Maria Bayard Dühring - Solid Mechanics

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18 Chapter 3 Topology optimization applied to time-harmonic propagating waves The two expressions (3.16) and (3.17) are equivalent, which is seen by taking the complex conjugate on both sides of expression (3.17). This means that when λj is computed by equation (3.16) it will also fulfill equation (3.17). Equation (3.16) is transposed and it is used that Skj is symmetric J T ∂Φ j=1 Skjλj = ∂u R k − i ∂Φ ∂u I k . (3.18) The expression in (3.18) can be solved as one system in a similar way as the discretized problem in (2.5). In the case where wk in the objective function (3.3) is equal to the field variable uk the right hand side of equation (3.18) is found by the expression ∂Φ ∂u R k,n − i ∂Φ ∂u I k,n = = Ωo Ωo ∂Φ ∂u R k ∂uR k ∂uR k,n In the case where wk indicates the derivative ∂uk ∂xk (3.18) is found by ∂Φ ∂u R k,n = Ωo − i ∂Φ ∂u I k,n = 2w R k − i2w I k ∂Φ Ωo ∂wR k ∂φk,n ∂xk − i ∂Φ ∂u I k ∂u I k ∂u I k,n dr (2u R k − i2u I k)φk,ndr. (3.19) − i ∂Φ dr = ∂w I k Ωo ∂φk,n ∂xk 2 ∂uR k ∂xk the right hand side of equation dr − i2 ∂uI k ∂φk,n dr. (3.20) ∂xk ∂xk The derivatives ∂uk can be calculated automatically by Comsol Multiphysics. The ∂xk adjoint variable fields are now determined such that the two last terms in (3.13) vanish and the derivative of the objective function reduces to dΦ ∂Φ = dξ ∂ξ + J 1 2 λT J ∂Skj k ∂ξ uj + 1 2 ¯ λ T J ∂ k ¯ Skj ∂ξ ūj , (3.21) k=1 and can be rewritten in the form dΦ dξ = ∂Φ ∂ξ + j=1 J ℜ k=1 λ T k J j=1 j=1 ∂Skj ∂ξ uj . (3.22) Finally, the derivative of the constraint function with respect to one of the design variables is ∂ 1 ∂ξn Ωd dr ξ(r)dr − β = Ωd 1 Ωd dr φJ+1,n(r)dr. (3.23) Ωd The mentioned vectors ∂Φ/∂ξ, Ωo (2uRk − i2uI k )φk,ndr, 2 Ωo ∂uR k ∂xk − i2 ∂uI k ∂φk,n ∂xk ∂xk dr and Ωd φJ+1,n(r)dr as well as the matrix ∂Skj/∂ξ are assembled in Comsol Multiphysics as described in [66].
3.5 Practical implementation 19 3.5 Practical implementation The actual update of the design variables in each step is done by employing the Method of Moving Asymptotes (MMA) by Svanberg 2002, see [67]. This is an efficient mathematical programming method to solve the type of smooth problems (meaning that the objective and constraint functions should at least belong to C 1 ) with the high number of degrees of freedom found in topology optimization. It is an algorithm that employs information from the previous iteration steps as well as gradient information - plus second order information if it is available. In order to get optimization problems that behave well during the optimization and to obtain applicable 0-1 designs, the optimization problems can be modified in different ways. First of all material damping can be applied such that the resonance frequencies are damped, which make the problems more realistic and well behaving during the optimization. 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 filter is employed. The filter makes the material properties of an element dependent on a function of the design variables in a fixed neighborhood around the element such that the optimized design is mesh-independent. In the present work, a close-type Heaviside morphology-based filter is applied to the problems, which has proven efficient for wave propagation problems [68]. The method results in designs where all holes below the size of the filter radius have been eliminated. To get a numerically stable optimization, the filter must usually be implemented as a continuation scheme where the property of the filter is gradually changed from a density filter to a morphology-based filter. A further advantage of these filter-types is that they tend to eliminate gray elements in the transition zone between solid and air regions. The problems studied here are non-unique with a number of local optima that typically originate from local resonance effects. To prevent convergence to these local optima a continuation method can be applied where the original problem is modified to a smoother problem. For wave problems it has been shown that a strong artificial damping will smoothen the response [39, 53], hence this method is applied to some of the problems in this project. A strong material damping is applied in the beginning of the optimization and after convergence of the modified problem or after a fixed number of iterations the problem is gradually changed back to the original one with a realistic damping. The continuation scheme of the filter has a similar effect. The iterative optimization procedure including the filter can finally be written in the pseudo code:
Page 1: Optimization of acoustic, optical a
Page 4 and 5: Title of the thesis: Optimization o
Page 6 and 7: Resumé (in Danish) Forskningsomr˚
Page 8 and 9: Publications The following publicat
Page 10 and 11: vi Contents 6 Design of acousto-opt
Page 12 and 13: 2 Chapter 1 Introduction waves. The
Page 14 and 15: 4 Chapter 2 Time-harmonic propagati
Page 20 and 21: 10 Chapter 2 Time-harmonic propagat
Page 22 and 23: 12 Chapter 3 Topology optimization
Page 32 and 33: 22 Chapter 4 Design of sound barrie
Page 38 and 39: 28 Chapter 5 Design of photonic-cry
Page 44 and 45: 34 Chapter 6 Design of acousto-opti
Page 66 and 67: 56 Chapter 7 Concluding remarks des
Page 68 and 69: 58 Chapter 7 Concluding remarks
Page 70 and 71: 60 References [14] E. Yablonovitch,
Page 72 and 73: 62 References [42] A. A. Larsen, B.
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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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566 ARTICLE IN PRESS Fig. 6. The di
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568 ~kðxÞ 1 ¼ k1 ARTICLE IN PRES
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570 ARTICLE IN PRESS M.B. Dühring
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572 ARTICLE IN PRESS M.B. Dühring
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574 frequency or the frequency inte
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Publication [P2] Design of photonic
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photonic-crystal fibers used for me
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2.2 Design variables and material i
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two air holes, is Λ = 3.1 µm, the
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Figure 4: The geometry of the cente
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performance will decrease and some
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[23] J. Riishede and O. Sigmund, In
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Surface acoustic wave driven light
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IDT GaAs AlGaAs 0.10 0.05 0.00 Ampl
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Publication [P4] Improving the acou
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083529-2 M. B. Dühring and O. Sigm
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Publication [P5] Design of acousto-
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Figure 1: The geometry used in the
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where ˜pijkl are the rotated strai
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where ∂Φ ∂w R 2,n − i ∂Φ
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Figure 3: The distribution of the o
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[12] J.S. Jensen and O. Sigmund, To
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Energy storage and dispersion of su
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093504-3 Dühring, Laude, and Kheli
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093504-5 Dühring, Laude, and Kheli
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Publication [P7] Improving surface
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FIG. 1: The geometry of the acousto
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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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