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Quasi-Arithmetic Filters for Topology Optimization
Umeå University, Faculty of Science and Technology, Department of Computing Science.
2016 (English)Licentiate thesis, comprehensive summary (Other academic)Alternative title
Kvasiaritmetiska filter för topologioptimering (Swedish)
Abstract [en]

Topology optimization is a framework for finding the optimal layout of material within a given region of space. In material distribution topology optimization, a material indicator function determines the material state at each point within the design domain. It is well known that naive formulations of continuous material distribution topology optimization problems often lack solutions. To obtain numerical solutions, the continuous problem is approximated by a finite-dimensional problem. The finite-dimensional approximation is typically obtained by partitioning the design domain into a finite number of elements and assigning to each element a design variable that determines the material state of that element. Although the finite-dimensional problem generally is solvable, a sequence of solutions corresponding to ever finer partitions of the design domain may not converge; that is, the optimized designs may exhibit mesh-dependence. Filtering procedures are amongst the most popular methods used to handle the existence issue related to the continuous problem as well as the mesh-dependence related to the finite-dimensional approximation. Over the years, a variety of filters for topology optimization have been presented.

To harmonize the use and analysis of filters within the field of topology optimization, we introduce the class of fW-mean filters that is based on the weighted quasi-arithmetic mean, also known as the weighted generalized f-mean, over some neighborhoods. We also define the class of generalized fW-mean filters that contains the vast majority of filters for topology optimization. In particular, the class of generalized fW-mean filters includes the fW-mean filters, as well as the projected fW-mean filters that are formed by adding a projection step to the fW-mean filters.

If the design variables are located in a regular grid, uniform weights are used within each neighborhood, and equal sized polytope shaped neighborhoods are used, then a cascade of generalized fW-mean filters can be applied with a computational complexity that is linear in the number of design variables. Detailed algorithms for octagonal shaped neighborhoods in 2D and rhombicuboctahedron shaped neighborhoods in 3D are provided. The theoretically obtained computational complexity of the algorithm for octagonal shaped neighborhoods in 2D has been numerically verified. By using the same type of algorithm as for filtering, the additional computational complexity for computing derivatives needed in gradient based optimization is also linear in the number of design variables.

To exemplify the use of generalized fW-mean filters in topology optimization, we consider minimization of compliance (maximization of global stiffness) of linearly elastic continuum bodies. We establish the existence of solutions to a version of the continuous minimal compliance problem when a cascade of projected continuous fW-mean filters is included in the formulation. Bourdin's classical existence result for the linear density filter is a partial case of this general theorem for projected continuous fW-mean filters. Inspired by the works of Svanberg & Svärd and Sigmund, we introduce the harmonic open-close filter, which is a cascade of four fW-mean filters. We present large-scale numerical experiments indicating that, for minimal compliance problems, the harmonic open-close filter produces almost binary designs, provides independent size control on both material and void regions, and yields mesh-independent designs.

Place, publisher, year, edition, pages
Umeå: Umeå Universitet , 2016. , 23 p.
Series
Report / UMINF, ISSN 0348-0542 ; 16.04
National Category
Computational Mathematics Computer Science
Identifiers
URN: urn:nbn:se:umu:diva-116983ISBN: 978-91-7601-409-7 (print)OAI: oai:DiVA.org:umu-116983DiVA: diva2:903713
Presentation
2016-02-19, Umeå universitet, Umeå, 09:00
Supervisors
Funder
eSSENCE - An eScience CollaborationSwedish Research Council, 621-3706
Available from: 2016-02-25 Created: 2016-02-16 Last updated: 2017-04-11Bibliographically approved
List of papers
1. On quasi-arithmetic mean based filters and their fast evaluation for large-scale topology optimization
Open this publication in new window or tab >>On quasi-arithmetic mean based filters and their fast evaluation for large-scale topology optimization
2015 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 52, no 5, 879-888 p.Article in journal (Refereed) Published
Abstract [en]

In material distribution topology optimization, restriction methods are routinely applied to obtain well-posed optimization problems and to achieve mesh-independence of the resulting designs. One of the most popular restriction methods is to use a filtering procedure. In this paper, we present a framework where the filtering process is viewed as a quasi-arithmetic mean (or generalized f-mean) over a neighborhood with the possible addition of an extra "projection step". This framework includes the vast majority of available filters for topology optimization. The covered filtering procedures comprise three steps: (i) element-wise application of a function, (ii) computation of local averages, and (iii) element-wise application of another function. We present fast algorithms that apply this type of filters over polytope-shaped neighborhoods on regular meshes in two and three spatial dimensions. These algorithms have a computational cost that grows linearly with the number of elements and can be bounded irrespective of the filter radius.

Place, publisher, year, edition, pages
Springer, 2015
Keyword
Topology optimization, Regularization, Filters, Fast algorithm, Large-scale problems
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-114034 (URN)10.1007/s00158-015-1273-5 (DOI)000366590800003 ()
Available from: 2016-01-11 Created: 2016-01-11 Last updated: 2017-12-01Bibliographically approved
2. Nonlinear filters in topology optimization: existence of solutions and efficient implementation for minimal compliance problems
Open this publication in new window or tab >>Nonlinear filters in topology optimization: existence of solutions and efficient implementation for minimal compliance problems
2016 (English)Report (Other academic)
Abstract [en]

It is well known that material distribution topology optimization problems often are ill-posed if no restriction or regularization method is used. A drawback with the standard linear density filter is that the resulting designs have large areas of intermediate densities, so-called gray areas, especially when large filter radii are used. To produce final designs with less gray areas, several different methods have been proposed; for example, projecting the densities after the filtering or using a nonlinear filtering procedure. In a recent paper, we presented a framework that encompasses a vast majority of currently available density filters. In this paper, we show that all these nonlinear filters ensure existence of solutions to a continuous version of the minimal compliance problem. In addition, we provide a detailed description on how to efficiently compute sensitivities for the case when multiple of these nonlinear filters are applied in sequence. Finally, we present a numerical experiment that illustrates that these cascaded nonlinear filters can be used to obtain independent size control of both void and material regions in a large-scale setting.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2016. 16 p.
Series
Report / UMINF, ISSN 0348-0542 ; 16.01
National Category
Computer Science
Identifiers
urn:nbn:se:umu:diva-115934 (URN)
Funder
Swedish Research Council, 621-3706Swedish Foundation for Strategic Research , AM13-0029
Available from: 2016-02-08 Created: 2016-02-08 Last updated: 2016-02-25Bibliographically approved

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