simplicity, no discretization of a i, i = 0, 3, f and y d is performed The finite element mesh in, E is as in (P h ε ) Theorem 54 The problem (R h ε ) has at least one optimal triple denoted [yh, p 1h, p 2h ] V h Uad h On a subsequence h 0, we have y h y ε p 1h p 1ε p 2h p 2ε weakly in H 1 (), in H 1 loc(), in H 1 loc(), where [y h, p 1h, p 2h ] is an optimal triple for (R ε), under assumption θ i k = 1, K compactly embedded, i = 1, 2, in (32) Sketch of Proof The existence of [yh, p 1h, p 2h ], for h > 0 fixed, follows as in 3, using the boundedness of U h ad in H2 loc ()2 Moreover p ih p iε, i = 1, 2, on a subsequence, in H 1 loc () by the Sobolev theorem We get [p 1ε, p 2ε] U ad (where θk i = 1, i = 1, 2 in (32)) 23

24 We get {yh } bounded in H1 () by fixing v h = yh in (514) written for [p 1h, p 2h ] On a subsequence, y h y ε weakly in H 1 () It is possible to pass to the limit in (513), (514) as in Section 3, to prove the optimality of [yε, p 1ε, p 2ε] for (R ε ) Remark It is also possible to take simultaneously ε 0, h 0 to show the approximation of (R) In the one dimensional case, for two materials, rates of convergence are established in [4] Remark Concerning condition (31), in dimension one, it is to be noticed that it is automatically satisfied by trigonometric functions, which may provide another advantageous approximation 6 Numerical Experiments The following numerical experiments were conducted in a setting that combines aspects of (11) (13) with aspects of (14), (15), namely 1 min y y d 2 dx + σ (61) y y d 2 dx, Ω, χ 2 E 2 E [ a1 χ + a 2 (1 χ) ] [ y v dx + b1 χ + b 2 (1 χ) ] y v dx (62) (63) Ω = f v dx, v H0(Ω), 1 Ω y ξ H 1 0(Ω), Ω where E O Ω R 2 ; E, being given and fixed as before; Ω, O to be optimized with χ denoting the characteristic function of O; σ 0, a 1, a 2, b 1, b 2 > 0, f L 2 (), ξ H 1 () all given As explained in [15], (61) (63) are motivated by the oil industry application studied in [29] The 24

25 regularized fixed-domain version of (62), (63) in discretized form is [ a1 H ε (p h ) + a 2 (1 H ε (p h )) ] y ε,h v h dx (64) (65) [ + b1 H ε (p h ) + b 2 (1 H ε (p h )) ] y ε,h v h dx + 1 ( 1 Hε (g h ) ) y ε,h v h dx ε = f h v h dx, v h V h H0(), 1 y ε,h ξ h V h H 1 0(), yielding the equation for the discretized state y ε,h Ṽh H 1 (), where f h Ṽh, f h f in L 2 (), and ξ h Ṽh, ξ h ξ in H 1 () are given; g h, p h Ṽh are discretized shape functions corresponding to discretizations of Ω and O, respectively; and V h, Ṽh are as is in the first part of Section 5, ie finite element spaces in constructed with piecewise linear continuous functions (with 0 trace on for elements of V h ) In particular, depending on ξ, (65) can mean homogeneous or nonhomogeneous irichlet conditions The shape optimization for (64), (65) takes the form min j(g h, p h ) := 1 y ε,h y d,h 2 dx (g h,p h ) Uad h 2 E (66) + σ y ε,h y d,h 2 dx, 2 (67) E U h ad := { [g, p] Ṽh Ṽh : g p on and p 0 on E }, with given y d,h y d in H 1 () Then one obtains the discretized adjoint equation for the discretized ad- 25

26 joint state q ε,h V h H 1 0() (cf [15, (34)]): (68) [ a1 H ε (p h ) + a 2 (1 H ε (p h )) ] q ε,h v h dx [ + b1 H ε (p h ) + b 2 (1 H ε (p h )) ] q ε,h v h dx + 1 ( 1 Hε (g h ) ) q ε,h v h dx ε = (y ε,h y d,h )v h dx + σ ( y ε,h y d,h ) v h dx, v h V h, E E and the descent directions (cf [15, Rem 9] and the first part of Section 5 above) (69a) (69b) w d (y ε,h, q ε,h ) := y ε,h q ε,h, v d (y ε,h, q ε,h ) := (a 1 a 2 ) y ε,h q ε,h + (b 1 b 2 ) y ε,h q ε,h Based on the above, we formulate the algorithm employed to obtain the numerical results presented below The algorithm is of gradient with projection type Algorithm 61 Step 1: Set n := 0 and choose admissible initial shape functions [g h,0, p h,0 ] U h ad Step 2: Compute the solution to the state equation y n := θ ε,h (g h,n, p h,n ), where θ ε,h : Ṽ h Ṽh Ṽh denotes the control-to-state operator corresponding to (64), (65) Step 3: Compute the solution to the corresponding adjoint equation q n := θ ε,h (y n ), where θ ε,h : Ṽ h V h, y ε,h q ε,h, denotes the solution operator corresponding to (68) Step 4: Compute descent directions w d,n = w d,n (y n, q n ), v d,n = v d,n (y n, q n ) according to (69) 26

27 Step 5: Set g n := g n + λ n w d,n and p n := p n + λ n v d,n, where λ n 0 is determined via line search, ie as a solution to the minimization problem (610) min λ 0 j(g n + λ w d,n, p n + λ v d,n ) Step 6: Set (g n+1, p n+1 ) := π h ( g n, p n ), where π h denotes the projection (611) π h : Ṽ h Ṽh U h ad, which is obtained by first setting g n+1 (x h i ) := max{0, g n (x h i )} and p n+1 (x h i ) := max{0, p n (x h i )} for each node x h i such that x h i E, and second setting of the triangulation T h p n+1 (x h i ) := min{ p n+1 (x h i ), g n+1 (x h i )} for every node x h i of the triangulation T h Step 7: RETURN (g fin, p fin ) := (g n+1, p n+1 ) if the change of g, p and/or the change of j(g, p) are below some prescribed tolerance parameter Otherwise: Increment n, ie n := n + 1 and GO TO Step 2 For all the numerical examples discussed subsequently, we stopped the iteration and returned (g fin, p fin ) := (g n+1, p n+1 ) if j(g n, p n ) j(g n+1, p n+1 ) < 10 8 AN g n g n+1 2 < 10 3 AN p n p n+1 2 < 10 3, where j(g n, p n ) j(g n+1, p n+1 ) / j(g n+1, p n+1 ) is used instead of j(g n, p n ) j(g n+1, p n+1 ) if j(g n+1, p n+1 ) > 1 and analogous for g n and p n The state equations as well as the adjoint equations that need to be solved numerically during the above algorithm are discretized linear elliptic PE with irichlet boundary conditions The numerical solution is obtained via 27

28 a finite volume scheme [19, Sect 4], [7, Chap III] More precisely, the software WIAS-HiTNIHS 1, originally designed for the solution of more general PE occurring when modeling conductive-radiative heat transfer and electromagnetic heating [9, 12], has been adapted for use in the present context WIAS-HiTNIHS is based on the program package pdelib [8], it employs the grid generator Triangle [24, 25] to produce constrained elaunay triangulations of the domains, and it uses the sparse matrix solver GSPAR [11] to solve the linear system arising from the finite volume scheme The numerical scheme yields discrete y n and q n (cf Steps 2 and 3 of the above algorithm), defined at each vertex of the triangular discrete grid, interpolated piecewise affine, ie affinely to each triangle of the discrete grid In consequence, the shape functions g n and p n are piecewise affine as well Where integrals of these piecewise affine functions need to be computed (eg in Step 7 of the algorithm), they are computed exactly A golden section search [22, Sect 102] is used to numerically carry out the minimization (610) Note that the minimization (610) is typically nonconvex and the golden section search will, in general, only provide a local min λ n For the regularized Heaviside mapping, we use 1 for r 0, ε(r + ε) (612) H ε (r) := 2 2r(r + ε) 2 for ε < r < 0, ε 3 0 for r ε Remark For some of the following examples, the stated initial shape functions [g, p] are merely piecewise continuous (cf the Introduction and [20]) and, thus, not in U h ad However, the stated [g, p] are only used to determine the values g h (x h i ), p h (x h i ), at the nodes x h i 1 High Temperature Numerical Induction Heating Simulator; pronunciation: hitnice of the triangulation T h, and 28

29 [g h, p h ] U h ad for the resulting affinely interpolated [g h, p h ] Example 62 The numerical computations of the present example employ the circular fixed domain (613) := { (x 1, x 2 ) : x x 2 2 < 1 } R 2 with fixed subdomain { E := (x 1, x 2 ) : x 1 > 3 4, x 2 < 1 } (614) 2 (note that E has two connected components) We use a fixed triangular grid provided by Triangle [24, 25], consisting of triangles The sets determined by the shape functions g, p are (615a) (615b) Ω g := int{x : g(x) 0}, Ω p := int{x : p(x) 0} Parameter settings: (616a) (616b) (616c) (616d) ε := 10 7, a 1 := 1, a 2 := 10, b 1 := 1, b 2 := 10, f(x 1, x 2 ) := 5, ξ(x 1, x 2 ) := 2 The cost functional j as in (66) depends on the given function y d,h For the first set of numerical results, we precompute y d,h := y ε,h numerically as the solution to the state equation (64), (65), using { 1 in E, (617) g(x 1, x 2 ) := 1, p(x 1, x 2 ) := 1 in \ E 29

30 y g p Figure 1: Precomputed y d,h used in Examples 62(a) 62(c) (left, isolevels spaced at 005), obtained as the solution to the state equation (64), (65); depicted together with the corresponding Ω g (middle, g 0 in gray) and Ω p (right, p 0 in gray) more precisely, the color of each triangle of the discretization is determined by the average of g (resp p) on that triangle The computed y d,h together with the corresponding Ω g and Ω p is depicted in Fig 1 Using the precomputed y d,h has the advantage that we actually know y d,h together with g, p as in (617) provides an absolute minimum in the following Examples (a) (c) (a) Setting σ := 1, the cost functional j is (618) j(g, p) := 1 y ε,h y d,h 2 dx y ε,h y d,h 2 dx E E with the precomputed y d,h from above Since j 0 and j(g, p) = 0 for g, p as in (617), these functions are optimal for j The initial shape functions are (619a) (619b) g 0 (x 1, x 2 ) := 1, p 0 (x 1, x 2 ) := 1 Results are depicted in Fig 2 In this example, we observe rapid convergence to the known minimum within two line searches, where the cost is reduced from the initial value of 118 to virtually 0 The employed 30

31 method is designed to facilitate topological changes of the shapes during the optimization process, and, in Fig 1, we see that such changes, indeed, occur (as they must, in this case, for the minimum to be found) Also note that the intermediate shapes shown in Fig 1 correspond to shape functions [g, p] / Uad h Such inadmissible shape functions are to be expected during line searches, where Step 6 of Algorithm 61 projects back into Uad h after each line search In a final note on Fig 1, we observe that the radial symmetry of the initial condition is broken by j (both E and y d,h are not radially symmetric), only retaining the symmetries with respect to the x 1 -axis and the x 2 -axis The rapid convergence to the precomputed optimum observed in the present situation is actually not typical, as the problem is strongly nonconvex, and the algorithm will typically converge to some different local minimum determined by the initial condition (cf Examples 62(b),(c) below) (b) Setting σ := 0, the cost functional j is (620) j(g, p) := 1 y ε,h y d,h 2 dx 2 E with the precomputed y d,h from above The initial shape functions are the same as in (619) Results are depicted in Fig 3 As compared to Example 62(a), the convergence is slower, the convergence criteria of Algorithm 61, Step 7, only being satisfied after the fifth line search ue to the absence of the gradient term in the cost functional, the initial cost is already much lower than in (a) The final state corresponds to a local min close, but not identical, to the precomputed absolute min 31

32 y y y g p g - p g - p Figure 2: Shape optimization of Example 62(a) Left: State isolevels spaced at 005 Middle: Shapes Ωg determined by g 0 (in gray) Right: Shapes Ωp determined by p 0 (in gray) Row 1: initial state, shapes, j(g0, p0 ) = 118 Row 2: intermediate state, shapes during line search #1, cost = 0102 Row 3: final state, shapes (after 2 line searches, j(gfin, pfin ) = , precomputed optimal state of Fig 1 has been recovered) (c) Returning to σ := 1 and the cost functional j as in (618), we now vary the initial condition, using (621a) (621b) ( 1 if x2 09, g0 (x1, x2 ) := 1 otherwise, p0 (x1, x2 ) := g0 (x1, x2 ) Results are depicted in Fig 4 Even though the initial shapes do not seem far from the ones used in Example 62(a), the present situation is 32

33 y y y g p g - p g - p Figure 3: Shape optimization of Example 62(b) Left: State isolevels spaced at 005 Middle: Shapes Ωg determined by g 0 (in gray) Right: Shapes Ωp determined by p 0 (in gray) Row 1: initial state, shapes, j(g0, p0 ) = Row 2: intermediate state, shapes during line search #4, cost = 115 Row 3: final state, shapes (after 5 line searches, j(gfin, pfin ) = 10 4, cost is reduced significantly, but precomputed optimal state of Fig 1 is not quite recovered) quite different: Convergence is reached only after 10 line searches and, thus, significantly slower than in Example 62(a) And even though the cost is reduced from 0861 to 0188 from initial to final state, the precomputed absolute min is not attained While Ωg does not change noticeably during the computation, the area of the difference between the final shape Ωpfin and the optimal shape is quite small (however, Ωpfin has more than the two connected components of the optimal shape) Comparing the isolevel pictures of the final states in Fig 2 and Fig 4, one observes 33

34 y y - - g p g - p Figure 4: Shape optimization of Example 62(c) Left: State isolevels spaced at 01 Middle: Shapes Ωg determined by g 0 (in gray) Right: Shapes Ωp determined by p 0 (in gray) Row 1: initial state, shapes, j(g0, p0 ) = 0861 Row 2: intermediate state, shapes after line search #1, j(g1, p1 ) = 0415 Row 3: final state, shapes (after 10 line searches, j(gfin, pfin ) = 0188, precomputed optimal state of Fig 1 has not been recovered) that, in E, where the cost functional is active, the pictures almost agree, whereas, outside of E, they are completely different Regarding symmetries, in contrast to previous examples, the x1 -axis symmetry is broken by the initial condition, and only the x2 -axis symmetry is retained (d) This example uses a different yd and a different right-hand side f More precisely, the parameter settings are still as in (616), except for (622) f (x1, x2 ) := 20(x21 + x22 ) + 20, 34

35 and the cost functional is as in (618), but now with (623) y d (x 1, x 2 ) := 2 Here, we are no longer in the situation of a known precomputed optimum As in (619), the initial shape functions are constantly 1 on Results are depicted in Fig 5 In this case, most of the cost reduction and shape change occurs already during the first line search, with convergence being reached after 4 lines searches While changes, including topological changes, occur in Ω p, the set Ω g remains virtually constant Example 63 For the following numerical result, the fixed domain is still the unit disk as in (613) However, the fixed subdomain E is now at the bottom of, (624) E := {(x 1, x 2 ) : x 2 < 07} The numerical computations employ a fixed triangular grid provided by Triangle [24, 25], consisting of triangles The parameter settings are as in (616), except for (625) f(x 1, x 2 ) := 10(x x 2 2) + 5 The cost functional is as in Example 62(d), ie (618) with y d (x 1, x 2 ) := 2 The initial shape functions are { 1 if x x , (626a) g 0 (x 1, x 2 ) := 1 otherwise, { 1 if (x 1, x 2 ) E, (626b) p 0 (x 1, x 2 ) := 1 otherwise 35

36 y y - - g p g - p Figure 5: Shape optimization of Example 62(d) Left: State isolevels spaced at 01 Middle: Shapes Ωg determined by g 0 (in gray) Right: Shapes Ωp determined by p 0 (in gray) Row 1: initial state, shapes, j(g0, p0 ) = 297 Row 2: intermediate state, shapes after line search #1, j(g1, p1 ) = 0265 Row 3: final state, shapes (after 4 line searches, j(gfin, pfin ) = 0262) Results are depicted in Fig 6 While, overall, one can observe similarities with the last case of Example 62, one now observes that the cost achieved during the first line search is actually lower than the final cost Of course, h this can occur only if [g, p] / Uad during the line search Another difference to previous examples lies in the x1 -axis symmetry now being broken by the shape of E Acknowledgement The second author acknowledges with thanks 36

37 y y y g p g - p g - p Figure 6: Shape optimization of Example 63 Left: State isolevels spaced at 01 Middle: Shapes Ωg determined by g 0 (in gray) Right: Shapes Ωp determined by p 0 (in gray) Row 1: initial state, shapes, j(g0, p0 ) = 275 Row 2: intermediate state, shapes during line search #1, cost = Row 3: final state, shapes (after 3 line searches, j(gfin, pfin ) = 0100) very useful discussions with Prof Enrique Zuazua on the analysis of such problems, and the financial support of Grant 145/2011, CNCS Romania References [1] Allaire, G, Shape optimization by the Homogenization method, New York, Springer-Verlag (2001) 37

38 [2] Bendsoe, M, Optimization of structural topology, shape, and material, Berlin, Springer-Verlag (1995) [3] Brezis, H, Analyse Fonctionelle Théorie et applications, Paris, Masson (1983) [4] Casado-iaz, J, Castro, C, Luna-Laynez, M, Zuazua, E, Numerical approximation of a one-dimensional elliptic optimal design problem, Multiscale Model Simul 9 (3) (2011), pp [5] Chenais,, Zuazua, E, Finite-element approximation of 2 elliptic optimal design, J Math Pures Appl 85 (9) (2006), pp [6] elfour, MC, Zolesio, J-P, Shapes and geometries Analysis, differential calculus and optimization, SIAM, Philadelphia, PA (2001) [7] Eymard, R, Gallouët, T, Herbin, R: Finite Volume Methods, pp in Ciarlet, P, Lions, J (eds): Solution of Equations in R n (Part 3); Techniques of Scientific Computing (Part 3), Handbook of Numerical Analysis, Vol VII North-Holland/ Elsevier, Amsterdam, The Netherlands (2000) [8] Fuhrmann, J, Koprucki, T, Langmach, H: pdelib: An open modular tool box for the numerical solution of partial differential equations esign patterns In: Hackbusch, W, Wittum, G (eds): Proceedings of the 14th GAMM Seminar on Concepts of Numerical Software, Kiel, January 23 25, 1998 University of Kiel, Kiel, Germany (2001) [9] Geiser, J, Klein, O, Philip, P: Numerical simulation of temperature fields during the sublimation growth of SiC single crystals, using WIAS- HiTNIHS J Crystal Growth 303 (2007), pp

39 [10] Grisvard, P, Elliptic problems in nonsmooth domains, Pitman, London (1985) [11] Grund, F, irect linear solvers for vector and parallel computers, pp in Vector and Parallel Processing VECPAR 98, Lecture Notes in Computer Science 1573 (1999) [12] Klein, O, Lechner, C, ruet, P-É, Philip, P, Sprekels, J, Frank- Rotsch, C, Kießling, F-M, Miller, W, Rehse, U, Rudolph, P: Numerical simulations of the influence of a traveling magnetic field, generated by an internal heater-magnet module, on liquid encapsulated Czochralski crystal growth Magnetohydrodynamics 45 (2009), pp Special Issue: Selected papers of the International Scientific Colloquium Modelling for Electromagnetic Processing, Hannover, Germany, October 27-29, 2008 [13] Mäkinen, R, Neittaanmäki, P, Tiba,, On a fixed domain approach for a shape optimization problem, pp in Computational and applied mathematics II, ifferential equations (Ames, WF, van der Houwen, PJ, eds), Amsterdam, North-Holland (1992) [14] Meyers, NG, An L p estimate for the gradient of solutions of second order elliptic divergence equations, Ann Sc Norm Sup Pisa 17 (3) (1963), pp [15] Neittaanmäki, P, Pennanen, A, Tiba,, Fixed domain approaches in shape optimization problems with irichlet boundary conditions, Inverse Problems 25 (2009), pp 1-18 [16] Neittaanmäki, P, Sprekels, J, Tiba,, Optimization of elliptic systems Theory and applications, New York, Springer-Verlag (2006) 39

40 [17] Neittaanmäki, Tiba,, Fixed domain approaches in shape optimization problems, accepted for publication in Inverse Problems In press [18] Osher, S, Sethian, JA, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J Comput Phys 79 (1) (1988), pp [19] Philip, P: Analysis, optimal control, and simulation of conductiveradiative heat transfer Mathematics and its Applications / Annals of AOSR 2 (2010), pp [20] Philip, P, Tiba,, Shape optimization via control of a shape function on a fixed domain: Theory and numerical results, pp in Numerical Methods for ifferential Equations, Optimization, and Technological Problems, Computational Methods in Applied Sciences 27, Springer (2013) (in press) [21] Pironneau, O, Optimal shape design for elliptic systems, Berlin, Springer (1984) [22] Press, W, Teukolsky, S, Vetterling, W, Flannery, B: Numerical Recipes The Art of Scientific Computing, 3rd edn Cambridge University Press, New York, USA (2007) [23] Sethian, JA, Level set methods, Cambridge MA, Cambridge Univ Press (1996) [24] Shewchuk, J: Triangle: Engineering a 2 quality mesh generator and elaunay triangulator, pp in Lin, MC, Manocha, (eds) Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science 1148 (1996) 40

41 [25] Shewchuk, J: elaunay refinement algorithms for triangular mesh generation Computational Geometry: Theory and Applications 22 (2002), pp [26] Sokolowsky, J, Zolesio, J-P, Introduction to shape optimization Shape sensitivity analysis, Berlin, Springer-Verlag (1992) [27] Sverak, V, On optimal shape design, J Math Pures Appl 72 (9) (1993), pp [28] Tiba,, Finite element approximation for shape optimization problems with Neumann and mixed boundary conditions, SIAM J Control Optim 49 (3) (2011), pp [29] Woo, H, Kim, S, Seol, JK, Lionheart, W, Woo, EJ, A direct tracking method for a grounded conductor inside a pipeline from capacitance measurements, Inverse Problems 22 (2006), pp