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Fixed-mesh curvature-parameterized shape optimization of an acoustic horn
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science.ORCID iD: 0000-0003-0473-3263
2012 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 46, no 5, 727-738 p.Article in journal (Refereed) Published
Abstract [en]

We suggest a boundary shape optimization approach in which the optimization is carried out on the coefficients in a boundary parameterization based on a local, discrete curvature. A fixed mesh is used to numerically solve the governing equations, in which the geometry is represented through inhomogeneous coefficients, similarly as done in the material distribution approach to topology optimization. The method is applied to the optimization of an acoustic horn in two space dimensions. Numerical experiments show that this method can calculate the horn's transmission properties as accurately as a traditional, body-fitted approach. Moreover, the use of a fixed mesh allows the optimization to create shapes that would be difficult to handle with a traditional approach that uses deformations of a body-fitted mesh. The parameterization inherently promotes smooth designs without unduly restriction of the design flexibility. The optimized, smooth horns consistently show favorable transmission properties.

Place, publisher, year, edition, pages
Springer, 2012. Vol. 46, no 5, 727-738 p.
Keyword [en]
Shape optimization, Material distribution approach, Acoustic horns, Helmholtz equation
National Category
Computer Science
Identifiers
URN: urn:nbn:se:umu:diva-61968DOI: 10.1007/s00158-012-0828-yISI: 000310426800008OAI: oai:DiVA.org:umu-61968DiVA: diva2:578825
Available from: 2012-12-19 Created: 2012-12-04 Last updated: 2017-12-06
In thesis
1. Boundary Shape Optimization Using the Material Distribution Approach
Open this publication in new window or tab >>Boundary Shape Optimization Using the Material Distribution Approach
2011 (English)Licentiate thesis, comprehensive summary (Other academic)
Place, publisher, year, edition, pages
Umeå: Department of Computing Science, Umeå University, 2011. 24 p.
Series
Report / UMINF, ISSN 0348-0542 ; 11.06
Keyword
Design optimization, Helmholtz equation, linear elasticity, FEM, fictitious domain methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:umu:diva-87584 (URN)
Presentation
2011-06-20, 10:00 (English)
Opponent
Supervisors
Available from: 2014-04-08 Created: 2014-04-04 Last updated: 2017-04-11Bibliographically approved
2. The Material Distribution Method: Analysis and Acoustics applications
Open this publication in new window or tab >>The Material Distribution Method: Analysis and Acoustics applications
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

For the purpose of numerically simulating continuum mechanical structures, different types of material may be represented by the extreme values {,1}, where 0<1, of a varying coefficient  in the governing equations. The paramter  is not allowed to vanish in order for the equations to be solvable, which means that the exact conditions are approximated. For example, for linear elasticity problems, presence of material is represented by the value  = 1, while  =  provides an approximation of void, meaning that material-free regions are approximated with a weak material. For acoustics applications, the value  = 1 corresponds to air and  to an approximation of sound-hard material using a dense fluid. Here we analyze the convergence properties of such material approximations as !0, and we employ this type of approximations to perform design optimization.

In Paper I, we carry out boundary shape optimization of an acoustic horn. We suggest a shape parameterization based on a local, discrete curvature combined with a fixed mesh that does not conform to the generated shapes. The values of the coefficient , which enters in the governing equation, are obtained by projecting the generated shapes onto the underlying computational mesh. The optimized horns are smooth and exhibit good transmission properties. Due to the choice of parameterization, the smoothness of the designs is achieved without imposing severe restrictions on the design variables.

In Paper II, we analyze the convergence properties of a linear elasticity problem in which void is approximated by a weak material. We show that the error introduced by the weak material approximation, after a finite element discretization, is bounded by terms that scale as  and 1/2hs, where h is the mesh size and s depends on the order of the finite element basis functions. In addition, we show that the condition number of the system matrix scales inversely proportional to , and we also construct a left preconditioner that yields a system matrix with a condition number independent of .

In Paper III, we observe that the standard sound-hard material approximation with   gives rise to ill-conditioned system matrices at certain wavenumbers due to resonances within the approximated sound-hard material. To cure this defect, we propose a stabilization scheme that makes the condition number of the system matrix independent of the wavenumber. In addition, we demonstrate that the stabilized formulation performs well in the context of design optimization of an acoustic waveguide transmission device.

In Paper IV, we analyze the convergence properties of a wave propagation problem in which sound-hard material is approximated by a dense fluid. To avoid the occurrence of internal resonances, we generalize the stabilization scheme presented in Paper III. We show that the error between the solution obtained using the stabilized soundhard material approximation and the solution to the problem with exactly modeled sound-hard material is bounded proportionally to .

Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2014. 46 p.
Series
Report / UMINF, ISSN 0348-0542 ; 18
Keyword
Material distribution method, fictitious domain method, finite element method, Helmholtz equation, linear elasticity, shape optimization, topology optimization
National Category
Computational Mathematics Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:umu:diva-92538 (URN)978-91-7601-122-5 (ISBN)
Public defence
2014-09-19, MIT-huset, MC413, Umeå universitet, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2014-08-29 Created: 2014-08-27 Last updated: 2017-04-11Bibliographically approved

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