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Nonlinear filters in topology optimization: existence of solutions and efficient implementation for minimal compliance problems
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
2016 (engelsk)Rapport (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Umeå: Umeå universitet , 2016. , s. 16
Serie
Report / UMINF, ISSN 0348-0542 ; 16.01
HSV kategori
Identifikatorer
URN: urn:nbn:se:umu:diva-115934OAI: oai:DiVA.org:umu-115934DiVA, id: diva2:901363
Forskningsfinansiär
Swedish Research Council, 621-3706Swedish Foundation for Strategic Research , AM13-0029Tilgjengelig fra: 2016-02-08 Laget: 2016-02-08 Sist oppdatert: 2018-06-07bibliografisk kontrollert
Inngår i avhandling
1. Quasi-Arithmetic Filters for Topology Optimization
Åpne denne publikasjonen i ny fane eller vindu >>Quasi-Arithmetic Filters for Topology Optimization
2016 (engelsk)Licentiatavhandling, med artikler (Annet vitenskapelig)
Alternativ tittel[sv]
Kvasiaritmetiska filter för topologioptimering
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.

sted, utgiver, år, opplag, sider
Umeå: Umeå Universitet, 2016. s. 23
Serie
Report / UMINF, ISSN 0348-0542 ; 16.04
HSV kategori
Identifikatorer
urn:nbn:se:umu:diva-116983 (URN)978-91-7601-409-7 (ISBN)
Presentation
2016-02-19, Umeå universitet, Umeå, 09:00
Veileder
Forskningsfinansiär
eSSENCE - An eScience CollaborationSwedish Research Council, 621-3706
Tilgjengelig fra: 2016-02-25 Laget: 2016-02-16 Sist oppdatert: 2018-06-07bibliografisk kontrollert

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