Maria Bayard Dühring - Solid Mechanics

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where ˜pijkl are the rotated strain-optical constants obtained by the expression ˜pijkl = MijmnMklpqpmnpq. (16) For a given optical mode of order ν and with the propagation constant βν the effective refractive index neff,ν is defined as neff,ν = βν/k0, (17) where k0 is the free space propagation constant. It is assumed that the propagating optical modes have harmonic solutions on the form Hp,ν(x1, x2, x3) = Hp,ν(x1, x2)e −iβνx3 , (18) where Hp,ν is the magnetic field of the optical wave. The governing equation for the magnetic field is thus the time-harmonic wave equation ∂ ∂Hp eijk bklblmemnp − k ∂xj ∂xn 2 0Hp = 0, (19) where eijk here is the alternating symbol and biknkj = δij. For a given value of k0 the propagation constant βν for the possible modes are found by solving the wave equation as an eigenvalue problem, whereby the effective refractive indices can be obtained. In this work the set of equations are reduced such that the model is only solved for the transverse components of the magnetic field H1 and H2. As the energy of the lower order optical modes is concentrated in the waveguides the calculation domain, where the eigenvalue problem is solved, can be reduced to a smaller optical area around the waveguide as indicated by the dashed lines in Figure 1. At the boundary of such a domain it is assumed that the magnetic field is zero as the energy density quickly decays outside the waveguide, thus the boundary condition is eijkHjmk = 0, (20) where eijk again is the alternating symbol and mk is the normal unit vector pointing out of the surface. The coupled model is solved by the commercial finite element program Comsol Multiphysics with Matlab [19]. 4.3. Design variables and material interpolation The problem of finding an optimized distribution of material is a discrete optimization problem (there should be air or solid material in each point of the design domain), but in order to allow for efficient gradient-based optimization the problem is formulated with continuous material properties that can take any value in between the values for air and solid material. To control the material properties a continuous material indicator field 0 ≤ ξ(r) ≤ 1 is introduced, where ξ = 0 corresponds to air and ξ = 1 to solid material. If µ describes any of the material parameters in the model, introducing the design variable means that µ(ξ) = µa, ξ = 0 µs, ξ = 1 , (21) where µa represents the material values for air and µs the material values for solid material. Although ξ is continuous, the final design should be as close to discrete (ξ = 0 or ξ = 1) as possible in order to be well defined. This is done by employing a morphology-based filter as described in subsection 4.6. The material parameters are interpolated linearly between the two material phases which clearly fulfills the discrete values specified in Eq.(21). µ(ξ) = µa + ξ(µs − µa), (22) 4.4. The optimization problem The purpose of the topology optimization is to maximize the acousto-optical interaction between the SAW and the optical wave. The fundamental mode in the waveguide is polarized in the x1-direction so it primarily senses the refractive index in this direction. Therefore it is mainly interesting that the 4
efractive index component n11 changes and the change depends on the different strain components - the more the strain components change the more n11 changes. The optimization problem is only stated for the piezoelectric model and the aim is to maximize an expression, which is dependent on one of the normal strain components. The objective function Φ is thus chosen as the squared absolute value of the normal strain in the vertical direction in the output domain Ωo and the formulation of the optimization problem takes the form max ξ log(Φ) = log Ωop |S2(r, ξ(r))| 2 dr , objective function, (23) subject to 0 ≤ ξ(r) ≤ 1 ∀ r ∈ Ωd, design variable bounds. (24) To obtain better numerical scaling the logarithm of the objective function is used. To check that the acousto-optical interaction has indeed improved by the optimization the optical model is solved for both the initial design and the optimized design for the case where a wave crest is at the waveguide and half a SAW phase later where a wave trough is at the waveguide. For these two cases the difference in effective refractive index for the fundamental mode ∆neff,1 will take the biggest value possible for the initial design as shown in [6]. ∆neff,1 is normalized with the squared applied electric power P and this is then defined as the measure for the acousto-optical interaction, see [6] for the expression for P . So the final goal of the optimization is to increase this measure. 4.5. Discretization and sensitivity analysis The mathematical model of the acousto-optical problem is solved by finite element analysis. For the piezoelectric model the displacement-electric potential vector is defined as w = [w1, w2, w3] = [u1, u2, V ]. The complex fields wi and the design variable field ξ are discretized using sets of finite element basis functions {φi,n(r)} wi(r) = N Nd wi,nφi,n(r), ξ(r) = ξnφ4,n(r). (25) n=1 The degrees of freedom corresponding to the four fields are assembled in the vectors wi = {wi,1, wi,2, ...wi,N } T and ξ = {ξ1, ξ2, ...ξNd }T . A triangular element mesh is employed and second order Lagrange elements are used for the complex fields wi to obtain high accuracy in the solution and for the design variable ξ zero order Lagrange elements are utilized. The commercial program Comsol Multiphysics with Matlab is employed for the finite element analysis. This results in the discretized equations n=1 3 Kjiwi = fj, (26) i=1 where Kji is the system matrix and fj is the load vector, both being complex valued due to the PMLs. To update the design variables in the optimization algorithm the derivatives with respect to the design variables of the objective function must be evaluated. This is possible as the design variable is introduced as a separate field. The objective function depends on the complex field vector w2, which is via Eq.(26) an implicit function of the design variables and can be written as w2(ξ) = w R 2 (ξ) + iw I 2(ξ), where w R 2 and w I 2 denote the real and the imaginary part of w2, respectively. Thus the derivative of the objective function Φ = Φ(w R 2 (ξ), w I 2(ξ), ξ) is given by the following expression found by the chain rule dΦ dξ = ∂Φ ∂ξ + ∂Φ ∂w R 2 ∂w R 2 ∂ξ + ∂Φ ∂w I 2 ∂wI 2 . (27) ∂ξ As w2 is an implicit function of ξ the derivatives ∂wR 2 /∂ξ and ∂wI 2/∂ξ are not known directly. The sensitivity analysis is therefore done by employing an adjoint method where the unknown derivatives are eliminated at the expense of determining an adjoint and complex variable field λ from the adjoint equation ∂Φ Kj2λ = − ∂wR 2 5 − i ∂Φ ∂wI , (28) 2
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Optimization of acoustic, optical a
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Title of the thesis: Optimization o
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Resumé (in Danish) Forskningsomr˚
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Publications The following publicat
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vi Contents 6 Design of acousto-opt
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2 Chapter 1 Introduction waves. The
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4 Chapter 2 Time-harmonic propagati
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10 Chapter 2 Time-harmonic propagat
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12 Chapter 3 Topology optimization
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22 Chapter 4 Design of sound barrie
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28 Chapter 5 Design of photonic-cry
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34 Chapter 6 Design of acousto-opti
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56 Chapter 7 Concluding remarks des
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58 Chapter 7 Concluding remarks
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60 References [14] E. Yablonovitch,
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62 References [42] A. A. Larsen, B.
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64 References [67] K. Svanberg, “
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Publication [P1] Acoustic design by
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558 In Ref. [3] it is studied how t
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560 2.2. Design variables and mater
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562 When the mesh size is decreased
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Page 117 and 118: IDT GaAs AlGaAs 0.10 0.05 0.00 Ampl
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Page 162: Suzuki, A. Onoe, T. Adachi and K. F
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Maria Bayard Dühring - Solid Mechanics

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