Maria Bayard Dühring - Solid Mechanics

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560 2.2. Design variables and material interpolation The problem of finding the optimal 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 0pxðrÞp1 is introduced, where x ¼ 0 corresponds to air and x ¼ 1 to solid material: 8 8 < 1 x ¼ 0; < 1 x ¼ 0; ~rðxÞ ¼ r2 : x ¼ 1; ~kðxÞ ¼ k2 : x ¼ 1: (5) r 1 Although x is continuous the final design should be as close to discrete (x ¼ 0 or 1) as possible in order to be well defined. The choice of interpolation functions may aid in avoiding intermediate (gray) material properties in the final design. In Ref. [19] it is suggested to find the interpolation function by looking at a 1D acoustic system where a wave with amplitude of unit magnitude propagates in air and hits an interface to an acoustic medium under normal incidence. Experience shows that good 0–1 designs in general can be obtained if the reflection from the acoustic medium in this system is a smooth function of x with nonvanishing slope at x ¼ 1. This is obtained by interpolating the inverse density and bulk modulus between the two material phases as follows: ~rðxÞ 1 ! 1 r2 ¼ 1 þ x 1 , (6) ~kðxÞ 1 ¼ 1 þ x which clearly fulfills the discrete values specified in Eq. (5). 2.3. The optimization problem r 1 k2 k1 1 k1 1 ! , (7) The purpose of the topology optimization is to minimize the objective function F which is the average of the squared sound pressure amplitude in the output domain, Oop. The formulation of the optimization problem takes the form 1 minx logðFÞ ¼log R Oop dr Z j ^pðr; xðrÞÞj Oop 2 ! dr objective function, (8) subject to R 1 Od dr Z Od xðrÞ dr bp0; volume constraint, (9) 0pxðrÞp1 8 r 2 Od; design variable bounds. (10) A volume constraint is included to put a limit on the amount of material distributed in the design domain Od in order to save weight and cost. Here b is a volume fraction of allowable material and takes values between 0 and 1, where b ¼ 1 corresponds to no limit. To obtain better numerical scaling the logarithm is taken to the objective function. 2.4. Discretization and sensitivity analysis The mathematical model of the physical problem is given by the Helmholtz equation (3) and the boundary conditions (4), and to solve the problem, finite element analysis is used. The complex amplitude field ^p and the design variable field x are discretized using sets of finite element basis functions ff i;nðrÞg ^pðrÞ ¼ XN ARTICLE IN PRESS M.B. <strong>Dühring</strong> et al. / Journal of Sound and Vibration 317 (2008) 557–575 n¼1 ^p nf 1;nðrÞ; xðrÞ ¼ XNd n¼1 x nf 2;nðrÞ. (11)
The degrees of freedom (dofs) corresponding to the two fields are assembled in the vectors ^p ¼ f ^p 1; ^p 2; ...; ^p Ng T and n ¼fx1; x2; ...; xNd gT . For the model in 2D a triangular element mesh is employed and tetrahedral elements are used in 3D. Quadratic Lagrange elements are used for the complex pressure amplitude ^p to obtain high accuracy in the solution and for the design variable x linear Lagrange elements are utilized. The commercial program Comsol Multiphysics with Matlab [25] is employed for the finite element analysis. This results in the discretized equation S^p ¼ f, (12) where S is the system matrix and f is the load vector which are both complex valued. To update the design variables in the optimization algorithm the derivatives with respect to the design variables of the objective and the constraint function must be evaluated. This is possible as the design variable is introduced as a continuous field. The complex sound pressure vector ^p is via Eq. (12) an implicit function of the design variables, which is written as ^pðnÞ ¼^p RðnÞþi^p IðnÞ, where ^p R and ^p I denote the real and the imaginary part of ^p. Thus the derivative of the objective function F ¼ Fð^p RðnÞ; ^p IðnÞ; nÞ is given by the following expression found by the chain rule dF dn qF qF q^p R qF q^p I ¼ þ þ . (13) qn q^p R qn q^p I qn As ^p is an implicit function of n the derivatives q^p R=qn and q^p I=qn 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 k from the adjoint equation where qF q^p R i qF q^p I S T k ¼ qF q^p R i qF q^p I T , (14) ¼ 1 R Oop dr Z ð2 ^p R i2 ^p IÞf1;n dr. (15) Oop The sensitivity analysis follows the standard adjoint sensitivity approach [26]. For further details of the adjoint sensitivity method applied to wave propagation problems, the reader is referred to Ref. [27]. Eq. (13) for the derivative of the objective function then reduces to dF dn ¼ qF qn qS þ Re kT ^p . (16) qn Finally, the derivative of the constraint function with respect to one of the design variables is q 1 R qxn Od dr Z ! xðrÞ dr b ¼ Od 1 R Od dr Z f2;nðrÞ dr. (17) Od The vectors qF=qn, R Oop ð2 ^p R i2 ^p IÞf1;n dr and R Od f2;nðrÞ dr as well as the matrix qS=qn are assembled in Comsol Multiphysics as described in Eq. [28]. 2.5. Practical implementation ARTICLE IN PRESS M.B. <strong>Dühring</strong> et al. / Journal of Sound and Vibration 317 (2008) 557–575 561 The optimization problem (8)–(10) is solved using the Method of Moving Asymptotes, MMA [29] which is an algorithm that uses information from the previous iteration steps and gradient information. To fulfill the volume constraint from the first iteration of the optimization procedure the initial design is usually chosen as a uniform distribution of material with the volume fraction b. It should be emphasized that the final design depends both on the initial design and the allowable amount of material to be placed and is therefore dependent on b. Here the best solutions out of several tries will be presented. To make the model more realistic and to minimize local resonance effects a small amount of mass-proportional damping is added.
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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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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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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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