Maria Bayard Dühring - Solid Mechanics

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two air holes, is Λ = 3.1 µm, the hole diameter over the pitch ratio is d/Λ = 0.92 and the diameter is d = 2.852 µm. The radius of the air core in the center is R = Λ. The core region Ωc where the objective function is optimized has the radius 0.5R, and the design domain Ωd is defined by the area between the two half circles with radius equal to 0.5R and 1.5R, respectively. The half circle at the center where the source is applied is equal to 0.05R. The maximum element size is 0.1 µm in Ωd and 0.8 µm in the rest of the domain. The refractive indexes for the chosen wavelength are na = 1 for air and ns = 1.43791 + 0.0001i for silica, where the imaginary part in ns causes absorption and is denoted as the absorbing coefficient αs. Before the optimization, a band-gap check is performed with the commercial program BandSolve, see [31]. The band diagram is calculated for the periodic structure with k0Λ as function of k3Λ, where k3 is the propagation constant in the x3-direction. The point (k3Λ, k0Λ) = (9.5127, 9.73894) for the fundamental mode was shown to be in the calculated band gap. The wave equation (2) is then solved as an eigenvalue problem with the absorption coefficient αs equal to zero and the propagating constant β for the guided mode is found, which is used together with the fixed value of k0 = 2π/λ0 to solve the model by employing the forcing term. 3.1 Study of the optimized design The optimization is performed with the filter radius r = 0.15 µm, a move limit equal to 0.05 for the maximum change in the design variables in each iteration step and the absolute tolerance 0.01 to terminate the optimization. The continuation method for the damping is applied such that αs is equal to 1 when the optimization starts and is reduced by a factor 10 after 100 iterations or if the tolerance is satisfied. This is repeated until αs reaches the value 0.0001. Introducing the damping lowers and widens the peak at resonance as seen on figure 2 where log(Φ/Φinit) is plotted as function of λ0 for increasing αs. Φinit is the value of the objective function for the initial design. The resonance is kept at the original wavelength λ0 = 2 µm for the low values of αs, but is slightly shifted to higher wavelengths for the two highest values of αs such that the optimization starts away from resonance. The optimized design is found in 316 iterations and the objective function Φ is increased 375% compared to the initial design. In figure 2, log(Φ/Φinit) as function of λ0 is compared for the initial and the optimized design. The wavelength at resonance is the same before and after the optimization and the peak value has increased for the optimized design. In figure 3(a) and 3(b) the initial and optimized design are plotted, respectively, and the distribution of the energy measure from the objective function normalized with the maximal value for the initial design, Φmax, is indicated with the contour lines. The optimized design is close to a 0-1 design and all parts are log(Φ/Φ init ) [−] 1 0 −1 −2 −3 −4 −5 −6 optimized design, α s =0.0001 initial design, α s =0.0001 initial design, α s =0.001 initial design, α s =0.01 initial design, α s =0.1 initial design, α s =1.0 −7 1.99 1.995 2 wavelength, λ [μm] 0 2.005 2.01 Figure 2: The logarithm to the normalized objective function Φ/Φinit as function of the optical wavelength λ0 is indicated both for the initial design with different values of the absorption coefficient αs and for the optimized design with αs = 0.0001. 6
Figure 3: The center region of the holey fiber where black indicates silica and white is air. (a) and (b): The contour lines show the distribution of the normalized energy measure from the objective function Φ/Φmax in the initial and the optimized design, respectively. (c) and (d): The normalized time average power flow P/Pmax in the x3-direction is indicated by the contour lines for the initial and the optimized design, respectively. connected, which is important due to fabrication issues. The design is almost symmetric around a vertical axis, which has not been enforced in the optimization. The small deviation from symmetry is most likely due to the unstructured mesh, which is not symmetric. Compared to the initial design the objective function in the optimized design has been redistributed, such that the modes shape is extending more in the vertical direction and the peak value in the center has increased. The time average power flow P in the x3-direction, normalized with the maximum value Pmax for the initial design, is plotted on figure 3(c) and 3(d) for the initial and the optimized design, respectively. The shape of the modes are in good agreement with the distribution of the energy measure from the objective function in figure 3(a) and 3(b), and their ratio of magnitude is also similar. Compared to the initial design the integral of P in Ωc has increased 378% for the optimized design. So the same improvement is obtained for Φ and P when Φ is used as the objective function. In order to explain that the objective function has increased, the absolute value of the magnetic fields H1 and H2 are considered. Both fields are symmetric around the vertical axis before and after the optimization and |H1|/|H1,max| is plotted with contour lines for the initial and optimized design in figure 4(a) and 4(b), respectively. |H1,max| is the maximum value of |H1| for the initial design. Like the distribution of the objective function, |H1| is confined by the vertical walls to the sides and is elongated in the vertical direction. In the initial design concentrations of |H1| are found where the 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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finer mesh. For the coupled model i
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FIG. 4: Results for the finite mode
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FIG. 6: Results for the acousto-opt
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Suzuki, A. Onoe, T. Adachi and K. F
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Maria Bayard Dühring - Solid Mechanics

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