Maria Bayard Dühring - Solid Mechanics

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where ∂Φ ∂w R 2,n − i ∂Φ ∂wI = 2 2,n Ωop ∂wR 2 ∂x2 − i2 ∂wI 2 ∂φ2,n R dr = 2S2 − i2S ∂x2 ∂x2 Ωop I 2 ∂φ2,n ∂x2 dr. (29) The sensitivity analysis follows the standard adjoint sensitivity approach [20]. For further details of the adjoint sensitivity method applied to wave propagation problems, the reader is referred to [12]. Eq.(27) for the derivative of the objective function then reduces to dΦ ∂Φ = + Re λ dξ ∂ξ T ∂Kj2 ∂ξ w2 . (30) The vectors ∂Φ/∂ξ and R 2S Ωop 2 − i2SI ∂φ2,n 2 ∂x2 dr as well as the matrix ∂Kj2/∂ξ are assembled in Comsol Multiphysics as described in [21]. 4.6. Practical implementation The optimization problem Eq.(23)-(24) is solved using the Method of Moving Asymptotes, MMA [22], which is an algorithm that uses information from the previous iteration steps and gradient information. When the mesh size is decreased the optimization will in general result in mesh-dependent solutions with small details, which make the design inconvenient to manufacture. To avoid these problems a morphology-based filter is employed. Such filters make the material properties of an element depend on a function of the design variables in a fixed neighborhood around the element such that the finite design is mesh-independent. Here a Heaviside close-type morphology-based filter is chosen [23], which has proven efficient for wave-propagation type topology optimization problems, see for instance [11]. The method results in designs where all holes below the size of the filter (radius rmin) have been eliminated. A further advantage of these filter-types is that they help eliminating gray elements in the transition zone between solid and air regions. 5. Results Results are now presented for the problem shown in Figure 1. The SAW is generated by 6 double electrode finger pairs. Each of the electrodes has a width equal to 0.7 µm and is placed 0.7 µm apart such that the wavelength of the generated SAW is 5.6 µm. The waveguide is placed at the surface and has the width 1.4 µm and the height 0.3 µm. The output domain Ωo consists of the waveguide and an area just below the waveguide with the same size, in order to increase the possibility that the waveguide will still be able to confine an optical mode after the optimization. The materials GaAs and AlGaAs are piezoelectric with cubic crystal structure. The material constants used in the piezoelectric model are given in Table 1 and are here given in the usual matrix form for compactness reasons. For the SAW to propagate in the piezoelectric direction the material tensors have to be rotated by the angle ϕ2 = π/4. The frequency is fsaw = 518 MHz (where fsaw = ωsaw/2π), which is the resonance frequency. The constant σj controlling the damping in the PMLs is set to 10 10 . Elements with maximum side length e1 = 0.6 µm are used in the domain except in the design domain where it is e2 = 0.1 µm and in the output domain where it is e3 = 0.02 µm. The optical model is solved both for the initial design, where the design domain is filled with solid material, and for the optimized design in order to compare the acousto-optical interaction. The free space wavelength of the optical wave is set to λ0 = 950 nm and the material constants for GaAs and AlGaAs used in the optical model are given in Table 2. For air simply the refractive index n0 = 1 is used. The logarithm to the objective function for the initial design is found to be Φ = −19.99 and the acousto-optical interaction measure is ∆neff,1/ (P ) = 1.500 · 10 −5 . Note that these values are low do the the limited numbers of electrodes, in practice several hundred electrode fingers are employed to generate a SAW. Table 1: The elastic stiffness constants, the density, the piezoelectric stress constants and the permittivity constants for the materials used in the piezoelectric model. Material c E 11,c E 22,c E 33 c E 12,c E 13,c E 23 c E 44,c E 55,c E 66 ρ e14,e25,e36 ε S 11,ε S 22,ε S 33 [10 11 Nm −2 ] [10 11 Nm −2 ] [10 11 Nm −2 ] [kgm −3 ] [Cm −2 ] [10 −11 Fm] GaAs 1.1830 0.5320 0.5950 5316.5 -0.160 9.735 AlGaAs 1.1868 0.5396 0.5938 5005.2 -0.152 9.446 6
Table 2: Refractive index and strain-optical constants for the materials used in the optical model. Material n0 p11, p22, p33 p12, p13, p23 p44, p55, p66 [-] [10 −12 Pa −1 ] [10 −12 Pa −1 ] [10 −12 Pa −1 ] GaAs 3.545 -0.165 -0.14 -0.072 AlGaAs 3.394 -0.165 -0.14 -0.072 Figure 2: The optimized design below the waveguide. Black color represents solid material and white color represents air. The objective function Φ is then maximized in the output domain Ωop by distributing air and solid material in the design domain Ωd where solid material refers to AlGaAs in the upper part of Ωd and GaAs in the lower part. The filter radius is rmin = 2.5e2, an absolute tolerance of 0.01 on the maximum change of the design variables is used to terminate the optimization loop and a move limit for the design variables equal to 0.2 is used. The optimized design was found in 692 iterations and the logarithm to the objective function was increased to -17.39 - i.e. more than 2 orders of magnitude. Figure 2 shows the optimized design where black represents solid material and white represents air. The design is almost a pure black and white design and compared to the initial design air holes around the design domain have appeared. On Figure 3 the distribution of Φ normalized with √ P is plotted close to the waveguide for the original design and the optimized design, respectively, together with the air holes represented with white color. It is seen that in comparison to the initial design the redistributed material in the design domain is influencing Φ such that it has a higher value in the output domain Ωo. This happens because the SAW gets trapped in the design and output domain because of reflections at the air holes. An air hole is created just below the output domain, which is creating some strain concentrations that are also extending to the output domain such that the strain here is increased. When solving the optical model for the optimized design the acousto-optical interaction is found to be ∆neff,1/ (P ) = 1.480·10 −4 , which is 9.9 times higher than for the initial design. As the fundamental optical mode is polarized in the x1direction it is important that the change in n11 is as big as possible and ∆n11/ √ P is plotted for the initial design and the optimized design in Figure 4. The change in n11/ √ P is an order of magnitude bigger for the optimized design compared to the initial design, which explains that ∆neff,1/ √ P has increased. In Figure 4 the optical mode is indicated with contour lines and it is seen that the mode is confined to the waveguide both before and after the optimization. It is observed that a side benefit of the optimization is that the optical mode tends to get more confined to the output domain because of the big contrast in refractive index at the air hole next to the output domain. This example shows that it is possible to improve the acousto-optical interaction between a SAW and an optical wave in a waveguide by topology optimization where the objective function is based only on 7
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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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564 objective function, Φ [dB] 130
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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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