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Boundary Shape Optimization Using the Material Distribution Approach
Umeå University, Faculty of Science and Technology, Department of Computing Science.
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 [en]
Design optimization, Helmholtz equation, linear elasticity, FEM, fictitious domain methods
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-87584OAI: oai:DiVA.org:umu-87584DiVA: diva2:710137
Presentation
2011-06-20, 10:00 (English)
Opponent
Supervisors
Available from: 2014-04-08 Created: 2014-04-04 Last updated: 2017-04-11Bibliographically approved
List of papers
1. Fixed-mesh curvature-parameterized shape optimization of an acoustic horn
Open this publication in new window or tab >>Fixed-mesh curvature-parameterized shape optimization of an acoustic horn
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
Keyword
Shape optimization, Material distribution approach, Acoustic horns, Helmholtz equation
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-61968 (URN)10.1007/s00158-012-0828-y (DOI)000310426800008 ()
Available from: 2012-12-19 Created: 2012-12-04 Last updated: 2017-12-06
2. Weak material approximation of holes with traction-free boundaries
Open this publication in new window or tab >>Weak material approximation of holes with traction-free boundaries
2012 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 50, no 4, 1827-1848 p.Article in journal (Refereed) Published
Abstract [en]

Consider the solution of a boundary-value problem for steady linear elasticity in which the computational domain contains one or several holes with traction-free boundaries. The presence of holes in the material can be approximated using a weak material; that is, the relative density of material rho is set to 0 < epsilon = rho << 1 in the hole region. The weak material approach is a standard technique in the so-called material distribution approach to topology optimization, in which the inhomogeneous relative density of material is designated as the design variable in order to optimize the spatial distribution of material. The use of a weak material ensures that the elasticity problem is uniquely solvable for each admissible value rho is an element of [epsilon, 1] of the design variable. A finite-element approximation of the boundary-value problem in which the weak material approximation is used in the hole regions can be viewed as a nonconforming but convergent approximation of a version of the original problem in which the solution is continuously and elastically extended into the holes. The error in this approximation can be bounded by two terms that depend on epsilon. One term scales linearly with epsilon with a constant that is independent of the mesh size parameter h but that depends on the surface traction required to fit elastic material in the deformed holes. The other term scales like epsilon(1/2) times the finite-element approximation error inside the hole. The condition number of the weak material stiffness matrix scales like epsilon(-1), but the use of a suitable left preconditioner yields a matrix with a condition number that is bounded independently of epsilon. Moreover, the preconditioned matrix admits the limit value epsilon -> 0, and the solution of corresponding system of equations yields in the limit a finite-element approximation of the continuously and elastically extended problem.

Place, publisher, year, edition, pages
SIAM Publications Online, 2012
Keyword
FEM, linear elasticity, material distribution approach, topology optimization, fictitious domain methods, preconditioning, surface traction
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-62007 (URN)10.1137/110835384 (DOI)000310214200001 ()
Available from: 2012-12-05 Created: 2012-12-04 Last updated: 2017-12-07Bibliographically approved

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