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Weak material approximation of holes with traction-free boundaries
Umeå University, Faculty of Science and Technology, Department of Computing Science.ORCID iD: 0000-0003-0473-3263
Umeå University, Faculty of Science and Technology, Department of Computing Science.
2012 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 50, no 4, p. 1827-1848Article 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. Vol. 50, no 4, p. 1827-1848
##### Keyword [en]
FEM, linear elasticity, material distribution approach, topology optimization, fictitious domain methods, preconditioning, surface traction
##### National Category
Computer Sciences
##### Identifiers
ISI: 000310214200001OAI: oai:DiVA.org:umu-62007DiVA, id: diva2:574492
Available from: 2012-12-05 Created: 2012-12-04 Last updated: 2018-01-12Bibliographically approved
##### 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. p. 24
##### 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)
##### 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 {$\epsilon$,1}, where 0<$\epsilon$$\ll$1, of a varying coefficient $\alpha$ in the governing equations. The paramter $\epsilon$ 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 $\alpha$ = 1, while $\alpha$ = $\epsilon$ provides an approximation of void, meaning that material-free regions are approximated with a weak material. For acoustics applications, the value $\alpha$ = 1 corresponds to air and $\alpha$ $\epsilon$ to an approximation of sound-hard material using a dense fluid. Here we analyze the convergence properties of such material approximations as $\epsilon$!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 $\alpha$, 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 $\epsilon$ and $\epsilon$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 $\epsilon$, and we also construct a left preconditioner that yields a system matrix with a condition number independent of $\epsilon$.

In Paper III, we observe that the standard sound-hard material approximation with $\alpha$ $\epsilon$ 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 $\epsilon$.

##### Place, publisher, year, edition, pages
Umeå: Umeå Universitet, 2014. p. 46
##### 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)
##### Supervisors
Available from: 2014-08-29 Created: 2014-08-27 Last updated: 2018-01-17Bibliographically approved

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