Shape optimization for interior Neumann and transmission eigenvalues
CChapter 1
Shape optimization for interior Neumann andtransmission eigenvalues
A. Kleefeld
Abstract
Shape optimization problems for interior eigenvalues is a very challeng-ing task since already the computation of interior eigenvalues for a given shapeis far from trivial. For example, a concrete maximizer with respect to shapes offixed area is theoretically established only for the first two non-trivial Neumanneigenvalues. The existence of such a maximizer for higher Neumann eigenvaluesis still unknown. Hence, the problem should be addressed numerically. Better nu-merical results are achieved for the maximization of some Neumann eigenvaluesusing boundary integral equations for a simplified parametrization of the boundaryin combination with a non-linear eigenvalue solver. Shape optimization for interiortransmission eigenvalues is even more complicated since the corresponding trans-mission problem is non-self-adjoint and non-elliptic. For the first time numericalresults are presented for the minimization of interior transmission eigenvalues forwhich no single theoretical result is yet available.
The task is to optimize the shape of a domain Ω ⊂ R with respect to the k -theigenvalue under the constraint that the area | Ω | of the domain is constant, say A .Here, the domain is an open and bounded set with smooth boundary ∂ Ω which isalso allowed to be disconnected. In the sequel, we consider two different problems.First, we deal with the maximization of interior Neumann eigenvalues (INEs).Precisely, one has to find numbers λ > ∆ u + λ u = Ω , ∂ ν u = ∂ Ω A. KleefeldForschungszentrum Jülich GmbH, Supercomputing Centre Jülich, 52425 Jülich, Germany,e-mail: [email protected] a r X i v : . [ m a t h . NA ] O c t A. Kleefeld is satisfied for non-trivial u , where ν denotes the normal pointing in the exterior.It is well-known that this problem is elliptic and the eigenvalues are discrete. Thecase λ = Fig. 1.1
Shape maximizer for the first six INEs obtained numerically. The recent optimal values λ k · A for k = ,..., .
66, 21 .
28, 32 .
90, 43 .
86, 55 .
17, 67 .
33 (see [AnOu17]).
The optimal values λ k · A for k = , . . . , .
66, 21 .
28, 32 .
79, 43 .
43, 54 . .
04 (see [AnFr12]) which have been improved recently to 10 .
66, 21 .
28, 32 . .
86, 55 .
17, 67 .
33 (see [AnOu17]). This paper reports improved values for thethird and fourth INE and at the same time the boundary of the shape maximizer isdescribed explicitly in terms of two parameters.The second problem under consideration is the interior transmission problem.Interior transmission eigenvalues (ITEs) are numbers λ ∈ C \{ } such that ∆ w + λ nw = Ω , ∆ v + λ v = Ω , v = w on ∂ Ω , ∂ ν v = ∂ ν w on ∂ Ω , has a non-trivial solution ( v , w ) (cid:54) = ( , ) , where n is the given index of refrac-tion. However, this is a non-elliptic and non-self-adjoint problem appearing first Shape optimization 3 in 1986 (see [Ki86]). Existence and discreteness for real-valued λ has been shownin [CaGiHa10]. But, the existence is still open for complex-valued λ except for spe-cial geometries (see [SlSt16, CoLe17]). The computation of ITEs for a given shapeis therefore a very challenging task (see [KlPi18] for an excellent overview of exist-ing methods). It is also noteworthy that neither theoretical nor numerical results areavailable for a shape optimizer of the first two ITEs. Within this paper we give nu-merical evidence for a shape minimizer of the first two ITEs and stating a conjecturewhich researcher in this field might want to prove in the future. Contribution of the paper
The contribution of this paper is twofold. First, improved numerical results for themaximization of some interior Neumann eigenvalues are presented using a simpli-fied parametrization of the boundary. Second, the previous concept is transferredin order to obtain numerical results for the minimization of interior transmissioneigenvalues for the first time for which no single theoretical result is yet available.
Outline of the paper
The paper is organized as follows: In Section 1.2, it is explained in detail how tocompute interior Neumann eigenvalues using a boundary integral equation followedby its discretization. Then, it is described how the resulting non-linear eigenvalueproblem is solved numerically. Further, the new parametrization is introduced andused to obtain improved numerical results for the maximization of some interiorNeumann eigenvalues. In Section 1.3, the concept of the previous section is appliedfor the minimization of interior transmission eigenvalues for which neither numeri-cal results nor theoretical results are yet available. Finally, a short summary and anoutlook is given in Section 1.4.
Recall that interior Neumann eigenvalues (INEs) are numbers λ = κ such that ∆ u + κ u = Ω , ∂ ν u = ∂ Ω is satisfied. Note that this problem is elliptic and it is well-known that the eigenval-ues are discrete and positive real-valued numbers. In the sequel, we ignore κ = Ω , we use a boundary integral equation approach. A single layer ansatz with un- A. Kleefeld known density ψ given by u ( X ) = (cid:90) ∂Ω Φ κ ( X , y ) ψ ( y ) d s ( y ) , X ∈ Ω is used, where Φ κ ( X , y ) = i H ( ) ( κ (cid:107) X − y (cid:107) ) / Ω (cid:51) X → x ∈ ∂ Ω , and using thejump condition yields the following boundary integral equation of the second kind12 ψ ( x ) + (cid:90) ∂Ω ∂ ν ( x ) Φ κ ( x , y ) ψ ( y ) d s ( y ) (cid:124) (cid:123)(cid:122) (cid:125) K ( κ ) = . (1.1)Note that the operator K ( κ ) : H − / ( ∂ Ω ) → H − / ( ∂ Ω ) is compact assuming aregular boundary (see [Mc00]). Hence, Z ( κ ) = I / + K ( κ ) is Fredholm of indexzero for κ ∈ C \ R ≤ and thus the theory of eigenvalue problems for holomorphicFredholm operator-valued functions applies to Z ( κ ) .The integral equation (1.1) is discretized via the boundary element collocationmethod. Precisely, we subdivide the boundary into n / (cid:103) ∂ Ω ), and defineon each piece a quadratic interpolation for ψ . This leads to (cid:18) I + M ( κ ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) Z ( κ ) ∈ C n × n (cid:126) ψ = (cid:126) , where the matrix entries of M are numerically calculated with the Gauss-Kronradquadrature (see [KlLi12] for details in the three-dimenensional case). The resultingnon-linear eigenvalue problem of the form Z ( κ ) (cid:126) ψ = (cid:126) κ including their multiplicities within any contour C ⊂ C which is based on Keldysh’stheorem. Precisely, one integrates the resolvent over the given contour whereas theintegral is approximated with the trapezoidal rule (see [Be12] for more details).Hence, we are now able to compute highly accurate INEs for a given shape Ω .Next, it is explained how to choose a parametrization for the boundary of Ω . Theidea is to use an implicit curve rather than an explicit representation of the curve.Equipotentials are implicit curves of the form m ∑ i = (cid:107) x − P i (cid:107) = c , (1.2) Shape optimization 5 where the parameter c and the centers P i are given. Here, (cid:107)· (cid:107) denotes the Euclideannorm. Precisely, all points x ∈ R satisfying (1.2) for given points P i , i = , . . . , m and parameter c describe the implicit curve. Example 1.
We choose three points ( −√ / , / ) , ( √ / , / ) , ( , − ) for m = ( − / , ) , ( / , ) , ( , −√ / ) , ( , √ / ) for m =
4. The edge length of thefollowing geometric shapes as shown in Figure 1.2 is √ Fig. 1.2
The choice of the points for m = ( −√ / , / ) , ( √ / , / ) , ( , − ) and for m = ( − / , ) , ( / , ) , ( , −√ / ) , ( , √ / ) shown as a red dot. The origin is shown as a blackdot. Next, we show the influence of the parameter c . As one can see in Figure 1.3 thelarger the parameter c gets, the more constricting the boundary gets. Additionally,one can see that we are almost able to obtain a possible shape of the maximizerfor the third and fourth INE. To add more flexibility, we introduce the additionalparameter α . The modified equipotentials are given in the form m ∑ i = (cid:107) x − P i (cid:107) α = c (1.3)We introduce the two in front of the parameter α in order to avoid the computationof the square root in the norm definition. In Figure 1.4 we show the influence of theparameter α fixing c =
2. As one can see, we have enough flexibility to obtain verygood approximations for a possible shape maximizer for the third and fourth INE.Thus, we have seen the influence of the parameters α and c . We shortly explain howto generate n points on the boundary for the given parameters α and c . This is doneas follows. First, the equation (1.3) is rewritten in polar coordinates. Then, n + φ in the interval [ , π ] are generated. Next, for each angle φ i theimplicit equation is solved for the unique r i via a root finding algorithm. Finally,the points given in polar coordinates ( r i , φ i ) , i = , . . . , n + ( x i , y i ) = ( r i cos ( φ i ) , r i sin ( φ i )) , i = , . . . , n +
1. Hence,we obtain n different points on the boundary of the scatterer (the ( n + ) -th point isthe same as the first point by construction). Those n points can now be used in theboundary element collocation method. A. Kleefeld
Fig. 1.3
The influence of the parameter c = .
75, 2 .
00, 2 .
25, 2 .
50, 2 .
75, and 3 .
00 for m = m = In order to calculate the value λ k · A , we need to numerically approximate the areaenclosed by the given implicit curve (see (1.3)). That is, we have n points distributedon the boundary ∂ Ω . With these points and the approximation via quadratic inter-polation, the domain (cid:101) Ω with the boundary (cid:103) ∂ Ω is defined. To approximate the areaof this region, we compute the area of the non-self intersecting polygon spanned bychoosing p (cid:29) n points including an additional point (the first point is the additional ( p + ) -th point). The approximate area is given by A ≈ A (cid:101) Ω = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ∑ i = ( x i − x i + )( y i + y i + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) which is an easy consequence of the formula ([Zw12, 4.6.1, p. 206]) Shape optimization 7
Fig. 1.4
The influence of the parameter α = .
5, 1 .
0, 1 .
5, 2 .
0, 2 .
5, and 3 . c = m = m = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x y y (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) x x y y (cid:12)(cid:12)(cid:12)(cid:12) + . . . + (cid:12)(cid:12)(cid:12)(cid:12) x p x y p y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The exterior normals on the boundary given implicitly by (1.3) are given by ν = ˜ ν / (cid:107) ˜ ν (cid:107) with ˜ ν = − α m ∑ i = ( x − P i ) (cid:107) x − P i (cid:107) ( α + ) . Now, we have everything together in order to optimize with respect to the twoparameter c and α . First, we consider the third INE. The reference value given byAntunes & Oudet is given by 32 .
90 using 37 unknown coefficients. The third eigen-value has multiplicity three. If we fix α = /
2, then the optimization with respect to c yields the result c = . . . . A. Kleefeld and fifth, respectively. As we observe, the reported numbers are more accurate. Ifwe fix α =
2, then we obtain c = . . . . .
90. But remember that we haveonly one unknown describing the boundary. If we choose α = /
2, then we have c = . . . . α = . c = . . . . .
86 with multiplicity three using 33unknown coefficients. If we use α =
2, we obtain c = . . . . α = / c = . . . . α = c = . . . . α and c gives α = . c = . . . . n =
512 for all numerical calculation to ensure that we have atleast six digits accuracy for the values λ k · A . This is guaranteed since we almost havea convergence of order four due to the fact that we have approximated the boundaryand the unknown density function by quadratic interpolation (refer to [KlLi12] fora superconvergence proof for three-dimensional scattering objects). Recall that interior transmission eigenvalues (ITEs) are numbers λ = κ ∈ C \{ } such that ∆ w + κ nw = Ω , ∆ v + κ v = Ω , v = w on ∂ Ω , ∂ ν v = ∂ ν w on ∂ Ω , has a non-trivial solution ( v , w ) (cid:54) = ( , ) . Here, n is the given index of refraction.This is a non-elliptic and non-self-adjoint problem. Existence and discreteness forreal-valued κ has already been established. However, the existence is still open forcomplex-valued κ except for special geometries. To compute such ITEs for a givenshape is therefore very challenging. We use the same technique as presented beforefor the numerical calculation of interior Neumann eigenvalues; that is, reduce theproblem to a system of boundary integral equations, discretize it via a boundaryelement collocation method, and solve the resulting non-linear eigenvalue problemvia the method of Beyn (see [Be12]). For more details, we refer the reader to [Kl13,Kl15] where ITEs for three-dimensional domains are computed and to [KlPi18] Shape optimization 9 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2-1.5-1-0.500.511.52 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2-1.5-1-0.500.511.52-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2-1.5-1-0.500.511.52-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2-1.5-1-0.500.511.52 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2-1.5-1-0.500.511.52
Fig. 1.5
The three eigenfunction of the shape optimizer for the third and fourth INE. The param-eters are α = . c = . . α = . c = . . for a good introduction for other methods to compute such ITEs. Straightforwardlylooking at real-valued ITEs using the index of refraction n = λ = A · κ . The values λ for eight different domains are given in Fig. 1.6But recall that there might be complex-valued ITEs as well which are not takeninto account. If we consider | λ | instead of λ using the same eight domains, weobtain the results as presented in Fig. 1.7.As one can observe, it seems that the circle is minimizing | λ | . Hence, if weconsider | λ | ≤ | λ | ≤ | λ | ≤ · · · , then we make the conjecture that the first absolute . . . . . . . . Fig. 1.6
The values λ for eight different domains using n = . . . . . . . . Fig. 1.7
The values | λ | for eight different domains using n = ITE is minimal for a circle for the index of refraction n >
1. If this is true, then itis also true for 0 < n < κ ( / n ) = √ n κ ( n ) . Further, since λ iscomplex-valued, it comes in complex conjugate pairs. Hence, the second eigenvaluewill be minimized by a circle as well.Further investigation of shapes that minimize higher interior transmission eigen-values is a very interesting and challenging topic. In this paper, it is shown how to efficiently compute interior Neumann eigenvaluesfor a given domain in two dimensions. Additionally, the value of the shape maxi-mizer for the third and fourth interior Neumann eigenvalue has been improved from32 .
90 and 43 .
86 to 32 . . Shape optimization 11 the same time, the number of parameters describing the boundary of a possiblemaximizer has been reduced to two parameters using modified equipotentials. Theconjecture is that the third and fourth interior Neumann eigenvalue might be givenby such modified equipotentials. This work presents very recent numerical resultsand a further investigation has to be carried out in order to validate whether theshape maximizer for higher interior Neumann eigenvalues can be found with mod-ified equipotentials. This idea can easily be used for extending this approach to thethree-dimensional case.Moreover, for the first time numerical results are presented for the minimizationof interior transmission eigenvalues in two dimensions although already the numer-ical calculation of those for a given domain is a very challenging task since theproblem is neither elliptic nor self-adjoint and hence complex-valued interior trans-mission eigenvalues might exist. From the theoretical point of view, this fact is stillopen. Additionally, it is open whether there exist a unique minimizer for the firstand second interior transmission eigenvalue. Here, we show numerically and henceconjecture that the first and second interior transmission eigenvalue is minimized bya circle. It remains to prove this observation, but it cannot be carried out by standardspectral arguments like for the Dirichlet, Neumann, Robin, or Steklov eigenvalueproblem. Moreover, one can now try to investigate the three-dimenensional case.Above all, one could also investigate the electromagnetic and/or the elastic scat-tering case in two and three dimensions.
Acknowledgement
I would like to thank the IMSE’18 steering committee for giving me the opportunityto present my recent results for the maximization of interior Neumann and mini-mization of interior transmission eigenvalues on July 19th, 2018. Further, I wouldlike to thank Paul Harris for the organization of this nice event at the University ofBrighton, UK.
References
AnFr12. Antunes, P.R.S. and Freitas, P.: Numerical optimization of low eigenvalues of the Dirich-let and Neumann Laplacians.
J. Optim. Theory Appl. , , 235–257 (2012).AnOu17. Antunes, P.R.S. and Oudet, E.: Numerical results for extremal problem for eigenval-ues of the Laplacian. In Shape optimization and spectral theory , A. Henrot (ed.), De Gruyter,Warzow/Berlin, (2017), pp. 398–412.Be12. Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems.
Linear AlgebraAppl. , , 3839–3863 (2012).CaGiHa10. Cakoni, F., Gintides, D., and Haddar, H.: The existence of an infinite discrete set oftransmission eigenvalues. SIAM J. Math. Anal. , , 237–255 (2010).CoLe17. Colton, D. and Leung, Y.-J.: The existence of complex transmission eigenvalues forspherically stratified media. Appl. Anal. , , 39–47 (2017).2 A. KleefeldGiNaPo09. Girouard, A., Nadirashvili, N., and Polterovich, I.: Maximization of the second posi-tive Neumann eigenvalue for planar domains. J. Differ. Geom. , , 637–662 (2009).Ki86. Kirsch, A.: The denseness of the far field patterns for the transmission problem. IMA J.Appl. Math. , InverseProblems , , 104012 (2013).Kl15. Kleefeld, A.: Numerical methods for acoustic and electromagnetic scattering: Transmis-sion boundary-value problems, interior transmission eigenvalues, and the factorization method .Habilitation Thesis, Brandenburg University of Technology Cottbus-Senftenberg (2015).KlLi12. Kleefeld, A. and Lin, T.-C.:. Boundary element collocation method for solving the exte-rior Neumann problem for Helmholtz’s equation in three dimensions.
Electron. Trans. Numer.Anal. , , 113–143 (2012).KlPi18. Kleefeld, A. and Pieronek, L.: The method of fundamental solutions for computing acous-tic interior transmission eigenvalues. Inverse Problems , , 035007 (2018).Mc00. McLean, W.: Strongly elliptic systems and boundary integral operators . Cambridge Uni-versity Press, Cambridge (2000).SlSt16. Sleeman, B.D. and Stocks, D.C.: Interior transmission eigenvalues of a rectangle.
InverseProblems , , 025010 (2016).Sz54. Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. Arch. Ration.Mech. Anal. , , 343–356 (1954).We56. Weinberger, H.F.: An isoperimetric inequality for the N-dimensional free membrane prob-lem. Arch. Ration. Mech. Anal. , , 633–636 (1956).Zw12. Zwillinger, D.: Standard mathematical tables and formulae . CRC Press, Boca Raton(2012). ndexndex